DIRECTIONS IN LARGE-SCALE SYSTEMS Many-Person Optimization and Decentralized Control

DIRECTIONS IN LARGE-SCALE SYSTEMS Many-Person Optimization and Decentralized Control

Edited by

y.e. Ho Harvard University

and

S. K. Mitter Massachusetts Institute of Technology

Plenum Press' New York and London

Library of Congress Cataloging in Publication Data

Conference on Directions in Decentralized Control, Many-Person Optimization, and Large-Scale Systems, Wakefield, Mass., 1975. Directions in large-scale systems.

Includes bibliographical references and index. 1. System analysis-Congresses. 2. Control theory-Congresses. 3. Mathematical

Optimization-Congresses. I. Ho, Yu-Chi, 1934- II. Mitter, Sanjoy. III. Title. QA402.C571975 003 76-10279 ISBN-13: 978-1-4684-2261-0 001: 10.1007/978-1-4684-2259-7

e-ISBN-13: 978-1-4684-2259-7

Proceedings of a Symposium on Directions in Decentralized Control, Many-Person OPtimization, and Large-Scale Systems held in Wakefield, Massachusetts, September 1-3, 1975

© 1976 Plenum Press, New York Softcover reprint of the hardcover 15t edition 1976 A Division of Plenum Publishing Corporation 227 West 17th Street, New York, N.Y. 10011

A" rights reserved

No part of this book may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, microfilming, recording, or otherwise, without written permission from the Publisher

Introduction

This book is the record of papers presented at the Conference on Directions in Decentralized Control, Many-Person Optimization, and Large-Scale Systems held at the Colonial Hilton Inn, Wakefield, Massachusetts from September 1-3, 1975. Our motivation for organizing such a conference was two fold. Firstly, the last few years have seen a great deal of activity in the field of Large-Scale Systems Theory and it has been certainly one of the dominant themes of research in the disciplines of Systems and Control Theory. It therefore seemed appropriate to try and take stock of what had been accomplished and also try to "invent"l the future directions of research in this field. Secondly, the 6th World IFAC Conference was being held in Cambridge, Massachusetts the week earlier and it provided an ideal opportunity for taking advantage of the presence of a large number of specialists from all parts of the world to organize a small conference where a free exchange of ideas could take place. It is left to the readers of this volume to judge to what extent we have been successful in our above­mentioned goals.

There is no accepted definition of what constitutes a "large­scale system" nor what large-scale system theory is. While this diversity does suggest that the field {whatever it may turn out to be} is in a state of flux, it does not necessarily imply chaos. There are a number of themes in the field and to some extent the differences are a matter of emphasis of particular viewpoints.

1 Borrowed from D. Gabor, Inventing the Future, Gollancz, London, 1965.

v

vi INTRODUCTION

There is, however, general agreement that large-scale systems (whatever definition one adopts) are not just straightforward but larger versions of "small-scale systems". Furthermore, new issues arise when one recognizes that most existing large-scale systems (such as an interconnected power system) consist of subsystems interconnected together, and due to economic, political, social and other constraints, as well as questions of reliability and complexity, decentralized operation of the system is mandatory.

Existing large-scale systems and new systems that will be designed exhibit (or are likely to exhibit) the following essential differences from a small-scale system. a) Presence of more than one controller or decision maker

resulting in decentralized computations. b) Correlated but different information available to the controllers

and possibly at different times. c) Requirements of coordination between the operation of the

different controllers resulting in hierarchical structures. d) Availability of only aggregated and highly imprecise models. e) Systems that may operate as a "team" or in an essentially

conflicting manner. Thus, there may be a single-objective function, multi-objective functions, or conflicting objective functions. 2

f) Moreover, it may be more reasonable to adopt a "satisfying" strategy rather than an optimizing strategy.

We are pleased that the papers presented in the conference were able to address themselves to many of these issues.

The problems of centralized and decentralized operation of economic and organizational systems have been of concern to economists since the time of Adam Smith. The paper of Groves is concerned with incentive compatible control of decentralized organizations. In the same area, Kulikowski studied problems of decentralized management and optimization of development in large production organizations.

Recent years have witnessed increasing interaction3 between control theorists and economists in attempting to apply the tools of optimal control theory to problems of the control of a national economy, certainly a prime example of a large-scale system. The idea here is to work with econometric models (possibly aggregated) and see the effects of various feedback strategies in trying to operate the economy about some desired paths.

2 H. Simon, Administrative Behavior, MacMillan, 1947.

3 G. K. Chow, Analysis and Control of Dynamic Economic Systems, Wiley, 1975.

INTRODUCTION

Westcott1s paper deals with these questions in the context of the modelling and control of the U. K. economy.

A fundamental concept in system theory is the concept of

vii

the llstate!! of a system. Roughly speaking, the !!state" of a system is an entity in which the past history of the system is summarized. Thus, knowing the system and the inputs to the system, it is pos sible (in pri nciple) to predict the future (pe r haps probabilistic) behavior of the system. Witsenhausen in his paper presents his initial thoughts on the definition of the !!state!! of a system when there are multiple controllers with different informa­tion structures controlling the system.

The theory of games certainly constitutes one of the scientific underpinnings of the study of large-scale systems. Over thirty years ago, von Neumann formulated the "games in extensive form" to describe dynamic decision making problems involving more than one person each with his own payoff and having access to generally different informatio n. However, main stream game-theoretic literature quickly turned to the more abstract form of games: Strategic form and the Characteristic Function form where !!what is the optimal decision? II is often the last question to be asked. It was left to differential games (a more modern development which is essentially a very specialized form of games in extensive form where the game tree is described by differential equations of motion) to rekindle our interest in extensive games. On the other hand, many of the concepts developed in traditional game theory serve as natural generaliza­tions of the established results in control and system theory which are one person extensive games. The papers of Blaquiere, Cruz, Olsder, Leitmann and Guardabassi deal with differential games, while the pape rs of Polak and Li n deal with computational con­siderations for multiple-objective optimization problems. They all represent the merging of ideas in these two disciplines.

Another branch of Large-Scale System Theory is Team Decis ion Theory. Here there is a single-objective function for the "team!! but the information available at the various control stations on which decis ions are to be made is different. No papers directly related to this theme were presented at the conference. However, the paper of Groves and that of Chu all had origins in team theory.

The role of information in large-scale systems has already been mentioned several times in this introduction. This question has been further investigated by Sandell when he discusses inform­ation flow in large-scale systems. In a somewhat different vein, Drenick presents a theory of organization based on concepts from Information Theory a la Shannon.

viii INTRODUCTION

Statistical mechanics is the physical theory which connects the observable behavior of large material objects with the dynamics of the invisibly small molecules constituting these objects. In statistical mechanics, there is an interplay between two different descriptions of the same object--the observational or macroscopic description and the dynamical or microscopic description. Both descriptions may be regarded as simplified models of reality that is more complex than either. It is the task of statistical mechanics to find and exploit the relationship between the two schemes of description. Aoki in his paper discusses fluctuations associated with the macroscopic state of a large system, terminology certainly reminiscent of statistical mechanics. Statistical mechanics is also alluded to by Astrom who develops a theory for large-scale flow systems.

The idea of decomposition was first investigated theoreti­cally in mathematical programming. The first paper was that of Dantzig and Wolfe when they investigated large linear programs with special structure. For linear programs, the structure of the system being studied is reflected in the pattern of zero and nonzero elements in the constraint coefficient matrix. When the program is large, the density of nonzero matrix elements is generally less than a few percent. Moreover, these are often arranged in an ordered way. For non-linear programs the situa­tion is less clear. There are two basic approaches to solving these large programs. The first approach is to use a centralized approach which, however, exploits the problem structure to perform certain computations efficiently. Examples of such efficient methods are the use of generalized upper bounding procedures and compact basis triangularization. The other approachis the decentralized approach where the problem is sub­divided into subproblems involving parameters, the subproblems are solved for fixed values of parameters, and then the parameters are adjusted by means of a coordinator in an appropriate way, and the subproblem is re-solved so that in the limit the solution to the original problem is obtained.

Attempts have been made to axiomatize this second approach, in particular, at Case Western Reserve University by Lefkowitz, Mesarovic and others who have called it the multi-level approach. In this approach the first level consists of independent subproblems, accounted for by defining one or more "second-level" subsystems, which influence in some way the original SUbsystems defined to be in the first level. The influence may take various forms depending on the initial decomposition of the problems. The goal of the second level is to coordinate the actions of the first level units so that the solution of the original problem is obtained. The origin of these ideas may be found in theories of organization.

INTRODUCTION ix

The multi-level approach has been used to solve dynamic optimization (control and filtering) problems with varying degrees of success (or failure). In the deterministic case work has been done by Pearson, Mesarovic and others and in the ·stochastic case by Athans, Chong, and others. Unfortunately, contrary to the mathematical programming situation, problems with special structures have not been discovered.

Computational experience with large linear programs seems to indicate that from the point of view of computational efficiency it usually is more advantageous to solve the problem in a central­ized fashion exploiting techniques like generalized upper bounding techniques. However, the important point perhaps is that one is often forced to solve the problem in a decentralized fashion, for example, due to difficulties in gathering data, to reduce fast memory requirements, more efficient debugging of programs, etc. This would be especially true for control problems which arise, for example, in power systems operation, where the con­trollers are likely to be in geographically separate locations due to ownership boundaries. Thus, large-scale system problems can be classified from an operational viewpoint. One distinguishes between the "planning and design of a large-scale system" and the actual "control of a large-scale system". The former is an off-line process while the latter is a real time control problem often with information processing constraints and may have to be done in a decentralized fashion.

These issues are discussed in the papers of Mayne and Chu while Davison addresses himself to the problem of robust decen­tralized control of a general servomechanisms problem.

The papers by Willems and Siljak properly belong to the qualitative theory of large-scale systems. They attempt to under­stand the global behavior of a large-scale system from knowledge about the behavior of the subsystems and the nature of the inter­connection. Willems and Siljak focus on stability and instability. Recent works of Morse and Corfmat (not reported in this Confer­ence) have discussed stabilizability of a system from a knowledge of qualitative (controllability, observability) properties of sub­systems.

It can be said that almost all of the work in control and system theory rests on the general framework of rational decision making. Typically, this framework is supported by a set of axioms (e. g. » those of the utility theory) which we seldom question or are conscious of. However, it has been argued that some of these axioms need to be re-examined from the viewpoint of operational feasibility. It is one thing to say that an individual shall order all consequences and quite another to actually carry out such an ordering. Often conceptually simple ideas are practically impossible to carry out. The most eloquent spokesman along such lines in recent years is undoubtedly L. Zadeh who has advocated

x INTRODUCTION

the idea of fuzzy sets to form both a descriptive and prescriptive theory of complex systems. Thus, his paper in this volume together with the criticisms of Mayne provide a counter point to balance our presentation. In addition, the theory of fuzzy sets need not always be considered as an alternative or in conflict with rational decision making. The former can be considered as a generalization of the latter. The paper by Aubin is one such example.

The works of Jacobson and MacFarlane cannot be strictly classified as belonging to the realm of large-scale systems. However, both represent a fresh look at classical problems. In the spirit of "inventing the future" by way of re-examining the past we have included them as part of the record.

The following table of contents is essentially the final program of the conference. The exception is the lecture by Hurwicz who did not submit a written version of the talk.

We would like to take this opportunity to thank Miss Marie Cedrone who took care of all the myriad details of organization so efficiently and ably. We also appreciate the assistance of M. Kastner, E. Mageirou, F. Schoute, D. Stein, and F. K. Sun in the preparation of this proceedings volume.

Finally, this conference could not have taken place without the financial assistance of the Office of Naval Research and the cooperation of Dr. Stuart Brodsky of ONR. We are grateful for their continuing support.

Cambridge, Massachusetts Christmas, 1975

Y.C. Ho S. K. Mitter

Contents

Information Flow in Decentralized Systems Nils R. Sandell, Jr.

Decentralized Control and Large Scale Systems D. Q. Mayne

Comparison of Information Structures in Decentralized

1

17

Dynamic Systems . . . . . . . . . . 25 Kai-ching Chu

On Fluctuations in Microscopic States of a Large System. 41 Masanao Aoki

Flow SysterIJs K. J. Astrom

Some Remarks on the Concept of State H. S. Witsenhausen

On Multicriteria Optimization E. Polak and A. N. Payne

57

69

77

Dynamic Games with Coalitions and Diplomacies. . . .. 95 A. Blaquiere

Three Methods for Determining Pareto-Optimal Solutions of Multiple-Objective Problems. . . . 117

Jiguan G. Lin

Stackelberg Strategies for Multilevel Systems . . . . . . 139 J. B. Cruz, Jr.

Incentive Compatible Control of Decentralized Organizations. . . . . . . .

Theodore Groves

xi

149

xii

Some Thoughts About Simple Advertising Models as Differential Games and the Structure of Coalitions

Geert Jan Olsder

Equilibrium Patterns for Bargaining under Strike: A

CONTENTS

187

Differential Game Model. . . 207 S. Clemhout, G. Leitmann, and H. Y. Wan, Jr.

Permanent Differential Games: Quasi Stationary and Relaxed Steady-State Operations 235

G. Guardabassi and N. Schiavoni

Modelling and Control of the U. K. Economy. J. H. Westcott

Decentralized Management and Optimization of Development . . . . . . . . .

Roman Kulikowski

Organization and Control R. F. Drenick

Decentralized Stabilization and Regulation in Large

253

265

279

Multivariable Systems. . . . . . . 303 Edward J. Davison

The Role of Poles and Zeros in Multivariable Feedback Theory .............. 325

A. G. J. MacFarlane

The Linguistic Approach and Its Application to Decision Analysis. . . . . . . 339

L. A. Zadeh

Fuzzy Core and Equilibria of Games Defined in Strategic Form. ..... 371

Jean-Pierre Aubin

Stabilization and Optimal Control of Nonlinear Systems Homogeneous -in-the -Input. . , . . 389

D. H. Jacobson

Stability of Large Scale Interconnected Systems Jan C. Willems

Large-Scale Systems: Optimality vs. Reliability D. D. Siljak and S. K. Sundareshan

List of Authors

Index .

401

411

427

431

* INFORMATION FLOW IN DECENTRALIZED SYSTEMS

Nils R. Sandell Jr.

Electronic Systems Laboratory

Massachusetts Institute of Technology

ABSTRACT

An important aspect of the design of large scale engineering systems is the specification of interfaces between various control­ler and measurement subsystems and of the information that must cross these interfaces. Examples drawn from aerospace, transpor­tation, power, and communications systems are given to illustrate this point. A formal theory of interface design is presently un­available, and so several heuristic approaches are described.

I. INTRODUCTION

Persons involved in the design and analysis of large scale engineering systems will attest to the ubiquity of the vexing problem of the specification of subsystem interfaces. Information must cross the boundary of e.g., a navigation or tracking sub­system and a fire control subsystem. Since these subsystems are often designed by separate groups of engineers, problems seem to inevitably arise at this juncture. Yet it is only recently, in the emerging area of decentralized control, that the attention of systems theorists has turned, at least implicitly, to this problem.

* This research has been conducted in the Decision and Control Sciences Group of the M.l.T. Electronic Systems Laboratory with support by NSF under grant NSF-ENG 75-10517, by NASA under grant NGL-22-009-124, and by ONR under grant ONR 041-496.

2 N.R. SANDELL, JR.

In general, a large scale engineering system is characterized by a multiplicity of system outputs (measurements) and inputs. A designer must determine a control system structure that assigns the choice of each system input to a given controller, and that specifies the interfaces between the various controllers and the information-gathering subsystems. Thus the information pattern of the system [1] is determined.

The desirability of a given control system structure is as­sessed in terms of its implementation cost and the performance of the best control system with the prescribed structure. Thus evaluation of control system structures involves the solution of an optimal control problem. This optimal control problem will in general be a non-classical stochastic control problem [1] when the specified control structure is decentralized. Since this lat­ter problem is unsolved even for the special linear-Quadratic­Gaussian (LQG) case [2-5], it is not surprising that a formal theory of specification of control system structure is presently unavailable. A notable exception to this statement is found in the case of static LQG teams [6,7].

In the absense of a formal theory, consideration of specific physical examples in which a decentralized control system structure seems natural is desirable. In Section II of this paper, a number of such examples drawn from aerospace, transportation, power, and communications systems are given. Section III contains discussion of several heuristic methods of determining control system structures, and Section IV contains a summary and conclusions.

II. EXAMPLES

In this section, selected physical systems are discussed for which decentralized control and estimation algorithms seem to be indicated on the basis of intuitive, historical.' and/or economic considerations.

A. Aircraft

The nonlinear equations of motion of an aircraft are of the general form

x = f (x, u) (1)

where x is the state vector of aircraft translational and rotational positions and velocities and u is the control vector of aircraft control surface deflections.

If equation (21) is linearized about a nominal state and control vector (i.e., a specified flight condition) a linear e­quation of the form

x(t) = Ax(t) + Bu(t) (2)

INFORMATION FLOW IN DECENTRALIZED SYSTEMS

is obtained, where x(t) and u(t) now denote the deviations from nominal of the state and control vectors. If the flight condition is steady symmetric flight, the system matrix has the form

A (3)

3

where Xl and x2 are the vectors of longitudinal and lateral states,

respectively [8]. Aircraft control systems (stability augmentation systems, autopilots, etc.) are ordinarily designed on the basis of the linearized equations.

Several points concerning equation (2) deserve emphasis. First, notice that the (linearized) lateral dynamics are unaffect­ed by the longitudinal dynamics. Moreover, it turns out that the coupling in the reverse direction is quite weak, so that it is common practice to take A12 = 0 and design separate controllers

for the lateral and longitudinal dynamics (see, e.g., [9]). Second, the phenomena expressed by equation (2) are naturally expressed in quite different time scales. For example, the longitudinal dynamics are comprised of two modes: the phugoid mode (about 130 sec time constant) and the short period mode (about 1 sec time constant). The separation of time scales can be further accentuat­ed if the rigid body assumption is relaxed and flexure modes are considered [10], and can be utilized to design separate control­lers for the various response modes.

B. Inertial Navigation Systems

The error equations for a slowly moving terrestrial navigator with a local level, north pointing platform can be written in the form [11]

O~ 0 1 0 0 0 0 0 ORE

OVE _w2 0 0 2r1 0 0 -g OVE s u ORN 0 0 0 1 0 0 0 O~

d OVN 0 -2r1 -lJl 0 0 0 OVN dt u s g

lJ1u 0 0 0 0 0 rlN 0 lJ1u

lJ1E 0 0 0 0 -ri 0 rI lJ1E N u

lJ1N 0 0 0 0 0 -ri 0 lJ1N u

4 N.R. SANDELL, JR.

+

o

bE

o bN

£ u

£E

£N

where

ov velocity error

OR = position error

~ platform tilt angle with respect to computer axes

Q earth angular velocity

( )E= east component

( )N= north component

( ) = up component u

g = gravitational force per unit mass at R

R = distance to center of earth

£ = gyro dr ift

b = accelerometer error

Ws = ~g/R, Schuler frequency

(4)

Notice that the ~-angle formulation is employed in equations (4) instead of the more usual ~~angle formulation [12]. The state variables of the *-angle formulation are related to those of the ~-angle formulation by a linear transformation that gives the relatively decoupled form of the equations (4).

Equation (24) can be written in the form

. Xl All A12 A13 xl wI

x2 A2l A22 A23 x2 + w2 (5)

x3 0 0 A33 x3 w3

where

INFORMATION FLOW IN DECENTRALIZED SYSTEMS

Isolated subsystems 1 and 2 are referred to as the north and east Schuler loops, and they are simple harmonic oscillators with an

5

84 minute period of oscillation. The Schuler loops are weakly coupled since the RMS level of the coupling term 2nuoV is ordi­narily at least an order of magnitude smaller than W!OR.Subsystem 3, rererred to as the earth rate loop, is not affected by subsystems 1 and 2. This matrix A33 has a pole at the orign, and a pair of undamped complex poles with 24 hr period of oscillation.

Thus the error dynamics of an inertial navigation system are seen to consist of weakly coupled subsystems, with fast and slow modes of oscillation. This fact can be utilized to design a sub­optimal Kalman filter consisting of a slow filter for the earth rate loop and a pair of decoupled filters for the Schuler loops. Variations of this idea are implicit in the literature - see, e.g. [13, 14].

C. Power Systems

The classical model for a system of N interconnected electric generators is given by the equations

P . e,~

N

P . - p . m,~ e,~

=1: sin j=l

(0. - 0.) + P n • ~ J )(" ~

[15], where

M. inertia constant of generator i ~

o. phase angle of machine i ~

P = mechanical power of machine i m,i

P e,i electrical power of machine i

(6)

(7)

(8)

6

v. = instantaneous terminal voltage, machine i l.

Iv.l= l.

magnitude of voltage, machine i

60 hz

N.R. SANDELL, JR.

Wo

X .• l.J

reactance of transmission line between generators i and j

P = local load of generator i R"i The classical model assumes that P ., Iv.1 and Po . are

m,l. l. N,l.

constant, and that the transmission system is loss less and in steady state. The assumption that P . is constant can be dropped m, l. by incorporating boiler-turbine-governor models. The assumption that Iv.1 is constant can be relaxed by modeling the exciter and

l.

the machine's electrical dynamics, and load models can also be used. But in any case the coupling between the variables will be of the form (8).

Thus dynamic models of interconnected power systems, when linearized have the general form

d dt

where A .. is l.J

( I v. I modeled l.

(Iv., and O. l. l.

~l x 1.

+

(9)

either zero (X .. = 00), has a single non-zero element l.J

as constant, 0i is a state) or two non-zero elements

are states).

Several points should be made with regard to the model above. First, the strength of the direct coupling between generators is dependent on the strength of the transmission line connecting them. Second, ignoring the coupling matrices A .. and A .. , as is

1J J 1 commonly done in design of power system controls, is equivalent to

INFORMATION FLOW IN DECENTRALIZED SYSTEMS 7

* assuming that i and j are connected through an infinite bus. Third, the various models comprising the overall power system model have dynamics with widely differing time constants. For example, the time constants of a boiler may be on the order of minutes and those of an exciter on the order of seconds. The systems average frequen­cy changes very slowly relative to the instantaneous frequency deviations of its generators [16].

These properties are utilized in the design of the various controllers for the power system. For example, interaction of slow boiler-turbine dynamics with other parts of the system is ignored in the design of boiler-turbine controls, and voltage regulators are ordinarily designed separately for each generator without consideration of the interactions between the voltage regulators of different machines. Interestingly, this latter method of design has led to power system oscillations in several cases [17,18].

D. Freeway Traffic Corridor

Hacroscopic models of traffic flow have been developed by analogy with the theory of the flow of compressible gases [19-20]. The resulting distributed parameter model can be discretized ap­proximately in space by sectioning the freeway into links. The state variables are the aggregate density (vehicles/mile) and aggregate velocity (miles/hour) of the vehicles on each link, and the control variables are the on and off ramp flow rates.

If the resulting nonlinear state equations are linearized about nominal profiles of density and velocity, Linear state equations of the form (10) are obtained [19-20]. More complicated equations are obtained for more involved traffic networks.

Under the assumption A .. = 0, i t j, the system decouples ~J

into a set of N independent equations. This assumption eliminates completely the coupling between the links, and since the significant dynamics are not contained within the links but result from their

* An infinite bus has voltage

Ivl sin(Wot+o)

with Ivl and 0 constant. Thus an infinite bus can absorb or generate any amount of power without a change in its voltage or phase.

8 N.R. SANDELL, JR.

xl All A12 0 0 0 0 0 xl

x2 A21 A22 A22 0 0 0 0 x2

d dt

o o o o o

+ (10)

coupling, is not too useful. A more reasonable assumption that has been advanced is to consider only the dynamics of adjacent links in designing a control system for a given link. E.g., the controller for link 2 is obtained by considering the equation

[::1 T" A12

:23l: J

[ " U1

] d 11'22 B2U2 (11) - A21 +

dt

0 A32 B3U3 33 x3

Success with this approximation has been reported in the literature [21] .

Another recent approach to the freeway corridor control problem is not based on an approximate model such as (11), but does employ a decentralized structure in which controllers have information only about links adj acent to the link they are controlling. After an optimization over the controller parameters, the performance was only 11% suboptimal as indicated by linear analysis with good performance in simulation [22]. Due to the decreased requirements for information flow, implementation of the decentralized control­ler structure is much simpler.

INFORMATION FLOW IN DECENTRALIZED SYSTEMS 9

E. Packet switching Data Networks

Recently, macroscopic [23,24] and microscopic [25,26] control­oriented models of packet switching data communication networks have been obtained. As discussed in [23], control of these net­works involves many difficult issues, including an extremely strong, explicit coupling between message and control information. An extremely simple model presented in [24] is discussed below.

The structure of the network is described by a capacity matrix C with components c .. :.. 0, i,j = l; ... ,N representing the capacity

1J in bits per second of the channel between nodes i and j. Defining the variables

x .. (t) 1J

a .. (t) 1J

k u .. (t)

1J

number of message bits at node i whose destination is node j.

external arrival rate in bits per second of messages with destination node j at node i.

fraction of C .. allocated to messages at node i 1J

destined (ultimately) for node k.

and neglecting the granularity of the message bits into messages, equation (12) is obtained

~ .. (t) 1J

The obvious constraints are

x .. (t) k > 0 u .. (t)

1J 1J

and

2: k < 1 u .. k 1J

k u .. (t)C .. +

1J 1J 2:2: k i

k u .. (t)C ..

J1 1J

> 0

Equation (12) has properties reminiscent of equation (10).

(12)

(13)

(14)

The dynamics of accumulation at each node are trivial - it is only the overall dynamics of the net that are significant. Thus there is no sense in which one can say that the nodes are weakly coupled with their neighbors. However, it may be possible, in a fashion analogous to (11), to neglect the dynamics of distant nodes in the design of local controllers.

Although analysis of dynamic routing in message switching data networks is just begun, such networks have been built. The ARPANET [27] utilizes an interesting decentralized routing

10 N.R. SANDELL, JR.

algorithm in which controllers located at the nodes of the network asynchronously share information with their neighbors [28].

III. APPROACHES TO STRUCTURING DECENTRALIZED SYSTE!vIS

In the previous section, a number of systems were described in which decentralized control structures seem feasible. Experience with these systems has indicated what dynamic interactions are critical, and therefore what degree of information flow is requir­ed. But more formal analytic tools are desirable both to sharpen knowledge of relatively well-understood systems and to approach new systems. In this section, a number of familiar methods from linear systems theory are reviewed with emphasis on the insight that these methods provide for choosing decentralized control structures.

A fundamental method in linear system theory is eigenanalysis. It is well-known that the eigenvalues of a linear system determine the relative stability of the modes of the system, but the role of the eigenvectors in determining the mode shapes is less often emphasized. Knowledge of mode shapes determines how the state variables of a large system interact in modal oscillations, and can therefore suggest what dynamic interactions are irnportantl The system designer can use this information to specify state variable information flows adequate to permit suppression of the oscillation by the controllers of the system. Another, related source of information is the sensitivities of the eigenvalues to system parameters [31].

Determination of Lyapunov functions is also usually thought of as a stability method, but can give considerable insight into large system dynamic interactions. Consider the linear system

x(t) = Ax(t)

and the Lyapunov equation

ATKi + KiA + Ei = 0

(15)

(16)

i where E colurnni,

is identically zero except for its element in row i, which is 1. If the system is stable, then the diagonal

elements

i k ..

JJ

i i k .. of K have the following interpretation: JJ

2 x.(t)dt

J (17)

1 Interesting illustrations of this statement have appeared recent-ly in the power system literature [29-301.

INFORMATION FLOW IN DECENTRALIZED SYSTEMS

where

x (0) k

k ':f i

k i i .

Thus k .. ~s the energy transfered

(18)

to state j by a unit initial

11

conditf6n disturbance in state i. sible for the equation

A similar interpretation is pos-

i iT' Al: + l: A + E~ o (19)

In this case, the diagonal element a~. of l:i is the steady-state power in the response of state j to ~Junit spectral intensity white noise disturbance affecting state i.

The last heuristic method for choosing decentralized control system structures to be discussed is, perhaps paradoxically, the solution of the usual centralized, steady-state, linear-quadratic­Gaussian (LQG) optimal control problem. solution of the centraliz­ed LQG problem is useful for the following reasons. First, the centralized solution provides a bound which can be used to compare the degradation in performance of various decentralized strategies (which, as discussed earlier, enjoy implementation advantages that cannot be incorporated into the optimization with the present state of the art.) Second, the centralized solution can provide considerable insight into desirable choices of decentralized system structure. This last point requires elaboration.

Consider a linear system of the form

[;;1 (t)] ;;2 (t) ][ :::::] + [:::::::]

(20)

with cost functional

(21)

It is well known (see, e.g. [32], that the solution of this problem is

] [Xl (t)] x 2 (t)

(22)

where the gain matrix in (22) is obtained by solution of an alge­braic Riccati equation. Note that the presence of the cross coupl-

12 N.R. SANDELL, JR.

ing gains G12 implies that the controller that chooses the input

ul must have access to the state variable x2 (or its estimate).

This may imply the need for communication links in a physically distributed system, but in any case requires the existence of an interface between the controller and the measurement and/or estimation subsystem associated with x2•

The nonzero gains G12 have two purposes:

(a) The input ul can be used to control x2•

(b) Disturbances in x2 propagate to xl. The resulting

disturbance in xl is controlled by ul •

Credance is lent to this statement by consideration of the special cases A12 = 0, and A21 = O. In the latter case the cross coupling

gains serve only purpose (b) (since x2 is uncontrollable by ul )

while in the former case the gains serve only purpose (a) (since disturbances in x2 do not propagate into xl).

On the basis of the preceeding considerations it is seems intuitive that if the systems are only weakly coupled, then the gains G12 and G21 will be small. However, this is only true if

Qll and Q22' Rll and R22 are comparable, since, e.g., if u2 is

penalized very heavily, then the optimal controller will tend to rely on ul even if this control is intrinsically less effective in

controlling x2• Solution of the Riccati equation for this problem

will blend

the system dynamic interactions

the effectiveness of the various controllers

the overall performance criterion

to give a gain matrix reflecting all these factors. Thus inspection of the centralized solution G matrix can give valuable information about preferable ways of decentralization of the control system.

In a dual fashion, solution of the Riccati Equation associat­ed with the filtering problem blends

the system dynamics

the dynamic uncertainty

the sensor characteristics

INFORMATION FLOW IN DECENTRALIZED SYSTEMS

to produce a filter gain matrix reflecting all the above factors. Thus inspection of the centralized filter gain matrix gives information about desirable way of decentralizing the estimation subsystem.

In this section, various heuristic methods for choosing decentralized control system structures have been presented. Existence of these methods does not detract from the need for a more scientific theory of interface specification. However, such

13

a theory will probably have to await considerable advances in non­classical stochastic control theory.

IV. SUMMARY AND CONCLUSIONS

Decentralized control system structures imply varying degrees of information flow. Greater communication of state variable information will in general permit better control system per­formance, but will require a more expensive and less reliable implementation. This tradeoff should ideally be resolved by optimization, but this would require solution of a nonclassical stochastic control problem which is not presently possible. Therefore, the designer must proceed on the basis of he.uristic and intuitive reasoning. A number of examples were given to illustrate the possibility of such reasoning, and the role of certain common methods in linear system theory in developing insight was stres­sed.

Acknowledgements

Many of the ideas in this paper arose in conversation with a number of individuals. The author would particularly like to thank Professors M. Athans, F.C. Schweppe and P. Varaiya, and Mr. W. O'Halloran Jr.

REFERENCES

[1). H.S. Witsenhausen, "Separation of Estimation and Control for Discrete Time Systems", Proc. IEEE, Vol. 59, No.ll, 1971.

(2). H.S. Witsenhausen, " A Counterexample in Stochastic Optimal Control, SIAM J. Control, Vol.6, 1968.

[3). Y.C. Ho and K.C. Chu, "Team Decision Theory and Information Structures in Optimal Control Problems - Part I", IEEE Trans. A.C., Vol. AC-17, No.2, 1972.

[4). K.C. Chu, "Team Decision Theory and Information Structures in Optimal Control Problems - Part II", IEEE Trans. A.C.,

14 N.R. SANDELL, JR.

Vol. AC-17, No.1, 1972.

[5]. N.R. Sandell Jr. and M. Athans, "Solution of Some Non-clas­sical LQG Stochastic Decision Problems," IEEE Trans. A.C., Vol. AC-19, No.2, 1974.

[6]. R. Radner, "The Evaluation of Information in Organizations", Proc. Fourth Berkely Symp. on Math. Statistics and Probabili­ty, pp. 491-530, Univ. of Calif. Press, Berkely, 1961.

[7]. J. Marshak and R. Radner, The Economic Theory of Teams, Yale Univ. Press, New Haven, Conn., 1971.

[8]. B. Etkin, Dynamics of Atmospheric Flight, Wiley, N.Y., 1972.

[9]. M. Athans, K.P. Dunn, C.S. Greene, W.H. Lee, N.R. Sandell Jr., 1. Segall, and A. S. Willsky, "The Stochastic Control of the F-8C Aircraft Using the Multiple Model Adaptive Control (IYIMAC) Method", Proceedings 1975 Decision and Control Conference, Houston, Texas, December 1975.

[10]. T.L. Johnson, M. Athans, G.B. Skelton, "Optimal Control Surface Locations for Flexible Aircraft", IEEE Trans. A.C. Vol. AC-16, No.4, 1971.

[11]. J. C. Pinson, "I,nertial Guidance for Cruise Vehicles", in Guidance and Control of Aerospace Vehicles, C.T. Leondes, ed., McGraw-Hill, New York, 1963.

[12]. C. Broxmeyer, Inertial Navigation Systems,McGraw-Hill, New York, 1964.

[13]. A.E. Bryson, Jr. "Rapid In-Flight Estimation of IMU Platform Misalignments Using External position and Velocity Data", Unpublished memo., Stanford University, January 1973.

[14]. W.F. O'Halloran Jr., and R. Warren, "Design of a Reduced State Suboptimal Filter for Self-Calibration of a Terrestrial Inertial Navigation System, AIAA Guidance and Control Conference, Stanford university, 1972.

[15]. E.W. Kimbark, Power System Stability: Synchronous Machines, Dover, N.Y., 1968.

[16]. M.L. Chan, R. Dunlop, and F.C. Schweppe, "Dynamic Equivalents for Average System Frequency Following Major Disturbances", IEEE Trans. P.A.S., Vol., PAS-9l, No.4, 1972.

[17]. R.T. Byerly and E.W. Kimbark, Eds., Stability of Large Electric Power Systems, IEEE Press, N.Y., 1974.

[18]. D.L. Bauer et. al., "Simulation of Low-Frequency Undamped Oscillations in Large Power Systems", IEEE Trans. P.A.S., Vol. PAS - 94, No.2, 1975.

[19]. M.J. Lighthill and G.C. Whitman, "On Kinematic Waves II: A

INFORMATION FLOW IN DECENTRALIZED SYSTEMS

Theory of Traffic Flow on Long Crowded Roads", Proc. Royal Soc. (London) Sere A, Vol. 229, 1955.

[20]. P.K. Houpt, Decentralized Stochastic Control of Finite State Systems with Applications to Vehicular Traffic Flow, Ph.D. Thesis, M.I.T., Nov. 1974.

[21]. L. Isaksen and H.J. Payne, "Suboptimal Control of Linear Systems by Augmentation with Application to Freeway Traffic Regulation", IEEE Trans. A.C. Vol. AC-18; 1973

[22]. D. Looze, Decentralized Control of a Freeway Traffic Cor­ridor, M.S. Thesis, M.I.T., August 1975.

[23]. N.R. Sandell Jr. and M. Athans,"Relativistic Control Theory and the Dynamic Control of Communication Networks", Proceed­ings 1975 Decision and Control Conference, Houston, Texas, December 1975.

[24]. A. Segall, "New Analytical Models for Dynamic Routing in Computer Networks", National Telecommunications Conference, New Orleans, December 1975.

15

[25]. L. Aldermeshian, Model Reduction Techniques in the Stochastic Control of Data Communication Networks, M.S. Thesis, M.I.T. November 1975.

[26]. L. Aldermeshian and N.R. Sandell Jr., "Model Reduction Tech­niques in the Stochastic Control of Data communication Networks", Proc. 9th Asilomar Conference, November 1975.

[27]. H.Frank, R.E. Kahn, and L. Kleinrock, "Computer Communication Network Design-Experience with Theory and Practice", Networks Vol.2, No.1, 1972.

[28] J.M. McQuillan, Adaptive Routing Algorithms for Distributed Computer Networks, Report No.283l, Bolt Beranek and Newman, Inc., May 1974.

[29]. R.T. Byerly, D.E. Sherman, and D.K. McLain, "Normal Modes and Mode Shapes Applied to Dynamic Stability Analysis", IEEE Trans. P.A.S., Vol. 'PAS - 94, 1975. --

[3D]. J. Szajner, Local Model Control of synchronous Generators, D.Sc~ Thesis, M.I.T., January 1975.

[31]. J.E. Van Ness, J.M. Boyle, and F.P. Imad, "Sensitivities of Large, Multiple-Loop Control Systems," IEEE Trans. A.C., Vol. AC-IO, No.3, 1965.

[32]. M. Athans, "The Role and Use of the LQG problem in Control System Design", IEEE Trans. AC., Vol.AC-16, No.6, 1972.

DECENTRALIZED CONTROL AND LARGE SCALE SYSTEMS

D.Q. MAYNE

Professor of Control Theory, Department of Computing and

Control, Imperial College of Science and Technology

London SW7 2BZ

INTRODUCTION

It is natural to hope that the large literature on the theory of large scale systems is relevant to the common control problem of synthesising regulators for complex linear or nonlinear multi variable systems. However there appear to be significant differences in the literature on these two topics. This paper attempts to explore these differences, briefly surveying the relevant literature, in the hope of drawing lessons for future advances.

Rosenbrock [lJ has emphasized elsewhere the untidy nature of many engineering problems. Firstly, they have many objectives such as (in our case) stability, speed of response, insensitivity to disturbance and parameter variation, resulting in the existence of more than one satisfactory solution, and requiring knowledge of trade-offs between objectives. Secondly, they have numerous constraints, often too many to list formally, although non­satisfaction is easily recognised. Thirdly, knowledge of the system is usually inexact, often because of simplification (for example, replacement of distributed parameter components by lumped approxim­ations). Certainly the design problem is complex, and the theory of large scale systems should help.

LARGE SCALE SYSTEMS

An important part of the literature on large scale systems is concerned with solving large sets of equations or large mathematical programming problems. The purpose of this literature is quite clear - reduction of computation - and this purpose provides the criterion by which the work can be judged. Despite its success for certain

17

18 D.Q. MAYNE

linear problems (e.g. decomposition of linear programs), this work is not directly applicable to the control of large noisy systems for many reasons, including (i) optimal feedback control (necessary because of random noise) is prohibitively difficult to calculate and to implement, and (ii) engineering problems typically have many objectives.

Certainly there do exist techniques for finding (an approximation to) the locus of non-inferior points in multicriteria optimization. In principle, since engineering problems typically have more than one objective such work should have some relevance. However, I am unaware of any application to multivariable control/decision problems.

Of more direct relevance is the large literature on decentral­ized control of stochastic systems. The set up is the usual one of· a set of controllers, each with the same performance criterion, but having its own set of permissible decisions, its own information and its own model of the system. As Varaiya [4J points out, there are three categories of relevant literature: (a) team theory; (b) competitive organizational forms; (c) hierarchical organizational forms.

The interesting feature of the competitive economy is its achievement of optimality through the decentralized action of numerous agents, each supplied merely with information of prices of various commodities. However this result is somewhat specialized relying heavily on special features (determinism, convexity) of the problem, and should not be too readily adopted as a desirable model in other fields; its overenthusiastic adoption in the area of non­linear programming (decomposition algorithms) has not achieved a substantial reward.

The team theory literature, on the other hand, gives due attention to the stochastic nature of many decision problems. In the static case under reasonable conditions (Gaussian disturbances, convexity of cost function) the optimal decision rule is linear. In the dynamic case, the classical information structure results, under the usual LQG conditions, in a linear optimal decision rule, but Witsenhausen's [2J well known result showed that for non­classical information structures the optimal decision rules need not be linear. Ho and Chu [6J, performed a valuable service by showing that linear optimal decision rules also result if the information structure is nested; the problem is reduced to an equivalent static problem. The resultant optimization problem, though feasible, is still severe. In fact, Varaiya [4J conjectures that the computational effort necessary for calculating optimal decision rules for a team is considerably greater than that for centralized decision making, so the former would only be chosen because of constrdints, or communication costs. Unless some result is obtained extending the class of problems admitting linear solutions, further progress is unlikely. In general the optimal decision rules are

DECENTRALIZED CONTROL AND LARGE SCALE SYSTEMS 19

nonlinear and prohibitively difficult both to compute and to implement. The difficulty of implementation is also an argument against the pursuit of optimality for nonlinear systems, since the resultant decision rules are bound to be nonlinear and complex. Our sights should perhaps be set lower, or at different objectives. The related work of decentralized stochastic control [7J achieves linear controls by imposing a fixed structure; again the resultant optimiz­ation problem is non-trivial, the controller is relatively complex and no attention paid to objectives other than optimizing an (artificial?) criterion.

One turns finally to the literature on hierarchical organisat­ional forms. The success these have undoubtedly achieved in social systems encourages our hopes, but it must be conceded that much of the control literature is disappointing. As Varaiya [3J emphasises there is no mathematical theory of such forms, but Hesarovic et al [2J do formalize the notion, classify possible designs and give five reasons for adopting a hierarchical structure: (1) one is often forced to design one level, fix this design, and coordinate at a higher level. (2) System descriptions are often only available on stratified basis. (3) Available modules (decision units) have limited capability, so that overall problem is formed into a multi­layer hierarchy of sub-problems. (4) Total resources are better utilised. (5) Flexibility and reliability is increased. Varaiya [3J, in his review of [2J, classifies the first three reasons as opinion, and states that there is little evidence for the fourth and fifth. The basic assumption in [2J is that there exists an overall performance function - an assumption which, as has already been pointed out, ill fits many engineering problems. The proposed hierarchical solution decomposes this optimization problem into a number of simpler problems - simpler in that certain constraints are ignored thus decoupling the local optimization problems. A co-ord­inator (second level controller), by adjusting certain parameters in the local optimizations, can ensure that the local solutions are co­ordinated, i.e. that the decisions obtained by solving the local optimization problems also solve the overall optimization problem. Basically, therefore, the contribution in [2J is more a contribution to mathematical programming than to systems theory - much of it directly related to standard duality theory. Apart from the fact that very few control problems can be viewed as a cooperative exercise in optimization, the claim of better utilisation of resources requires comparison of different structures, and their costs, and necessitates consideration of computational complexity, and of information trans­mission cost. Equally the claim for reliability requires comparison with other structures, and is unlikely to be justified in a design which does not include reliability as a design objective. A typical control application by a disciple of [2J deals with a determ­inistic control problem, and yields, via hierarchical optimization, an (open-loop) control schedule. The coordinating parameter is also an (open-loop) schedule, the lack of feedback being due to the

20 D.Q. MAYNE

unrealistic neglect of randomness. The coordinator is not superior to its local controllers either by virtue of task complexity (the reverse is probably true) or in the time scale of its operations, thus appearing unlike its counterpart in organizations etc.

Finally one must observe that in [2J the hierarchical structure is imposed a-priori; there is no evidence that this structure is, in any sense, optimal. This a-priori imposition of structure is also a feature of the more realistic work (in that randomness is taken into account) of Athans et al [7J. Athans considers a structure in which a set of decentalized controllers solve their own well formulated stochastic control problems, neglecting inter-actions, and apply the resultant control. A coordinator can apply a control of the same type as that of the looal controller. If the coordinator has the same information as that of the local controller he, of course, "takes over", cancelling the local controls and applying the globally optimal control. This "take over" also occurs, as Athans shows, even if each local controller transmits only the control, and not its measurements, to the coordinator. Chung and Athans [8J obtain meaningful solutions only by preventing instantaneous error-free transmission of information to the coordinator, as for example in their "periodic coordinator". The periodic coordinator still has to solve a formibable stochastic organization problem; this fact, and the a-priori imposition of structure, inhibits acceptance of this approach.

The recurrent a-priori imposition of structure, hierarchical or otherwise, highlights the lack of examples for which a hierarchical solution is optimal. It is interesting to speculate on the ingred­ients of such a problem; it appears that the problem should be stochastic, communication should be casted, and that agents should have different abilities. Consider the following problem, due to M.H.A. Davis [9J. Two agents u and v possess the capability for adding two numbers and determining sign but not for testing the equality of two numbers. A third agent w can perform the latter task. u receives two signals Xl, YI and v similarly receives X2, Y2. Xl and X2 are independent normal random variances with the same (unknown) mean ~ (which can be +1 or -1) and variance 1. YI and Y2 take value 1 or 2 and YI = Y2 only if ~ = 1; if YI ~ Y2 ~ = -1. Agents u, v are required to guess the value of~; the cost for each agent, of failure is $1. The total cost J, to be minimized, is the sum E of the error probabilities of u and v, and C, the cost of communication. Consider the following three cases: 1. No communication: u chooses ~ = 1 if Xl > 0; v chooses ~ = 1 if X2 ~ 0, E = 2 X .16 ~ .32; J = .32. -2. u, v communicate with each other with cost CI: u, veach choose ~ = 1 if Xl + X2 ~ 0, E = 2 X .08 = .16, J = .16 + CI. 3. u, v communicate with w with cost C2: w chooses ~ = 1 if YI = Y2 and communicates this choice to u, v. Hence E = 0, J = C2. Obviously any of the structures can be made optimal by choice of

DECENTRALIZED CONTROL AND LARGE SCALE SYSTEMS

CI, Cz. Note, if u, v cannot add, mutual communication has no value. Structure 3 is hierarchical in the sense that 'advice is sought' from an agent with "suPerior" abilities.

21

A common theme of all the large scale systems literature discussed above is, firstly, its concern with optimality as the sole criterion, and, secondly, its concern with the structure of the controller and not of the system being controlled. This contrasts strongly with the 'in-house' control literature.

MULTIVARIABLE CONTROL SYSTEMS

A substantial part of the control literature is concerned with objectives other than optimality and this, in turn, forces a concern with the structure of the system being controlled. A relevant example is the literature on stability of interconnected systems [10]. Here the concern is, of course, with stability, rather than optimality, and the literature is typically concerned with structural problems, such as determination of bounds on interactions in order to ensure stability of an interconnected system of stable subsystems. The recent literature [11-13] on synthesis of controllers for linear multivariable systems emphasises the classical concern with multiple objectives (stability, performance, sim9licitly of controller, loop­by-loop implementability, ease of tuning, integrity, low interaction) and, hence, trade-off, in realistic engineering situations. Now it is interesting to observe that many of these objectives, including comprehension of controlled system behaviour to facilitate tuning, loop-by-loop implementability, and integrity, require that the system being controlled has a "decentralised" structure, and that a major part of the design is devoted to "pre-compensation" to reduce interaction by making the system "appear" approximately diagonal [11] or approximately triangular [13]. The control literature on other topics, such as model simplification, decoupling etc., relevant to multi variable control problems shows a similar concern for structural properties of systems being controlled.

CONCLUSIONS

One might be forgiven for drawing the conclusion that the literature on large scale systems is dominated by a concern for optimality, perhaps due to the powerful influences of the decomposit­ion theorems in (linear) programming and the theories of competitive economics, teams and optimal control. But the concepts are much richer, as the thoughtful paper by Simon [14] on "The Architecture of Complexity" shows. The theme of the paper is that complexity, in social, biological, physical and symbolic systems frequently takes the form of hierarchy. Several important points are made: (1) the evolution of complex systems from simple systems is aided by the existence of stable intermediate forms - in which case the resultant complex form will be hierarchic. (2) Hierarchies are generally "near decomposable", with the effect that the "internal dynamics" of

22 D.Q. MAYNE

subsystems are high-frequency whereas the "interaction dynamics" are low frequency. (3) Hierarchical systems are "comprehensible", mainly because of their "near-decomposability". (4) To achieve comprehens­ibility, "recoding" may be necessary.

Optimality is not mentioned~ It is perhaps not too fanciful to compare the existence of stable intermediate forms with the property of loop-by-Ioop implementability (which requires the controlled system to remain stable as each feedback loop is closed in sequence) or with the property of integrity (maintenance of stability in face of certain component failures); near decompensability is, of course, directly comparable with' diagonal dominance' [11] or "generalised diagonal dominance" [13J, a necessary property of the compensated system in certain design procedures, while the precompensator, which achieves this, may perhaps be regarded as a form of recoding. Certainly low interaction (near-diagonal or near-triangular) systems are 'comprehensible', facilitating tuning of controller parameters etc. In any event, it is interesting to note the main virtues of the hierarchical structure are comprehensibility, and stability of inter­mediate forms (integrity). An example (of a symbolic system) is the recent work on programming [1SJ which employs hierarchical structuring to facilitate the comprehension, documentation, testing, debugging and development of large, complex programs.

To summarise it seems legitimate to criticise the large scale system literature on the following grounds: (1) it is obsessively concerned with optimality. This enthronement of optimality immediat­ly reduces its relevance to many engineering problems. Determination of optimal solutions is difficult even for small or medium scale systems; it would appear to be prohibilively difficult for large scale systems. Optimal controllers, even if they could be determined would, in most cases, be prohibitively difficult to implement and their complexity would inhibit comprehension, tuning, implementation, etc. (2) Where a decentralised or hierarchical structure is employed, this structure is imposed only on the controller, and not on the system being controlled. Since the system may have highly interactive subsystems, this procedure has major disadvantages: (a) it is well known in the multi variable control literature that the performance of a totally decentralised control scheme is severely limited if strong interactions between subsystems exist. (b) If a coordinator is intro­duced, with instantaneous error free access to subsystems measurements the coordinator will act as a central controller, making the decent­ralised controllers redundant; this take over is necessary if an improvement over the performance of decentralised controllers is required. (c) If an (artificial) attempt to prevent takeover is made, by restricting the rate at which the coordinator can act (e.g. period­ic coordination) a performance will be obtained intermediate between that of the totally decentralised controllers and that of a central­ised controller.

It should be explicitly recognised that decentralization of the controller reduces performance unless the system being controlled

DECENTRALIZED CONTROL AND LARGE SCALE SYSTEMS 23

has low interaction (is structured). In many cases the system does have this property; in others the system may be made to 'appear' almost non-interacting by 'recoding' (e.g. precompensation [11]) •

(3) The literature overlooks the fact that hierarchical systems have largely been introduced for reasons, such as comprehensibility, other than optimality, and these are relevant to applications. In large programs, for example, comprehensibility is a necessity, overriding the claims of optimality.

REFERENCES

1. H.H. Rosenbrock, "The Future of Control", IFAC 1975.

2. M.D. Mesarovic et al, "Theory of Hierarchical Multilevel Systems" •

3. P. Varaiya, Review of [2], IEEE Trans. AC-17, 1972.

4. P. varaiya, "Trends in the Theory of Decision-Making in Large Systems", Annals of Ecomonic and Social Management, 1/4, 1972.

5. H.S. Witsenhausen, "Separation of Estimation and Control for Discrete Time Systems", Proc. IEEE, Vol. 59, 1971.

6. Y.C. Ho and K.C. Chu, "Team Decision Theory and Information Structures in Optimal Control Problems", IEEE Trans. AC-17, 1972.

7. M. Athans, "Survey of Decentralized Control Methods", MIT Report ESL-P-555.

8. C.Y. Chang and M. Athans, "On the Periodic Coordination of Linear Stochastic Systems", MIT Report, July 1974.

9. M.H.A. Davis, Private communication.

10. J.P. LaSalle, "Vector Lyapunov Functions", Technical Report 75-1, Brown University, February 1975.

11. H.H. Rosenbrock, "Design of Multivariable Systems by the Inverse Nyquist Array", lEE Proc. 1969.

12. A.G.J. MacFarlane, "A Survey of Some Recent Results in Multi­variable Feedback Theory", Automatica, Vol. 8, 1972.

13. M. Araki and 0.1. Nwokah, "Bounds for Closed-Loop Transfer Functions of Multivariable Systems", IEEE Trans. AC-19, Oct. 1975.

14. H.A. Simon, "The Architecture of Complexity", Proc. American Phil. Society, Vol. 106, December 1962.

15. O.J. Dahl, E.W. Dijkstra and C.A.R. Hoare, "Structured Programming", Academic Press, 1972.

COMPARISON OF INFORMATION STRUCTURES IN DECENTRALIZED

DYNAMIC SYSTEMS

Kai-ching Chu

IBM Thomas J. Watson Research Center

Post Office Box 218, Yorktown Heights, New York 10598

ABSTRACT

This paper deals with the optimization problems we face when operating and designing a decentralized dynamic system. The system is controlled by a set of controllers-each having different information and control variables. Data acquisition and communication mechanisms of the controllers are specified by the information structure, which plays a very important role in the system performance. Through the study of a class of linear-quadratic regulator problems, we have demonstrated the relation of the merit of a structure versus its complexity, and have discussed various concepts in designing an efficient system.

I. INTRODUCTION

When modern control theory is applied to dynamic systems, either for simplicity or idealization reasons, it is usually handled in a centralized way. Historically, it is generally assumed that there is in the system a single controller who observes the status of the environment and then makes the best possible decisions. In spite of many success stories, numerous researchers have come to realize the limit of the theory when applied to large-scale physical systems or any societal systems. Usually involved in these systems are many controllers or decision makers who (i) command different actions at different times and (ii) have access to different information concerning the environment and the respon­ses of other decision makers. An expansion of the theory in a more general framework, to account for the decentralized control, is necessary.

During the past couple of years, a number of researchers have attempted to attack the decentralized formulation from a team-theoretic approach as first introduced by Marschak1 and Radner.2,3 The controllers are considered as members of a team organized

25

26 K.C. CHU

to achieve a certain common goal. Information structures of a decentralized dynamic system define what each controller knows for decision making and the mechanism with which this information is made known to the individuals. Such Information may include not only parameters of the environment, but also, as the systems are dynamic, decisions made earlier by other controllers. This latter feature makes the relationship between information structure and control actions extremely complicated for decentralized dynamic systems.4,5 So far, the theoretic solutions of only a few special structures have been worked out. 6-11

In this paper we shall consider a large class of decentralized control systems with complexity constraints for their optimization. Such simplification will enable us to solve many problems explicitly, and hence, identify the relations between various information structures and their performance. We shall examine certain design concepts of decentrali­zation through the evaluation and comparison of many related structures. It is hoped that, through the study of some elementary problems, we can have a better understanding of the role the information structure plays in decentralized dynamic systems.

II. LINEAR-QUADRATIC DECENTRALIZED REGULATION PROBLEM WITH COMPLEXITY CONSTRAINT

Consider a time-invariant deterministic linear dynamic system with N controllers

(1)

with the distribution of xt=o given, where x ( n -vector) is the state variable for the entire system, and u; (m; -vector, i = 1, ... , N) is the control variable handled by controller i. The state x=O is considered as the goal desired by all the controllers.

The information available to controller i at any time t is assumed to be

Y; = H; x, i = 1, ... , N (2)

where we assume H; -:F Hi when i -:F j and H; has k; independent rows. The variable Y; in (2) includes the ith controller's measurement of the state and any communications from other controllers concerning the state at the time t.

The control u; is assumed to be a direct feedback from Y;, namely

= 1, ... , N (3)

where F; is a time-invariant m; xk; matrix.

The common objective of all the controllers is to drive x quickly towards 0 as t -00 while using as little total control energy as possible. A simple performance index for this purpose is

INFORMATION STRUCTURES IN DECENTRALIZED DYNAMIC SYSTEMS 27

(4)

where Q ~ 0 is n x nand R; > 0 is m; x m; .

Information available to the i th controller in the past (time < t) is not directly reflected in y; at time = t and will not be used for constructing the control at t ; i.e., the decision makers are considered to be of zero-memory in terms of utilization of their information as specified by (3). Such simplification is designed to construct easily implement able real-time feedback controls for potential applications. Later in the paper, we shall present many evidences that limiting u; in terms of fixed feedback gain F; and fixed dimension y; usually still provides fairly good performance in terms of optimizing the objective. Certain time varying nonlinear controllers may help to achieve the ultimate limit of performance and the amount of transmissible information among the controllers,12,13 but their construction can easily get so involved and impractical that they may very well lose their attractiveness when it comes to real applications.

If we assume the initial system disturbance x 0 was uniformly distributed over an n -dimensional sphere, it can be shown that (4) is equivalent to a performance index

(5)

where

(6)

and

(7)

is restricted to being a stable matrix with the choice of F;. Expression (5) can be ob­tained from (4) by substituting (2) and (3) into 0) and solving for x.

The objective of the entire decentralized system is to decide the feedback gains F; for all i in order to minimize (5). Following the approach of the single regulator problem with constant output feedback,14 with a little effort, it can be shown that

Theorem 1 (optimal conditions); IS

J = 0/2) trace K , (8)

where

KD + D TK + M = 0 . (9)

The optimal F*; should satisfy

= 1, ... , N (10)

28 K.C. CHU

where

LDT + DL + In = O. (11)

We do not know that the F~ satisfying the above conditions is unique within the stable domain of D. The computation experience of this author has suggested that this may well be true, and we would like to pose it as a proposition.

The above conditions on optimal Fi have been used in a few applications. 16-19 In this paper, we will use these conditions extensively to analyze a few structure concepts, particularly from the viewpoint of evaluating their information values and determining the best designs.

III. INFORMATION STRUCTURES AND THEIR VALUES

To understand the mechanism and value of information transmission in a decentral­ized system, it would be useful to compare alternative structures in terms of their complex­ity and the optimal J* that can be derived from their use. In a team-theoretic frame­work, better information for all controllers implies better performance for the entire system, provided the planning and operation of the structure associated with the 'added information incur no costs. However, practically, we should account for such costs. In the designing stage, it is important to make trade-offs between the simplicity of the structures and their associated optimal J*.

Generally, the information structure of a system lies between two extremes:

Complete information - Every controller know everything, or Hi in (2) has full rank n for all i. In such a case, control u i is practically a function of x. The well­known optimal solution is

or

i =1, ... , N (13)

and

J*" = (1/2) trace K", (14)

where the subscript c stands for complete information, and

(15)

Null information - No controller has access to any information on x. This means

INFORMATION STRUCTURES IN DECENTRALIZED DYNAMIC SYSTEMS 29

matrices H; in (2) has null rank for all i. The optimal control for stable A under such circumstances is a constant or what Radner called a routine. 3 We have

u70 = 0 i =1, ... , N (6)

and

J*o= 0/2) trace Ko, (7)

where subscript 0 stands for null information and

(8)

For general information structure { H; I i= 1, ... ,N}, results in Section II tell us what { F*;} to choose, and the value of J* should lie between that of (14) and (17). In the rest of the paper, we will address some aspects of the problem of picking appropriate structures {H;li=l, ... ,N} so the payoff in performance is worthy of the complexity and the costs of the structure. We shall pursue this end from the viewpoints both of collective planning and of individual selection. First, we shall take a direct approach by evaluating J* with (8)-(11) for alternative structures. Then we shall discuss an indirect approach

by comparing the individuals' {H;} with those of a special case such as the complete

information or null information structures.

IV. STRUCTURES WITH VARIOUS SIZES OF INFORMA nON POOLS

We assume that each controller within a system first has certain basic measurement h/x about x. The controllers are then organized into groups of various sizes, with

complete information exchange within groups, and no information exchange at all between groups. For example, if controllers 1, 2 and 3 form a group, the total information availa­ble to each of them is

Controllers I, 2, 3 are said to form a pool to exchange information. The entire system might be considered organized in partial information decentralization. As the size of an information pool is increased, the communication with it becomes more extensive through more complex links. The controllers within it are able to know the state x better from the pool and, supposedly, to make better decisions.

In order to see more clearly the effect on system performance of grouping controllers into information pools, it will be of interest to consider an example in scalar terms.

30 K.C. CHU

Xj = -xj-c "'2:. j xj+u j +0.01 "'2:.j uj , i,j =1, ... , N . (I9)

j # j ;ki

Parameter c represents the identical dynamic interactions among different components of vector x. The weights in the performance index (4) are given as

and

qjj = j 1 if i = j

1 q if i;k j

rj = 1 for all i,

where the performance interaction q satisfies -1/ (N -I) <q < I .

(20)

(21)

The basic observation of the i Ih controller is x j • Suppose the controllers are partitioned into a set of information pools Ip , and all pools have k controllers, where k divides N. If controller belongs to pool Ip, or i € Ip, its information then is

Yj = {x j }. I J € p

i =1, ... ,N. (22)

Using symmetry in state feedback, we assume the control form

Uj =fcfj+f,"'2:.j Xj' i € Ip (23) j ;ki i = 1, ... , n . j € Ip

Figure 1 shows the optimal J* as a function of the pool size k and three different values of q, for N = 100 and c =0.01. Figure 2 shows the same for c =0.1 .

20 20 t-q=OI

q = 0.1

15 t-

J*

q = 05 q =05

10-

c = 001 c = 0 I

't 5 q=09

q =09 -1 I I I I I .. I -.1.._ .. L L I I ._1 __

0 20 40 60 80 100 0

20 40 60 80 100

Figure I Figure 2

INFORMATION STRUCTURES IN DECENTRALIZED DYNAMIC SYSTEMS 31

It is "interesting to note that, as expected, the performance of the system is enchanced as the pool size increases. However, after a certain pool size has been reached, the marginal performance improvement will become so slow (or negligible) that it may no longer be worthwhile to build up the communication complexity within the system.

As we compare the curves with different values of q or c, it appears that an increase in communication within the system will impact J* differently. The curves become steeper when q is large or c is small. Generally speaking, the larger the performance interaction q, the greater the need for coordination among the controllers and the more helpful the communication (increase k). On the other hand, the larger the dynamic interaction c, the more each controller will be able to infer other controllers' information from his own observation, and hence the need of explicit communication may be less.

Ideally, we would like to appraise an information structure on the basis of its net value, namely J* minus the cost of information and the associated computation. Although it is impossible to define the information cost here explicitly, a rough relation between such'cost and the structure complexity can be established. In our system, if there are N/k pools of size k, a two-way communication link is constructed between any two controllers of the same pool. Then there will be a total of (N / k) • k(k -1)/2 = N(k -0/2 links. If the cost is, to a first degree of approximation, proportional to the number of such links, this cost is roughly linear in the pool size. Supposing certain assumptions on the cost can indeed be made like above, we can see that it is possible to make a meaningful comparison of structures and pick one with proper complexity to fulfill a task.

V. OVERLAPPED COMMUNICATION NETWORKS IN SUBSYSTEMS

In the last section, we described the complexity of various structures when controllers can be organized into a set of disjoint information pools. Sometimes in large systems, none of the controllers has complete information about everything. Nevertheless, these controllers do not form clearly disjoint local groups to handle their information. The real situation is rather that of many subsystems interconnected by information and control links. The interactions by such links generally are strong when the distance between the subsystems is small, and weak or non-existent when the distance is long, where the distance is a certain measure of the physical or temporal closeness of the subsystem controllers. In other words, each individual controller has a certain limited field of information (and, reciprocally, a certain limited territory of control), which overlaps with that of other controllers. The information structure of the entire system is then a network formed by the aggregation of all such overlapped linkages.

Figures 3a-c serve as examples to describe the situation pictorially. Each block there indicates a subsystem associated with a controller u i and subs tate Xi' There are certain dynamic interactions or performance interactions between these blocks. In Fig. 3a, no information is made available to controller i except its own substate Xi' In Fig. 3b, every controller receives the substate information from four "neighboring" subsystems via communication links. The thick links in the figure represent the field of information of a

32

DDD D@D DDD a) STRUCTURE I

J*

b) STRUCTURE n c) STRUCTURE III

1 1

1

1

I I 1 1/ I /1 j,/ 1

,,1 I I I n

Figure 3

I STRUCTURES

ill TIl etc.

Figure 4

K.C.CHU

typical controller i. In Fig. 3c, substate information from eight neighboring subsystems are transmitted, etc. As we proceed from Fig. 3a to Fig. 3c, information structures become increasingly complex. However, the performance J* vs. such structure com­plexity is typically as shown in Fig. 4. Similar to Figs. 1 and 2, the complexity of the structure has" decreasing marginal value" in improving the performance.

An earlier paper of this author18 described a detailed analysis done on the decentral­ized regulation for a string of identically structured subsystems. The feedback regulation is generated by various information structures on neighboring subsystems. The results were later applied to the automatic control problem of high-speed vehicular strings. 19,20 It is recognized that remote input data (those not from the immediate adjacent subsystems) do not contribute significantly to the stabilization and optimization problem. Such results, though intuitively not surprising, if obtained quantitatively by systematic methods, can provide a guideline for specifying the communication requirement for system design.

VI. INDIVIDUAL CONTROLLER'S INFORMATION OPTIMIZATION

Number ki' the rank of Hi' is the number of independent measurements controller i may have. We may say that controller i has an information device of capacity k i or of k i channels. Each of these channels is responsible for observing a particular combina­tion of components of x.

INFORMATION STRUCTURES IN DECENTRALIZED DYNAMIC SYSTEMS 33

Roughly speaking, what we have discussed in the last two sections is the design problem when the allocation of total information devices among all controllers is consid­ered. In this section and the one that follows, we would like to discuss choosing the best information device for each controller with a given capacity.

Geometrically, each row of H; can be considered as an observation vector in x space for controller i. There are k; independent ones, where k;Sn (=rank x). One interesting design problem is the pointing of observation vectors in the best admissible directions so as to optimize the performance index J. Such a situation arises, for example, when one has to decide the mounting angles for rate gyros relative to the axis of a flying craft. In certain real problems, we may not have the freedom of pointing the observation vector anywhere we like, because certain quantities are just not observable. This limitation will not be considered here.

If we suppose k; ~m; (=rank u; ), it is possible to orient the set of observation vectors for the controller in such a way that the optimal feedback control for i with complete information is realizable. Although the complete information of x may not be available to the controllers, they are said to have optimum-sufficient information. The partial information that the controllers have is performance-relevant and is no less valuable a priori than the complete information.

On the other hand, there may be certain orientations of the observation vectors where absolutely nothing is valuable toward optimization. Such structures, though nontrivial, are just as useless as the null structure, and hence can be called null-equivalent.

Consider the following example of two controllers.

X 2 = .5 x I - X 2 + .5 u I + u 2

where x I ' X 2' U I and u 2 are all scalars. Performance index is

Complete information solution

u* Ie = -.5091 x 1-.4269 x 2

U*2C = -.4269x l -·5091 x 2

and

J\ = .3942.

(24)

(25)

(26)

(27)

34 K.C. CHU

Null information solution

U*lO = U*20 = 0 (28)

and

J* 0 = 5/6 = .8333. (29)

Decentralized partial information solution

Suppose both controller 1 and controller 2 are able to observe only one piece of scalar data. Let us assume their data symmetric so

y 1 = cos 11 Xl + sin 11 x 2

y 2 = sin 11 XI + cos 11 x 2 (30)

The value J* for various observation angles 11 is plotted in Fig. 5 in polar coordinates, with J* as the distance from the curve to the origin. It is noted that, at direction 11=- tan -1 (41/40) = -45.7° or 134.3°, J* is null-equivalent (J* = J* 0)' At 11=39.98° or -140.02°, the information is optimum-sufficient (J* = J*c)' because the observation vector is just aligned with that required to construct u * 1 c and u * 2 c' One thing worth our attention is that the best 11 and the worst 11 are not exactly perpendicular to each other.

Generally, when rank ui > rank Hi' J*c is not achievable. We should have the following relation under all circumstances:

J* 0 ~ J* worst directions ~ J* best directions ~ J* c . (31)

The following theorems give conditions under which the information is optimum­sufficient or null-equivalent.

Theorem 2. If the space spanned by the rows of Hi contains the one spanned by the rows of (B/I(,) for all i, where Kc is the solution of the Riccati equation (15), then the information structure {Hi} is optimum-sufficient.

Proof. Under the given condition, we can always find Fi for all = -Ri -IB/Kc and u* = U*ic .

Theorem 3. If B /K 0 L 0 H:' = 0 for all I

where K 0 is the solution of (18) and L 0 satisfies

Lo AT + ALo + In = 0, (32)

then the information structure {Hi} is null-equivalent.

INFORMATION STRUCTURES IN DECENTRALIZED DYNAMIC SYSTEMS 35

8

" J

Figure 5

Proof. With the given condition, F*i = 0 for all i satisfies equations (9), (10)

and (11) Under the assumption that the solution is unique and we know that u *i 0 = 0

for all i.

The condition in Theorem 3 simply means that all observation vectors of i (rows of

Hi) are perpendicular to all the row vectors in B /K 0 L o. Row vectors of B /K 0 L 0 seem

to represent the desired directions for controller to start looking for useful informa-

tion.

VII. INDIRECT COMPARISON OF INFORMATION STRUCTURES

So far, our study on various information structures and the associated controls has

been done through direct evaluations. The control functions are designed to be executed

by individuals in an on-line decentralized way. But the off-line design problems, such as

computations using (8)-(11) in Theorem 1, are actually still done centrally. One of the

36 K.C. CHU

troubles is that computations of (8)-(11) may turn out to be very involved, if not impossi­ble, when the system is very large. Hence, there may be a need to deleg!lte even the design problem to local decision makers. Thus, both the control execution and structure design are, to some extent, decentralized. The system is then both self-regulating and self-organizing by the joint efforts of the decision makers.

One situation when this is possible is in taking the ideal complete information struc­ture as a reference and comparing the individuals' control and information with those of the reference. Different structures are appraised indirectly by comparing their closeness to the reference structure.

The problem faced by each individual in this indirect approach is to bring his control function as close as possible to what he is supposed to have in the complete information case. When optimizing a system, the basic objective is to minimize 1 J* - J*" I. However, the indirect approach suggested here is to minimize 1 U*i - U*i" 1 for all i instead. Such an alternative is based on the fact that function J is continuous in ui ' and the closeness in u i for all i implies the same in J .

Specifically, we already know that

i = 1, ... ,N . (33)

where the miX n matrix Si = -Ri-IB/K" and K" satisfies (15). The best Fi which minimizes II Fi H - Si lit for a given full rank Hi is

i = I , ... ,N. (34)

A

Solution Fi is of course suboptimal with respect to F*i of (8). But it may pay to use it for large systems for the following reasons:

(i)

(ii)

A Off-line computation of Fi itself is decentralized.

Comparison of various information structures are broken down to independent efforts of individuals. The only thing they need from a "consultant" is the reference (33).

When controller i is free to choose the information matrix Hi as well as the feedback gain F i , the problem is even more interesting if the indirect approach is used. We assume that ank Hi is limited to be k i , where k i ~ n. Controller i is asked to place k i observation vectors in the best directions so that, with the corresponding Fi ' II FiHi - Si II is minimized. This problem essentially is to approximate one matrix by another of lower rank, which has been solved by Eckart and Young. Z1

til M liZ = trace(MIM).

INFORMATION STRUCTURES IN DECENTRALIZED DYNAMIC SYSTEMS 37

Of course, when n ~ k; ~ m;, exact S; can be constructed. When k; < m; , the procedure of finding (F;H;) is the following: (There is no unique way to keep F; and H; apart.)

1. Express S; in the canonic form

u 1\ v (35) n x n

where U and V are orthogonal matrices, and 1\ is a diagonal matrix with non-negative diagonal elements A, ~ A2 ~ A3 ~ ... ~ O.

2. Replace all but k; of the diagonal elements of A by zeros, beginning with the smallest and continuing in order of increasing magnitude. The resulting matrix is M.

3. F;H; is given as

(36)

4. The solution is unique unless Ak . = Ak . +1' and the minimum of II F;H; - S; II is I I

II F H - S 112 = ~R k ,A2 I I f I=' i + .I

(37)

where R is the rank of 1\ or S;.

From a functional viewpoint, H; transmits data from state x to the measurement, and F; transmits the measurement to control actions. The combined (F;H;) is then suppos­ed to transmit data from state x directly to the control actions with "channel capacity" k;. Orthogonal matrices U and V change the coordinates of the action space and the state space respectively, so that independent channels between the two new spaces can be visualized by 1\. The choice of M is simply to pick up the k; most significant chan­nels of 1\.

" Solution (34) is applied to the example of the last section. Suboptimal pe~formance J is plotted in Figure 5 against various observation directions. It shows that J coincides with J* at those best and worst directions as it should, and these special directions can

1\ also be derived from (36). Indeed, the differences between J and J* for other directions are very small for this particular example.

VIII. SUMMARY

Direct evaluation and indirect comparison methods have been used to study various information structures of a class of decentralized control problems. System operation and

38 K.C.CHU

design problems have been addressed from both the standpoint of the individual controller and of the entire system. Relations between system performance and structure complexity are discussed through suggestive examples. We have demonstrated that information structure plays an extremely important role in organizing a system. Efficient decentrali­zation depends on controllers having relevant information through personal observation or a proper communication mechanism.

IX. REFERENCES

\. Marschak, J. Elements for a theory of teams, Management Sci. I (1955) pp. 127-137.

2.

3.

Radner, R. Team decision problems, Annals of Math. Stat. 857-88 \.

Radner, R., The evaluation of information in organizations. Berkeley Symposium on Mathematical Statistical Probabili~y.

sity of California Press (1961) 491-530.

33, no. 3 (1962)

Proceedings 4th Berkeley: Univer-

4. Witsenhausen, H.S. A counterexample in stochastic optimal control. SIAM J. Control 6 (1968).

5. Witsenhausen, H.S. On information structures, feedback and causality. SIAM J.

6.

Control 9 (1971) 149-159.

Chong, C.Y. and Athans, M. differential information sets (October 1971) 423-430.

On the stochastic control of linear systems with IEEE Trans. on Auto. Control AC-16, no. 5

7. Ho, Y.C. and Chu, K.C. Team decision theory and information structures in optimal control problem. Part I. IEEE Trans. on Auto. Control AC-17, no. 1 (February 1972) 15-2 \.

8. Chu, K.C. Team decision theory and information structures in optimal control problems. Part II. IEEE Trans. on Auto. Control AC-17, no. I (February 1972) 22-28.

9. Sandell, N., Jr. and Athans, M. Solutions of some nonclassical LOG stochasic decision problems. IEEE Trans. on A uta. Control AC-19, no. 2 (April 1974) 108-116.

10. Ho, Y.c. and Chu, K.C. Information structure in dynamic mUlti-person control problems. Automatica 10, no. 4 (July 1974) 341-35\.

II. Chong, C.Y. and Athans, M. On the periodic coordination of linear stochastic systems. Proceedings 1975 IFAC. Boston, Massachusetts (August 1975).

INFORMATION STRUCTURES IN DECENTRALIZED DYNAMIC SYSTEMS 39

12. Bismut, I.M. An example of interaction between information and control. The transparency of a game. IEEE Trans. on Auto. Control AC-18, no. 5 (October 1973) 518-522.

13. Sandell, N.R., Jr. and Athans, M. Control of finite state finite memory stochastic systems. Proceedings 1974 Princeton Conference on Information Sciences and Systems. Princeton, New Jersey (March 1974).

14. Levine, W. and Athans, M. On the determination of the optimal constant output feedback gains for linear multivariable systems. IEEE Trans. on Auto. Control. 15 (February 1970) 44-48.

15. Chu, K.C. Decentralized regulation of dynamic systems. Large-scale Dynamic Systems. ed. C. F. Martin. NASA publication SP-371 (1974).

16. Davison, E. J., Rau, N.S. and Palmay, F.V. The optimal decentralized control of a power system consisting of a number of interconnected synchronous machines. Int. J. Control 18, no. 6 (1973) 1313-1328.

17. Linton, T.D., Fischer, D.R., Tacker, E.C. and Sanders, C.W.,Jr. Decentralized control of an interconnected electric energy system subject to information flow constraints. Proceedings of the IEEE Decision and Control Conference. San Diego, California (December 5-7, 1973) 728-730.

18. Chu, K.C. Optimal decentralized regulation for a string of coupled systems. IEEE Trans. on Auto. Control AC-19, no. 3 (June 1974) 243-246.

19. Chu, K.C. Decentralized control of high-speed vehicular strings. Transportation Science 8, no. 4 (November 1974) 361-384.

20. Sarachik, P.E. and Chu, K.C. Real-time merging of high-speed vehicular strings. Transportation Science 9, no. 2 (May 1975) 122-138.

21. Eckart, C. and Young, G. The approximation of one matrix by another of lower rank. Psychometrika 1, no. 3 (September 1936) 211-218.

ON FLUCTUATIONS IN MICROSCOPIC STATES OF A LARGE SYSTEM

Masanao Aoki

Department of System Science

University of California Los Angeles

ABSTRACT

We consider a system composed of many small interacting units so that we have a microscopic as well as some aggregated or macro­scopic description of the system. We consider specific examples to illustrate the relation between the macrovariab1es and micro­variables with particular attention paid to fluctuations of micro­variables compatible with macroscopic states of the systems.

1. Introduction

In treating a large system involving many interacting agents, attention is usually focused on macroscopic or aggregate variables of the system as a whole and its microscopic variables or local subsystem structures are usually ignored or not treated in any detail. For example, in discussing equilibria of such a system, the question of fluctuations or possible variations on the micro­scopic subsystem level seems never to be addressed. The reason for this state of affair is mainly that of analytical convenience, since the time path of macroscopic system behavior usually turns out to be described by a deterministic dynamic equation under a set of plausible assumptions. This equation is usually much easier to derive and solve than (stochastic) equations for the time behavior of any individual agents in the system. See for example Mortensen [10J.

The purpose of this paper is to point out that the fluctua­tions of microscopic or subsystem variables experienced by indiv­dual agents in the system, such as firms, industries or sectors making up the total system or economy, are important and to call for closer examination of them.

41

42 M.AOKI

The notion of equilibria of systems under imperfect informa­tion assumption and the concept of self-fulfilling expectation, both of which figure prominently in some recent economic literature [2-13] can be seen to be closely related to this question of fluctu­ations on the subsystem level although this connection is not brought out in the literature. Consider for instance constructing a short-run macroeconomic dynamic theory from the microeconomics for agents in the economy who choose their decisions with imperfect information. Economic agents may be price-setting firms. Under imperfect information assumption, they base their decisions on prices, money wages and the amount of labor to employ, etc., on their expectations (estimates) about the current as well as future economic conditions, i.e., futuremacrostates of the dynamic economy they are in.

Hahn states in his inaugural address [7]" ••• an economy is in equilibrium when it generates messages which do not cause agents to change the theories they hold or the policies they pursue •.•• and agent abandons his theory when it is sufficiently and systema­tically falsified ••• ". Message agents receive are composed of data pertaining to macroeconomic variables data pertaining mostly to agents' own activities. For example in case of firms, GNP, various price indices, unemployment level, etc. are the macro­economic data. In other words, Hahn implicitly requires that agents ~ust be able to solve sequential statistical decision or hypothesis testing problems of deciding whether the received messages (signals) reject whatever (subjective) beliefs they hold about the state of the economy i.e., about some macroeconomic variables. To this end, agents must know how much fluctuations or variations in messages they receive are within statistically reasonable bounds before they decide to revise or reject their subjective beliefs. For example, A. Leijonhufvud raised a question whether an industry wide constant output over time implies that outputs of the individual firms making up the industry remain cons­tant over time in a simple Marsha11ian model [8]. The answer is not necessarily so as shown in [1, pp. 199-202].

We must realize, as the above Marsha11ian model indicates, that a macrostate is compatible with a number of microstates. On a priori ground, under suitable assumptions, the a priori (uncond­itional) variance of a macrovariab1e, which is a sum of microvari­ables, is proportional to the square root of the number of parti­cipants. What is more significant, however, is the magnitude of a posteriori (conditional) variances of the microvariab1es, given the value of a macrovariab1e which is a function of these microvar­iab1es. In other words, how much variation in microvariab1es are compatible with the observed value of the macrovariab1e? The vari­ance turns out to be large for the examples in this paper. In view of rather large variances of microvariab1es that are potentially and possibly present in microstates, decision makers' expectations ought to include their subjective estimates of the variances of

FLUCTUATIONS OF A LARGE SYSTEM 43

local variations as well as the macrovariables. Another way of stating this is to say that in deciding whether their expectations were fulfilled or cheated, economic agents must have some idea of the conditional variances compatible with the observed values of macroeconomic variables. Without this knowledge of conditional variances, economic agents can not really say that an unexpected event has occurred.

One could also draw some implications of the fluctuations in microvariables from somewhat different point of view. One distin­guishing feature and a potential source of difficulty for decision makers is that they cannot sometimes distinguish local disturbances affecting only small percentage of them, i.e., participants in local markets, from global disturbances which affects the whole system hence every decision maker in the system. Decision makers can obtain local information about his immediate environment or microscopic state relatively more eaSily than global information or information about the aggregate or macroscopic state, which they jointly determine due to time delay and cost in information gathering and so forth.t This leads to an important question of whether individuals' expectations are consistent with aggregate behavior of the whole system. There is no strong reason a priori to assume that they are consistent. This possible inconsistency of expectations held by microunits with the macrosystem behavior often leads them to erroneous or suboptimal and sometimes des tab­lizing decisions, and to suboptimal or destablizing overall system performances. Information on fluctuations of microvariables would be definitely useful in such situations.

In economic literature several authors dealt with this aspect of economic systems. For example, Lucas, in several of his papers, advanced and tested the so-called aggregate supply hypothesis in which economic agents such as firms interpret as the general demand shifts observe changes in prices in local markets [9J. Iwai discusses an analogous decision problem of economic agents who base their decisions on their subjective estimates of aggregate or market average prices and local prices which they must set [llJ. Aoki [1, Chapter l3J discusses a model of a middleman in which he must try to disentangle local disturbances form global one of demand shifts since these two types of disturbances lead him to two entirely different courses of actions in general.

In this paper we shall discuss a fundamental question which lies in the back of these problems, i.e., how large the variations of microscopic variables observed locally are possible or compati­ble with a given macroscopic state? We use a simple search model and show that a rather large fluctuation of microscopic states is consistent with a given macroscopic state. It is important to bear in mind that we are not concerned with a priori bounds but rather a posteriori bounds. As argued above, failure to pay adequate

tThese agents often must decide knowing current and past local inf­ormation and outdated information on macroscopic variables.

M.AOKI

attention to local variations may have many serious implications both for microunits and overall decision maker such as the govern­ment in the case of the economic systems.

In section 2, a simple search model is described and its macro­and microvariables are identified. A couple of other examples are shown to be reducible to this basic model. Section 3 contains the description of the most probable (maximum likelihood) estimate of the microstates, and the conditional and unconditional means are computed. These are generally all different. We compute the conditional variance of the microvariables, i.e., the variance of the microvariables, given the value of the macrovariable and show that it is of the order of the conditional mean. The last section provides some additional comments on the main theme of the paper.

2. Basic Model We treat a very simple search model as a basis of our discu­

ssion. The model is such that a microstate has a multi-nomial dis­tribution. Many situations are reducible to this basic model. This is illustrated by a couple of simple examples later in this section.

Suppose there are k markets (labor exchanges) to which a worker could go each period (a market day). A worker can visit only one market per day. There are a total of N workers. We assume is fixed for the sake of simpler presentation. wel1re interested in the situation where N is much larger than k.

Suppose Ni of the workers shows up in market i, i=l, ••• ,k. In market 1, the market clearing price w, is established as a function of Nl • Similarly w2 is the market clearing price in market 2, and so on. We assume that wi = fi(Ni ) ,21 i=l, ••• ,k • (Realistically, the workers do not know the nature of f i (') pre­cisely but in this paper we assume that the functional form f i (') is known so that N.'s are the only unknowns at the beginning of each day.) They d01 not know the market clearing prices (wages) prevailing in other markets on the same day. Assume that workers are identical in every respect. Suppose also that each adopts a

k search strategy P = (Pl, ••• ,Pk), .21 Pi=l, Pi ~ 0, i=l, ••• ,k, where

1=

Pi is the probability of a worker going to market i.

Suppose the daily average price W = (wl+ ••• +wk)/k is broad­cast at the end of each day so that each worker knows the left hand side quantities of (1) exactly, one day later

N = Nl + ••• + Nk (1)

; = {fl(Nl ) + ••• + fk(Nk)}/k.

For the sake of definiteness and simplicity assume that the

FLUCTUATIONS OF A LARGE SYSTEM

market clearing price is determined by the formula

fl..(Ni ) = y. - 8.N .• i=l ••••• k l. l. l.

and rewrite (1) as

where

N = Nl + ... + Nk

kw-y=8N + 1 1

k y I y.

1 l.

45

(2)

(3)

Variables on the left in (2) and (3) are macroeconomic (or macro­scopic) variables. the total number of employed in the country and the "average wage" prevailing in the country.

Denote the probability that N. appears in Market i. i=l ••••• k by P(N1 , ••• ,Nk) • They are given %y the mu1ti-nomia1 distribution.

For example. with k=2 , we have

We now ask that given N , w (of yesterday), and (P1 , ••• ,Pk ) of the search strategy k-tup1et,_what (N1 •... ,Nk) is consisten~ with the observed data. Nand w? For example. we may say that (Nl, .•.• Nk) which maximizes P(N1 •.••• Nk) is a way of defining a representative microeconomic state of this model. The rational for that is the maximum likelihood estimate of N1 , ••.• Nk , i.e., the combination of N1 •.•. ,Nk with the highest probabi1l.ty. is most likely to be the one actually realized. On the other hand, we may want to use the mean of (N1 , ... ,Nk) as a typical microeconomic state. We shall compute the mean and the variance of N, to show that a large variation in N, is still consistent with fhe observed data. l.

There are other models that can be reduced to the same or similar framework of the above model. We give two examples before we proceed with our analytical illustration of the model.

Example 1 Suppose there are k

the total of N workers,

related to N' s i

by Q

industrial sectors in a country employing k I Ni = N. The total output Q is

i=l k I f,(N,) , for example by 1 l. l.

46

We are given that workers have probabilities for the industries l ••• k,

Then given the macroscopic data Nand micro economic states compatible with Nand E(NIIN,Q) var(NIIN,Q) one could expect?

Example 2

M.AOKI

Q , what are the Q and what are

In this example one supposes that there are k firms producing homogeneous nonstorable goods of an industry with linear technology.

k Let Qi be the output of Firm i, L Qi = Q. Choose the units

I _ _ of output small enough so that Qi = Liq , where q is the common unit. Then we have LLi = L where L'S and L are integers. Because of nonstorable nature of the good, a market clearing price is determined as a function of Q or as g(L) with some g. Suppose L is uniquely determined as a function of the market

k clearing price. Call this p. Then we have L L = p. Let

I i PI",Pk be the relative share of the market of these k firms,

PI ~ 0 , LPi = 1. Assume that they hire workers proportional to

Qi or Li so that ciLi is the number of workers Ni , i=l, ... ,k

LciLi = N ,

where N is the total number of workers. Then the macroeconomic variables p and N are related to microeconomic variables Ni's by

LNJ.. = N and L~ N. = P . c i J.

The variables Q or p and N are the industry-wide data while L. or N. are the data for the firm i.

J. J. Example 1 is further discussed in section 5,

3, Maximum Likelihood State: Most Probable Microeconomic State We now return to the first search model. Assume k-2 merely

for ease of exposition and perform max [P(Nl ,N2)], or equivalently Nl ,N2

max [~n P(Nl ,N2)] subject to (2) and (3). Let

L = ~n P(Nl ,N2) - A(N - Nl - N2) - ~(2w - Y - 0lNl - 02N2)'

where A and ~ are Lagrange multipliers. Maximize L with

respect to N. J. dL . 3/ by computing 0 = aN J.=1,2.- .

i

FLUCTUATIONS OF A LARGE SYSTEM

These lead to

1 tn Ni + 2Ni = tn Pi + A + POi' i=1,2

or 1 2Ni POi A

N. e = P. e e, i=1,2 • l. l.

Again using the assumption that N. l.

1

2Ni ( 1 1 Ni e = Ni 1 + 2N + --2 +

i 8Ni

= N + 1. + 0 (1:....) i 2 N.

l.

is large, we have

47

(5)

or from (5), Ni denoted by N~

that maximizes P(Ni,N2) are given by

subject to (2) and (3), . l.

N~ ~ P. l. l.

POi A 1 e e - 2 ' i=1,2 •

We see that

- 1 where m = 2 w + 2(01 + 02) - y is a kn~wn constant.

(6)

The maximum likelihood estimates N. , i=1,2 consistent with - l. the macroeconomic variables Nand ware then,

°iP N* = NP e +

i i 0lJl 02Jl PIe + P2e

, i=1,2

When there are k markets instead of 2, we obtain

N~ = NP. l. l.

°iJl e +

k 0iJl L Pie

i=1

(7)

(8)

48 M.AOKI

instead. Note that N7 is not the same as the unconditional or a

priori mean whichlis N. = NP .. We show next that N~ is also different from conditiofia1 or1a posteriori mean when tfie constraints (3) is incorporated. Denote this mean by N., and call it ex post mean.

1

4. Ex Post Mean and Variance of Microeconomic Variables We now go back to the original model with k markets, and

compute the variance of N. about N. , as a measure of f1uctu-t · I' 1 1 a 10ns. t lS convenient to introduce a notation.

k 0.]1 Z = I P.e 1 (9)

i=l 1

Then, the above may be written compactly as

* a.]1 1 Ni NP i e 1 /z + O(N)

d 1 N -- ~n Z = m + O(-N) d]1 (10)

where m is a known macr~scopic variable. Consider a large number of market days on which wand N hence Nand m also remain constant. Therefore, we may say the system of k labor markets as a whole is in a stochastic equilibrium. What are the microeco­nomic states compatible with N and observ~d w? The average number of workers reporting to market i, N. then is

1

LNi P(N1 ,··· ,Nk)

LP(N1 ,···,Nk)

where the summation is over N1 ... N such that L.a.N. constant, i.e., w constan~. 111

Define the generating function

N! N1 g(zl,···,zk)=L N1 N' zl

l' •.. k'

(11)

N and

(12)

where the summation ranges over equality constraints as above.

N1 •.. Nk , subject to the same zls are evaluated at z. P.

i=l, ••• ,k. Thing

Similarly,

g( ••• ), (11) may

(L&n g\ zl\az;:/z.

1

1 1

be written as

(13)

FLUCTUATIONS OF A LARGE SYSTEM

Since

1 d = g dZl

2" we can express Nl as

N2 = 1

49

all evaluated at Z. = 1 , i=l, •• "k • Therefore, evefi when the macroeconomic variables are constant,

the variance of a microeconomic variable, the number of workers in Market 1 is not zero and is given by

where from (13) we see that

N2 1 (tV " "1 + f-i a: :~J

L 1 J Zi Pi

= Nl[l+O(l)]

P. ~

(14)

50 M.AOKI

where

_ m+ 1 tf-="~( u...,:O-;:-) _ f' 2( uO) ~ P"(u ) + N o - 2 f(u ) Uo 0 f (uO) zi = Pi

O(N)

The constant Uo is determined by 0 = p'(uO) where p(u) 01 ok

= - (m+l) ~n + N ~n f(u;z) , f(u;z) = (zlu + ••• + zk u ).

Comparing Nl with N~, we see t~ey are generally different. When the average number of_workers is Ni , then the conditional variance is of the order Nl also. (The differences between Nl and Nt disappear as N + ~ .)

5. Discussion The maximum likelihood estimates

o - a;i {tn NI - it b Ni' -- "(Q - i :iNi ai)} ,

which leads to (dropping 2N. term) l.

~n N. ].lei

~n P. + A l. I-cd l. N.

l.

where

e. = A.a. > 0 .

for Example 1 are obtained by k

(N - A.Ni ) + 2 Ni ~n Pi l. i=l

l. l. l. (A+L'.. ) l. Let L'.. be defined by

l.

].lei L'.. I-a l.

(P.eA) i l.

Suppose 0 < a ~ 1 i

Denote it by 0 .• l.

Ni = Pi e Then L'.i satisfies

-(l-ai)L'.i 0 e

Then, this equation has a unique solution.

FLUCTUATIONS OF A LARGE SYSTEM

Then the maximum likelihood estimate of Ni is, as before,

N~ Pi e (HOi)

= l.

where

A N/Z e =

and where Z is given by (9). One may proceed analogously as in Section 4 to compute' its variance.

In above analysis, the probabilities pIS are assumed to be given constants. More realistically, pIS would be endogenous, changing, for example, with observed sample variances in each market, depending on attitudes towards risk and so forth. This aspect has not been discussed in this paper.

51

Although the paper derives its motivation largely from econo­mic considerations of a model with many geographically separated local markets, fluctuations in microscopic variables are also important in noneconomic context such as in ecology or limnology in which the distribution of patches or local clusters have impor­tant implication for observation (sampling) schemes and for the purpose of management since local behavior could be markedly diff­erent from the average or aggregate values lead one to believe.

APPEND]{ We evaluate the second term of (14) and show that it is at

most of the order Nl • If the summation were to

(12)7 the generating function N

range over all possible ENi = N g(zl"",zk) would be given by

in

(zl+" ,+zk) • To sum the right hand term of (12) only over Nl, ••• ,Nk such

that Eo.N. = const., we use the following well-known artifl.ce [14J. l. 1. Choose the unit of ai's small enough so that oils as well as m are integers. We can assume that o's and m have no common fact­ors by dividing through the common factor in the constraint equation. Define

f(u;z)

Then

g(Z1"",Zk) = I °INI + + °kNk m

is the coefficient of m in f(u)N u , i.e., if we select the terms

52 M.AOKI

with the power urn, we have g(zl, •.• ,zk) • This can be expressed by the residue method as

where u -is the complex variable. The contour integral is taken around u=O in a region in which feu) is analytic. We evaluate next this integral approximately using the steepest descent method [14J. Rewrite the integrand as

where

Define

Then

Thus

-m-1 f( )N p(u) u u;z = e

p(u) = -(m+1)~n u + N ~n f(u;z) •

Uo by (m+1) f' (uO;z)

o = p' (uO) - -- + N • Uo f(uO;z)

= (m+1) + N ~f" (uO) _ f' (uo) 21 p"(uo) 2 f( ) 2'

U o U o f (uO)

1

I2np"(u ) o From (13), (14), and (A.2), we have

1

~n g ~ -(m+1)~n Uo + N ~n f(u O) - 1 ~n 2n

<\ dUO NuO --+---dZ 1 f(uO)

-i d:1 ~n p"(uO)

(A.1)

(A.2)

FLUCTUATIONS OF A LARGE SYSTEM

°1 NuO 1 a

= f(uO) - 2" aZl in p"(uO)

where the first term vanishes by (A.l). Then

From (A. 3) ,

aNI N uo °1

N zluO °1

(,01 +~) -= f(uO;z) aZl 2 auaz1 f (uo;z)

~ a in p" (uo~ .. 12 aZ l J Use the relation

/ = (m+l) f'(uO;z) f(uO;z) N Uo in the above to reduce it to

Let

<0>

°i L:P.o.uO 1. 1.

m+l N

53

54

Then

We have

and

f"(u ) o -f-=

f" f' 2 y--y-

M.AOKI

r'N + 1 a"n pJ L 1 "2 Zl aZl N J

FLUCTUATIONS OF A LARGE SYSTEM

°i 2 (EPioiuO ) 1

r02> - <0> - <O>~ 1

°i 2 "2 = 2" (EP iUO )

Uo Uo

Note that 5l.n p"(u ) 0 is at most of order 5l.nN • Thus we have

REFERENCES 1. M. Aoki, "Dynamic Economics: A System Theoretic Approach to

Theory and Control, North-Holland/American Elsevier, Forth­coming, 1976.

55

2. Kenneth J. Arrow and Jerry R. Green, "Notes on Expectations Equilibria in Bayesian Settings," Working paper No. 33, August, 1973.

3. Daniel S. Christiansen and Muku1 K. Majumdar, "On Shifting Temporary Equilibrium," Discussion paper 74-12, March, 1974.

4. Jean-Michel Grandmont and Werner Hildenbrand, "Stationary Proc­esses of Temporary Equilibria," Working paper IP-203, April, 1974.

5. J. Green, "Temporary General Equilibrium in A Sequential Trading Model with Spot and Future Transactions," Core Discussion paper No. 7127, August, 1971

6. Sanford Grossman, "Self-Fulfilling Expectations and the Theory of Random Markets, Department of Economics, University of Chicago, February, 1973.

7. F. Hahn, On the Notion of Equilibrium in Economics, Cambridge University Press, 1973.

8. Axel Leijonhufvud, "The Varieties of Price Theory: What Micro­foundations for Macrotheory," Discussion paper No. 44, Depart­ment of Economics, UCLA, January, 1974.

9. R. E. Lucas, "Some International Evidence on Output-Inflation Tradeoffs," American Economic Revision 63, pp. 326-34, 1973.

10. Dale T. Mortensen, "Rational Price Dispersion, Search and Adjustment, presented at 1974 meeting of the Econometric Society, San Francisco, December, 1974.

11. K. Iwai, "On Disequilibrium Economic Systems", Part I and II Cowles Foundation Discussion paper No. 385, Yale University, December, 1954.

12. J. F. Muth, "Rational Expectations and the Theory of Price Movements," Econometrica, 29, pp. 315-335, 1961.

56 M.AOKI

13. Thomas J. Sargent, "Rational Expectations, the Real Rate of Interest, and the 'Natural' Rate of Unemployment," Paper presented at the Brookings Panel on Economic Activity, September, 1973.

14. E. Schrodinger, Statistical Thermodynamics, Cambridge University Press, 1944.

FOOTNOTES

1. Loosely speaking, N/k Stirling's formula for Footnote 3.

should be large enough so that the (N/k) { is a good approximation. See

2. It is possible to include some exogenous random disturbances ~i in wi = fi (Ni'~i)'

3. We use the Stirling's formula

1 R.n N! ::: N(J/,n N-l) + Z(J/,n N + J/,n 2n)

aL in evaluating aNi' The Stirling's formula provides:

approximation for large N. For example with N = 10 ratio of the left and the right hand side differs from

less than 10-4 •

good

, the 1 by

FLOW SYSTEMS

K J !strom

Department of Automatic Control Lund Institute of Technology S-220 07 LUND 7, SWEDEN

1. INTRODUCTION

An attractive idea when investigating large systems is to mi­mic the success of statistical mechanics, i e to consider a large system composed of many copies of identical subsystems. A funda­mental difficulty is that it is hard to find meaningful systemthe­oretic problems which can be obtained by interconnecting identical subsystems. This paper therefore considers a class of linear sys­tems called flow systems. Although the systems are not identical many of their properties remain invariant for different intercon­nections. Flow systems can therefore be used as a starting point for analysing certain large systems. Flow systems have been used as models for industrial and biological processes.

2. TANK SYSTEMS AND FLOW SYSTEMS

A collection of tanks connected by pipes is called a tank sys­tem. Such systems are common in industry. They have also been ex­tensively used as models for biological and ecological systems. Tank systems are often explored by tracer analysis. A traceable substance, which propagates through the system in the same way as the fluid, is introduced at some point of the system. The tracer concentration at another point in the system is then measured. One key problem is to analyse what properties of a tank system that can be found from such an experiment.

Assume that the flows and volumes are in equilibrium, the tra­cer propagation can be described as a linear time invariant dynami­cal system. The dynamical systems describing tracer propagation

57

58 K.J. ASTROM

have, however, some special properties which motivates that they are given a special name flow systems. The impulse response of a flow system is non-negative which reflects the fact that the tracer concentration is never negative. Moreover, if the tanks system is open which means that all tanks are connected to an outlet (possib­ly indirectly through other tanks) all tracer will eventually leave the system. The corresponding open flow systems then have the pro­perty that the integral of the impulse response is unity.

Open flow systems will be investigated in this paper. They have many interesting properties which have largely been found in connection with impulse response analysis of tank systems. The re­sults are widely scattered in literature. Important contributions are found both in engineering and medical literature. This paper is an attempt to present a unified approach.

Two simple examples corresponding to a tank with pure m~x~ng and a tank with pure plug flow are first investigated. A formal definition of an open flow system is then given and interconnec­tions of open flow systems are introduced. The so called Stewart­-Hamilton equation which can be used to determine the total volume of an open tank system is then derived. The volume obtained is the part of the volume which participates in the flow also called the volume of distribution.

3. EXAMPLES

Two simple examples of flow systems will first be given.

EXAMPLE 1 (IDEAL MIXING) Consider a tank with volume V and constant inflow and outflow q (volume flow). Assume that there is perfect mixing in the tank and that the fluid is not compressible. Let ci be the concentration of a tracer in the inflow and c the tracer concentration in the tank and at the outflow. A mass balance for the tracer gives

dc V - = q(c.-c)

dt ~

The propagation of the tracer through the system can thus be de­scribed as a linear time invariant dynamical system whose input output relation is characterized by the impulse response

h(t) = (v/q)e-qT / V (1)

EXAMPLE 2 (PURE TRANSPORT OR PLUG FLOW) Consider a pipe where there is a pure material transport with uni­form velocity and no mixing. Let the volume of the tube be V and

FLOW SYSTEMS 59

the flow q. Let ci denote the concentration of some substance in the inlet and c the concentration of the same substance at the out­let. The concentrations are related by

c(t) = ci(t-V/q)

and the impulse response of the system becomes

h(t) = t5(t-V /q) (2)

where 6 is the Dirac delta function. The propagation of a tracer through a tank with ideal mixing and for a pipe with pure plug flow can be described by linear time invariant dynamical systems.

In both cases the impulse re~ponses have tbe properties.

h(t) ~ 0 (3)

co f h(t)dt = I (4) 0

and co

f th(t)dt - V/q (5) 0

The equation (3) means that the tracer concentration is never ne­gative and the equation (4) implies that all tracer will finally leave the system. If the impulse response is measured by injecting a tracer in the inlet and measuring the tracer concentration in the outlet the volume to flow ratio V/q can thus be determined from the equation (5) both for an ideal mixing tank and for a pipe with pure plug flow.

4. AN AXIOMATIC APPROACH

The theory of flow systems will now be developed systematical­ly. The analysis will be carried out for systems with one inlet and one outlet. There are, however, no difficulties to extend the re­sults to more general situations. In analogy with the simple exam­ples the systems will be characterized by their impulse responses. Introduce

DEFINITION I A single-input single-output time invariant linear system is called a flow system if the impulse response has the property (3). It is called an open flow system if the impulse response also has the property (4).

It follows from the previous examples that the transportation of a substance through a tank with perfect mixing and through a

60 K.J.ASTROM

pipe with pure mass transport without mixing can be described by flow systems.

Notice that the quantity

tz f h(t)dt tl

can be interpreted as the probability that a particle entering the system at time 0 will exit in the interval (tl,tZ)' The impulse response of a flow system can thus be interpreted as a probability density. It is, therefore, also called the residence time distri­bution or more correctly the density of the residence time distri­bution. The properties (3) and (4) are far reaching. A flow system is e g always input-output stable. To explore the properties fur­ther we analyse the transfer function H defined by

H(s) = f e-st h(t)dt o

The equation (4) implies that

00

H(O) = f h(t)dt = I o

For Re s ~ 0 we have

IH(s) I = If e-st h(t)dtl ~ j le-stl h(t)dt ~ o 0

00

~ f h(t)dt = I o

Re s :: 0

(6)

(7)

The magnitude of the transfer function of a flow system is thus less than or equal to one in the closed right half plane.

Let w· be arbitrary real numbers and xi arbitrary complex numbers. Tten

-iw t e 2 h(t)dt

00 iwkt 2 f I Lxke I h(t)dt::: 0 o

(8)

FLOW SYSTEMS 61

It follows from a famous theorem of Bochner (1932) that the condi­tions (7) and (8) also imply (4) and (3).

An open flow system can thus also be defined as a linear time invariant system whose transfer function satisfies (7) and (8). This is not done because the conditions (3) and (4) are much more appealing to physical intuition.

4. INTERCONNECTION OF FLOW SYSTEMS

There are several ways to interconnect flow systems. They can e g be connected in series, parallel or in feedback connections in the same way as ordinary linear systems are interconnected. More interesting and more useful results are, however, obtained if the interconnection is done in a different way. Since flow systems are used to describe the propagation of a tracer in a tank system we will first consider different ways to connect tanks together. In­terconnection of flow systems will then be defined by considering the flow systems which describe the propagation of a tracer in the interconnected tanks.

Tanks can be connected in many different ways. The outflow of one tank can be sent to another tank (series connection). A flow can be split up in different parts which are sent through tanks and again continued (parallel connection). Part of the outflow of a tank can be mixed with the inflow and sent to the tank again (feedback connection).

It seems intuitively clear that if the tracer propagation in two tanks is described by flow systems in the sense of Definition 1, then the propagation of a tracer in the interconnected tanks is also a flow system. It will now be shown formally that this is in­deed the case.

By a series connection of two tanks we mean the system ob­tained by letting the outlet of one tank be connected to the inlet of the other tank as illustrated in Fig 1.

Assume that the tracer propagation in Sl and S2 can be de­scribed by flow systems with the transfer functions HI and H2 . Let ci' c l and c denote the tracer concentrations at the inlet of Sl' the outlet of Sl and the outlet of S2 respectively. Then

Cl(s) = Hl(S)Ci(s)

C(s) = HZ(s)C l (s)

Elimination of Cl gives

62

r-----' q. Cj I : q,Cj

-7 V ---7

I I 1 I I I 5 I L ____ ll

r---­ -, I I

I I ! q,c V2 ---7

: I I I ~I L ____ :..1

K.J.ASTROM

Fig 1. Series connection of the flow systems Sl and SZ'

and we thus find that the propagation of a tracer in a series con­nection of two tanks can be described by a linear system with the transfer function

(9)

To show that the transfer function H corresponds to a flow system we introduce the corresponding impul~e responses, i e

00

h (t) = f hZ(t-s)hl(s)ds s 0

It is clear that if hl and hZ are non-negative then hs is also non­-negative. Furthermore it follows from (8) that

Taking (9) as the definition of a series connecting of two flow systems it has thus been shown that the series connection of two flow systems is a flow system.

We will now proceed to other ways of connecting flow systems. A parallel connection of two tanks is obtained by splitting the inflow q into two flows ~lq and ~zq where 0 ~ ~l ~ 1 and ~l + ~Z = = 1. These flows are then taken as inflows to the tanks Sl and Sz whose outflows are then combined assuming perfect mixing. The pa­rallel connection is illustrated in Fig Z.

To analyse the propagation of a tracer through two tanks Sl and Sz in parallel it is assumed that the tracer propagation through S~ ana Sz can be described by flow systems with the transfer func­t~ons Hl and HZ' Let ci denote the tracer concentration at the in­let and cl and Cz the tracer concentrations at the outlets of the

FLOW SYSTEMS

I I I S I L ____ L q,c

--7

Fig 2. Parallel connection of the flow systems Sl

and S2'

tanks. Then

HI (s)Ci(s)

H2(s)Ci(s)

Since the output flow is obtained by ideal mixing of the flows

alq and a2q , with tracer concentrations cl and c2, the concentra­

tion at the outlet becomes

The propagation of a tracer through a parallel connection of two

tanks can thus be described by a linear system with the transfer

function

To verify that this is a transfer function of a flow system the

impulse responses are introduced. Hence

63

64 K.J.AsTROM

It is clear that if hI and h2 satisfy (3) and (4), then hp will also satisfy the same equations.

The feedback connection Sf of two tanks or two flow systems Sl and S2 is illustrated in Fig 3. Let the inflow to S3 be q and the tracer concentration ci' Furthermore let the proportion a of the outflow of Sl be the inflow to S2' It is assumed that the out­flow of S2 is perfectly mixed with the system inflow.

If aql is the flow through S2' a flow balance then gives

Hence

Let c2 denote the concentration at the outlet of S2 then

C2(s) = H2(s)C(s)

,-----1

q,c j ,-... q,c1 l L--..I-' __ q_,C_ ~ ~ V1 )

l' 'I I' t I I 511

L ____ ..!.J

1-----'

: I I ~ ~ __ ..L............I

Fig 3. Feedback connection of the flow sys­tems Sl and S2' The inflow q is perfectly mixed with the outflow of S2, and the mixture is fed to Sl' The outflow of Sl is split into two streams, one of which goes to S2 and the other part is the outflow of Sf.

FLOW SYSTEMS 65

The input to Sl is a mix of two flows q and aq/(l-a), having con­centrations ci and c2 respectively. The concentration cl at the in­let of Sl is thus

Furthermore

which gives

C (s) (l-a)Hl (s)

C. (s) 1

The tracer propagation through a feedback connection of two tanks can thus be described by a linear system with the transfer func­tion

a ::; a < 1 (11)

Assuming that HI and H2 are transfer functions of flow systems it will now be shown that Hf is also such a transfer function. We have

(l-a)Hl (0) 1 - a 1

1 - a

Furthermore introduce H = HlH2' Since Sl and S2 are flow systems, it follows from the equation (7) that

!H(s)! ~ 1 for Re s ~ 0

The series expansion

thus converges uniformly for a S aa < 1 and Re s ~ O. The corre­sponding impulse response then satisfies

where * denotes convolution. Since Sl and S2 are flow systems, we have hl(t) ~ a and h2(t) ~ 0, and we find hf(t) ~ O.

66 K.J.ASTROM

Summing up we get

THEOREM 1 Let Sl and S2 be open flow systems with the transfer functions HI and H2' The series Ss' parallel Sp and feedback Sf connections of Sl and S2 whose transfer functions are defined by

H H2Hl (9) s

H alHl + a 2H2 o ~ 0.1' 0. 2 ~ 1, 0.1 + 0.2 1 (10) p

(l-a)Hl 0 a < 1 (11) Hf ::;

1 - aHl H2

are then also open flow systems.

Remark. Notice that the series connection of two flow systems is identical to the series connection of two linear systems. The parallel and feedback connections of flow systems are, however, not the same as the parallel and series connection of linear systems.

Using Theorem 1 the propagation of a tracer through a tank system can be studied in the same way as signal propagation is ana­lysed in an ordinary linear system.

5. THE STEWART-HAMILTON EQUATION

The analysis of the simple tank systems corresponding to a tank with ideal mixing in Example 1 and to a tank with pure plug flow in Example 2 shows that the following equation

'J th(t)dt = V/q o

(12)

holds in both cases. Compare wi th the equation (5). Recalling the probabilistic interpretation of the impulse response h as the resi­dence time distribution the equation (12) simply says that for a tank system with one inlet and one outlet the ratio of volume to flow equals the mean residence time. The equation (12) was first used by the physiologists Stewart (1897) and Hamilton (1932) who developed methods to determine the blood volume of the heart. The equation (12) will therefore be called the Stewart-Hamilton equa­tion. The equation has been widely used both in biology, physiolo­gy and engineering. It has also been misinterpreted and therefore the cause of much controversy.

The equation (12) can be derived by the following heuristic

FLOW SYSTEMS 67

argument. Consider an open tank system with inflow q. The fraction h(t)dt of the particles which enter the system at time zero will exit in the interval (t,t+dt). These particles have traversed the volume dv = t·q. Integrating over all particles now gives (12). The validity of the equation (12) can also be shown formally in many cases. We have the following result:

THEOREM 2 Let 51 and S2 be tank systems with one inlet and one outlet and vo­lumes Vl and V2' Let the tank system S3 be a series, parallel or feedback connection of Sl and S2' Assume that the Stewart-Hamilton equation holds for Sl and S2 then it also holds for S3.

Proof. Let Hl and HZ be the transfer functions which characte­rize the tracer propagat~on in 51 and S2' The different ways to interconnect the systems will be discussed separately.

First consider a series connection. It follows from Theorem 1 that the tracer propagation in S3 then is characterized by the transfer function H3 = H1H2' The mean residence time of 53 is then given by

The third equality follows from the fact that the flows through Sl and S2 are the same in a series connection.

Now consider a parallel connection. See Fig 2. Since the flow through 51 is alq and that through S2 is u2q, we get

and

The mean residence time of S3 ~s given by

and the result is thus established also for a parallel connection.

For a feedback connection, Fig 3, the flow through Sl is ql = q/(l-a) and the flow through S2 is aql = aq/(l-a). Hence

Vl/ql = (l-a)Vl/q

V2/(aq) = (1-a)V 2/(aq)

68

The equation (10) gives

H' 3

The mean residence time is then given by

i th (t)dt = - H'(O) = - __ 1 __ H' - ~ H' o 3 3 I-a 1 I-a 2

and the proof is now complete.

K.J.AsTROM

2 (1-a)(H'+aH H') 1 1 2

Remark 1. Combining Theorem 2 with the results of Example 1 and Example 2, it is thus found that the Stewart-Hamilton equation holds for systems which are obtained by series, parallel or feed­back connections of simple flow systems with pure transport or' with ideal mixing.

Remark 2. The Stewart-Hamilton equation has been derived only for systems which are open flow system. Internal recirculations are allowed provided that only a fraction of the flow is recirculated (a < 1 in Theorem 1). All fluid particles must, however, sooner or later leave the system, or formally the equation (4) must hold. This will not be the case if all the flow is recirculated.

6. REFERENCES

Andersson, B J (1957): "Studies on the circulation in organic systems with applications to indicator methods". Trans Roy Inst of Technology, No 114 (1957), Stockholm, Sweden.

Bochner, S (1932): "Vorlesungen tiber Fouriersche Integrale". Leipzig 1932.

Danckwerts, P V (1953): "Continuous flow systems". Chem Eng Sci l (1953) 1-13.

Hamilton, W F, Moore, J W, Kinsman, J M, Spurling, R G (1932): "Studies on the circulation". Am J Physiol 22. (1932) 534.

Sheppard, C W (1962): "Basic principles of the tracer method". John Wiley, New York 1962.

Stewart, G N (1897): "Researches on the circulation time and on the influences which affect it". J Physiol 22 (1897) 159-183.

SOME REMARKS ON THE CONCEPT OF STATE

H.S. Witsenhausen

Harvard University, Cambridge, Massachusetts and Bell Laboratories, Hurray Hill, New Jersey

1. THE DETERHINISTIC STATE

Consider a system which is subject to a succession of T inputs ul,uZ, ... ,uT where ut belongs to a given set Ut , for each t. Outputs

Yt in given sets Yt are generated for t = 1, ... ,T. The input-output description of the system is given by functions St as follows.

t = 1, ... ,T

These relations exhibit the familiar triangular structure corres­ponding to causal generation in the temporal order defined by the indexing.

By a state variable description of this system one means a sequence of sets Xt called state spaces, and functions

such that the equations

f (x l'u t ) t t-t z, ... , T

69

70 H.S. WITSENHAUSEN

t 2, •.. ,T

describe the same input-output mapping as above.

Note that for systems in which the input determines the output uniquely, the analysis requires no initial state, the physical initial conditions of the system being either given, hence not variables of the problem, or else adjustable, hence actually inputs and thus included in u1 .

There is a trivial way to obtain a state description, by letting Xt = U1 (9 U2 ~ ... Q) Ut' taking f t to be the concaten-

ation function "append" and gt = St 0 ft. This takes the state to be the input history. To go beyond this simple and unsatisfactory device it is necessary to introduce a notion of minima1ity. This is done as follows [2]. For given t, the input and output sequences can be cut into two parts u f = (u1 ' ... ,u ),

ront t

Yfront = (Yl'···'Yt) and u tail = (ut+l,···,uT), Ytai1 =

(Yt+1""'YT)' By the triangular structure Yfront only depends

upon ufront' but y 01 is dependent upon both u f and u '1' tal. ront tal. which can be written y 01 = F(uf ,u 01)' Now an equivalence

tal. ront tal.

relation - is defined in U = fl U b u(l) - u(2) when front 1 e y front front

the sections F(uf(l) ,.), T ront

F(u (2) .) which are functions from front'

Utai1 = IT Ue into Ytai1 t+l

n Ye, are the same. Then Xt is taken t+1

as the quotient space Uf /-. It can be easily shown that the ront

equivalence class x 1 and the input u determine uniquely the t- t

equivalence class x t and output Yt'

An equivalence class can also be viewed as an atom of the partition which it induces, or as an atom of the algebra of sets generated by this partition. It should therefore not be too sur­prising if in stochastic problems 0-a1gebras would take the part of state spaces.

Note that in the original paper [2] of Nerode, the index t ranges over all positive integers. In that case tails obtained by cuts at two different times can be compared after a translation aligning the starting points. This permits equivalences among front sequences of different lengths. In the cases we are consider­ing here, "time to go" is an une1iminab1e part of the state.

THE CONCEPT OF STATE 71

2. PARALLELISM

If two branches of an administration operate independently and the next level waits for reports from both, then the relative time at which decisions are taken in one branch as compared to the other is irrelevant, these decisions are parallel and cannot influence each other. In physics, it is known that two events whose interval is space-like cannot be assigned an invariant time order. For these and other reasons, a system's inputs and outputs can sometimes be indexed in several different ways, each of which yields a triangular structure hence a sequence of minimal state spaces. A state space in one such sequence cannot in general be identified with any of the spaces in another sequence. The chief purpose of our remarks is to describe this situation more closely, in the hope that this will make less strange some of the diffi­culties that arise in stochastic control.

3. FEEDBACK

Suppose there are n inputs u. E U., i = l, ... ,n and m outputs 1 1

y. E Y., j = l, ... ,m. Each output y. is a function S. of the J J J J

inputs. While y. will generally depend only on some of the ui one can always formally consider it as a function of all input vari­ables, a function that may be independent of some of its arguments. At this stage it is entirely possible that all inputs be applied in parallel and all the outputs produced in this way could then actually depend on all inputs. In such a case there is effectively only one time stage. As the role of the state is to carry data from one stage to the next, there is no place for a state in that situation.

However this total parallelism is ruled out when inputs can depend upon outputs, that is, when there are feedback possibilities. It is essential to specify what information about the outputs is available for each input. This information is provided in general by functions of the outputs which determine a partition of the output product space. For simplicity, let us consider only the case where an input has either complete knowledge of a given output or none. Then one need only specify for each input the set of (indices of) outputs that are observed. (Actually, this simpli­fication implies no loss of generality.)

4. THE SEQUENTIAL CASE

Can one (totally) order the inputs in such a way that, for each input, the outputs which it observes depend only on preceding inputs? That this need not be the case has been long known in the theory of games in extensive form. The value chosen for an input

72 H.S. WITSENHAUSEN

may influence the order in which other inputs are applied, and because of this, there may be no fixed order of the inputs for which the above condition is satisfied. When such an order does exist the system is called sequential and the next question is whether this order is unique. In general, because of parallelism, there will be a set of such total orders.

It turns out that this set is precisely the set of all total orders compatible with a certain partial order.

5. THE PARTIAL ORDER

To analyse the ordering problem, one can use the technique developed for the stochastic case in [4]. First, it is clear that the outputs that a certain input observes cannot depend on that input (absence of self-information). Now a subset of the inputs is called closed when any output observed by any input in the set is dependent only upon inputs in the set. Now if i and j are two input indices, the relation i R j is declared to hold when any closed set of inputs containing j also contains i. The relation is reflexive and transitive, i.e., a quasi-order. The sequential case obtains if and only if the relation is also antisymmetric, hence a partial order, and in that case the triangular structure is obtained for just those total orderings of the inputs that are compatible with this partial order. All these facts are shown in Theorem 1 and Lemma 3 of [4].

For example, suppose that there are 7 inputs u., i=1, ... ,7 and 5 outputs Yj' j = 1, ... ,5. Inputs ul and u2 observ~ no outputs

while for i = 3, ... ,7 input u. observes output y. 2' The functions 1 1-

giving the outputs in terms of the input show that outputs Yl""'YS depend respectively on the following sets of inputs {ul }, {ul ,u2},

{u2}, {u3 ,u4}, {u4 ,uS}. The resulting partial order of the input

indices is given by the diagram

and there are 42 total orders, among the 7! = 5040 possible ones, that are compatible with this partial order and yield triangular structure.

Any of these 42 orders permits the formation of a sequence of state spaces, consisting of equivalence classes in the product

THE CONCEPT OF STATE 73

of the preceding input spaces, in the usual manner. The result is a plethora of state spaces.

It is important to note that the above partial order is obtained from the original input-output description by analysis of parallelism. This is in sharp contrast with the role partial orders play in certain extensions of algebraic system theory [1], where it is assumed that a certain partial ordered set (e.g. multi-dimensional "time") is given first and system properties are described with respect to it, ab initio.

6. THE LATTICE OF CUTS

A cut of a partially ordered set is a partition of the set into two complementary subsets, the anterior set and the posterior set, such that no element of the posterior set precedes any element of the anterior set under the partial order. Cuts are partially ordered by inclusion of their anterior sets, and under that order­ing the cuts form a lattice. That is two cuts a and S have a least upper bound a~S whose anterior set is the union of those of a and S, and a greatest lower bound a",S whose anterior set is the intersection of those of a and S. The ascending paths from the minimum of the lattice (empty anterior set) to the maximum (empty posterior set) correspond to the total orders compatible with the original order; along such a path the elements are taken one by one into the anterior set.

The lattice of cuts derived from the partial order of a system yields a lattice of state spaces. For each cut, the inputs are divided into anterior and posterior inputs. Outputs are called anterior if they depend only upon anterior inputs, otherwise they are posterior. Now, two systems of values for the anterior inputs are declared equivalent if they determine the same mapping of posterior inputs into posterior outputs. Then, dividing the cartesian product of the anterior input spaces by this relation yields a state space X for cut a. a

In the example, the lattice of cuts has 18 elements of which 2, the minimum and maximum, give trivial degenerate one-point state spaces, leaving 16 distinct genuine state spaces. Along each ascending path, 6 of these are encountered.

7. THE REDUCTION THEOREM

Suppose that one wishes to calculate the states for given inputs. If all states, for all cuts, are desired, the task can be somewhat lightened by the use of relations between the states that hold regardless of the inputs. This is the motivation for

74 H.S. WITSENHAUSEN

stating the following easy result, Reduction Theorem: If a and (3 are two cuts, then the states x , ~R and xa (3 determine the state xa (3 without reference to thg ~fiputs, ~ 'f

Proof: Let U1 denote the product space of the inputs that are anterior un~er both cuts, likewise U2 for those anterior under a only, U3 for those anterior only under (3, U4 for those posterior under both cuts,

What needs to be shown is that, if

(1) (ui,ui) - (u" u") l' 2 under the equivalence at cut a on U1 0 (2) (ui,u3) - (u" u") l' 3 under the equivalence at cut (3 on U1 0 (3) (u ' ) - (u") under 1 1

the equivalence at cut aA(3

then

Let Y4 be the product space of the outputs posterior under av(3 hence under both a and (3, The system input-output description gives these outputs as functions of the inputs by f : U1 Q9 U2 0 U3 (9 U4 + Y4 , Proving (4) means showing that

By the equivalence at cut a, one has

(6) f ( I I ) f (" II ) f 11 U ,.. U u1 ,u2,u3 ,u4 = u1 ,u2,u3 ,u4 or a u3 £ 3' u4 ~ 4'

in particular, choosing u 3 = u3, one has

( 7) f ( ' , I ) f (" II , ) f 11 U u1 ,u2,u3,u4 = u1 ,u2,u3 ,u4 or a u4 £ 4'

By the equivalence at cut (3, one has

(8) f ( , ') f ( II ") f 11 C" U C" U u1 ,u2,u3 ,u4 = u1 ,u2,u3 ,u4 or a u 2 ~ 2' u4 ~ 4'

in particular, fixing u2 at Uz gives

(9) f( ' II , ) - f( II II II ) f 11 U u1 ,u2,u3,u4 - u1 ,u2,u3,u4 or a u4 £ 4'

By the equivalence at cut a A(3 one has

(10) f(ui,u 2,u3,u4)

u4 £ U4,

U2

U3

THE CONCEPT OF STATE

in particular, fixing u = u" u = u' one has 2 2' 3 3

( 11) f ( ' " , ) - f (" " , ) f 11 c- U ul ,u2,u3,u4 - ul ,u2,u3,u4 or a u4 ~ 4·

For all u4, the right hand side of (7) equals the right hand side of (11), while the left hand side of (11) equals the left hand side of (9). Thus (7), (9), and (11) imply (S) which completes the proof.

75

Returning to the example, the reduction theorem determines the state at all cuts given the state at just 7 cuts, those with the anterior input index sets {1},{2}, {1,3}, {2,S}, {1,2,4}, {1,2,3,4,6} and {1,2,4,S,7}. Notice that 7 is the number of inputs, and that none of the 42 ascending paths encounters more than 3 of these 7 cuts.

8. CONCLUS IONS

The difficulties that can arise in defining states for deter­ministic systems are naturally compounded in the stochastic (and game) case. The state should be a summary ("compression") of some data (the "past") known to someone (an observer or a controller) and sufficient for some purpose (input-output map, optimization, dynamic programming). In addition one would like a concept of minimality. At this point, a notion of state has only been proposed for a few classical special cases. The utility of a state notion in the most general case is highly questionable. In the intrinsic model [3], the umpire's a-fields can be considered as providing a notion of state, however this does not yield a dynamic programming algorithm for optimization. The absence of a useful state is per­haps most evident in the case of 2-person zero-sum stochastic games: the whole history of data available to one player contains clues about the opponents strategy that can seldom be compressed without possible regrets in the future.

REFERENCES

1. Mu1lans, R.E. and Elliott, D.L., Linear Systems on Partially Ordered Time Sets, Proc. 1973 Conf. on Decision and Control, IEEE 1973.

2. Nerode, A., Linear Automaton Transformations, Proc. Am. Math. Soc., Vol. 9, pp. 541-544, 1958.

3. Witsenhausen, H.S., On Information Structures, Feedback and Causality; SIAM J. on Control, Vol. 9, pp. 149-160, 1971.

4. , The Intrinsic Model for Discrete Stochastic Control: Some Open Problems; Lecture Notes in Economics and Mathem. Systems, Vol. 107, pp. 322-335, Springer-Verlag, 1975.

ON MULTICRITERIA OPTIMIZATION

E. Polak and A. N. Payne

Department of Electrical Engineering and Computer Sciences and the Electronics Research Laboratory University of California, Berkeley, Calif. 94720

ABSTRACT

This paper examines the state of the art in multicriteria opti­mization. For this purpose, multicriteria problems are classified in terms of complexity as finite and small, finite and large, and infinite. The relative merits of typical methods for solving each of these classes are discussed and some suggestions for future work are made.

1. INTRODUCTION

Multicriteria optimization problems are almost always two or three phase problems. The first phase is common to all of them and can be stated as follows. Given (i) a set n of (feasible) decision variables (also referred to as alternatives), (ii) a set of criteria

fi : n + lR, i 1,2, ••• ,m,t (also referred to as attributes) which

produce values f(x) (with f = (f1,f2, ••• ,fm», and (iii) a partial m * order ~ (2) in lR, the space of values, construct the set of ~-

inferior decision variables ~ and the set of noninferior values V defined as follows: N

tThe fi are usually defined on a space X with n c X.

*In this paper we shall assume that v1 ~ v2 if vi ~ v~ for i = 1,2,

••• ,m; v1 ~ v2 if vI I v2 and v1 ~ v2 ' This is the most commonly

used partial order.

77

78 E. POLAK AND A.N. PAYNE

(1)

where

N (v) t:,. {v' E lRm I v' ~ v} (2) .

and

(3)

The second phase consis~s of the following: given a set of accept­

able performance values V C lRm, find all the points in a

(4).

and in

(5)

The third phase may consist of either selecting a point in ~a or imposing a total order on the elements of nNa •

In this paper we shall examine the difficulty involved in solving multicriteria problems, as well as some of the techniques suggested for their solution. For this purpose, it will be con­venient to group multicriteria problems into three distinguishable classes: (i) when n consists of a small number of elements, (ii) when n consists of a large, but finite, number of elements, and (iii) when n is a subset of a normed space.

2. THE "SMALL" MULTICRITERIA DECISION PROBLEM

We begin with the simplest case, when n contains a small number of elements, say less than 20, and the number of criteria is fairly small, say less than 6. This is a common situation when one is buying a car, a radio, a saw, etc. In this case, the construction of VN and nN is quite trivial so that phase 1 poses no difficulties, while in phase 2 the number of decision variables to be considered is further restricted by V , the acceptable performance set. In fact, the set V is often nsed to convert criteria into constraints and thus reduceathe number of criteria. The final choice (phase 3) is often facilitated by the fact that, usually, the criteria can be ordered in terms of their relative importance. Such an ordering is is called a lexicographic ordering of the criteria. It is used to impose, successively, more stringent acceptable performance require­ments. Also, it is not uncommon to reorder the relative importance of criteria after the process of elimination of unacceptable alter­natives has reached a certain stage.

MULTICRITERIA OPTIMIZATION 79

Let us illustrate this progressive elimination process by showing how a hypothetical consumers analyst might arrive at a rec­ommendation for a radial-arm saw out of a field of 8 alternatives; i.e., the cardinality of Q is 8. The following attributes are commonly considered to be pertinent to this selection (in the initial order of decreasing importance): (a) depth of cut at 900 , (b) rip width, (c) motor type, (d) depth of cut at 45 0 , and (e) price. The alternatives and attributes are displayed in Table 1. Note that the attribute (c) is not a numerical value, but it could be con­verted to one by assigning value -1 to an induction motor and value o to any other type motor since induction motors are preferred. Also, since the set (1) is implicitly specified in terms of minimiza­tions, the negatives of the depth of cut and rip width must be used to convert this problem to standard form. The acceptable perform­ance values are specified as follows: (a) the depth of cut at 90 0

should be at least 3", and (b) the rip width should be at least 25".

Now, in this case, rather than first construct VN and QN' it is more expedient to first eliminate all alternatives which do not yield acceptable performance. Thus, alternatives 2, 5, 7, and 8 are immediately eliminated, leaving only alternatives 1, 3, 4, and 6 for censideration. Denoting alternative i by ai' we see that f(a4) ~ f(al) and hence al $~. By inspection, the analyst now obtains that QNa = {a3,a4,a6}· Next, the analyst reorders hi~ priorities to :et motor type take precedence over the other criter1a and hence e11m­inates a60 Since a depth of cut of 3 1/8" at 90 0 does not represent a useful improvement over 3", the analyst considers a3 and a4

Attributes

(a) (b) (c) (d) (e) Depth of Rip Motor Depth of Price

Alternative Cut at 90 0 Width Type Cut at 45 0

1 3 in. 2s% in. induction 2t in. $265

2 3 24.l 4

induction II 8

293

3 :J. 25 induction 2 220 8

4 3 2~ 4 induction 2.!

2 215

5 21 2

24 2 induction II

8 175

6 ~ 4 24 8 universal 11

4 271

7 3 l~ 8

universal II 8

123

8 zl 8 24 induction 1 300

Table 1. Data for rad1al-saw select10n [6].

80 E. POLAK AND A.N. PAYNE

indistinguishable under this criterion. Hence the final decision is made on the basis of depth of cut at 45°, yielding our analyst's recommendation to buy a4 •

Note that the specification of acceptable performance does not always remove a criterion from the list of functions one desires to m1n~m1ze. For example, in the selection of a saw, an acceptable performance requirement may be that the price should be less than $300, but one still wishes to minimize the sum to be paid for a saw.

Thus, not infrequently, in the "small" problem case, as in our example of selecting a saw, we do not have much difficulty in making a decision since the process is considerably simplified due to the reduction of the number of alternatives first by the acceptable per­formance requirements and subsequently by lexicographic elimination~ The final sequential reduction of alternatives by the lexicographic approach appears to be "natural" in that people tend to adopt it without special training [18], [22].

3. THE LARGE, BUT FINITE, MULTICRITERIA DECISION PROBLEM

We now consider the case where n consists of a finite number, v, of alternatives, but v is too large for the construction of nNa to be possible by inspection. This is obviously a combinatorial problem involving sorting and one would expect a sizable bibliography on the subject. However, this is not so, and we have only been able to find [14]-[16], [27]. The latest work dealing with this problem appears to be that of Kung, Luccio, and Preparata [15] who obtain bounds on Cm(v), the number of scalar comparisons necessary for com­puting the sets VN and ~l' where m is the number of criteria and v is the cardinality of f(n). Specifically, they show that

C (v) m ~ O(v 10g2v) for m = 2,3 (6)

C (v) m

m-2 < 0(V(10g2v) ) for m> 4 (7)

In [26], Yao shows that

Cm(v) ~vlog2v + v - 1 for m > 2 (8)

Kung et al. exhibit algorithms which satisfy the bound (6) for m = 2,3, but there appears to be no explicit algorithm that achieves (7) for m ~ 4. However, as we shall shortly show, it is quite easy to specify a simple algorithm which, for any m, requires at most

- A C (v) = mv(v+l)/2 m

scalar comparisons. Now suppose that m 10 and v

(9)

103• Then the

MULTICRITERIA OPTIMIZATION 81

bound in (7) is

m-Z o (v(logz v) ) (10)

while

(11)

Thus, the bound (7) is not very sharp, at least not until v gets to be very large. Consequently, for moderate size problems the algo­rithm below should be quite acceptable. This algorithm requires that the vectors v E fen) be indexed as a set {vi}~ l' v. I v .• l.= l. J The algorithm first examines Vy with Y = v. If Vy $ Va' then Vy is

removed from the list. If v E V , then v is compared with v., yay J j = y - 1, ••• ,Z,1, until either (a) v ~ v., in which case Vy is rejected, or (b) v < vJ., in which ca~e v.Jis rejected and the list is

y - J . renumbered so that (vl ,v2'··· ,Vj_l'Vj +l '· •. vy) -+ (vl ,v2,.·. ,vy_l ), or (c) the top of the list, namely vl ' is reached, in which case Vy E VN and is removed as such. The process then begins again with y, th~ length of the list, decreased by one.

General Purpose Multicriteria Optimization Algorithm

v Data: {vi}i=l' Va' Step 0: Set k = 1, ~ = 1, Y = v.

Comment: k is the index for the rejected values (Vi = rk) and ~ is the index for noninferior values (Vi = v~); Y is the total number of values under immediate consideration.

Step 1: If Vy E Va' go to step 3; else, go to step 2.

Step Z: Set r k = vy ' k = k + 1, Y = Y - 1. If Y = 0, stop; else, go to step 1.

Step 3:

{vi}~=1 v = vy '

Step 4:

Step 5:

step 6;

Y = Y -

SteE 6: step 7;

SteE 7:

If Y = 1, set v~

and corresponding

Y = Y - 1, j = y,

= v 1' print the set of noninferior values . {A}~ alternatl.ves x. . 1 and stop; else, set l. l.=

and go to step 4.

If v < v. or v. ~ V , go to step 5; else, go to step 8. - J J't" a

Set r k = vj ' k = k + 1. If j = y, set Y = Y - 1 and go to

else, renumber (Vj+1,Vj+2""'Vy) -+ (Vj,Vj+1"",vY_l)' set

1,and go to step 6.

Set j = j - 1. If j = 0, set v~ = v, ~ = ~ + 1, and go to else, go to step 4.

A ~ If y = 0, print the set of noninferior values {vi}i=l and

82 E. POLAK AND A.N. PAYNE

corresponding alternatives {xi} ~=l and stop; else, go to step 3.

Step 8: If v > v., set r k = V, k = k + 1, and go to step 3; else, - -Il J go to step 6.

Since the superior algorithms for the casesm = 2 and m = 3 are easy to state, we give below our extension of the schemes proposed in [15] for these cases. Our extension accounts for V and for the fact that vi = VI is possible, whereas in [15] it was ~ssumedvi ~ v~ for i ~ j. J Bicriteria Algorithm (m=2)

Data: {Xi}~=l' {Vi}~=l' Va· Step 0: Sort the vectors v., i = 1,2 •.•. ~ on the basis of the first component and renumber so tRat vi ~ V2 ~ .•• ~ v~. Set i = 1, Y 1.

Siep 1: Find the largest integer ~(i) such that v! = v~+l vi+~(i) • Step 2: Find an integer ~(i) E I(i) ~ {i,i+l, ••. ,i+~(i)} such that

v~(i) = min{v~lk E I(i)}; set uy = v~(i)' ~y x~(i)' and y = y + 1. Step 3: If i + ~(i) < v, set i i + ~(i) + 1 and go to step 1; else, go to step 4.

Comment: The set {u.}! 1 consists of all the values in f(Q) satis­

fying ui < u; < ••• < ~~~= A value Vi not included in {ui}I=l cannot

be in VN•

Step 4: Set i = 1, ~ 1, bO = 00.

2 Step 5: If u. > b. 1 or u. ~ V

- ~ - ~- ~ 'f a go to step 6; else, set ~~ ui ' go to step 6.

(i.e., u. ~ VN ), set b. = b. 1 and A ~ 'f a ~ 2~-Xn = ~., ~ = ~ + 1, b. = u., and ~ ~ ~ ~

Step 6: If i < y,

set of noninferior

set of noninferior

set i = i + 1 and go to step 5; els~ print the

acceptable values {v.}~-l and the corresponding A ~ J J-

alternatives {x.}. 1 and stop. J J=

The tricriteria algorithm described in [15] is more complex. It is based on the following argument. Suppose we are given a set

of noninferior values V. = {v.}~ 1 in lR3, ordered so that VII < v12 < •• 1 J ~ ~= - ~l

~ vj ' and suppose that we wish to determine if the set Vj +l = {Vi}i=l

where v:+l > v:, also consists of noninferior values only. Let D. 2J 3 - J 2 . _ ~

Wi = (vi,vi ) E lR for ~ - 1,2, •.. ,j + 1, and let Wj = {Wik}k=l be the f . f· 1 . {}j L A _ (A2 A3)

se~ 0 3non~n er~or va ues ~n Wi i=lo et wk - vk,vk wik (v. ,v. ) and assume,without loss of generality, that ~k ~k

MULTICRITERIA OPTIMIZATION 83

v~ ~ V~ ~ ••• ~ v~. Then Vj +1 consists of noninferior values if

and only if the set W. U {w~. 1} consists of noninferior values only. Furthermore, Wj U {Wj !l} cont~sts of noninferior values if and only

if v~ 1 < v~*' where j* is the largest integer in {1,2, .•• ,~} such J+ ~2 J 2 .

that v < v. 1 for p = 1,2, ••. ,J*. p - J+

Tricriteria Algorithm (m=3)

Data: V 'V

{x'}i l' {v.}. l' V • ~ = ~ ~= a

Step 0: Sort the set {v.}~ 1 on the basis of the first component, ~ ~=

and renumber {vi}~ 1 and {x.}~ 1 ~= ~ ~=

1 1 accordingly so that vI ~ v2 <

1 < v •

- 'V 1 1 Step 1: Find the largest integer 0 such tQat Vo = vI and apply the bicriteria algorithm to {wl ,w2 , .•. wo}(w. ~ (vi,vt), i = 1,2, .•. ,0)

. ., ~~. 1 1 1 to f~nd all the non~nfer~or values {w. }k-l (~th w. < wi < ••• <w. )

~k - ~l 2 ~~ in {w'}~=l' Renumber the set (vi ,vi , •• ,vi ,vo+l,··,vv) ~ (vl ,v2 ' .. .,Vyt. 1 2 ~ Step 2: for j -

Step 3:

Step 4: •..• j*.

Set i = ~ + 1, k 1,2, .... ~.

2 3 Set w = (v .• v .) • ~ ~

~. 2 3

(v .• v.) J J

1 1 Find the largest index j* such that w. < w for J' = 1.2.

J -

Step 5: If w2

(wl .w2 ••.. wj *.

i = i + 1. and

2 ~ ~ < wj * and vi E Va' set vk+l vi' xk+l = xi' renumber

w.wj*+l.···.wk) ~ (wl .w2 •.••• wk+l ). set k = k + 1.

go to step 6; else. set i = i + 1 and go to step 6.

Step 6: If i ~ y. go to step 3; else. print the noninferior values

{Vj}~=l and the corresponding noninferior alternatives {X}~=l and

stop. tI

Note that in the three algorithms which we have presented the test for vi E Va is carried out simultaneously with the determina­tion of whether vi E VN• If it is relatively easy to determine

Va n {vi}~=l' then it may be more efficient to do this first. Similarly; it may be more efficient to first determine VN completely and then find the set VNa = Va n VN• In fact. in many decision

problems this may be the only way to proceed since the decision maker may use information provided by a knowledge of VN to establish the set of acceptable performance values.

84 E. POLAK AND A.N. PAYNE

Thus, the construction of the noninferior, acceptable perform­ance value set VNa is a tractable task in the finite alternative case. If the third phase is to select a single point, then this task can be accomplished as follows. (i) Scan the range of values for each criterion. Those criteria in which the values vary little

(e.g., (m~x v~ - m~n v~)/ake(v~) is small) can be removed from

further consideration since these criteria offer little discrimina­tion. (ii) For the remaining criteria, establish an order of im­portance with appropriately narrow bands of acceptable performance. A successive application of these bands should reduce the subset of noninferior alternatives under consideration until a final choice is made.

When the final task is not to select a single alternative but to select several or to order the alternatives linearly, as is the case in university admissions or in the processing of fellowship applications [8], it is not uncommon to use a weighting method con-

sisting of the assignment of the scalar value ~ Aifi(x.), where . f=i J Al > 0, i = 1,2, ... ,m, are certain weights, to the alternative j. When there is some experience available, these weights can be com­puted by regression. For example, suppose a fellowship committee awards annually graduate fellowships to entering, deserving students. The attributes which the committee takes into account are grade point average (G.), rating of letters of recommendation (L.), GRE scores (E.), andla rating of the school of undergraduate eaucation (S.). Su~pose there have been N applicants for the fellowship over a ffumber of preceding years and the committee ranking rj' 0 ~ rj ~ N, of each of the N applicants has been recorded. Then, to automate their selection process, the committee chooses weights a,6,y,0 > 0 which minimize

N 2 L (aG .+6L .+yE .+OS .-r.) i=l l l l l l

(12)

In the current year, the number aGo + 6L. + yEo + oS. is the ith student's score to be used in the lfinallrankiffg. lThis process is not beyond criticism since a value which is not noninferior can be given higher ranking than a noninferior one. Nevertheless, for lack of anything better, the weighting process is used quite com­monly when a ranking must be produced and is usually applied to all alternatives in ~ rather than to noninferior alternatives only.

4. THE INFINITE ALTERNATIVES 11ULTICRITERIA DECISION PROBLEM

We shall now consider the case where ~ consists of an infinite number of points, usually defined by equality and inequality con­straints. It is no longer a problem which can be solved by

MULTICRITERIA OPTIMIZATION 85

n enumeration. We shall concentrate on the case where n C lR. Let us review the most important facts about this case (see, for example, [7], [20]).

- n m-l - 1 m-l Let f : lR + IR. be defined by f (x) = (f (x), ••• , f (x) ) , where the fi, i = 1,2, •.. ,m - 1, are the first m - 1 criterion functions. Let

(13)

Let s : V + lR defined by

s(v) ~ min{fm(x)Ix E Q, f(x) ~ v} (14)

Let r denote the graph of s(o), i.e.,

t, ml - m - - m -r = {vE IR. v = (v,v), vE V, v = s(v)} (15)

and let

V ~ f(Q) (16)

Proposition 1: The set of noninferior values, VN,is contained in r, the graph of s(o). Furthermore,

- m I m - m VN = ~ = (v,v ) Ern V v = s(v) < v' , Vv' E V I) N (v) } (17)

where N(v) is defined by (2). ~

The relation (17) can be reinterpreted as follows.

Corollary: An alternative x E_QN(!(x) E VN) if and only if x is a global minimizer of (14), for v = f(x) and, in addition, fm(x) < fm(x') Vx' E Q satisfying f(x') 2 f(x). ~

Proposition 2: Th~ sen~itivity function_s(o) is_monotonically decreasing; i.e., v' > v implies that s(v') < s(v). Furthermore, suppose s(o) is piecewise continuously differentiable. If v t: r () V satisfies 'ils(v) < 0, then v E VN' ~

Now suppose that

Q = {x E lRn I g (x) = 0, h (x) ~ O} (18)

n k n!/, where g : IR. + lR -' h_: lR + lR are twice continuously differen-tiable. For any v E V, let

(19)

86 E. POLAK AND A.N. PAYNE

If n- satisfies the Kuhn-Tucker constraint qualification for almost all v v E V and if the corresponding minimizers of (14) satisfy second order necessary conditions of optimality (see [17, p.234]), then by interpreting the results in [24], [25], [17, p.236], we conclude that s(·) is piecewise continuously differentiable and that Vs(v) = - v(v), where V is the Lagrange multiplier associated with the constraint f(x) ~ v in (14).

These observations lead to the following conclusions. (i) To compute a noninferior alternative solve (14) for some v E V to ob­tain_a sol~tion x(vl an~ a cor!esponding multiplier ~(v). I! f(x(v» = v, then (f(x(v», s(v»E r n V, and if Vs(v) = -v(v) < 0, then v = (v,_s(v» E VN and xCv) E nN• If these conditions fail, try another v. In principle, almost all points in nN can be found in this manner. (ii) Since nN consists of an infinite number of alternatives, the entire set nN cannot be constructed.

As in the finite alternative case, we find that problems with two or three criteria are special because the sets VN can be dis­played as curves or parametrized families of curves when m = 2 or m = 3. This fact is utilized in the algorithms described in [20], [21] which produce an efficient, piecewise cubic approximation to VN• Once such an approximation to VN is obtained, one can proceed as follows. First, making use of a set of acceptable performance values Va one can reduce the set of noninferior values VN to be con­sidered to VNa = VN II Va. Then one can establish bands of equivalent performance for each criterion, which are ordered with respect to their relative importance. One can then apply these bands succes­sively to eliminate a large number of alternatives and narrow down the final selection to a sufficiently small region so as to make the final choice fairly easy. This process is essentially the same as the one described in the finite alternative case.

When there are more than three criteria, the information dis­play problem makes the approach described above, to say the least, quite cumbersome, if not impossible. However, since s(·) is likely to be piecewise continuously differentiable, one may try to impose a total order on n by means of an aggregation function a : ~n+ ~ of the form

12m a(x) = a(f (x), f (x), .•. , f (x» (20)

and one can choose to minimize either a(x) over x E n or a(v) over v E r. Since on the manifold r V-a(v) = (I lIVs(v»va(v) may have

v m-discontinuities and since both a(v) and V-a (v) may be difficult to compute, it is common to make a final sel~ction by solving the aggregated problem

min{a(x) I x E n} (21)

MULTICRITERIA OPTIMIZATION 87

In the next section we shall discuss the manner in which aggregation has been approached by several authors. However, before proceeding with this, we must point out the two serious drawbacks of aggrega­tion. The first and most obvious drawback is that in many cases a solution of (21) will not yield an alternative in ~ or value in VN· The second drawback is that aggregation is based on the assumption that a total order can actually be imposed on Q. Now there is con­siderable evidence [1] that, in a multiattribute situation, humans quite commonly will prefer alternative a to alternative b, alterna­tive b to alternative c, but they may not prefer alternative a to alternative c. This shows that the imposition of a total order is often contrary to our intuitive aims and hence is quite likely to lead to less than ideal selections. Thus, aggregation should be used with extreme caution.

5. IMPOSITION OF A TOTAL ORDER: AGGREGATION

There are basically three schemes, with endless variations, for imposing a total order on the partially ordered set of values V = f(Q). The first and oldest (which was used even by Pareto in the last century) consists of imposing a weighting pattern on the

m . . . criteria; i.e., one minimizes LX-f1.(x), with x E Q and 1t1. > O. We

1.=.l have seen already an example of this technique in Sec. 3. As it was shown in [7], when V is directionally convex, any solution of

m .. min{i~lt1.f1.(x) Ix E Q} is noninf~ri~r. Furthermore, by using all the

It in the set {It E lRm lit ~ 0, i~llt1. = I}, all the noninferior alter­

natives can be computed. Thus, starting with a weighting pattern (ltl, ..• ,ltm), we can co~pute at least one noninferior alternative and then perturb the 1t1. to see whether some other noninferior point in the vicinity of the first one is more desirable. This is quite common practice in linear-quadratic regulator design [2].

The second scheme is based on the hypothesis that there are indifference surfaces and that these surfaces are equi-cost sur­faces for some unknown aggregate cost or utility function u. Thus, for example, H~nieski [12] assumes that we can specify trade off coefficients a1. such that given any v = f(x), v lies on an indif­ference surface if

m . d i du = .E a1. ~ = 0 (22)

1=1 v1.

When all a i = 1, (22) describes a point at which the fractional changes in all criteria cancel each other out. Integrating (22), Hanieski then proposes to minimize

m. . u(v) = .E a1.R,nv1.

i=l (23)

88 E. POLAK AND A.N. PAYNE

i i subject to v f (x), x E Q. Since this is a very simple scheme it is also easy to analyze it. Generally, in the context of a heuristic selection scheme for a final point v E VN, one tends to agree that (22) describes a point of confusion or indecision, i.e., a point v at which the decision maker finds it almost impossible to decide on a preferable small perturbation in v since any fractional gain in one criterion is offset by an approximately equal loss in the other criteria. So one may accept this as being a point on a surface of confusion or indecision, but.a surface of confusion need not be a surface of indifference. For example, suppose m = 2 and

1 2 I 2 -v1 1 2 VN = {(v ,v ) v = e }. Also suppose a = a = 1. Then on VN 1 2 2 1 1 1

u(v ,v ) = tnv + tnv = -v + tnv (24)

and we see that u has a maximum but certainly no minima (u(v) + _00

as v1 + 0) or local minima. Thus, u does not appear to be an ap­propriate function to minimize. Furthermore, the smaller we make u(v), the closer v approaches a "dictatorial" solution, i.e., either v1 ::: min{f1(x) Ix E Q} or v 2 ::: min{f 2 (x) Ix E Q}, which is inconsistent with a desire for a trade off or "compromise" between criteria. Our simple example shows that one must be very cautious in trans­lating well defined local trade off considerations into a global aggregate utility function.

Before leaving the second scheme of aggregation, we shall describe two other schemes for obtaining an aggregate utility func­tion from local trade off considerations, so as to illustrate the range of ingenuity that has been devoted to this subject. The first is due to Geoffrion [10] and Geoffrion, Dyer and Feinberg [11]. They assume that the decision maker (DM) has a global preference function u in mind but that he can only furnish local trade off information in the form of trade off ratios. Thus, they consider the problem of solving

min{u(f(x)) Ix E Q} (25)

in a man-machine interactive process. Their method consists of two subprocedures: one for determining a usable feasible direction and one for step size determination. Given a value v = f(x) E V, the decision maker specifies the tradeoff ratio wi between the ith criterion fi and a selected reference criterion, say fm; i.e.,

~i w dU(~) jdU(V)

dV1 dVm i = 1,2, ... ,m-1 (26)

These m - 1 trade off ratios describe indifference (equi-cost) surface of u

the tangent hyperplane to the passing through v; i.e.,

~l 1 ~1 ~2 2 ~2 ~m-l m-l ~m-1 m ~m w (v -v ) + w (v -v ) + ... + w (v -v ) + (v -v ) = 0 (27)

MULTICRITERIA OPTIMIZATION 89

Ai To obtain the w , the DM answers the query: "How much of a change ~vi are you willing to permit in the value of vi to obtain a change

i ... m of ~v in v , assuming that all other criteria do not change?" Then

i = 1,2, ••• ,m - 1 (28)

Thus, with w = (w,l),

" u(f(x» = au(v) af(x)T w x avm ax

(29)

and even though au(v)/avm is not known, w is all we need to solve the Frank-Wolfe [26] direction finding problem

min{("xu(f(x», x - x)lx En} (30) af'(~) T

since we can substitute ~ w for "xu(v) in (30) without affecting

the resulting solution h = x - x Once the direction h is computed, Geoffrion et ale require the DM to specify a step size. Thus, they have invented a well-inspired heuristic method which works well in some cases. Obviously, the method inherits all the possible pathologies that we have mentioned in conjunction with Hanieski's scheme.

To conclude our discussion of aggregation methods which extend local trade off information to a global preference function, we describe a scheme due to Briskin [3], [4]. who assumes that the DM can specify all the criteria in terms of the units of a single criterion. say fm. that the aggregate utility function is of the form

m - 1 m-l u(v) = v + u(v ••••• v )

and that u can be specified by a set of differential equations:

au(v) = h j (v) a)

j 1.2, ..•• m-l

(31)

(32)

Briskin then integrates this system of equations to obtain u(o) and then minimizes u(f(x» subject to xE Q.

To see how Briskin's scheme works. we reproduce one of his examples which involves hauling freight [4]. Suppose m = 3. with fl measuring time,f2 weight. and f3 dollar cost for a given set of alternatives. The problem is to deliver as much weight as quickly as possible and with as little cost as possible. Briskin assumes that the criteria fl and f2 can be expressed in terms of dollars. Next. he supposes that (a) the DM is willing to spend $30 to gain

90 E. POLAK AND A.N. PAYNE

one hour when the travel time is 30 hours but only $5 when the travel time is 20 hours, (b) the rate of exchange of money for time varies exponentially with the time taken, (c) the DM is willing to spend $20 to deliver 100 lbs. more if the delivered weight is 180 lbs., (d) the rate of exchange of money for weight varies inversely with the weight delivered, and (e) the willingness to spend money to gain time is independent of weight and vice versa. From all of this we get that u must satisfy partial differential equations of the form

du(vl ,v2)

dvl

du(vl ,v2)

dV2

(33)

(34) .

From the given data and (33) and (34) we get enough algebraic equations to solve for c, k1 , and k2, which can then be substituted

into the obviously assumed form of u: 1

_ 1 2 klv u(v ,v ) = -ce (35)

Briskin's method probably shares the faults inherent in Hanieski's method and other methods which transform local trade off preferences into a global preference function. In addition, it. requires much more sophistication in specifying the functions hJ than the coefficients in the Hanieski and the Geoffrion et ale schemes.

A totally different approach to aggregation is represented by the compromise solution or goal programming methods [5], [9], [13], [23], [28]-[31]. In fact, these methods are not based on the desire to construct a global preference function but on a desire to com­pute an alternative x whose value f(x) is close to the ideal value v defined as follows. Let

i 1,2, •.• ,m (36)

The compromise program then is defined as

min{IIW(f(x)-v)II Ix E n, f(x) E V } , £ a

£ > 1 (37)

where W is a positive definite weighting matrix and, as befor~ V represents the set of acceptable performance values. A solutiona (x*,f(x*» of (37) is called a compromise solution. Yu [28] has shown that, under mild assumptions, x* E nN for any norm 11 0 11£ with

MULTICRITERIA OPTIMIZATION 91

1 < ~ < 00. Compromise solutions are appealing, but, as pointed out by Yu [28] and Leitmann and Yu [29], compromise solutions are quite sensitive both to the units in which the criteria are expressed and to the particular norm II'II~ used. This is obviously a drawback and consequently Salukvadze [23] and Zeleny [30] have independently proposed that scaling be utilized to reduce this effect. Thus they suggest that the problem

~ > 1 (38)

. jA1 jA2 lAm be solved instead of (37), with S = d~ag(l v ,Iv , ... ,1 v ), pro-vided no vi = O. It is clear that while this eliminates the sen­sitivity of the solution of (38) to the units used, it makes the solution sensitive to the values of vi. Thus, while the compromise solution approach is an obviously attractive way for selecting a single noninferior alternative without constructing the entire set ~N' it is still not quite a perfect tool either.

The obvious conclusion is that the infinite alternative multi­criteria decision problem is orders of magnitude more difficult than the finite alternative multicriteria decision problem and that methods for its solution still leave much to be desired. Thus, because of its great practical importance, the infinite alternative multicriteria problem represents an area of challenging research.

6. CONCLUSION

As we have seen, as long as the number of alternatives is finite (but not astronomically large), the multicriteria optimiza­tion problem is tractable. However, when the set of alternatives is a continuum and the number of criteria is larger than 3, the multicriteria optimization problem becomes tremendously more dif­ficult and one's confidence in the soundness of the choice made by the various schemes proposed is nowhere near as great as in the finite case. In a way, the difficulty of the infinite alternative multicriteria optimization problem can be attributed to the fact that there are currently no entirely satisfactory methods for approximating such a problem by means of a finite discretized version of the problem. We can expect a good deal of future work to be devoted to the question of selecting a "grid" for the approxi­mation of a continuum-type multicriteria optimization problem. Perhaps a connection of this new work with the old (such as in [20], [19]) will be based on the acceptance of points of confusion, which were discussed in Sec. 5, as a basis for selecting such a grid.

ACKNOWLEDGEMENT

Research sponsored by U.S. Army Research Office Durham Contract

92 E. POLAK AND A.N. PAYNE

DAHC04-73-C-0025, the National Science Foundation Grant ENG73-OS2l4AOl and the Joint Services Electronics Program Contract F44620-7l-C-00S7.

REFERENCES

[1] Abelson, R. P., "The Choice of Choice Theories," in S. Messick and A. H. Brayfield (eds.~ Decision and Choice, New York: McGraw-Hill, pp. 257-266, 1964.

[2] Athans, M., "The Role and Use of the Stochastic Linear-Quad­ratic-Gaussian Problem in Control System Design," IEEE Trans. Automat. Contr., vol. AC-16, pp. 529-552, Dec. 1971.

[3] Briskin, L. E., "A Method Unifying Multiple Objective Func­tions," Management Science, Vol. 12, pp. B406-B4l6, June 1966.

[4] Briskin, L. E., "Establishing a Generalized Multi-Attribute Utility Function," in J. L. Cochrane and M. Zeleny (eds.), Multiple Criteria Decision Making, Columbia, S. C.: USC Press, pp. 236-245, 1973.

[5] Charnes, A. and W. W. Cooper, Management Models and Industrial Application of Linear Programming, Vol. I, New York: Wiley, 1961.

[6] Consumer Reports, VoL 39, pp. 164-171, Feb. 1974.

[7] Da Cunha, N. O. and E. Vector-Valued Criteria Balakrishnan and L. W. of Control, New York:

Polak, "Cons train ted Minimization Under in Linear Topological Spaces," in A. V. Neustadt (eds.), Mathematical Theory Academic Press, pp. 96-l0S, 1967.

[S] Dawes, R. M., "A Case Study of Graduate Admissions," American Psychologist, Vol. 26, pp. ISO-ISS, Feb. 1971.

[9] Dinkelbach, W. and H. Isermann, "On Decision Making Under Multiple Criteria and Under Incomplete Information," in J. L. Cochrane and M. Zeleny (eds.), Multiple Criteria Decision Making, Columbia, S. C.: USC Press, pp. 302-312, 1973.

[10] Geoffrion, A., "Vector Maximal Decomposition Programming," Western Management Science Institute, Univ. of California, Los Angeles, Working Paper No. 164, Sept. 1970.

[11] Geoffrion, A. M., J. S. Dyer and A. Feinberg, "An Interactive Approach for Multi-Criterion Optimization with an Application to the Operation of an Academic Department," Management

MULTICRITERIA OPTIMIZATION 93

Science, Vol. 19, pp. 357-368, Dec. 1972.

[12] Hanieski, J. F., "Technological Change as the Optimization of a Multidimensional Product," in J. L. Cochrane and M. Zeleny (eds.), Multiple Criteria Decision Making, Columbia, S. C.: USC Press, pp. 550-569, 1973.

[13] Huang, S. C., "Note on the Mean-Square Strategy of Vector Valued Objective Functions," JOTA, Vol. 9, pp. 364-366, May 1972.

[14] Kung, H. T., "On the Computational Complexity of Finding the Maxima of a Set of Vectors," Proc. 15th Annual IEEE Symp. on Switching and Automata Theory, pp. 117-121, Oct. 1974.

[15] Kung, H. T., F. Luccio and F. P. Preparata, "On Finding the Maxima of a Set of Vectors," J. ACM, Vol. 22, pp. 469-476, Oct. 1975.

[16] Luccio, F. and F. P. Preparata, "On Finding the Maxima of a Set of Vectors," Istituto di Scienze de1l'Informazione, Universita di Pisa, Pisa, Italy, Dec. 1973.

[17] Luenberger, D. G., Introduction to Linear and Nonlinear Programming, Reading, Mass.: Addison-Wesley, 1973.

[18] MacCrimmon, K. R., "An Overview of Multiple Objective Decision Making," in J. L. Cochrane and M. Zeleny (eds.), Multiple Criteria Decision Making, Columbia, S. C.: USC Press, pp. 18-44, 1973.

[19] Meisel, W. S., "Trade off Decisions in Multiple Criteria Decision Making," in J. L. Cochrane and M. Zeleny (eds.), Multiple Criteria Decision Making, Columbia, S. C.: USC Press, pp. 461-476, 1973.

[20] Payne, H. J., E. Polak, D. C. Collins and W. S. Meisel, "An Algorithm for Bicriteria Optimization Based on the Sensitivity Function," IEEE Trans. Automat. Contr., Vol. AC-20, pp. 546-548, Aug. 1975.

[21] Polak, E., "On the Approximation of Solutions to Multiple Criteria Decision Making Problems," Presented at XXII Inter­national Meeting of The Institute of Management Sciences, Kyoto, Japan, July 24-26, 1975.

[22] Russ, F., "Consumer Evaluation of Alternative Product Models," Ph.D. Dissertation, Carnegie-Mellon University, 1971.

[23] Sa1ukvadze, M. E., "Optimization of Vector Functionals. 1. The Programming of Optimal Trajectories," Automation and

94 E. POLAK AND A.N. PAYNE

Remote Control, Vol. 32, pp. 1169-1178, Aug. 1971.

[24] Smale, S., "Global Analysis and Economics 1. Pareto Optimum and a Generalization of Morse Theory," in M. Peixoto (ed.), Dynamical Systems, New York: Academic Press, pp. 531-544, 1973.

[25] Wan, Y. H., "Morse Theory for Two Functions," Ph.D. Disser­tation, Univ. of California, Berkeley, 1973.

[26] Wolfe, P., "Convergence Theory in Nonlinear Programming," in J. Abadie (ed.), Integer and Nonlinear Programming, New York: American Elsevier, pp. 1-36, 1970.

[27] Yao, F. F., "On Finding the Maximal Elements in a Set of • Plane Vectors," Comput. Sci. Dep. Rep., University of Illinois

at Urbana-Champaign, Urbana, Ill., July 1974.

[28] Yu, P. L., "A Class of Solutions for Group Decision Problems," Management Science, Vol. 19, pp. 936-946, April 1973.

[29] Yu, P. L. and G. Leitmann, "Compromise Solutions, Domination Structures and Salukvadze's Solution," JOTA, Vol. 13, pp. 362-378, Mar. 1974.

[30] Zeleny, M., "Compromise Programming," in J. L. Cochrane and M. Zeleny (eds.), Multiple Criteria Decision Making, Columbia, S. C.: USC Press, pp. 262-301, 1973.

[31] Zeleny, M., Linear Multiobjective Programming, Berlin: Springer-Verlag, 1974.

DYNAMIC GAMES WITH COALITIONS AND DIPLO}~CIES

A. BLAQUIERE

Laboratoire d'AutomatiQue TheoriQue Tour 14, Universite de Paris 1 2 Place Jussieu, 15005 Paris

INTRODUCTION AND NOTATION

The alm of this paper is to introduce, or to make more precise,

a few concepts. Part of the theory has been already developed, and

results have been obtained in (1)-(3). We shall leave it to the

reader to put the new concepts in its frame.

The definition of a diplomacy has been introduced In (4). It

lS glven here In a more elaborated form. Roughly speaking this

concept takes care of the fact that, in a dynamic game with coali­

tions, any subset of the set of players can try to improve (in a

sense to be defined) its pay-off by switching from one set of coa­

litions to another as time evolves. In the simplest case, any

player can try to improve his pay-off by switching from one coali­

tion to another as time evolves. According to that, diplomacies

can be defined for dynamic games only.

The notation and the nomenclature will be the ones of (1)-(3).

Here is a short list of the main ones.

Set of players : J = {J 1 , J 2 ,···, I N}

Sets of strategies: Ss ' S = 1, 2, ... , N

95

96 A. BLAQUIERE

strategy N-tuple s = (s1' s2, ... sN)' s E S

S = 8 1 x 82 x ... x SN

ne , e = 1,2, ... N, linear spaces

Preference relation of J e , e = 1,2, ... N, (reflexive, not ne­

cessarily transitive): (;;a.)e' (;;a.)e C n~

Comparison relation of J e , e = 1,2, ... N, (reflexive and sym­

metric): Ce , Ce C S2

Set of states of the game: G

state of the game: x, x E G

In order to simplifY the statement, the following notation

will be gi ven ~n a simplified form; that is, we shall suppose that

the initial state of the game is prescribed and that every strategy

N-tuple in S generates one and only one terminating path from that

state.

Then the performance index of J e' e = 1,2, ... N, is

S -+

sl-+

Ve(s) is the pay-off of J e , computed along the terminating path

generated by s from the given initial state.

At last the definition of an optimal strategy N-tuple rewrites

Definition 1. s* is an optimal- strategy N-tupZe iff

Illustrati ve examples are given in (3).

DYNAMIC GAMES WITH COALITIONS AND DIPLOMACIES 97

COALITIONS AND COALITlVE PARTITIONS

Let there be associated with any non-empty subset R of J

R = { J l' J_.~, ..• J }, a u.::: ar a1 < a2 < ••• < ar

( i ) the r-tuple

VR = (Val' Va2 ,··· Var )

where Val' Va2 ,··· Var are the performance indices of players J a1 ,

J 2, ... J , respectively; and a ar

( ~~).p . () () 2 ........ a pre.lerence relat~on ~ R' ~ R C QR' where

QR = Q 1 x Q 2 x ••• x Q . a a ar

Furthermore, let there be associated with R and any strategy

N-tuple s, the r-tuple

7T(s,R)=(Sl,s2""s )ES1xS x ... xS. a a ar a a2 ar

For a non-empty subset Q of J such that card Q < N, we shall 2

define a relation CQ, CQ C S , by

sCQs' ~ n(s, J-Q) = n(s', J-Q)

and for Q = J we shall let

Note that CQ is reflexive, symmetric and transitive. Then s

and s' in S are CQ-equivalent if and only if sCQs' and S is parti­

tionable into CQ-equivalence classes such that two strategy N-tuples

are in the same class if and only if they are CQ-equivalent. This

parti tion is the CQ-equivaZence partition of S, denoted by S/CQ.

Definition 2. A non-empty subset Q of J is a coalition if and only

if every merriber of Q has the comparison relation CQ•

If Q is a coalition, VQ and (~)Q are the performance index and

the preference relation of the coalition, respectively.

98 A. BLAQUIERE

Defini tion J. A partition P of J is a coaUtive pda'tition if and

only if every member of P is a coalition.

Let Z be a given non-empty set of coali ti ve partitions of J.

Now, let there be associated with any coalitive partition P of

J in Z, and any non-empty subset R of J, the set rr(p,R) of all non­

empty intersections R n K for all K E P.

For a non-empty subset Q of J such that card Q < N, we shall

define a relation CQ, CQ c Z2, by

pc~' <=> rr(p,J-Q) = rr(p',J-Q)

and for Q = J we shall let CQ = Z2. (see Figure 1)

cQ is reflexive, symmetric and transi ti ve. Then P and P' in

Z are CQ-equivalent if and only if PC~" and Z is partitionable

into CQ-equivalence classes such that two coalitive partitions in

Z are in the same class if and only if they are CQ-equi valent. This

partition is the CQ-equivalence partition of Z, denoted by Z/CQ.

Clearly, there is a similarity between the definitions of CQ and CQ. A few words about it, as an introduction to the following

paragraph, may be in order. First let us consider the case where

the pay-off of a player J 6 , 6 = 1,2, ..• N, 1S a real number and does

not depend on the coalitive partition in which the N players are

involved, if any. If we look for a Nash equilibrium, we are faced

with the following question:

Starting from some st~tegy N-tuple (and a set of N pay-off)

can a player J 6' 6 E {1,2, ... ,N}, increase his pay-off by changing

his strategy s6' while the other players keep their strategies un­

changed ?

* Then a strategy N-tuple s 1S a Nash equilibrium if and only

if

DYNAMIC GAMES WITH COALITIONS AND DIPLOMACIES

• • • 0

0000 .. Q

P

• • • 0 0 •

0 •

• • • • •

I1(p,J-Q) I1(P' ,J-Q)

••••• '----v-------'

Q'

Fig. 1

P'

• • 0 0

0

0

------- p ---------------------------------- p ----------4 5

Fig. 2

99

100 A. BLAQUIERE

This is a special case of Definition 1, when Ca = C {J }'

a = 1,2,ooo,N, and when the pay-off are real numbers 0 a

Now, let P be any coalitive partition of the set of players J, and

suppose that the pay-off of player J a' a = 1, 2,00 oN, for strategy

N-tuple s, is Va(s) = Wa(s,P)o Then ask the following informal

question:

Starting from the coaZitive partition P and from st~te9Y N­

tupZe s .. can a pZayer J a .. a E {1,2,ooo,N}, increase his pay-off by

switching from one aoaZition to another .. whiZe the other pZayers do

not change the way they coaZide in their set.. s being kept un­

changed ?

* More precisely, for given strategy N-tuple s, P in Z is an

equilibrium (or optimal) t if and only if

* Wa(s,P ) ~ Wa(s,P)

Vp, P E Z, PC{Ja}p* } a = 1,2,00 oN

At last, by putting together the two definitions above, one

obtains the following definition:

(p*,s*) ()t in Z x S is an equilibrium or optimal if and only if

* * Wa(s ,P )

Vs, SES,

~ Wa(s,P)

* sC{J}s ; a

'riP, P E Z, PC{Ja}p*

and

t From an individual point of view

a = 1,2,00 oN

DYNAMIC GAMES WITH COALITIONS AND DIPLOMACIES 101

These arguments can be easily extended to the case where,

starting from some strategy N-tuple s and some coaliti ve partition

P in Z, not only one player but a subset R of J tries to obtain a

better (in a collective sense) pay-off by deviating.

Roughly speaking, the main idea in what follows 1S not to

change everything at once, but to see what happens when "a subset

of J deviates, while the other players are frozen. This point of

Vlew lays the stress upon what that subset of players is able to

choose and upon what it is unable to choose. For instance, in a

3-player game where J = {J" J 2 , J 3}, if J 2 and J 3 do not cooperate,

player J can choose to cooperate with J 2 , or with J 3 , or to play

alone; and if J 2 and J 3 cooperate, J, can choose to cooperate with

both J 2 and J 3 , or to play alone. He has no influence on the coo-t peration or non-cooperation of J 2 and J 3 .

Then, another question arises: if J, chooses to cooperate with

say J 2 , will J 2 accept that cooperation? this question is approached

in an illustrative example, at the end.

OPTIMALITY WITH RESPECT TO A SET OF PLAYERS

Now, for any given coalitive partition P of J in Z, let the

performance indices of the players be WS(o ,p), S = ',2, ... N, where

{ S x Z -+ \IS

W S ( s ,p) l--+ W s ( s ,p)

Again let us associate with any non-empty subset R of J,

R = {J " J 2, ... J } , a' < a2 < ••• < ar, a a ar

the r-t uple WR = (W l' W 2"" W ) . a a ar

.. , 4 (~~)' .. 1 • h Def1nit1on . P,s 'l-n Z x S 'l-S opt'l-mal-- unt respect to a non-

empty subset R of J if and only if

t Here, we do not treat negociation

102

v s, * s E S, sCRs

"jp, PEZ, PC~*

and

Note that sand P play similar roles.

A. BLAQUIERE

. (p*,s*) . . In some problems we may w1sh that be opt1mal w1th respect

to every non-empty subset of J in a given class, which leads to

Definition 5. (p*,s*) in Z x S is optimal on K, K * ¢, K ~ P(J),

where P(J) is the collection of all non-empty subsets of J~ if and

only if

Vs, sES, and

'tJ P , P E Z, PC~*

FUZZY OPTIMALITY WITH RESPECT TO A SET OF PLAYERS

In what follows, we shall consider a collection Z of fuzzy

sets 1n Z, the members of which will be denoted by p(~). For given

membership function ~,

~ { Z -+ M

where M is the membership space, p( ~) 1S the set of ordered pairs

P E Z

~(p) is the grade of membership ofP inP(~).

Let M be a given set of membership functions; that is, a set of

mappings of Z into M; and let

~ E M}

DYNAMIC GAMES WITH COALITIONS AND DIPLOMACIES

For any given ~ in M, let the performance indices of the players

be U 13 ( • , ~), 13 1 ,2, . .. N, where

Again let us associate with any non-empty subset R of J,

R = {J a l' J a2' ... J ar}' a 1 < a2 < ••• < ar, the r-tuple

UR = (U l' U 2 ' ••• , U ). a a ar

103

Next we shall consider two cases, namely M =[0,1] in Case (a),

and M = ~ 0,1 } in Case (b). In both cases we shall suppose that M

1S such that

't;j~ E M PEZ

. • -CR -CR C M2, and, for any subset R of J, we shall deflne a relatlon ,

by

~CR~, <=> I ~(p) = I 11' (p) PEz PEz

Clearly we have In both cases (a) and (b)

and

v(z) = I ~(p) EM PEz

I v( z) = 1 z E Z/CR

\! ~ E M

Then v(z) can be looked upon as the grade of membership of z In the

fuzzy set

-;-(v) = {(z,v(z))} R z E Z/C

. . . CR . . From the deflnltlon of It follows that the functlon v

depends on the players In the set J-R only, whereas, for each given

104 A. BLAQUIERE

z, the grades of membership ll(P), P E z, depend on the players in

the set R only. Indeed, they must satisfY

I ll(P)/V(Z) = 1 PEZ

For instance, in the 3-player game of figure 2, let R = { J 1 } •

Then

and

P E z1 = J 2 and J 3 do not cooperate

P E Z <=> 2

J2 and J 3 cooperate.

Clearly, V(Z1) (Salf for instance -8) and V(Z2) (say for instanoe -2)

characterize the "degree of cooperation" between J 2 and J 3 only.

Now, provided that J 2 and J 3 do not cooperate, J 1 can choose

to cooperate with J2 , or with J 3 , or to play alone. Likewise, pro­

vided that J 2 and J 3 cooperate, J 1 can choose to cooperate with

them or to play alone. For instance, for

Z = z1 = {P 1 ,P 2 ,P 3} , let ll(P 1 ) = - 1 , 1l(P2 ) = -3,

ll( P 3) = -4, and for Z = z2 = {P4 'P5} , let

1l(P4 ) = - 1 , ll(P 5) = - 1 .

The following defini tions will hold for both cases ( a) and (b).

Definition 6. (/,s~) in Mxs is optimal with respect to a non­

empty suhset R of J if and only if

~ ~ uR(s ,ll )(~)R UR(S,ll)

v s, s E S, * sCRs ; and

II E M, -CR * II II

- * ~ We shall say that (P(ll ),s ) is fuzzy optimal with respect to R if

and only if (ll*'S*) is optimal with respect to R.

DYNAMIC GAMES WITH COALITIONS AND DIPLOMACIES

Definition 7. (/\s*) in Mxs is optimal- on K, K =f. rp, K b P(J),

if and only if

* * UR(s,].1)

} UR(s ,].1 )(;;;')R

'ds, sCRs * 'dREK s E S, ; and

\}].1, M, -R * ].1 E ].1C ].1

Likewise, we shall say that (P(].1*),s*) is fuzzy optimal on K

if and only if (].1*,s*) is optimal on K.

In Case (b), given].1 in M, there is one and only one P In Z

such that ].1(p) = 1. One can easily verify that Definitions 6 and

4 on one hand and Definitions 7 and 5 on the other hand are equi­

valent, with ].1*(P*) = 1 (and indeed ].1*(P) = 0 for all P =f. p*).

DIPLOMACIES

Definition 8. A diplomacy is a mapping ~

, {:: :(X) Let Z be a given set of diplomacies.

For any given diplomacy ~ in this set, let the performance

indi ces of the players be W S ( • ,~), S = 1,2, ... N, where

{ S x Z ~ llS

Ws (s,~) 1-+ WS(s,~)

105

For any non-empty subset R of J, WR will have a similar mean-

ing as before. Moreover, we shall define a relation CR CR C Z2 , - , by

~CR6.' <==> ~(x)CR~, (x) V x E G

Definition 9. (~*,s*) in ZxS is optimal with respect to a non­

empty subset R of J if and only if

106 A. BLAQUIERE

'its, s E S,

Definition 10. * '* (b. ,s ) in Z xS is optimal on K, K =1= t/>, K C P(J),

if and only if

* * WR(s ,b. )(~)R WR(s,b.)

} 't;j s, s E S, sCRs '* 't;jR E K ; and

't;j b. , b. E Z, b.CRb.'*

FUZZY DIPLOMACIES

Definition I I. A fuzzy diplomacy is a mapping -;;

Let there be given a set M of mappings ~

and consider the set of fuzzy diplomacies P(~)

p(]l) { G -+ Z

x 1-+ p(~( x) )

for all ~ in M, namely

Z = {P(~): ~ EM} For any given ~ in M, let the performance indices of the players be

Ur/e,):;"), i3 = 1,2, ... N, where

DYNAMIC GAMES WITH COALITIONS AND DIPLOMACIES 107

{ S x M ~ QS

Us (s;iT) J-..+ US(s,li)

For any non-empty subset R of J, UR will have a similar meaning as -R -R -2

before. Moreover, we shall define a relation C , C ~ M , by

't/X E G

Defini tion 12. CjI* , s *) in M x S is optimal UJi th respeat to a non­

empty sUbset R of J if and only if

UR(s*,Jt)(~)R uR(s,l1)

'ds, s E S,

- *) *) Then we shall say that (p(~ ,s is optimal with respect to R if

and only if (~*,s*) is optimal with respect to R.

Definition 13. (li*,s*) in Mxs is optimal on K, K =I=!p, K ~ P(J),

if and only if

UR(S*,ii*)(~)R UR(s,li)

} 'lis, s E S, seRs * VR E K ; and

'tiii , li E M, -R _*

~ C ~

Likewise, we shall say that (P(ii*),s*) is optimal on K, if and only

~f (-* *)" t" 1 K ~ ~,s ~s op ~ma on .

Now, let us consider the special case where M is such that

~(P)E{O,1}

I ~(p) = 1 PEZ

'dP E Z ; and } \/. E M

Moreover, let us suppose that Z and M are related by the following

condi tion, namely 'rJ t:, E Z, 311 EM, and 'tj)i EM, 3 t:, E Z such that

l1(X) (t:,(x)) = 1 'tix E G

108 A. BLAQUIERE

Then one can easily verify that Definitions 12 and 9 on one hand,

and Definitions 13 and 10 on the other hand, are equivalent, with

Vx E G

Furthermore, if Z is a set of constant diplomacies; such that

'riP E Z, 311 E Z, and 'illl E z, 3 P E Z

such that

ll(x) = P 'r/XEG

then Definitions 9 and 4 on one hand, and Definitions 10 and 5 on

the other hand, are equivalent, with p* = ll*(x).

Ex.ample. 1. Consider a two-player differential game in which the

set G of states of the game is En, and let the sets of strategies

of players J 1 and J 2 be prescribed sets of functions

and

respectively .

Let J = {J 1 ,J 2} and Z = {P, ,P 2} where P 1 and P2 are the

coalitive partitions whose members are

{J 1} and {J2 } for P,; and

{J 1 ' J 2 } for P2.

Let M be the set of all functions )l: Z + M

(M = [0,1] or M ='{O,'}) such that )l(P 1) + )l(P2 ) = 1, and let M be

the set of all functions]1: En + M. As ]1(x) , for x E En, is a

function of P defined on Z, we shall denote by p ,(x) and P2(x) the

values of ~(x) at P = P, and at P = P2 , respectively. Indeed, P1

and P2 are mappings from En into M and, furthermore,

p,(x) + P2(x) =1, 'rix E En, so that we shall let P, = P and

P2 = , - P. where p is a mapping from En into M. We shall denote

by Q the set of all such mappings.

DYNAMIC GAMES WITH COALITIONS AND DIPLOMACIES

Let the performance indices of J 1 and J 2 be U1(o,p) and U2(o,p), respectively, where

{ 8 x Q -+ R

U1 (p, ,P2'P) ~ U,(p, ,P2' p) and

U2 {

8 x Q -+ R

(P"P2'P) ~ U2(P1 ,P2'P) (R set of real numbers)

Now, let us rewrite Definition '3 for that example.

First let K = ({J,}, {J2 })· Then * * (P"P2' *) . . P 1.S opt1.mal on K if

and only if

* * p*) ;;;'U,(p" * p ) , U,(pl' P2' P2' and

* * * * p) U2(P1' P2' P );;;'U2(p" P2'

\tp" P, E 8, and \t P2' P2 E 82 , and V p, P E Q.

This set of conditions may be separated into 3 conditions, namely

* * * * U,(p" P2' p' );;;. U,(p" P2' pI)

V(p" p'), (p" p') E 8, x Q

2) * * * * p" ) U2 (p 1 ' P2' p" ) ;;;. U2(p" P2 ,

\t(p2 , p ") , (P2 , p It) E 82 x Q

3) * p ,( x) * = p" (x) 'i/x E En

109

so that, the problem of finding such an equilibrium is separated

into two maximization problems together with a verification condi­

tion. Indeed, there may be no solution. However, there may exist

a subset Q' of Q such that the new problem obtained by replacing Q

by Q' in the above definition has a solution.

Let us make that point more precise by considering a diffe-

110 A. BLAQUIERE

rential game in which G = E, the dynamic equation is

x = the initial state is x(o) = 0, and the target is x = 1. There is

a single terminating path, from the initial state, namely x = t, tE[0,1) .

Let Q' be the set of all piece-wise continuous functions from

[0,1) into [0,1) , and let U1 and U2 be defined by

J1 + J01 U1(p) - OP(t)f1(t)dt (1-p(t))gCt)dt

P E QI

where f l' f2 and g are given continuous functions froI!l [ 0,1) into

R, such that f 1(t) > ° and f 2(t) > ° and g(t) > 0, 'fit E [0,1).

We have

f 1 (t) > g(t) => p'*(t) = 1

f 1 (t) < g( t) => p'*(t) = °

f 2(t) > g(t) => p 11*( t) = 1

f 2(t) < g(t) => p 11* (t) = °

Case 1. Now suppose that f 1, f2 and g are such that there exists

t , c

t E (0,1), such that c

f 1 (t) > g(t) and f 2(t) > g( t) , 'fit E (O,t c ); and

f 1 (t) < g(t) and f 2( t) < g( t) , 'fit E (t ,1). c

Then

p'*(t) * = p" (t) * = p (t) = 1 'fit E (o,t c ); and

p'*(t) * * 'fit E (t ,1). = p" (t) = p (t) = ° c

that is, the optimal behavior for J 1 and J 2 is not to coalide on

(O,t ) and to coalide on (t ,1). c c

DYNAMIC GAMES WITH COALITIONS AND DIPLOMACIES 111

Case 2. Now suppose that there exist tc and td in (0,1) such that

tc < td and

f'1(t) > g( t) and f 2(t) > g( t) , \It E (O,t ); c

and

f 1(t) < g( t) and f 2(t) > g( t), \It E (tc,td ); and

1'1 (t) < g( t) and f 2 ( t) < g( t), \It E (td ,1).

Then there does not exist an equilibrium in the sense of the

above def'inition; that is, on the interval (tc,td ), J 1 has advanta­

ge by coaliding with J2 whereas J2 has advantage by not coaliding

with J l'

In that situation, let p be a prescribed piece-wise continu-o

ous function from (tc ' t d) into [0,1 ], and let Q" be the set of all

piece-wise continuous functions from [ 0,1] into [0,1] which coin­

ci de wi th p on (t , t d) . o c

Then, if we replace Q by Q" in the above definition, we obtain

an equilibrium in which

* p (t)

* = 1 \It E (O,t); and

c p (t) =

* po(t) tit E (tc,t d ); and

p (t) = ° 'Vt E (td ,1).

In other words, in a situation where J 1 has advantage by coa­

liding with J2 whereas J 2 has advantage by not coaliding with J 1

(or conversely), we are led to introducing a rule such as :

J 1 and J 2' will not coalide, or they will be forced to coalide, or,

if' we deal with fuzzy coalitions, they will partly coalide.

Now let us consider the case where K has one element only, na­

mely {J1,J2 }. Then, in Definition 13, we have R = {J 1,J2 }, and

~ = R2. Let us define (~)R by

2 (a,b)ER, (a',b')

(a,b) (~)R (a',b')

a > a' or b > b'

or (a = a' and b = b')

112 A. BLAQUIERE

Then Definition '3 rewrites: (p~, only if

* p*) .. . P2' ~s opt~mal on K ~f and

U, (p, ,p 2' p) ;;.. U, (p ~, p ~, p *) and

U2(P"P2'P) * * p*) ;;.. U2(p" P2'

implies that

U, (p , ,p 2' P ) * * p *) = U,(p" P2' and

U2(P"P2'P) * * p*) = U2(p" P2'

that is, (* * *). P" P2' p ~s a Pareto equili bri um..

One can easily verifY that, in the example above, any piece­

* wise continuous function p : [0,'] -+ [ 0, '] such that

* p (t) = for af,(t) + Sf2(t) > g(t); and

* p (t) = 0 fo r af, ( t) + Sf 2 ( t) < g( t )

where a and S are non-negative constants such that a + S = " is a

solution.

In the pages before we used the term cooperate, or non­

cooperate, with the meaning beZong to, or does not beZong to, a

same coalition. Now we see that this meaning should be extended

since, in the last part of Example', we meet with a situation in

which J, and J 2 are supposed to agree on a Pareto equilibrium.,

though they do not coalide for all t in [ 0,' ].

Example 2. Now consider a three-player differential game in which

the set G of states of the game is ~, and J = {J"J2 ,J3} and

Z = {P"P2 ,P3 ,P4 ,P5} , where P"P2 , ... P5 are the coalitive parti­

tions whose members are

{J, ,J2 } and {J 3} for P 1

{J,}, {J2} and {J 3} for P2

{J, ,J 3} and {J2 } for P3

{J,} and {J 2 ,J3} for P4

{J"J2 ,J3 } for P5

DYNAMIC GAMES WITH COALITIONS AND DIPLOMACIES 113

Let us suppose that the performance indices of J 1, J 2 and J 3 ,

namely u1 ' U2 and U3 respectively, are real valued filllctions which

do not contain the strategies in their arguments. As before, for

x E En, let us denote by P1(x), P2(x), ••• P5(X) the values of ll(x)

at P = P 1, P = P2 ,· .. P = P5 , respectively. P 1 , P2 ,,,,P5 are map­

pings from En into M (M = [0,1] or M = {O,n) and

p,(x) + P2 (X) + ... + P5(X) = 1, 'dxEEn.

We shall leave it to the reader to verifY that, for . * lie * lit K = ({J,}' {J2 }, {J3}) Definition 13 rewntes: P = (P 1 , P2 , ... P5 )

is optimal on K if and only if

lie U1(p) ~U,(p)

'dp, P = (p"P 2 ,·· P5 ), such that

and

and

* U2 (p ) ~ U2 (p)

'tj p, such that

and lie

U3(p )~U3(P)

'tj P, such that

lie '* '* * * P, + P5 = P, + P5 and P2 + P3 + P4 = P2 + P3 + P4

3 For K = ({J"J2 ,J3 }), R = {J"J2 ,J3}, QR = R , and (~)R defined by

( a, b , c) E R3, (a I , b' , c ,) E R3} <= fora> a' or b > b' or c > c'

l (a=a' and b=b' and c=c') (a,b,c) (~)R (a',b',c')

De fi ni tion 13 rewrite s :

* * lie '* P = (P l , P2 ,···P 5) lS optimal on K if and only if

114 A. BLAQUIERE

U 1 (p) * r

u 1 (p ) * ;;;. U 1 (p ) and = U 1 (p ) and

U2( p ) * U2(p) * ;;;. U2(p ) and

1 = U2( p ) and

~

* U3( p) * U3(p) ;;;. u3(p ) = U3( p )

p = (P1,P2,···P5)

is, * is a Pareto e quili bri um. that P

CONCLUDING COMMENTS

A few concluding comments seem in order.

First, we have supposed at the begining that every strategy

N-tuple in S generates one and only one terminating path from the

prescribed initial state. As a direct consequence, the way the

players coalide has no influence on the path, it has an influence

on the pay-off only. There is no difficulty extending the theory

by assuming that the dynamic equations depend on both s, s E S, and

11, 11 E Z (or Y;, Y; E Z) .

In (1 )-( 3), S is the cartesian product S1 x S2 x ••• x SN; that

is, the members of S are strategy N-tuples. That assumption can

be relaxed without changing the results obtained. Then S is a giv­

en non-empty set, whose members s are better termed cottective stna­

tegies. A collective strategy need not be a strategy N-tuple.

In the present paper, Z x S (or Z x S) is a set of collective

strategies (l1,s) (or (~,s». Accordingly the results obtained in

(1)-(3) can be readily rewritten with just a small change in the

notation.

REFERENCES

(1) A. Blaquiere, Quantitative Games: Problem Statement and

Examples, New Geometric Aspects, in The Theory and Application

of Differential Games, (Edited by J.D. Grote) D. Reidel Pu­

blishing Company, 1974.

DYNAMIC GAMES WITH COALITIONS AND DIPLOMACIES

(2) A. Blaquiere, Vector-Valued Optimization In Multi-Player

Quantitative Games, in Multicriteria Decision Making (Edited

by G. Leitmann and A. Marzollo) Springer, Wien New York,1976

(3) A. Blaquiere, Une generalisation du concept d'optimalite et

115

de certaines notions geometriques qui s'y rattachent, Colloque

International de Theorie des Jeux, Institut des Hautes Etudes

de Belgique, Bruxelles, 29 et 30 Mai 1975 (to appear).

(4) A. Blaquiere et K.E. Wiese, Jeux qualitatifs multietages a N personnes, coalitions, C.R. Acad. Sc., Paris, t.270,

p.1280-1282, Serie A (11 Mai 1970)

(5) G.J. Olsder, An Advertising Model as a Differential Game with

Changing Coalitions, Memorandum Nr 66, Technische Hogeschool

Twente, February 1975.

(6) G.J. Olsder, Some Thoughts about Simple Advertising Models

and the Structure of Coalitions, Conference on Directions in

Decentralized Control, Many-Person Optimization and Large­

Scale Systems, September 1-3, 1975.

(7) L.A. Zadeh, Toward a Theory of Fuzzy Systems in Aspects of

Network and System Theory (Edited by R.E. Kalman and

N. DeClaris) Holt, Rinehart and Winston, Inc., 1971.

THREE METHODS FOR DETERMINING PARETO-OPTIMAL

SOLUTIONS OF MULTIPLE-OBJECTIVE PROBLEMS

Jiguan G. Lin

Department of Electrical Engineering & Compo Sci.

Columbia University, New York, N. Y. 10027

1. INTRODUCTION

Typical problems in system design, decision making, decentralized control, etc., and most multi-person games bear the following general formulation of multiple -objecti ve (MO) optimization problems:

maximize zl=Jl(xl>"" x n ), ... , and ZN =IN (xl, ... , xn)

subject to (xl>"" xn)E X. (1)

Where xl>"" xn are variables (or functions) under control; X is a nonempty constraint set; Jl>"" I N are real-valued objective functions; Nand n are finite integers, 2~N<C'Xl, l~n<oo.

zi represents, depending on the specific problem under study, the ith system performance index, the index of ith objective or criterion, the ith controller's performance index, or the ith player's payoff. An n-tuple x':'=(xi', ... , x'i) is a Pareto:::: optimal solution of the MO problem iff x':'E X and there exists no n -tuple x in X such that (i) J i (x)~ J i (x':') for all i and (ii) J.(x) >J.(x':') for at least one j.

J J

In general, there are many Pareto-optimal solutions, but any of them is by definition a good alternative to any other so far as the given multiple objectives are concerned. Therefore, it is useful to obtain the set P of Pareto-optimal solutions. For instance, it was proposed in [1] that the design of a

117

118 J.G. LIN

system be a two-stage process: determine the set of Pareto= optimal systems fir st, then choose a single system in the set. In analyzing a multi-person cooperative game or differential game, a first step is to determine the "core" .. which is, rough­ly speaking, the set of all Pareto-optimal "imputations" [2]-[4].

Since it was introduced in 1896, the general interests in Par eto I s optimum have mainly been in the us e of the concept in various contexts in different forms and the analytical charac­terization of Pareto-optimal solutions to various problems in different ways (e.g., see [5]-[11], or Refs. I-II, 16-17 cited in [11 ]). Computational methods proposed [1], [3], [7] for deter­mining the set P have virtually been confined to linear scalari­zation, namely, the method of linear combinations (LC). The usefulness of the method is quite limited, since in most cases the Pareto-optimal solutions thereby obtained form a tiny and extreme subset of P [9]-[10] and offer no "middle ground" for compromise or tradeoff.

In an investigation [8] into the method of LC, it was discovered that the concepts of directional shadow and directional extremum play useful roles in the determination of P. This discovery made possible (i) the relaxation of the convexity assumption on which the method of LC was usually based, and (ii) the invention of two general methods, the method of proper equality constraints (PEC) and the method of proper inequality constraints (PIC), which rely on no convexity assumption at all [8]-[15]. Recently, other methods also using inequality con­straints but in somewhat different ways have been proposed in [16] and [17], independently of this author.

The purpose of this paper is to discuss the method of LC, and to introduce both methods of PEC and of PIC.

II. MAXIMAL INDEX VECTORS

Visualizing multiple -objecti ve optimization problems on the space of indexes often enables one to obtain deeper insights into many of the difficulties encountered. Let Z denote the image of the constraint set X in the space of indexes, namely, Z=[(Jdx), ... , IN(x))lxEX}. An index vector zO=(J1(x°)' ... ,IN(XO)) is called [11] a maximal vector in Z if zOEZ and there exist no z' in Z such that z'5z. Where, z'5'z denotes the ordering

PARETO·OPTIMAL SOLUTIONS OF MULTIPLE·OBJECTIVE PROBLEMS 119

relation~<: zi ~zi for all i and zj > Zj for at least one j. It

follows immediately that x is a Pareto-optimal solution iff xE X and Z = (J1(x), ... , IN(x)) is a maximal index vector.

Observe that the set M of maximal vectors in Z is exactly the set of Pareto-optimal solutions of the following elementary multiple -objecti ve optimization problem:

maximize zl,"" and ZN

subject to (Zh"" zN )E Z. (2)

Thus, methods for determining the set P of problem (1) can be derived from those for determining the set M of problem (2). In the sequel, we shall therefore consider Z to be a general nonempty set in RN.

III. METHOD OF LINEAR COMBINATIONS

Let C(w)=[zEzlwTz~WTzl for all Zl in Z}, and

K(w)=[xEXlwTJ(x)~WT J(xl ) for all Xl in X},

h ) N T Q were W=(Wh""WN is a vector in R, wZ-WIzI+"'+WNzN, b. and J(x) = (Jl(x), ... , IN(x)). And consider separately the case

of strictly positive weights and that of positive weights.

A. Using Strictly Positive Weights

One can show, say, using the arguments in the proof of Theorem 4.l(a) in [11], that

CS~U{C(w)lw>O}CM, and KS~U[K(w)lw>O}cP. These general facts mean that one can obtain some maximal vectors by maximizing wTz =wIzI+'" +wN zN-and some Pareto= optimal solutions by maximizing wT J(x) = WIJl (x)+ ... +WNJN (x)­with weights Wh.'" WN ranging over strictly positive numbers. Figs. 2 -3, 5-8 in Appendix A illustrate, however, that the entire set M and P cannot always be determined this way.

Suppose Z is a convex set determined by a finite number of points in R N, like a polyhedron or a polyhedral cone.

':' ZI is definitely greater than Z if ZI:> z. Z is positive if z ~ 0, definitely positive if z>O. and strictly positive if Z > O.

120

Then, it follows from Theorems 6.1 and 6. 2 in [11] or Theorem 4.3 in [5] that M = CS. Therefore, we have

J.G. LIN

Proposition 1. Suppose X is a convex set determined by a finite number of points in a linear space and each J i is an affine or linear function on the linear space. Then P= KS'

But, suppose Z is a general closed convex set in RN. Then, according to Theorem 1 in [6], we have CSlcMC CSl '

where CSl~U[c(w)lw>o and Wl+ ... +wN=l} and CSI is the

closure of CSI ' Similarly, suppose X is a general closed convex set in Rn and each J i is a continuous concave function on Rn. Then, according to Theorem (41) in [7], we have KSI C P CKSI provided that every Pareto-optimal solutions is "regular", where KSI ~U[K(w) Iw>o and Wl+'" +wN =l} and KSI denotes the closure of KSI ' Using strictly positive weights, one may obtain the set M if Z is convex, and the set P if X is convex and Ji's are concave.

But, is convexity of Z really required so that MC CS? Is convexity of X really required so that pc KS?

B. Using Positive Weights

Unlike CS, the relationship between CD ~u[C(w)1 w> O} and M varies. MCCD in Figs.2-5; CDcM in Fig. 6; Mej:CD and CDej: M in Fig. 7; and both CD and Cs are empty while M is nonempty in Fig. 8. The relationship between KD ~U[K(w) \w>O} likewise varies.

Zadeh [1] asserted that if X is a convex set in Rn and if each Ji is a linear function on R n (that is, if the image [J(x) IxEX} is a convex set in R N), then pCKD . This assertion obviously implies that MCCD if Z is convex.

Again, is convexity of Z and X really required so that MC CD and pc KD respectively?

C. Directional Shadows and Directional Convexity of Z

Figs. 2 - 5 clearly illustrate that convexity of Z is not really necessary for the relation MC CD to hold. But a certain weak kind of convexity is sufficient to ensure the

PARETO·OPTIMAL SOLUTIONS OF MUL TIPLE·OBJECTIVE PROBLEMS 121

desired relation. Like in [18], call the set Zp ~{z-ep IZE Z and ~ ~ O} the p-directional shadow of Z, where p is a nonzero vector in RN and ~'s are real numbers. Observe that the (1, 0) -directional shadows of the sets in Figs. 2 and 3 are convex sets.

In terms of directional shadow, we have the following Theorem l(a) using the facts stated in Lemmas 1-3. Their proofs are given in Appendix B except that of Lemma 3, which can be found inside the proof of Theorem 4.2 in [11].

Lemma 1. Suppose p is a definitely positive vector in RN. Then z is a maximal vector in Z if and only if it is a maximal vector in Zp.

Lemma 2. Let CP(w) ~ (zE zPlwT z~ wT z' for all z· in ZP},

and C};=U(CP(w)\w>oL Suppose p is a definitely positive

vector in RN and Zp is convex. Then CD = zn Cb .

Lemma 3. Let MP be the set of maximal vector in Zp. Suppose p is a definitely positive vector in RN and Zp is convex. Then MPcCB.

Theorem l(a). Suppose Zp is convex for some definitely positive vector p in RN. Then MC CD.

Given a nonzero vector p in RN , Z is said to be p-directionally convex [18] if given any zl and zl. in Z and any positive numbers Al and Xl. with Al +Al.= 1, there is a positive number ~ such that AIZI+Al.Zl.+~pEZ. By definition, the set in Fig.2 is p-directionally convex for all p; the set in Fig. 3 is (l,O)-directionally convex.

By the contact theorem [II, Theorem 3.1] and the standard separation theorem on convex sets, it was proved in [11] that if Z is p-directionally convex for some definitely positive vector p, then for a vector ZO to be a maximal vector in Z, there must exist a definitely positive vector w such that the scalar product wT z attains its maximum on Z at zo. Therefore, we have

Theorem l(b). Suppose Z is p-directionally convex for some definitely positive vector p in RN. Then MCCD .

122 J.G. LIN

Both Theorems l(a) and l(b) are in fact equivalent by the following fact (see Appendix B for the proof).

Lemma 4. Given a nonzero vector p in R N, Z is p-directional­ly convex if and only if Zp is convex.

To summarize, one can obtain the entire set M by the method of LC using positive weights (not all zero), provided that Z is p­directionally convex for some definitely positive vector p. It is noteworthy that the magnitude of the vector p is immaterial.

D. Directional Convexity of X or of its Image

Now, let Z be specifically defined by tJ(x) IxEX}. Then Theorem I can be restated as follows.

Theorem 2. Suppose the image fJ(x)lxEX} is p-directionally convex for some definitely positive ve~tor p in RN. Th~n pCKD.

This means that under such a weak condition, the totality of Pareto-optimal solutions can be obtained by maximizing non­zero positive combinations of the multiple objective functions subject to the given constraints. Since directional convexity is much weaker than convexity, the class of MO problems for which the method of LC is useful is thereby enlarged. The following are two useful consequences, whose proofs are given in Appendix B.

Theorem 3. Suppose J(x)=Jx+b for some real Nxn matrix J and some con,stant vector b in RN , and X is q -direc­tionally -convex for some vector q in R n with Iq> O. Then the image {Ix+b \xEX} is p-directionally convex for p = Iq, and

PCKD ·

Theorem 4. Suppose the graph G~{(x, J(x)) !xEX} of the function J=(Jh ••• , I N) is q -directionally convex for some vector

. n+N . ( )- 0 q=(qh •.. , qn' qn+l' ... ', qn+N) III R such that qn+l'···' qn+N > . Then PCKD .

IV. DIRECTIONAL EXTREMA

Most nonlinear multiple-objective optimization problems still do not satisfy such a weak convexity condition. Can one devise a general method for determining the entire set P,

PARETO-OPTIMAL SOLUTIONS OF MULTIPLE-OBJECTIVE PROBLEMS

without relying on any convexity condition or on any other condition at all? The answer is yes.

Given a nonempty set Y and a nonzero vector p in RN. call z a p-directional extremum of Y if pT z=SUp[pT Y I yE Y}.

123

Let us consider the set Z in Fig. I for example and examine the relationship between directional extrema and maximal vectors. Note that the set M is given by segments ~ (without end point n) and Qq.

First. let Y=Z and vary p. One readily sees that the set of p-directional extrema of Z. with p ranging over all definitely positive vectors in RN. contains only segments be and hi. Thus not a single maximal vector is included.

Next. let p=eN=(O •••.• O.I) and vary Y instead. Assume Y is a lineal subset of Z. defined by [zEZ Iz=zo+PeN and PE Rl} for some zOEZ. One can see that the set of eN-directional extrema of Y, with Y ranging over all lineal subsets of Z, contains the segment *ghmnoq. Hence all maximal vectors are included.

Fig.l A set Z in RZ. All boundary points except the solid curve ghmnoqg are not included. solid lines show a typical lineal subset; the area shows a typical sectional subset.

those on The wide

crosshatched

124 J.G. LIN

Similarly, assume Y is a sectional subset of Z, defined by [zEZ Iz=zo+v and vERN with vi;?: 0, i:fN} for some zOE Z. One can also see that the set of eN-directional extrema of Y, with Y ranging over all sectional subsets of Z, contains the segment hmnoq. Hence all maximal vectors are also included.

Since the method of LC using positive weights is actually to determine the set of p-directional extrema in Z, with p ranging over all definitely positive vectors, it thus produces no maximal vectors in this set, while plenty of them actually exist.

On the other hand, if we determine the set of eN-direc­tional extrema of various subsets of Z, we can determine the entire set M without relying on directional convexity or any specific nature of the set Z. The following method of PEC and method of PIC are based on these observations.

V. METHOD OF PROPER EQUALITY CONSTRAINTS

Call a vector z an eN-linearlly supremal vector for Z if z is an eN-directional extremum of a nonempty lineal subset defined by {z'EZ !z'=z+PeN and PE Rl}; call it an eN-lineally maximal vector in Z if, in addition, zEZ. It follows that a maximal vector must be an eN -lineally maximal vector. Thus, the set L of eN-lineally maximal vectors in Z contains uncondi­tionally the entire set M. This means that if the set L can be determined and those eN-lineally maximal vectors which are unqualified to be maximal vectors in Z can be detected, then we can obtain the entire set M from the set L by retaining the qualified, or discarding the unqualified.

A. Determining Maximal Vectors via Lineally Maximal Vectors

Let A=[a.ERN- l l Za. is nonempty}, where Za.~ [zEZ Iz.=a.i for i:fN} . Define a function ¢ on A by 1

¢(a.)=sup[zNlzEZcJ, if Za.has an upper bound; ¢(a.)=+oo, otherwise. Trivially, the hypograph of ¢, [(a.,S)la.EA and S~¢(a.)}, contains the closure Z of Z, and the graph of ¢, {(a.,¢(a.))la.EA}, contains all eN -lineally supremal vectors for Z. Furthermore, let BQ {a.EA l¢(a.):f +"" and (a., ¢(a.))E Z}. Then the portion {(a.,¢(a.))la.EB}

of the g~aph of ¢ determines exactly the set L of all eN -lineally maximal vectors in Z. In Fig. 1, the segments -a"bCd (without

PARETO.QPTIMAL SOLUTIONS OF MULTIPLE-OBJECTIVE PROBLEMS 125

endpoint d), er (without define the graph of if:>. set L.

f), ~, and rsfU' (without rand u) The segment ghmn"Oq determines the

Using arguments similar to those in the proof of Theorem 3.1 in [12] or Theorem 1 in [14], we have the following necessary and sufficient condition. (See Appendix B for the proof. )

Theorem 5. Given o.oEB. Zo =(0.0, if:>(oP» is a maximal vector in Z if and only if

(a) if:> (a.)~(a.u) for any a. in A such that 0; ~ 0.0 and

(b) if:> (0.)< if:> (0.°) for any a. in B such that 0.50.°_

Therefore, we can obtain the set M by first determining the function if:> and the sets A and B, then discarding the unqualified vectors (a.0,if:>(a.0» according to conditions (a) and (b).

B. Parametric-Equality-Constrained

Single-Objective Optimization Problem

Now, we can develop a general method for determining the set P. To do so,denote by Xa. the set txEXl J-(x)=a.., ijN} and

1 1 define the sets A and B, and the function if:> as follows.

A ~ ta.ERN-1lxa. is nonempty]

if:>(a.) = suPtJN (x) IxEXa.} if IN(x) is bounded above on Xi if:>(a.)=+t;Q, otherwise.

B~ta.EAIif:>(a.)j+(l) and IN(~)=if:>(a.) for some ~ in X a.} •

For determining the function if:> , one may solve analytically or numerically the following associated parametric-equality­constrained single-objective (PECSO) optimization problem:

maximize ZN = I N (x)

subject to J1(X)=a.h ••• , IN_1(X)=a.N-l

and xEX.

Denote the optimal solutions of the associated PECSO problem by x(a.) and call them PECSO-optimal solutions, for short. Then ¢(a.)=JN(~(a.» for a.E B. Therefore, the totality of PECSO­optimal solutions ~(a.), with a. ranging over the set B, contains

126 J.G. LIN

the entire set P. Moreover, if o.oEB, then for a PECSO­optimal solution ~(a.0) to be a Pareto-optimal solution, it is necessary and sufficient that conditions (a) and (b) are satisfied.

C. The Method of P roper Equality Constraints

Conditions (a) and (b) clearly imply that (i) converting all but one objectives to equality constraints with arbitrary constants, say a.~, ••. , a.°N_l' as the constraint levels and (ii) maximizing the retained single objective function subject to the objective-converted equality constraints (in addition to the given) does not always yield a Pareto-optimal solution. In other words, if these constants are improper, even an optimal solution of the resultant single-objective problem cannot be a Pareto-optimal solution of the MO problem. 0.0 = (a.~, ••• ,a.°N -d is therefore called a proper vector of eguality constants iff (i) a.°EB and (ii) both conditions (a) and (b) are satisfied. The objective-converted equality constraints Jdx)=a.~, .•• , IN_dx)=a.°N_l are said to be proper iff the vector (a.~, •.. , a.~_l) of equality constants is proper.

To determine the set P is thus to determine those PECSO­optimal solutions which arise from proper objective-converted equality constraints. Let B~'< denote the set of proper vectors of equality constants. Then P=f~(a.)' a. EB~'< and x(a.) is a PECSO­optimal solution}. This is called the method of proper eguality constraints (PIC). Example.2 illustrates an application of the method to a nonconvex, nonconcave, nonlinear, triple-objective optimization problem.

VI. METHOD OF PROPER INEQUALITY CONSTRAINTS

The discussions in this section will be relatively brief, since they are similar to those in Section V. Call z in RN an eN-sectionally supremal vector for Z if z is an eN-directional extremum of a nonempty sectional subset defined by {z'EZ !ZI=Z+V and vERN with vi:2:0, i#N}; call it an eN -sectionally maximal vector for Z if. in addition, z+VEZ for some v in RN such that v:2: O. It follows from definition that the set S of e N_ sectionally maximal vectors for Z contains unconditionally the entire set M.

PARETO-OPTIMAL SOLUTIONS OF MULTIPLE-OBJECTIVE PROBLEMS

A. Det ermining Maximal Vectors via

Sectionally Maximal Vectors

Let A = to,ERN-1lzo, is nonempty}, where zo, ~ tzEZ Iz.:.::: o,l- , - 1

127

if:. N} • Define a function 1\1 on !:: by 1\I(0,)=SUPtZN IzEZo,} if zo, has

an upper bound; I\t(o,)=+c:o, otherwise. The hypograph of 1\1 contains the closure Z, and the graph of 1Ir contains all eN-sectionally suprema1 vectors for Z_ Moreover, let B ~to,EA I 1\1 (0,) f:.+eo and ~(o,)=~N for some 'Z in Zo,}. Then, the p-;rtion-t(a.,I\I(o,»lo,E~} of the graph of '¥ determine exactly the set S. In Fig. 1, the segments bIb, bcd' (without endpoint d), hlh, hrnn;;q, SIS (with­out stl and 'Sti'l (without u) define the graph of W. The segments -- -=----hih and hmnoq together determine the set S.

It follows from the definition of zo, that I\t is decreasing on A. Using this property and arguments similar to the proof of Theorem 3.1 in [13], we have the following necessary and sufficient condition useful for qualification. (See Appendix B for the proof.)

Theorem 6_ Given o,°E~. zO=(o,0, W(o,°» is a maximal vector in Z if and only if (c) 1\I(0,)<,~(o,O) for any 0, in ~ such that 0,>"0,0 .

Therefore, one can obtain the set M by first determining the function ~ and the set B, then discarding those unqualified vectors (0,0,1\1 (0,0) ) according-to condition (c).

B. Parametric-Inequality-Constrained Single-Objective

Optimization Problem

Now we can develop another general method for determin­ing the set P. Denote by xo, the set txEX I J i (x):'::: o,i' if:.N} , and define the sets ~ and la, and the function ~ -as follows.

~ ~ {o,ERN-1Ixo, is nonempty},

1\! (0,) = suptJN (x) \xEXo,} if IN(x) is bounded above on Xa ;

,(o,)=+rx>, otherwise.

~ ~to,E~ I $ (o,)f:.+o:> and IN(~)= t(o,) for some ~ in xo,} .

128 J.G. LIN

Likewise, for determining the function $, one may solve the following associated parametric-inequality-constrained single= objective (PICSO) optimization problem:

maximize ZN =IN(x)

subject to JdX)~a.h ...• IN-1(x)~a.N_1

and xEX.

Denote the optimal solutions of the associated PICSO problem by x(a.) and call them PICSO-optimal solutions. Then $ (a.)=JN(x(a.) ) for a.E~. Therefore, the set of PICSO-optimal solutions x(a.), with a. ranging over the set B, contains the entire set P. And if a.°EB, then for a PICSO-optimal solution x(a.°) to be a Pareto-opti~al solution, it is necessary and sufficient that condition (c) is satsified.

C. The Method of Proper Inequality Constraints

With a similar reason, a.0=(a.~, .•. , a.~-d is called a proper vector of inequality constraints iff (i) a.°E~ and (ii) condition (c) is satisfied. The objective-converted inequality constraints J 1 (x) ~a.~, .•. , I N -1 (x) ~ a.°N -1 are said to be proper iff the vector (a.~, ... , a.~_d of inequality constraints is proper.

To determine the set P is thus to determine those PICSO= optimal solutions which arise from proper objective-converted inequality constraints. Let B':' denote the set of proper vectors of inequality constants. The-; p=[x(a.) la.EB;~ and x(a.) is a PICSO= optimal solution}. This is called the mclhod of proper in­equality constants (PIC). The same triple-objective optimization problem is solved also by this method in Example 3 for illustration.

VII. CONCLUDING REMARKS

1. The usual convexity assumption under which relation pc KD is ensured has been weakened to directional convexity in a single direction. The arguments and results in Secs. III. C and III. D can easily be adapted to ensure that pcRs. The class of MO optimization problems for which the method of linear combinations (using either positive weights or

PARETO-OPTIMAL SOLUTIONS OF MULTIPLE-OBJECTIVE PROBLEMS 129

only strictly positive weights~ are useful is thereby enlarged.

2. It is still an open question: how to detect the non­Pareto-optimal solutions in KD or 1<S so that one can deter­mine exactly the set P?

3. By either method of proper eguality constraints or method of proper ineguality constraints, one can determine exactly the entire set P of any given MO optimization problem without assuming convexity of any kind (see Secs. V-VI).

4. The latter two methods are twins having natural re­semblance, but neither is an extension of the other. Each does have its own distinctive characteristics and merits, though not apparent (see [12] and [13]).

5. Note that the order 5" instead of the order > is used in conditions (b) and (c). For problems with more than double objectives (i. e. ,- N> 2), :;; should not be replaced by > (since the former is weaker than the latter). Intuitively speaking, condition (b) says that ¢ is absolutely right-decreasing on B from Cio. If :;: were replaced by>, it would be undesirably weakened and say only that'!p is strictly right-decreasing on B from Cio. Thus, any Ci with -J2 :$;Ci1< 0 and Ciz= 4 in Example 2 would be mistaken as a proper vector of equality constants, and any Ci with _0$ <Ci1< 0 and Ciz=4 in Example 3, as a proper vector of inequality con­tants. The interested reader is referred to [12]-[14] for more discussions on pointwise, directional, decreasing properties like the above.

6. Various necessary and/or sufficient conditions useful for testing or verifying, either directly or indirectly, the properness of given vectors of equality (or inequality) constants can be found in [12]-[15].

7. A common idea behind these three methods is the conversion of the given multiple objectives to a single one so that plenty of well-developed numerical methods and algorithms for single-objective nonlinear programming or optimal control (e. g .• see [19]. [20]) can be applied.

ACKNOWLEDGEMENT

This work was supported by the National Science Foundation under Grant GK-32701.

130 J.G. LIN

APPENDIX A: EXAMPLES

Example 1. Various relations of Cs and CD to M.

Zz

a

b -t--~--+----.. Zl

Zz

Zz

a b

Fig. 2

M = as

Cs= ab' (without endpoints a and b)

Fig. 3 CSc MC CD. where

M =lJC

Fig.4

Cs = be (without endpoint c)

CD= [a} UbC

Cs = MC CD' where

M =bC

Cs=be

C D = [a}UkU{d)

PARETO-OPTIMAL SOLUTIONS OF MULTIPLE-OBJECTIVE PROBLEMS 131

Zz

Zz

Fig. 5 Cs c: M = CD' where

M=ab u"ci Cs= at; u Cd (without end­

points a and d)

CD= ail u 7cr

Fig.6 CSC: M and CDC: M, where

M= abUbCU Cd

CS=ab UCd (without end­points a and d)

CD =abUcl

Fig.7 CSC: M, but Mrt CD and

CDrt M, where

M=bcUCdU&

CS=bcUde

CD = {a}U~U de U{f}

Fig. 8 M = a:b (without endpoints a and b), but

both Cs and CD are empty.

(Dash lines indicate that boundary points are not included. )

132

Example 2. Determine the set P of Pareto-optimal solutions of the following triple-objective problem:

subject to xl+xZ+X4=O

l-x~+~ -X~+2XIX3+2xzX3~ 0

and

J.G. LIN

by the method of PEC. The associated PECSO problem is:

subject to xl-x3=al> xZ+x3+3=aZ

xl+xZ+x4= 0

1-~+~-~+2xIX3+2xzX3~O

and 3-~-~-2xzX3-2xz-2x3:2:0.

The PECSO-optimal solution ~(a) for a=(al' az}, the function cf>, the sets A and Bare:

~1 (a)=al +[l-a~ +(az - 3)z t~Z ~z (a)= az -3 - [l-a~+ (az - 3)z J1/z,

'" [Z Z JI/Z x3 (a) = l-al +(az - 3) '~4(a) = -al -az+ 3; (3)

cf>(a)= [1-ai+(az-3)zt/l

A=B=[(aJ, az) la~-(az-3)z:S;1 and O:s;a,z:O::4}. See Fig. 9.

Condition (a) implies that for aO to be proper, it is necessary that o+cf>(aO)/3ai:!OQ when the right-hand partial deri- -lIz vative [12] ~+cf>(aO)/~ai exists. Since a+cf>(a)/~al=-al[1-ai+(az-3)zJ, those a in B with a1< 0 sh<;mld be discarded. The remainder is B I =[(a1, az) 1 o:s;al~[l +(az - 3)z J1/z and O:s;az:S;4}. Since C. +cf> (a) loaz=

(az-3)[1-a~+(az-3frlk, those a in BI with 3<az<4 should be dis­carded. (Note that the right-hand partial derivative a+cf>(a)/aaz does not exist at a in B I with az=4.) The remainder is

BII=f(av 4) I O:s;a1:!OJ2}U [(alo a z) IO~al:!O [1+(az-3)Zfz and O~az:O:: 3}.

PARETO-OPTIMAL SOLUTIONS OF MULTIPLE-OBJECTIVE PROBLEMS 133

a.z

4 I

3 I , ,\

Z

1

o

/1

,

Fig.9 The set B given by

[(a.ha.Z) \a.fs:(a.z-3)z+1 and Oso'zS:4}

Fig.lO The set B':' given by the

the union of

[( a.h 4) I OS:a.1 s:J2} and

[(a.ha.Z)I Os:a.1s:[(a.z-3)z+l ]1/z and

Os:a.z<Z}.

Fig.11 The set ~ given by

the union of

[(a.h a.z)I-"'<a.1s:j2 and ZS:a.zS:4},

{(a.l>a.z)l- co<a.1s:[(a.z-3)z+lt/z and

Os:a.zS:Z}, and

t(a.loa.ill-'" <a.l~JiO and -"'<a.zS: oj.

134 J.G. LIN

l{l Now, since ¢(aloaZ)~¢(a104)=[2-af] for aEB" with2~<lz";;;3,

condition (b) implies that any such a should be disqualified. I' - Z IZ

The remainder is B"I= {(alo 4) \O~al~J2 }U{(aloaz)IO$alS [l+(az-3) ] and Osaz<2}. Applying Theorem 5, one ~an verify that each a in BIll is proper. Therefore, B':'= B"I. Hence P=[2(a)laEB'~), with ~(a) defined by (3).

Example 3. Determine the set P of the triple-objective problem of Example 2 by the method of PIC. The associated prcso problem is:

maximize z3=x3

Xl+XZ+X4=0

l-xi+x~-x~+2xlX3+2xzx3~ 0

and 3 -x~ -x~ -2xzx3 -2xz -2x3~ O.

The prCSO-optimal solution 'X(a) for a=(a1>az), the function *, the sets A and Bare:

r rJ [Z ]l/Z '" [2 ]l/Z ..., [ Z ]1/2 ,... ( xda)=al+ 2-0.1 ,xz(a)=l- 2-0.1 ,x3 (a)= 2-CL ,x4 CL)=-a1-1,

l ~(a)=[2-af]1/z, for 0~al:f.J2 and 2~azs;4;

{ ~da)=~, ~z(a)=1-J2, 'X3 (a)=fi, x4(a)=-1,

*(CL)=J2, for alsO and 2:f.azM;

,., 2 l/Z ,.. Z 1/2

{

X1(CL)=[l+(aZ-3)] ,xz(a)=az-3-[l+(az-3)],

N [ 2 ]l/z "" x3(a)= 1+(CLz-3) ,x4 (a)=-a 2 +3,

l\r(a)=[l+(az-3)2]1/z for O~al and 0 ~a2~ 2;

PARETO-OPTIMAL SOLUTIONS OF MULTIPLE-OBJECTIVE PROBLEMS 135

{ ~da )=JlO, :X:z(a)=-3-JIO, x3(a)=JlO, x4(a)= 3,

l\r(a)=/lO, for 0$a1 and O~az.

A =B = [(a1oaZ) I-Ill <a1~J2 and 2~az::;; 4}

U £{a.1,az)[-('O<a1~[1+(az-3)ztIZ and 0~az~2} iJt(a1' az) \- f.O < a 1::;; jiO and -110 <: az~ oj. See Fig. 11.

Theorem 6 implies that vector (aI, az) is not proper if a 1< 0, if az< 0, or if a1~j2 and 2:s::az<4. Discarding these, we have the remainder 1?' = £(a1o 4) I O~ a1~j2} U[(ai,a~10:S;a1:!l:[{az-3)z+l JV2and O~ a2< 2}. One can verify by Theorem 6 that B*=B '. Consequently, P=t)((a) \ aEB*}, with x(a) defined-by -(4) • Note that B':';;;B;". -

APPENDIX B: PROOFS

Proof of Lemma 1. Since zc ZP, a maximal vector in Zp is necessarily a maximal vector in Z. Assume z is not a maximal vector in Zp. Then, z'S;z for some z, in Zp. By definition, z'=z"-13p for some z" in Z and some 13;;':0. Conse­quently, z"5 z and z cannot be a maximal vector in Z.

Proof of Lemma 2. We prove that CD C Clb first. Let zECD, that is, zEC(w) for some wS;O. Then WTz;;':WTz' for all z.- in Z. Since w T p;;':O, we have wT z;;':wT (z' -13p) for all z, in Z and all 13;;,: O. This Obviously implies that zECP(w) CCB. There­fore, CDC~. Consequently, CD;;; znCDcznDb' Next, let

zEznCt,. By definition, for some w>O, we have WTZ;;':WTz" for all z" in ZP, hence for all z" in Z. It follows that zEC(w)CCD .

Consequently, Z nc~c CD' Therefore, CD = znc~.

Proof of Theorem l(a). From Lemma 1 and 2, M=MP

and cD=zncE. Applying Lemma 3, we then have

M;;;znM=znMPcznc~ = CD'

Proof of Lemma 4. Since the necessity has been proved in [18, Theorem 2.1J, we prove only the sufficiency here. Suppose Zp is convex. Then given any zl and z2 in Z and any A1;;':0 and A2;;':0 with A1+A2=1, we have A1Z1+A2Z2EZP. By definition, there are z3 in Z and 13;;,:0 such that z3 -13p=A1z1+A2z2. Thus

136 J.G. LIN

Proof of Theorem 3. Given two arbitrary points zl and zZ in the image Z and two arbitrary positive numbers Al and Az with Al+AZ:::l. By definition, zl=Jxl+b and zZ=Jxz+b. for some Xl and x Z in X. Let q be a v;ctor such th~t p~Iq>O and X is q-directionally convex. Then AIXI+AzXz+i3q=x3 for some i3~ 0 and some r in X. Consequently, Al (Ix1+b)+Az(Ixz+b)+i3Iq=Ix3+b, or AlZl +AzZz+i3pEZ. Hence the image is p-directionally convex. It follows from Theorem 2 that PCKD .

Proof of Theorem 4. Let J be an Nx(n+N) matrix defined by I =[0:1], where 1 is the NxN identity matrix. Consider the maximization of z=Iy subject to yEG. Since

p~ Jq=(q , ... , q ) 5> 0 and G is q -directionally convex by - n+l n+N

hypothesis, the image fIyl yEG} must be, by Theorem 3, p­directionally convex. But Uj lyEG} =[J(x) IxEX} , the image f J(x) I xEXJ is p-directionally convex for some pSO. Thus PCKD by Theorem 2.

Proof of Theorem 5. (If) Assume there is z 1 in Z such that

1- 0 Th 1~( 1 1 )-( 0 0)_ ° d 1> O_n-( 8) z >z. en a. - Zlo •.. , zN-l > Zl,"" zN-1 -a. an zN-zN-'P a. • If a?-EB, then ¢(a.1)~z~~¢(a.°); and if a.l€A but a.1~ B, then ¢(a.1»z~~¢(a.O). Obviously, no such zl can exist if both conditions (a) and (b) are satisfied. (Only if) Let Zo be a maximal vector in Z. Then for any a. in A with a. >0.0, we must have zN < z~ for any z in Z such that zi=a.i , i;tN. Thus, ¢(a.)~ SUp[ZN IzEZaJ~z~~¢(a.O). Hence condition (a) must be satisfied. Now if a.EB, then ¢(a.)=z~ for some zl in Za.. Consequently, ¢(a.)=z~ <: z~ = ¢(a.0 ). Hence condition (b) must also be satisfied.

Proof of Theorem 6. (If) Assume there is zl in Z such that zl:>zO. Then a.1~(zL ... ,~_tl>(Zl'" .,Z~_l)=a.° and z~~z~=W(a.°). If ctE;I2, then W(a.1)~z~ ~¢(a.0) and condition (c) is violated. But if a.1EA but a.1~B, then '11 (0.1» z~ ~ '11(0.0), contradicting the decreasing property of '11-:- Thus ZO is a maximal vector in Z if condition (c) is satisfied. (Only if) Let ZO be a maximal vector in Z. Then, for any a. in A with 0.>0.0, we must have ZN <z~ for any z in Z such that zi~ a.i: i;tN. Now, if a.E.~, then 1\r(a.)=z~ for some zl in Za.. Consequently, 1\r(a.)=z~< z~= ~(a.0). Hence condition (c) must be satisfied.

PARETO-OPTIMAL SOLUTIONS OF MULTIPLE-OBJECTIVE PROBLEMS 137

REFERENCES

[1] L. A. Zadeh, "Optimality and Non-Scalar- Valued Perfor­mance Criteria," IEEE Trans. Automat. Contr., vol. AC-8, pp.59-60, Jan. 1963.

[2] J. Von Neumann and O. Morgenstern, Theory of Games and Economic Behavior, 3rd ed. Princeton: Princeton Univ. Press, 1953.

[3] Y. C. Ho, "Differential Games, Dynamic Optimization, and Generalized Control Theory, II J. Optimiz. Theory Appl. vol. 6, no. 3, pp.179-209, 1970.

[4] J. G. Lin, "On N-Person Cooperative Differential Games, II Proc. 6th Princeton Conf. Inform. Sci. & Syst., Princeton, N. J., pp. 502 -507, Mar. 1972.

[5] T. C. Koopmans, IIAnalysis of Production as an Efficient Combination of Activities, " Activity Analysis of Production and Allocation, T. C. Koopmans, Ed. New York: Wiley, pp. 33 - 97, 1951.

[6] K. J. Arrow, E. W. Barankin, and D. Blackwell, IIAdmissible Points of Convex Sets, II Contributions to the Theory of Games, vol. II, H. W. Kuhn and A. W. Tucker, Eds. Princeton: Princeton Univ. Press, pp.87-91, 1953.

[7] N. O. Da Cunha and E. Polak, "Constrained Minimization under Vector-Valued Criteria in Finite Dimensional Spaces, II J. Math. Anal. and Appl. vol. 19, no. 1, pp.103-124, July 1967.

[8] J. G. Lin, IIMultiple-Objective Optimization, 11 Columbia Univ., Dept. of Elect. Eng. & Compo Sci., Syst. Res. Gr. Tech. Rept~, Dec. 1972.

[9] G. A. Katopis and J. G. Lin, "Non-Inferiority of Controls under Double Performance Objectives: Minimal Time and Minimal Energy, II Proc. 7th Hawaii Int. Con£. Syst. Sci., Honolulu, Hawaii, pp.129-13l, Jan. 1974.

[10] J. G. Lin, "Circuit Design under Multiple Performance Objectives, 11 Proc. 1974 IEEE Int. Symp. Circuits & Systems, San Francisco, Calif., pp. 549-552, Apr. 1974.

138 J.G. LIN

[11] J. G. Lin, "Maximal Vectors and Multi-Objective Optimization," J. Optimiz. Theory Appl., vol. 18, no. 1, pp. 41- 64, Jan. 1976.

[12] J. G. Lin, "Proper Equality Constraints (PEC) and Maximi­zation of Index Vectors, " J. Optimiz. Theory Appl. vol. 20, no. 4, Dec. 1976.

[13] J. G. Lin, "Proper Inequality Constraints (PIC) and Maximi­zation of Index Vectors" to appear in J. Optimiz. Theory Appl .•

[14] J. G. Lin, "Multiple-Objective Problems: Pareto-Optimal Solutions by Method of Proper Equality Constraints (PEC)," to appear in IEEE Trans. Automatic Control.

[15] J. G. Lin, "Multiple-Objective Programming: Lagrange Multipliers and Method of Proper Equality Constraints, " to be presented in 1976 Joint Automatic Control Conf., July 1976.

[16] P. L. Yu, "Cone Convexity, Cone Extreme Points, and Nondominated Solutions in Decision Problems with Multi­objectives, " J. Optimiz. Theory Appl. vol.14, no. 3, pp.3l9-377, Sept. 1974.

[17] H. Payne, E. Polak, D. C. Collins, and W. S. Meisel, "An Algorithm for Bicriteria Optimization Based on the Sensitivity Function, " IEEE Trans. Automat. Contr. vol. AC-20, no. 4, pp.546-548, Aug. 1975.

[18] J. M. Holtzman and H. Halkin, "Directional Convexity and the Maximum Principle for Discrete Systems, " J. SIAM Contr., vol. 4, no.2, pp.263-275, 1966.

[19] D. M. Himmelblau, Applied Nonlinear Programming, New York: McGraw-Hill, 1972.

[20] A. E. Bryson and Y. C. Ho, Applied Optimal Control, Waltham, Mass.: Blaisdell, 1969.

STACKELBERG STRATEGIES FOR MULTILEVEL SYSTEMS*

J. B. Cruz, Jr.

Coordinated Science Laboratory

University of Illinois, Urbana, Illinois 61801

1. INTRODUCTION

In this paper we will present a brief sketch of how the leader­follower or Stackelberg strategy concept for two-person dynamic nonzero-sum games might be extended to the coordination of several subsystems. We will also explore how hierarchical control could be formulated in the Stackelberg sense.

Differential games has a long history rooted on game theory as initiated by von Newmann and Morganstern and on control theory. There are several books on the subject as well as several survey papers, the most recent of which are by Ho [1,2]. We will discuss the Stackelberg solution concept [4] originally suggested for static duopoly, and extensions to dynamic nonzero-sum games [5-llJ. One of the earliest papers on many-player differential games is that by Case [3]. A special N-player Stackelberg differential game has been previously described also [8]. In this presentation we will further extend the Stackelberg solution concept to M-person optimization in the context of a large scale system.

2. COORDINATED NONINTERACTING SUBSYSTEMS

In this section, we will sketch how the Stackelberg strategy concept [4-7] can be adapted for formulating an approach to the

* This work was supported in part by the National Science Foundation under Grant GK-36276, in part by the Joint Services Electronics Program under Contract DAAB-07-72-C-0259, and in part by the U.S. Air Force under Grant AFOSR-73-2570.

139

140 J.B. CRUZ, JR.

coordination of several subsystems. We assume that there is no direct interaction among the subsystems but each subsystem is coupled to a coordinator. For the purpose of the development, we consider M discrete-time subsystems, each modeled by

iii i i x (k+l) = f [x (k),u (k),v (k)],

i=l, ••• ,M (1)

i where x is the n.-dimensional state vector of the ith subsystem,

i ~ u is the m.-dimensional local control vector for the ith system,

i ~ v is the p.-dimensional coordinating signal from the coordinator

~ . to the ith subsystem, f~ is an n.-vector function on n. mi p. ~

R ~ x R x R ~, and k is an integer variable representing discrete

time. The control ui is to be chosen by decision-maker i, DM., ~

with respect to the cost function

i i J [u ] =

N •. • • ~ ~ ~ ~

~ L [x (k),u (k),v (k),k], k=O

i=l, ••. ,M. (2)

i The coordinating signal v comes from a coordinator subsystem

which is modeled by o 0 0 0

x (k+l) = f [x (k),u (k)] (3)

where XO is the n -dimensional state vector of the coordinator o

subsystem, UO is an m -dimensional control vector chosen by the o

coordinator, DM , fO is an n -dimensional vector function on n mOO

R 0 x ROThe coordinating signals are generated by mappings i 0

C from the space of control sequences [u (k)} to the space of

signals vi(k). For example vi(k) might be a function of xO(k),

and xO(k) depends on the sequence [uo(j)} for j < k through Equation (3).

The control UO is chosen with respect to the cost function

N o 0 0 1 M

~ L [ x (k), u (k), w (k), ••. , w (k)] k=O

(4)

where wiCk) is a q.-dimensional feedback signal from subsystem i. ~ .

These feedback signals are generated by mappings F~ from the

space of control sequences [ui(k)} to the space of feedback signals

STACKELBERG STRATEGIES FOR MULTILEVEL SYSTEMS

wi(k). For example wi(k) might be a function of xi(k), which is

related to the sequence ui(j), for j < k, through Equation (1).

The Stackelberg approach to the coordination of the M sub­

systems is to consider OM as a leader and DM., for i=l, ••• ,M o ~

141

as followers. We imagine that DM provides OM. with a coordinating i . 0 ~ •

signal v and OM. minimizes J~ with respect to u~ for each given i ~ 0 0

v. The coordinator then minimizes J with respect to u ,

considering that the feedback signals wi(k) from the subsystems

result from choices of ui which minimize Ji for i=l, ••• ,M. The

coordinator supplies vi to each OM. which result from the optimum uO• ~

For each subsystem from i=l to i=M let us assume that there . iii is a mapp~ng T from the space of v to the space of u such that

iii i iii J [T (v ), v ] ~ J [u ,v] (5)

i i for each admissible v and for all admissible u. For each

UO we have

i w =

i Ti. Ci i i where F is the composition of the mappings F , T ,

and ci • Finally, we assume that in the admissible set of uO,

there is a UO such that

o AO 1 1 1 ~o M M M AO J [u ,F. T· C (u ), •.. , F' T· C (u)] ~

J O[uo , r! 1 1 o_M ~ M 0 T' C (u ), ... , p- yo C (u)]

(6 )

(7)

for all uO in the admissible set. The optimum coordinating signals i i ~o

are v C (u ), and the optimum local controls for the subsystems i i AO. AO

are u = T (u ), for ~=l, •.• ,M. In general u will depend on all 01 M h f h' 1 the states x ,x , ... ,x so t at a ter t e opt~mum contro s are

implemented the subsystems are interacting through the coordinating . 1 i

s~gna s v .

According to the problem formulation above, each decision­maker i faces a one-person optimization problem, once the entire

coordinating signal vi for all remaining stages is declared by the coordinator. Thus the principle of optimality will hold for each subsystem i provided there is no change from the declared sequence

142 J.B. CRUZ, JR.

of coordinating signals. If the coordinator declares a new sequence of coordinating signals starting at stage k, each DM. must recompute . ~

the optimal sequence for u~ starting at stage k, but if there are no further modifications in the declared sequences of coordinating signals, the new control sequences satisfy Bellman's principle of optimality. That is, the cost-to-go for any number of stages rema~nlng is optimal. For the coordinator, however, several possible variants of the Stackelberg concept are possible. In general, the principle of optimality will not hold for the coordinator. Since the principle of optimality is desirable to preserve for many reasons, we constrain the admissible control space ~f the coordinator so that the resulting Stackelberg strategy satisfies the principle of optimality. For two-person nonzero sum dynamic games such a strategy is called feedback Stackelberg strategy [7], or more generally, Stackelberg equilibrium strategy [lOJ since the control does not have to be feedback in form. By definition, the Stackelberg equilibrium strategy sequence, starting at stage k, and the Stackelberg equilibrium strategy sequence starting at a later stage j > k are coincident beginning with stage j. Thus one feature of this Stackelberg equilibrium strategy is that once declared, all the sequences of coordinating signals will remain invariant with respect to the number of remaining stages. Consequently the strategies for the decision-makers of the M subsystems will remain invariant.

If decentralized control is desired, it would be necessary

to restrict each ui to be realized as a mapping from xi and vi. Further implementation constraints might be imposed, such as

restricting ui to be a linear function of xi plus a time function.

In general, determination of the Stackelberg equilibrium strategies for the problem formulated in this section would be extremely difficult to obtain, However, if the vector functions

o 1 M . 'f h ' i d i I' f ,f ""f are l~near, 1 t e mapp~ngs C an Fare lnear,

and l' f LO Ll LM d t' 1 t' lIt' , , ... , are qua ra lC, ana y lca so u ~ons are possible to obtain, analogous to those for two-person games [5-7].

A stochastic formulation could be made by introducing

disturbances in fO,fi,c i and Fi , and taking expectations for JO i and J. The basic philosophy remains the same but the problem is

much more difficult. Except for linear-quadratic-gaussian situations and other very special cases, with special information structures, no general results are known. For the LQG case, results analogous to those for two-person games are possible [11].

STACKELBERG STRATEGIES FOR MULTILEVEL SYSTEMS 143

Finally, the nature of information exchange depends on the iii i. operators C and F. For example, C and F m1ght be such that

only slow or long term phenomena are communicated. Fast phenomena might be largely the responsibility of load controllers but long term phenomena might be of interest to both coordinator and the individual subsystem. This structure may be investi­gated in the framework of Stackelberg strategies for coordinated subsystems. The theory of singular perturbations in large scale systems [12] in conjunction with the suggested formulation of this section appear to be natural bases for investigating coordinated large scale systems.

If the M subsystems are interacting in addition to the presence of coordinating signals, the subsystems may playa subgame using the Nash solution concept for example, and the group would then be treated as a follower playing a Stackelberg game against one leader [8]. Alternatively, an imbedding parameter may be introduced such that when the parameter is zero, no interaction is present. The strategies are then obtained as expansions in the parameter. For the use of imbedding for decoupling large-scale systems see [13,14].

3. HIERARCHICAL M-LEVEL STRATEGIES

In the previous section we considered M noninteracting subsystems at one level and one coordinator at another level. In some situations it may be necessary to have more than two levels. In this section, we will outline how the Stackelberg concept may be generalized to M-level strategies. However, to keep the discussion as simple as possible, we consider only one decision-maker and one subsystem for each level, and only a total of three levels.

Consider a discrete-time dynamic system described by

x(k+l) = f[x(k+l),u l (k),u2 (k),u3 (k)] (8)

I 2 With three decision-makers DMI , DM2 , DM3 , who choose u , u ,

and u3 respectively. The detailed structure of three subsystems and their interconnection are not explicitly exhibited to focus attention on the control strategy hierarchy. The controls are chosen with respect to the cost functions

i123 Ni 123 J [u ,u ,u ] = ~ L [x(k),u (k),u (k),u (k),k]

k=O

Notice that all three controls appear in each cost function and in the composite state equation.

(9)

144 J.B. CRUZ, JR.

We now propose a hierarchy of control strategies, where u 3

. h . .. 3 f . i d 2 Th f 1S C osen to m1n1m1Ze J or a g1ven u an u us so ar as DM3 is concerned DM1 and DM2 are leaders and DM3 is a follower.

DM2 then chooses u2 to minimize J 2 for a given u1 and for u3

constrained to be that chosen by DM3 • So far as DM2 is concerned, 1

DMI is a leader and DM3 is a follower. Finally, DM1 chooses u

so as to minimize J 1 under the constraint that u3 is that chosen

by DM3 and u2 is that which is chosen by DM2 • Thus from the

point of view of DM1 it is the leader followed by DM2 who is

in turn followed by DM3 •

Consider a mapping T3 from the space of fu1} x fu2} to [u3}

such that

312 12 3123 J [u ,u ,T3 (U ,u )] ~ J [u ,u ,u ] (10)

for all u3 and for each u1 and each u2 of the control spaces.

A . h' f 3. (1 2) n opt1mum c 01ce or u 1S T3 u ,u • Next consider a mapping

T2 from the space of fu1} to fu2} such that

2 1 1 1 1 J [u ,T2 (U ),T3(u ,T2 (U »] ~

2 1 2 1 2 J [u ,u ,T3 (U ,u )] (11)

for each u1 and for all u2 in the admissible control spaces.

An optimal choice for u2 is T2 (u1). Finally consider a control A1 [ 1} u from the set u such that

1 A1 A1 A1 A1 J [u ,T2 (U ),T3 (u ,T2 (U »] ~

1 1 1 1 1 J [u ,T2 (U ),T3 (U ,T2 (u »]

For the hierarchical sequence DM1, DM2 , DM3 , the optimal

Stacke1berg strategies are defined as

All [1 (1) (1 1) J u = arg min J u ,T2 u ,T3 u ,T2 (U )

A2 A1 u = TZ (u )

... 3 u T ( ,,1 A2)

3 u ,u

(12)

(l3)

(14)

(15)

STACKELBERG STRATEGIES FOR MULTILEVEL SYSTEMS

The extension of the definition to M levels on the control hierarchy should be clear. DMi , for i=2, ••• ,M-l, is neither

a follower nor a leader. So far as DM. is concerned, the order 1-

of the hierarchy for DMl, •.. ,DMi _l is immaterial. However,

DMi+l, ••• ,DMM must announce their strategies in the order

given herein. DMM is a follower, and DMl is the leader.

As in the coordinated set of subsystems of Section 2, the strategy spaces may be further restricted so that Bellman's principle of optimality holds. When this is imposed we have a Stackelberg equilibrium strategy.

145

It should be noted that in defining the control hierarchy, no special structure in the subsystems and no special structure in the interconnections are assumed. Naturally, in specific calculations, special structures could result in tremendous simplifications but the hierarchical control concept remains the same.

When f is linear and Li for i=l, ••• ,M are quadratic the determination of the strategies could be expected to lead to Riccati type matrix equations. A stochastic formulation of control hierarchy could also be outlined. Unless special information structures are assumed, the problem of determining the Stackelbery strategies could be quite formidable.

We have merely outlined a Stackelberg rationale to the multilevel control of a large scale system. This approach appears to be natural to consider in studying organizations where there is already a chain of command or hierarchy of decision-making. In situations where there is no clear hierarchy it would be advisable to investigate all permutations in ordering analogous to that for two-person games [9].

4. CONCLUDING REMARKS

In this brief presentation, we have sketched how the Stackelberg concept might be modified and extended in the context of multilevel large scale systems. In one section we explored how we might organize the control of M noninteracting subsystems using one coordinator. The coordinator sees a larger system-wide cost function and determines what coordinating signals to send to the subsystems, noting that each outsystem has its own optimization to carry out for each given coordinating signal. This structure is two-level. If there is interaction among the subsystems, one might consider introducing a parameter which decouples the system when the parameter is zero. In order to compute the cost-to-go for each subsystem, each subsystem decision-maker must either forecast the coordinating signal for all remaining stages, or it must be told what strategy

146 J.B. CRUZ, JR.

the coordinator employs to obtain the coordinating signal. This key step is inherent in all dynamic game situations where controls other than the ones made by decision-maker i should be forecast, estimated, or computed according to an assumed rationale of behavior of the other decision-makers.

In another section we sketched how the Stackelberg concept extends to a multilevel or hierarchical control strategy. Intermediate decision-makers regard all decision-makers who act before him as leaders, the order of action being immaterial. Decision-makers who act later must act in a particular hierarchy. No structure is specified for the arrangement of subsystems, the focus being on the control hierarchy. As in any dynamic game­theoretic situation, the cost-to-go for each intermediate decision­maker must be predicated on the strategies of all decision-makers who have acted previously.

REFERENCES

1. Y. C. Ho, "Commentator's Report on Session on Differential Games," Proc. of the Fifth World Congress of IFAC, Paris, 1972.

2. Y. C. Ho, "Decisions, Control, and Extensive Games," Proc. of the Sixth World Congress of IFAC, Boston, 1975.

3. J. H. Case, "Toward a Theory of Many Player Differentia 1 Games," SIAM Journal on Control, Vol. 7, 1969, pp. 179-197.

4. H. von Stackelberg, The Theory of the Market Economy, Oxford University Press, Oxford, England, 1952.

5. C. 1. Chen and J. B. Cruz, Jr., "Stackelberg Solution for Two-Person Games with Biased Information Patterns," IEEE Trans. on Automatic Control, Vol. AC-17, No.5, 1972, pp. 791-798.

6. M. Simaan and J. B. Cruz, Jr., "On the Stackelberg Strategy in Nonzero-Sum Games," Journal of Optimization Theory and Applications, Vol. 11, No.5, 1973, pp. 533-555.

7. M. Simaan and J. B. Cruz, Jr., "Addi tiona 1 As peets of the Stackelberg Strategy in Nonzero-Sum Games," Journal of Optimization Theory and Applications, Vol. 11, No.6, 1973, pp. 613-626.

8. M. Simaan and J. B. Cruz, Jr., "A Stackelberg Strategy for Games with Many Players," IEEE Trans. on Automatic Control, Vol. AC-18, No.3, 1973, pp. 322-324.

9. T. Basar, "On the Relative Leadership Property of Stackelberg Strategies," Journal of Optimization Theory and Applications, Vol. 11, No.6, 1973, pp. 655-661.

STACKELBERG STRATEGIES FOR MULTILEVEL SYSTEMS 147

10. J. B. Cruz, Jr., "Survey of Nash and Stackelberg Equilibrium Strategies in Dynamic Games," Annals of Economic and Social Measurement, Vol. 4, No.2, 1975, pp. 339-344.

11. D. Castanon and M. Athans, "On Stochastic Dynamic Stacke1berg Strategies," Proc. Sixth World Congress of IFAC, Boston, 1975.

12. P. V. Kokotovic, R. E. O'Malley and P. Sannuti, "Singular Perturbations and Order Reduction in Control Theory-An Overview," Proc. Sixth World Congress of lFAC, 1975, Boston.

13. P. V. Kokotovic, "Feedback Design of Large Linear Systems," Chapter 4 in Feedback Systems, J. B. Cruz, Jr., editor, McGraw-Hill Book Co., New York, 1972, pp. 99-137.

14. P. V. Kokotovic, W. R. Perkins, J. B. Cruz, Jr., and G. D'Ans, lie-Coupling Method for Near Optimum Design of Large Scale Linear Systems," Proc. lEE, Vol. 116, pp. 889-892, 1969. Also in System Sensitivity Analysis, J. B. Cruz, Jr., editor, Dowden, Hutchinson, and RoSS, Stroudsburg, Pa., 1973, pp. 313-316.

INCENTIVE COMPATIBLE CONTROL OF DECENTRALIZED ORGANIZATIONS

Theodore Groves

Northwestern University

I. THE BASIC ORGANIZATIONAL DECISION PROBLEM

Many organizational decision problems may be usefully modeled as the programming problem:

P: Max F (x) x

subject to G(x) ~ 0

N N N K where x E:R ,F : :R "':R, and G : :R "':R .

(1.1)

However, for large organizations in which the numbers of de­cisions N and constraints K are very large it is often either in­feasible or prohibitively expensive to accumulate under a single individual's control complete information regarding the functions F(·) and G(.) and for a single individual to solve problem P. In such cases, what one individual is unable to do may be possible for many, working together, to accomplish. The organizational design problem, in part,l is concerned with how to organize a multi-person organization to solve such a problem.

However, for multi-person organizations the problem of ~ ganizational control arises. This problem may be described, following Arrow [1], in two parts: (1) the definition of the operating rules or the specification of rules of behavior for the different agents to follow in making the decisions assigned to their control, and (2) the definition of the enforcement rules or

149

150 T. GROVES

the specification of rules for evaluating the individual agents that provide appropriate incentives for them to follow the pre­scribed operating rules.

Much research has been devoted to the elaboration of operatin~ rules for such large organizations, including work in team theory, decentralization theory,3 and decomposition a1gorithms.4 Far less work has addressed the specification of enforcement rules or the incentive prob1em. S In this paper I describe a general approach for solving the incentive problem in the organizational decision problem context or more generally for solving the control problem and summarize some of the major results obtained by myself and others following this approach. 6

II. THE CANONICAL DIVISIONAL FORM OF THE DECISION PROBLEM

Given the organizational decision problem P and any specified number of organization members, an organizational form? is defined by an assignment of the N decisions to the various members and a specification of the ~ priori information available to each member--that is, the information each member has prior to any communication among the members. In this paper we assume the organization consists of 1+1 members (agents)--I divisional man­agers, i=l, •.. ,I, and a headquarters manager or Center, i=o--; that each agent i controls the decisions xi where x = (xo ,x1"" ,xI) E lRN; and that with the assignment of de­cisions toe problem P can be written as:

I fi(xi,xo) + f (x ) D: Max t. 1 (2.1) tx .} ~= o 0

~

subject to g. (x. ,x ) < 0 ~ ~ 0 i 1, ... ,I

g (x ) < 0 o 0

N· where x. E lR~

~

L;I i=o Ni

No Ko go : lR .... lR ,

I and Ei=o Ki = K.

We also assume a priori information is dispersed or decentralized such that each agent i knows only the functions f.(·) and g.(.).

~ ~

Although any problem of form P can be written in the canonical form D by merely setting No = Nand Ko = K, i.e. assigning all decisions to the Center, the ~ priori information assumption clearly restricts the model. Given a particular structure of the ~ priori information it may not be possible to assign the decisions

CONTROL OF DECENTRALIZED ORGANIZATIONS 151

to the agents in a manner implied by form D. Also of considerable importance is the requirement that if any of the N decisions enters as a non-trivial argument of more than one of the functions fi or gi, then it must be assigned to the Center, i.e. be a component of the vector xO' Although ,2. priori information may be completely decentralized, this requirement limits the degree of decentraliza­tion of decision making.

Despite the restrictions imposed on the model by canonical form D, the model is quite general and, by augmenting the number of decisions and constraints, may be applicable for situations not appearing to satisfy the restrictions at first glance. For examplS' consider the widely discussed simple additive decomposition model:

I I A: {~~} Ei=l Fi(x i ) subject to ~i=l Gi(xi ) ~ K

L

(2.2)

where xi is an activity vector of division i, K is a vector of resources available to the organization, Gi(xi ) is the resource requirements of activities xi' and Fi(xi) is the i th division's profitability at the level xi' Typically it is assumed that ~ priori the Center knows only the available resources K and each division i knows only the functions Gi(') and Fi(·)' Since the decisions xi and Xj enter the same constraint, the requirement of form D would imply that all decisions be assigned to the Center, i.e. that is, Xo = (xl, ... ,x1)' But this would violate the ~ priori information assumption since the Center does not know Fi(') or Gi ( .) .

However, by adding I new decisions, xoi' i = 1, ... ,1, (where each xoi is of the same dimensionality as the vector K) and adding

I the constraint ~i=l xoi ~ K, it is easy to show that problem A is

equivalent to:

I B: Max Ei=l {x.,x 1

L 0

subject to

F. (x.) = L' L

Gi (Xi) -

I Ei=l x .

OL

I f. (x.) ti=l L L

x oi - gi(xi,xoi )

- K == g (x ) < 0 o 0 -

(2.3)

< 0

where Xo = (xol, ... ,xo1)' Furthermore, problem B is precisely of form D and satisfies the ~ priori information restrictions. The additional I decisions are, of course, allocations of the available resources K to the I divisions.

152 T. GROVES

III. DEFINING THE CONTROL PROBLEM

As mentioned above, the multi-person organizational control problem consists of two parts: (1) the choice of operating rules and (2) the choice of enforcement rules. However, given any set of enforcement rules or, more specifically, rules for evaluating the I divisional managers, the divisional manager's operating rules are automatically defined by assuming the managers strive to take whatever decisions will maximize their evaluation measures. Thus, the fundamental control problem is to find enforcement rules for the divisional managers and operating rules for the Center such that, when the managers max~m~ze their evaluation measures and the Center follows its operating rules, all the agents' decisions solve the decision problem D.

Now under the ~ priori information assumptions for Problem D, since the Center knows only the functions f o (·) and go(·), in order to compute evaluation measures and its own decisions Xo non­arbitrarily, it must acquire some information from the divisions. This requirement complicates the control problem since the evalua­tion rules must not only induce the managers to take optimal de­cisions xi but also to communicate optimally. This is frequently referred to as the "revelation problem" or the problem of inducing the managers to communicate "truthfully" or "honest1y"-Le. to send messages the operating rules of the Center require for optimal decisions to be taken.

Thus, to define the control problem a communication process must be formalized. In this pa~er we consider a very general abstract communication process. Let M denote an abstract set called a language; each element mi in M denotes a possible message division manager i can send the Center. An element mi may de­note a single message sent at one time to the Center or an entire sequence of messages sent in a lengthy iterative communication pro­cedure. 10 In any case, the Center acquires an I-tuple m = (m1, •.. ,mI) of divisional messages which it uses to compute its decisions xo.

Concerning the sequencing of message, decision, and evaluation operations, in this paper we assume that the messages mi from the divisional managers are sent first; next, the Center computes its decisions xo; then, the division managers compute their decisions xi; and finally, the Center computes the divisional evaluation measures. The reason for specifying that the Center makes its decision first is that this permits the division managers to base their own decisions on the value of x selected by the Center. 11

o

At the time the Center takes its decisions xo ' its only infor­mation is the I-tuple of divisional messages m and its ~ priori

CONTROL OF DECENTRALIZED ORGANIZATIONS

knowledge of f o{') and go{')' Thus, any decision rule for the Center is a function xo{') of mE MI (supressing f o(') and go(')

153

as arguments). However, at the time the Center computes the evaluation measures all agents have taken their decisions and hence it is reasonable to allow the divisions' evaluations to depend on the realized values of the divisions' contributions to total payoff, fi{Xi'xo), i = 1, •.• ,1, as well as the other information of the Center--the messages m E MI and the functions f o (') and go(')'

Although the evaluation measure for any division i may depend on the realized contributions to total payoff of other divisions, fj{xj,xo), j 1 i, accountants have emphasized that for motivational and other reasons a manager's evaluation should be based on controllable performance on1y--that is, on performance attributable or responsive to the particular manager!s decisions and not other managers' decisions. 12 While this is a vague pre­scription and is achievable in the strictest sense only when the Center has no decisions to make {i.e. the divisions are completely independent),l3 we will interpret the dictum by restricting the evaluation measures to depend only on the individual divisions' realized contributions and the joint message of all division man­agers. Thus, any evaluation measure for division i is a rea1-valued function Ei of the joint message m = (m1"" ,mI) E MI and the i th division's realized contribution to total payoff fi(xi,xo)' Not allowing a division's evaluation measure to depend on other divisions' realized contributions eliminates using "profit-sharing" as an evaluation measure. 14

Summarizing, a control mechanism is defined to be a triple

C [M'Xo(·),(Ei(·»~=l} consisting of:

a) a language M, I No b) a decision rule for the Center Xo : M .... lR ,and c) I divisional evaluation measures

Ei : lR x MI .... lR where the first argument of Ei (.) is the

i th division's realized contribution to total payoff fi(xi,xo)'

Given a control mechanism C = (M,xo('),(Ei('»i}, the i th division's evaluation depends on its own decisions Xi and the I-tuple of all divisions' messages m; Ei[fi(xi,xo(m»;m]. The i th

manager is assumed to choose his message mi and decision Xi in an effort to maximize this quantity.

Now, a priori the ith manager knows fi(') and gi(')' Addi­tionally we assume he knows the control mechanism C or at least the

154 T.GROVES

language M, decision rule xo(')' and his own evaluation measure Ei(')' Further we assume that once the Center chooses its de­cisions xo(m) = Xo all divisions are informed of the Center's choice. (This information is, of course, not available to the divisions when they choose their messages.)

Since an arbitrary evaluation measure Ei will depend on the messages of all the divisions, mj' j = 1, •.. ,1, the i th division manager's best message mi and decision xi may depend on which messages the other divisions choose. However, for an optimal con­trol mechanism, we require that each division's best decisions xi depend only on the decisions taken by the Center, xo(m), and his best message, mi, be independent of the other divisions' messages. Thus, for an optimal control mechanism, a division manager needs no information about the other divisions' messages (nor, of course, . about their decisions either).15

Now it is possible to find control mechanisms for which a divisions' best message is not unique--even though they all lead to optimal decisions. Although choosing one of these best messages rather than another will have no effect on the particular division, it may affect the choice of the Center's d~cisions (in cases of multiple optima) and the value of the other divisions' evaluation measures. Since there would be no way to know which among multiple best messages a division manager would send nor to induce him to choose one rather than another, such control mechanisms are capri­cious in terms of the evaluations of the divisions. To eliminate such capriciousness, we require, for an optimal control mechanism, that if there are multiple best messages for the divisions then all lead the Center to pick the same decisions and lead to the same value of the evaluation measures for all divisions.

.. h d f· . 1 1 h· by·.16 Summarlzlng, t en, we e lne an optlma contro mec anlsm

Definition: An optimal control mechanism, denoted C, is a control

mechanism tM'~o(·),(Ei(·»i} such that:

a) it is decisive: for each division i = 1, ... ,1, there exists a deci~ion rule ~i(') (a function of xo) and a message mi E M that maximize

~l.[xl.(·),m] ~ i.[f.(x.(; (m»,; (m»,m] 1 1 1 0 0 ,. ,.

subject to g.(x.(x (m»,x (m) < 0 1 1 0 0 -

CONTROL OF DECENTRALIZED ORGANIZATIONS 155

b) it is efficient: for all I-tuples «;i'~i(·»)i=l satis­fying a), the resulting decisions (xo(m),(xi(xo(m»)i) solve the organizational decision problem D; and

c) it is impervious: if «mi,xi(·»)i and «mi,x{(·»)i both satisfy a), then

x (m) = x (m ') o 0

and w.[x.(·).m] = w.[x~(.),m/] for all i. ~ ~ ~ ~

The organizational control problem may be then simply stated as the problem of finding an optimal control mechanism given the decision problem D.

In order to have a meaningful problem we assume henceforth that problem D satisfies the following regularity conditions that are sufficient to guarantee the problem has a solution.

Regularity Conditions

A.l: The functions fj' j = 0,1, ...• 1 are upper semi-continu­ous functions and the functions gj' j = 0,1, ... ,1 are continuous functions;

A.2: The sets X == [x \g (x ) ~ OJ, X. == ex. \g.(x.,x ) < ° o 000 1 ~1~0-

for some x EX} for all i, and X.(x )=={x.\g.(x.,x )<O} G 0 ~o 1~~0-

for every Xo and all i are compact; . N·

A.3: Xl == ex \g. (x. ,x ) < ° for some x. E lR 1} is closed for o 0 ~ ~ 0 ~

all i;

A.4: ~ " 0. i=l

Under (A.l) - (A.4) problem D has a solution:

Proposition 1: Under (A.l) - (A.4)

TT.(X) == Maxt£.(x.,x )\x. E X.(x)} 10 ~10 110

is an upper semi-continuous function on xi n X . o 0

Proof: It is straightforward to verify that Xi (·) is a non-empty upper semi-continuous mapping on X~ n XO. Then, since f i (·) is upper semi-continuous, TT.(.) is also, by Berge's theorem [3,

1 Theorem 2, p. 116].

156 T. GROVES

Proposition 2: Under (A.1) - (A.4), problem D has a solution.

I Proof: By Proposition l,.ti=l TIi(xo) + fo(xo) is an upper semi-

continuous function on nx~ n Xo which is a compact set. Thus, " i l:iTfi (.) + f o (') at.tains a maximum, say xo ' on ~Xo' Furthermore,

by definition of TI i , fi(xi,xo) attains its maximum on Xi(xo) at, say xi' Note that (Xj) satisfies the constraints of problem D.

satisfying the constraints.

)n

Let (x j ) be any other decisions

Thus, since gi(xi,xo) ~ 0 for all i, i g (x ) < 0, x EX. Thus x E n X

Xo E xi for all i. Also, since o

00- 0 0 0 i 0 nx • In addition, since

o gi(xi'xo) ~ 0, xi E Xi(xo)' But, by

TIL' (xo) :> f. (x. ,x ). - 1. 1. 0

the definition of TI.(·), 1.

Thus

1: ~ Tf. (x ) + f (x ) ~ ~. f. (x. ,x ) + f (x ) . .1.:1. 0 00-1.1.1.0 00

Also, by the definition of x and x. o 1.

E. f . (x. ,x ) + f (x ) = 1:. ff • (X ) + f (x ) > I:.". (X ) + f (x ). 1. 1. 1. 0 0 0 1. 1. 0 0 0 - 1. 1. 0 0 0

Hence

l:.f.(x.,x) + f (x ) > t.f.(x.,x ) + f (x ) 1. 1. 1. 0 0 0 - 1. 1. 1. 0 0 0

for all (x.) satisfying the constraints. Thus (i.) solves problem D. J J

IV. A GENERAL SOLUTION OF THE CONTROL PROBLEM

IV.l A Class of Optimal Control Mechanisms

In this section a particular control mechanism is defined which is then proved to be optimal. Further, we prove following Green and Laffont [10] that given the mechanism's language and decision rule for the Center the evaluation measures are, in a sense, unique. Finally some extensions and limitations of these results are dis­cussed.

* To begin, the language M of the control mechanism is defined

CONTROL OF DECENTRALIZED ORGANIZATIONS

as:

* M

157

( ,.."i ""i No m. : X -+ JR \ 'it EX, a closed subset of JR ,mi ~ 0 0 0 (4.la)

where x o

IYi} is an upper semi-continuous function on X o

I En i=l

(see Regularity condition A.4).

Thus to specify the language it is necessary to know.!!. priori any feasible decision vector Xo for the Center. This seems a weak re­quirement. A message mi E M* is interpreted as a reported maximal divisional profit function, i.e. mi(xo) is the amount of divisional profit i reports he will contribute to the organization if the Center takes the decisions.xo and i takes his maximal decisions xi given xo' The domain ~ is interpreted as the set of Center's decisions Xo that admit a feasible and maximizing decision x ..

~

If we define a division's "true" maximal divisional profit function 'IT. by:

~

~.(x ) - Max(f.(xl,x )\x. E X.(x )} ~o ~ 0 ~ ~o

i Proposition 1 ensures that 'ITi is defined (at least) over Xo n Xo

- i and is also U.S.c. on this set. Furthermore Xo E Xo n Xo' Thus,

interpreting x~ as the largest closed domain over which 'IT. is u.s.c., 'ITi is thus an element of M''<, ~

* Given this language, the Center's decision rule xo(') is de-fined by:

* x (m) o

* x o I maximizes I:~=l m. (x ) + f (x ) +constant L ~ 0 0 0

(4,lb)

subJ'ect to x E X = {x \g (x ) < o} , o 0 . 0 0 0 -

for any constant,

Since Xo En Xi n X , Xi is closed, and Xo is compact, Qxin Xo is i 0 0 0 L

a non-empty compact set. Further, since mi, all i, and fo,2.re U,S.c. functions, timi(xo) + fo(xo) attains a maximum on ox~nxo' Although this maximum may not be unique, the rule x~(m) pIcks a specific maximizer, and furthermore picks the same maximizer for all m' such that mj = mj + constant. The decision rule x~(·) is obviously interpreted as the rule maximizing the total organiza-

158 T. GROVES

tion's reported profits.

Suppose, in particular, that the divisions all send their "true" maximal profit functions 'lTi and the Center takes the de-.. A_* .. '" i .. A

C1S10ns Xo = xo('IT). S1nce Xo E Xo n Xo for every 1, fi(xi'xo) attains its maximum at some point (not necessarily unique), say, xi in Xi(xo); that is 'lTi(Xo) = fi(Xi,Xo)' The proof of Proposi­tion 2 then establishes immediately that the decisions

(Xj)~=o are a solution to problem D.

Thus, if the divisions send their "true" frofit functions 'IT. and then, given the Center's decisions ~o = ~~(TI), maximize thetr own profits fi(Xi,X o) (subject to feasibility), the resulting de­cisions solve problem D. rt is the role of the evaluation meaSures E1 to induce this behavior, i.e. give the division managers an in­centive to send these messages and take these decisions. Note again that because of the sequencing of~decisions, manager i will know the Center's decisions Xo = x~(m) at the time he must make his decisions xi and thus is able to maximize fi(xi'x o) sub­ject to xi E Xi(xo) if Xi(xo) is non-empty.

* Finally, the evaluation measures Ei (·) are defined in terms of:

o * - * E.(f.,m) =f. +L: . ..,l.[m.(x (m» -m.(x )] +[f (x (m» -f ex)], 1 1 1 JT1 J 0 J 0 0 0 0 0

all i (4.lc)

The measure E~ thus evaluates the i th manager on the basis of his division's realized profits plus the sum of the deviations of expected reported profits from reported profits at Xo of all the other divisions and the deviation of the expected profits of the Center from profits at xo when the Center takes the decision x*(m). However, as is shown below, this particular evaluation measureo

is only one of many' optimal evaluation measures. Specifically, we define the class a~ of evaluation measure <E~('» by:

1

if< - [(E:('»iIE:(fi,m) =ai(m\mi)E~(fi,m) +Si(m\mi )} (4.ld)

where a. (.) is any strictly positive and 1

S0°) is any arbitrary function of all divisions' h . ttl h . messages except t e 1 t at 1S constant on

the sets [m/\m~lm~ = m. + constantt, 1 J J

where m\m. = (m., ... ,m. l,m.+l , ... ,mr ). An interesting particular 1 1 *1- 1 \ member of the class a' is given by setting a.(m m.) = 1 for all

1 1

CONTROL OF DECENTRALIZED ORGANIZATIONS

m\m., and ~

where

Then

L . ..,t. (m. (X- ) JT~ J 0

i i m. (x » + f (x ) - f (x )

J 0 0 0 0 0

ii, Xo xo(m\mi ) maximizes ~j"li mj(xo) + fo(xo)

subject to g (x ) < 0. 00-

< i i * Ei(fi,m) =fi - tEj"li mj (x o) +fo(xo) -[Lj"li mj (xo(m»

159

+ f (/c (m) )] }, (4.2)

o 0

and it is easy to see that Ei is a member of Ilc i since Xo maximizes

~j"li mj(xo) + fo(xo) + constant, subject to go(xo) ~ ° for any

constant. The measures Ei(f,m) thus evaluate the i th division on the basis of his division's realize.d profits less the total ex­pected impact on all the other div:t'sions' reported and the Center's realized profits attributable to the i th division's message. In the special case when division i's.message does not affect the Center's decisions, they will be x~(m\mi) and division i will be evaluated solely on its own realized profits. Otherwise it is assessed for the expected reported impact on the rest of the organ­ization by causing the Center to choose x*(m) instead of xi(m\m.).

o 0 ~

Given a control mechanism C';C = jM'\x:(.),<E~(·»d defined by (4.la-d) where (E~('» is in the class a* of the evaluation measures, the decision problem confronting the i th division manager is:

D':: Choose a decision rule x~(.) as a function of the ~ ~ * Center's decisions Xo and a message mi such that

"k ~':'k ",;'( wi[xi('),m/mi ] = Ei[fi(xi(xo(m/mi» xo(m/m i »]

is maximized subject to g.[x.(/c(m/m.»,x"c(m/m.)] ::: 0, ~ ~ 0 ~ 0 ~

i.e. x. (x';c (m/m.» E X. (x * (m/m.», for every m\m. E M,cI-l. ~o ~ ~o ~ ~

As shown above, if m. = TI, and x.(xo) maximizes fi(xo,xo) ~ ~ . ~ -'-

subject to xi E Xi(xo) for all Xo E X~, then the decisions (x~(m), <Xi(X~Cm»)i) solve problem D. Thus if (mi,xiCo» is the only solution of the i th manager's decision problem, then

160 T. GROVES

f'k* (*)} AA

t M ,xo('), Ei(') i would be optimal. However (mi,xi('» as de-fined above are not the unique solutions of Pkoblem Df. There­fore we first characterize all solutions of D~ (Theorem 1) and then show that all such solutions yield decisions that solve problem D (Theorem 2). These results will then establish the optimality of any C* in e* = [[M'\x~(·),(E!(·»dl(E!(·»i E a*} (Theorem 3) :

,,< The following theorem characterizes all solutions of D. :

~

Theorem 1: 17 * * The pair (mi,x i (·» maximizes

··k -k * OJ.[x.(·),ml subject to x.(x (m» E X.(x (m» ~ ~ ~ 0 ~ 0

f \ E M''< I -1 . f d 1 . f or every m mi ~ an on y ~

(a) * x.(x ) maximizes f.(x. ,x ) on X.(x ) for every ~o. ~~o ~o

x E X~ n X 000

and

i.: i (b) m. (x ) = ff. (x ) + constant for

~ 0 ~ o. every Xo E Xo n Xo and is

(Xi)c. h 1 undefined on (X~)c n X , where • 0 0

o ~s t e comp ement

of the set X~. o

To prove Theorem 1 we first establish two Lemmata:

,,< Lemma 1: The decision rule x.(·) maximizes

~

(Jt[ x. (.) ,ml subject to X. (x''< (m» EX. (x''< (m» ~ ~ ~ 0 ~ 0

for every m such that Xi(xo(m» is non-empty, i.e.

if x:(~o) max~m~zes fi(xi,x o) subject to xi E Xi(xo) Xo E X~ n X .

o 0

* xi(') = ~.(.) ~

for every

Furthermore, if x1(~o) do~s not max~m~ze fi(xi'xo) subject A A ~ A *I

to Xi E Xi (xo) .Jor some Xo E Xo n XO ' then there exists some m EM such that X. (x" (m» is not empty and

~ 0

* A A * * A OJ. [ x. ( . ) ,ml }> OJ. [ x . ( . ) ,m/m.l ~ ~ ~ ~ ~

* * A for all m. EM such that X.(x (m/m.» is not empty. ~ ~ 0 ~

Proof of Lemma 1: (first pa~t - if) is not empty, i.e. xo(m) E X~ n X .

o Then

Let m be such that X. (x*(m» ~ 0

CONTROL OF DECENTRALIZED ORGANIZATIONS 161

ok "k "';~

W.[x.(o),m] - W.[x.(o),m] 1 1 1 1

* *"k * * = a.. (m\m.)[ f. (x. (X (m», X (m» - f. (x. (X (m», X (m»] 1 1 110 0 110 0

\ Max * * >'< = a.. (m m.)[ EX ( >'< ( » f. (x. ,x (m» - f. (x. (x (m», x (m»]" 0

1 1 Xi i Xo m 1 1 0 1 1 0 0 -

i( * for every xi(o) such that xi(xO(m» E Xi(xo(m» 0

(second part) 0 Let m. = TT. and let ';(. maximize TTi (xo) + fo(xo ) i ,....,1. ~ ~ 0,.. .....

on Xo n XO. Define A = TTi(xo) + fo(xo) and let (m\mi) be any ( 1) . *1-1 1- -tuple 1n M such that

1 0 + f (x ) + E where E > 0 for x

o 0 0

2: . ..,1. m. (x ) = JT1 J 0

A

X o

{

'Ii. (x )

-A i for a 11 x E X n X , x of X o 0 0 0 0

(Clearly there exists such an (m\m.) in M*I-l)0 1

It follows simply that xo maximizes uniquely. Hence x~(m) = XOO Let xi(xo)

* ... 1.

Now, for any mi E M such that Xi(x~(m/mi» is not empty,

= a.. (m\m. ) t f. (x. , x ) + ~ . ..,1. m. (x ) + f (x ) 1 1 1 1 0 JT1 J 0 0 0

ok "k,.. "k" ,... "k,.. - L[x.(x (m/m.»,xo(m/m.)] - 2: • ..,1. m.[x (m/m.)]

1 1 0 1 1 JT1 J 0 1

~~ ,.. * There are two cases to consider: Either (1) xo(m/m.) = x maximizes 1 0

TT.(x) + 2: . ..,1. m.(x ) + f (x ) on X , or (2) it does not. 1 0 JT1 J 0 0 0 0

Case 1: In this case x~ = xo as Xo is the unique maximizer. But then

.,~ "k "k ,... w'1:[x1.(o),m] - W.[x.(o),m/m.] =

1 1 1

= a.(m\m.)[f.(x.,x) - f.(x.(x ),x)] :> 0 111101100

since x.(x ) does not maximize f.(x.,x ) over X.(x ) but x. does. 10 110 10 1

Case 2: In this case

162 T. GROVES

* '" ,.. * ~"': AI W 0 [ x. ( 0 ) ,m] - w. [ x. ( 0) ,m m.] 1 1 1 1 1

= a.(~\~.)['Ii.(X ) +2: . ..". ~.(x ) + f (X ) 1 1 1 0 JT1 J 0 0 0

f (x.(x'~),x*) -t . ..". ~.(x'~) - fo(x~)] i 1 0 0 JT1 J 0 A ,.. "k ,.. "'k

> a.(m\m.)[n.(x ) +L: . ..". m.(x ) + 1 1 1 0 JT1 J 0

,,(

f (X ) o 0

,~* ,.. * - f.(x.(x ),x) -I: . ..". m.(x)

1 1 0 0 JT1 J 0 * - f (X )] o 0

.... \ ,.. Max "k -k o.J( = a.(m\m.)[ X ('~) f.(x.,x) - f.(x.(x ),x)] > O. 1 1 Xi E i Xo 1 1 0 1 1 0 0

Thus, in either case w~[x.(.),~] :> 1.1t[X>'~(o),rrJ/m.].\ 1 1 1 1 1

Lemma 2: The message m. maximizes 1

* ,.. ,.. ,,\ .;: 1.1;'[x.(·),m] subject to x.(x (m» E X.(x (m»

1 1 1 0 1 0

\ -'·1-1 for every m m. E Mft if and only if 1

m~(x ) = 'Ii. (x ) + constant for every x E Xi n X 1010 000

and is undefined on (Xi)c n X . o 0

Proof of Lemma 2: (if)

w'l:<[X 1.(o),m/m'l:] = a.(m\m.)[n.(x'\m/m>'~» +L: . ..". m.(x'\m/m':» 1 1 1 0 1 JT1 J 0 1

+ f (/< (m/m':») + 13. (m\m. ) 0

o 0 1 1 1

But since m~(x ) 1 0

,~ "k ".(x) + constant, X (m/m.) maximizes

1 0 0 1

i 'Ii. (x ) + L: . ..". m. (X ) + f (X ) on X n X .

1 0 JT1 J 0 0 0 0 0

Thus

IJ/:[x.(.),m/m':] = a.(m\m.)[Max (ft. (x ) +t: . ..". m.(x) 1 1 1 1 1 Xo 1 0 JT1 J 0

+ f (x ») + S.(m\m.) o 0 1 1

~ a.(m\m.)[n.(/«m/m.» + t: . ..". m.(x'\m/m.» - 1 1 1 0 1 JT1 J 0 1

+ f (/< (m / m . ) )] + 13. (m ,m. ) o 0 1 1 1

"k A

w.[x.(.),m/m.) for any m. such that 1 1 1 1

for any m. such that X.(x*(m» is not empty. 1 1 0

CONTROL OF DECENTRALIZED ORGANIZATIONS 163

of<

(oyly if): Suppose m. ~ ff. + constant on X. Then there

exist Xo and x~ in Xo such I.that 1. 0

'~l '~2 1 2 m. (x ) - m. (xO) :> n. (x ) - n. (x ). 1.01. 1.01.0

Now let A be any number satisfying

and B be defined by:

= Minfn. (xl) Max

B -xoEx~nxo

[ff.(x) + f (x )], . 1. 0 1. 0 o 0

,~ 2 Max * m. (x ) +A - -i [m. (x ) + f (x )]} 1. 0 xoEXonxo 1. 0 o 0

A M,,<I-l

Let (m\m.) be any (I-I)-tuple in such that 1.

1 if 1

-f (x ) x x o 0 0 0

E .=/. m. (x ) 2 2

-f (x ) +A if x x J 1. J 0 o 0 0 0

B otherwise for all x E X o 0

[Clearly such an (m\m.) exists in M'kI - l ] . 1.

It is easy to verify that x:(m/ffi ) = x~ and x:(m/m:)

Hence

"k"k A -k-k Ai":

w.[x.(·),m/orr.] - W.[x.(·),m/m.] 1. 1. 1. 1. 1 1.

_ ~\A) [ 2) L: " 2) ( 2) - a. (m m. orr. (x o + . ../.. m. (x + f x

1. 1 1 JT1 J 0 0 0

- TT. (xl) - L: . ../.. m.(xol ) - f (xl)] 1 0 JT1 J 0 0

" ,,2 1 1 Q,. (m\m. ) [TT. (x ) + A - orr. (x) > 0

1 1 1 0 1 0

which contradicts

ject to x.(x;Cm» 1

the assumption that m: maximizes w':[/«·),m] .'_ '~I-l1 1

E X.(x;(m» for every m\m. E M 1 1

sub-

Finally, if m:(xo) is defined for some ~o E (X~)c n Xo, then

for (ffi\mi) defined as above with ~o substituted for x~ (in defini-~ "k""k ,,.,.,.. ,........ ··k",·k.

t10n of B also), x (m/m.) = x and X.(x ) = X. (x (m/m.» 1S empty. 01.01010 1.

164 T. GROVES

We can now prove Theorem 1:

Proof of Theorem 1: (if) Follows immediately from the if parts of Lemmata 1 and 2.

(only if). Suppose not

(a) If x~(xo) does . 1

Xo E X~ n Xo ' the second cannot be a solution for

not maximize fi(xi'xo ) on Xi(xo) part of Lemma 1 establishes that

*

for every * (mi,xi('»

any mi EM.

maximize fi(Xi,xo) on Xi(xo) for eve~y f ni(xo) + constant for every Xo E X; n Xo

A\ A ... I-l then by the second part of Lemma 2, there exists some m mi E M" such that

"k~~,. "i,("k,..* W.[x.(·),m/n.]· > lI).[x.(·),m/m.] . 1 1 1 1 1 1

,,< Also, by Lemma 2, m.(x o) cannot be defined for any

1 A\" 'k " otherwise there exists an m mi such that Xi(xo(m» QED Theorem 1.

-!( *

i c Xo E (Xo ) n X o is undefined.

or

Next we show that any I-tuple of pairs (mi,xi('» characterized in Theorem 1 yield decisions solving the organization decision problem D.

* ok Theorem 2: If for~every i, (mi,xi('» satisfy a) and b) of Theorem 1, then (x;(m*),(xi(x;(m""»)i) is a solution of problem D.

Proof: /";'(I _ 'k"it: 'k 'k "k

Let \x.>._ = (x (m ),(x.(x (m »).). J J-o 0 1 0 1

Si .. nce m'1: (xo) is defined on Xoi n Xo and undefined on (Xi) c n X * ~ * 0 0 Xo = xo(m') is defined and satisfies the constraint g (x ) < 0, "i':: • -J~ • ";'( o .. ~ 0,,;," -

i.e. Xo E XO' Also, S1nce Xi(Xo) 1S not empty, xi = xi(xo) is de-fined and satisfies the constraint g. (x. ,x''<) < 0, for all i.

1 10-~ I

Let (x.) be any other decisions also satisfying the con-straints. J j=o Then

"k 'k 'k l,. f. (x. ,x ) + f (x ) - 1:. f. (~.,~ ) - f (~ ) 1 1 1 0 0 0 1 1 1 0 0 0

> ~.'f!.(x''<) + f (/') - r:. ,..,MaExX ("'"') f.(';('.,';(') - f (~) - 1 1 0 0 0 1 X. . X 1 1 0 0 0

1 1 0

r:.'f!. (x*) + f (x*) E.'f!.(i) - f (~ ) 1100011 0 0 0

Max [~.n.(x) +f (x)] -~.n.(~) -f (~)'> 0. Q.E.D. xoEXo 1 1 0 0 0 1 1 0 0 0 -

CONTROL OF DECENTRALIZED ORGANIZATIONS

* * Theorem 3: Any control mechanism C in the class e is optimal where by (4.la-d)

e* £(M*,x*(.),(E*i(·».}\(E~(·», Eci*} (4.3) o ~ ~ ~

* Proof: Theorem 1 establishes the decisiveness of C (a) and Theorem 2 establishes the efficiency of C*(b). Thus, we need only its imperviousness (c).

165

Let «m.,x.(·»). and «m~,i~(·». satisfy a) of the definition ~ ~ ~ ~ ~ ~

of optimality. Then, by Theorem 1,

mi = m~ + a i on X~ n Xo for some a i constant

and

i.(x) and ~ 0

x~(xo) maximizes fi(xi,xo) on X.(x ) for every

x E X~ n X • a8d 0

* ~ 0* ~ * ~I * Thus, by the definition of x (.), x (m) = xo(m) =x , o 0 0

* ~ * * ~, ~, lll.[x.(·),m] - l1l i [x.(.),m ] ~ ~ ~

""'\"\ '" * * ,.. * ,.. - * -= a..(m\m.)[f.(x.(x ),x) +!: . ..,l.[m.(x ) -m.(x )] +f (x) -f (x )] 1 1 1 1 0 0 JTl J 0 J 0 0 0 0 0

~ '\"'" I A I * * ,.. * A -- a..(m m )[fi(x.(x ),x ) + 2: • ..,l.[m.(x ) + a. - m.(x ) - a.] 1 ~ 0 0 JTl J 0 1 J 0 ~

* + f (x ) - f (x )] = 0 o 0 0 0

since a.. (m\m.) is constant on £m\m.\m. = m~ + constant} and ~ 1 1 J J

f.(x.(x ),x ) = f.(x~(x ),x ) for every x E xi n X and llO 0 llO 0 000

* i x E X n X. Q.E.D. 000

IV.2 A Partial Characterization of Optimal Control Mechanisms

* * Although any control mechanism C in the class e is optimal by Theorem 3, one might wonder if e* contains all optimal control mechanisms. A partial answer to this question can be given based on some recent work of Green and Laffont [10] .18 Denote by '"" e the class of a 11 control mechanisms C = tM,x (.), :,'E. (.) > . 1 such

o 1 1 that

166 T. GROVES

E.(f.,m) = y.(m\m.)[f. + T.(m)] ~ ~ ~ ~ ~ ~

(4.4)

I I-I where T. : M ... JR is an arbitrary function and y. : M ... JR-'-L

~ . ~ TO

is an arbitrary strictly positive function. It is easy to see that e* is a subclass of mechanisms in~. Now, one can show tha~ if C in ~ is optimal, then there exists a control mechanism in C in e* that is equivalent to C in the sense that C* leads to the same optimal decisions Xo for the Center and the same evaluations of the divisions as C. Thus, the class e* spans the equivalence classes of optimal control mechanisms of the form C.

This result is given as Theorem 4 below. However to establish it, several characteristics and properties of control mechanisms are defined and several preliminary propositions established.

Definition: Given any control mechanism C in which the language M is the space M* [c.f. (4.la)], the mechanism is called truth­inducing if and only if the pairs (m!,x!('» [c.f. Theorem 1] are the unique message-decision rule pairs maximizing Wirxi(xo(m»,m]

subject to x.(x (m» E X.(x (m» for every m\mi € M*I-l. ~ 0 ~ 0

* * Theorem 1 establishes that every control mechanism C in e is truth-inducing.

Another pr£Perty of mechanisms is defined for a subset e of mechanisms in ~ where

e = (C E ~Ia) M = M*, b) x (.) = x*(')' and o 0

c) y.(m\mi ) and T.(m) are constant on the sets ~ ~

(m l e M*Ilm~ = m. + constant for all j}} J J

The property is defined as:

Definition: A control mechanism C in ~ satisfies property 0 if and only if, for all i

a)

b)

* Ti(m) is independent of mi at xo(m); *1 ,., * * (m/mi ) E M and mi EM, xo(m/mi )

T.(m/m.) T.(m/m.). ~ ~ ~ ~

i.e. if, for

x~(m/mi)' then

T.(m/m.) - T.(m/m.) * * Lj~imj[xO(m/mi)] + fo[xo(m/mi )] ~ ~ ~ ~

- tj~imj[x~(m/mi)] - fo[x~(m/mi)]

CONTROL OF DECENTRALIZED ORGANIZATIONS 167

f 11 I E M*I ~ E M*. or a m m. , m. 1. 1.

'1< Lemma 3: A control mechanism C in ~ is in ~ if and only if it satisfies Property O.

Proof: This is straightforward to verify.

Using shown it is

the characterization of ~;< provided by property 0, it can be that any mechanism C in ~ is truth-inducing if and only if in ~''<.

~

Proposition 3: A control mechanism C in ~ is truth-inducing if and only if C E ~* .

Proof: (if). This is just Theorem 1. ~

(only if). By Lemma 3 if C satisfies property 0, then it is in ~''<. Thus, we consider the negation of both parts of the definition of property O.

1) Suppose for some i that Ti(m) is not independent of ;< I ""(I A 't~ mi at xo(m); then there exist (m mi) E M' ,m. E M

"" ";,........ -k A~ with mi t mi suc~ that x~ = x~(~/mi) = xo(m/mi) but Ti(m/mi) > Ti(m/mi)' Let TIi = mi' (Clearly there exists some problem D satisfying A.l-A.4 such that TI. = m .. ) But then

1 1.

"k'k ··k -k "'k 'k y.(m\m.)[f.(x.(x ),x ) +T.(m/m.) -f.(x.(x ),x) -T.(m/ff.)]

1 1. 1. 100 1. 1. 1. 1. 0 0 1 1.

= y.(m\m.)[T.(m/m.) - T.(m/n.)] » 0, 1. 1. 1. 1. 1. 1

A

which contradicts the fact that C is truth-inducing.

2) Suppose for some i that C does not satisfy b); then there exist (I ) E weI A E M"< such that m mi ' mi

T. (m) - T. (m/m.) = 2:: • ...,£. [m. (/( (m» - m. (/< (m/m. ) )] 1 1. 1. JT1. J 0 J 0 1

+ f (x*(m» - f (x*(m/m.» + E for E > O. o 0 0 0 1.

168 T.GROVES

Now suppose

TT. (x ) = 1. 0

* * -L: • ..,t.m.(x (m» -f (x (m» JT1. J 0 0 0

*,.. *,.. -t . ..,<.m.(x (m/m.» -f (x (m/m.» +6 JT1. J 0 1. 0 0 1.

- Max [L: • ..,t.m.(x) -f (x)] - E x X JT1. J 0 0 0

o 0

* for x = x (m) o 0

* /" for x = x (m m.) o 0 1.

* for x 'f x (m), o 0

* /" x 'fx (m m.) o 0 1.

* where 0 < 6 < E. Since TT. (.) is upper-semi continuous, TT. EM. 1. 1.

(Also, clearly there exists a problem D satisfying A.l-A.4 such that this holds.)

* "*,, It is easy to verify that xo(m/TT.) = x = xo(m/mi) and thus, 1. 0 * * by part 1 above, T.(m/TT.) = T.(m/m.). Thus, letting Xo = xo(m).

1. 1. 1. 1.

* * * ,,* w.[x.(x ),m] - udx.(x ),m/TT.] L L 0 1. L 0 L

\ *" A = y.(m m.)[TT.(X) - TT.(X) + T.(m) - TL.(m/mL.)] L L 1.01.0 L

+L: . ..,t.m.(x*)+f (x*) -I: . ..,<.m.(i) -f (i )+E] =y.(m\m.)(E-6)>0 JTL J 0 0 0 JTL J 0 0 0 L 1.

which again contradicts the fact that C is truth-inducing. \

Proposition 4: Given~any optimal control mechanism C in ~ such that the language M=M~, it is truth-inducing if and only if it is also in C*.

* Proof: (if). By Theorem 3, any mechanism in C is optimal and by Theorem 1 it is truth-inducing.

* (only if). We need only show that xo (·) = xo (·)' i.e. satisfies (4.lb), and that Yi(m\mi) and Ti(m) are constant on the sets

£m' E M*I\m~ = m. + constant for all j}, since then C is in ~ and thus by Pro~ositlon 3 is in C* since it is truth-inducing.

~ * Since C is truth-inducing, mi = TTi + constant are i's best messages. And, since the mechanism is optimal

x (TT) maximizes L:. TT . (x ) + f (x ) over X for every TT E M*I o LL 0 00 0

CONTROL OF DECENTRALIZED ORGANIZATIONS

(Le. for every problem D). Also, since the mechanism is imper­vious xo(m*) = xo(m*') for all best messages m* and m*' i.e. for

* mt = TIi + ai and m1' = TIi + ai· Thus xo (·) = xo(·)·

That Yi(m\mi) and Ti(m) are constant on the sets

(m' E M*Ilm! = m. + constant for all j} follows from the imper­viousness 01 an ~ptimal mechanisms and that these sets are the sets of the division's best messages. I

Theorem 4: If C = (M,xi·),(Ei(·»i} mechanism in the class e, then there and a mechanism C* in e* such that:

is any optimal control * exists a function \\f:M .... M

x (m) o

E. (f. ,m) 1 1

* x ('l:' (m» o

* E. (f. , 'l:' (m) ) 1 1

I "" for a 11 m E X M.

i=l 1

169

where Hi = (mi E MI (mi,xi(·» solves i's decision problem for some orga izational problem D with (f(.),g(.» such that TIi (·) E M*} and 'Y (m) == [1\1 (ml ),.··, HmI )]·

Proof: Note first of all that since C E e, Ei(fi,m) = Yi(m\mi)[f i + Ti(m)] so that (mi,xi('» solves i's decision problem only if xi(xo) maximizes f.(x.,x ) on X.(x ).

1 1 0 1 0

* Now, for every TIi EM, let ~ (TIi) be defined as the set of i's best messages under C. That is, since C is optimal, and mi E ~ (TIi), then (mi,xi(·» solves i's decision problem when i is characterized by the functions (f·(·),gl·(·» such that

~ . 1 fi(xi(xo),xo) = TIi(xo) on X~.

* Let Hm ) = (~ (m~), ... ,~ (m~» and define

* * x (m ) o

* x (Hm » o

* * * T. (m ) = T. (91 (m » 1 1

* * * * * yi(m \m i ) = y i (9l(m )\~ (mi »

for every m* E M*I .

* Note that xo(') is well-defined since C is optimal and thus im-pervious which implies that if mi and mt are in ~ (mt) for all i, then xo(m) = xo(m') == xo. Also, since C is impervious, if m. and mi E ~ (m~) for all i, Wi[xi(·),m] - Wi[xi(·),m'] = 0 w~ich implies that:

170 T.GROVES

y.(m\m.)[Tf.(x) + T.(m)] = y.(m'\m~)[TT.(X ) + T.(m')] L L LO L L L LO L

But note that since both m. and m! maximize W[xi(x (m)),m] L L 0

= Yi(m\mi)[Tfi(xo(m)) + Ti(m)], they also maximize Yi(m\mi)[TTi(xo(m))

+ ai + Ti(m)] for any constant ai. Thus imperviousness of C implies that for any constant a.

L

y.(m\m.)[TT.(X) + a. + T.(m)] =YL·(m'\m~)[TT.(X) +a. +T.(m')l. L L LO L L L LO L L

* Hence Ti (.) and

Now let 1\1 (mi) be any element of Then, since both mi and any m/. E ~.I\I ni = 1\1 (mi ) , i.e. mi and mi € ~ (ni)'

-1 ~ ~ (mi) for every mi e Mi· (mi) are best messages for

and

Thus

* * x ('1' (m) = x [ ~ ·If (m»] = x (m), T. ['1' (m)] o 0 0 L

* y.['¥(m)\ 1\1 (mi )] = y.(m\m.). L L L

E. (f. ,m) = y. (m\m L·)[ f. + T. (m)] 1 L L L L

= T. (m) L

* * = y. ['1' (m)\ 1\1 (m.)][ f. + T. ('1' (m)] - E. [ f. ,'1' (m)]. L 1 L L L L

Claim: * * * * * C = (M ,x (.), (E. ( • » .} is in e • o L L

* Proof: If we show C is both truth-inducing and optimal, then by Proposition 3 it will be in e*.

Note first of all that (mi,xi(·» solves i's decision problem under C* only if Xi(Xo) maximizes fi(xi'xo) on Xi(xo). Thus, if C* is not truth-inducing, then for some i, there exists an mi 1 TTi + constant E M* and m*\mi E M*I-1 such that

* ~ * ~ * ~ * ltd x. ( .) ,m 1m.] > w.[ x. ( • ) ,m In.], or L 1 1 1 1 L

* * *~ *" * * * * E.[TT. (x (m Im.»,m 1m.] > E.[TT. (x (m ITT.»,m/n.] or LLO L L L10 1 1

E. [TT. (x (Hm *) I~(mi))) ,Hm *) I~(m.)] L 1 0 1

* * > E.[n.(x (\f1(m )/~(TT.»).,~(m )/~ (TT.)] 1 1 0 L L

CONTROL OF DECENTRALIZED ORGANIZATIONS

which contradicts the fact that any (m[,ii('» where m[ E ~ (TIi ) is a best message-decision rule pair for i under C. Thus, C* is truth-inducing.

171

Sin£e it is truth-inducing C* is decisive. Also C* is efficient since x~[~(m)] = xo(m) maximizes ~ini(xo) + fo(xo) over Xo since C is e~fici;nt and ii(xo) maximizes fi(Xi,Xo) on Xi(xo) for all Xo EX;. C" is, finally, impervious since C is. This establishes the claim and hence Theorem 4.

Although by Proposition 3*a control mechanism C E 6 is truth­inducing if and only if C E ~, it is not true that if C is merely optimal that it is also in 6*. Since all mechanisms in 6* are both optimal and truth-inducing, this means that there exist optimal mechanisms C in 6 that are not truth-inducing even though the language of any C is the set of professed profit functions and the Center's decision rule is to maximize the sum of reported (plus its own) profits (subject to feasibility).

A particularly simple optimal, yet non-truth-inducing, member of 6 may be defined by:

and

1 * 1 * M = M , x (.) = x (.)

1 E. (f. ,m) ~ ~

o 0

1 1 f. + h. (x (m» + I: . .../. . [ m. (x (m» ~ ~ 0 JT~ J 0

+ f (xl(m» - f (x ) o 0 0 0

m. (;Z )] J 0

where hi:Xo -+ lR is any arbitrary function such that Li=l hi(xo)= 0

for all xO' It is easy to show that Cl = [Ml,x~(.),<Ei(·»} is optimal and that its best messages are all of the form mi(') = [TI. (.) + h i (')]+ constant. Thus Cl is not truth-inducing.

~

Of course, for this example, it is obvious that knowing any mi(') one c~n determine ni(') up to a constant by subtracting h i (·) from mi(')' An interpretation of Theor~~ 4 is thatAthis is generally true. Given any optimal mechanism 6 (not just 6), by applying the function ~ to any mi one can determine the true TIi(') up to a constant. Thus the mechanism might as well be based on asking divisions to report their true profit functions TIi, maximizing re~orted profits, and then using some evaluation measures in a' to evaluate the divisions.

172 T. GROVES

IV.3 Total Profit Allocating Mechanisms

* A noteworthy feature of any (optimal) control mechanism C in e* is that in general the sum of the divisions' evaluations, even when the divisions are reporting "honestly" (as they have an in­centive to do under C*), will not equal the organization's total realized profits:

That is, in general

!:iE.(f.,TT) = L:.Yi(TT\TT.)[f.(x.,x ) + Ti(TT)] l. l. l. l. l. l. 0 (4.5)

~ t.f.(x.,i ) + f (x) or for that matter to tl..fl..(xl..'xo)' l. l. l. 0 0 0

where x o

* '" * ,. \ ) x (TT), x. = x.(x), y.(TT\TT.) = a..(TT TT. , and o l. l. 0 l. l. l. l.

T.(TT) = !: . ..,(.[TT.(x) -TT.6Z)] +[f (x) -f (;Z)] +~.(TT\ft.)/a..(ff\TT.) l. JTl. J 0 J 0 0 0 0 0 l. l. l. l.

[c.f. (4.1 c-d) and (4.4)]. In particular, in those cases in which y. (m\m.) = a.. (m\mi ) = 1 for all m\m., this means that l. l. l. l.

t.T.(TT) l. l. (n-l)[L. .[TT. (x ) - TT. (;Z )]

J J 0 J 0

+ [f (x ) - f (;Z)] + L:.~.(TT\TT.) ~ f (x ) or O. 00 00 l.l. l. 00

Now it turns out that, based on a theorem of Hurwicz,19 it is not possible to find arbitrary functions ~i(m\mi) such that equality in (4.5) holds for all m E M*I.

This result is important since many standard divisional accounting procedures and new SEC disclosure regulations requiring line-of-business reporting require that total organizational pro­fits be distributed or allocated or attributed to the various divisions for reporting purposes. Since it is not possible to find any C* with the total profit allocating property (i.e. such that tiEi(fi,TT) + fo(x) = L.ifi(xi'xo) + fo(xo»' then, by Theorem 4, whatever accounting measures are used to report the separate contribution of each division, these measures should perhaps not be used as evaluation measures since they will not be part of any optimal control mechanism.

It seems that the interest in having measures for allocating total profits among all divisions (and perhaps the Center too) originates in the view that total organizational achievement must

CONTROL OF DECENTRALIZED ORGANIZATIONS

be just the sum of the achievements of the organization's con­stituent parts. This view may be contrasted with the view that total achievement may be greater than (or less than) the sum of the individual achievements.

173

Such considerations, in any event, raise the question of the intrinsic meaning of the evaluation measure for an optimal control mechanism. That is, what significance, if any, does the mag­nitude of the evaluation measure in any particular instance have and can one meaningfully compare the values of the measures for two different divisions? This question is important since, pre­sumably, in order to motivate a divisional manager to maximize his division's evaluation measure some type of compensation scheme that is increasing in the evaluation measure must be used. Furthermore, it would seem desirable for equity considerations if there were some objective basis for distinguishing among the per­formance of different divisions on the basis of their evaluation measures.

While a full exploration of this issue is beyond the scope of this paper and has not yet been completely developed (c.f. how­ever, Groves and Loeb [16])* it is interesting to note that one of the optimal mechanisms in e may yield measures with some intrinsic meaning. Specifically, consider the mechanism

C = (M*, X~(.), (Ei(·»iJ where Ei is defined in (4.2) as:

E.<£.,m)=f. -(~ . ...t.m.(xi) + 1 1 1 JTl J 0

i * *} f (x ) - [r . ...1 • m. (x (m» + f (x (m»] o 0 JTl J 0 0 0

(4.6)

i . where Xo x~(m\mi) maXlmlzes ~jiimj(xo) + fo(xo) subject to go(xo) ~ O. Now, when all divislon manager's are responding to the incent!ves so that mj (.) = TTj (.) + constant for all j :f i: the measures Ei(fi,m) measures the opportunity cost to_th~ org~nlza­tion of having the i th division. In other words, Ei(fi,TT/mi» is the profit the organization would lose if the i th division could be and were abandoned. 20 Thus the measure Ei(fi,m) have an intrinsic opportunity cost meaning even though (except when no division can affect the Center's decision),

~.E.[f.(X.,x )] < ~l.fl·(xl·'xo) 1 1 1 1 0

for all m including TI and Xi' where Xo realized decisions taken by the agents.

Another approach to the issue of total profit allocating mechanisms has been developed by Hurwicz [21, 22] and Groves and Ledyard [13] in a quite different context. For the organiza-

174 T.GROVES

tional control problem the thrust of their results may be expressed as follows:

Consider the class ~+ of all control mechanisms

c+ = (M,x (·),(E.(·».} in which E.(f.,m) = f. + T.(m) o J. J. J. J. J. J.

and such that

!: E . (f. ,m) = I:. f. + f (x (m» J. J. J. J. 0 0

or

!: T . (m) = f (x (m» J. 0 0

I for all m E M

i.e. the evaluation measures are total profit allocating. 2l Now, since there does not exist any mechanism in ~+ that is optimal in general, we look instead for a mechanism in ~+ that we shall call satisfactory:

Definition: Any control mechanism C is satisfactory if and only if

a) it is stable: there exist an I-tuple «mi,xi('»)i of message-decision rule pairs such that for each i

w.[x.(o),m] == E[f.(x.(x (m»,x (m»,m] J. J. J. J. 0 0

:> w.[x.(o),m/mi ] - J. J.

for all (m.,x.(o» such that mJ.' EM and J. J.

x. (x (m/m.» E X. (x (it/m.» J. 0 J. J. 0 J.

b) it is efficient: 'for all I-tuples «mi'x i ('») satis­

fying a), the resulting decisions (x (m),«x.(x (m»).) o J. 0 J. solve the organizational decision problem D; and

c) it is impervious: if «~i,Xi(o»i and «mi,xi(o»)i both satisfy a), then

x (it) = x (m ') o 0

" [" ( ) "] - " ['" )" '] and Wi Xi . ,m - Wi Xi (. ,m for all 1.

In essence, a satisfactory mechanism is one for which a Nash (or non-cooperative) equilibrium of divisional strategies (message­decision rule pairs) exists and that is efficient and impervious.

CONTROL OF DECENTRALIZED ORGANIZATIONS

In contrast, an optimal mechanism is one for which a dominant strategy equilibrium exists and that is efficient and impervious. It is clear that an optimal mechanism is satisfactory but not necessarily vice versa.

Whether or not satisfactory mechanisms exist in e+ (that is, satisfactory mechanisms that are total profit allocating for problem D) in general is an open question. However, for some special classes of problems one can find such mechanisms. To illustrate such a mechanism, consider the following problem:

I Max 2:::1.'=1 TT.(x ) + f (x )

1. 0 0 0 x o

subject to g (x ) < 0 o 0 -

175

Problem DO is clearly a "reduced form" of problem D arising the division managers take the decision x~(xo) for every xo

when. in Xl..

o

Now consider the following restrictions on problem DO:

Restrictions on DO:

B.l: The functions TIi are continuous concave functions No ~

defined on 1R+ ,the nonnegative orthant of lR •

B.2: The function f (x ) is a linear function: o 0

B.3:

f (x ) = a - b 'x o 0 000

where b E 1R~ is strictly positive. o -t+

No' The set X = (x E lR I g (x ) < o} o 0 1 0 0 -

No the nonnegative orthant of lR

o Under these restrictions, problem D may be written as:

Max t 'I'!. (x ) - b No 1. 0 0

x ElR+ o

x • o

The restrictions on DO are, of course, very severe, especially with reference to the Center's functions f o (') and g (.). How-ever, these restrictions can be relaxed.22 0

Now for the problem Dl we can exhibit a satisfactory mechanism c+ that is total profit allocating. The language M+ is defined as:

176

+ The Center's decision rule x (.) is defined as: o

T. GROVES

(4.7a)

(4.7b)

Given the language and decision rule, a division's message may be interpreted as the increments (or decrements since mi may be negative in any component) in the levels of the decisions Xo that division i wishes the Center to add to (subtract from) the aggregate levels requested by the other divisions. Note that at a Nash equilibrium of messages (m)i=l' each division must be

implicitly requesting the same aggregate level of the decisions

x!(m) = ~mi or, in other words, they must all agree on the level.

It is,the role of the evaluation measures to ensure that this will be possible.

where

The evaluation measures E:(') are defined by: 1

(as required since C+ E e+) (4.7c)

= aib oL:.m. + y. r1 - l [m _ "(m\m )]2 _ o(m'\m )2} o JJ 211 it"" i i

with Y > 0, L:.a. = 1 and 1 1

1 Il i == Il(m\mi ) - 1-1 Ej:fi mj

2 _ 2 1 °i = oo(m\m i } == 2(1-1)(1-2)

= 1" ( -I 2 w.~. m. - ,JTl J

2 Il(m\m.)) •

1

An interpretation of the evaluation measures Et(·) is as follows: Let us call bo·x!(m) the "cost" of decisions xt(m) and Tt(m) the i th division's "cost" share. The evaluation measure thus subtracts from the realized "profits" of each division, TTi' a "cost" share consisting of a proportional share of the total "cost" plus a

CONTROL OF DECENTRALIZED ORGANIZATIONS 177

positive mUltiple (t) of the difference between the squared devia­tion of the division's message from the mean of the other divisions' messages (corrected for small sample bias) and the squared standard error of the mean of the others' messages (for small samples). Thus, given the aggregate level of the decisions requested by all divisions, l.jmj' division i's cost is larger as the amount it re­quests deviates from the average of the others' request and smaller the greater the squared standard error of the mean of the others' messages.

alternative interpretation of the control mechanism C+ from the alternative sPecification: For each message

No define a function hi :lR+ ... lR by:

(4.8)

Since each message mi defines such a function, a division's message mi can be interpreted as communicating the function hi(';mi)' Thus,

an alternative specification of the language M+ of C+ is:

H+ = {h'(';m.)\h.(x ;m.) satisfies (4.8)} 1 1 1 0 1

We call any message h.(·) in H+ a (quadratic) reported profit function. 1

Next, note that given the messages m E M+1 , the decision x~(m) = ~imi maximizes Eihi(xo;mi) - boxo' Finally, in terms of the functions h1·(·;mi), the evaluation measures Et(n.,m) may be

1 1 expressed as:

+ T. (m) 1

where

T +. (m) == a.. b • x +(m) + Max {r: [h (x 'm) a b x)) 1 1 0 O le • .J... ,. - •• •

-u JT1 J 0 J J 0 0

+ + Y 2 - ~J',.,L.[h.(x (m);m.) - a..b ·x (m») - -2 cr i T1 J 0 J J 0 0

(4.9)

With this specification, the control mechanism C+ may be viewed as a parametric representation of a mechanism in the class e* [see (4.3) and especially (4.ld»). Since C+ can be shown to be a satisfactory mechanism, a division's best message roi given the messages of the other divisions can be shown to be the parameters of a quadratic aPtroximation to the division's true profit function n. at the level x (m) = r.m. of the Center's decisions

1 0 J J •

178 T.GROVES

The cost share Tt(m) in the form (4.9) can be interpreted as assessing the i th division (i) its proportional cost share, plus (ii) the total reduction in the aggregate net reported profits (gross reported profits hj(·;mj) less proportional cost shares a·bo'xo) caused, in effect, by division i's request for a level ot decisions different from the average of the other's requests, less a positive multiple (y/2) of the squared standard error of the mean of the others' messages. [Compare this with the mechanism C, c.£. (4.6).J

. h f 23 h ' 1 h +. . We state, W1t out proo, t e resu t t at C LS a satLs-factory control mechanism and is total profit allocating:

Theorem 5: Under the restrictions B.l-B.3, the control mechanism C+ defined by (4.7a-c) is a satisfactory mechanism and is total profit allocating as well, i.e.

I + I + + ~i=lEi[TIi(xo(m)),mJ = ~i=lTIi(xo(m)) + fo(xo(m))

I + for every m E M such that x (m) > O. o -

It is important to recognize that under a satisfactory (but not optimal) control mechanism such as C+ a division's best message mi will depend on the messages of the other divisions. Thus to implement such a control mechanism some type of iterative adjust­ment process seems to be called for. However, by appending an itera­tive adjustment process to a satisfactory control mechanism opens up the question of what self-interested behavior of division managers would be. Essentially, under an iterative adjustment process, a division manager's strategy is a response rule--that is, a function describing the message to be sent by the manager at each stage of the iterative procedure, given the information acquired by the manager up to that stage. Under suitable conditions adjustment pro­cesses can be found for problems such as problem Dl such that, if the division managers follow Cournot behavior (i.e. send their best message assuming the others messages remain fixed), a Nash equili­brium will be arrived at, in the limit at least.

However, Cournot behavior has been frequently criticized as unrealistic, especially in games with relatively few players (as we might suppose to be the case for even large organizations such as a divisionalized firm). Thus, what constitutes reasonable behavior remains to be defined and furthermore, given a behavioral assumption other than Cournot behavior, whether or not a control mechanism with an iterative adjustment process can be found that leads to solutions solving the underlying optimization problem such as Dl remains an interesting problem.

CONTROL OF DECENTRALIZED ORGANIZATIONS 179

Two results in this area have been obtained by Dr~ze and de 1a Va11e~ Poussin [8] and Hurwicz [22]. Roughly speaking Dr~ze and de 1a Vallee Poussin were able to exhibit an iterative adjust­ment process and total profit allocating control mechanism such that the prescribed behavior leading to an optimal solution of the optimization problem is a maximin strategy for a player.

Hurwicz's results are largely negative. Viewing the problem as an n-person game where the strategies are response functions (not messages) he, in effect, asks if a Nash equilibrium of response functions leads to optimal decisions where the decision rule is given by some control mechanism (as defined in this paper) and the argu­ment of the decision rule is a fixed point of the Nash equilibrium of response functions. His results show that except for very special cases, if one confines the search to total profit allocating control mechanisms then!!.£ "non-manipulable" mechanism exists--that is, one such that a Nash equilibrium of response rules leads to optimal decisions .24

However, it should be noted that Hurwicz's results depends crucially on the requirement that the control mechanism be total profit allocating. Dropping this requirement completely changes the results since for any optimal control mechanism such as any c* in ~*, a division manager's best message is independent of the messages of the others and thus the constant response function giving this message dominates any other response function.

180 T. GROVES

Footnotes

1. We are concerned in this paper with the problem of designing the organization for decision making rather than decision implementation. That is, even for problems of type P that are small enough for one person to solve, implementation of the optimal decisions may require many persons. In its broadest scope the organizational design problem would be concerned with the implementation problem as well.

2. See Marschak and Radner [26] and the references cited therein.

3. See, for example, Arrow and Hurwicz [2], Koopmans [24], and Malinvaud [26], and Reiter [2B], a highly arbitrary sample.

4. See, for example, Dantzig [4], Dantzig and Wolfe [5,6], Geoffrion [9], and Jennergren [23], also an arbitrary sample.

5. Arrow, in his 1964 paper [1], mentions only Goode and McCarthy (complete references are not provided). The accounting litera­ture frequently refers to incentive problems as does the econo­mics literature in discussing both the competitive market system and the "free rider" problem in public goods and externality models. However, as far as I am aware, only Arrow [1] and Hurwicz [19,20,21] have discussed this issue in a general form with the orientation taken in this paper.

6. The relevant papers are Groves [11, 12], Groves and Ledyard [13,14], Groves and Loeb [15], Green and Laffont [10] and Loeb [25].

7. This definition of an organizational form is to be distinguished from that of team theory, c.f. Marschak and Radner [27, p. 124].

B. See Jennergren [23] for a detailed study of this model. Jennergren also discusses the incentive difficulties with the Dantzig-Wolfe decomposition algorithm which may be applied to a special form of problem A in which the functions F.(.) are linear and the constraints ~.G.(x.) < K are represenEed by

h 1 · . 1" 1 1 1 -t e 1near 1nequa 1t1es:

A.x. < b. 1 1 1

~.A .x. < b 1 01 1 0

9. See Hurwicz [lB]; a seminal paper formalizing concretely communication processes. Team theory and, more generally, statistical decision theory also formulate models with explicit, though abstract, informational processes.

CONTROL OF DECENTRALIZED ORGANIZATIONS

10. As required by various decomposition algorithms and de­centralized planning procedures; see reference at note 4

181

supra. Of course, for an iterative process there is another side--namely messages from the Center to the division managers. It is unnecessary for this paper to formalize this aspect. See however Groves [11].

11. In Groves [11] a similar model is discussed within the team theoretic framework which allows for simultaneous decision making under uncertainty. The effect of the decision sequencing assumption made here is to allow sufficient information to be exchanged to solve the decision problem D. If decisions were to be made simultaneously or more generally if communi­cation is restricted it may be impossible to solve D under any conditions. In such a situation the decision problem may be viewed as a problem under uncertainty and an objective of expected payoff maximization adopted. This is the approach of Groves [11].

12. See, for example, Horngren's text [17] or Demski [7].

13. At least in such a way as to motivate optimal decisions.

14. See Groves [11] and Loeb [25] for a discussion of "profit­sharing." See also discussion at note 15 below.

15. Under profit-sharing, a managers best message depends on the messages and decisions of the other divisions. Thus, profit­sharing is not compatible with an optimal control mechanism as defined here.

16. Although the title of this paper refers to "incentive compatible" control, the term "optimal" is used here since we assume the division managers maximize their evaluations.

17. This theorem is a combination and generalization of the results in Groves [12].

18. All the new results of this section are essentially contained in Green and Laffont [10]. However, since they do not require imperviousness of an optimal mechanism their version of Theorem 4 requires that there exist a unique dominant strategy equilibrium. In cases of non-unique dominant strategy e~uilibIia, such as arise here for any optimal control mechanism C' in C/', they define an "extended" mechanism in which the evaluation measures are not functions but correspondences. They also only prove their theorems for the very special case in which Xo contains only two points. The modifications and gen­eralization contained here are, however, only rather trivial extensions of their basic ideas.

182 T. GROVES

19. See footnote 24 below and discussion in the text.

20. Assuming no fixed costs and also that by abandoning the ith division the constraint gi(xi'xo) ~ 0 could be avoided.

21. Only slight modifications would be involved if ~iTi(m) = 0

for all m E MI were required.

22. A rather complicated way of relaxing restrictions B.2 and B.3 is to separate the Center's function into two parts--one "communicating" like the divisions and the other choosing Xo and computing the evaluations Ei' Since this approach would require elaborate respecifications of the model and new definitions of a control mechanism and the relevant properties. it will not be persued here.

23. Proofs are contained in Groves and Ledyard [13,14] .

* 24. Note that if there exists an optimal control mechanism C that is total profit allocating. then the constant response function giving the best messages of the divisions would be a Nash equilibrium of response rules leading to optimal decisions. Thus this optimal control mechanism would be "non-manipulable" and total profit allocating contradicting Hurwicz's Theorem.

CONTROL OF DECENTRALIZED ORGANIZATIONS

Bibliography

[1] Arrow, K., "Control in Large Organizations," Management Science, Vol. 10, No.3 (1964).

[2] and L. Hurwicz, "Decentralization and Computation in Resource Allocation," in Pfouts, R. (ed.), Essays in Economics and Econometrics, University of North Carolina Press, Chapel Hill, North Carolina (1960), pp. 34-104.

183

[3] Berge, C., Topological Spaces, Macmillan Co., New York (1963).

[4] Dantzig, G., Linear Programming and Extensions, Princeton University Press, Princeton, New Jersey (1962), Chapter 23.

[5] and P. Wolfe, "The Decomposition Algorithm for Linear Programs," Econometrica, Vol. 29, No.4 (October 1961), pp. 767-778.

[6] , "Decomposition Principle for Linear Programs," Operations Research, Vol. 8, No.1 (1960).

[7] Demski, J., "Evaluation Based on Controllable Performance," unpublished manuscript (1975).

[8] Dreze, J. and D. de la Vallee Poussin, "A Tatonnement Process for Public Goods," Review of Economic Studies, Vol. 38, No.2 (April 1971), pp. 133-150.

[9] Geoffrion, A., "Elements of Large-Scale Mathematical Pro-gramming: Parts I and II," Management Science, Vol. 16, No. 11 (July 1970), pp. 652-701.

[ 10] Green, J. and J. J. Laffont, "Characterization of Strongly Individually Incentive Compatible Mechanism for the Revelation of Preferences for Public Goods," Discussion Paper No. 412, Harvard Institute of Economic Research, (May, 1975).

[11] Groves, T., "Incentives in Teams," Econometrica, Vol. 41, No.4 (July, 1973), pp. 617-631.

[12] , "Information, Incentives, and the Internalization of Product Externalities," in Lin, S. (ed.), Theory and Measurement of Economic Externalities, Academic Press, New York (1975 forthcoming).

[13] and J. Ledyard, "Optimal Allocation of Public Goods: A Solution to the Free Rider Problem," Working Paper No. 144, Center for Mathematical Studies in Economics and Management Science, Northwestern University (May, 1975).

184

[ 14]

T. GROVES

___________ , "An Existence Theorem for an Optimal Mechanism for Allocating Public Goods," (1975, in prepa­ration) .

[15] and M. Loeb, "Incentives and Public Inputs," Journal of Public Economics, Vol. 4, No.3 (August, 1975), pp. 211-2 2 6 .

[16] , "Goal Congruent Evaluation of Divisional Management" (in preparation).

[17] Horngren, C., Cost Accounting: A Managerial Emphasis, 3rd ed., Prentice-Hall, Englewood Cliffs, New Jersey (1972).

[18] Hurwicz, L., "Optimality and Informational Efficiency in Re­source Allocation Processes," in Arrow, K., Karlin, S., and Suppes, P. (eds.), Mathematical Methods in the Social Sciences, Stanford University Press, Stanford (1960), pp. 27-46.

[19] , "The Design of Mechanism for Resource Allocation," American Economic Review, Vol. 63, No.2 (May, 1973), pp. 1-30.

[20] , "Organizational Structures for Joint Decision Making: A Designer's Point of View," in Tuite, M., Chisholm, R., and Radnor, M. (eds.), Interorganizational Decision Making, Aldine, Chicago (1972), pp. 37-44.

[21] , "On Informationally Decentralized Systems," in McGuire, C. B. and Radner, R. (eds.), Decision and Organization, North-Holland, Amsterdam (1972), Chap. 14.

[22] , "On the Existence of Allocation Systems Whose Man-ipulative Nash Equilibria are Pareto Optimal", unpublished paper given at 3rd World Congress of the Econometric Societ~ Toronto, August 1975.

[23] Jennergren, L., Studies in the Mathematical Theory of Decen­tralized Resource-Allocation, unpublished Ph.D. Thesis, Stanford University (1971).

[24] Koopmans, T., "Analysis of Production as an Efficient Combin­ation of Activities," in Koopmans, T. (ed.), Activity Analysis of Production and Allocation, Wiley, New York (1951), pp. 33-97 (esp. pp. 93-95).

[25] Loeb, M., Coordination and Informational Incentive Problems in the Multidivisional Firm, unpublished Ph.D. Thesis, North',Jestern University (1975).

CONTROL OF DECENTRALIZED ORGANIZATIONS

[26] Malinvaud, E., "Decentralized Procedures for Planning," in Malinvaud, E. and Bacarach (eds.), Activity Analysis in the Theory of Growth and Planning, Macmillan, London (1967), pp. 170-208.

[27] Marschak, J. and R. Radner, Economic Theory of Teams, Yale University Press, New Haven (1972).

185

[28] Reiter, S., ''Formal Modeling of Organizations," in Tuite, M., Chisholm, R., and Radnor, M. (eds.), Interorganizationa1 Decision Making, Aldine, Chicago (1972), pp. 87-93.

SOME THOUGHTS ABOUT SIMPLE ADVERTISING MODELS AS

DIFFERENTIAL GAMES AND THE STRUCTURE OF COALITIONS

Geert Jan Olsder

Twente University of Technology

P.O. Box 217, Enschede, The Netherlands

ABSTRACT

Two simple dynamic advertising models are discussed. In both models three companies compete through advertising for the trade of a fixed pool of customers.

Several criteria, to be maximized by the companies, are consider­ed. Nash optimal solutions - open loop as well as closed loop - are studied.

It is assumed that two of the three companies can cooperate against the third one. Such a coalition, which can change in time, is formed according to specific rules. It is then seen that, if one allows coalitions, they do indeed appear with respect to one of the models. With respect to the other model a coalition is disadvantage­ous to both participants and hence will not be formed.

1. INTRODUCTION

The theory of differential games can be considered to be an extension of optimal control theory as well as of game theory. Both game and optimal control theory have been well developed, but in the combination of both, the theory of differential games, new features appear which are neither present in optimal control theory nor in game theory. Some of these features will be discussed with respect to some simple advertising models. These (dynamic) models are des­cribed by ordinary differential equations. The components of the state vector are the number of customers of the companies (players) which compete in the game. Each company can influence its number of

187

188 G.J.OLSDER

customers - and those of the other companies - by advertising. In section 2 each company wants to maximize its own profit within a prescribed time interval and in section 3 the problem is considered in which each company wants to maximize its own number of customers at a certain instant of time.

In this paper two concepts of modelling the feature of adver­tising and its influence on the numer of customers are given which give rise to two different mathematical models. Many models exist in this area, all of which could be defended in some way; see for in­stance [IJ. In section 2 and 3 the first of these models is used, whereas in section 4 the second one is used. Also several definitions and interpretations of "profit" exist, see [IJ; here we will use only one such a definition. In some problems to be considered state and/ or control constraints will be present, an example of which is for instance that borrowing money is not allowed.

Several concepts of optimal solutions exist [2J; here Nash solutions, open as well as closed loop [3J, will be studied.

Coalitions will also be considered. Two companies may form a coalition, according to specific rules, and together compete against the third one. These coalitions may change in time. It is clear that the optimal solutions become different if one allows the companies to cooperate. Blaquiere[4J, [10J has made a theoretical study of time-dependent coalitions and the rules in this paper, according to which coalitions are formed, are a specific case of Blaquiere's more general definitions. Mixed coalitions will be defined too. Within the theory of mixed coalitions, coalitions are possible to a certain extent. For the static case mixed coalitions have been considered by Aubin [5J. It is possible to treat mixed time dependent coalitions within the theory of ordinary control variables.

It seems to be extremely hard to obtain (analytic) solutions in the case of coalitions being allowed. Sometimes only the character­istic features of those solutions will be given. However, the basic problem is important and has been frequently discussed in some form. In [6J for instance the same kind of problems is discussed with res­pect to big powers in the world and in [7] the economies of some less developed countries are modelled as a (static) game in which coalitions can occur.

2. MAXIMIZATION OF THE PROFITS

2.1 A One Person Game

In the game discussed in this section only one player acts and therefore the problem to be stated belongs to the theory of optimal control.

ADVERTISING MODELS AS DIFFERENTIAL GAMES 189

A company C has x(t) customers at time t. Up to time t C has made a profit of

t f (t) f (cx(t) - u(t»dt,

a

where c is a positive constant. The function u(t), restricted to

u(t) ~ a, t ~ a,

represents the money put into advertising. Advertising increases the number of customers according to

d~~t) = u(t), x(a) = xa > a.

It is assumed that borrowing money lS not allowed, i.e.

f(t) ~ a, t ~ a.

The company C will choose u(t) in such a way as to

maximize f(T),

l.e. maximize the total profit during a given time interval o ~ t ~ T; T is fixed.

(I)

(2)

(3)

(4)

(5)

The optimal solution to the optimal control problem (1)-(5), which is rather obvious and can for instance be obtained by applying the maximum principle [8J, is

--1 CXa(t) u(t)

a ~ t < T -c

(6)

provided that T - ~ > a. This kind of solution is well known in economics; first oge reinvests as much as possible and during the last part of the process one reaps one's profits.

2.2 A Two Person Game

Two companies, to be called C1 and C2 ' operate on a market with a fixed total number of customers (note that in subsection 2.1 the number of customers was not constant), From now on we assume that the customers have been scaled in such a way that this total number

190

equals one. Up to time t company C. has made a profit of 1

t f.(t) = f (c.x.(t) - u.(t»)dt,

1 0 1 1 1 i = 1,2,

G.J.OLSDER

(7)

where x. is the number of customers of C .• The constant c. is posi-111

tive and can be interpreted as an efficiency parameter for company C.; the more efficient the company, the larger c .• Advertising in-

1 1

creases the company's number of customers and decreases the number of those of the other company at the same rate. The corresponding model is

dX I (t)

dt

The following constraints should be satisfied:

u. (t) <:: 0 t <:: 0 i I ,2, 1

x. (t) <:: 0 t <:: 0 i I ,2, 1

f. (t) <:: 0 t <:: 0 i 1,2. 1

Company C. chooses u. (t) in such a way as to 1 1

maximize f.(T), i = I ,2, 1

(8)

(9)

(10)

(II)

(12)

where T is a fixed positive number. The game defined in this way is clearly nonzero sum. Only Nash-optimal solutions will be considered [3J. The problem described in (7)-(12) will be denoted by PI' Problem P2 is defined by (7)-(9), (12) and (13);

u.(t) ~ c. x.(t), t <:: 0, i = 1,2. 1 1 1

(13)

Problem P3 is defined by (7)-(10), (12) and (13).

Problem P2 is the simplest and hence will be treated first.

Define

u. (t) = U. (t)x. (t) 1 1 1

i I ,2, (14)

and reformulate problem P2 in terms of ui ;

ADVERTISING MODELS AS DIFFERENTIAL GAMES 191

dX I - u 2(1-x\). F = ulxl

° ~ u. ~ c i ' ~

'T

max f (c\-uI)xI dt, u l °

T max f (c2-U2) (l-xl)dt,

u2 ° (15)

where the fact has been used that, because of x\ + x2 = I, we have

essentially one state equation. According to the theory in [2J or [3J, the Hamiltonians

(c\-UI)x\ + PI {ulx l - (I-x l )u2},

(c2-u2)(I-x l ) + P2 {ulx l - (l-x l )u2}, (16)

must be maximized with respect to u l and u2 respectively. The func­tions Pj(t) are the adjoint functions, def~ned by

dp. _J = dt

p.(T) = 0, J

j I ,2. (17)

For the time being we are looking for open loop solutions. For closed loop solutions eqs. (17) have to be replaced. A straightforward calcu­tion shows that the optimal solution to problem P2 is:

{:i , ° ~ t ~ t. ~

U. (t) i 1,2, ~

, t. ~ t ~ T ~

(18)

where, if we assume c I ~ c 2 '

I c\ l+c2 tl = T t2 = tl - In(- .~).

c\ c2 c 1 (19)

* Here it is assumed that the corresponding optimal trajectory x\(t)

* * * satisfies xl(t)E(O,I), i.e. x\(t), x2(t) > 0, for all t E [O,TJ.

Otherwise the solution to the problem P2 as described in formula (IS)

192 G.J.OLSDER

becomes slightly different (the problem P2 as described by (7)-(9), (12) and (13) does not make sense then). Hence the company with the higher c, i.e. the company which makes a bigger profit from each client, goes on reinvesting for a longer time than the other company. This is what we would expect.

If we add condition (10) to problem P2 , we get problem P3 and

the optimal solution of P3 may be different from (18). The solution

(18) is only a possibility for the optimal solution. The other pos­sibilities are the bankrupt-policies, to be defined.

Each company, say CI , can secure a certa~n minimu~ positive. profit by playing ul = O. The other company w111 play u2 = I unt11

CI is out of business, i.e. has no customers left (xl(t) = 0), and

then play u2 = O. This is called the bankrupt-policy. Both companies

compare the profit made by playing (18) and the bankrupt-policy. If for one of the companies the bankrupt-policy is favorable, then this will be the optimal solution to P3 ; otherwise it will be (18).

The difference between PI and P3 is that in PI money can be saved up and reinvested later whereas in P3 money can be reinvested only at the moment it is received. However, saving up money and re­investing it later is worse than reinvesting immediately and hence the freedom of saving up money in P will not be used at all. Thus it is easily argued that the oPtimal solution to PI equals the op­timal solution to P3•

Let us consider problem P? but this time we are looking for

closed loop solutions. We now assume that the optimal open loop . solution xt(t) satisfies xt(t) E (0,1), for all t E [O,TJ.

Instead of formula (17), the adjoint functions now satisfy

aH I aH I aU2 (cl-u l ) - PI (u l +u2) + PI (I-XI)

aU2 - --- dU2 --= -

aX I aX I aX I ,

aH2 aH2 aUI (c2-u2) - P2 (ul +U2)

au} (20) - --- aU l

--= - P2 xI aX I ax) ax)

The derivatives au2/ax1 and au)/axl need not to exist everywhere

(one expects the solutions ui(x1,t) to be 1 or 0 depending on where

the current point (x1,t) is situated in the plane spanned by the x)- and t-axis); because of this the necessary conditions in [3J cannot be applied straightforwardly. However, a sufficiency theorem

ADVERTISING MODELS AS DIFFERENTIAL GAMES 193

in [9J (theorem 3), can now be applied. The essential part of of this theorem states - in connection with our example - the following. A sufficient condition for ul(t) and u~(xl,t)(with corresponding tra­jectory x7(t» to be optimal is that continuous, piecewise different­iable ~I(t) and ~2(t) exist such that

(cl-~)X~ - (cl-ul)x l + ~I(t) {u~x~ - u2(I-X~)}

- ~I(t) (~>I - ~(I-xI)}+ UI(t)(x~-xl) ~ 0, ~I(T) = 0 (21)

for all xI E (0,1), u l E [O,cIJ and almost all t E [O,TJ,

-* * - -* - * (c 2-u2)(I-x l ) - (c 2-u2)(I-x l ) + ~2(t) {ulx I - u 2(I-x l )}

- ~2(t) {~xI - u 2(I-x l )} + ~2(t)(x~-xI) ~ 0, ~2(T) = 0, (22)

for all xI E (0,1), u2 E [O,c 2J and almost all t E [O,TJ.

It is now easily shown that the closed loop solution equals the open loop solution; substitute ~.(t) = p.(t), i = 1,2, where p.(t)

~ ~ ~

are the solutions of eq. (17) and substitute for x~ the trajectory

which corresponds to the open loop optimal solution, in the ineqs. (21) and (22).

2.3 A Three Person Game Without Coalitions

Three companies; C , C2 and C3 , operate on a market again with a fixed total numbers of customers. Up to time t C1 has made a profit of

t f.(t) = J (c.x.(t) - u.(t)dt, i = 1,2,3, ~ 0 ~ ~ ~

(23)

where x.(t), c. and u.(t) are defined exactly as in subsection 2.2. ~ ~ ~

The advertising model is now

dX I xl xl , xl (0) d"t= u 1 - ---u - ---u x l +x3 2 x 1+x2 3

dX2 x2 x2 x2(0) d"t= u - ---u - ---u , 2 x2+x l 3 x2+x3 1 X 20 > 0, (24)

194 G.J.OLSDER

which is to say that, if the number of customers of one company in­creases, the numbers of customers of the other two companies decrease proportionally to their respective numbers of customers. The following constraints should be satisfied:

Company C. 1

profit at

u. (t) :2: 0, t :2: 0, 1

x. (t) :2: 0, t :2: 0, 1

f. (t) :2: 0, t :2: 0, 1

will choose u. (t) in 1

a given

max u. (t)

1

time T,

f. (T) , 1

i.e.

i =

i 1,2,3, (25)

i 1,2,3, (26)

i = 1,2,3. (27)

such a way that it maximizes its

1,2,3. (28)

Only Nash-optimal solutions will be considered for the differential game (23)-(28), to be called problem P4' No coalitions will be allowed in this section.

As in subsection 2.2, the optimal solution to P4 will be derived

via two other problems, to be called P5 and P6 . The mathematics be­

comes rather cumbersome now. The main difficulty is that eqs. (24) are nonlinear in x.(t) which is the reason that some of the calcula-

1

tions cannot be done explicitly any longer. However, it is to be ex­pected that the kind of solution obtained in subsection 2.2 will ex­tend to this subsection. Therefore we will restrict ourselves to the tools and kinds of solutions obtained in subsection 2.2 and extend these to this subsection.

In order to solve problem P5 , defined by (23)-(25), (28) and (29) :

u.(t) ~ c.x.(t), t :2: 0, i 1 1 1

1,2,3, (29)

u i is replaced with u i ;

u. (t) = u. (t)x. (t). 1 1 1

In terms of ui problem P5 is:

ADVERTISING MODELS AS DIFFERENTIAL GAMES

X IX2 x l x 3 - ----- u - ----- u x I+x3 2 x I+x2 3'

cyclic

o ~ u. ~ C. 1 1

T max J ~. (t) 0

(c. 1

~. (t) )x. (t)dt 1 1

i 1,2,3.

1

Hamiltonians are defined as follows:

(CI-~I )x I + PII (~Ixi x l x2 x l x3 _

HI - ---- u - --u) x I+x3 2 x I+x2 3

+ Pl2 (~2x2 x2x3 x 2x I _

----u - --u) x2+x I 3 x2+x3 I

+ p 13 (u3x3 x3x I x3x2 _

- ---- u ------u ); x3+x2 I x3+x I 2

for H2 and H3 similar expressions hold.

The Hamiltonian H. is to be maximized with respect to u .. 1 1

The adjoint functions - for the open loop case - are defined as

dp.. dH. -2:1.. = _ 1

dt ax. J

, p .. (T) = 0 , i,j = 1,2,3; 1J

195

(30)

(31 )

these differential equations are difficult to solve analytically because the state variables x. appear in (31). It seems, however,

1 that the solution to Ps is

where t., i 1

then tl = T

c. 0 ~ t ::; t i , 1

U. (t) 1 "{ (32)

0 t. ~ t ~ T, 1

1,2,3, depend on c i and x 1P ' i = 1,2,3. If c 1 ~ c 2 ~ c3 '

1 ,while t2 and t3 must be obtained numerically. c 1

196 G.J.OLSDER

No attempt has been made to obtain the closed loop solution. Here the closed loop solution will probably differ from the open loop solution.

Problem P6 is defined as problem Ps with the addition of (26). If the solution (32) gives x.(T) > 0, i = 1,2,3, then (32) is a

~

candidate for the optimal solution to P6 . Other candidates are (and

they are the only ones if x.(T) ~ 0 for some i) the bankrupt-solutions ~

to be defined in the next paragraph.

As in subsection 2.2, each company, say CI , can secure a minimum

positive profit by playing ul = 0, while the others try to minimize

the profit of C1 by playing u.=c.,i=2,3. This strategy goes on until ~ ~

CI has no customers left. For the remaining time interval a two per­son game between C2 and C3 remains. Here it is assumed of course that at the moment at wnich CI goes bankrupt,C 2 and C3 both have a positive number of customers.

If for at least one company the bankrupt-solution is preferable to (32) - if admissible - , then this bankrupt-solution is optimal. Otherwise (32) is the optimal solution. In certain situations two companies instead of one may prefer to go bankrupt.

Exactly as in subsection 2.2 it can be argued now that the op­timal solution to P4 equals the optimal solution to P6 .

For the remainder of this subsection we will assume that (32) constitutes the optimal solution. Some trajectories x.(t) correspon-- . ~ ding to ui(t) = ci have been sketched on the s~mplex xI + x2 + x3 =

constant, xi ~ 0, in fig. Ib, where it is assumed that c l = c2 = c 3 .

For c 1 # c 2 and/or c l # c3 the figure will be similar, but not longer

symmetric.

If the initial position xiO ' i 1,2,3, is at point C, see fig.

Ib, then a stalemate arises, i.e. the solutions x.(t), i = 1,2,3, ~

remain constant. If the initial position is situated within the quad­rangle ABCD, then the state vector x(t) will move in the direction of point A. For t sufficiently large and u.(t) = c. the solution

~ ~

passes through point A. Note that, if the initial point is at point I, see fig. Ib, x (t) is an increasing function of time, x (t) is decreasing and x2tt) first increases and then decreases. T~is means that although company C2 seems to be doing well in the beginning, it eventually goes out of business.

ADVERTISING MODELS AS DIFFERENTIAL GAMES 197

Fig.la.The simplex xI + Xz + x3

constant

F · lb. 1.g. Parts of optimal trajectories on the simplex. In the figure: c l = Cz = c3 •

Z.4 A Three Person Game \-lith Pure Coalitions

Problem P6 of ssect.Z.3 will be considered again, with the dif­ference that coalitions between two players are allowed. When two companies decide to cooperate, i.e. form a coalition, they put all their customers together and act as a single player. Thus a two per­son game arises. It is assumed that if the cooperating companies have already made some profit, these profits are kept separated. Each com­pany may terminate the coalition at any time at its discretion; no binding agreements can be made. It is supposed that at termination the customers (and the profits obtained during the coalition) will be divided proportionally to the amount of customers each company brought into the coalition initially. The efficiency parameter c .. of a coalition between C. and C. is defined as 1.J

c .. 1.J

1. J

c.x.+c..x. 1. 1. J J x.+x.

1. J

which is easily seen to be constant during the coalition. If for instance C) and Cz form a coalition, then the following two-person game results:

198 G.J.OLSDER

(33)

If one thinks of player xl 2 as two players anyhow - having one ac­tion variable u l ,2 - the tesult is:

dX l xl ~ = x l +x2 (u l ,2-u3)'

dX3 dt = u3-u l ,2' (34)

It is easily seen that x./{x l +x2) , i = 1,2, does not change with time. As in the previous~subsection the way results will be obtained is somewhat intuitive because the mathematics necessary to justify all the steps made becomes too complicated.

Note that here the definition of a coalition is different from the one in the general N-person games as for instance described in [11 J. In the general theory the total profit (utility) can be divided among the members of the coalition in any way possible. Here, however, the distribution of the total profit is fixed. This causes some well­known theorems, as for instance in an essential constant-sum game the core is empty, not to be valid here.

If the initial point xIO, x20 , x30 lies within quadrangle ABCD,

see fig. 2, then, if companies C2 and C3 do not cooperate and Cl plays

u l = c l ' company Cl will acquire all the customers in a finite time.

If, however, companies C2 and C3 cooperate, the situation becomes different.

If the initial position is within triangle ABD and Cl plays

u l - c l ' the game will still end up at point A, whatever C2 and C3 do (in case c l = c2 = c3 this means that initially Cl has more than

half of all the customers). If, however, the initial point is in triangle BCD, say at point I in fig. 2, and if C2 and C3 form a coali-

tion according to the rules mentioned above, then essentially a two person game results. According to the theory in subsection 2.2 and especially formula (18), we expect the two companies to put all the money they obtain back into advertising immediately.

ADVERTISING MODELS AS DIFFERENTIAL GAMES

B G~x 2

199

Fig. 2. Trajectories if coalitions are formed. In this figure: c 1 = 3, c2 = 2, c3 = 1.

The corresponding trajectory on the simplex becomes part of a straight line through A and I; the state moves away from point A.

The coalition of C2 and C3 will not last forever because, if

the state passes line BF, C2 can force the state to go to point G

in finite time. Hence C3 ends the coalition with C2 before point E

is reached and starts a coalition with C). Company C) is eager to

accept this coalition with C3 because otherwise it loses anyhow. Be­

cause of this new coalition the state will now move along the line GE, away from G. Of course this second coalition will not last for­ever. Before the state arrives at point H, C3 will terminate the coalition with C) and start a new one with C2 ' etc., etc.

In this way the state will remain within the triangle BFD. With the strategy of changing coalitions, however, none of the companies makes any profit. Hence within this triangle none of the companies will really bother about the actual state as long as it is away from the sides. When the state approaches one of the sides, a coalition will certainly be formed.

The companies will realize that, without the help of the others,

200 G.J.OLSDER

they cannot le~ve triangle BFD. Near the end of the game, when the companies start to think about reaping their profit instead of re­investing, they would like the state vector to be in the most favor­able position within triangle BFD. For C1 and C2 the most favorable

position will be near point B, whereas for C3 it will be point D (this is of course only true as long as c 1 > c2 > c3). Hence C1 and

C2 will probably make an agreement to keep the state near their com­

mon point of interest, point B.

The company which stays alone, say C3 , may realize,if the other

two companies decide on an agreement such as described above, that it ~s only a puppet in the hands of C1 and C2 because C1 and C2 use

C3 in order to keep the state "vibrating" near point B, within tri­

angle BFD. Hence C3 will make hardly any profit. Therefore it is

more advantageous for C3 to choose for a bankrupt-solution instead

of being a puppet in the hands of C I and CZ' C 1 and C2o• aware of

this, will leave some profit to C3 , so that C3 will play the game

of the changing coalitions. This can for instance be done to keep the state vector at a modest x3-level (> 0) instead of at point B, where x3 = O.

With the three company game described, assuming that coalitions are allowed, neither set of two companies will want the third one to go out of business. because that may cause their own undoing (note that the points B, D, F are unstable equilibrium points and a slight perturbation will cause only one player to acquire the cus­tomers). Thus three companies together form a "stable" situation. This is no longer true when four or more comp~nies are involved. In case of four companies for instance, with c. = I, i = 1,2,3,4, where

~

it is assumed that initially no company has more than half of all the customers, each combination of three companies can force the fourth one to go out of business and then an stable situation with three companies remains.

3. MAXIMIZATION OF THE NUMBER OF CUSTOMERS

In this section the same model is considered as in the previous section. The difference is that the companies now want to maximize their respective numbers of customers, i.e.

max u. (t) ~

x. (T) , ~

i = I,Z,(3,), (35)

ADVERTISING MODELS AS DIFFERENTIAL GAMES 201

where T again is a fixed positive number.

We will consider the analogues of some of the problems treated in section 2. Problem P7 is defined by (7)-(11) and (35); problem Pa is defined by (7)-(9), (13) and (35) and problem P9 is defined by

(7)-(10), (13) and (35).

Problem Pa is the simplest and a straightforward calculation,

similar to the one in subsection 2.2, shows that the optimal solution is:

where

ti". (t) ~

c. ~

o ::; t ::; T, i

u. (t) = ti". (t) x. (t) , i = 1,2. ~ ~ ~

1 ,2, (36)

Here it is again assumed that the corresponding optimal trajectory x~(t), x~(t) satisfies x~(t), xi(t) > 0 for all t E [O,TJ.

Eq. (36) is the open as well as the closed loop solution. These so­lutions are the same in this case because the additional terms in a~2/axl and a~l/axl in the adjoint equations, if we are looking for

the closed loop solutions, are zero for the solution (36).

In problem Pa we assumed that T, c 1' c 2 and x)(O) have values

such that X~(T), X;(T) > O. This means that neither company can force

the other company to go out of business during the time interval [O,TJ. With these assumptions the solution to Pa will also be the

solution to P9 . If, however, T, c 1' c2 and xl(O) have such values

that one company, say C1, can force C2 to go out of business, i.e.

x2(t) = 0 for some t E [O,TJ, then the solution to P9 will not be

unique. C2 may in the beginning put some money into advertising. This

will be immaterial because X;(T) will be zero anyhow.

As in subsection 2.2, it is easily shown that the solution to P7 equals the solution to P9'

Also in the three-company-game, with or without coalitions, the solutions are very similar to those of the previous sections. A dif­ference is that the companies will go on investing until the final time T (unless they become bankrupt earlier on in the process of course).

202 G.J.OLSDER

4. ANOTHER MODEL WITH RESPECT TO WHICH COALITIONS WILL NOT OCCUR

In the model in this section it is assumed that the increase in customers is directly proportional to both the rate of advertising and the customers not already committed to the company. Thus in the case of two companies the model becomes

dX 1 ~ = xZu 1 - x1uZ '

and the case of three companies the model becomes

dX I dt = (xZ+x3)u I - xluZ - x lu3 '

The criteria in this section will be

max u. (t) ~

x. (T) , ~

and the constraints will be

o~u.(t) ~ I, ~

(37)

(38)

(39)

(40)

which express the fact that the advertising capacity may be limited.

It is easily seen that the optimal solution for the two - as well as for the three-company case is u~(t) = I, 0 ~ t ~ T. The cor­

~

responding trajectories tend to go to an equilibrium point. For the Z-company-case this equilibrium point is xI Xz = i and for the 3-

. . I. 3 d company-case ~t ~s xI = Xz = x3 = 3' Hence ~n the -company-case an

for T sufficiently large, x:(T), i = I ,Z,3, will be about ~, indepen­

dent of the initial values. Now suppose that CI and Cz form a coali­

tion, according to the same rules as in section Z, then the equations become

ADVERTISING MODELS AS DIFFERENTIAL GAMES 203

dx) x) {x3 u),Z - (x)+xZ)u3} , cit = X)+XZ

dXZ Xz {x3 u),Z - (X)+XZ)U3} cit = X)+XZ

,

(4) )

which is essentially a two-company game. For this two-company game the trajectories also tend to go to the equilibrium point; x3 = ! , X)+XZ = i. Hence for T sufficiently large X)+XZ is about i ' whi~h is worse than X)+XZ ~ f which could be obtained without coalitions.

Even if C) and Cz cooperate during a short time-interval it is easi­

ly shown that they will not be as well off as when they do not co­operate at all.

5. MIXED COALITIONS

In this section it will not be necessary for two companies to completely cooperate and thus to lose their own identity within such a coalition. Coalitions up to a certain degree will be defined.

If C) would like to cooperate with Cz for )OO.a Z % and Cz would

like to cooperate with C) for )OO,Sj %,a 2 and Sl may depend on time

and 0 :::; az"~):::; ), then the degree of cooperation will be d)Z~ min

(aZ'S)). Similar definitions are made for the cooperations of C)

with C3 and Cz with C3 , the extent of which will be d)3 and dZ3 res­

pectively.

If there are no coalitions, the equations are

x. ~

i = ),Z,3;

if there is a full coalition (in the sense of section Z) of Cj and

CZ' the equations are

Xi = gi(x, v)' v Z' v 3) i = ),Z,3;

if Cj and C3 cooperate fully the equations are

204 G.J.OLSDER

x. = hi(x, wI' w2 ' w3) i = 1,2,3; ~

and if C2 and C3 cooperate fully the equations are

x. = k. (x, zi ' z2' z3) i = I ,2,3. ~ ~

Here u., v., w. and z. are the decision variables of company c .. ~ ~ ~ ~ ~

In the construction of the coalitions in subsection 2.4, it was required that vI = v 2 ' wI = w3 and z2 = z3' but here we will con-

sider the more general case. The equations above can be written in a compact form as

XI {fl(l-dI2-dI3) + gl d l2 + hi d I3}(I-d23) + kl d23 ,

x2 {g2 d l2 - f2(I-dI2-d23) + k2 d 23}(I-d I3 ) + h2 d 13 ,

X3 {h3 d l3 - k3 d23 + f3(I-dI3-d23)}(I-dI2) + g3 d 12 ,

(42)

where fi' gi ' hi' k i are shorthand notations for f i (x,u l ,u2,u3),

gi(x, vI' v 2 ' v3), hi(x, wI' w2 ' w3) and ki(x, zl' z2' z3)' and

where

The (time dependent) decision variables for CI are u I ' vI' wI' zl'

a 2 , a 3 the decision variable for C2 are u2 , v2 ' w2 ' z2' SI and S3

and the decision variables for C3 are u3 ' v3 ' w3 ' z3'YI and Y2 . The

restrictions on the variables ai' Si' Yi are that they are all non­

negative and

(43)

With the construction of mixed coali hons it is clear that no binding agreements are made; each company can terminate a coalition at its own discretion. If we restrict a., S., y. to the values 0 and 1, sub-

~ ~ ~

ject to (43), then we again get the pure coalitions of subsection 2.4.

ADVERTISING MODELS AS DIFFERENTIAL GAMES 205

ACKNOWLEDGEMENT

I would like to acknowledge the discussions with Professor Jim Case of the Johns Hopkins University on the subject of this paper.

REFERENCES

[1J Sethi, S.R., "Optimal Control Problems in Advertising" in "Optimal Control Theory and its Applications", Springer Verlag Lecture Notes in Econ. and Math. Systems, no. 106, 1974, pp. 301-337.

[2J Starr, A.W., Ho, Y.C., "Nonzero-Sum Differential Games" and "Further Properties of Nonzero-Sum Differential Games", JOTA, vol. 9,1969, pp. 184-219.

[3J Leitmann, G., "Cooperative and Non-Cooperative Many Players Differential Games", International Centre for Mechanical Sciences courses and lectures no. 190, Udine, 1974, published by Springer.

[4J Blaquiere, A., "Dynamic Games with Coalitions and Diplomacies", this volume.

[5J Aubin, J.P., "Fuzzy Games", Proceedings of the 7th IFIP Conference on Optimization Techniques, Nice, France, 1975.

[6J Haas, E.B., "The Balance of Power: Prescription, Concept or Propaganda?", World Politics, vol. 5, 1953, pp. 442-477.

[7J Gately, D., "Sharing the Gains from Customs Unions among Less Developed Countries", Journal of Development Economics I, 1974, pp. 213-233.

[8J Lee, E.B., Markus, L., "Foundations of Optimal Control Theory", John Wiley, 1967.

[9J Stalford, H., Leitmann, G., "Sufficiency Conditions for Nash Equilibria in N-Person Differential Games", in Blaquiere, A., (ed.), "Topics in Differential Equations", North Holland, 1973.

[10J Blaquiere, A., Wiese, K.E., "Jeux qualitatifs multH!ages aN personnes, coalitions", C.R. Acad. Sc. Paris, t.270, pp. 1280-1282 (mai 1970), Serie A.

[11J Owen, G., "Game Theory", Saunders, 1968.

EQUILIBRIUM PATTERNS FOR BARGAINING UNDER STRIKE:

A DIFFERENTIAL GAME MODELl

S. C1emhout, G. Leitmann and H. Y. Wan, Jr.

Cornell University and University of California

Ithaca, New York and Berkeley, California

ABSTRACT

A class of N-person, general-sum differential games is consi­dered for which non-cooperative Nash equilibria can characterize situations such as bargaining during a strike. A subclass of strate­gies is shown to be playable equilibria. The existence and unique­ness of this subclass, as well as the possibility of profitable re­contracting, are analyzed and illustrated with examples. Equilibrium implies that any player will eventually fare as well as if he had accepted the claim of all other players at any earlier time. Possi­ble generalizations are outlined or discussed.

I. INTRODUCTION

The bargaining process is instrumental in resolving a widening spectrum of social and economic conflicts, e.g., international talks on disarmament, exchange parities and tariff reduction, management­labor negotiations on pay, working conditions and labor participa­tion, producer-consumer discussions on environmental issues, products' quality and prices. The bargaining process presents interesting theoretical ramifications. Economists are increasingly dissatisfied

IS. Clemhout and H. Wan, Jr. acknowledge beneficial discussions dur­ing the Spring of 1970 with P. Varaiya concerning a four-party bar­gaining model. They are also grateful for helpful suggestions by J. Harsanyi and R. Selten. Special thanks are due to Dean R. B. ~1cKersie for stimulating exchanges of ideas. This research has been supported by the National Science Foundation (GP-24205) and the Office of Naval Research (N00014-69-A-0200-l0l2).

207

208 s. CLEMHOUT, G. LEITMANN, AND H.Y. WAN, JR.

with the Walrasian fiction that everyone is a price-taker, that any equilibrium can only be asymptotically approached with an infinite delay and that an equilibrium, once reached, leaves no room for mu­tually beneficial recontracting. Realistic bargaining theories must allow all participants to make price-offers, reach settlement in finite time and end up with potentially inefficient terms, e.g., in­stalling coal-stokers on electric trains.

The sizable existing literature on bargaining concentrates on the bilateral monopoly situation of management-union bargaining. Partly this is because strikes are socially costly and their set­tlement terms have a significant impact on the inflation/unemploy­ment trends. Partly it is also because theories worked out for collective bargaining can be useful in other contexts. Past studies2 are insightful, yet incomplete. In this section, we review the existing literature. A differential game model will be set up in the next section. In Section III, we prove that under certain con­ditions guaranteeing the eventual settlement, there exists a class of equilibrium bargaining strategies -- not necessarily unique -­leading to respective settlement terms. All of these equilibria imply that each player will fare as well as he would have, had he accepted all the other players' initial offers. Many equilibrium settlement terms are "inefficient" in that recontracting can be beneficial to all. Section IV develops an alternative theory for "efficient" settlement terms. Three examples are provided in Sec­tion V and some concluding remarks are contained in the last section.

For expository convenience, we shall synthesize from the litera­ture a composite bargaining theory. Two players, 1 and 2, engage in bargaining. Their payoffs can be represented by a point in Figure 1, where FF represents the Pareto-optimal frontier, i.e., the collection of payoff pairs over which there is no way to increase the payoffs of both players. These positions can be achieved only by co-operating players eschewing the strike. The composite bar­gaining theory includes four principal components:

(15) . a. The Exchange of Threats. As Nash pOlnted out, under-

lying each co-operative game, there is a non-co-operative game, in which players pose threats to each other. Each player may possess only one single fixed threat (e.g., to quit) or an entire arsenal of threats (e.g., slow-down, walkout, sitdown strike, etc.) out of which some threat is chosen according to some criteria to attain the "best" str:ategical advantage against the position of his adversary. In Figure 1, the point ~ corresponds to the pair of payoffs to both players if the threats are carried out against each other. In the

2The reader is referred to the appropriate sections taining surveys of the literature (5,7,9,17,18,19). references offers sizable bibliographies.

of works con­Each of these

EQUILIBRIUM PATTERNS FOR BARGAINING UNDER STRIKE 209

F

~2 ····-··········r~I.~21

o ~I F

Figure 1. Bargaining Process - conventional view

210 S. CLEMHOUT, G. LEITMANN, AND H.Y. WAN, JR.

composite theory, threats are never carried out.

b. The Terms of Settlement. This represents one particular payoff pair on the FF locus. Actually which point on the FF locus will be chosen depends upon how potent each player's threat looms against the other. For instance, the Nash solution N=(V~,V~) is defined by the condition(16):

* * (Vl-~l) (V2-~2) ~ (Vl-~l) (V2-~2)

for all (Vl 'V2) on FF.

As Luce and Raiffa(14) pointed out, such solutions represent "fair" solutions "prescribed" for compulsory arbitration (a system widely in use only in Australia) rather than likely outcomes in real-life bargaining situations. The same remark applies to Harsanyi's n­person generalization of the Nash model(12).

c. The Bargaining Sequence. To modify the prescriptive model of Nash into a descriptive model where demand and counter-demands are traded, one usually constructs some "behavioral" model postu­lating how the players would behave rather than a "game theoretic" model proving why the players must so behave. To begin with, the initial demands of the players imply a payoff pair (V~,V~) impossible to be realized simultaneously (i. e., "beyond" the FF frontier). Various heuristic justifications are supplied for the respectiv{ll) behavioral patterns, e.g., the "Zeuthen-N?sh" model of Harsanyi , and the "Zeuthen-Hicks" model of Bishop(lJ. These theories specify who should concede at what time, with the process terminating at a point on FF. For instance, the N~N path in Figure 1, leading to N, represents such a bargaining path. Such theories appear to be some­what less convincing than a game-theoretic model. The latter aims at deriving concession rates from optimality conditions, the former assumes how players concede without explaining why.

d. The Imperfection of Knowledge, etc. Most of the existing theories (except Bishop) postulate "group rationality" (i. e., Pareto Optimality) which precludes strikes. If "rational" players cooperate, certainly the socially wasteful strikes should never happen. To explain the occurrence of strikes, the most common hypothesis used is the imperfection of knowledge of the players vis-a-vis each other. Through some 1earning-by-bargaining processes, players modify their demands according to ac~uired knowledge in some presumedS but unex­plained, manner. Case(), Cross(6), Dussaix and Haurie( ), reason along these lines. Only in Harsanyi and Se1ten(10) do the players adopt Bayesian strategies to modify their demands according to some criterion after optimal play. However, in the latter model, players either "make-up" or "break-up" in a once-for-a11 manner which does not correspond to the intermittent phenomenon of real-life strikes.

EQUILIBRIUM PATTERNS FOR BARGAINING UNDER STRIKE 211

Alternative theories for the explanation of strikes aim at the four-party bargaining model where rank-and-file workers and stock­holders also play their roles. Suffice to say, each of such theories captures some essence of truth. No definitive synthesis has been reached to date.

What we shall present is a bargaining model under strike. Since threats are being carried out as negotiation proceeds, this is a realistic situation inexplicable with the group rationality axiom. Since the strike is usually costly to all players, a non-constant sum model is most appropriate. Since it is desirable to avoid hard­to-verify behavioral postulates, a game-theoretic framework is adopted which tries to explain observed behavior from some optimality cri­terion. In this context, the Leitmann-Liu(13) differential game model meets all the above requirements, and it is a special case of this study.

In (20) playability is assumed for the class of equilibria dis­cussed. In this paper a more general class of equilibria is esta­blished and playability is proved under mild conditions. We also outline extensions for future work in several significant directions.

II. THE MODEL FORMULATION

We shall now consider a model with N-person bargaining over n issues. The questions to be answered are:

(1) How long will the strike last? (2) Under what conditons will the settlement be reached? (3) What are the values of the game to each player?

We shall define the concepts involved in this differential model and then characterize an entire class of equilibria. special examples will illustrate the economic implications

game Later,

in depth.

Definitions: Players: Issues : State Variables:

Excess Claims:

i 1, •.. ,N j l, ••• ,n N 1 . x. (~, ••. ,~~, .•. ,~) where. ~ ( ~ ~ ~)' h ~ . x = xl""'x., ... ,x , w~t x. represent~ng

the portion 6f "progrietory rights," etc., over issue j claimed by player i. E(x)=~xi-J, with J=(l, •.• ,l), measures

i the degree of incompatibility of the claims for each and every issue.

212

Residual:

Playing Space:

S. CLEMHOUT, G. LEITMANN, AND H.Y. WAN, JR.

. k R~(X)=J-L x represents the residual

k~i rights on all issues for player i if he agrees with the claims made by every other player.

XO=xx [0,00) where X={x!E(x)e: R:,

Ri(x) e: (R~\{O}) , for i=l, ••. ,N},

R: being the non-negative orthant of the

real n- dimensional Euclidean space.

Remark. By definition,

k J- L x kofi

i x -E(x) , i x

i R (:x)+E(x)

Hence, for any xe:X and for all i,k,k~i, ink

x e:(R+\{O}), wx ~J ,

The economic rationale for these conditions is as follows. In a situation in which any player can veto settlement, the residual or the claim of any player i on any issue j must be a non-negative fraction:

l1oreover, neither the residual nor the claim of any player should be zero on all issues:

i i x ~O~R (x) •

An individual claiming nothing is no player. A player being promised nothing would always veto such a settlement.

Terminal Manifold:

Remark

eO=ex[o,oo)

where e={xe:x!Lxi=J}. Play ends at the smallest value of

t such that (!,t)=(y,T)e: eO. y=(yl, .•. ,yN) will be called the settlement terms and T the terminal instant.

i i xe: e implies x =R (x) for all i.

EQUILIBRIUM PATTERNS FOR BARGAINING UNDER STRIKE 213

Payoff for Player i:

I The C function Ui (')

i i aU i (y ,T)

Uij(y ,T) ~ i ay.

J

> 0 for all j£{1,2, ••• ,n}

< 0

for all (y,T)£Xo , is the payoff function for player i.

Strategies and Controls:

pi ~ {All bounded, continuous functions pi(.) :

(Xo\So)+Rn } is the strategy set for i, with xi=pi(x(t),t)

as the control of i at (x(t),t). Let:

Remark

Given (x ,t ) £ ~, p(') £ P, a solution for: o 0

x(t)

need not be unique.

p(x(t),t), x(t ) = x (fl) o 0

Playable Strategies at (x ,t ), P(x ,t ): o 0 0 0

p(·)£P is defined as playable at (x,t )£(Xo\So) iff o 0

for some T£(t ,00) there is at least one continuous o

function, ~(.) : [t ,T]+X, satisfying (fl), such that: o

~(t) £(X\S) for all t£[t ,T) o

(fl. ) ~

$(T) £ S

Let:

214 S. CLEMHOUT. G. LEITMANN. AND H.Y. WAN. JR.

Outcome Function:

V ( .) : {(x , t , P ( . ) ,H' » I (x ,t ) e:xo , p ( • ) e: P (x ,t ), H . ) o 0 0 0 0 0

satisfies (~), (~i) and (~o.)}+RN 11

i where Vi(x ,t ,p(·),~(·»=Uo(~ (T),T)

o 0 1

Equilibrium Strategy N-tup1e: * 1* N* p (.) = (p (.), ••• ,p (.» is an equilibrium strategy

N-tup1e iff it is an element in P satisfying the o

i e: {l, ••• ,N}, o

Nash Conditions:

for all

for all

for all

(x ,t ) e: X i ~ 0 h i-1* i

P (.) !4, (p (·),···,P. (.),p (.),

i+1* N* p (. ), ••• ,p ( • » e:P (x ,t ), o 0

for all ~*(.) satisfying (~), (~i) and (~ .. ) with 11

respect to (x ,t ) and p*(.), and o 0

for all i~*(.) satisfying (~), (~i)

i * respect to (x ,t ) and p (.), o 0

and (~ .. ) with 11

* * i* i* V.(x ,t ,p (.),~ (·»~V.(x ,t, p (.), ~ (0». 1 0 0 -1 0 0

Let:

p* !4, {All equilibrium strategy N-tup1es}

III. A CLASS OF EQUILIBRIA

We shall introduce two sets of sufficient conditions for play termination before we state and prove the main proposition. Economic interpretations will follow.

Define: I ii B.(x ,t )~{(x,t) x and t are such that U.(R (x),t)~Uo(R (x ),t )}

100- 1 -1 0 0

N B (x ,t )1l.. n Bi(x ,to) 000-1 0

EQUILIBRIUM PATTERNS FOR BARGAINING UNDER STRIKE 215

Ti(X ,t ) A sup{tl~ x such that (x,t)EB.(x ,t )} o· Q - 1 0 0

Ti(X ,t ) ~ sup{tl~ x such that (x,t)EB (x ,t )} 00= 000

We now introduce two conditions:

The Uniform Terminability Condition (UTC) on (Ul(·), ..• ,UN(·»:

o -For any (x,t) EX , T(x,t)<oo •

The Finite Patience Condition (FPC) on U.(·): 1

For any (x,t) E XO, Ii (x,t)<oo

Remark o - -

Clearly, for any (x,t)e;X , T(x,t)~T.(x,t) ; hence, FPC for any i implies UTC. The reverse is not tru~. These are illustrated in Example 1, Section IV below. Our proposition is stated in terms of the weaker condition, UTC. However, FPC can be more readily in­terpreted in economic terms; i.e., for any situation (x,t), player i prefers an immediate settlement by accepting his residual Ri(x) rather than prolonging the strife beyond a time I.(x,t), no matter how favorable for him the eventual settlement can1 become after the prolonged strife.

The Characterization of an equilibrium class:

Define: ~ 'U I 0 0 l' = {p(')EP at any (x,t) E (X \9 ) ,

(i) E. (x) 'Ui

for any j, = O""'>~P.(x,t) ~O; J i J -

(ii) for i 'Uk any i,j, R. (x) = 0 ""'> ~ p.(x t)< O' and

J J ' = , k#i

i 'Uk i (iii) for any i, ~ U .. (R (x) ,t) ~ p. (x,t) = U. (R (x), t) }. j 1J k#i J 10

Proposition

If (Ul (·), .•• ,UN(·» satisfies the uniform terminability

216 s. CLEMHOUT,G. LEITMANN, AND H.Y. WAN, JR.

'" condition, then P is a non-empty class of equilibrium strategy N-tuples:

Moreover, the equilibrium outcome is such that each individual receives a pay~ff equal to what he would receive, had he accepted as settlement his initial residual at the initial instant:

For all i, all (x ,t )£Xo, all p(.) £ ~, and all $(') satisfying 00",

(6). (6.). (~ .. ) with·respect to (x ,t ) and p(.) • 1 11 0 0

'" i V. (x ,t • P (. ) , cp (.» = U. (R (x ). t ) 1 0 0 1 0 0

(*)

(The Conservation of Outcome)

Proof of the Proposition:

Define:

Non-emptiness. We shall construct a non-empty subclass '" of P.

G ~ {All bounded, continuous functions

II f4. {n(·)

i i . U. (R (x),t)gh(E(x»Eh(x) 1 10

nh (x. t) = -n--'----------....;;.;;..--i i

E U •• (R (x),t)g.(E(x»E.(x) 1 1J J J

for all iE {l, ...• N}, hE {l, ...• n}}

/\ /\ P f4. {p(')

. 1 N ~~(.) = - E

J N-I 1 k i

n.(.) - n.(.) J J

for all iE{l, .•.• N}, j£{l, .•. ,n}}

Obviously, the constant function g(.) defined by g~(v) 1.

v£(Rn\{O}), belongs to G. G # $ implies II # cp and ~ # $. To show +

EQUILIBR.IUM PATTERNS FOR BARGAINING UNDER STRIKE 217

'" 6 '" P ~ ~, we need only to show that pCp by checking the three condi-tions defining~. Before doing so, we shall consider the following two identities:

i i U. (R (x),t)gh(E(x»

10 (I)

n . . E U .. (R1 (x),t)g:(E(x»E.(x) I 1J J J

I\k E Ph(x,t)

k~i

(II)

o 0 k k Since for all (x,t) e:: (X \6 ), Ukj(R (x),t»O, gj(E(X»>O, and Ej(X)~O

with strict inequality holding at least for some j, all the denomi­nators in the fractions in (I) and (II) are positive. Hence, the fractions remain finite.

From (I), Eh(x) = ° implies that p!(x,t) = ° for all i, so that

condition (i) is met. From the last expression of (II), the frac­tion is positive and U. (Ri(x),t) is non-positive. Hence,

10

E ~hk(x,t)~O for all (x,t)e::(Xo\6 0 ) and condition (ii) is automati­k~i cally fulfilled. By inspection, (II) implies condition (iii), so

1\ '" '" p~p and P is non-empty.

the Equilibrium and the conservation of outcome. First we establish

Lennna o For any (x ,t )e::X ,

o 0

218 s. CLEMHOUT, G. LEITMANN, AND H.Y. WAN, JR.

'" '" for any p(')£P,

i"'() ("'l() "'i-l() i() "'i+l() for any p' = p . , .•. , p " p . ,p " •.• ,

~N(.))£P(x ,t ) where pi(.) mayor may not equal ~i(.), o 0 .", '" and hence 1 p (') mayor may not equal p(.),

for any cf> (.) :[t ,t ]+X satisfying (ll) and (1l1·) with respect o I

to i~(.) and (x ,t ), and o 0

one has: i i U.(R (cf>(t)),t)=U.(R (x ),t ) •

1 1 0 0

Proof of the Lemma

i i U.(R (cf>(t)),t)=U.(R (x ),t ) + 1 1 0 0

t

f {U. (R i (cf>{ s) ) ,s) t 10

o

n i k - L U .. (R (cf>{s)),s) L ~.(cf>{s),s)}ds

I 1J k"'i J

(**)

By condition (iii) of the definition of ~, the integrand is identically zero, which proves (**). If for all ~(.)£~, p(.) is playable over (XO\SO), then the above Lemma implies both the Nash condition (hence, P'C;P*) and the conservation of outcome rule, so that the proof of the proposition is complete. Suppose the supposition is false; then there must be at least one ~(.)£P' which is not playable at some (x ,t ); i.e., all of the functions cf>(.) associated with ~(.) ando(xo ,to) through ill). and (lli) fail to terminate in S at some fi~ite time. Still for all such cf>(.), condition (**) must hold.

We shall show that the failure of cf>(.) to terminate in S in finite time may take several forms. Refering back to the definition of X, we observe that some boundary points of X are not in X, others are in (X,S). Hence, three alternative forms of failure must be considered:

Case a. There is t>t o

such that cf>(')\ [ ): [t t)+(X\S), to,t 0,

but Lim cf>(s) i X. That is, for some i, s+t

EQUILIBRIUM PATTERNS FOR BARGAINING UNDER STRIKE 219

Remark

Case b. There is t>to such that $(')\ [t ,t): [to ,t)+(X\6), o

but for any £>0.

$(t)+£~(t)iX. That is. either for some i and j,

d k o and dt E $.(t»O, or for some j, k~i J

E.($(t» = 0 and dd E $~(t)<O • J t k J

That is, play never ends.

In view of the Lemma, UTC rules out Case c. lhe conditions, (i) and (ii) in the definition of P preclude the occurrence of Case b. Turning to Case a, we observe that as x £X , Ri(x )£(Rn\{O}).

00+

By the Mean-Value Theorem, there is a u£(O,l) such that

i i U.(R ($(t».t) - U.(R (x ),t )

1 1 0 0

= Uio«l-U)Ri(xo)'

i - E U .. «l-u)R (x ), j 1J 0

ut + (l-u)t )(t-t ) o 0

i ut + (l-u)t )R.(x )<0

o J 0

This is in contradiction with (**). Thus, every $ (.) generated by any ~ ( .) for any (x , t ) must terminate in 6 at some finite time. °Th~s completes the proof.

It is an open question whether or not there exists any equi­librium strategy N-tuple not in class ~.

Remark

In this model, if

N=2 (Two players) iii

U.(y ,T) = R, • (y -ol.J)-k.T, i=l,2 (Linear payoffs) 1 1 1

i n, where R, £ R+ {a} ,

220 S. CLEMHOUT, G. LEITMANN, AND H.Y. WAN, JR.

and

{1 if i=l °li = 0 if i=2

k.>O 1

g(.)=g(·)£G, (Concession rates proportional to excess)

then the resulting ~(.) corresponds to the equilibrium in Leitmann-Liu(13).

Economic Interpretations of the Equilibria

1. Existence. The possibility of settlement depends upon how much players individually (FPC) or collectively (UTC) favor early settlements. If players are very demanding over settlement terms relative to settlement date, it is possible that no equilibrium exists for achieving settlement.

2. Characterization. In any equilibrium, the outcome for any one player will be no better or worse than what he could have achieved if initially (at to) he had accepted the claims of all the other players and contented himself with the residual left over for him, Ri(xo)' At any instant prior to settlement, the adoption of equilibrium strategies causes all the other players - i.e., that particular (N-1)-member syndicate facing him - to make such collec­tive concessions that the improvement of his bargaining position, Ri(x(t», exactly balances his momentary loss due to the strife. Collectively, the other players may not want to concede faster. Faster concession might induce him to hold out even longer in anti­cipation of even better outcomes. These other players may also doubt the wisdom of slowing down concessions. After all, this par­ticular player did decline to settle with an equally favorable out­come. Since every player now feels indifferent about when to settle, the strife goes on.

1\ ~ 1\ The following points distinguish the subclass P from the rest,

(P\P). First, under ~, once issue j is settled, it is never reopened. Second, the residual claim of any player i on any issue j, Rl(x), never decreases. Finally, the speed of improvement of the residual, iLRj(x(t», depends only upon the instant, t, the excess vector, E(x), dt . and his own residual, R1 (x) , Differences in the vectors x, unless accompanied by differences in E(x) or Ri(x) , do not affect d i dtRj (x).

3. Non-uniqueness. Under equilibrium strategies, any player finds his residual claim improving at such rates as to offset exactly his instant cost of strife. But if there exist more than

EQUILIBRIUM PATTERNS FOR BARGAINING UNDER STRIKE 221

one issue, the improvement of his residual claim can take alternative forms: Rapid improvement on one issue but slow progress on another, or alternatively, the opposite can happen. Hence, there can be al­ternative equilibrium strategy N-tuples, leading to non-uniqueness regarding both the settlement terms and the duration of the strife.

4. Group Irrationality. The equilibrium is not group rational in two ways.

a. Wasteful strifes have occurred. The industrial strike is the most eminent example. To carry out threats is basically in­compatible with group rationality.

b. Inefficient settlement terms may be perpetuated. The well-known real-life examples include the posting of stokers on electric trains as well as the exclusion of certain labor-saving inventions. In principle, some "recontracting" may lead to arrange­ments profitable to all parties. In practice, such readjustments are often not carried out.

Analytically, such a phenomenon means that there are two players (i=I,Z) and two issues (j=I,Z) such that for some q>O,

> q >

Z UZI (y ,T)

Z UZZ(y,T)

For some E~min(Yi,y~/q), if player I yields Eq to player 2 and

receives E on issue I in return, both players will be better off. Such rearrangement should be "recontracted" under group-rationality. While real-life seems to allow "inefficient" settlement terms to occur, some researchers may prefer an alternative theory in which unique, "efficient" settlement terms will always emerge. Such a theory is presented in (20), where:

the settlement terms are

Y1 + Z 1 y =

the outcomes are

222 S. CLEMHOUT, G. LEITMANN, AND H.Y. WAN, JR.

If r l =r2=r, closed form solutions can be obtained for:

the duration of the strike 1 2 T=t -£n(2-x -x )/r

o 0 0

1 2 1 2 Y =(l-x )/(2-x -x )

the settlement terms { 0 0 0

2 1 1 2 y =(l-x )/(2-x -x )

o 0 0

The parametric variations in 1 2 x , x , r l and o 0

r 2 lead to changes in 1 2

T, Y , Y , VI and V2 as summarized below.

+ - + -sign [

+ + - -] = - + - +

IV. EXAMPLES

o - - 0 - 00-

To illustrate the earlier discussions, three examples will be presented below.

Example 1. (N-2, n=l). Two players bargain over the single issue of sharing a fixed income. We shall consider two alternative cases, depending upon the forms of the payoff functions assumed.

Case 1. i i U.(y,T)=y

1 e -r.T

1 i i Clearly, U. (y ,T)<O<U. 1 (y ,T). 10 1

Note that this payoff is proportional to the discounted integral of an income stream with size yi at discount rate r i :

i Y

-r T i

e /r. 1

Following the results in the last section, a unique optimal strategy pair in ~ can be found:

~ i / i p.(x,t)=U. (l-x ,T) U. 1 (1-x ,T) 1 JO J

In other words, having made the initial demand, xi, player i will modify his terms such that if player j accepts i'~ terms, the pay­off to j is independent of the timing of such an acceptance.

The intriguing conclusion one can obtain from this model is

EQUILIBRIUM PATTERNS FOR BARGAINING UNDER STRIKE 223

that, either player can affect the payoff of his adversary, none has any influence upon his own payoff. In our model, the initial demands

xl, x2 are historically given and their determination lies beyond the d~eorY we offer.

Finally, one should point out that both the settlement terms and the duration of the strike are uniquely determined in a one­issue bargaining model. Once more than one issue is involved, such determinacy is lost as we shall see below.

In Figure 2a, X is the rectangular triangle with vertices (1,0), (1,1) and (0,1). The graphic derivation of Rl(xo)=l-x~ and R2(xo~1-x~ is also illustrated. 6 is the hypotenuse between (1,0) and (0,1). 6° is the half-strip with 6 as its base. The surface 82 , consisting

of points (xl ,x2 ,t) where U2(x2,t)=U2(R2(xo),to)' is shown as a

collection of linear segments parallel to the xl-aXiS. The surface

E2 , consisting of points (xl ,x2 ,t) where U2 (1-xl ,t)=U2 (R2 (x),t) =

U2 (R2 (Xo),to) is shown as a collection of linear segments parallel

to the x2-axis. Figure 2b shows E2 and a surface El , similarly

defined. The intersection between El and E2 is the trajectory of

x(.) leading from Xo to x(T) in 6. Regions below El and E2 are Bl

and B2 , respectively. Tl , T2 and T can all be read off. We define

now "the offer locus" 0. as (E.neo) for i,j=1,2, i;'j. All informa-J 1

tion on yl, y2 and T can be gathered from 01 and 02 in 60 • FPC on

U2 (·) is represented by the meeting of 01 and the half-line through

(0,1) at T2 (x ,t). For any (x ,t ) in XO, T2 (x ,t ) is finite. o 0 0 0 0 0

The rationale for excluding (1,0) (similarly (0,1» from X is also

clear. In general, T. depends on (x ,t ) and there may be no finite 1 0 0

upper bound over XO •

iii -ri T . Case 2. U.(y ,T)=y +a.(l-y )e , O<a.<l, 1=1,2. Again i 1. 1 1

Uio(y ,T) and Uil (yl,T) have the postulated signs. If a l =a2 ~!, UTC

is satisfied although FPC is not met for either player. For 1 . [ 1 122 a l =a 2 <-2' UTC falls. For example, if a =a =- x =x =- and t =0

1 2 4' 0 0 3 0'

Figure 2c shows that 01 and 02 will never meet.] No settlement can ever materialize.

224 S. CLEMHOUT, G. LEITMANN, AND H.Y. WAN, JR.

t

.........

........

CT, •.•.••.••.•.•.

()«(T) ,T)

(I, I )

Figure 2a. Example 1 - Derivation of offer curve

EQUILIBRIUM PATTERNS FOR BARGAINING UNDER STRIKE 225

t

Figure 2b. Example 1 - Graphic solution

226

(-~. ~ ) (~.~) (~. ~)

S. CLEMHOUT, G. LEITMANN, AND H.Y. WAN, JR.

Offer curve of 2

Offer cu rve of I

Figure 2c. Example 1 - Non-termination case

EQUILIBRIUM PATTERNS FOR BARGAINING UNDER STRIKE 227

Exampl~~. (N=3. n=l). Three countries bargain over sharing a petroleum reserve on the continental shelf. Suppose that

i i U.(y ,T)=y - S.T, S.>O, i 1 1 1 1,2,3

Ui(·) satisfies the sign restrictions on its first order partial derivatives as well as the FPC.

Conservation of outcome now means:

i Y -S T i

i R (x )-S.t , i = 1,2,3 .

o 1 0

Since E yi = 1, ERi(x ) i i 0

i l-2E(x ) and x o 0 Ri(x) + E(x ), summing

o 0

up and rearranging results in:

(T-t ) = 2E(x )/E Sk o 0 k

"-

i Y

i x - E(x )

o 0

E Sk-S , k;'i 1

E Sk k

In the class P, there is only one equilibrium. The fact that the remaining duration of the strife, (T-to), varies proportionately with the excess claim and inverse-proportionally with the "collective time preference," E Sk' is quite expected. What is surprising is the fact that for one kplayer - but never more than one player - it is possible to end up with a settlement share, yi, which exceeds his initial claim, x~. The necessary and sufficient condition for this to happen is S.> E Sk' Better interpretation can be obtained

1 k;'i as follows. Since, under the equilibrium play,

we have: iiI

x (t) = x +-2 (S.- E Sk)(t-t ) • o 1 k;'i 0

This result implies that settlement under equilibrium play is still possible even if player i increases his demand all the time, pro­vided he is more eager to settle early than all other players com­bined in the sense that S.> E Sk.

1 k;'i I i In Figure 3a, 8 is the simplex {x ~ x = I}, spanned by the

1

three unit vectors. X is the tetrahedron bounded by 8 and the three planes characterized by Ri(x)=O, i=1,2,3, respectively. From such as xo=(.S,.3,.4), interior to X, one can determine x~ and Rl(xo) graphically. The plane through Xo perpendicular to the xl-axis marks off x~. The projection of Xo along the xl-axis on18 y~eldS x;.

The plane through x' perpendicular to the x -aX1S marks o

228 S. CLEMHOUT, G. LEITMANN, AND H.Y. WAN, JR.

Xl

Figure' 3a. Example 2 - Claims, excess claim and residuals

EQUILIBRIUM PATTERNS FOR BARGAINING UNDER STRIKE

229

t

Figure 3b. Example 2 - Graphic SOlution

230 S. CLEMHOUT, G. LEITMANN, AND H.Y. WAN, JR.

The distance between xol and Rl(x ) is E(x ). o 0

In Figure 3b, eO is represented as a prism with e as its base.

Three planes of the form 0i ~ eO n{(X,t)IRi(x)=Ri(xo)+8i(t-to)}

are shown. These may be called "syndicate offer surface to i" showing what i will receive if he accepts the collective claims of the other two players. The intersection of the three planes deter­mines both T, the duration of the strike, and y=(yl,y2,y3), the terms of settlement.

Example 3. (N=2, n=2). Management and labor bargain over both profit-sharing ("pay") and power-sharing ("control"). The payoffs

Ui(y~,y~,T), i=l,2, satisfy the sign restrictions on first partial

derivatives and UTC. Consider a subclass of equilibrium strategy f\

pairs in P, where

1 2 g.(E(x))=M.=g.(E(x)) , j=1,2 ,

J J J O<M. <l=L M.

J j J

Then we have:

Hence,

k . i Uko (R (x), tH\Eh (x) ~ =-2 k

LUk.(R (x),t)M.E.(x) 1 J J J

i,k = 1,2 , i + k , h = 1,2

k Uko(R (x),t)

(~ -:-2----''''-'k'-------)~Eh (x) . L U (R (x),t)M.E.(x) 1 kj J J

El(x)/El(x) Ml

E2 (x)/E2 (X) M2

1 If Ml = 2 = M2 , simultaneous settlement on both issues will occur.

1 If l-1l >2>M2 ,bargaining over pay will end before the settlement on

1 control. If M2>2>Ml , bargaining over control will end before the

settlement on pay. This shows the non-uniqueness of equilibrium. For a more detailed analysis of this example, see (20).

EQUILIBRIUM PATTERNS FOR BARGAINING UNDER STRIKE 231

v. POSSIBLE GENERALIZATIONS - A SKETCH

1. Integrative Bargaining. In the previous discussions, each issue represents a prize of fixed size to be distributed to the various players. But in reality, the propositions according to which the prize is distributed may influence the size of the prize itself. If y denotes the shares of all issues in the settlement: y=(y1, ... ,yN) and Fj(~) d:notes the size.of ~rize j~ the reward for

1 1 1 1 1 i on issue j is now z.~y. F.(y). Set z =(z1' ... ,zn); then the

J J J '" i . 1 i-1 payoff for i may be written as U.(z ,T). Defining l X=(x , ... ,x ,

i i+1 Nil k .. i R (x),x , ... ,x) where R (x)=J- L x as before and deflnlng x

k#i 0

analogously, the trajectory x(·) under a class of equilibrium stra­tegy N-tup1es may be characterized by the conditions:

'" iii i U.(R1 (x(t»F1 ( x(t», ... ,R (x(t»F ( x(t») 1 n n

'" iii i - U.(R1 (x )F1 ( x ), ... ,R (x )F ( x » 1 a 0 non 0

for all i .

Taking the time derivative, we get conditions for the "syndicate· collective concession rates" facing each player. From the concession rates of these N syndicates, each containing (N-1) players excepting the i-th, individual concession rates may be deduced.

2. Pressure Tactics. So far we deal with payoff functions of fixed form and the control of players only relates to the claims of shares in distribution. ~10reover, there is no other cost of strife besides the delayed settlement. Real-life bargainers may either (i) influence the attitudes of their rivals through persuasion or pro­paganda, or (ii) impose costs on their foes through the organization of boycotts, pickets, etc., or (iii) raise claims on issues embar­rassing their opponents but having no intrinsic interest to themselves. All these three types of situations can be handled through the intro­duction of "person-specific" issues which only affect the payoffs of some of the players but not those of the others. Consider for instance, a management which is interested in high entrepreneurial reward, solid control of the firm and good reputation of the product. The labor union is only interested in high wages.

The union may demand "co-determination" to create a favorable bargaining position but trade such demand away later for a higher wage hike. Or else the union may launch a picketing operation against the firm's sales division, mount a boycott campaign against the firm's product. Analytically, these types of operations shift parametrically the payoff function of the management, if one regards

232 s. CLEMHOUT, G. LEITMANN, AND H.Y. WAN, JR.

the good will of the product as a parameter. Or else, one may re­gard the goodwill as a person-specific issue. Whatever the union does is equivalent to claiming some share of that issue. Such an issue is irrelevant to the worker, but if the management does not bribe the union to cease and desist, then the reduction of the management's share will reduce the latter·s payoff. In our model, aggressive claims do not affect the equilibrium payoff of the oppo­nent. It may work to improve the settlement terms yi by resorting to pressure tactics. However, any such gain would be offset by a lengthened strife.

3. Bargaining differential games as components for super games.

(a) Sequential Games. Example 1 of the last section shows that many of the results in our model depend upon the initial position, yet the initial position is left unexplained in our theory. One approach to handle this problem is to set up a multi-stage game, each stage corresponding to a negotiation-contract cycle. The pay­off at each stage depends partly upon certain parameters - say, public image, etc.- which depend upon the levels of initial claims made in the past. Partly the payoff depends upon the current re­ward, which is controlled by other players' initial claims in the current cycle (as indicated by our earlier analysis). A differential game describing the bargaining process in a cycle is now imbedded in a multi-stage super game.

(b) Coalition Games. In many real-life situations, not every single player can block settlement. Consequently, winning coalitions may be formed which exclude some ineffective opponents, settle the issue and divide the spoils among the coalition members. Once a winning coalition is singled out, the bargaining among its members each of them can now block the coalition - precedes as in our bar­gaining differential game. Which alternative winning coalition will actually emerge will be explained by a (yet to be constructed) super game.

4. History-dependent games (21) . In a differential game frame­work, if a player has chosen a specific strategy, then his current action ("control") can only depend upon the current state, irrespec­tive of the particular history leading up to such a state. Enlarge­ment of the strategy space - in recognition of history - can only be accommodated by introducing new state variables summarizing certain salient features of past history. A more fundamental reformulation appears also possible. One may specify controls at time t to be dependent upon the entire history up to time t. This can be done through an "instantaneous strategy at t" which is a vector of func­tionals mapping the space of histories of length t-to into a vector of controls at t. As time proceeds, histories lengthen. Hence, instantaneous strategies for different values of t are defined over

EQUILIBRIUM PATTERNS FOR BARGAINING UNDER STRIKE 233

functional spaces of ditferent lengths, A strategy is a particular continuum of instantaneous strategies, each component of which deals with histories of a specific length.

VIr. REFERENCES

(1) Bishop, R. L., "A Zeuthen-Hicks Theory of Bargaining," Econometrica 32, 1964, 410-7.

(2) Case, J., "Applications of the Theory of Differential Games to Economic Problems," in Kuhn, H.W. and G.P. Szego eds. Differential Games and Related Topics, New York, Elsevier, 1971.

(3) Clemhout, S., G. Leitmann and H. Y. Wan, Jr., "A Differential Game Model of Duopoly," Econometrica, Vol. 39, No.6, November 1971, 911-38.

(4) Clemhout, S. and H. Y. Wan, Jr., "A Class of Trilinear Games," Journal of Optimization Theory and Application, 14, 4, October 1974, 419-24.

(5) Coddington, A., Theories of the Bargaining Process, Chicago, Aldine Publishing Co., 1968.

(6) Cross, J.G., "A Theory of the Bargaining Process," American Economic Review, 55, 1965, 67-94.

(7) deMenil, G., Bargaining: Monopoly Power Versus Union Power, Cambridge, MIT Press, 1971.

(8) Dussaix, A. and A. Haurie, "Un Mod~le Dynamique de Negociation Sous Forme d'un Jeu Semi-Differentiel," Rapport de Recherche No.5, Ecole des Hautes Etudes Commerciales, Montreal, September 1971.

(9) Fouraker, L. E. and S. Siegel, Bargaining Behavior, New York, McGraw-Hill Book Co., Inc., 1963.

(10) Harsanyi, J.C. and R. Selten, "A Generalized Nash Solution for Two-Person Bargaining Games with Incomplete Information," Working Paper no. 285, Center for Research in Management Science, University of California, Berkeley, October 1969.

(11) Harsanyi, J.C., "Approaches to the Bargaining Problem before and after the Theory of Games," Econometrica, 24, 1956, 144-57.

234 S. CLEMHOUT, G. LEITMANN, AND H.Y. WAN, JR.

(12) Harsanyi, J.C., "Bargaining in Ignorance of the Opponent's Utility Function," Journal of Conflict Resolution, 6, 1962, 144-57.

(13) Leitmann, G. and P. T. Liu. "Collective Bargaining: A Differential Game," Journal of Optimization The~~ A~plications, 13, 4, April 1974, 427-435 and 14. 4. October 1974, 443...,.4.

(14) Luce. R. D. and H. Raiffa, Ga~~~an~ Deci~~. New York, Wiley, 1957.

(15) Nash. J., "Non-Cooperative Games," Annals of Mathematics, 54. 2. September 1951. 286-95.

(16) Nash. J •• "The Bargaining Problem," Econometrica, 18, 1950. 155-62.

(17) Rapoport, A .• N-Person Game Theory, Concepts and Appl!cations. Ann Arbor, The University of Michigan Press, 1970.

(18) Siegel, S. and L. E. Fouraker, Bargaining and Group Decision Making. Experi~ents in Bilatera~~opoly. New York. McGraw­Hill Book Co., 1960.

(19) Walton, R.E. and R. B. McKersie. A Behavioral Theory of Labor Negotiations. New York. McGraw-Hill-- Book Co .• 1965.

(20) Clemhout, S .• G. Leitmann and H. Y. Wan, Jr., "Bargaining under Strike: A Differential Game View." Journal of Economic Theory, 11. 1. August 1975. 55-67.

(21) Basar. T .• "A Counterexample in Linear Quadratic Games: Existence of Nonlinear Nash Solutions." Journal of Optimi­zation Theory and Applications. 14, 4. October 1974. 425-30.

PERMANENT DIFFERENTIAL GAMES:

QUASI STATIONARY AND RELAXED STEADY-STATE OPERATIONS

G. Guardabassi and N. Schiavoni

Istituto di Elettrotecnica ed Elettronica Politecnico di Milano, Milan, Italy

ABSTRACT

This paper deals with permanent differential games, i.e. dif­ferential games the state of which must satisfy a periodicity constraint.

To any given permanent differential game a conventional (non­differential) game of obvious interest can be associated in a straightforward way by simply considering the equilibrium states only of the dynamical system under consideration. In this situa­tion a comparison between the solutions of the former (dynamical) and the latter (static) game is of obvious interest. In particu­lar, the main concern here can be expressed by the following que­stion: Under what conditions (henceforth called dynamic dominan­ce conditions), for a given Pareto-optimal solution of the asso­ciated static game, a nonconstant control exists such as to do­minate, in the Pareto sense, the constant solution above (which is Pareto-optimal within the class of the constant solutions only)?

The approach taken in the present paper to takle such a kind of question can be considered a standard one in Periodic Optimi­zation Theory [lJ and consists in restraining the attention to spe­cial classes of control functions (namely quasi-constant and chat­tering controls) leading to relatively simple and general dynamic dominance conditions, each one of which calls for nothing more than the analysis of a suitable static game.

235

236 G. GUARDABASSI AND N. SCHIAVONI

1. PROBLEM STATEMENT AND PRELIMINARY DEFINITIONS

Consider a finite-order continuous-time dynamical system described by

and time-invariant

x f (x,u) y = hex)

where, for any t € t, while

u(t) E U c Rm

x(t) G: Xc Rn

yet) Ie Y = RP

Rl . is , x the derivative of x with

(l-a) (l-b)

respect

(l-c)

(l-d)

(l-e)

are the control, state and output vectors, respectively.

to

The set U and the functions f(.,.): X x U--X and h(.) : X-+Y are assumed to be given. Furthermore for any ~ in a given subset T of the positive real numbers, let.f2. (.t:") be the set of all piece-wise continuous func­tions mapping [O,~) into Rm.

In what follows, only the permanent regimes of system (1) will be taken into consideration; hence, the state periodicity con­straint

x ('t) = x(O) (2)

is introduced, 1 thanks to which, for any 1:: € T, the 't: -periodic ex­tension over R of x(.) : [0,1::] ~ X is a -r -periodic motion of system (1) generated by (the 1:' -periodic extension of) u(.)e n(~). The pair (u(.), x(.)) will henceforth be referred to as a ~-perio­dic regime of system (1).

For system (1), under condition (2), a vector valued perfor­mance index of the form

, 1 f't' J [1:'; u(.) l!.:t 0 g(y,u) dt 1:' € T (3)

is considered, where g(.,.) : Y x U..."..Rq is a given function. Pro­blem (1)-(3) will now onwards be referred to as a permanent diffe­rential game (the term periodic differential game has been discar­ded in order to avoid any possible confusion with differential ga­mes defined on periodically time-varying systems).

To be consistent with the adopted notation, the assumption must be made [2J that f (. ,.) is such as to yield a unique ~ -perio-

PERMANENT DIFFERENTIAL GAMES 237

dic motion of system (1), for any (1.'; u(.» G Txll('1:"). t-.Thenever such an assumption fails to be true, the set of all possible ~-pe­riodic motions of system (1), corresponding to u(.)e.Il(~), should be parametrized through an additional decision variable w e W(~ ; u(.» C Rk, k~n. However, the new decision variable w seems not to play any essential role within the afore stated problem, but for an increased formal and computational complexity.

As for the overall performance vector (3), it has to be point­ed out that, even if the average value of a given current return function is often a simple and natural choice, yet it may well hap­pen that a non-integral measure of performance, like

max g.(y(,9.),u( .. S).)) ,901££0,,1:') 1.

ie-{1,2, ... ,q} ,

is more significant in particular situations.

Furthermore, note that problem (1) - (3) is a time-invariant permanent differential game. Actually, a more general formulation (not considered here, for the sake of simplicity) is the one where both the system and the current return function are periodically time-varying, with a fundamental period Y . If this is the case and the set T of the admissible periods isoa subset of the multi­ples of ~ , we have a periodic permanent differential game.

o

An almost trivial, yet fairly consequential property of time­invariant permanent differential games is that the value of the overall performance vector is time-shift invariant. This property can easily be exploited in a fairly interesing way in dealing with performance measures of a non-integral type. As an illustration, consider a simple problem charachterized by a two-dimensional per­formance vector of the form :

J L't:;u(.)1 It a

11't" - gl (y,u) dt 1:'" 0

In view of the above made remark, it should be apparent that the same problem, with the performance J substituted by

a

~fo'l: gl (y,u) dt

is equivalent to the preceding one in the sense that all the solu­tions of the first problem can be obtained from the solutions of

238 G. GUARDABASSI AND N. SCHIAVONI

the second one by a (possibly trivial) time-shift operation. Thanks to such an argument, however, no more than one max opera­tor can be cancelled in the performance vector, so that formal and computational difficulties may arise whenever more than one component of the overall performance vector is not of an integral type.

In any permanent differential game, a class of periodic re­gimes which is a priori of a great technical and conceptual inte­rest is constituted by the so called constant regimes, i.e. the regimes corresponding to standard equilibrium states (in a sy­stem theoretical sense) of system (1) under constant control ac­tions.

Whenever the attention is restrained to this basic subclass of periodic regimes, problem (1)-(3)apparently reduces to the following one

f(~, ii) = 0

y - h{jt) 0

GeU

J C ii] ~ g (y , 6)

(4-a)

(4-b)

(4-c)

(5)

Notice that problem (4), (5) is a standard stationary op­timization problem characterized by a vector valued performance index (Multicriteria Mathematical Programming Problem). To sol­ve it, nothing more than a static model of system (1) is requi­red, so that eqs'. (4-a) and (4-b) may occasionally be sustituted by some equivalent expression of the input-output equilibrium re­lationship, namely

y = if (IT)

or

'P (y, 6) = 0

Thus, it can be argued that the most natural way of exsten­ding a stationary multicriteria optimization problem, namely a game, in order to take into consideration the system dynamics is just to properly formulate a corresponding permanent differential game. As a matter of fact, by the common rationale which is be­hind the formulation of a standard differential game reference is basically made to the analysis of a transient, i.e. of a tran­sition from some given initial state (or set) to one (or more) terminal target set (sets). In other words the constant solutions of standard differential games do not constitute, in general, a significant subset of the solutions set.

PERMANENT DIFFERENTIAL GAMES 239

Thus, since it is hard to s~e a standard differential game as a way of describing a permanent situation; it is all but surprising that essentially the same problem has seldom been given in the past both a differential and a "nondifferential" game theoretic formula­tion, despite of the undeniable relevance that the notion of a per­manent regime takes on, even in a dynamic environment.

The inherent capability of permanent differential games of establishing a direct bridge, within the framework of the same prob­lem, between game theory and generalized control theory [3/ is one of the peculiar features of this new kind of generalized con­trol problems and, at the same time, one of the main motivations for further efforts in this area.

In order to precisely state the particular problem dealt with in the present paper, it is convenient to introduce some further definitions and notations.

Definition 1

Given a (function) set V and a vector valued function (func­tional) G(.) : V ~Rq, a Pareto-optimal input relative to G(.) and V is any V O e V such that

for no VEV.

Finally, the following notation is introduced. For any integer V , v e RV and 1:' 1ST, the- constant function over [O,'t'] the value of which is equal to v will be denoted by v(.).

So far, with reference to problem (4), (5), let fio be a Pareto­optimal input relative to J[.) and U. Then, going back to problem (1) - (3)fa natural and basic question is : under what conditions the control function 6°(.) can be proved not to be a Pareto-optimal input relative to J ['t';.J and.n. ("t'), for some 't'c T? In other words, under what conditions the constant solution ijO(.) can actually be improved in some definite way. by taking into consideration the sys­tem dynamics? In order to specifically state the problem raised by the question above, the following definition is now introduced.

Definition 2

If

i) n° is a Pareto-optimal input relative to j [.] and U

240 G. GUARDABASSI AND N. SCHIAVONI

ii) there exist 1:' e T and u(.) e n. (1:') .such that

o of J (1:'; U(.)] -J('1::; U(.)] ~ 0 •

then. UO(.) is said to be dynamically dominable.

In standard Periodic Optimization Theory. two basically dif­ferent approaches have been followed in order to get simple and useful dynamic dominance conditions. The first one [4] - [9J con­sists in restraining the class of the control functions to some suitable subset ofl1 so as to get. in a relatively simple way. suboptimal solutions to problem (1)-(3). Of course. each one of these solutions is a good condidate to be a dynamic dominant of GO(.). The second approach i101 - L147 consists in performing a local analysis around the considered steady-state in order to check for the possibility of improving (in some suitable sense) GO(.) by means of a properly designed control function variation. The for­mer approach will be followed also in the present paper. where the quasi-stationary and the relaxed steady-state techniques are suitably adjusted in order to encompass the multivalued performan­ce index case.

2. QUASI-CONSTANT CONTROL

First of all a regularity property in the control set has to be introduced through a suitable controllability concept.

Let

x ~ {x: f(x,u) = 0 e

for some u E U}

and \f> (. ; ••••• ) be the set of the (standard) equilibrium states and the transition function of system (1). respectively. then the following definitions can be given.

Definition 3

System (1) is controllable in X relative to g(.,.) if, for any pair xr, Xs e X. there exists a time t(xr, XS ) and a control u*(.; xr, X S): [0, f(xr , xs» .... U such that

i)

ii)

(-( r s) 0 r u*( •• , xr, x S » If t x,x ; , x

J rs is finite.

PERMANENT DIFFERENTIAL GAMES

Defini don 4

A subset U of U is a regular control set of system (1) rela­

tive to g(.,.) if there exist X c X such that e

i) system (1) is controllable in X relative to g(.,.)

ii) U = {ii eU : f(x,ii) = 0 , for some x ex} 1

As for the periods set T, assume that for any t E R there

241

exists t'ET such that 't'~t. Then, for any sufficiently large 1;:GT

a control function ll(.; 't')c t1 (1;) can be defined in the following

way.

Let {ul , u2 , •.• , uPr be a finite subset of a regutar control

set U of system (1) relative to g(.,.) and denote by x E X the

unique (in view of a ~reviouskassumption) equilibrium stat~ of

system (1) correspond1ng to u (.), k = 1,2, .•. , p. Furthermore,

let

and define

1 u

it (t; 1 2

U X ,x ) 2

u 2 ':l

G(t;"!:') ~ u* (t; x x~) ,

" o ~ t < tl A

tl ~ t < tl ...

tl ~ t <t2

t .( t 2- < t2

;..

t '" t<t p-l P ~ =' t <t = ~

]l Jl

(6)

A ) A ~ -( k k+l) where tk = (0(1 + 0(2 + •.• + O<k 't and tk tk - t x , x ,

for any k = 1,2, ... , p.

Now, in view of the adopted definitions, it turns out that

242 G. GUARDABASSI AND N.SCHIAVONI

A l[ -12 11 J [1:; u(.;'!:')] == ~ (o{l~ -t(x ,x ))g(h(x ),u )+J12 +

- 2 3 2 2 + (~2~ - t(x ,x)) g(h(x ),u ) + J 23 +

••• +(o{ 't" - t(xp,xl )) g(h(xJl), uP) + J ]= n )11

= (<Xl - t(x~. x2)) J LulJ+ (c(2 _ t(x~.x3))J [u2J+ ...

( N "t(xP. x~ J- [-uP] 1 ( ) ••. + "1)1- 1: -') + ~ J 12+J23+·· ·+Jpl

hence

lim J [1:'; G(.;1:")] .-E. k ~ o(k J [u ] (7)

't" -+- oD k=l

so that the values of the performance index obtainable by quasi­constant operation, i.e. by controls of the kind one obtains from G(.;~) as ~ goes to the infinity, does coincide with the convex hull of J (U]. For any set S, the convex hull of S will henceforth be denoted by co (S).

Theorem 1

If U is a regular control set of system (1) relative to g( ... ) and there exists J}f E co (J [ IT]) such that

then UO is dynamically dominable and a dominating control can be found within the class of quasi-constant controls.

Proof

If J *" E co (J [ ij 1 ), then, in view of a Caratheodory' s theo­rem [lSi , tj:ere2must exist _a posi ti ve integer p '" q + 1,;;( E A_ and a set -tll , u ,"', o.PJ cD such that P

-* J

H~nce, if_IT (.;~ is t~e c~~trol function o~tained from (6) by let­tlllg p = p, 0... =o(and u = u ,k 1,2, ... , p. then from (7) it follows that

o f. lim '2;'-'>- 00

PERMANENT DIFFERENTIAL GAMES 243

and the theorem is proved.

The dynamic dominance test given by Theorem 1 can directly be used in a relatively simple way whenever the number q of the performance indices is very small (say 2 or, at most, 3); other­wise, an equivalent result may be useful, thanks to which the abo­ve test is led to the analysis of a suitable Multicriteria Mathe­matical Programming problem. Specifically, consider the game

k k f(x , u ) = 0

l h(xk ) = 0

uk E U c U

0( € A 1 q+

} k 1,2, ... , q+l

1 2 1 q+l k k j\. [ uq+ ] ~ ~ ( J d..; u , u , ..• , L- 0( kg Y , u ).

k=l

Then, the following result can be established.

Theorem 2

A vector J * E co (J [ lJ ]) such that

o # J * - J [ \iO] ~ 0

(8-a)

(8-b)

(8-c)

exists if and only if ( /10 ... 0 I' ; u , 0, ..• , 0 ) is not a 1 ~ ~~ Pareto-optimal input relative to J [.;.,., ..• ,.] and A 1 XU. q+

Proof

Again in view of the Caratheodory's theorem, the feasible solution set of problem (8) does coincide with coO I~ fi :i ). Since

J [110 ..• 01 '; \io,O, •.. , oj

the theorem follows immediately from Definition 2.

3. RELAXED STEADY-STATE

By relaxed steady-state analysis (4], /6) - {9/ reference is generally made to the analysis of those states of system (1) which are equilibrium states in the Filippov sense Ll6J (nonstandard equi­librium states),

244 G. GUARDABASSI AND N. SCHIAVONI

One of the basic results in relaxed steady-state analysis is here derived in a way which is very simple and quite similar to the one adopted in the preceding section to introduce the notion of quasi-constant operation.

1 F2r any 't"ET and any positive integer p, let c( GoA and {u , u , ... , uP} cUbe such that Jl

Jl ... k :Eo(kf(x, u) 0 k=l

for at least one X € X. Then, letting

\:1(t;'t) ~

1 u

2 u

t 1~ t<t =1:', r P

(9)

where t k , k = 1,2, ••• , p, is the same as in (6), it is rather apparent that the 't -periodic motion of system (1) corresponding to u(.; 't:) tends to the relaxed equilibrium x(.) as 1: goes to zero, since

j t" p lim ! x(t) dt = 2: c( kf(x,uk) = 0 • 1:'.0 t' 0 k=l

Then,

lim J ['t; ti(.; 1:')] 1::' ..po 0

The control one obtains from u(.; t) as "t' goes to zero is often referred to as a chattering control.

Now, recalling that the order of system (1) has been denoted by n, consider the following (static) game :

n+q+l 2: k=l

k o(kf(x, u ) = 0

y - hex) k

u EO U, k

c( e An+q+l

o

1 , 2 , ... , n +q + 1 ,

(lO-a)

(lO-b)

(lO-c)

(lO-d)

PERMANENT DIFFERENTIAL GAMES

N [ 1 2 n+q+ 1] ~ J 0<; u , u , ... , u (lO-e)

Assuming that, for any t > 0, there exists 1: e T such that 1: < t, a dynamic dominance criterion can now be obtained which is just the counterpart of the criterion one gets, in the quasi-constant case, when Theorem 1 and Theorem 2 are suitably combined.

Theorem 3

245

If ( 110 ... 01' ; lio, 0, .•• , 0 ) is not a Pareto-optimal in-put relative to j [ .; ., ., ... ,.J and A 1 x Un+q+l , then lio (.) is dynamically dominable and a dominatin~+g6ntrol can be found within the class of chattering controls.

Proof

For any positive integer p, consider the game

P k 2:o(k f(x,u) 0 k=l

Y - h(~) = 0 k

u &U , k = 1,2, ... , p,

o<eA P

N

J [0<.; }l

1 u , 2 uP] ~ u , ••• ,

P: ,.J k 2:.oI.. kg(y,u) . k=l

If ( I 10 ... 0 I' ; lio, 0, .•. , 0) is not a Pareto-optimal input relative to j [.;., ... ,.J and A x UP, then there exist 01 IS A and ijl, li2 , .1;1., aPe u such tha~ P

-PJ •• ', u -

Hence, an obvious consequence of the above discussion is that ti° (.) is dominat~d by th~ chftte:tng control a(o;"t') obtained from (9) by lett~ng 0{ =0{, u = u , k = 1,2, •.. , }l. Ey noting that the vector

P k ~o(k f(x,u ) k=l

~ ('" k L....<Xk g y,u ) k=l

246 G. GUARDABASSI AND N. SCHIAVONI

~ n+q anyhow belong to the convex hull of 't' (x,U) c R , where

f(x,u)

g(h(x), u)

the conclusion can be drawn, in view of the Caratheodory's theorem, that it is completely useless to have p greater than n+q+1. On the other hand, if (110 ••• 01'; uO, 0, ••• ,0) is not a Pareto-optimal input relative to J [.;.,., ... ,.] and A x UP, it is not either a Pareto-optimal inpu~ relative to Jv [.;.,~, ••• ,.] and AI' x uV , for all V~ p. Thus, the most general condition is apparently ob­tained by letting p to take on a value equal to its upper bound, namely p = n+q+1.

Remark 1

If some of the components of f(x,.) and/or g(h(x),.) are con­stant on U, for all x e X, then the number n+q+ 1 in (10) can obvious­ly be reduced to n*+q*+l, where n* and q* are the numbers of the components of f(x,.) and g(h(x),.), respectively, which are non­constant on U, for some x E X.

Notice that, ~ being "a priori" unknown, the dynamic domi­nance condition of Theorem 3 cannot be given an easy to use geome­tric interpretation. A geometric dynamic dominance condition can instead be obtained by restraining the attention to a suitable spe­cial class of chattering controls. Letting XO and yO be the con­stant values of x and y, respectively, corresponding to uo(.), the following result can in fact be stated.

Theorem 4

If

i) f(.,.) and h(.) are continuous at (XO, nO) together with their first partial derivatives with respect to x.

ii)

iii) g(.,.) is continuous on £Y1 x U together with its first partial derivative with respect to y.

iv) It-

there exists a € co (t (U» such that

f/J I: l: - ~ (UO) ~ 0 ,

PERMANENT DIFFERENTIAL GAMES

where the function follows

t(·) : is defined as

a (u) = g(yO ,u)-g/yO ,UO) h)xO) [f)XO ,GO)] -If(xO ,u),

then crO(.) is dynamically dominable and a dominating control can be found within the class of chattering controls.

Proof

247

Since J'~ co (d (U», there must exist (in view of the Caratheo­dOfY'sztheorem~ a positive integer ji~q +1,0< € A- and a set {u , u , ... , i1} c U such that Jl

JI - k *" L 0(. Y(u ) = y • k=l k() (J

Then, for any j3e[o,l], let tk ~ «;{l+O(Z+'" + O(k)'t' and consider a piece-wise constant control defined as follows

ul 0 ~ t < (-'tl

'" ijZ f-'tl~ t < /-' £Z

u(t;"t!'f)= •...••••••..•....•...•.•..•

uJl, ft}l-l~ t<ftJl _0

u (->'t' ~ t< 't'

so that TI(.;'t' ,(3) becomes a particular chattering control, as 't' goes to zero. Correspondingly, if there exist x Y;) e X such that

then

(l-f) f(x (f)' ti°) +~ t~kf(X (~), uk) = 0

lim Jrt:; u(.; 't' 'f)] = (1 - f) g(h(x(f»' UO) + t'--o

+~ fakg(h(x(f»' uk). k=l

Now, notice that x(O) = XO (h(x(O»= yO) and, in view of assumpt­ions (i) and (ii),

dx (fb)

df-> f= 0

248 G. GUARDABASSI AND N. SCHIAVONI

Hence, in view of

d J~1'l1 d{J ~=O

assumption (iii),

_g(yO, fiO) + ~ O<kg(yO,uk ) -k=l

Therefore, the continuity assumptions and

o " ~O

(!= 0

.... imply the existence of ,se[o,l) such that lim IT(.;'t',f) does dominate iiO(.), for all fE(O,"fo)' 't'-O

By pursuing exactly the same line of reasoning adopted to pro­ve Theorem 2, the following result (given without proof) can also be established.

Theorem 5

~ If assumptions (i) - (iii) of Theorem 4 hold, then a vector o G co (t(U» such that

o " (* - t (ITO) ~ 0

exists if and only if, letting

N q+l V (~; }, u2 , ... , uq+l ) ~ 2:.o<'k Y(uk) , () k=l 0

in turns out that (/10 ••• 01'; ITo, 0, ••• ,0) is not a Pareto-opti mal input relative to l(';""""') and Aq+lx U(q+l). -

4. CONCLUDING REMARKS

The notion of permanent differential game has been introduced in this paper as the most natural way of including the system dy­namics into a problem originally described by a standard (nondif­ferential) game.

Thus, the (control) problem of finding a suitable permanent

PERMANENT DIFFERENTIAL GAMES 249

(not necessarily constant) regime for the given system turns out to be sharply separated from the one of taking its state from a given initial value to the desired regime in some acceptable way (transient design). Despite of the fact that in most practical applications the first kind of problems is by far more important than the second one,it has to be pointed out that only the latter is generally discussed in the theory of differential games, where permanent regimes (when­ever analyzed) may be obtained in all but a simple way by considering the limit form taken on by the "intermediate part" only of the solu­tion,as long as the lasting of the control interval goes to the infinity.

As far as the comparison between a permanent differential game and its stationary version is concerned, two kinds of dynamic do­minance conditions have been obtained in this paper by restraining the attention to special classes of control functions.

These classes, namely quasi-stationary and chattering controls, were so chosen as to cut out, in a sense, the system dynamics and reduce the search for a dynamically dominating periodic regime to the analysis of a suitable static game.

Among the many problems which still are open in this poorly explored area, particular attention seem to deserve the possibility of improving (in some suitable sense) any constant solution of the considered game by control variations the spectrum of which be nonzero also at intermediate frequencies or the discussion of the several (and to some extent similar) problems arising when a not completely cooperative situation is dealt with.

ACKNOWLEDGMENT

The work reported here has been supported by the Centro di Teoria dei Sistemi (CNR).

REFERENCES

1. G. Guardabassi, A. Locatelli, S. Rinaldi: "The Status of Periodic Optimization of Dynamical Systems", Journal of Opti­mization Theory and Applications, vol. 14, no. 1, pp; 1,20, 1974.

2. G. Guardabassi "The Periodic Optimization Problem", to appear in Journal A - The Benelux Journal of Automatic Control.

250 G. GUARDABASSI AND N. SCHIAVONI

2. Y.C. Ho: "Differential Games, Dynamic Optimization and Genera­lized Control Theory", Journal of Optimization Theory and Applications, Vol. 6, no. 3, pp.179-209, 1970.

4. F.J.M. Horn, J.E. Bailey: "An Application of the Theorem of Relaxed Control to the Problem of Increasing Catalyst Selecti­vity", J.ournal of Optimization Theory and Applications, Vol. 2, no. 6, pp. 441-449, 1968.

5. A. Locatelli, S. Rinaldi: "Optimal Quasi-Stationary Periodic Pro­cesses", Automatica, vol. 6, pp. 779-785, 1970.

6. S. Rinaldi: "High-Frequency Optimal Periodic Processes" IEEE Transactions on Automatic Control, vol. AC-15, no.6,pp. 671-673, 1970.

7. J.E. Bailey, F.J.M. Horn: "Comparison Between Two Sufficient Con­ditions for Improvement of an Optimal Steady-State Process by Periodic Operation", Journal of Optimization Theory and Applica­tions, vol. 7, no. 5, pp. 378-385, 1971.

8. J.E. Bailey: "Optimal Periodic Processes in the Limits of Very Fast and Very Slow Cycling", Automatica ,vol. 8, pp. 451-454, 1972,

9. M. Fjeld: "Quasi-Stationary and High-Frequency Periodic Control of Continuous Processes", 5th IFAC Congress, Paris (France) ,1972.

10. G. Guardabassi: "Optimal Steady-State Versus Periodic Control: A Circle Criterion", Ricerche di Automatica, vol. 2, no. 3, pp. 240-252,1971.

11. S. Bittanti, G. Fronza, G. Guardabassi: "Periodic Control: A Frequency Domain Approach", IEEE Transaction on Automatic Con­trol, vol. AC-18, no. 1, pp. 33-38, 1973.

12. M. Matsubara, Y. Nishimura, N. Takahashi: "Optimal Periodic Con­trol of Lumped Parameter Systems", Journal of Optimization Theo­ry and Applications, vol. 13, no. 1, pp. 13-31, 1974.

13. G. Guardabassi, N. Schiavoni: "Boundary Optimal Constant Control Versus Periodic Operation", 6th IFAC Congress, Boston (USA), 1975.

14. S. Bittanti, G. Fronza, G. Guardabassi: "Optimal Steady-State Versus Periodic Operation in Discrete Systems", to appear in Journal of Optimization Theory and Applications.

15. R.T. Rockafellar: "Convex Analysis", Princeton University Press, Princeton, 1970.

PERMANENT DIFFERENTIAL GAMES

16. A.F. Filippov: "Differential Equations with Many-valued Discountinuous Right-hand Side", Soviet. Math. Dok1., no. 4 pp. 941-945, 1963.

251

MODELLING & CONTROL OF THE U.K. ECONOMY*

J. H. Westcott

Department of Computing & Control

Imperial College, London, England

Abstract

A small model of the U.K. Economy has been estimated from official statistical records utilising extreme parsimony in the equations and hence considerable aggregation of variables. The size of the model is determined by the need to say something about the four major policy problems facing the U.K., namely, unemploy­ment, price stability, economic growth and balance of payments equilibrium. It has 16 behavioural equations and 11 identities, and is conveniently divided into 3 main sectors of Expenditure, Distribution and Trade. Simultaneous estimation procedures are employed for determining the parameters and disturbances within each sector.

A number of control variables have been included in the model and the equations have been transformed into state-variable form in order to utilise the techniques and results of optimal control theory in its linear quadratic Gaussian form. The weights attached to the terms of the quadratic performance index condition the character of the performance obtained and their choice therefore remains a matter of political considerations. While the economy is certainly not linear it is possible that the resulting linear controller, using as it does the feedback principle, would give a very tolerable approximation to the desired performance in control­ling the real economy.

An experiment is described by means of which it is hoped to demonstrate this by utilising the large-scale forecasting model of the London Business School as a simulation of the true economy. The paper describes the design considerations that arise in

*POST-IFAC/75 Seminar

253

254 J.H. WESTCOTT

applying the linear controller from the small model for purposes of controlling the larger one.

Introduction: The "Programme of Research into Econometric Methods" (PREM) represents an unusual conjunction of interests between control engineers and economists with the support of the Social Science Research Council. To economists it is not at all obvious that the linear continuous dynamics, which engineers have used so successfully, is at all applicable to analysing the behaviour of the economy. Even to an engineer it is clear that the economy represents a 'difficult' system in that it represents a very 'noisy' one to which statistical analysis is essential.

A further feature which is distressing to an engineer is the extreme brevity of records; only 17 years of quarterly samples are available. In addition there is next to no possibility of measuring responses to test signals since such cavalier treatment of the economy is not currently socially acceptable. Finally there is one other factor which complicates the use of standard identification procedures. The system is not really ever an open loop one since officials are always trying to control some aspect of its performance by methods which are often very crude, not to say brutal.

However, there is one consolation in the fact that the sample period is quarterly. This means that the amount of computation that can be undertaken between samples is very great. In this sense there are new possibilities for the control expert since very sophisticated control algorithms can be used.

The problem: It is required to derive a relatively simple dynamic model of the economic system whose complexity shall be such as to provide a realistic context for consideration of the four main policy problems confronting the U.K. These are the problem of controlling unemployment, while at the same time aiming at some degree of price stability. This has implications for economic growth with the attendant difficulties of maintaining balance of payments equilibrium.

Derivation of the model: Development of the model commences from the Keynesian statement that aggregate supply is equal instantaneously to aggregate demand and proceeds by decomposing both supply and demand into components of particular interest. Supply consists of GDP(Y) and Imports (M) while demand is divided into five components: Consumption C, Investment I, Change of stocks ~S, Government expen­diture G and Exports X.

Thus Y + M = C + I + ~S + G + X

MODELLING & CONTROL OF THE U.K. ECONOMY

A popular move in economics is to express these quantities in % growth form thus

y =<¥)c + <V)i + cf'.i)f'.s + <y)g + <~)x - <~)m where the lower case notation corresponds to % growth rates.

255

From the control point of view this move has advantages; it ensures that the series have stationary characteristics allowing the use of standard estimation techniques, the noises tend to be whiter and there are fewer equations necessary with more simple dynamics involved.

From this basic statement a set of dynamic behavioural equations can be written down in terms of the economic variables. There are 16 output variables of interest, 12 of them being expressed in terms of a number of other dynamic variables together with a noise term. There is also a set of 11 identities which are relationships of an accounting type without dynamics. The full details of the equations are given in refs. 4 and 5.

Estimation of parameter: Each parameter in the equations has had to be estimated from the available data given in the official blue book of economic data for the U.K. Initially the AstrBm-Bohlin estima­tion technique was utilised to obtain the maximum likelihood esti­mates for each equation taken on its own. This results in a residual noise term at the end of each equation some of which have dynamic terms of their own. That is to say they consist of filtered or 'coloured' noise rather than white noise. However, strictly speaking these noise terms are not independent of one another and a more com­plicated simultaneous estimation procedure needs to be utilised if these coupling effects which are significant are to be taken into account. The technique for doing this is described in detail in ref. 3 together with a simple worked example. It is interesting to note that the simultaneous estimation procedure tends to produce a more simple set of equations than the one-equation-at-a-time estima­tions.

Control of the System: The interest in these equations centres on the possibility of controlling outputs more closely and for this purpose it is necessary to identify those variables which are at our command and those which are not. In Fig. 1 is given a classification of variables distinguishing the outputs from the identities and signifying \vhich of the remaining variables may be regarded as controls in this sense and which must be regarded as exogenous without any possibility of influence internally. World trade index and import prices clearly come into this category but this still leaves six control variables, three of which are not surprisingly taxes. The incidence of the controls in the various equations, it

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MODELLING & CONTROL OF THE U.K. ECONOMY 257

will be observed, is far from a one-for-one situation, so that the orchestration of control variables to achieve any given manipulation of a particular output variable will not be simple.

Optimal Control: This seemed a natural opportunity to apply the ideas and techniques of optimal control. However in order to do this in the most direct way it is necessary to express the dynamic equations in state-vector form.

Thus the equations that have been derived are in the normal structural form with which economists are familiar as follows:-

-1 -1 -1 A(z ) y(k) = B(z ) u(k) + C(z ) w(k)

where z-l is the backward shift operator w is the noise disturbance.

In order to apply optimal control theory it is necessary to transform this into a minimum state-space realisation as follows:-

x(k+l) = Fx(k) + G u(k) + G w(k)

y(k) = Hx(k) + D u(k) + Ew(k)

An attempt was made to do this using observable Pair Realisa­tions with the unwelcome result that a greatly increased number of parameters resulted not easily related to the previous ones. Motivated by Desoer and Mayne a block observable canonical form was utilised with very satisfactory results as illustrated in ref. 2.

The most difficult task now remaining is to agree a form for the quadratic measure of performance. A number of views exist on how this should be done. It is in any case a political decision as to how much emphasis to place on different desirable outcomes. As to how one weights the desirability of full employment against the drawback of a poor balance of payments is a choice with heavy poli­tical overtones. Most workers have been content to select a small number of outputs and to weight these in a 'rather' arbitrary way. Some examples of this applying to PREM models are given in ref. 4. An interesting suggestion by Bray (ref. 5) is that a complete set of weights should be selected for variables and regarded as a type of 'shadow' price as in L.P. optimisation. Naturally the results obtained are strongly coloured by the particular choice of weights and here one has to be guided by the motivations of the user, but once this is agreed the control strategy that results cannot be improved upon within the limitations of the assumptions made of a linear system with quadratic performance index disturbed by per­turbations having Gaussian distributions.

258

I~

,

7 • I

~50 inputs

I 1)

I '~1 1<1= Con troIs .'::. (Order 6)

L.B.S. Model 350 NIL equations

Optimal Controller <1 for PREM Model .....

L.Q.G.

KJ Solution

~ = Lk~ + hk

~150

outputs; use

"'- ~ 50 I

J.H. WESTCOTT

Aggregator ~ & Differen-

j-i"j-nr

16 for

Observer for PREM states

(30) ~

(~ 16 r 14

states direct states by observer

PREM Model

outputs PREM

(16 Dynamic equations ill Identities

Figure 2. Optimal Controller for Prem 2.2 Model Driving the U.K. Economy Represented by the L.B.S. Large Scale Model

MODELLING & CONTROL OF THE U.K. ECONOMY 259

The L.B.S. Model

The result is that we have the specification for a linear feedback controller that would be ideal if the economy were really the linear approximation that our model is. Unfortunately it is not such a simple system when looked at in sufficiently fine detail. On the other hand many technological processes have similar characteristics when looked at closely, but they are nevertheless very successfully controlled by linear control systems designed on assumptions of linearity. The well-known virtue of linear feedback control lies in its ability to m~n~nnse the effects in the overall performance of such departures from the assumptions.

So the question arises of how successful would the linear controller be in controlling the real economy. Since it is not currently allowable to do an experiment on the real thing it will be necessary to make as realistic a simulation of this as possible and to try the experiment this way. To design such a simulation from scratch would be a major task, but there is no need for this. A number of large-scale forecasting models of the U.K. economy have been established and it has been possible to take advantage of one of the most consistently accurate of these by a collaboration with a team working at the London Business School (6). The model con­sists of approximately 350 non-linear equations, having about 50 inputs and 150 outputs. It is constantly being up-dated as new data become available using techniques of parameter estimation that have proved in the past to give the greatest consistency and stability of performance in forecasting.

As is well-known to control engineers a model required for pure prediction purposes will have different characteristics to one used as a reference for control purposes. It is a point which economists find difficult to appreciate, since their thinking has been mainly in terms of open-loop correction of output variables. The powerful stiffening effect that feedback control has is unfamiliar to them. It is for this reason that a convincing demonstration of this effect in credible terms assumes such importance.

Matching the Models

a) The Outputs

However there are problems of technical interest in the task of matching the feedback controller designed on the basis of a highly aggregated model to a large-scale simulation of the economy in which a much lower order of aggregation has been utilised throughout. The situation is illustrated in Fig. 2 where it will be observed that while the L.B.S. model has about 150 outputs, the PREM model is based on a total of 16 output variables. Furthermore the outputs of the L.B.S. model are expressed in terms of levels

260

% Tax 100% 90

67%

33%

,J. intercept d

----- --- - --- ---~--~ ..... ~Cx

'I' b

¥

I

intercept e I

~I

J.H. WESTCOTT

+ d

x, Taxable Income

5 Parameters involved;- a, b, c, d, e.

Figure 3. Example of Disaggregation in Relation to a Control Instrument Based on Personal Taxable Income

MODELLING & CONTROL OFTHE U.K. ECONOMY 261

while the PREM variables are % growth variables. Two things are needed in order to match them; a number of outputs from the L.B.S. model (about 50) need to be aggregated in groups to represent the aggregated variables represented in PREM and then these variables need to be differentiated to form the % growth variables employed in PREM.

The next problem arises due to the fact that the optimal controller derived by optimal control theory from the PREM model provides a controller connecting the state variable x to control variables u. The state-vector has order 30 and has to be esti­mated from-the 16 output variables y. An 'observer' is designed for this. Of the 30 states, 16 are-available directly, leaving 14 to be estimated by the 'observer'. This is fairly straight forward, using standard observer theory.

b} The Inputs

The more difficult problem arises at the control variable interface. The L.B.S. model has about 50 input variables: some of these are exogenous inputs which cannot be regarded as the basis of control instruments, but even so there are many more that can be so regarded than the 6 control instruments utilised in the PREM model. The problem then is to find a way of disaggregating the control variables used in PREM so as to match those utilised in the L.B.S. model. Unfortunately there is no unique way of doing this; to meet a given value of an aggregated control variable from PREM may be satisfied in many different ways using the inputs available in the L.B.S. model.

The difficulty is perhaps usefully exemplified by taking as an example that of personal income tax. As is customary, this is a progressive tax in the sense that the larger the personal income the greater in proportion the rate of taxation, so any idea of a linear approximation disappears as a possibility immediately. In fact there is also a dead band since below a sufficiently small income no tax is exacted at all. The tax rates are also constant over a band of income and then jump to another rate suddenly. The staircase curve in Fig. 3 shows the % tax plotted against taxable income. The concept of taxable income involves further complica­tions, since individuals with the same gross income may attract quite different scales of allowances before taxable income is arrived at. The bold curve in Fig. 3 shows an approximation to the true one which allows the curve to be defined by 5 parameters a. to e. shown on the diagram. When this is associated with figures representing the spectrum of incidence of taxable incomes of given sizes in the economy it is possible to calculate the total tax yield from personal incomes and it is this quantity which assumes the role of one of six control variables resulting from the PREM model (see Fig. 1 for the others).

262 J.H. WESTCOTT

Clearly any given aggregated value of the total tax yield from personal incomes can be arrived at by a number of different choices in the values for the parameters a. to e. in Fig. 3, but there might be very considerable differences in political terms between them and so once more, as with the performance index weights, decisions have to be made on political grounds, a matter which lies outside the scope of technical problems. Reliance is placed in the known skill of Civil Servants in the Government in securing a specified tax yield that a Minister decides he wants to "take out II of the economy. It is not an exact science as some well-publicised miscalculations show, but it may be noted that in the past the exercise has always been performed on an open-loop basis with only infrequent and often too-long delayed corrections being made after things have obviously gone wrong.

The experiment that is proposed here will illustrate the bene­fits of continuous feedback control derived from all outputs and in consequence the sensitivity of the disaggregation exercise may well be much reduced. Similar problems arise with the other two tax instruments among the control variables. Even more contentious is the case of Government personal welfare payments where, as with taxes, the rates of payment are at the choice of the Government, but calculating the aggregated magnitude of payments resulting from these is a difficul·t and complex calculation. The remaining two controls of minimum deposit rate and exchange rate are happily already in rate terms and more straight forward.

An agreed solution to the matching problem is being actively sought and it is hoped very shortly to obtain the first results from this interesting control experiment.

References

(1) Wall, K.D., Preston, A.J., Bray, J. and Peston, M.H., "Estimates for a Simple Control Model of the U.K. Economy", Chapter 1 in Modelling of the U.K. Economy, Heinemann Press, London 1973.

(2) Preston, A.J. and Wall, K.D., "Some Aspects of State Space Models in Econometrics", Proceedings of IFAC/IFORS International Conference on Dynamic Modelling and Control of National Economies, University of Warwick, 9-12 July 1973.

(3) Wall, K.D. and Westcott, J .H., "Macro-Economic Modelling for Control", IEEE Transactions Automatic Control Vol. AC-19 No.6, December 1974.

MODELLING & CONTROL OF THE U.K. ECONOMY

(4) Wall, K.D., and Westcott, J.H., "Policy Optimisation Studies with a Simple Control Model of the U.K. Economy", Proceedings of the 6th IFAC Congress, Boston 1975.

263

(5) Bray, J., "Optimal Control of a Noisy Economy with the U.K. as an Example" in Journal of the Royal Statistical Society, Series A, Vol. 138, Part 3 (1975) pp. 339-373.

(6) The L.B.S. Quarterly Econometric Model of the U.K. Economy.

DECENTRALIZED MANAGEMENT

AND OPTIMIZATION OF DEVELOPMENT

Roman Kulikowski

Institute of Organization and Management Polish Academy of Sciences .

1. STATEMENT OF THE PROBLEM

The present paper deals with a normative model of long term development of a centrally planned economy. The productive subsystem of the economy consists of n sectors Si' i = 1,2, ••• ,n,shown in Fig. 1. Each sector

X: xl X: " '\,,~/

';j

X·· ~. "

s· s- . , . , .. to

ZVb •• /I Zv; 'J

7Js~ ••. '"

Xj& Xfi

Fig. 1. The n-sector productive subsystem

265

266 R. K,ULlKOWSKI

produces Xii goods per year and purchases Xji ,j=1, ••• ,n goods from sectors Sj in such a way that the net profit

n

Di ,= Yii - I: Yji , Yii = PiXii' Yji = pjXji , (1) j=1 j~i

where Pi price attached to the commodity Xii ,i=1, ••• ,n, is maximum.

The first-level decision center (D.C.) allocates the gross product (generated by the production system be­fore the base year):

n n n

YO = L Di = L Yi ; Yi = Yii - L Yij , (2) i=1 i=1 j=1

j~i

among the production factors and government expendi­tures: Z1 - capital investments, Z2 - labour (equal to the consumption), Z"" (v =3, ••• ,m) - government expendi­tures in education, health services,research and devel­opment, etc., in such a way that the given utility func­tion U(Z1"",Zm) is maximum subject to the constraints

-IlL L Zv ~ YO' Z V ~ 0, v = 1, ••• , m • ( 3)

v=1 It is assumed that the sector production

~i' i = 1, ••• ,n, depend on the inputs Yji,j as well as the production factors Zvi' i.e.

where

Yii = ~i[Z1i'···'Zmi'Y1i'···'YniJ,

)n: Z vi $. Z,., f Y = 1, ••• , m. i=1

functions = 1, ••• ,n,

The main idea behind the decentralized management of development consists in decomposition of decision vari­ables in such a way that sectorial decisions (regarding Y .. t j=1, ••• ,n t for each sector i) can be made independ-~J

end of the higher level decisions (regarding Z;i' V= 1, ••• ,m) •

DECENTRALIZED MANAGEMENT AND OPTIMIZATION 267

As shown in Refs. [2,3J the decomposition of deci­sions can be achieved for typical production functions such as Cobb-Douglas, C.E.S., etc.

In the case of generalized Cobb-Douglas production functions

[ 1 q. 1..!!..1 0(. .• Yii = Fi (Z1i, ••• ,Zmi) ~. YjiJ~, i = 1, ••• ,n (5)

J=1 j;ti

under optimum (sectorial) strategy Yij = f ij ,j=1, ••• ,n, the following relations hold:

Yji = Di.ji'Iii , j,i = 1, ••• ,n, (6)

= nn (oG ji) OG'ji/qi p1/qi 'Iii Fi (Z1i,···,Zmi) p. i'

j=1 J . j~i ~=1, ••• ,n, (7)

Di(Yji) = Di = (1 - qi)'Iii , i=1, ••• ,n. (8)

The two important results follows from the relations (6) - (8):

1. As follows from (6) the normative n-sector non­-linear model (1)-(5) behaves, under optimum strategy in a similar way to the linear Leontief model with thj technological coefficients (X ji' i, j=1, ••• ,n, i ~ j ~ • However, the outputs 1ii' i=1, ••• ,n, in the non-linear ~odel are specified in.a.unique manner by prices Pj' J=1, ••• ,n, and Fi coeff~c~ents.

2. When prices are fixed the sector net profits Di , i=1, ••• ,n, do not depend on sector interflows 'I ji , j,i= =1, ••• ,n, i~j, and the gross net product Y = ~Di is a linear function of Fit i=1, ••• ,n.

The result 1 can be used for a simple estimation procedure of ~ji coefficients. Assume for that purpose that the input-output statistical tables for an n-sec­tor economy be given.,Assume also that each sector op-

*) As shown in [3] a similar relation can be obtain­ed for n-sector C.E.S. function model.

268 R. KULIKOWSKI

timizes the net profit (1) so that the relation (6) be valid. Then it is possible to derive the numbers ~ji(t) = Yij(t)/Yii(t) = t ij(t)/tii(t),j=1, ••• ,n, for

each sector i=1, ••• ,n, provided the input-output tables for the past time t = 0,-1,-2, ••• are known. Then using the well-known statistical estimation methods it is pos­sible to find the estimates ~ji of ~ji which fit the statistical data. That method has been used in partic­ular for the estimation of 'Xji coefficients in the long term development model (MRI) of Polish economy which has been recently constructed at the Institute of Organization and Management of Polish Academy of Sci­ences.

The result 2 is very helpful when one considers the allocation of Zy' Y =1, ••• ,m, among the Si sectors of the production system. Before atacking that problem it is necessary to introduce the time variable explic­itely. For that purpose it is convenient to deal with the output production intensities Yi(t),rather than the integrated (within 1 year) quantities Yi ,i=1, ••• ,n. For the same reason we shall deal with the produotion fac­tor intensities X,>i (t), instead of Zvi t v =1, ••• ,m, i = = 1, ••• ,n. The relation between Yi and zvi is assumed to be nonlinear and inertial. A possible approximation of that relation is an integral operator

Yi (t) = rJ kli (t)jf.>" , t ~Y = 1, (9) ,,;,=1 v=1

where t

5 ~y ~)

k~i(t-'l1) [zVi(1:)] d1:, 0 <<¥y< 1 -0) (10)

kVi(t) = given non-negative funotions which become zero for t < 1: ; \::JJ.,)' (!>y, v =1, ••• ,m = given positive numbers.

A typical example of kVi(t) is the delayed exponent­ial function

*) The continuous variables are used here instead of discrete (changing once a year), which is a matter of convenience rather than general methodology.

DECENTRALIZED MANAGEMENT AND OPTIMIZATION 269

- S'Yi(t-Tyi) k Vi ( t) = KYi e , t > T))i (11 )

= 0, t < T))i' where KYi , SVi,Tvi = non-negative parameters.

The delay T'Yi expresses the construction delay (in the case of capital investment), or delay be"tween re­search and development etc. The parameters hY,i specify the depreciation in time of capital, R and D,educatio~ etc., while ob y are responsible for nonlinear saturation effects (i.e. the decreasing return to scale with re­spect to capital investments and other government ex­penditures).

As shown in [2,3] the parameters TVi' bYi,KYi of (11) can be estimated by least r.m.s. method provided the past data concerning the expenditures zVi(t), t = = 0,-1,-2, ••• , v= 1, ••• ,m, i = 1, ••• ,n, are known as well as the estimates of Fi(t), i = 1, ••• ,n, t~O.

Assuming that all the Fi (Z1i' ••• 'Zmi)' i = 1, ••• , n, functions have been identified one can formulate the problem of optimization of long-range development. The gross product generated within the planning interval [O,Tl has the present value

n T Y = L S wet) Yi (t) dt, (12)

i=1 ° where wet) = (1 + E. )-t, £ - the given discount rate.

The functions Yi(t) in (12) depend on the expendi-ture intensities ~Yi(t), t" [O,T] , Y= 1, ••• ,m, i = 1, ••• ,n, which should satisfy the following contraints:

n T I 5 w»(t) Z)Ji (t) dt ~ Zv' y = 1, ••• ,m, (13)

i=1 ° ZYi(t»,.O, t6[0,T], y= 1, ••• ,m, i = 1, ••• ,n, (14)

wv(t) = weight attached to the v-th expenditure (when the expenditure is financed by the bank or foreign loan and should be paid back not later than t = T, one can assume wv(t) = (1 +~)T-t, where? - interest rate),7v= given amount of production factors (government expendi­tures).

270 R. KULIKOWSKI

])eels/on (eve l

-i

/1/

Fig. 2. Allocation of Gross National Product

The development optimization problem can be formulat­ed as follows. Find the non~negative functions zYi(t) : : ZVi(t), 'J: 1, •.• ,n, i : 1, ... ,n, t e[o,T], such that the functional (12) attains maximum subject to the con­straints (13), (14).

It should be observed that the total value of pro­duction factors and government expenditures Zv' v: 1, .•• ,rn, depends on the GNP (YO) generated by the economy before each base year as shown in Fig. 2. The income of labour force (Z2) is being spent on private consump-tion. The supply of commodities generated by the pro­duction system should be equal to the demand. It is possible to show [3J that a non-negative system of prices exists which takes case of readjustment of the supply to the demand. In order to derive the long-range development projections it is necessary to construct a model of consumption structure and technological change. It is also necessary to take into account the foreign trade, environment, limitations of natural resources etc. Since the volume of the present paper is bounded we shall restrict our analysis however to the solution of the optimization problem (12)-(14).The problems con­nected with modelling of prices, technological change,

DECENTRALIZED MANAGEMENT AND OPTIMIZATION 271

utility and consumption structure change had been stud­ied elsewhere (1,2,3J.

2. OPl'IMlZATION OF DEVELOPMENT

Consider first of all the single sector model with the production functional

T m

y = 5 n fy(t) dt, (15) _N 0 ~~ l~

f)J(t) ={s k,)(t-'t) x,,('C') d1:J ' if:J'J= 1, o ))=1

.~v Xy(tt') = [zv(t)] ,0 (06»<1, v= 1, ••• ,m,

ky(t) = given functions, ky(t) = 0, t < 0, having "in­ertial" Laplace transforms Ky(p). We assume that v = 1 corresponds to the most inertial term. When, e.g. Ky(p)= = eTl>P,))= 1, ••• ,n, one can set T1~T2~ ••• ~Tm.

It is.necessary to find non-negative strategies x.,>(t) = x.y(t), y = 1, ••• ,m, t t: (O,TJ such that (15) at-tains maximum subject to the constraints

T 1/~v

5 Wy('t) [Xy( t)] d't' ~ Zy, ').l = 1, ••• , m. (16)

o In order to solve that problem one can apply the gen-

eralized H~lde; inequality m {T 1/f> ~ ~v

y= 1 D f,,(t) dt~ 0 ~ If" (t:)\d1:j J

which becomes an equality if (almost everywhere) 1/01 1~

f1 (t:) = C.,ly (t') , 't' 6: [0, T], v = 2, ••• ,m (17)

c~ = const, y = 2, ••• ,me The eqs. (17),which can be treated as necessary con­

ditions of optimality, express the following principle of "proportional expenditure effects".

In order 12 maximize the gross product ill expendi-

272 R. KULIKOWSKI

ture intensities in each sector of economy should be Cllo'Sen in such a way -'t1iat in eachtime instant t tne expenditureseffects f. Vi (,1) -are proportional 12 the ~ inertial expenditure effect f.1i(!).

According to that principle it does not pay to de­velop the production capacity if a corresponding level of education, R and D, etc. has not been achieved. In other words, if the education (or scientific level) is the most inertial process one should coordinate the gov­ernment expenditures in such a way that the increase of production capacity is proportional to the education (scientific) level.

In the present case one obtains m ~ T t

Y ~ n c: y 5 dt 5 k 1(t -t') x1 ('7:) d't' = "=1 0 0

-~» ~ ? cy J x1(~) dt: J

o

T 1/atA

~ w1 ('t') [x1 (1:)J d't' o

Then

DECENTRALIZED MANAGEMENT AND OPTIMIZATION 273

In order to solve equations (17) in an explicit way it is convenient to use Laplace transforms of (17):

K1 (P) X1(p) = ci Ki(p) Xi(p), i = 2, ••• ,m.

{ T J~ r 1 S J\-d" = c1 ~(p), ~(p) = ~ w1 (~) ~ k1(t-t)dt

Since

X (t)

where

The numerical values of cy ' v = 2, ••• ,m, can be de­rived by (16) and one gets

~y _ T 1.ht" x'y( t) = (Zy/X).) xJ t), Xv = j Wy (1::) ['i ('t)] dt"

o y= 1, ••• ,m. (19)

(20)

when

274 R. KULIKOWSKI

Example. Maximize A

~ ) ~ }\." 1-~ Y = h ~ e ->. (t- 'l: )[ X('l:r /2 dt' ry( t)j dt, (21)

subject to T

5 x(t) dt ~X, o T

5 yet) dt ~ Y. '1:

By (19) one obtains

(22)

(2~)

T x(~) = 1 e- .A(t-,", )dt = ; [1 - e- ~(T-tt )], (24)

7: t

yet) = ~ x(~) e- A(t-'t)d't' =

= 12 [1 - e- i\t _ e- ~T(1 - e-.Atl

~ (25)

o

It is interesting to observe that i(~) is monoto­nously decreasing to zero when t ~ T. At the same time yet) increases (starting from z~ro) _monotonously along with t. When T is large enough x(t),y(t) attain satura­tion levels when t --:r co , or t ~ 0 respectively. It can be explained assuming (21) describes the integrated economy with limited supply of capital stock (22) and labor (2~).The best investment strategy (24) should de­crease in time when capital stock increases while the labor intensity (25) should increase in order to use the increasing capacity.

It should be explained why it is necessary to assume that K1(P) should be most "inertial" among K...,(p) , = = 1, ••• ,me For that purpose assume K.~(p) = eTyP , v = 1, ••• ,m. Then

X~(p) = e-p(T1 - Tv) ~(p),

and when T1 < T)) , the inverse of X (p) does not exist.

For the same reason when K1(p)/Ky(p) is a ratio of poly-

DECENTRALIZED MANAGEMENT AND OPTIMIZATION 275

nomi~s with nu~erato; of higher order than denumerator the Xy(t) conta1ns bet) functions and the integral in (16) will not converge.It is now obvious also why it is important to have 0 < 1Xy<1 when one wants to have a u-nique solution.

Since the amount of resources (16) is bounded, the solutions obtained are very sensitive to the time horizon T.As a result the model (15),(16) is especially useful for the analysis of concrete projects, where the expenditures and revenues terminate in time.

This is the usual situation also at the macro-level where one takes into account only these investment pro­jects which incur the benefits (i.e. which contribute to the output production) within the planning interval. Then the planning should be organized as a continuous process with the moving (each year) time horizon while the optimization strategies should be readjusted each year according to the changing objectives.

For a continuous development process the integral constraints (16) may be replaced by the amplitude con­straint

m 1/cXy

L [X»(t)] = Z(t), t € [O,T]. (26)

y=1

Since the equations (17) can be solved for iy(t) , V = 2, ••• ,m, so that the operators xy(t) = Ry[x1J ex­ist one should try to solve the equation

1/IXA ~ [ ,1/\)(y [X1(t)] + L ~(x1)J = Z(t), t€ [OtT]

v=2 for X1(t). The constants c'v' y = determined by the conditions:

T 1/oc~ T

5 w/~) [xy(~il d~ = tv J ° °

where m

2, ••• ,m, should

wet) Z(t) dt,

.y = 1, ••• ,m,

L .~ '» = 1, t» - given posi ti ve numbers. v=1

be

(27)

276 R. KULIKOWSKI

The present consumption model (15),(26),(27) corre­sponds to a situation when the existing resources Z(t) should be completely utilized at each t E. [0, T] with the given consumption structure determined by '(V ' )) = 1, ••• , m.

The single sector (consumer) model (15), (16) can be extended easily to the n sector case (12)-(14).

Suppose the n production functions

nm Sv Yi = Gf Zvi' i = 1, ••• , n,

v=1 m

q = 1 - 2: by > 0,

~=1

be given. Assume also that the expenditures in each v­-th sphere of activity (Zy) be given, i.e.

n

.2: Zvi ~ Zy' Y = 1, ••• , m. i=1

(28)

The problem of optimum allocation of Zv among the n sectors canAbe formulated as follows.Find the non-nega­tive Zvi = Zyi' Y = 1, ••• ,m, i = 1, ••• ,n, such that

y = ~ Gf n z~. (29)

i=1 "1)=1

attains maximum subject to (28). Since Y is a strictly concave in the convex set (28)

a unique solution exists and (as can be easily shown) becomes

Z.,i (Gi/G)Zv ' ,

1 , ••• , m, i = 1, ••• ,n (30) = y=

where n G = L Gi ,

i=1 and m by

Y(Zyi) = Gq n Zy .. (31) v=1

DECENTRALIZED MANAGEMENT AND OPTIMIZATION 277

The function (31) can be ity function with respect to ••• ,Zm. The government tries

also regarded as the util­government expenditures Z1' to maximize the utility qnm b."

U(Z1' ••• , Zm) = G Zy v=1

by optimum allocation of resources Z (generated by the economy) among the m different spheres of activity. In other words, _?- t is necessary to find Z" = ~y ,Y =1, •• __ , ••• ,m, such that maximize (31) subject to the constraint

m

2: Zy ~ Z. ))=1

The optimum strategy is unique and (as can be shown) becomes

, Y=1, ••• ,m, (32) »=1

while

REFERENCES

1. Kulikowski, R. "Modelling and Optimum Control of Complex Environment Systems". Control a. Cybernetics 2, 1/2 (1973).

2. Kulikowski, R. "Modelling and Optimization of Com­plex Development". Bull. Pol. Acad. Sci.~, 1 (1975).

3. Kulikowski, R. "Modelling of Production, Utility Structure)Prices and Technological Change". Control a. Cybernetics (to appear).

ORGANIZATION AND CONTROL

R. F. Drenick

Polytechnic Institute of New York

333 Jay Street, Brooklyn, N.Y. 11201

1. INTRODUCTION

A more descriptive title for this article might have been "control in organizations and the organization of control." For one, it reports on a current attempt [3J at developing a mathemati­cal theory of organization in which the use of feedback and of other concepts taken from control theory seems inevitable. And conversely the article speculates on whether or not some of the ideas that are coming out of this organization theory might not be useful also to control theory. Points of contact seem to exist already in the field of hierarchical control [7J and in human factors considerations recently discussed by Rijnsdorp [9J.

Organization theory is not a new subject. It has on the con­trary been an active field of study for sociologists and social psychologists. Its history is long enough for it to have its own historians, and its literature is correspondingly large. The idea of treating organizational problems by mathematical methods is relatively new, however. It seems to have occurred first to Morgenstern [8J but it was not realized until the development of the theory of teams by Marschak and his co-workers [6J.

The theory which is described below has some points in common with team theory but differs from it in a basic assumption, namely the main reason for which organizations are set up in the first ulace. The view that is taken here is broadly speaking that an or­Ganization is a system that is superior in performance to that of its components. In order to make this statement more specific, one must define what is meant by "performance" and in what way the or­ganization members fall short of achieving what would be desirable.

279

280 R.F. DRENICK

This is done in Sections 2 and 3 of this paper. Some detail is in particular devoted to the discussion of the mathematical model of the individual organization member. The model that was desired here should be general enough to apply within reason, to organizations made up of human members but also to others consisting of machines, such as computers and robots, or for that matter organizations that include men as well as machines.

The remaining sections describe the kind of results regarding organizations to which one is led, starting from the assumptions regarding their members. The discussion is kept qualitative, part~y in order to keep it reasonably brief but more importantly because the qualitative results are likely to be of greater interest. After all, most of what is known regarding organizations is of a qualita­tive nature. The comparison between theory and practice is there­fore best done on that level. In fact, the assumptions which are needed as a basis for a mathematical theory are inevitably much more detailed than the data that are, or are likely to be, known in practice. A quantitative verification therefore seems improbable, at least in the near future.

Organizations in practice however do exhibit many qualitative characteristics which should be borne out by a valid theory, such as their structures and the nature of their operating procedures. One should therefore want a theory to lead to fairly specific con­clusions regarding the best structures for organizations and of their sub-units, as well as to general statements regarding the operating principles and procedures which the organization members should follow. The conclusions regarding structures were considered particularly important. In fact it was felt that a theory that failed in that respect could not really claim to be a theory of organization to start with. The theory discussed below seems to live up to this requirement, and the conclusions to which it leads are quite consistent with observations in practice.

ACKNOWLEDGMENT

The work reported in this paper was supported by the Office of Naval Research under contracts No. NOOOI4-67-A-0438-o0l2 and N00014-75-C-0993. This support is gratefully acknowledged. In carrying out the work, the writer has collaborated with Dr. A.H. Levis of Systems Control Inc., and more recently with Dr. D. Rothschild of the Polysystem Analysis Corp. Their help was essen­tial.

ORGANIZATION AND CONTROL 281

2. BASIC ASSUMPTIONS

The point of departure for an organization theory is inevitably an assumption of why organizations should exist in the first place. The assumption on which the present theory is based is roughly speaking this: organizations are set up to execute a task which is too complex or too time-consuming to be satisfactorily performed by an individual. It then follows immediately that the structure and operating procedure of an organization must be so designed that the abilities of its members are as well utilized, and their limitations made as inconsequential, as possible.

These are of course merely qualitative statements which must be rephrased in a quantitative way if they are to form the basis of a mathematical theory. In the one to be described here the assumptions were made as follows.

The task that is to be performed by the organization was taken to be an input-output conversion, with a reward function indicating the relative desirability of every output to a given input. In fact, in order to avoid complications initially, the conversion task was drawn up especially simple: once during every time unit, an "input symbol" x. is acquired by the organization; if it responds with the "output symbol" Yk it is rewarded with the sum Rjk . The num­ber n of possible input symbols and the corresponding number m of output symbols (the "input and output alphabets") are assumed finite. The reward function is therefore an (n x m) - matrix.

The input symbols Xj are assumed to turn up with known prob­abilities Pj, and their occurrence in different unit time inter­vals to be mutually independent. The performance of the organiza­tion is rated by the mean reward it manages to secure. (These as­sumptions have also been used in team theory [6J.)

Compared with these, the assumptions regarding the strengths and limitations of the organization members had to be made more carefully, and even though an effort was made to keep them simple, also, considerable complexity quickly grew unavoidable. (It might be mentioned that the formulation of these assumptions was the main difficulty experienced in the development of the theory: several sets were drawn up and found too inadequate or too unmangeable to be useful, typically after more than a year's work.) These assump­tions are described in more detail in the next section. The fol­lowing remarks are adequate here.

The basic capability of an organization member is in effect determined by the fact that there are single-member organizations. It follows that a member must be assumed capable of executing the same kind of task as the organization as a whole, namely the

282 R.F. DRENICK

conversion of any of a finite number of input symbols, Uj say, into corresponding output symbols vk'

The basic limitations of a member on the other hand are as­sumed to lie roughly speaking, in the speed and accuracy with which he performs his conversion task. The speed is more particularly characterized by his "processing times," i.e., by the times t· k which needs for the acquisition and interpretation of the inpui symbols Uj, as well as for the production and delivery of the subsequent output symbols vk' The accuracy with which he works is defined by a similar set of quantities, namely the probabilities p(vkluj) for the response vk to the input Uj.

This characterization is similar to that of a discrete noisy channel in information theory. One can also look on it as a kind of sto­chastic service system in which the waiting times, queue lengths, etc., are ignored. Either view however induces a final assumption, namely, that each member of the organization be required to keep up with the mean processing rate expected of him. His mean processing time 7, in other words, should obey

T:: L:. L t.k p(u.) p(vklu.) < 1 J K J J J -

where the time unit on the right is the interval between organiza­tional input. This requirement precludes the presence in the or­ganization of members who, in the steady-state, operate with infi­nite delays between input and output and can make no positive con­tribution to the overall performance of the organization.

3. CHARACTERISTICS OF ORGANIZATION MEMBERS

The main properties that characterize the performance of an organization member in the organization were described briefly in the preceding section. They were the speed and accuracy with which he executes his task. To be more specific, they were two sets of quantities namely the processing times tjk and the probabilities p(vkluj) for conversion of an input symbol Uj into an output symbol vk- A fast working member will of course be characterized by short processing times, and one who is totally incompetent by infinite ones. An accurate performer will have a probability near unity for the conversion of Uj into the output vk that is desired of him, and probabilitles near zero for all others.

These two sets of quantities have been called "primary perform­ance variables," to distinguish them from others which affect per­formance also but which do so parametrically, by way of the primary ones. The latter are called "secondary" or, to use a term from the vocabulary of social psychology "intervening."

ORGANIZATION AND CONTROL 283

The secondary variables seem unavoidable if organization mem­bers are to have any of the characteristics which, according to experimental and social psychology, are exhibited by human beings who operate in organizations. In fact, some of those variables appear to be convenient even for the characterization of certain machines, computers for example.

Consider the processing times tjk which measure the time an organization member needs to acquire and interpret an input symbol, Uj' as well as to produce and deliver the corresponding output symbol vk. In many situations in practice the first group of tasks is completely different in nature from the second. It is therefore appropriate to assume that tjk is in fact the sum of two terms, the "input and the output processing times." It then seems reason­able, and is in fact fairly well consistent with experimental evi­dence, to assume that the first depends only on the input symbol Uj and the second only on the output symb 01 vk. I.e.,

t(i} + t(o} j k (3. I)

The channel which represents the organization member is thus best visualized as a tandem arrangement of two channels, the "input and the output channels."

The human input processing time t(~), or the "reaction time" in the language of experimental psychology, is known to be a mono­tone increasing function of the size n of the input alphabet. It is furthermore known that it also depends on the probability p(Uj) with which the input symbol is expected: the human input channel more particularly adapts its processing times in such a way that the most frequent symbols are processed most rapidly. Evidence accumu­lated about twenty years ago indicated more specifically that

k log p(u.} J

(3. 2a)

(see e.g. [5, p. 39J), which makes the mean input processing time

'T(i} = - k 1:. p(u.) log p(u.), J J J

(3.2b)

proportional to the input entropy. This remarkable result created considerable excitement at the time because this kind of channel is the most efficient processor of information, according to information theory. More recent evidence [1, p. 294J does not Seem to be en­tirely consistent with those early observations. (~n the theory described here it was accordingly assumed that 1) is more gen­erally a concave function of the vector p of input probabilities p(Uj) .

284 R.F. DRENICK

Other parameters characterizing the nature of the input have been considered as further possible secondary variables, the number of sources from which the input must be collated being a fairly obvious one. They have not been used in the theory so far, however.

There are on the other hand certain other variables which appear to be highly pertinent to the performance of human organiza­tion members, and these are all of a strictly psychological nature. In the present theory, they are collectively referred to as the ~ -vector. Thus,

t (i) = t(i) ( . . n, J J

p, ~). (3. 3)

The phychological variables have of course been the subjects of intensive study by sociologists and social psychologists (see, e.g., [10, p. 36]). It becomes clear from reading their literature that there is no one generally accepted way of defining the compo­nents of ~ nor, for that matter, an agreement on what a reasonable choice is of the dimension of ~ (i.e., how many components have an appreciable effect on performance). One criterion, however, which is convenient at least for mathematical treatment does not seem to have been considered and that is that the components of ~ be chosen in such a way, so far as possible, that no one of them is a function of any other. A set of components that appears to have this prop­erty, and that furthermore seems to be pertinent to the design of an organization, is the following.

(a) Feeling of autonomy (b) Feeling of being appreciated by the organization ("self-

worth appreciation") (c) Load-induced stress (d) Relations vis-a-vis co-workers (e) Self-appraisal

There are no doubt many others, and many other equivalent choices. These however are suspected to be those most relevant to organiza­tional design, at least as of this writing.

The output processing times of the generic organization member are assumed to be functions of a similar set of secondary perform­ance variables,

(3.4)

but n is understood to be the numbers of destinations among which he must distribute his outputs and p the vector of the distribution probabilities. The dependence on n is more specifically monotone increasing, and that on p concave.

ORGANIZATION AND CONTROL 285

Less is known, at least to this writer, regarding yecondary variables that enter the conversion probabilities p(vkluj)' One can no doubt as~ume that these probabilities depend on the five components of the ~-vector listed above. Furthermore, it is well established by psychological experimentation that they depend on the mean processing time in a very striking way [e.g. 5, p. 69J,

p(vklu.i = p(vklU.i 7, ~). J' J

In fact, if 7~ 1, i.e. if the organization member is not overloaded and manages to keep up with the rate at which his tasks are assigned to him, he works with a very low incidence of error. For 7 > 1, on the other hand, that incidence increases radically and mor particu­larly in such a way that the situation 7 = 1 is re-established. The member in other words substitutes erroneous tasks that take shorter processing times and performs them in place of those which he actu­ally should perform. When that happens the member is said to be "overloaded. II

This is a piece of information which entered into the theory in a very crucial way, as will be seen presently.

The two sets of primary performance variables, namely the pro­cessing times and conversion probabilities that have been intro­duced here need not be the only ones to be employed in the theory. They are merely those that appeared of chief pertinence to the results that were desired. Others can be, and no doubt eventually should be, introduced. The cost of the member to the organization is obviously one that should be injected soon. If the member is a computer, its various memory capacities may be important. If he is a robot, his mobility, flexibility, and perhaps energy consumption may matter. If he is a human being slated for a supervisory posi­tion, some other and very subtle parameters may matter.

4. BASIC ORGANIZATIONAL FEATURES

As mentioned in the Introduction, the qualitative results that were obtained from the theory may be more interesting and more use­ful than the quantitative ones. This section describes some of the former, and more particularly those which pertain to the gross features of an organization.

The first, and perhaps most pervasive, conclusion of this kind was reached promptly and non-mathematically, directly from the view of an organization as an input-output processing system. It is that the conventional organization chart may not be overly useful as a characterization of organizational functioning. In many cases, a complete re-arrangement may be advantageous.

286 R.F. DRENICK

Board of Directors I

I President

I I I I I Vice Pres. Vice Pres. Vice Pres. Vice Pres. Director Assistant Manufacture Finance General Personnel to Pres. & Eng'g. Sales

J I 1 1 1 I I

Manager Manager Manager Director Vice Pres. Plant A Plant B Plant C Field Product

Sales Sales

Figure I

Traditional Form of an Organization Chart

To be more specific, consider a chart [4, p. 37] such as the one in Fig. I which is typical of innumerable others to be seen in business and government. It is a block diagram representing the positions of certain officers and their departments, connected by lines. These are the so-called "lines of authority." There does not seem to exist a clear definition of what that term means. (In fact there does not seem to be a clear definition of what the term "authority" means.)

In the view of an organization as an input-output processing system, the same positions are interconnected by lines also, but they are now transmission links over which messages, money, mate­rials, services, etc., are sent. They therefore have a clearer-cut interpretation, and quite specific statements are possible of what should b e transmitted over those links. On the other hand, they induce a thorough re-arrangement of the organization chart, namely the one shown in Fig. 2. One arrives at this diagram by the follow­ing reasoning.

Suppose that the designer of the organization finds that the task that needs to be executed is too difficult for one member. In other words, if this member were required to execute it by himself he would be overloaded. As was explained in the preceding section, he would commit errors and the performance of his organization would fall short of the optimum. Let him be called the "Executive" here. The designer can then provide relief for him in only the following ways. He can assign to him a group of assistants who preprocess the inputs for him, and a second who take over from him the production and the delivery of the outputs. They are called

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the "staff" and the "line" in the theory. It is possibe that these two groups together can remove the overload on the Executive. If they do so, without incurring overload on themselves, error inci­dence throughout will be drastically reduced, and organizational performance correspondingly improved. If not, i.e. if an appreci­able performance degradation remains a third unit can be added. Its function is roughly equivalent to that of the feedback branch in a control system and it is called the "control branch" in the theory. This is by contrast to the so-called "executive branch" which con­sists of staff, line; and the Executive.

All of these units are identifiable in the chart of Fig. 2. The Executive can be equated with the President. The line in Fig. 2 consists of the three plant managers and, of course, everyone else in those plants. This is no doubt as all organization theorists would view it, too. The Vice President in Charge of Manufacture and Engineering would probably be considered part of the line also but, as will be explained below (Section 6), he often deserves a better treatment.

The sub-units of the staff may be more controversial. The personnel department will perhaps be generally acceptable as a staff unit, but the sales department may not be, and the Board of Directors almost certainly will not be. The staff, organization theorists argue, is generally considered to have some kind of an advisory function and neither of these two units has that. It has however been found convenient in the theory to generalize the staff concept and to inClude in it every organization member who SUbmits inputs to the Executive: there does not seem to be any way of distinguish­ing advisory inputs from others.

With this convention, the sales department is absorbed into the staff since it injects into the organization one of its most impor­tant inputs, and so is the Board of Directors whose function is partly to provide advice to the Executive and partly to raise capi­tal, both being in the nature of inputs.

The control group is shown as being made up by the financial department whose main mission presumably is the exercise of fiscal control, and by the Assistant to the President. The second is on the assumption that he acts as a kind of roving trouble shooter throughout the organization.

Figure 2 indicates by a dotted line that certain communications may pass from the staff directly to the line, from the sales depart­ment for instance or from the personnel department. An organization in which the Executive can be by-passed in this fashion is roughly what is called "decentralized," by organization theorists. In a "centralized" one, such by-pass links are forbidden: all trans­missions must in one way or another pass through the Executive or,

ORGANIZATION AND CONTROL

to put this more precisely, the Executive must be a cut point in the digraph that connects the sources of the organization to its destinations.

289

Most of the work done so far on the theory discussed in this article treats centralized organizations. They are conceptually simpler than the more general decentralized ones. On the other hand, they are somewhat old-fashioned. Centralization was considered to promote efficiency forty or fifty years ago, when ideas like "work measurement" and "scientific management" dominated much of the writings on the subject. The organization members were then treated rather like robots and given as little discretion as possible with how to do their jobs, let alone what jobs to do. The mathematical theory of such organizations is simpler for various reasons. One can for instance defend the neglect of the ~ -factor which, ac­cording to the discussion in the preceding section, can affect per­formance: an organization member that is used largely like a robot might as well be one, and might then also be represented as one in the theory.

A further assumption that was made in much of the work so far is that an organization member operates in an error-free fashion unless he is overloaded. The organization designer can then hope to achieve optimal performance by avoiding overload throughout the organization.

The next two sections describe in a ~ualitative manner the reasoning he can follow in the design of the executive branch, i.e. of staff and line. It will be noted that this procedure will acutally lead to a suboptimally performing organization. To achieve the optimum he can follow a procedure of the kind sketched in Section 7 which discusses decentralization. Section 8 deals briefly with the design of the control branch.

5. DESIGN OF THE LINE

The function of the line is to relieve the Executive of as much of his output load as possible, without suffering overload among its members. If it achieves that, it transforms the outputs of the Executive into organizational outputs in a one-to-one manner and hence can avoid degrading the performance.

It is conceivable that a single line member can achieve that. He may be more skilled in the job of producing the organizational outputs, i.e. have processing times that are shorter than those of the Executive, or he may be able to deliver the outputs to their destinations while the Executive cannot.

In most organizations however this will not be good enough:

290 R.F. DRENICK

the Executive may be relieved o~ his overload but it will merely be shi~ed ~rom him to the line. In that case, the next thing to do is to expand the line. Suppose that the organization is restricted to being centralized, and that the line is expanded to two members, such as Charlie and Delta in Fig. 3. The principle by which they relieve the Executive remains unchanged but they themselves may avoid overload by a suitable distribution between them o~ the output production.

It is easy to convince onesel~ that there are only two basic ways o~ distributing this task. One is to assign jobs alternately to Charlie and Delta or, more generally, assign a job to Charlie with a certain probability and to Delta with the complementary one. The second is to split the job and have Charlie do one part o~ it and Delta the other. In general, a probabilistic rule can be used ~or these assignments, also.

The ~irst mode o~ operation is called "alternate processing" in the theory and the second "parallel processing." The latter may not be possible o~ course because there is no way o~ splitting a job. (Automobile repair shops for instance use alternate processing primarily because repair jobs are not conveniently divisible in general.) In the theory, it is the organization designer who is assumed to make the choice between the two modes and who also chooses the assignment rule. In the terminology used, he lays down the "standard operating procedure (SOp)" for the line. He can make it "pure" (Le. deterministic) or "mixed" (Le. stochastic). His choice of the SOP is governed by two considerations: one is to minimize the output load of the Executive, as measured by the mean Executive out­put processing time 7(~), and the other is to avoid overload on Charlie and Delta, i.e. to maintain

(5. 1)

Mathematically this is patently a programming problem which may however be quite complicated if one assumes dependence of the proc­essing times on the various secondary variables introduced in (3.3) and (3.4), even if ~ is disregarded (which may be fairly well defensible in a centralized organization, as was explained in the preceding section). The programming problem, first of all, may have no feasible solution. If it has some, and there is a pure SOP among them, th~t will be typically best in that it minimizes the output load 7(~) of the Executive: the concavity assumption for (3.4) assures that. This was a desired result. Human organization mem­bers, and for that matter machines, have difficulties with the ex­ecution of mixed SOP's and tend to avoid severely mixed ones. A theory, in other words, that fails to favor pure SOP's would be somewhat unrealistic. The total exclusion of mixed SOP's on the other hand is impossible, also. It is easy to construct situations in which no pure ones exist.

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292 R.F. DRENICK

If the problem has no feasible solution, i.e. if the constraint (5.1) cannot be satisfied, overload on either Charlie or Delta or both is unavoidable. The line must then be further expanded. It develops that overload among the line members can always be elimi­nated if their number can be made large enough. In fact, one of the results that is rather typical of several others is the following.

Proposition 5.1 [3, p. 48]: consider an alternately ~~9cessing M-member line. Suppose that the input processing times t~t) of all members are constants and that

tRk = t(i) + t(o) < t < 00 Rk Rk- m.ax

for R=1,2,.. M and k=1,2,.. n, as M - 00 • The SOP for the line can always be so chosen that no member is overloaded, provided only M is made large enough.

This result incidentally exhibits a rather disagreeable feature that shows up in many others. It is often necessary to make assump­tions regarding the pool of individuals from which the organization members can be chosen. In this instance it assumed that, no matter how poor the candidates for line membership are, their processing times are always bounded by some fixed upper bound t max .

The indefinite expansion of the line will always eliminate overload among an alternate processing line, according to the above proposition, but it will in general re-introduce overload on the Executive. For, according to (3.4), his output processing times increase monotonely with the number of destinations. One must, in other words, expect that there will be a maximum number of line mem­bers who can receive their assignments directly from the Executive. This number has been called the "span of control" or "of supervision" by organization theorists. Considerable effort has been devoted by them to the search for a span that would be a universal optimum but without discernible success. This theory indicates that the search would not have been successful because the maximum depends on many factors, and confirms more recent socialogical studies which came to the same conclusion.

In order to avoid Executive overload, the organization designer can interpose an echelon of junior executives between the Executive and those line members who produce the organizational outputs (the "operators"). This stratagem reduces the number of line members who receive their instructions from the Executive directly. It may in other words eliminate his overload without re-introducing it among the line.

If one echelon of junior executives turns out to be insuffi­cient, i.e. if the Executive still has too many immediate

ORGANIZATION AND CONTROL 293

subordinates the number of echelons can be increased, if necessary, until neither the Executive nor any of his juniors has more than two subordinates.

The design procedure patently leads to the hierarchical line structure which is generally associated with the line of central­ized organizations in practice. One can also expect it to lead to a line that is free of overload throughout, unless the Executive or one of the junior executives is unable to cope with the task of making job assignments to only two subordinates. This is a mathe­matical but hardly a realistic possibility. The best SOP's for all executives, senior as well as junior, will be pure if pure ones exist in the first place. This, too, is in rough agreement with observations in practice.

One can therefore perhaps say that agreement between theory and practice is ~uite good, as far as the lines of centralized organiza­tions are concerned.

6. DESIGN OF THE STAFF

One might suspect, from the symmetry between line and staff in Fig. 3, and also in Fig. 2, that their function, structure and oper­ational procedures would be very similar. This is not so, however, according to the theory. The staff function in a centralized or­ganization is more ambiguous, and the staff structures and operating procedures more varied and more complex.

To see this, suppose that the design of the line has been completed along the lines suggested in the preceding section, and that output load of the Executive has been reduced to a minimum. Suppose that he is nevertheless overloaded because his input load is too large. The designer can then attempt to bring him relief by providing him with a staff. As will be seen, this move can always be successful, though sometimes in a rather anti-climactic fashion.

There are more specifically three ways in which the staff can be used. For one, it may be used like the line, i.e. to realize a one-to-one transformation of the organizational inputs into others which are more easily processed by the Executive. This mode of op­eration would be of interest for input processing which would be too time consuming for the Executive or which he is not very competent to carry out himself. Salesmen, recruiters, intelligence agents, or interpreters, function largely in this way. It is possible that such a one-to-one transformation is sufficient to remove Executive overload. If it is, the staff functions like the line and the symmetry between the two, in structure and SOP, in fact is realized.

It can however happen, and in practice is often does happen,

294 R.F. DRENICK

that this is not enough. The staff can then in principle resort to a second mode of operation which is also one-to-one and which may reduce the Executive input load: according to information theory, it can encode its inputs in such a way that they are "matched" to the Executive input channel. In practice, however, little of that is done (and, if the channel actually has processing times of the form (3.2a) none is necessary). In the theory, at any rate, coding has not been considered so far.

Under the circumstances, if the Executive is still overloaded, errors are unavoidable. The best that the staff can then do is to make the errors in place of the Executive but to make them in such a way that their effect on the performance of the organization is minimized. Its function then ceases to be one-to-one. A staff that operates in this way is said to "eqUivocate," using a term borrowed from information theory. An equivocating staff responds to an or­ganizational input X· with a message Uk to the Executive but does so in general with an "equivocation probability" P(uk I x j ).

The determination of the best equivocation SOP for the staff develops to be again a mathematical programming problem which may or may not be linear, depending on the characteristics of the pro­cessing times. A typical result is the following.

Proposition 6.1 [3 7' 78J. Consider an Executive whose input processing times t~x~ for the symbols u j are constants such that J

. t (i) < 1 nllnj Xj _ '

There always exist equivocation SOP's for the staff which avoid overload on the Executive. The optimal ones among them may be pure or mixed. If there is a SOP under which he is not fully loaded, then there is a pure SOP that is optimal. If all optimal SOP's are mixed, there is one for which no more than two equivocation proba­bilities P(uk Ixj ) 4 0, for each Xj'

This kind of result indicates that overload on the Executive is avoidable by suitable staff equivocation. The same cannot be said for the staff itself. On the contrary, one can construct con­ditions which do not appear overly unrealistic and under which over­load among the staff is inevitable. One can construct others in which overload can be avoided but only by allowing it to follow sub-optimal equivocation policies.

The staff, can operate in principle with alternate or parallel processing. This is therefore as on the line. Most staffs in practice are parallel processing. The structures which are optimal for such staffs, according to the theory, are however much more

ORGANIZATION AND CONTROL 295

varied than those of the line. The hierarchy may be optimal again but this seems to be so merely for the borderline case in which the output load on the staff members is negligible. In the opposite case, the best structure develops to be more committee-like, with many transmission links among the members.

The most general structure for a three-member paralles process­ing staff is shown in Fig. 4. The figure exhibits a feature which has been found common to most optimal staff structures so far. It is that a kind of ranking appears among staff members and that one member in particular (Alpha, in the figure) has clearly a distin­guished position. The temptation to call him the "Chief of Staff" was irresistible, and that is what he is called in the theory. This Chief in effect decides for the staff as a whole what message is to be transmitted to the Executive. In the figure it is the composite symbol uk = (uf' u~, u~). This choice must in general be trans­mitted by way of the intra-staff links to the remaining members, along with the information of what components of uk have already been sent to the Executive. Echo in the figure for instance must be told that u~ and u~ have been pre-empted and that he has no choice but to transmit u~.

The position of the Chief of Staff is evidently crucial to the performance of the organization. One can accordingly expect it to be occupied by the highest officer. In the organization of Fig. 2 in particular this would be the position occupied by the President, in which case the Vice President for Manufacturing and Engineering would be the Executive of the theory. The two officers together form a two-member axis which is fairly t~rpical of many organizations in practice. They then often bear the titles of President and

Figure 4

Parallel Processing Staff

296 R.F.DRENICK

Executive Vice-President. This result of the theory ~ therefore be confirmed in practice. The same ~ be true of the various staff structures. Sales departments for instance whose members typically experience mostly input load tend to be hierarchical, as indicated by· the theory, while general military staffs who prepare elaborate plans of action must cope with output loads primarily. They there­fore tend to operate in a more diffused committee-like fashion.

So, qualitative agreement between the theory and the practice of staff operation seems satisfactory, also.

7. DECENTRALIZED ORGANIZATIONS

A decentralized organization, as explained in Section 4, is one in which transmission links by-pass the Executive and go directly to the line. Figure 2 shows one such link going from the staff to line, and Fig. 3 shows two. Neither figure shows by-pass links going from outside sources to the line directly, without pre-processing by the staff. These are quite common in practice, however; line members often cannot do their jobs effectively unless they use their own sources. A field soldier for example is quite free to use his own judgment in his choice of the hole in which to hide from enemy fire.

The mathematical theory of decentralized organizations is quite incomplete, at this writing. Nevertheless, certain basic features have already emerged and a few preliminary results can be reported.

The first thing to be said perhaps is that the separation in the treatment of staff and line which was used in the preceding two sections to explain the theory of centralized organizations cannot be carried over to decentralized ones. It is not strictly defensible for centralized ones either (it leads to suboptimality in general) but it fails quite irretrievably in the more general case. This makes for greater mathematical difficulty.

A second complication is that the l\J -vector of psychological components can no longer be neglected. In fact it appears that the incentive for decentralization can under certain circumstances come from the beneficial effect decentralization can have, by way of the l\J -vector, on the processing times and error probabilities of the line members.

The main incentive however is that the by-pass links around the Executive increase the processing capacity of the organization. The equivocation that needs to be injected into the transmissions from staff to line can typically be reduced, and the performance accordingly improved.

ORGANIZATION AND CONTROL 297

The kind of result that is being derived may be illustrated by the following

Proposition 7.1. Consider a design drawn up for a centralized organization with a single-member staff (Alpha) and a single-member line (Charlie). and which requires rqUiVocation in the inputs to the Executive with probabilities P(uk xj ). Suppose that Charlie's input processing times are constants and that

7 = E. l~ t k p(uklx.) p. < 1 C J K C J J

(i.e .• that Charlie is not fully loaded). In that case. the per­formance of the organization can always be improved by decentral­ization. i.e. there exists joint equivocation probabilities p(u~ IXj) and p(u~ IXj. u~). with uk = {u~. u~}. which lead to a performance that is at least as good as that of the centralized design.

The most striking consequences of decentralization appears to be the effect on the structure of the line. Since the line is no longer restricted to inputs from the Executive alone. the best structures for it are often similar to those of the staff. In par­ticular. a resolution of the line into groups each of which operates in a committee-like fashion. seems optimal under certain circum­stances. at least by all present indications. This may be in agree­ment with the recent trend in practice toward the "open system" in the line organizations of industry.

Certain problems associated with decentralization that were expected to have simple solutions have so far proven quite stubborn. Thus. the concept of coordination which played a crucial role in the theory of Multi-Level Hierarchical Systems [7, p. 109ff] has so far refused to assume an equally crucial rule in the theory described here.

8. DESIGN OF THE CONTROL BRANCH

The control branch of an organization was introduced in Section 4 as being a rough analog of the feedback branch in a control sys­tem. Its mission is to detect errors in the operation of the organ­ization or more generally, of a degradation of performance due to errors or other causes. It is to diagnose the reasons for it and to institute corrective action. if such action is indicated.

In order to execute its mission. the control branch can in principle tap all transmission links in the executive branch. as well as its inputs and outputs. It can actually do more than that: it can also query each member in an effort to establish the "values"

298 R.F. DRENICK

of the components of their lJ.!-factors. Is he satisfied with the amount of automony he has; with the amount of time and attention he gets from his supervisor; etc.?

It is perhaps evident from this that the flow of information into the control branch can be much greater and more varied than into the feedback controller of a control system. The corrective action in principle is similarly varied. Thus the control branch can function in the manner of a feedback controller which is, broadly speaking, to intercept the input symbols to the organization and to substitute for them others which elicit the desired outputs whenever the actual inputs fail to do so. It can in fact do the same for every member or group members. The corrective action may also go beyond a symbol-by-symbol substitution and change the SOP for some or all of the organization members. It may even change the structure. The control branch then acts, so to speak, as a designer-in-residence and redesigns the organization whenever that is desired.

The variety of information available to the control branch, and the corresponding variety of responses, makes its theory rather difficult. As of this writing in fact only partial results have been obtained.

It has become clear that in general, the control branch must have its own staff, line, and "Control Executive." It mayor may not be centralized. In view of the incisive actions that are potentially taken by the Control Executive, the position may often be best filled by the highest officer of the organization.

The following has become known regarding the operational pro­cedures of the control branch. In its effort to alleviate perform­ance degradation due, first of all, to error incidence in the executive branch, it has two options, as is well known from systems theory: redundancy and feedback. In the case of organizations, however, the distinction between these two becomes rather blurred: a control branch that attempts to reduce the effect of error inci­dence often functions in a manner that can apparently be interpreted either way.

Performance degradation may however be due to causes other than errors. It may be due to the fact that the organizational design was based on faulty assumptions, for instance regarding the charac­teristics of the organization members, or regarding the input prob­abilities, or for the matter regarding the reward matrix itself. This kind of trouble apparently can be alleviated only be feedback and more particularly by one of an adaptive kind. The feedback must in general also have a feature that might be called "error toler­ance." The mere indication of performance degradation should not, in other words, necessarily precipitate corrective action. The

ORGANIZATION AND CONTROL 299

reason for this is, roughly speaking, that it may be either illogical or impractical, or else merely uneconomical, to take at its face value every indication of a discrepancy between actual and desired performance. The best reaction of the control branch may actually often be to leave well enough alone. The kind of control principle, if confirmed by further study, may be useful to control theory in general [2J.

9. THE ORGANIZATION OF CONTROL

The preceding section presented a brief account of present thinking regarding a theory of control in organizations. This section is in a way a counterpart. It discusses the organization of control and its relationship to the theory explained above.

One can, first of all, expect such a relationship to exist. The organizations developed here are charged with a decision-making task of the simple kind described in Section 2. The control of a plant is a decision-making task also, if a more complicated kind. Offhand, one can therefore expect that the controller of a plant should be organized roughly along the lines suggested by the theory, at least if it is made up of subsystems that are limited in the speed and/or accuracy with which they work. (On the other hand, one should not necessarily expect such a parallel if, e.g., the cost of communication is the main consideration in how to organize the sub-systems; team theory would be the one to use in that case.)

Indications are that a hierarchical controller in fact is organized along the lines suggested by the theory, even though the structure that is displayed in the block diagrams of such controllers is quite different at first glance.

The simplest hierarchical controller is usually shown [7, p. 86J as consisting of three sybsystems arranged in two echelons, two unit controllers in the lower echelon, and a "coordinator" in the higher one. This structure can be considered a re arrangement of the one in Fig. 3. To see this, imagine the figure "folded over" on itself so that the sources are placed next to the destinations, the staff member Alpha next to the line member Charlie, and Bravo next to Delta. Interpret the destinations of the original arrangement as being part of the plant, which is of course what they would be in a control system. In fact, the organizational outputs yl and y2 would be the control signals that drive the plant. The sources from which the organization derives its inputs lie partly outside the system and supply the reference signals xl and x2 . Partly, how­ever, they lie within the plant and supply the feedback signals x3 and x4 to the organization. Charlie and Alpha together can be thought of as making up one unit controller, as shown in Fig. 5, and Bravo and Delta the other. The Executive assumes the role of the coordinator of the hierarchical controller.

300

CHARLIE

1 Y

1 x

COORDINATOR

PLANT

REFERENCE INPUTS 2 x

Figure 3

The Organization of a Hierarchical Controller

R.F. DRENICK

In this rearrangement the dotted link leading from Alpha to Charlie in Fig. 3 is contracted to the short segment shown between them in Fig. 5. This transmission link no doubt exists in the second arrangement since Alpha and Charlie presumably are part of the same piece of equipment. In that case, however, it by-passes the Executive. The organization is therefore "decentralized" in the sense in which the term is used in organization theory. It is decentralized also in the less specific sense of the term in con­trol theory where it apparently merely means that the controller consists of several sUbsystems.

The theory further indicates that another link should in general exist, namely the one from Alpha to Bravo shown in Fig. 4, which is accordingly inserted in Fig. 5. An additional link that is not shown in Fig. 5 may often be necessary however. It is one that goes from Bravo to Alpha and carries the signals x2 and x4 which are received by Bravo, but may be required also by his fellow staff member Alpha. The transmissions between these two staff mem­bers could of course be routed also over the Coordinator but, according to the theory, either Alpha or Bravo or both can be dispensed with unless the coordinator is overloaded without them.

ORGANIZATION AND CONTROL 301

If he is, and if Alpha and Bravo are needed for his relief, the routing through him of additional signals will memely aggravate his overload. Alpha and Bravo, in other words, should be expected to have to communicate with each other, for best performance.

The nature of the signals that pass among all three, should, according to the theory, be determined as solutions to certain programming problems which take into account the limitations of the subsystems and the reward function that underlies the problem.

One might argue that the marriage that has just been described between hierarchical control and organization theory is a bit forced. Thus, it could be pointed out that the latter relies on discrete alphabets instead of the continua that are always used in the former, or that the latter is "static" rather than "dynamic." These arguments are correct but indications are that they may not matter greatly. As far as the discretness of the signals goes, it is an assumption of convenience at the moment. The crucial assump­tion is that one can define a mean processing time for every organ­ization member which is finite and which must not exceed a certain maximum for error-free (or at least nearly error-free) performance. Some such assumption seems to be implicit in hierarchical control as well for, without it, there presumably would be little point in resolving a controller into sUbsystems.

The inclusion in organization theory of dynamical effects has not been attempted so far because of the considerable complications that are expected from it. The list of components of the ~-vector,

for instance, would have to be expanded to include such factors as training, learning, history of success and failure, tec. The programming problems would have to be generalized as well, to include planning, job se~uencing and scheduling. The reward func­tions would have to be allowed to penalize the organization for delays. The mathematical optima for structure and SOP might develop to be time-variable a.nd point to an organization that is much more fluid and ephemeral than those that now exist. Thus, it is possible that little would be gained, at least towards a verification of the theory.

302 R.F. DRENICK

10. REFERENCES

1. D.F. Broadbent, Decision and Stress, Academic Press, N.Y. 1971

2. R.F. Drenick, Control of Partly Unknown Systems, submitted for publication

3. R.F. Drenick and A.H. Levis, A Mathematical Theory of Organization, submitted for publication

4. R. Carzo and J.N. Yanouzas, Formal Organization, Irwin-Dorsey Press, Homewood, Ill., 1967

5. H.W. Garner, Uncertainty and Structure as Psychological Concepts, Wiley, N.Y., 1962

6. J. Marschak and R. Radner, The Economic Theory of Teams, Yale Univ. Press, Conn. 1969

7. M. Mesarovic et. al., Theory of Hierarchical Multi-level Systems, Academic Press, N.Y., 1970

8. o. Morgenstern, Prolegomena of a Theory of Organization, Rand Rpt. RM734 (unpublished)

9. J.E. Rijmsdorp, et. al., Man's Role in Control Systems, Paper P. 4, Proc. Sixth IFAC Congress, Boston, 1975

10. J.L. Franklin, A Path Analytic Approach to Describing Causal Relationships Among Social-Psychological Variables in Multi­Level Organization, Unpub. Thesis, U. Michigan, 1973

DECENTRALIZED STABILIZATION AND REGULATION IN LARGE MULTIVARIABLE

SYSTEMS

Edward J. Davison

Department of Electrical Engineering

University of Toronto, Toronto, Canada

ABSTRACT

A general description of the decentralized control problem , with particular emphasis on large scale systems, is made. In this problem, constraints on the structure of the information flow between the manipulated inputs of the system and measured outputs of the system are imposed. Typically, it is desired to find a controller for this system (subject to the above constraints), so that stability of the resultant closed loop system is obtained, and so that regu­lation/tracking of the outputs of the system occurs, independent of any input disturbances occurring in the system. A motivation for dealing with the decentralized problem is made, classical ways of solving the problem are described, and recent results obtained on the problem are surveyed. Some numerical examples are included, in particular, a power system consisting of three interconnected syn­chronous machines; it is shown in this power system that there isno real advantage is using a (more complex) centralized control system over the coventional (and more simple) decentralized control system which is normally applied.

1. INTRODUCTION

There is increasing interest in the study of large scale systems e.g. electric power systems [1], socioeconomic systems [2], [3] chemical process control [4], flow problems [5] etc. Quoting R. Bellman [6] "we need theories of large scale systems. This is particularly so since society is composed of large systems and since without applications, control theory will be sterile. Some typical large systems are economics, education, urban systems, transporta-

303

304 E.J. DAVISON

tion, energy, environment". When control theory is applied to solve problems of such large systems, an important feature called decentralization often arises. Such systems have several local control stations, of which at each station, the controller observes only local system outputs and controls only local inputs. All of the controllers however are involved in controlling the same large system. The reason why such a decentralization constraint usually arises in large systems is due to the fact that a centralized con­troller, i.e. a single controller which observes all outputs of the system to control all inputs of the system, usually will require excessive computational requirements and excessive information gathering networks to make such a controller realistic; the decen­tralization constraint makes control of the large scale system realistic.

There has been a large amount of research devoted to the topic of controlling large scale systems the last few years. The optimal control approach [1], [7] - [18] adopts an optimal control formula­tion of the problem, e.g. control inputs typically are found to mini­mize a given performance index of the system over a finite time in­terval subject to certain information flow constraints (such as de­centralization). Alternately, theories of hierarchial systems are being developed [19] - [23] in which composite systems consisting of smaller subsystems are studied.

This paper shall not try to review the above literature but concentrate only on some aspects of the decentralization control problem; in particular on the stability problem (pole assignment problem) and the servomechanism problem of a decentralized system [24] - [28].

2. PROBLEM FORMULATION

There are two problems this paper is concerned about. The first problem is concerned about finding under what conditions there exis~a set of appropriate decentralized feedback control laws that will stabilize the complete system. In the case of centralized control, the result is well known - namely that any modes of the system which are not both controllable and observable must be stable [29]. In the case of decentralized control, a generalization of this condition is necessary; this is accomplished by introducing the idea of "fixed modes" of a system [25].

The second problem this paper is concerned about is finding under what conditions there exists a solution to the robust servo­mechanism problem for a decentralized system. In the case of cen­tralized control, the conditions are expressed in terms of the "transmission zeros" [37] of the system. In the case of decentra­lized control, the conditions are expressed in terms of the fixed

DECENTRALIZED STABILIZATION AND REGULATION 305

modes of a certain augmented system [28].

The following stabilization problem and servomechanism problem will be considered:

The Decentralized Stabilization (Pole-Assignment) Problem

Consider a linear time-invariant multi variable system (plant) with v local control stations described by:

x = Ax +

y. = C.X 1 1

v

I i=l

B.U. 1 1

(i 1,2, ... ,v) (1)

m· r· where xERn is the state, u.ER 1 and y.ER lare the input and output respectively of the ith lo~al controllstation (i=1,2, ... v). The decentralized stabilization problem is to find v local output feed­back control laws with dynamic compensation for (1) to stabilize the resultant closed loop system. The set of local feedback laws are assumed to be generated by the following feedback controllers:

z. S.z. + R.y. 11111

u. Q.z. + K.y. + v. 11111 1

, i = 1,2, ... ,v (2)

n' m· where z.ER lis the state of the ith feedback controller, v.ER 1 is the ithllocal external input and S., Ql" R., K. are real c6nstant

111 matrices of appropriate size.

Let S be any nonempty symmetric open subset of the complex plane C (i.e. if AES, then its complex conjugate A*ES); then the decentralized pole assignment problem is to find v local output feedback control laws with dynamic compensation such that all poles of the resulting closed loop system are in S.

The Decentralized Servomechanism Problem

A linear, time-invariant system (plant) is described by: v

x = Ax + I i=l

y. C.x + F.w 1 1 1

ref e. y. y. 1 1 1

yl!! Cl)lx + F~w 1 1 1

B.u. + Ew 1 1

Ci 1,2, ... ,v) (3)

306 E.J. DAVISON

where we:Rn is an unmeasurable disturbance vector, e. , i = 1, •.. , \I is the eFror in the system which is the differenre Between the out­put Yie:R i and the specified reference input y:e, i=l, .•. ,\I,and

1

y~e:RrT , i = 1, ... ,\1 are the only outputs which are available for m~asurement at the ith control station respectively.

~t Yred t~:J q[U ' e ~ [U and assume that

\I = m. > 1, rank C. = r. > 1, i = 1, ... , \I.

1 - 1 1-

Let r ~ I r i , i=l

equation:

rank B. m \11

r ~ I i=l

m r. w is assumed to satisfy the following 1

(4)

where (C1 ,A1) is observable and zl (0) to satisfy the following equation:

is unknown. Yref is assumed

n2 A2z2 z2e:R Go

(5) o C2z2

where (C2,A2) is observable and z2(0) is specified.

It is assumed for non-trivality that rank[El = rank Cl = n, +

rank G = rank C = dim(o), maxin, dim(o)} ~ 1, otAl ) C (, 0(A2) c C where 0(') denotes the set of eigenvalues of (.) and C denotes the closed right half complex plane. Let C' = (CT' , ... ,Cm') and let all other quantities be defined as before iW (1). \I

The decentralized robust servomechanism problem is to find \I local output feedback control laws with dynamic compensation for (3) so that the resultant closed loop system is s~able, and such that e + 0 as t + 00 Vx(O)e:Rn, Vz (O)e:Rnl, Vz 2 (0)e:R 2 for any perturba­tions in the plant data: tC. ,A, B.), i = 1, ... , \I which do not cause the resultant closed loop perrurbed system to become unstable. In this problem, it is seen then that it is desired to find a decentralized compensator for (3), so that asymptotic tracking occurs independent of any external disturbances in the plant and any plant parameter variations.

3. CONVENTIONAL APPROACH TO THE PROBLEM

Large technological systems have been present in society for

DECENTRALIZED STABILIZATION AND REGULATION 307

decades (in particular in chemical engineering, electric power systems, and process control) and decentralized control systems have often been found successfully to control these systems. It is clearly of interest to discuss how such controllers have been obtained.

The basic idea in the conventional way of solving the above two problems is to find a controller for each station so that the closed loop system is stable and so that the output of each station is "satisfactorly regulated", neglecting entirely any interactions existing between the different control stations. Such an approach works well, if in fact, there is little interaction occurring between control stations. It is of obvious interest to know, a priori, if there will be severe or favourable interaction existing between stations, and criteria (so-called "interaction-indices") for predicting when severe or mild interaction are to occur have been proposed, e.g. [30] - [34]* . Unfortunately, such criteria are based on heurestic arguments, and so cannot be used to develop a theory of decentralization. However, often deep insight into the reason why severe interaction occurs, may be obtained by using such interaction indices (e.g. see [33]) and this insight can then be used to avoid problems in which severe interaction would normally occur.

4. SOLUTION OF THE STABILIZATION PROBLEM

The following definition of "fixed-modes" is a generalization of the centralized idea of uncontrollable modes and unobservable modes of the triple (C,A,B), and is basic to the problem of decid­ing whether a decentralized system can be stabilized.

4.1 Fixed Modes of (1)

In (1) and (3) let

(6)

Definition (Wang, Davison [25])

rXn nXn J1 xm Consider the triple (C,A,B)sR x R xK and the two sets of

* Note that this problem is not to be confused with the so-called "decoupling problem" [35], in which it is desired to find a centra­lized controller for the system so that the resultant controlled system has the property that each input of the system affects one and only one output (or set of outputs) of the system.

308 E.J. DAVISON

v V integers ml, ..• ,mv; rl, ... ,rv with m = L m., r = L r. which

i=l 1 i=l 1 specifies (1). Let I( be the set of block diagonal matrices as follows:

{ m.xr.} K = K I K = block diag (Kl , ..• , K), Ki £R 1 1, i = 1, ••. ,V (7)

then the set of fixed modes of (C ,A, B) with respect to K is as follows:

defined

A(C,A,B)() = KQKcr(A+BKC) (8)

where cr (A + BKC) denotes the set of eigenvalues of A + BKC.

Remark 1

The set of fixed modes of a given triple (C,A,B) with respect to K can be calculated as followsthLet K ~ block dia~(Kl' .. • ,Kv)£K and let k(i,r,s) denote the (r,s) element of K., (1=1, ... ,v) (r = 1, .•• ,m.) (s = 1, ..• ,r.). Notice that K contains the null matrix, hence A.(C,A;B,K) S;.cr(A~1= {~l ... A ~. For each A.w(A), det (\1-A-BKC) 1S a polynom1a11n k(1,r,s)~1=1, •.• ,v), tr=l, ... ,m.), (s=l, ... ,r.). If det (A.1 - A-BKC) is identically zero, th~n clearly A.~ A(C,A,B,K).1Qn the other hand, if det(A.1 - A-BKC) is not identihlly zero, then there exists K £ K such tfiat det (A. 1-A-BKC) f 0, hence A. ~ A(C,A,B, K). This procedure determines the set of fixed modes 6f A(C,A,B,K).

The above remark forms the basis of the following algorithm to determine the fixed modes of (C,A,B) with respect to K .

Algorithm to Find Fixed Modes of (C,A,B) with respect to K

1.

2.

3.

4.

5.

Find the eigenvalues of A.

Select an arbitrary matrix K£K(by usin~ a pseudo-random number generator, say) so that II A II ~ II BKC II . Find the eigenvalues of A+BKC.

The fixed modes of (C,A,B) with respect to K are contained in those eigenvalues of A+BKC which are common with the eigenvalues of A(if there are any). Moreover for "almost any*" K chosen, the fixed modes will be identically equal to those eigenvalues of A+BKC which are common with the eigenvalues of A.

Repeat steps 2 to 4 using a different K, if there is any doubt which eigenvalues of A are the fixed modes of A.

* i.e. the class of matrices K which does not result in the fixed modes being identically equal to the common eigenvalues of A+BKC and A is either empty or lies on a hypersurface [36] in the parameter space of K.

DECENTRALIZED STABILIZATION AND REGULATION 309

Remark 2

In the case of centralized control, i.e. when K = Rmxr , then the fixed modes of the system (1) A(C,A,B,Rmxr) are just the modes of the triple (C,A,B) which are not both controllable and obser­vable [25].

4.2 Solution of the Stabilization (Pole-Assignment) Problem

The following theorem gives a solution to the decentralized stabilization problem previously stated in section 2.

Theorem 1 (Wang, Davison [25])

Consider the system (C,A,B) of (1). Let K be the set of block diagonal matrices defined in (7). Then a necessary and sufficient condition for the existence of a set of decentralized controllers (2) such that the closed loop system is asymptotically stable is that:

A(C,A,B,K) C C (9)

where C denotes the open left-hand complex plane.

An algorithm for finding a decentralized stabilizing compensa­tor for (1) assuming that theorem 1 holds is given in [25].

The following corollary gives a solution to the decentralized pole assignment problem.

Let S be any nonempty symmetric open subset of the complex plane C, then:

Corollary 1 [25]

Under the same assumptions as in theorem 1, the necessary and sufficient conditions for the existence of a set of decentralized controllers (2) such that all poles of the resulting closed loop system are in S is that:

A(C,A,B,K) C S (10)

4.3 Numerical Examples [25]

The following simple numerical examples illustrate the applica­tion of theorem 1 and corollary 1.

Example 1

Consider the following system (see Figure 1):

310 E.J. DAVISON

r---------------. I I

1 I I t~+ I --

I 5-1 1 I I I 1 I 1 I --I 5-2 I I 1 I I I 1 I

u 2 1 --5+1 I Y2

I L ____________ .....I

Figure 1 Decentralized control problem - example 1

x 0 G LD x + G c:) 0 C : ~) x

with K 0 {G' ~}" k,' R}

(11)

In this case (11) has a fixed

mode A = 2 and hence the system (11) cannot be stabilized using decentralized control (with dynamic compensation).

Example 2

Consider the following system:

x = (~ ~) x + (~

C:)·C :)X with K '{G' ~)I k"k,'R}

(12)

In this case (12) has no fixed

modes and hence the system can be stabilized using decentralized control as is done in Figure 2. Note also, that in this example, a

DECENTRALIZED STABILIZATION AND REGULATION 311

r------ -------, I I

I '---;--..--- Y2

U2: 5 I I 1 1- _____________ _

35+1 ~------------~-----5+3

Figure 2 Solution of decentralized control problem - example 1

decentralized controller can always be found so that the poles of the resulting closed loop system are "arbitrarily fast" (from corollary 1).

Example 3

Consider the

. (0 x = ~

1 o o

following

D x +

system (see Figure 3) :

G ~) (:~) (13)

In this case (13) has a fixed mode

A = 1 and hence the system cannot be stabilized using decentralized control.

4.4 Example of a Large Scale Interconnected Power System

The following example is a power system consisting of three interconnected synchronous machines, and is important in the sense that it shows in large scale systems there often is no real advan-

312 E.J. DAVISON

r------------, I I

5-1 r---:-I _ .... 52 1 YI

I I I I

Figure 3 Decentralized control problem - example 3

tage is using a (more complex) centralized control system over a (relatively simple) decentralized control system.

The following power system (see Figure 4) consisting of 3 in­terconnected synchronous generators is taken from [1]. All the three machines are assumed to be of the same capacity, viz 1.0 p.u. Machine No.2 is modelled to be a hydraulic machine while machines No.1 and 3 are modelled to be thermal machines. The values of power, voltage and angle obtained by a standard load flow analysis are also indicated in Figure 4. The values of the line impedances are such that Machine No.2 can be considered to be a remote machine while machines Nos. 1 and 3 can be considered to be local machines. Machine No.3 is chosen as the reference machine.

It is shown in [1] that the above multi-rnachin~ network may be described by the following equations (for small disturbances about the system's operating pOint):

x Ax + Bu (14)

y ex

where xER26 is the state, UER6 is the input and YER3 are the out­puts of the system:

!'IE refl !'IE ref2 !'Iwl !'IE ref3

!'Iw2 u = !'ITingl

y (15)

!'Iw3 !'IT. 2 lng !'IT. 3 mg -

DECENTRALIZED STABILIZATION AND REGULATION 313

~l~ 2 o 0

:; _--r~-_'·0245/0·5· "T'"--r...L...,--'·0 ).0·,· ...--___ .....,

0·05,i 0204

Figure 4 Power system consisting of 3 interconnected synchronous machines

where the superscript refers to machine 1, 2 or 3 and ~T. refers to the change in gate (valve) position, ~E f refers to tRg change of exciter voltage and ~w refers to the ch~ge of speed of the machine. Numerical values for A,B,C are given in [1].

In this case, it is desired to investigate the advantages of using a centralized control structure to control (14) over a decen­tralized control structure. The purpose of a controller in this study is to "speed up the response" of the closed loop system "as much as possible" ,by shifting the dominant eigenvalues of the system.

The following decentralized controller is assumed - it is widely applied commercially in the control of such power systems:

['Erefl] [kIt 0

k3~J ['W'] ~Eref2 k2 (s) ~w2 (16)

~E f3 0 ~w3 re

The following centralized controller will also be considered:

[Erefl] [kll (5) k12 (s) k13(S)] [::~] ~E f2 k12 (s) k22 (s) k23 (s) (17)

re ~E f3 k3l (s) k32 (s) k33 (s) re

314 E.J. DAVISON

Table 1 summarizes the effect of using the decentralized con­troller (16) in contrast to the centralized controller (17). Inthis case, it can be seen that there is no advantage in using the more complex centralized controller (17) over the decentralized con­troller (16) (which is conventionally used to control the system) with respect to eliminating the dominant fixed modes of the system,

Table 1 Comparison of Decentralized and Centralized Control (16), (17) for Interconnected Power System

Open loop e1genva1ues o~, A given by (14)

-0.175 -0.183 + j 10.5 -1.14 ~ j 6.13 -1.24 -2.33 -3.33 -3.33 -16.6 + j 3.72 -19.8 ~ j 23.1 -30.0 -31.6 -34.2 -43.0 + j 1.65 -56.7 -93.5 -106 -Ill + j 20.0 -120 :; j 19.7 -293

Fixed Modes for Decentrali zed Controller (16)

-1.24

-3.33 -3.33

-106 -Ill + j 20.0 -120 ~ j 19.7 -293

Fixed Modes for Centralized Controller (17)

-1.24

-3.33 -3.33

-106 -Ill + j 20.0 -120 ~ j 19.7 -293

i. e. the dominant time con~tant of the closed loop system cannot be made any faster than 1.24 sec using either a decentralized or centralized control structure. For interest, the dominant eigenvalues of the closed loop system using the decentralized control (obtained in [1]):

[lIErefl] liE f2 re liE f3 re

are as follows:

-0.367 -1. 24

o 0.093

o

-1. 88 ~ j 1.44 -2.97 + j 16.5 -3.33 -3.33

(18)

DECENTRALIZED STABILIZATION AND REGULATION 315

showing that the simple decentr~lized controller (18) gives a dominant time constant for the closed loop system which is only approximately a factor of 3 times slower than the fastest dominant time constant achievable.

For completeness, the following decentralized controller:

LIE fl re LIT. 1 lng lIEref2 LIT. 2 lng LIE f3 re LIT. 3

lng -'

kl (s) 0

k2 (s) 0

o k 3 (s)

o k4 (s)

o 0

o 0

and centralized controllers:

LIE fl re LIT. 1 lng LIE f2 re LIT. 2 lng LIE f3 re LIT. 3 lng

kll (s)

k2l (s)

k3l (s)

k4l (s)

kSl (s)

k6l (s)

k12 (s)

k22 (s)

k32 (s)

k 42 (s)

kS2 (s)

k62 (s)

k 13 (s)

k 23 (s)

k 33 (s)

k43 (s)

k S3 (s)

k 63 (s)

(19)

(20)

will also be considered. In this case, it is assumed that the control gate valve can also be used to control the synchronous machine (not conventionally done). Table 2 summarizes the effect of using the decentralized controller (19) in contrast to the cen­tralized controller (20). It is seen now that no dominant modes of the power system are now fixed, which means that the decentralized controller (19) should be much more effective than the decentralized controller (16) in controlling the power system. It is again seen that there is no real advantage in using the more complex cen­tralized controller (20) over the decentralized controller (19).

Table 2 Comparison of Decentralized Control (19), (20) for Interconnected Power System

Fixed Modes for Fixed Modes for Decentrali zed Centralized Controller (19) Controller (20)

-106 -106

-111 + j 20.0 -111 + j 20.0 --120 + j 19.7 -120 + j 19.7 - --293 -293

316 E.J. DAVISON

5. Solution of the Servomechanism Problem

The following definitions are required in the solution of the decentralized robust servomechanism problem.

Definition

Let the minimal polynomial of AI' A2 be denoted by Al (s), A2 (s) respectively, and let the zeros of the least common multiple of Al(S), A2 (S) (multiplicities included) be given by:

Let

(AI' A2 ,···,Aq)

m Let C* E: R(r +r) x (n+r) be given as follows:

o o

o

o o

o

o o

o o

(21)

(22)

TIl m.xr. block diag(Kl, ... ,Kv); Ki ER 1 1 i 1, ... ,V}

(23)

r.) } 1 , i= l, ... ,v

(24)

The statement z contains the actual output y means there exists a nonsingular transformation T so that Tz = [;] where y is

the output of the plant which is physically measured.

The following theorem gives a solution to the decentralized robust servomechanism problem previously stated in section 2.

DECENTRALIZED STABILIZATION AND REGULATION 317

Theorem 2 [28]

A necessary and sufficient condition that there exists a robust decentralized controller for (3) such that e + 0 as t + 00

for all unmeasurable disturbances w described by (4) and for all specified reference inputs y f described by (5), and such that the controlled system is stable thas a specified closed loop pole lo­cation) is that the following conditions all hold:

(a) (C , A, B) has no unstable fixed modes (has no fixed modes) .mh Km. Wlt respect to

* (b) The fixed modes with respect to K of the q systems

[~J} j == 1,2, ... ,q not contain A. , ]

(c)

j == 1,2, ... ,q respectively. m

Yi must contain the actual output Yi' i == l, ... ,v respectively.

A characterization of all controllers which solve the decen­tralized robust servomechanism problem is given in [28].

Discussion of Theorem 2

Condition (b) of theorem 2 can only be satisfied provided m. > r., i == 1,2, .. . ,v and provided rank B > rank C. This means tnat f6r there to be a solution to the problem, it is essential that at each control station of the system, there be at least the same number of independent local controls as there are local out­puts. Condition (c) of theorem 2 simply states that if the decen­tralized controller is to be robust, it is essential that all local outputs which are to be regulated must be measurable at the corres­ponding local control station. Condition (a) of theorem 2 is just a stabilizability condition for the system (3) to be stabilizable via decentralized control.

If m. r. , i 1,2, ... ,v then condition (b) becomes: 1 1

rank [A -:i I :] n+r, j 1,2, ... ,q (25)

which is equivalent to the condition that the transmission zeros [37] of (C,A,B) not coincide with A., j == 1,2, ... ,q corresponding to the class of disturbance/referenc~-inputs to (3). It should be noted however that if m. > r., i == 1,2, ... ,q then condition (b) of theorem 2 is not in gen§ral §quivalent to this condition. (Note that in the case of centralized control, i.e. when I(m == Rmxrm, condition (b) becomes equal to condition (25).)

318 E.J. DAVISON

Figure 5 Solution to decentralized robust servomechanism problem

5.1 Numerical Example

The following numerical example illustrates the application of theorem 2.

Example 4

Consider the following system:

x = [~ ~J x + [~ ~ ] [~~J + [~;J w

(26)

[~:J = G lJ + G:J w ° x

where ~~] 0 G:] and [::] e Gj -[~F:] where w is a

step disturbance i.e. w satisfies the equation zl = 0, w = zl' 2

zlsR , and y f corresponds to the class of step functions, i.e . . re 2" d"" ( ) Yref = 0 , z2 = 0, 0 = z2' z2 sR In thls case con ltlons a

and (c) of theorem 2 are satisfied (see example 2 re condition (a)), and it is only a matter of checking condition (b). In this example q= 1, (A., j = 1,2, ... ,q) = (0) and so condition (b) becomes:

J

DECENTRALIZED STABILIZATION AND REGULATION 319

{C* , [~ ~J, [~J}not have 0 as a fixed mode with respect to

K' where K' ={(~1 ~ )IK, ,K2,R1X2} • (27)

On substituting numericai values, {c*, [~ ~J, [~J}becomes:

[ ~---~---~---~] [~ ~ ~ ~] [~!~] 100 0 ' 0 1 00' 0 I 0 0001 1000 0:0

* which has no fixed modes with respect to J(. It is therefore seen that there exists a solution to the decentralized servomechanism problem for (26), and that in particular, a decentralized controller can always be found so that the closed loop system has arbitrary pole assignment. aigure 5 gives a solution to the problem; in this figure, S is a 2n order compensator (non-unique) designed to stabilize the following controllable-observable system:

n = [1 1 -1 o 0 1 0 o 0

o o

o o ~J n

(28)

Consider now the case when (3) arises from interconnecting various subsystems together to produce a composite system; this corresponds to the case of many real large scale systems arising in practise, e.g. in process control. It is of interest to know how likely will a solution exist to the composite system, given that a solution exists to each subsystem of the composite system. The following result is obtained.

5.2 Interconnected Composite Systems

Consider the special case of (3) when v subsystems are inter­connected together, each subsystem being described by:

x. 1

y. 1

m Yi

A.x. + B.u.+ E.w 1 1 111

C.x. + F.w 1 1 1

C~x. + F~w 1 1 1

v + I

j=l ~i

E .. x. 1J J

(i 1,2, ... ,v) (29)

then it is shown in [28] that if there is a solution to the robust servomechanism problem for each subsystem of the composite system

320

v (29) (with the term I

j=l f.i

E.J. DAVISON

E .. x. put equal to zero) then there 1J J

always is a solution to the decentralized robust servomechanism problem for (29) provided the interconnection matrices E .. are

1J "small enough". However if (CI!!, A., B.) is not just stabilizable

111 and detectable but is controllable and observable for i = 1,2, .. . ,v then there "almost always"*is a solution to the decentralized robust control problem for l29) for any magnitude of the interconnection matrices. Thus this result gives a justification for the wide­spread use of applying only local controllers about each subsystem of large scale systems which arise, for example, in chemical process control and electric power systems.

6. CONCLUSIONS

A survey of some particular problems ar1s1ng in large scale systems has been made in this paper. In particular, a study of the stabilization (pole-assignment) and servomechanism problem has been made for the case of linear time-invariant systems subject to decentralized control.

It should be remarked that other classes of feedback control configurations, which correspond to a broader interpretation of the term decentralized control, may also be considered in the problem formulation. For example, if we assume that some local control stations cannot observe their outputs in the usual decentralized way, then this can be considered in the problem formulation of theorems 1,2 by letting K be the class of matrices with some zero blocks in the diagonal. Another example is the case when the given system consists of an ordered set of subsystems in which all output information preceding any given subsystem is available to that sub­system. This constraint can be dealt with by letting K in theorem 1, 2 be the class of matrices in block triangular form.

The subject of large scale systems is relatively new and not well understood. Much work is required, for example, on determin­ing the minimum order compensator required to satisfy a problem statement, on relating the results of information structures with decentralized control etc.

ACKNOWLEDGEMENT

The author wishes to thank Mr. W. Gesing and N. Tripathi for assistance in the numerical examples. This work was supported by the National Research Council under grant No.A4396. *See [28] for a precise technical definition of this.

DECENTRALIZED STABILIZATION AND REGULATION 321

REFERENCES

1. Davison E.J., Rau N.S., Palmy F.V., "The Optimal Decentralized Control of a Power System Consisting of a Number of Inter­connected Synchronous Machines", Int. J. Control, vol.18, No.6, 1973, pp. 1313-1328.

2. Varaiya P., "Trends in the Theory of Decision-Making in Large Systems" Ann. Economic and Social Measurement, vol.l/4, 1972, pp. 493-500.

3. Athans M., Kendrick D., "Control Theory & Economics in a Survey, Forecast & Speculations", IEEE Trans. on Automatic Control vol. AC-19, No.5, 1974, pp. 518-523.

4. Foss A.S., "Critique of Chemical Process Control Theory", IEEE Trans. on Automatic Control, vol. AC-18, No.6, 1973, pp. 646-652.

5. Lau R., Persiano R.M., Varaiya P., "Decentralized Information and Control: a Network Flow Example", IEEE Trans. on Automatic Control, vol. AC-17, 1972, pp. 466-473.

6. Bellman R., "Large Systems", IEEE Trans. on Automatic Control, vol. AC-19, No.5, 1974, p.465.

7. Arrow K.J., "Optimization, Decentralization and Internal Pricing in Business Firms", in Contributions to Scientific Research in Management, University of California, Los Angeles, 1959.

8. Marschak J., Radner R., The Economic Theory of Teams, Yale University Press, New Haven, 1971.

9. Chong C.Y., Athans M., "On the Stochastic Control of Linear Systems with Different Information Sets", IEEE Trans. on Automatic Control, vol. AC-16, No.5, 1971, pp. 423-430.

10. Ho Y.C., Chu K.C., "Team Decision Theory and Information Structures in Optimal Control Problems - Part I", IEEE Trans. on Automatic Control, vol. Ac-17, 1972, pp. 15-22.

11. Chu K.C., "Team Decision Theory and Information Structures in Optimal Control Pr.oblems - Part II", IEEE Trans. on Automatic Control, vol. AC-17, 1972, pp.22-28.

12. Aoki M., "On Decentralized Linear Stochastic Control Problems with Quadratic Cost", IEEE Trans. on Automatic Control, vol. AC-18, 1973, pp. 243-250.

322 E.J. DAVISON

13. Sandell N .R., Athans M., "Solution of Some Non-Classical LQG Stochastic Decision Problems", IEEE Trans. on Automatic Control, vol. AC-19, 1974, pp. 109-116.

14. Aoki M., Toda M., "Parameter Adaptive Resource Allocation Problem for a Decentralized Dynamic System", IEEE Trans. on Automatic Control vol. AC-20, No.2 1975, pp. 192-199.

IS. Athans M., "Theoretical and Practical Problems in the Design of Stochastic Large Scale Systems", Iranian J. Sci. & Tech., vol. 3, No.4, 1975, pp. 211-219.

16. Athans M., "Survey of Decentralized Control Methods", Report No. ESL-P-sss MIT Electronic Systems Laboratory, MIT, July 1974, Annals of Economic and Social Measurement, submitted.

17. Chu K.C. "Optimal Decentralized Regulation for a String of Coupled Systems, IEEE Trans. on Automatic Control, vol.AC-19, No.3, 1974, pp. 243-246.

18. Chong C.Y., Athans M., "On the Periodic Co-ordination of Linear Stochastic Systems", 6th Congress of IFAC, Boston, Aug. 1975, paper No. 19.4.

19. Mesarovic M.D., Macko D., Takahara Y., Theory of Hierarchical Multilevel Systems, Academic Press, NY, 1970.

20. Rosenbrock H.H., Pugh A.C., "Contributions to a Hierarchical Theory of Systems", Int. J. Control, vol. 19, 1973, pp. 845-867.

21. Rosenbrock H.H., "Structural Properties of Linear Dynamical Systems" Int. J. Control, vol. 20, 1974, pp. 191-202.

22. Rosenbrock H.H., "Recent Results in Large Scale Systems" 12th Allerton Conference on Circuit and System Theory, 1974, pp.s74-579.

23. Ozguner V., Perkins W.R., "Controllability, Pole Placement & Stabilizability in Large Scale Composite Systems", 12th Allerton Conference on Circuit and System Theory, 1974, pp. 441-449.

24. Aoki M., "On Feedback Stabilizability of Decentralized Dynamic Systems", Automatica, vol. 8, 1972, pp. 163-173.

25. Wang S .li., Davison E.J., "On the Stabilization of Decentralized Control Systems", Control System Report No. 7213, Dept. of Electrical Engineering, University of Toronto, IEEE Trans. on Automatic Control, vol. AC-18, No.5, 1973, pp. 473-478.

26. Davison E.J., "The Decentralized Stabilization and Control of a

DECENTRALIZED STABILIZATION AND REGULATION

Class of Unknown, Nonlinear Time-Varying Systems", Automatica, vol. 10, 1974, pp. 309-316.

323

27.Corfmat J.P., Morse A.S., "Decentralized Control of Linear Multivariable Systems", 6th Congress of IFAC, Boston Aug. 1975, paper 43.3.

28.Davison E.J., "The Robust Decentralized Control of a General Servomechanism Problem", Control System Report No. 7423, April 1975, Dept. of Electrical Engineering, University of Toronto, IEEE Trans. on Automatic Control, Feb. 1976, to appear.

29.Brasch F.M., Pearson J.B., "Pole Placement using Dynamic Compen­sators", IEEE Trans. on Automatic Control, vol. AC-15, 1970, pp.34-43.

30.Mitchell D.S., Webb C.R., "A Study of Interaction in a Multi­loop Control System", Proc. 1st IFAC Congress, Moscow 1960, pp. 1009-1018.

31.Rijnsdorp J.E., "Interaction in Two-variable Control Systems for Distillation Columns",Automatica, vol. 3 1965, pp. 15-51.

32.Davison E.J., "A Nonminimum Phase Index and its Application to Interacting Multivariable Control Systems", Automatica, vol.5, 1969, pp. 791-799.

33.Davison E.J., "The Interaction of Control Systems in a Binary Distillation Column", Automatica, vol. 6, 1970, pp. 447-461.

34.Davison E.J., Man F.T.,. "Interaction Index for Multivariable Control Systems" Proc. lEE, vol. 117, No.2, 1970, pp. 450-462.

35.Morse S .A., Honham H .M., "Status of Noninteracting Control", IEEE Trans. on Automatic Control, vol. AC-16, No.6, 1971, pp. 568-581.

36.Davison E.J., Wang S.H., "Properties of Linear Time-Invariant Multivariable Systems Subject to Arbitrary Output and State Feedback", IEEE Trans. on Automatic Control, vol. AC-18, 1973, pp. 24-32.

37.Davison E.J., Wang S.H., "Properties and Calculation of Trans­mission Zeros of Linear Multivariable Systems", Automatica, vol.lO, 1974, pp. 643-658.

THE ROLE OF POLES AND ZEROS IN MULTIVARIABLE

FEEDBACK THEORY

A.G.J.MacFarlane

Engineering Department, University of Cambridge

Cambridge, England

1. Introduction. The increasing interest in large-scale systems with complex control structures, together with the widespread use of state-space models as the basic form of system description, naturally leads one to wonder what relevance the basic ideas of classical control theory (poles, zeros, transfer functions, Nyquist diagrams, root loci) have to such problems. Classical single-variable feedback theory revolves round the properties of poles and zeros of scalar-valued functions of a complex variable. Rosenbrock's pioneering work (Rosenbrock, 1970,1974) showed that algebraic definitions could be given for multivariable poles and zeros, and that multivariable frequency-response design methods could be developed. Work by MacFarlane (1975), Kouvaritakis (1975a,1975b), Karcanias (1975) and Shaked (1975)has shown that generalisations exist of the Nyquist (1932)-Bode(1945) frequency response approaches and of the root locus method (Evans 1954). Almost all of this work however is either algebraic,using concepts such as the Smith­McMillan (McMillan,1952)! (Rosenbrock,1970) form of a transfer function matrix, or geometric, using concepts such as the null-space of the output map of a state­space description. What is required to tie it all together, and to relate it directly to the complex­variable base of classical control theory, is the establishment of a link between multivariable poles and zeros and complex function theory. This brief paper has two purposes

325

326 A.G.J. MacFARLANE

(i) to show how the poles and zeros of a matrix-valued function of a complex variable relate to the standard theory of algebraic functions (Bliss,1966) and

(ii) to briefly discuss the significance of this link for feedback theory and design.

2. Algebraic definitions of poles and zeros for multivariable systems

The Smith-McMillan form definition of the poles and zeros of a matrix-valued function G(s) of the complex variable s can be shown (Kontakos, 1973) to be equivalent to the following simple pair of rules.

Pole polynomial: The pole polynomial pes), whose roots are the poles, is the least common denominator of all non-identically zero minors of all orders of G(s).

Zero polynomial: The zero polynomial z(s), whose roots are the zeros, is the greatest common divisor of the numerators of all minors of G(s) of order r, where r is the normal rank of G(s) (that is the rank of G(s) for almost all values of s), provided that these minors have been adjusted in such a way as to have the pole pOlynomial as their common divisor.

3. Complex-variable definitions of poles and zeros for multivariable systems

An extremely important consequence of the fact that the multivariable zeros are determined by the greatest common divisor of all minors of order r, where r is the normal rank of G(s), is that zeros are only "generically defined" for square-matrix-valued functions. That is to say th:it if G( s) is non-square and has zeros, then a small perturbation of any element of G(s) will cause these zeros to vanish. Thus zeros are only "naturally" defined for square-matrix-valued functions G(s). For such functions, we can associate with G(s) an algebraic function g(s) (Bliss,1966).The link we wish to establish then follows by showing that the poles and zeros of g(s), defined by standard complex­variable arguments, are the same as the poles and zeros of G(s), defined by algebraic arguments.

3.1 The characteristic function

Given an m x m transfer function matrix G(s)we can, in some sense to be made precise in what follows, consider its eigenvalues {g.(s) i = 1,2, ... ,m}to be

l

functions of a complex variable given by the solution

POLES AND ZEROS IN MULTIVARIABLE FEEDBACK THEORY

of the equation

det [g(S)I m G (s)] = 0

This may be expanded as

327

(1)

gm(s) + ales) gm-l(s) + .......... + ames) = 0 (2)

where the coefficients {a.(s) : i = 1.2 •...• m} are rational functions in S.l If b (s) is the least common denominator of the coeff~cients a.(s). we may

l put this in the form

m m-l bo(s) g (s) + bl(s) g (s)+ ... +bm(s) = 0 (3)

where the coefficients polynomials in s.

{b . ( s ) : i = I, 2, ... ,m} are now l

The function of a complex variable g(s) whose values are defined by the equation (3) is called an algebraic function. Thus the set of eigenvalues g.(s) of a square matrix-valued rational function of a l complex variable G (s) are as socia ted with an algebraic function g(s) which we will call the characteristic function of G(s).

3.2 Ordinary points and critical points

Let m m-l

~(s,g)=bo(s)g (s) + bl(s) g (s) + ... + bm(s) ( 4- )

For any specific value of the complex variable s,~(s,g) becomes a polynomial in g whose discriminant (Bliss, 1966) can be calculated; let D (s) denote the discrimin­and of ~(s,g). Ordinary point gof g(s): An ordinary point of the characteristic function g(s) is any point of the complex plane such that

b (s) ~ 0 and D (s) ~ 0 o g

Critical point of g(s): The remaining points of the complex plane at which either

b (s) = 0 o

or D (s) = 0 g

are called critical points of the charactertistic function.

328 A.G.J. MacFARLANE

3.3 Riemann surface of characteristic function

At every ordinary point. equation (3) has m distinct roots. since the discriminant does not vanish. and the theory of algebraic functions then shows that in any simply-connected region of the complex plane contain­ing only ordinary points the values of the character­istic function g(s) form a set {g.(s):i=1.2 •...• m} of distinct analytic functions. Ea6h of these distinct analytic functions g.(s) is called a branch of the characteristic functi~n g(s). Arguments based on the standard techniques of analytic continuation together with the properties of algebraic equations show that the various branches can be organised into a single entity. the appropriate algebraic function.which can then be regarded as a natural generalisation of the familiar elementary funetion of a complex variable. An elementary function of a complex variable has the set of complex numbers (fas both its domain and its range. An algebraic function has the complex number set ~ as its range but has a new and appropriately­defined domain which is called its Riemann Surface ~ (Springer. 1957).

3.4 Poles and zeros of the characteristic function g(s)

Let us now consider the values of the character­istic function g(s) in the vicinity of a critical point. Suppose a is a critical point. and that a small curve on the RIemann surface is run round a so as to enclose no other critical points. We can start on one sheet of the Riemann surface. corresponding to a branch gl(s) say. and then analytically continue gl(s) along ~he curve which surrounds the cri tical point. Since whenever a cut is introduced we always find that the values of the algebraic function on one side are simply a permuta­tion of the values on the other, the function gl(s) which is being analytically continued will go back into itself after a finite number. say p. of encircle­ments of the critical point. If we therefore set

(s - a) = (s')p

so that gl(s) = gl [(s')p + a] = 0 l (s')

( 5 )

( 6 )

then the function 0(s') will be an analytic single­valued function of s' in the neighbourhood of the point s'=O, excluding the critical point itself. Consequently the function 0(s') can be developed in a Laurent series expansion.

POLES AND ZEROS IN MULTIVARIABLE FEEDBACK THEORY

o (Sl) = 1

n = +00

C (sl)n n

n = - 00

It follows from this that, in a neighbourhood of the critical point, gl(s) will have an expansion

n

C (s - a)p n

In fact, all the values of the various branches of the algebraic function may be obtained from this one expression simply by giving

n

(s-a)p all its possible alternative values. There are now two cases to consider, according as b (s) in

o equation (3) is, or is not, zero.

329

( 7 )

( 8 )

If b (a) t 0, then ~(s,g) will have precisely m roots (someoof which may be multiple roots of course). These roots will all be continuous at the point s = a, so that in this case no negative powers of (s-a) can appear in the expression (8).

th If b (a)=O, say because b (s) has a q order o 0 zero at s=a, we may put

b (s) = (s a)qh ( s) 0 0

where h (a) 0

t O. If we now form

(s - a)(m-l)q ~(s, g)

then we get

(s - a)(m-l)q ~(s, g)

= (s - a)(m-l)q[bo(S)gm + bl(S)gm-l+ ... +bm(s))

= b (s) o

(s-a)q

= h (s) o

where

m m-l v + h 1 ( s ) v + ..•. + h m ( s )

v(s) = (s - a)q g(s)

( 9 )

(10 )

(ll )

330 A.G.J. MacFARLANE

and v(s) satisfies an equation of the form

m m-l ~(s,v) = h (s)v + hl(s)v + .... +h (s)=O

o m

where h (a) 1 0, so that the roots of ~(s,v) must be continu8us at s = a. Thus the Laurent expansion of the branch vl(s) will contain no negative powers at s = a. Since the corresponding branch gl(s) will be given by

vl(s) g (s)= --

1 (s-a)q

(12 )

(13 )

it immediately follows that the expan~ion of gl(s) will have at most a finite number of negatlve powers at s = a and thus have a pole at s = a. Hence the zeros of b (s) define the poles of the characteristic function -----o~~--------------~----------------------------------------------g(s).

Now let us consider the zeros of the characteristic function g(s). We have that

m a ( s ) = II g. ( s )

m i=l l (14)

and so

b ( s ) m m II g. (s )

b (s) = l 0 i = 1

(15 )

Obviously a sufficient condition for s' to be a zero of g(s) is that a (s') should vanish. This would not however cover the ~ase where a zero coincided with a pole, in which case a cancellation could take place between b (s) and b (s). We therefore define the zeros of ~(s) to beOthe zeros of b (s)

m

3.5 Poles and zeros of the matrix G(s)

With every square m ~ m matrix-valued function of a complex variable G(s) is associated an algebraic function of a complex variable g(s), the characteristic function, and associated with this algebraic function is a set of poles and zeros. It is natural to make a direct association of these poles and zeros with G(s) and to call them the poles and zeros of G(s). Of course, for this to be a meaningful and useful thing to do, one must show that this way of defining the poles and zeros of the matrix G(s) gives the same result as the methods

POLES AND ZEROS IN MULTIVARIABLE FEEDBACK THEORY 331

using standard matrix theory concepts which have already been treated, such as the Smith-McMillan form of G(s).

We have that:

(i) the poles of g(s) are the roots of

(ii)

b (s) = 0 o

the zeros of g(s) are the roots of

b (s) = 0 m

Now the coefficients a.(s) in the expansion: l

det [gIm -

m m-l = g +a1(s)g + a 2 (s)

G(s)]

m-2 g + .... a l(s)g+a (s) m- m

(16 )

are all appropriate sums of minors of G(s) since it can be shown that

det [gI m-G(s)l

m = g + (trace G) gm-l+ ( Iminors of G of order 2)gm-2+

...... + det G(s) (17)

and thus b (s) must be the least common denominator of all minorsoof all orders of G(s).

However it has already been noted that :

(i) the pole polynomial pes) of a square matrix G(s) is the least common denominator of all minors of all orders of G(s);

(ii) the zero polynomial z(s) of a square matrix G(s) is the numerator of the rational function obtained from det G(s) after adjusting det G(s) in such a way as to obtain an equivalent rational function with pes) as its denominator.

Since b (s) has been shown to be the least common denomingtor of all minors of all orders of G(s) we thus have that

b (s) o = pes)

Furthermore since

a (s) = det G(s) m

(18 )

332 A.G.J. MacFARLANE

and a (s) m

we have that

b (s) m

pes)

b (s) = m = b(S)

o

= det G(s)

b (s) m

pes)

and so we must have that

b (s) = z(s) m

(19)

(20)

We thus reach the important conclusion that the poles and zeros of the characteristic function g(sr­are the same as the poles and zeros usually defined for the square matrix G(s) via the Smith-McMillan form. We may therefore regard the square matrix G(s) as unambiguously associated with an algebraic function g(s) whose poles and zeros may thus be called the poles and zeros of G(s), and thence, by a normal abuse of language, the poles and zeros of the system whose transfer function matrix is G(s).

4. Significance of multivariable poles and zeros for feedback system studies

The intimate relationship between zeros and feed­back is best introduced by considering a simple basic multivariable feedback arrangement. Suppose we have a system with the same number of inputs and outputs so that we may connect all the outputs directly back to the correspondingly-numbered inputs. Now the essential feature of a set of feedback connections of this sort is that one set of variables has forcibly been made the same as another set of variables, and so the difference between the two sets of variables must vanish identically. The operator which generates the difference between input and output vectors, when operating on the input vector, is obviously

F(s) = I - G(s) (21)

if G(s) is the transfer function matrix of the original system. Since the feedback connections force the difference between input and output to be identically zero, any non-zero input signal transform vector, u(s) say, which exists under these feedback conditions must be such that

F(s)u(s) = 0 (22)

POLES AND ZEROS IN MUL TIVARIABLE FEEDBACK THEORY 333

As we are considering the free motion of a linear system, the components of u(s) will consist of the transforms of exponential modal functions, and we can thus associate the set of values of s for which this relationship holds directly with the system closed­loop characteristic frequencies. Equation (22) shows that the appropriate values of s are those for which the columns of F(s) become linearly dependent and thus are defined by

det F(s) = 0 (23)

Hence the closed-loop characteristic frequencies must be associated with the zeros of the matrix F(s), which is called the return-difference matrix (MacFarlane,1970) for the set of feedback loops which is being closed. Since the poles of the matrices G(s) and F(s) are obviously the same, we see that the poles of the return­difference operator are associated with the system open­loop characteristic frequencies and its zeros are associated with its closed-loop charaeteristic frequenc­ies. A detailed examination of this situation gives the following result which is of basic importance in multivariable feedback theory.

Change of characteristic polynomial under feedback

Suppose that we have a linear dynamical system 11 and feedback connect another system 12 to it. Let L (s) be the m x:n loop transmittance matrix corresponding to the set of feedback connections made and let

F(s) = I - L(s) m

be the corresponding return-difference matrix for the set of connections made. Let

(24)

OLCP(s) = characteristic polynomia~ of the composite system formed by PI and ;2 before the feedback connections are made;

CLCP(s) = characteristic polynomial of the complete system formed after the feedback connections are made;

then it can be shown that (Hsu and Chen, 1968)

Closed-loop characteristic polynomial Open-loop characteristic polynomial

= det F(s) (25) de t F (CIO )

334 A.G.J. MacFARLANE

This shows that the stability or instability of any system created by making a set of feedback connections to any given system is completely determined by the location of the zeros of the appropriate return­difference matrix.

If we add a controller represented by a matrix K(s) to a system represented by G(s) in the usual feed­back arrangement then the return-difference matrix will assume the form

(26)

where the plus sign is due to the assumption of negative feedback. As the norm of IG(S)K(S) becomes arbitrarily large then the zeros of F (s). and thus the system's closed-loop characteristic frequencies will tend to the zeros of G(s)K(s). This simple result is the essential basis from which generalised root-locus methods may be developed. when taken together with the geometric theory of zeros which ties together zeros and state-space structures. (Kouvaritakis and Shaked. 1975). Thus a natural extension of classical root locus ideas may be made to the multivariable domain.

Furthermore the algebraic function approach to multivariable system poles and zeros provides a bridge to the generalised Nyquist-Bode design techniques (MacFarlane. 1975) as well as giving a firm theoretical foundation for the proof of the generalised Nyquist stability theorem. (MacFarlane. 1975)

The zeros represent the nature of the couplings between a system's characteristic modes and its environ­ment; they depend on the matrices Band C of the standard state-space model and on the way in which these matrices are related to the eigenframework of the matrix A of the standard state-space model. Associated in a one-to-one correspondence with the invariant zeros of a system are a set of zero directions (Karcanias. 1975). In the state space these correspond to rectilinear motions taking place in the kernel of the output map C. There will thus be no observed motion in the output space. but there may be a corres­ponding rectilinear motion in the input space.

There is an interesting inversion of the roles of pole and zero with respect to possible corresponding forms of rectilinear motion in the input. state and out­put spaces. We have that (Karcanias. 1975): (i) corresponding to a pole we have with zero input a rectilinear motion in the state space of the form xoept

POLES AND ZEROS IN MUL TIVARIABLE FEEDBACK THEORY 335

generating a rectilinear motion Cx e pt in the outuut space, where p is a pole and als8 an eigenvalue of A, and Xo is the corresponding eigenvector.

(ii) corresponding to ~ zero we have a rectilinear motion of the form ge Z in the input ~pace generating a rectilinear motion of the form x e Z in the state space with zero output. Here g agd x are the "zero directions" corresponding to the z~ro z.

To get some further physical insight it is helpful to consider a trivial illustrative example. Suppose we have a system with one input, one output and two states. Let the eigenvalues of the A-matrix be Al and A2 and assume Al i A2 · Let {u l ,u2 } and {vI,v~} be the eigen­vectors and dual eigenvettors of A respectively. Let the output map be C = c , a row vector, and the input map be B = b, a column vector. Then a straightforward calculation shows that this system has one zero given explicitly by the formula:

t[ t t ] zero = c u l v1 A2 + u2v2Al b

c tb

It is obvious that by a suitable choice of either of the maps Band C and by variation of the other, the zero can be placed at any point on the real axis in the complex plane. (It goes to the point at infinity when the vector b is orthogonal to the vector c.) This clearly indicates, in the context of this simple example, how zeros represent the couplings between the system's modes and its external environment. Difficulties in the feedback control of a system simply arise from limita­tions of power or of information. Different arrange­ments for coupling power into the system, and for extracting information from it, lead to different patterns of system zeros; the number and location of these zeros gives an immediate indication of the difficulty of imposing feedback control. The only fundamental way to change the pattern of zeros and hence, implicitly, to change the nature of the feedback control problem is to change the arrangements for taking output measurements or for providing power inputs. It can be shown that the addition of extra measurements or, more difficult in practice, the provision of extra inputs, can always be done in such a way as to remove unwanted zeros. Furthermore the combination of outputs by the designer can always be carried out in such a way as to create new zeros in suitable locations. Right-half-plane zeros represent particularly unfortunate forms of input

336 A.G.J. MacFARLANE

and/or output coupling which result in severe control difficulties. Since, from one point of view, zeros result from an irrecoverable loss of information about states, associated with the existence of a null space for the output map C, no amount of serial dynamic compensation can recover this information. Thus the only practical cure for the acute situation associated with a system right-half-plane zeros in a feedback loop transmittance is the modification of the system itself by taking a fresh set of outputs and/or inputs. Suitably placed left-half-plane zeros can be used to help in providing acceptable locations for closed-loop poles under high-gain feedback. In designing a regulator therefore, the combination of suitable sets of outputs in forming multi variable feedback loops should be done in such a way as to create an appropriate set of left­half-plane zeros for a loop transmission matrix. Synthesis techniques for doing this have been developed by Kouvaritakis (Kouvaritakis and MacFarlane, 1975a, 1975b). and Karcanias (1975). Zeros are the primary design tool for the selection of variables in forming feedback loops, and their systematic use could prove particularly helpful in the early stages of feedback system development when freedom may still exist to choose what sets of variables are to be manipulated and what sets are to be measured for control purposes. It could well be that in future feedback system designers will be more concerned with selecting suitable system variable sets for manipulation and measurement in such a way that subsequent control is relatively easy to achieve than with devising elaborate compensation schemes for systems whose structure has previously been arbitrarily fixed.

REFERENCES

BLISS, G.S., 1966 (reprint of 1933 original): Algebraic functions (New York: Dover) BODE, H.W., 1945, Network Analysis and Feedback Amplifier Design (New York: Van Nostrand) EVANS, W.R.,1954, Control System Dynamics (New York: McGraw-Hill) HSU, C.H. and CHEN, C.T.,1968, A proof of the stability of multivariable feedback systems, Proc.IEEE,56,2061-2062 KARCANIAS, N. ,1975, Geometric theory of zeros and its use in feedback analysis, Ph.D. Thesis, University of Manchester KONTAKOS,T.,1973 Algebraic and geometric aspects of multivariable feedback control systems,Ph.D. Thesis, University of Manchester

POLES AND ZEROS IN MUL TIVARIABLE FEEDBACK THEORY 337

KOUVARITAKIS. B. and MACFARLANE.A.G.J.,1975a. Geometric method for. computing zeros of square matrix transfer functions, to be published in Int.Jnl.Control. KOUVARITAKIS, B. and MACFARLANE,A.G.J .• 1975b. Geometric method of computing and synthesizing zeros for general non-square matrix transfer functions, to be published in Int.Jnl.Control KOUVARITAKIS, B. and SHAKED, U.,1975, Asymptotic behaviour of root loci of linear multivariable systems, to be published in Int.Jnl.Control MACFARLANE, A.G.J.,1970. The return-difference and return-ratio matrices and their use in the analysis and design of multivariable feedback control systems, Proc.IEE, 117. 2037-2059 MACFARLANE, A.G.J.,1975, Relationships between recent developments in linear control theory and classical design techniques, Measurement and Control. 8, 179-18~, 219-223, 278-284, 319-324 and 371-375 McMILLAN, B .• 1952, Introduction to formal r.e_aJizabili ty theory - II, Bell Syst.Tech.J.,31, 541-600 NYQUIST, H., 1932, Regeneration theory, Bell Syst.Tech. J., II, 126-147 ROSENBROCK, H.H., 1970, State space and multivariable theory (London: Nelson) ROSENBROCK, H.H., 1973, The zeros of a system, Int.J. Control, 18 297-299 ROSENBROCK. H.H., 1974, Correction to 'The zeros of a system', Int.Jnl.Control, 20, 525-527 ROSENBROCK, H.H., 1974, Computer-aided control system design (London: Academic Press). SPRINGER, G., 1957, Introduction to Riemann Surfaces (Reading, Mass: Addison-Wesley). WOLOVICH, W.A., 1973, On determining the zeros of a state-space system, IEEE Trans. on Aut. Control,AC-18, 542-544 ZADEH, L.A., and DESOER, C.A., 1963, Linear system theory (New York: McGraw-Hill)

THE LINGUISTIC APPROACH AND ITS APPLICATION TO DECISION ANALYSISt

L.A. Zadeh

Division of Computer Science Department of Electrical Engineering and Computer Sciences University of California, Berkeley, CA 94720

ABSTRACT

In a sharp departure from the conventional approaches to deci­sion analysis, the linguistic approach abandons the use of numbers and relies instead on a systematic use of words to characterize the values of variables, the values of probabilities, the relations between variables, and the truth-values of assertions about them.

The linguistic approach is intended to be used in situations in which the system under analysis is too complex or too ill-defined to be amenable to quantitative characterization. It may be used, in particular, to define an objective function in linguistic terms as a function of the linguistic values of decision variables.

In cases in which the objective function is vector-valued, the linguistic approach provides a language for an approximate linguis­tic characterization of the trade-offs between its components. Such characterizations result in a fuzzy set of Pareto-optimal solu­tions, with the grade of membership of a solution representing the complement of the degree to which it is dominated by other solutions.

1. INTRODUCTION

The past two decades have witnessed many important theoretical

-1"Research sponsored by Naval Electronics Systems Command Contract N00039-76-C-0022, U.S. Army Research Office Contract DAHC04-75-G0056 and National Science Foundation Grant ENG74-0665l-AOl.

339

340 L.A. ZADEH

advances in decision theory [1-18] as well as in such related fields as mathematical programming, statistical analysis, system simula­tion, game theory and optimal control. And yet, there are many observers who would agree that it is by no means easy to find con­crete examples of successful applications of decision theory in practice. What, then, is the reason for the paucity of practical applications of a wide-ranging theory that had its inception more than three decades ago?

Although this may not as yet be a widely accepted view, our belief is that the limited applicability of decision theory to rea1-world problems is largely due to the fact that decision theory -­like most other mathematical theories of rational behavior -- fails to come to grips with the pervasive fuzziness and imprecision of human judgment, perception and modes of reasoning. 1 Thus, based as it is on the foundations of classical mathematics, decision theory aims at constructing a model of rational decision-making which is quantliative, rigorous and precise. Unfortunately, this may well be an unrealizable objective, for real-world decision processes are, for the most part, far too complex and much too ill-defined to be dealt with in this spirit. Indeed, to be able to cope with rea1-world problems, the mathematical theories of human cognition and rational behavior may have to undergo an extensive restructuring -­a restructuring which would entail an abandonment of the unrealis­tically high standards of precision which have become the norm in the literature and an acceptance of modes of logical inference which are approximate rather than exact.

The linguistic approach outlined in the present paper may be viewed as a step in this direction. In a sharp break with deeply entrenched traditions in science, the linguistic approach abandons the use of numbers and precise models of reasoning, and adopts instead a flexible system of verbal characterizations which apply to the values of variables, the relations between variables and the truth-values as well as the probabilities of assertions about them. The rationale for this seemingly retrograde step of employing words in place of numbers is that verbal characterizations are intrin­sically approximate in nature and hence are better suited for the description of systems and processes which are as complex and as ill-defined as those which relate to human judgment and decision­making.

It should be stressed, however, that the linguistic approach

lIn fact, far from being a negative characteristic of human think­ing -- as it is usually perceived to be -- fuzziness may well be the key to the human ability to cope with problems (e.g., language translation, summarization of information, etc.) which are too com­plex for solution by machines that lack the capability to operate in a fuzzy environment.

THE LINGUISTIC APPROACH TO DECISION ANALYSIS 341

is not the traditional non-mathematical way of dealing with human­istic systems. Rather, it represents a blend between the quantita­tive and the qualitative, relying on the use of words when numerical characterizations are not appropriate and using numbers to make the meaning of words more precise [19,20].

The central concept in the linguistic approach is that of a linguistic variable, that is, a variable whose values are words or structured combinations of words whose meaning is defined by a semantic rule [20]. For example, Age is a linguistic variable if its values are assumed to be young, not young, very young, not very young, ~ or less young, etc., rather than the numbers 0,1,2, ••. , 100. The meaning of a typical linguistic value, say not very young, is assumed to be a fuzzy subset of a universe of discourse, e.g., U = [0,100], with the understanding that the meaning of not very young can be deduced from the meaning of young by the application of a semantic rule which is associated with the variable Age. In this sense, then, young is a primary term which plays a role akin to that of a unit of measurement. However, it is important to note that (a) the definition of young is purely subjective in nature; and (b) in contrast to the way in which the conventional units are used, the semantic rule involves nonlinear rather than linear opera­tions on the meaning of the primary terms. These issues are dis­cussed in greater detail in Section 2.

An important part of the linguistic approach relates to the treatment of truth as a linguistic variable with values such as true, very true, not very true, more ~ less true, etc. The use of such linguistic truth-values leads to what is called fuzzy logic [21] which provides a basis for approximate inference-rroffi possibly fuzzy premises whose validity may not be sharply defined. As an illustration, an approximate inference from (a) x is a small number, and (b) x and yare approximately equal, might be (c) y is more or less small. Similarly, an approximate inference from (a) (x is a­small number) is very true, and (b) (x and yare approximately equal) is very true, might be (c) (y is more £!. less small) is true. In these assertions, small is assumed to be a specified fuzzy sub­set of the real line R ~ (_00,00); approximately equal is a binary fuzzy relation in R x R; and true and very true are fuzzy subsets of the unit interval [0,1].2 Because of limitations on space, we shall not discuss the applications of fuzzy logic to decision analysis in the present paper.

2A brief exposition of the basic properties of fuzzy sets is con­tained in the Appendix. A more detailed discussion of various aspects of the theory of fuzzy sets and its applications may be found in [22]. The most comprehensive treatise on the theory of fuzzy sets is the five-volume work of A. Kaufmann [23]. Some of the applications of the theory of fuzzy sets to decision analysis are discussed in [24-32].

342 L.A. ZADEH

Insofar as decision analysis is concerned, the linguistic approach serves, in the main, to provide a language for an approxi­mate characterization of those components of a decision process which are either inherently fuzzy or are incapable of precise mea­surement. For example, if the probability of an outcome of a deci­sion is not known precisely, it may be described in linguistic terms as likely or not very likely or very unlikely or more ~ less likely, and so forth. Or, if the degree to which an alternative a is preferred to an alternative B is not well-defined, it may be assigned a linguistic value such as strong or very strong or mild or very weak, etc. Similarly, a fuzzy relation between two varia­bles x and y may be described in linguistic terms as "x is much lar­ger than y" or "If x is small then y is large else x is approximate­.!l. equal to y," etc.

As will be seen in Section 2, a linguistic characterization such as "x is small" may be viewed as a fuzzy restriction on the values of x. What is important to realize is that the assertion "x is small" conveys no information concerning the probability dis­tribution of x; what it means, merely, is that "x is small" induces an elastic constraint on the values that may be assigned to x. Thus, if small is a fuzzy set in R whose membership function takes the value, say, 0.6 at x = 8, then the degree to which the constraint "x is small" is satisfied when the value 8 is assigned to x, is 0.6.

In what follows, we shall outline the main features of the lin­guistic approach and indicate some of its possible applications to decision analysis. It should be stressed that such applications are still in an exploratory stage and experience in the use of the linguistic approach may well suggest substantive changes in its implementation. 3

2. LINGUISTIC VARIABLES AND FUZZY RESTRICTIONS

As stated in the Introduction, a linguistic variable is a variable whose values are words or sentences which serve as names of fuzzy subsets of a universe of discourse. In more specific terms, a linguistic variable is characterized by a quintuple (X,T(X),U,G,M) in which X is the name of the variable, e.g., Age; T(X) is the term-set of X, that is, the collection of its linguistic

3The linguistic approach has been applied to various problems in situation calculus by Yu. K1ikov, G. pospe1ov, D. Pospe10v, V. Pushkin, D. Shapiro and others at the Computing Center of the Academy of Sciences, Moscow, under the direction of N.N. Moyseev. Other types of applications of the linguistic approach have recent­ly been reported by P. King and E. Mamdani [33], R. Assi1ian [34], G. Retherford and G. B100re [35], F. Wenstop [36], L. Pun [37], V. Dimitrov, W. Wech1er and P. Barnev [38], and others.

THE LINGUISTIC APPROACH TO DECISION ANALYSIS 343

values, e.g., T(X) = {young, not young, very young, not very young, ••• }; U is a universe of discourse, e.g., in the case of Age, the set {0,1,2,3, .•. }; G is a syntactic rule which generates the terms in T(X); and M is a semantic rule which associates with each term, x, in T(X) its meaning, M(x), where M(x) denotes a fuzzy subset of U. Thus, the meaning, M(x) , of a linguistic value, x, is defined by a compatibility -- or, equivalently, membership -- function ~x: U + [0,1] which associates with each u in U its compatibility with x. For example, the meaning of young might be defined in a particular context by the compatibility function

1 for 0 < U < 20 (2.1)

1 for u > 20

which may be viewed as the membership function of the fuzzy subset young of the universe of discourse U = [0,00). Thus, the compati­bility of the age 27 with young is approximately 0.66, while that of 30 is 0.5. The variable u E U is termed the base variable of X. The value of u at which ~x(u) = 0.5 is the cross~r ~oint of x.

If X were a numerical variable, the assignment of a value, say a, to X would be expressed as

X = a (2.2)

In the case of linguistic variables, the counterpart of the assign­ment equation (2.2) is the proposition "X is x," where x is a lin­guistic value of X. From this point of view, x may be regarded as a fuzzy restriction on the values of the base variable u. This fuzzy restriction, which is denoted by Rx(u) (or simply R(u», is identical with the fuzzy subset M(x) which is the meaning of x. Thus, the propositi9n "X is x" translates into the relational assignment equation4

R(u) = x (2.3)

which signifies that the proposition "X is x" may be interpreted as an elastic constraint on the values that may be assigned to u, with the membership function of x characterizing the compatibility, ~ (u), of u with x.

x

As an illustration, consider the proposition "Edward is young." The translation of this proposition reads

4As will be seen later, a relational assignment equation involves, more generally, the assignment of a fuzzy relation to a fuzzy restriction on the values of a base variable [39].

344 L.A. ZADEH

R(Age(Edward» = young (2.4)

where Age(Edward) is a numerical variable ranging over [0,00), R(Age(Edward» is a fuzzy restriction on its values, and young is a fuzzy subset of [0,00) whose membership function is given by (2.1). To simplify the notation, a relational assignment equation such as (2.4) may be written as

Age (Edward) (2.5)

with the understanding that young is assigned not to the variable Age(Edward) but to the restriction on its values.

In this sense, each term, x, in the term-set of a linguistic variable X corresponds to a fuzzy restriction, R(u), on the values that may be assigned to the base variable u. A key idea behind the concept of a linguistic variable is that these fuzzy restrictions may be deduced from the fuzzy restrictions associated with the so­called primary terms in T(X). In effect, these fuzzy restrictions play the role of units which, upon calibration, make it possible to compute the meaning of the composite (that is non-primary) values of X from the knowledge of the meaning of the primary terms.

As an illustration of this technique, we shall consider an example in which U = [0,00) and the term-set of X is of the form

T(X) = {small, not small, very small, very (not small), (2.6)

not very small, very very small, ••• }

in which small is the primary term.

The terms in T(X) may be generated by a context-free grammar [40] G = (VT,VN'S,P) in which the set of terminals, VT' comprises (, ), the primary term small and the linguistic modifiers very and not; the nonterminals are denoted by S, A and B and the production system is given by:

S + A S + not A A+B B + very B B + (S) B + small

Thus, a typical derivation yields

S + not A + not B => not very B => not very very B

=> not very very small

(2.7)

(2.8)

THE LINGUISTIC APPROACH TO DECISION ANALYSIS 345

In this sense, the syntactic rule associated with X may be viewed as the process of generating the elements of T(X) by a succession of substitutions involving the productions in G.

As for the semantic rule, we shall assume for simplicity that if ~A is the membership function of A then th5 membership functions of not A and very A are given respectively by

~not A = 1 - ~A (2.9)

and

(2.10)

Thus, (2.10) signifies that the modifier very has the effect of squaring the membership function of its operand.

Suppose that the meaning of small is defined by the compati­bility (membership) function

2 -1 ~sma11 (u) = (1 + (O.lu)) ,

Then the meaning of very small is given by

2 -2 ~very small = (1 + (O.lu) )

u > 0

while the meanings of not very small and very (not small) are expressed respectively by ---- ----- ----

1 -2 -2

~not very small (1+ (O.lu) )

and 2 -1 2

~very (not small) = (1 - (l+(O.lu)) )

(2.11)

(2.12)

(2.13)

(2.14)

In this way, we can readily compute the expression for the member­ship function of any term in T(X) from the knowledge of the member­ship function of the primary term small.

In effect, a linguistic variable X may be viewed as a micro­language whose syntax and semantics are represented, respectively, by the syntactic and semantic rules associated with X. The sen­tences of this language are the linguistic values of X, with the meaning of each sentence represented as a fuzzy restriction on the values that may be assigned to the base variable, u E U, of X.

SA more detailed discussion of the effect of linguistic modifiers (hedges) may be found in [41], [42], and [43].

346 L.A. ZADEH

In the characterization of a decision process, we usually have to deal with a collection of interrelated linguistic variables. In this connection, it is helpful to have a set of rules for transla­ting a proposition involving two or more linguistic variables into a set of relational assignment equations. The rules in question are as follows. 6

Let X and Y be linguistic variables associated with possibly distinct universes of discourse U and V, and let P and Q be fuzzy subsets of U and V, respectively. Then, the conjunctive proposi­tion p defined by

p ~ X is P and Y is Q

translates into the relational assignment equation

R(u,v)=PXQ p

(2.15)

(2.16)

where R(u,v) is the restriction on the values that may be assigned to the ordered pair (u,v), u E U, V E V, and P x Q denotes the car­tesian product of P and Q. Equivalently, (2.16) may be expressed as

R(u,v) = :P () Q (2.17)

where P and 9 are the cylindrical extensions of P and Q, respec­tively, and P n Q is their intersection. (See Appendix.)

Similarly, the disjunctive proposition

translates into

p ~ X is P or Y is Q

R (u,v) = P U Q p

(2.18)

(2.19)

where P U Q is the union of the cylindrical extensions of P and Q.

The conditional proposition

translates into

p ~ If X is P then Y is Q

R (u,v) p

(2.20)

(2.21)

6Such rules will be referred to as semantic rules of Type II when it is necessary to distinguish them from the semantic rules which apply to individual variables (i.e., semantic rules of Type I).

THE LINGUISTIC APPROACH TO DECISION ANALYSIS 347

where pI is the complement of P and ffi denotes the bounded sum. (See A36). lfore generally, the conditional proposition

p ~ If X is P then Y is Q else Y is R

translates into

R (u,v) = (pI ffi Q) n (P ffi R) p

(2.22)

(2.23)

Eq. (2.23) follows from (2.21) by the application of (2.15) and the fact that

p ~ X is not P

translates into

R (u) P

where pI is the complement of P.

pI

(2.24)

(2.25)

In cases where a linguistic truth-value, T, such as true, very true, more or less true, etc. is associated with a proposition, as in

p ~ (X is small) is very true (2.26)

the following rule of truth-functional modification may be used to translate p into a relational assignment equation:

translates into

p ~ (X is A) is T

-1 Rp(U) = ]JA H

(2.27)

(2.28)

where ]JA1 is the inverse of ]JA and * denotes the composition of the binary relation ]JA1 with the unary fuzzy relation T. (See A60.) It can readily be verified that the membership function of ~(u) is given by

]JR (u) = ]JT(]JA(u)) , p

u € U (2.29)

where ]JT is the membership function of the linguistic truth-value T and ]JA is that of A.

The basic translation rules stated above may be employed, in combination, to translate more complex propositions involving

348 l.A. ZADEH

relations between two or more variables. As an illustration, con­sider the following proposition:

~ ~ X is large and Y is small or (2.30)

X is not large and Y is very small

which may be regarded as a linguistic characterization of the table shown below:

X Y (2.31)

large small

not large very small

For simplicity we shall assume that U = V = {0,1,2,4} and that small and large are fuzzy sets defined by (see Appendix)

small = 1/1 + 0.6/2 + 0.2/3 ,

large = 0.3/2 + 0.7/3 + 1/4

(2.32)

(2.33)

In this case, the application of (2.15) and (2.18) leads to the following expression for the restriction on (u,v) which is induced by the proposition in question:

R (u,v) = large x small + not large x very small p -- -- --- -----

where x and + represent the cartesian product and the union, respectively. Now, from (2.9) and (2.10) it follows that

and hence

R (u,v) P

not large = 1/1 + 0.7/2 + 0.3/3

very small = 1/1 + 0.36/2 + 0.04/3

0.3/(2,1) + 0.7/(3,1) + 1/(4,1)

+ 0.3/(2,2) + 0.6/(3,2) + 0.6/(4,2)

+ 0.2/(2,3) + 0.2/(3,3) + 0.2/(1,3)

(2.34)

(2.35)

(2.36)

(2.37)

in which a term such as 0.6/(3,2) signifies that the compatibility of the assignments u = 3 and v = 2 with p is 0.6.

As a further illustration, consider the proposition

q ~ If (X is large and Y is small or X is not large

and Y is very small) then Z is very small

(2.38)

THE LINGUISTIC APPROACH TO DECISION ANALYSIS 349

in which the proposition in parentheses is that of the preceding example and the universe of discourse associated with Z is assumed to be the same as U.

In this case, using (2.20), we have

R (u,v,w) = R'(u,v) EB very small q p

(2.39)

where

R (u,v) p

0.3/(2,1,1) + (2,1,2) + (2,1,3) + (2,1,4)) + ... (2.40)

+ 0.2/(4,3,1) + (4,3,2) + (4,3,3) + (4,3,4))

very small = 11(1,1,1) + (1,1,2) + (1,1,3) + (1,1,4) + + (1,2,1) + (1,2,2) + (1,2,3) + (1,2,4) + + (1,4,1) + (1,4,2) + (1,4,3) + (1,4,4)) +

+ 0.04/((3,1,1) + (3,1,2) + (3,1,3) + (3,1,4) + + (3,4,1) + (3,4,2) + (3,4,3) + (3,4,4)) ;

(2.41)

R'(u,v) is the complement of R (u,v) and EB is defined by (A36). p p

To illustrate the rule of truth-functional modification, con­sider the proposition

p ~ (X is small) is very true

where small is defined by (2.32) and

true ~ 0.2/0.6 + 0.5/0.8 + 0.8/0.9 + 1/1

In this case,

very true = 0.04/0.6 + 0.025/0.8 + 0.64/0.9 + 1/1

and (2.29) yields

jl (1) 1 p

jl (2) 0.2 p

jl (3) jl (4) = 0 p p

which means that the compatibility of the assignment u is 0.2, while those of u = 3 and u = 4 are zero.

(2.42)

(2.43)

(2.44)

(2.45)

2 with p

The above examples serve to illustrate one of the central features of the linguistic approach, namely, the mechanism for

350 L.A. ZADEH

translating a proposition expressed in linguistic terms into a fuzzy restriction on the values which may be assigned to a set of base variables. Once the translation has been performed, the resulting fuzzy restrictions may be manipulated to yield the restrictions on whichever variables may be of interest. These restrictions, then, are translated into linguistic terms, yielding the final solution to the problem at hand.

In what follows, we shall illustrate this process by a few simple applications which are of relevance to decision analysis.

3. LINGUISTIC CHARACTERIZATION OF OBJECTIVE FUNCTIONS

In the literature of mathematical programming and decision analysis, it has become a universal practice to assume that the objective and utility functions are numerical functions of their arguments.

In most real-world problems, however, our perceptions of the consequences of a decision are not sufficiently precise or consis­tent to justify the assignment of numerical values to utilities or preferences. Thus, in most cases it would be more realistic to assume that the objective function is a linguistic function of the linguistic values of its arguments, and employ the techniques of the linguistic approach to assess the consequences of a particular choice of decision variables.

To be more specific, consider a simple case of a decision pro­cess in which the objective function G(ul""'un) takes values in a space V while the decision variables ul, ... ,un take values in Ul,""Un , respectively. To simplify the discussion, we shall assume that Ul = U2 = ... = Un = U.

The linguistic values of decision variables as well as those of the objective function are assumed to be of the form {low, not low, very low, not very low, ... , medium, high, not high, very high, not very high, not low and not high, not very low and not very high, ~r:-It can readily be verified that these linguistic values can be generated by a context-free grammar whose production system is given below:

S -+ A C -+ very C (3.1) S -+ S and A D -+ very D A -+ B C -+ low A -+ not B D -+ high B -+ C B -+ D B -+ medium

THE LINGUISTIC APPROACH TO DECISION ANALYSIS 351

in which S, A, B, C, D are non-terminals, S is the starting symbol, and and, not, very, low, medium and high are terminals, with low, medium and high playing the role of primary terms.

The simplicity of this grammar makes it possible to compute the meaning of various linguistic values by inspection. For exam­ple, the meaning of the value not very low and not high is given by

(3.2)

h 1 2. f were ow 1S a uzzy of that of low, and ' respectivelY:-

set whose membership function is the square and n denote the complement and intersection,

It is important to note that the assumption that all of the decision variables and the objective function have the same term­set does not imply that the corresponding primary terms are also identical. Thus, low, for example, in the case of i-th decision variable need not have the same meaning as low for j-th decision variable (j f. i) or G. To illustrate this point, suppose that U = {1,2,3,4}. Then low for u1 might be defined as

low = 1/1 + 0.8/2 + 0.2/3 (3.3)

whereas low for u2 may be

low = 1/1 + 0.6/2 + 0.1/3 . (3.4)

Typically, a tabulation of the linguistic values of G as a function of the linguistic values of the decision variables would have a form such as shown below (low2 ~ very low, ~ed ~ ~edium)

u1 u2 G (3.5)

low low low 2

low med low

low -2 low not low ----

It should be noted that, in general, not all of the possible com­binations of the linguistic values of decision variables will appear in the tableau of G.

The definition of G by a tableau of the form (3.5) induces a fuzzy restriction on the values that may be assigned to the deci­sion variables and G. More specifically, let £ij denote the

352

linguistic value of tableau, and let vi objective function. expressed by

L.A. ZADEH

j-th decision variable in the i-th row of the be the corresponding linguistic value of the

Then the fuzzy restriction in question is

9., x··.x9., Xv + 11 In 1 (3.6)

+9., x"·x9., Xv ml mn m

where m is the number of rows in the tableau, and X and + denote the cartesian product and union, respectively. The fuzzy restric­tion RG(ul""'un,v) on the values of ul, ... ,un and v may be viewed as the meaning of the tableau of G in the same sense as the trans­lation rules (2.15-2.29) express the meaning of various proposi­tions as fuzzy restrictions on the values of the base variables. 7

As a very simple illustration of (3.6) , assume that the tableau of G is given by

ul u2 G (3.7)

low low low 2

low low' low'

where low for ul and u2 is defined by (3.3) and (3.4), respective­ly, and low for G has the same meaning as for u2'

In this case, we have

RG(ul ,u2 ,v) = (1/1+0.8/2+0.2/3) X (1,1+0.6/2+0.1/3) (3.8)

x (1/1+0.36/2+0.01/3)

+ (1/1+0.8/2+0.2/3) x (0.4/2+0.9/3+1/4)

x (0.4/2+0.9/3+1/4)

1/(1,1,1) +0.6/(1,2,1) +0.36(1,1,2) + ...

+ 0.9/(1,3,3) + ... +1/(1,4,4) .

Note that the restriction defined by (3.8) is a ternary fuzzy relation in Ul x U2 x V.

An important aspect of the linguistic definition of G is that it provides a basis for an interpolation of G for values of the decision variables which are not in the tableau. Thus, since the meaning of the tableau is provided by the fuzzy (n+l)-ary relation RG(ul""'un,v), we can assert that the result of substitution of

7A more detailed discussion of this point may be found in [44].

THE LINGUISTIC APPROACH TO DECISION ANALYSIS 353

arbitrary linguistic_values ~il, ... ,Iin for ul, ... ,un is the com­position of Rc with £il""'£in' This implies that the value of G corresponding to the prescribed values of ul, ... ,un is given by

G(I·l,···,I. ) = RG(ul,···,u ,v)*£'l*"'*£' ~ ~n n ~ ~n

(3.9)

where * denotes the operation of composition. 8

As a very simple illustration of (3.9), assume that RG(ul,u2'v) is expressed by (for simplicity, Ul = U2 = V {1,2})

RG(ul ,u2 ,v) = 0.8/(1,1,1) +0.9/(1,2,1) +0.3/(2,1,1) (3.10)

+ 0.7/(2,2,1) +0.3/(1,1,2) +0.2/(1,2,2)

and that

+ 0.6/(2,1,2) +0.5/(2,2,2)

0.3/1 + 0.5/2

0.9/1 + 0.2/2

(3.11)

(3.12)

The ternary fuzzy relation (3.10) may be represented as two matrices

A ~ [0.8 0.9]

0.3 0.7

B ~ [ 0.3

0.6

0.2]

0.5 (3.13)

Forming the max-min products of A and B with the row matrix [0.3 0.5] we obtain the matrix

[0.3

C ~ 0.5

0.5]

0.5 (3.14)

and forming the max-min product of this matrix with the row matrix [0.9 0.2] we arrive at

D = [0.3 0.5] (3.15)

which implies that the interpolated value of G is

(3.16)

8In the terminology of relational models of data, (3.9) may be viewed as an extension to fuzzy relations of the operation of disjunctive mapping [45].

354 L.A. ZADEH

To express this result in linguistic terms, it is necessary to approximate to the right-hand member of (3.16) by a linguistic value which belongs to the term-set of G. The issue of linguistic approximation is discussed in greater detail in [20] and [36].

In summary, if the objective function is defined in linguis­tic terms by a tableau of the form (3.5), the fuzzy restriction on the values of the decision variables which is induced by the defi­nition is an (n+l)-ary fuzzy relation in Ul x ••• x Un x V which is expressed by (3.6). By the use of this relation, the objective function may be interpolated for values of the decision variables which are not in the original tableau.

4. OPTIHIZATION UNDER HULTIPLE CRI.TERIA

The linguistic approach appears to be particularly well­adapted to the analysis of decision processes in which the objec­tive function is vector- rather than scalar-valued. 9 The reason for this is that when more than one criterion of performance is involved, the trade-offs between the criteria are usually poorly defined. In such cases, then, linguistic characterizations of trade-offs or preference relations provide a more realistic con­ceptual framework for decision analysis than the conventional methods employing binary-valued preference relations.

A detailed exposition of the application of the linguistic approach to the optimization under multiple criteria will be pre­sented in a separate paper. In what follows, we shall merely sketch very briefly the main ideas behind the method.

To simplify the notation, we shall assume that there are only two decision variables and two real-valued objective functions Gl and G2' The values of Gl and G2 at the points (ui,u~) and (ur,u~) are denoted by GI and G~, respectively.

In the conventional formulation of the problem, the partial ordering defined by

1 1 2 2 G~ > G2 Gl 2 (Gl ,G2) > (Gl ,G2) ¢} and ~ G2 1 2

(4.1)

induces a pre-ordering in Ul x U2 defined by

1 1 2 2 1 1 2 2 (u l ,u2) > (u l ,u2) ¢} (Gl ,G2) > (Gl ,G2) (4.2)

9The literature on the optimization under multiple criteria is quite extensive. Of particular relevance to the discussion in this section are the references [46]-[48].

THE LINGUISTIC APPROACH TO DECISION ANALYSIS

With each point (u~, u~) in Ul x U2' we can associate the set of points D(u~,u~) which dominate it; that is,

(4.3)

355

If C is a constraint set in Ul xU2 , then a point (u~,u~) in C is undominated if and only if the intersection of C with D(u~,u~) is the singleton {(u~,u~)}. The set of all undominated points in C is the set of Pareto-optimal solutions to the optimization under the objective functions Gl and G2 .

Generally, additional assumptions are made to induce a linear ordering in the set of undominated points in C or, at least, to disqualify some of the points in this set from contention as solu­tions to the optimization problem. The main shortcoming of these techniques is that the assumptions needed to induce a linear order­ing tend to be rather arbitrary and hard to justify.

In the linguistic approach to this problem, the Pareto-optimal set is fuzzified and its size is "reduced" by making use of what­ever information might be available regarding the trade-offs between Gl and G2. Since the trade-offs are usually poorly defined, they are allowed to be expressed in linguistic terms. Generally, the trade-offs are assumed to be defined indirectly via fuzzy preference relations [49], although in some cases it may be possible to define an overall objective function directly as a lin­guistic function of the linguistic values of Gl and G2 .

As a simple illustration, a linguistic characterization of a fuzzy preference relation might have the following form.

Assume that the strength of preference is a linguistic varia­ble whose values are strong, very strong, not strong, not very strong, weak, not very strong and not very weak, ... in which the primary terms strong-and weak are fuzzy subsets of the unit inter­val. The meaning of such linguistic values may be computed in exactly the same way as the meaning of the linguistic values of X in Example 2.6.

2 ~et p denote the degree to which (ui,u~) is preferred to (ul,u2). Then, a partial linguistic characterization of p may be expressed as:

356

1 2 If (Gl is much larger than Gl and

G~ is approximately equal to G~ or

G~ is much larger than G~ and

Gl . . 1 1 G2) 1 ~s approx~mate y equa to 1

then p is strong

L.A. ZADEH

(4.4)

In this expression, the terms much larger and approximately equal play the role of linguistic values of a fuzzy binary rela­tion in V x V, while strong is a linguistic value of p. By the use of appropriate semantic rules, the expression in question can be translated into a fuzzy restriction on 5-tuples of the form (ul,u~,ui,u~,v). Combined with similar fuzzy restrictions result­ing from whatever other linguistic characterizations of p might be available, (4.4) yields a fuzzy preference relation p in Ul x U2 x Ul x U2 x V which provides a basis for fuzzifying the Pareto­optimal set and thereby reducing the degree to which some of the points in this set may be regarded as contenders for inclusion in the set of optimal solutions.

More specifically, let uO ~ (u~,u~) be a point in Ul x U2 • Furthermore, let D(uO) be the fuzzO set of points in Ul x U2 which results from setting ul edual to u in the fuzzy preference rela­tion p. As in (4.3), D(u ) is the fuzzy set of points which domi­nate uO•

° It will be recalled that when D(u ) is a non-fuzzy set, the point uO is undominated and hence an elemeat of the Pareto-optimal set if and only if the in8ersection of D(u ) with the constraint set e is the singleton {u}. More generally, if D(uO) is a fuzzy set then the degree to which uO belongs to the fuzzy Pareto­optimal set, P, may be related to the height 10 of the fuzzy set D(uO) fie - {uO} by the relation

(4.5)

In this sense, then, the Pareto-optimal set is fuzzified, with each point uO assigned a grade of membership in the fuzzy Pareto­optimal set by (4.5).

The fuzzification of the Pareto-optimal set has the effect of reducing the degree of contention for optimality of those points

lOThe height of a fuzzy set is the supremum of its membership func­tion over the universe of discourse.

THE LINGUISTIC APPROACH TO DECISION ANALYSIS 357

which have a low grade of membership in the set. In general, the extent to which the size of the Pareto-optimal set is reduced in this fashion depends on the linguistic information provided by the trade-offs. Thus, if the fuzzy restrictions which are associated with the translations of the linguistic statements about the trade­offs are only mildly restrictive -- which is equivalent to saying that they convey little information about the trade-offs -- then the reduction in the size of the Pareto-optimal set will, in general, be slight. By the same token, the opposite will be the case if the restrictions in question are highly informative --that is, have the effect of assigning low grades of membership to most of the points in Ul x U2 x V.

In sketching the application of the linguistic approach to optimization under multiple criteria, we have side-stepped several non-trivial problems. In the first place, the preference relation p which results from translation of linguistic propositions of the form (4.4) is a fuzzy set of Type 2 (i.e., has a fuzzy-set-valued membership function), which makes it more difficult to find the intersection of D(uO) with the constraint set as well as to com­pute the grade of membership of uO in the fuzzy set of Pareto­optimal solutions. Secondly, the preference relation represented by p may not be transitive (in the sense defined in [49]), in which case it may be necessary to construct the transitive closure of p. And finally, it may not be a simple matter to apply linguistic approximation to ~p(uO). Notwithstanding these difficulties, the linguistic approach sketched above or some variants of it may even­tually provide a realistic way of dealing with practical problems involving decision-making under mUltiple criteria.

5. CONCLUDING REMARKS

In the foregoing discussion, we have attempted to outline some of the main ideas behind the linguistic approach and point to its possible applications in decision analysis. The specific problems discussed in Sections 3 and 4 are representative -- but not exhaus­tive -- of such applications. In particular, we have not touched upon the important subject of the manipulation of linguistic proba­bilities in problems of stochastic control nor upon the problem of multistage decision processes and inference from fuzzy data.

At this juncture, the linguistic approach to decision analysis is in its initial stages of development. Eventually, it may become a useful aid in decision-making relating to real-world problems.

358 L.A. ZADEH

REFERENCES

1. L.J. Savage, The Foundations of Statistics, Wiley, New York, 1954.

2. R.D. Luce, Individual Choice Behavior, Wiley, New York, 1959.

3. G. Debreu, Theory of Value, Wiley, New York, 1959.

4. K.J. Arrow, Social Choice and Individual Values, Wiley, New York, 1963.

5. R.D. Luce and P. Suppes (eds.), Handbook of Mathematical Psychology, Wiley, New York, 1965. --

6. R. Radner and J. Marschak (eds.), Economic Theory of Teams, Yale University Press, New Haven, 1972.

7. J. Marschak, "Decision Making: Economic Aspects," in Encyclo­pedia of Social Sciences, McMillan, New York, 1968. ----

8. M. Tribus, Rational Descriptions, Decisions and Designs, Pergamon Press, New York, 1969.

9. G. Menges (ed.), Information, Inference and Decision, Reidel, Dordrecht, 1974.

10. W. Edwards and A. Twersky (eds.), Decision Making: Selected Readings, Penguin Books, London, 1967.

11. D.V. Lindley, Making Decisions, Wiley, London, 1971.

12. H. Raiffa, Decision Analysis, Addison-Wesley, Reading, 1970.

13. R.L. Winkler, An Introduction to Bayesian Inference and Decision, Holt-,-Rinehart and Winston, New York, 1972.-

14. H. Simon, "Theories of Bounded Rationality," in Decision and Organization, C.B. McGuire and R. Radner (eds.), North­Holland, 1972.

15. N. Rescher (eds.), The Logic of Decision and Action, Pittsburgh, 1967.

16. P.C. Fishburn, Mathematics of Decision Theory, Mouton, The Hague, 1974.

17. H.W. Gottinger, Bayesian Analysis, Probability and Decision, Vandenhoeck and Ruprecht, Gottingen, 1975.

THE LINGUISTIC APPROACH TO DECISION ANALYSIS 359

18. P.C. Fishburn, "A Theory of Subjective Expected Utility with Vague Preferences," Theory and Decision 6, pp. 287-310, 1975.

19. L.A. Zadeh, "Outline of a New Approach to the Analysis of Complex Systems and Decision Processes," IEEE Trans. on Systems, Man and Cybernetics SMC-3, pp. 28-44, 1973.

20. L.A. Zadeh, "The Concept of a Linguistic Variable and its Application to Approximate Reasoning," Inf. Sciences, Part I, 8, pp. 199-249; Part II, 8, pp. 301-357; Part III, 9, pp. 43-80, 1975. - -

21. L.A. Zadeh, "Fuzzy Logic and Approximate Reasoning (In memory of Grigore Moisi1)," Synthese 30, pp. 407-428, 1975.

22. L.A. Zadeh, K.S. Fu, K. Tanaka, M. Shimura (eds.), Fuzzy Sets and Their Applications to Cognitive and Decision Processes-,-­Academic Press, New Yor~ 1975.

23. A. Kaufmann, Introduction to the Theory of Fuzzy Subsets, vol. 1, Elements of Basic Theory, 1973; vo1~App1ications to Linguistics, Logic and Semantics, 1975; vol. 3, Applica­tions to Classification and Pattern Recognition, Automata and Systems, and Choice of Criteria, 1975, Masson and Co., Paris.

24. R.E. Bellman and L.A. Zadeh, "Decision-Making in a Fuzzy Environment," Management Sciences 17, B-141-B-164, 1970.

25. K. Tanaka, "Fuzzy Automata Theory and its Applications to Control Systems," Osaka University, Osaka, 1970.

26. R.M. Capoce11i and A. De Luca, "Fuzzy Sets and Decision Theory, Information and Control 23, pp. 446-473, 1973.

27. L. Fung and K.S. Fu, "An Axiomatic Approach to Rational Decision Making Based on Fuzzy Sets," Purdue University, Lafayette, Indiana, 1973.

28. H.J. Zimmermann, "Optimization in Fuzzy Environments," Institute for Operations Research, Tech. Univ. of Aachen, Aachen, 1974.

29. T. Terano and M. Sugeno, "Microscopic Optimization by Using Conditional Fuzzy Measures," Tokyo Institute of Technology, Tokyo, 1974.

30. N. Tanaka, T. Okuda and K. Asai, "A Forma1ation of Decision Problems with Fuzzy Events and Fuzzy Information," Dept. of Industrial Eng., University of Osaka Prefecture, Osaka, 1975.

360 L.A. ZADEH

31. D.N. Jacobson, "On Fuzzy Goals and Maximizing Decisions in Stochastic Optimal Control," Nat. Res. Inst. for Math. Sci., Pretoria, 1975.

32. R.G. Woodhead, "On the Theory of Fuzzy Sets to Resolve 111-Structured Marine Decision Problems," Dept. of Naval Archi­tecture, University of Newcastle, 1972.

33. P.J. King and E.H. Mamdani, "The Application of Fuzzy Control Systems to Industrial Processes," Warren Spring Lab., Stevenage, and Dept. of Elec. Eng., Queen Mary College, London, 1975.

34. E.H. Mamdani and S. Assilian, "An Experiment in Linguistic Synthesis with a Fuzzy Logic Controller," Int. Jour. of Man-Mach. Studies 7, pp. 1-13, 1975.

35. G. Retherford and G.C. Bloore," The Implementation of Fuzzy Algorithms for Control," Control Systems Center, University of Manchester, Manchester, 1975.

36. F. Wenstop, "Application of Linguistic Variables in the Analysis of Organizations," School of Bus. Ad., University of California, Berkeley, 1975.

37. L. Pun, "Experience in the Use of Fuzzy Formalism in Problems with Various Degrees of Subjectivity," University of Bordeaux, Bordeaux, 1975.

38. V. Dimitrov, W. Wechler and P. Barnev, "Optimal Fuzzy Control of Humanistic Systems," Inst. of Math. and Mechanics, Sofia, and Dept. of Math., Tech. Univ. Dresden, Dresden, 1974.

39. L.A. Zadeh, "Calculus of Fuzzy Restrictions," in Fuzzy Sets and Their Applications to Cognitive and Decision Processes, L.A. Zadeh, K.S. Fu, K.-ranaka and M~himura (eds.), Academic Press, New York, 1975.

40. A.V. Aho and J.D. Ullman, The Theory of Parsing, Translation and Compiling, Prentice-Hall, Englewood Cliffs, 1973.

41. L.A. Zadeh, "A Fuzzy-Set Theoretic Interpretation of Linguis­tic Hedges," Jour. of Cyber. ~, pp. 4-34, 1972.

42. G. Lakoff, "Hedges: A Study in Meaning Criteria and the Logic of Fuzzy Concepts," Jour. of Phil. Logic ~, pp. 458-508, 1973.

THE LINGUISTIC APPROACH TO DECISION ANALYSIS 361

43. H.l.f. Hersh, "A Fuzzy Set Approach to Modifiers and Vagueness in Natural Language," Dept. of Psychology, The Johns Hopkins University, Baltimore, 1975.

44. L.A. Zadeh, "A Fuzzy-Algorithmic Approach to the Definition of Complex or Imprecise Concepts," Electronics Research Lab., University of California, Berkeley, 1974. (To appear in the Int. Jour. of Man-Machine Studies.)

45. R.F. Boyce, D.D. Chamberlin, M.M. Hammer and W.F. King III, "Specifying Queries as Relational Expressions," SIGPLAN Notices, ACM, No.1, 1975.

46. J.L. Cochrane and M. Zeleny (eds.), Multiple Criteria Deci­sion-Making, University of South Carolina Press, Columbia, 1973.

47. M. Zeleny, Linear Mu1tiobjective Programming, Springer-Verlag, New York, 1974.

48. A.M. Geoffrion, "Proper Efficiency and the Theory of Vector Maximization," Jour. of Hath. Anal. and Appl. 22, pp. 618-630, 1968.

49. L.A. Zadeh, "Similarity Relations and Fuzzy Orderings," Inf. Sciences 3, pp. 177-200, 1971.

362 L.A. ZADEH

APPENDIX

FUZZY SETS - NOTATION, TERMINOLOGY AtID BASIC PROPERTIES

The symbols U,V,W, ... , with or without subscripts, are gener­ally used to denote specific universes of discourse, which may be arbitrary collections of objects, concepts or mathematical con­structs. For example, U may denote the set of all real numbers; the set of all residents in a city; the set of all sentences in a book; the set of all colors that can be perceived by the human eye, etc.

Conventionally, if A is a fuzzy subset of U whose elements are u1 , ... ,un ' then A is expressed as

(A1)

For our purposes, however, it is more convenient to express A as

or

with the understanding

and

A u + ... +u 1 n

A

that, for

u. + u. 1 J

n

I i=l

all

U. 1

u. 1

i,

(A2)

(A3)

j,

(A4)

(AS)

As an extension of this notation, a finite fuzzy subset of U is expressed as

F lJu +"'+lJu 1 1 n n (A6)

or, equivalently, as

(An

where the lJi, i = 1, ... ,n, represent the grades of membership of the ui in F. Unless stated to the contrary, the lJi are assumed to lie in the interval [0,1], with ° and 1 denoting no membership and full membership, respectively.

Consistent with the representation of a finite fuzzy set as a linear form in the ui' an arbitrary fuzzy subset of U may be expressed in the form of an integral

THE LINGUISTIC APPROACH TO DECISION ANAL VSIS 363

(A8)

in which ~F: U + [0,1] is the membership or, e~uivalently, the compatibility function of F; and the integral JU denotes the union (defined by (A28» of fuzzy singletons ~F(u)/u over the universe of discourse U.

The points in U at which ~F(u) > 0 constitute the support of F. The points at which ~F(u) = 0.5 are the crossover points of F.

Example A9. Assume

U a+b+c+d. (AlO)

Then, we may have

A a + b + d (All) and

F 0.3a + 0.9b + d (A12)

as nonfuzzy and fuzzy subsets of U, respectively.

If U = 0 + 0.1 + 0.2 + + 1 (Al3)

then a fuzzy subset of U would be expressed as, say,

F = 0.3/0.5 + 0.6/0.7 + 0.8/0.9 + 1/1 (A14)

If U [0,1], then F might be expressed as

F = flo ~ / u l+u

(Als)

which means that F is a fuzzy subset of the unit interval [0,1] whose membership function is defined by

1 ~ (u) =--

F 1+u2 (A16)

In many cases, it is convenient to express the membership function of a fuzzy subset of the real line in terms of a standard function whose parameters may be adjusted to fit a specified mem­bership function in an approximate fashion. Two such functions are defined below.

364 L.A. ZADEH

S (u;a, 13, y) 0 for u < a (Al7)

2 (u-ar y-a for a < u 2.13

1 - 2(~r y-a for S2. u 2.y

= 1 for u2:Y

'IT(u;S,y) S (u;y-S, y-~, y) for u ~y (A18)

13 1 - S(u;y,y+z,y+S) for u2:Y

In S(u;a,S,y), the parameter 13, 13 = a+y , point. In 'IT(u;S,y), 13 is the bandwidth, t~at between the crossover points of 'IT, while y is

is the crossover

is unity.

is the separation the point at which 'IT

In some cases, the assumption that ~F is a mapping from U to [0,1] may be too restrictive, and it may be desirable to allow ~F to take values in a lattice or, more particularly, in a Boolean algebra. For most purposes, however, it is sufficient to deal with the first two of the following hierarchy of fuzzy sets.

Definition A19. A fuzzy subset, F, of U is of ~ 1 if its membership function, ~F' is a mapping from U to [0,1]; and F is of type n, n = 2,3, .•• , if ~F is a mapping from U to the set of fuzzy subsets of type n-1. For simplicity, it will always be understood that F is of type 1 if it is not specified to be of a higher type.

Example A20. Suppose that U is the set of all nonnegative integers and F is a fuzzy subset of U labeled small integers. Then F is of type 1 if the grade of membership of a generic ele­ment u in F is a number in the interval [0,1], e.g.,

~sma11 integers(u) u 2 -1

(1+(5» , u=0,1,2, ... (A21)

On the other hand, F is of type 2 if for each u in U, ~F(u) is a fuzzy subset of [0,1] of type 1, e.g., for u = 10,

~sma11 integers(10) = low (A22)

where low is a fuzzy subset of [0,1] whose membership function is defined by, say,

1 - S(v;0,0.2S,0.S) , V E [0,1] (A23)

THE LINGUISTIC APPROACH TO DECISION ANALYSIS 365

which implies that

low = J>l- S(v;0,0.25,0.5))/v (A24)

If F is a fuzzy subset of U, then its a-level-set, Fa' is a nonfuzzy subset of U defined by

for ° < a < 1.

F a (A25)

If U is a linear vector space, the F is convex if and only if for all A E [0,1] and all u1 , u2 in U,

(A26)

In terms of the level-sets of F, F is convex if and only if the Fa are convex for all a E (0,1].11

The relation of containment for fuzzy subsets F and G of U is defined by

U E U (A27)

Thus, F is a fuzzy subset of G if (A27) holds for all u in U.

Operations on Fuzzy Sets

If F and G are fuzzy subsets of U, their union, F U G, intersection, F (I G, bounded-sum, F ffi G, and bounded-difference, F e G, are fuzzy subsets of U defined by

11Th , d f' .. f . d . 1 1S e 1n1t10n 0 convex1ty can rea 1 y sets of type 2 by applying the extension to (A26).

(A28)

(A29)

be extended to fuzzy principle (see (A75))

366 L.A. ZADEH

where V and A denote max and min, respectively. The complement of F is defined by

F'

or, equivalently,

F' = U e F •

It can readily be shown that F and G satisfy the identities

(F n G)'

(F U G)'

(F ffi G)'

(F e G)'

F' U G'

F' n G'

F' e G

F' ffi G

and that F satisfies the resolution identity

(A32)

(A33)

(A34)

(A35)

(A36)

(A37)

(A38)

where Fa is the a-level-set of F; aFa is a set whose membership function is ~aF = a~F ' and fOl denotes the union of the aF, with a a a E: (0,1].

Although it is traditional to use the symbol U to denote the union of nonfuzzy sets, in the case of fuzzy sets it is advanta­geous to use the symbol + in place of U where no confusion with the arithmetic sum can result. This convention is employed in the following example, which is intended to illustrate (A28), (A29) , (A30) , (A3l) and (A32).

Example A39. For U defined by (AlO) and F and G expressed by

F = 0.4a + 0.9b + d (A40)

THE LINGUISTIC APPROACH TO DECISION ANALYSIS 367

G = 0.6a + O.Sb (A4I)

we have

F + G 0.6a + 0.9b + d (A42)

F()G 0.4a + O.Sb (A43)

FEBG a + b + d (A44)

FeG 0.4b + d (A4S)

F' 0.6a + O.lb + c (A46)

The linguistic connectives and (conjunction) and or (disjunc­tion) are identified with () and +, respectively. Thus,

F and G /:, F () G (A47) and

F or G /:, F + G (A48)

As defined by (A47) and (A48) , and and or are implied to be noninteractive in the sense that there is no-"trade-off" between their operands. When this is not the case, and and or are denoted by <and> and <or>, respectively, and are defined in a way that reflects the nature of the trade-off. For example, we may have

(A49)

(ASO)

whose + denotes the arithmetic sum. In general, the interactive versions of and and or do not possess the simplifying properties of the connectives defined by (A47) and (A48), e.g., associativity, distributivity, etc.

If a is a real number, then Fa is defined by

(ASI)

For example, for the fuzzy set defined by (A40) , we have

F2 = 0.16a + 0.81b + d (AS2)

368 L.A. ZADEH

and

1/2 F = O.63a + O.95b + d . (A53)

These operations may be used to approximate. very roughly. the effect of the linguistic modifiers very and ~ or less. Thus.

and very F {; F2

more or less F ~ Fl/2 -----

(A54)

(A55)

If Fl •••.• Fn are fuzzy subsets of Ul •..•• Un • then the carte­sian product of Fl"" .Fn is a fuzzy subset of Ul x ... x Un defined by

Fl x ••. x Fn f (llFl (ul ) 1\ ••• 1\ llFn (un») /(ul •••• .un ) (A56)

Ulx .. ,xUn

As an illustration. for the fuzzy sets defined by (A40) and (A4l). we have

FXG (O.4a+O.9b+d) x (O.6a+O.5b)

O.4/(a.a) +O.4/(a.b) +O.6/(b.a)

+ O.5/(b.b) +O.6/(d.a) +O.5/(d.b)

which is a fuzzy subset of (a + b + c + d) x (a + b + c + d) .

Fuzzy Relations

(A57)

An n-ary fuzzy relation R in Ul x .. , x Un is a fuzzy subset of Ul x ... x Un' The projection of R on Ui x··· x Ui • where

",,-.:...I.-.~-"-'-- -- -- 1 k (il •...• i k ) is a subsequence of (l •...• n). is a relation in U . x... XU· def ined by 11 1k

Proj R on Uil x ••• x Uik

{; fv llR(ul.···.u )/(ul·····u ) Ujl.··· .UjJl, n n

Uilx",xUik

where (jl •...• jJl,) is the sequence complementary to (il •...• i k ) (e.g .• if n=6 then (1.3.6) is complementary to (2.4.5». and Vu. u' denotes the supremum over UJ'l x ... x UJ' n •

Jl·· .. • JJI, IV

(AS8)

THE LINGUISTIC APPROACH TO DECISION ANALYSIS 369

If R is a fuzzy subset of Uil, ..• ,Uik' then its cylindrical

extension in Ul x ... x Un is a fuzzy subset of Ul x ••. x Un defined by

R J~R(U. , ••• ,u. )/(ul'···,u) 11 1k n

(AS9)

U x·· .XU 1 n

In terms of their cylindrical extensions, the composition of two binary relations Rand S (in Ul x U2 and U2 x U3' respectively) is expressed by

Ro S = Proj RnS on Ul XU3 (A60)

where Rand S are the cylindrical extensions of Rand S in Ul x U2 x U3 • Similarly, if R is a binary relation in Ul x U2 and S is a unary relation in U2, their composition is given by

R 0 S = Proj RnS on Ul . (A6l)

Example A62. Let R be defined by the right-hand member of (AS7) and

S = O.4a + b + O.8d . (A63) Then

Proj R on Ul(~ a+b+c+d) = O.4a + O.6b + O.6d (A64)

and R 0 S = O. 4a + O. Sb + O. Sd . (A6S)

The Extension Principle

Let f be a mapping from U to V. Thus,

v = f(u) (A66)

where u and v are generic elements of U and V, respectively.

Let F be a fuzzy subset of U expressed as

(A67)

or, more generally,

F (A68)

370 L.A. ZADEH

By the extension principle, the image of F under f is given by

f(F) ~ f(u1) + ... + ~ f(u ) 1 n n

(A69)

or, more generally,

(A70)

Similarly, if f is a mapping from U x V to loJ, and F and G are fuzzy subsets of U and V, respectively, then

f(F,G)

Example A72. Assume that f is the operation of squaring. Then, for the set defined by (A14), we have

f (0.3/0.5 + 0.6/0.7 + 0.8/0.9 + 1/1)

0.3/0.25 + 0.6/0.49 + 0.8/0.81 + 1/1

Similarly, for the binary operation V (~max), we have

(0.9/0.1+0.2/0.5+1/1) V (0.3/0.2+0.8/0.6)

= 0.3/0.2+0.2/0.5+0.8/1+0.8/0.6+0.2/0.6

(A7l)

(A73)

(A74)

It should be noted that the operation of squaring in (A73) is dif­ferent from that of (A51) and (A52).

FUZZY CORE AND EQUILIBRIA OF GAMES DEFINED

IN STRATEGIC FORM

Jean-Pierre Aubin

Universite Paris-IX Dauphine

75775 - PARIS CEDEX 16 - FRANCE

Introduction

Let us consider a n-person game

{X(A),FA}A E Q described in strategic form by

1) a family a of coalitions A C N, where

2)

3)

Let x

N = {1,2, ..• ,n} is the whole set of players

multistrategy subsets X(A) of XA ~ xi

of the coalition A i E A

A {A} multiloss operators F = fi i E A where

f~ : xi ~ R is the loss function of player i 1.

behaving as member of coalition A.

-1 -n {x , ..• ,n } E X(N) be a multi strategy of the

whole set of players.

We shall say that it is a weak equilibrium if

there exists A E 1ttn = {A E IR n* such that Vi E N,A i ~ 0 n .

and L AI. = 1} such that, for all coalitions A, i=l

L i - I A if A ( A) A f. (x) ~ A inf i y i E A 1. E X(A) i E A Y

where we set f~ f. and FN = F. 1. 1.

371

372 J.-P.AUBIN o

It is an eguilibrium if A E~n = {A EnLn such that

Ai > 0 for all i EN}. The ~ C({X(A),FA}A E Q)

is the set of multistrategies x E X(N) which are not

rejected by any coalition A E a, where, by definition,

a coalition A rejects (or blocks) x E X(N) if it can

find yA E X(A) yielding to each player i participating

in A a loss f~(yA) (strictly) smaller than the loss

fi (x) (see [3] for instance).

It is clear that any equilibrium belongs to the

core. The converse statement is generally false. For

that purpose, we will "shrink" the core by allowing

more coalitions to form and reject strategies. This is

the reason why we embed the set ~of coalitions A

(subsets of N) into the set J of fuzzy coalitions T

(fuzzy subsets of N in the sense of Zadeh [9} ) •

Afterwards, we will "extend" the game {X(A), FA}A E ~ to a "fuzzy game") {XO(T), FT}T E1 and thus, define

how a fuzzy coalition T rejects some multistra~

x E X(N). Then we define the "fuzzy core" as the set of

multistrategies which are not rejected by any fuzzy

coalitions.

We will prove, by using the general results of

[2] about fuzzy games, that any equilibrium belongs to

the fuzzy core and the fuzzy core is contained in the

set of weak equilibria. This is a result analogous to

the equivalence property of economic games (see [ 2] ) ,

[4J, [6J). Existence of weak equilibria is proved in

[ 1], [S], [B].

1. Fuzzy coalitions

Let us notice that we can identify a coalition

A with its "characteristic function" TA E {O,l}n

defined by its rates of participation

GAMES DEFINED IN STRATEGIC FORM

f 1 if i E A

lOififf A

In other words, a player i either participates wholly

373

in the coalition A or does not participate at all in it.

Definition 1 ============

We shall say that any T E [O,l]n is a "fuzzy

coalition" and that its ith component T, is the "rate ~

of participation of player i in this fuzzy coalition.

Remark ======

From now on, we shall associate with

T = {Tl, ... , Tn} E [O,l]n the map T. from lRN into iRN

defined by

(1) (T.C), = T,C,. ~ ~ ~

We shall denote by

(2 ) A = {i E N such that T, > O} T ~

the support of the fuzzy coalition T, i.e., the set of

active players of the fuzzy coalition. We shall set

= T (3 )

= T

2. Fuzzy games described in strategic form and fuzzy

core

We introduce fuzzy games {X(T), FT} described

in strategic form and, in particular, extend a coopera­

tive game {X(A), FA} described in strategic form to a

374 J.-P.AUBIN

fuzzy game {X(T), F T}. Since the loss functions f~ of l

players i of the coalition A can depend upon strategies

of all players i, we are led to use a multiloss

correspondence FT : XT ~ RT of the fuzzy coalition T

instead of a multiloss operator. This slight

modification does not complicate the situation too much.

(4 )

We begin by defining

Q(T) = {m E. IRQ such that +

i.e., the set of {m(A)}A E Q where

(5 ) VA E a, m(A) ) 0 and Yi E N, I A3i

m(A) = T. l

Let us notice that if m E Q(T), then A C A whenever T

m(A) > o.

coalition

We assume that

E a, X(A)

TT U i

iEA

is a closed convex subset of ( 6)

we set

(7 )

A U T

Let us cons ider the map:

(8 ) { m , {x A} A } E a ( T ) x IT x (A) 1-4 AEQ.

where we define T x

T x

GAMES DEFINED IN STRATEGIC FORM

(9)

Definition 2 ============

A , T

T,i x

375

I A3i

We define the "canonical multi strategy subset" AT

XO(T) C U of fuzzy coalitionAT to be the set of

multistrategies x T = I m(A)x A AEQ. T

when m ranges over Q(T)

and x over X(A) for all A E Q. ~..;;;;...-"--"'-

b) ~~~~~~~~~~~~=~~=~g~=~~!~~!~~~=~~~~~~g~~~~~~~ FT ~~ ~=~~~~~=~~~!~~~~~

Definition 3 ============ A

The multiloss correspondence FT from U T into

tRT is defined by

( 10) { I AEQ

Definition 4 ============

A A m(A)F (x )}

I AEq

T x

We shall say that {XO(T), FT}T E J describes

the "canonical extension" of the cooperative game A

{X(A), F }A EQ into a fuzzy game described in

strategic form.

We set

{ i) F:(XT ) FT (xT) + /R T ( 11 )

+ 0

FT(xT) °T i i) F:(XT) = + IR+

Let us consider now a fuzzy game {X(T), FT}T E J

376 J.-P. AUBIN

described in strategic form, where, for any T E J,

A X(T) CUT is a multistrategy set

( 1 2 ) FT : X(T) I-f-'>:RT is a multiloss

correspondence.

Definition 5 ============

We shall say that a multistrategy x E X(N) is

rejected by a fuzzy coalition T if there exists T - °T T x E X(T) such that T.F(x) E F (x ).

+

The core C(X(T), FT) of the fuzzy game is the

subset of multistrategies x E X(N) which are not

rejected by any fuzzy coalition T.

In particular, by using the canonical

extension {Xo (T), F T}, we will use the following

vocabulary.

Definition 6 ============

A Let {X(A), F } be a game described in

strategic form. We shall say that x E X(N) is

"canonically rejected" by a fuzzy coalition T E J T °T T

if there exists x E XO(T) such that T.F(x) E F+(x), A i.e., if there exists m E Q(T) and x E X(A) such that

( 13) Y1.· E A T'

T.f. (x) > 1. 1. L

A3i

A A m(A)f. (x )

1.

The "canonical fuzzy core" of the game {X(A), FA} is

the subset of multistrategies x E X(N) which are not

canonically rejected by any fuzzy coalition T.

The characteristic form (~,J) of the fuzzy

GAMES DEFINED IN STRATEGIC FORM 377

game {X(,), F'} is obviously defined by

( 14) J (,) = F' (X (,» = U x' E XC,)

(see [2]). We shall show now how convexity, continuity A

and compactness properties of a game {X(A), F }A are

conveyed :

Let us assume that for any A ~~

{l X(A) is convex and compact ( 15)

:i) Vi fA E A, is convex, lower semi-i

continuous.

Let {Xo (')' F'} be the canonical extension

of {X(A), FA} A· Then, for any fuzzy coalition,

( 16)

( 1 7)

:Ls convex.

Proof =====

i) Xo (') is convex and compact

ii) Vx'E Xo (')' F:(X') is closed and

convex

iii) F:(XO ('» is closed and convex,

inf L c E F't(x,) iEA ,

a) Let us begin by noticing that aCT) is a convex

378

compact subset of RQ. +

J.-P. AUBIN

k A

b) Therefore, if x~ = L AEQ

m (A)xk , E Xo (') for any

k = 1, .•• , K, we can write any convex combination

, x

K

L aJ....x~ k=l

, x

in the form

K

L k=l

K k where meA) = L akm (A). Futhermore Xo (') is compact

as continuouski~age of the compact subsets aCT) and

X (A) •

c) Let us prove that F:(X') is convex; it is sufficient

to prove that

d = K k A A L a k (L m (A) F (x k » = L m (A)

k=l AEGl AEa

, , belongs to F+(x ), whenever

, x for all k,

K and where we set meA) L

k=l

A Since the functions f. are convex, we deduce ~

that k K akm (A) L meA)

k=l k

K akm (A) L meA)

k=l where yA A

x k belongs to the convex subset

X (A) . Therefore, since L meA) A , inequality y x

AEQ ,

d ~ L A A meA) F (y ) implies that d E F:(X')

AEq

d) F' (x') is closed +

since the subsets aCT) and X(A)

GAMES DEFINED IN STRATEGIC FORM 379

are compact and since the maps {m,{xA}A} ~ L m(A)f~(xA) A~i l.

are lower semi-continuous, then F:(X') is closed.

e) A proof analogous to the proof of the convexity of

F:(X') shows that F:(XO ('» is convex.

f) F: (Xo (,» is closed since the subsets a (,) x Tf x (A)

are compact and since the maps AEcr

are lower semi-continuous.

K , g) Let x L a k x ; be a convex combination. For any

k=l

e: > h . mk E Q(..-) and A E () h h o t ere eXl.sts L x k X A suc t at k A m (A)X k

X~ and l: AEq

\ i A A , , L A fi(xk)~f (Xk,A)+e:.

iEA l:

AEa

Therefore, by setting meA)

K k

K l: mk(A) and

k=l A

x l: k=l

akm (A) A meA) Xk E X(A), we obtain

f'(x',).), l: meA) AEQ

(18 )

Indeed,

l: iEA

U mEQ(,)

I m(A)FA(X(A». AEQ

x

L AEQ ,

A A m(A)F (x )

+ e:.

u u I m(A)FA(xA)= U L m(A)FA(X(A». mEt.:H') xAEX(A) 1\ea mEQ(,) AEq

380

Remark ======

(19 )

J.·P. AUBIN

Let us consider the case where, for all i EN,

i) Ri is a closed convex subset of a vector

space Ui

ii) Y(i) is a closed convex subset of a

vector space V

iii) Li E £(U i , V) is a continuous linear

operator

and where the multistrategy sets X(A) are defined by

( 20) A A

X(A) = {x E R such that L iEA

L A, i E i~ L Y (i)

iEA

The mulstistrategy set Xo (') of the fuzzy coalition.

is contained into the multistrategy sets X(.) defined

by

Indeed, any x' defined by

A x E X(A) satisfies

L A i

n L ., i \' ( \' A, i E \' () \' ( . ) 'iLl..X = L m A) L L.x L mAL Y l.

i=l AEa iEA l. AEQ i~

Remark ======

where

n L Y (i)

i=l L m(A)

A3i

n L 'i Y (i) .

i=l

h X(A) = XA and Let us consider the case were

GAMES DEFINED IN STRATEGIC FORM

(22)

Then

(23)

A the loss functions f. = f. are convex

]. ].

functions from Ri into R.

T T F+(X )

Indeed, on one hand, {Tifi(xi)}iEA belongs to FT(x T) T

since T. f. (xi) ]. ].

= m({i})f. (m(~)xi when m E Q(T) is ]. Ti

381

defined by m(A) = Ti if A {i} and m(A) o otherwise.

On the other hand,

since for all i E N

I. A3].

A,i m(A)f.(x )~

].

+ IR T +

thanks to the convexity of the loss functions. •

We shall extend this remark to the case where

are the loss functions associated with the

loss functions f. ].

(24)

N : X ~ R by the following formula

1\ A A

f.(x,x) ].

/\ (we set A

1\ A 11 N - A and x = {x , x } E X (N)

/'\ A

x X ).

Let us set also

382

(25)

and

(26)

(27)

A# A ={ fi (x )}:lEA

Let us assume that

T

I iEA

T

and

A Im(A)X

T

{the subsets X

i and the functions fi

convex.

J.-P_ AUBIN

T x

are

Then, for any x T E X (T) and for all i E A , we obtain .. 0 T

n. . (28) L \~T.f~(x ) ~ f (x T ;\)

i=l ~ ~ T

We need to begin by proving a lemma, the proof of which

is inspired by [7] .

Lemma 1 =======

i E A , T

(29)

Proof =====

we

Let m E a(T) and x A E XA be given. For any

can find y(~) E 1lr x j such that jEA j~iT

1\

1:. A3J

\' m(A){ A A (0)}j A~i ~ x ,~ y ~ •

It is trivial j i. If j ~ i, we can write,

GAMES DEFINED IN STRATEGIC FORM 383

1\

L (A) A A A '

_m_ {X ,1T y(i)}J = L Ti A~{i,j}

meA) i

T

A, j X + L meA)

A:}i T i

Ah

1'\.' Y (i) J

A:Ji

~

Let us define y(i) by

(30)y(1)j

Since T, 1

T, J

(31)

Therefore,

L (_1 --.!..)m(B)xB,j + L m(B) -B~{i, j} T j T,

1 B3j B~i

I (_1 1 - -)m(B) + L B~{i, j} T j T,

1 B3j BtH

L meA) + I meA) and A~{i,j} AJi

A;t$j

meA) + L meA), we deduce that A3j

L A.:>i Aflj

m (A.)

T, 1

A/H

L m (A) (_1

A~{i,j} T j

1\

_1) + T,

1

T, J

m(B) T

j

L A~j

A~i

X B, j

meA) T,

J

r \' m (A) {A A (~)} j L -- X ,1T Y 1 A::>i T i

\' A,j 1 1 1 L m(A)x (-+---) +

A~{, '} T, T, T,

(32)

+ L meA) xB,j

A '}j T j

Aih

Let us write x T

. 1,J 1 J 1

) A, j L meA. x

A3j T j

L AEQ

A m(A)x T

•

where m E aCT)

384 J.·P. AUBIN

XA E E Q and X(A) for all A •

(33)

A By lemma 2, there exists y(i) such that

T x I A;)i

Hence, the convexity of the loss function fi implies

(33)

/\

= f (I meA) {xA,TIAy{~)}) , i A3i T i

Therefore, T for all representations of x

n

I i=l

i T ~ i \ A# A A T.f. (x )' L A L m(A)f. (x ) 1 1 i=l A~i 1

<A, I m(A)FA'(xA» AE

Thus, by taking the infimum, we obtain inequality (28).

3. Equivalence between the fuzzy core and the set of

equilibria

A Let {X(A),F }A E a be a game described in

strategic form by multistrategy sets X(A) and multiloss

operators FA satisfying

(15 )

X(A) is convex and compact i" i)

ii) Vi E A, f~ is convex and lower 1

continuous.

semi-

GAMES DEFINED IN STRATEGIC FORM 385

Any strong canonical cooperative equilibrium x E X(N)

belongs to the core C({XO(T) ,FT}T E J) of its canonical

extension. Conversely, the canonical fuzzy core is

contained in the subseL of weak canonical cooperative

equilibria.

Before proving Theorem 1, we need to prove

Lemma 2 below which characterizes the canonical fuzzy

core. Let us introduce the function a defined on X(N) by

(34) a(x) = sup T E J

inf [<A,T.F(x» - W(T,A)] A E ?n(T)

where ~(T) is the set of probabilities on the support

AT of T and

(35) W(T,A) inf fT(xT,A) x E XO(T)

Lemma 2 ======= Let us assume that the multistrategy sets X(A)

and the loss functions f~ are convex. Then x E X(N) ~

belongs to the canonical fuzzy core if and only if

a(x) .;; o.

Proof

First, let us assume that a(x) .;; o. Then, for

any fuzzy coalition T, inf [<A,T.F(x»-W(T,A)] .;; o. A E }n(T)

Sin c e 1h. ( T ) is co mp act and A -+ < A) T • F (x) > - W (T , A) is

semi-continuous, there exists A E ~(T) which achieves

the minimum. Therefore, VT E J, 3A E /'fz(T) such that

<A,T.F(x» (W(T,A). This implies that x is not rejected

by T. [For,

dEFT(x T)

T if it was, there would exist x E XO(T) and

such that T,f, (x) > d, for all i EA. Then ~ ~ ~ T

we would obtain <A,T.F(x» > <A,d> ~ fT(xT,A) ~ W(T,A),

386 J.-P. AUBIN

i. e., a contradiction.l .

Conversely, les us assume that x belongs to rt 01' the canonical fuzzy core. Since T.F(x) - F+(XO(T»

(which is convex by proposition i), the Hahn-Banach

separation theorem states that there exists A E ~T, A to 0, such that <A,TOF(X»" W(T,A). Since W(T,A) > _00,

this implies that A E RT. Then, by dividing by . +

L A~ > 0 the two sides of this inequality, we can say iEA. that A E ~(T). Therefore, we have proved that a(x) , 0 .•

Now, we prove lemma 3 which relates the

functions {T,A} ~ W(T,A) and {A,A} ~ W(A,A) where we set

(36)

Lemma 3 =======

(37)

W(A,A) inf

x A E X (A)

For any A, we have

W(T,A) = inf m E aCT)

L m(A)w(A,A) AEQ

Then the function {T,A} ~ W(T,A) is convex with respect

to l' and concave with respect to A.

Proof ===== Indeed, we can write

n i A A fT (xT ; A) infA iL A A~i m (A) f. (x )

L !!U...A~ ~

xT AEa

T

= inf L m (A) .L Aif~(xA) A ~

;: m(A)x A£ Q ~E.A

xT A<:' Q T

GAMES DEFINED IN STRATEGIC FORM

Thus, we obtain

W(T,A) = T

X

inf I m(A) inf I Aif~(xA) E Q(T) AEQ A iEA m E X(A) x

inf I m(A)w(A,A) m E Q (T) AEa

Proof of Theorem 1 ================

We deduce from (37) that <A,T.F(x» ~ w(T,A)

fuzzy coalition TEl. Since T E~(T) I

iEA T

~

for any T,

we deduce that a(x) ~ O. Then x belongs to the

canonical fuzzy core by lemma 2.

Conversely, let x belong to the canonical

fuzzy core; By Lemma 2 and assumption (15), we deduce

that a(x) ~ O. Therefore

(38) inf [<A,T.F(X» - W(T,A)] ~ a(x) ~ O.

AEmn

By lemma 3, we can use the minimax theorem: there

exists I Em n (which is compact and convex) such that

387

SUp[<I,T.F(X»-W(T,I)] TEJ

sup in£[<A,T.F(x»-W(T,A)} (0. TE! AEvf

By taking T = TA, we deduce that for any coalition A, - A A

<A,T .F(x» t; W(A,A) (Since W(T ,A) "W(A,A» .

Hence x is a weak equilibrium. •

388 J.-P.AUBIN

REFERENCES

[1] J.-P. Aubin. Equilibrium of a convex cooperative

game. MRC Technical Summary Report = 1279 (1972).

[2] J.-P. Aubin. Fuzzy games. MRC Technical Summary

Report = 1480 (1975).

[3] R. Aumann. The core of a cooperative game without

side-payments. Trans. Amer. Math. Soc. 98 (1961),

539-552.

[4] R. Aumann. Markets with a continuum of players.

Econometrica. 32 (1964) 39-50

[5] E. Baudier. Competitive Equilibrium in a game.

Econometrica. 41 (1973) 1049-1068.

[ 6] G. Debreu and H. Scarf. A limit theorem on the

core of an economy. Int. Econ. Rev. 4 (1963)

235-296.

[7] H. Scarf. On the existence of cooperative

solution for a general class of n-person games.

J. Econ. Th. 2 (1971), 169-181.

[8] L. Shapley. Utility comparison and the theory

of games; La Decision (1968).

[9] L. Zadeh. Fuzzy sets. Information and Control 8

(1965), 338-353.

STABILIZATION AND OPTIMAL CONTROL OF NONLINEAR SYSTEMS

HOMOGENEOUS-IN-THE-INPUT

D.H. Jacobson

National Research Institute for Mathematical Sciences

Pretoria, South Africa

ABSTRACT

Stabilizing and control strategies are presented for nonlinear and bilinear systems which are homogeneous-in-the-input. For this class of systems it is demonstrated that nonlinear controllers can be synthesized, which optimize a wide variety of performance criteria. These controllers produce asymptotic stability also in the interesting cases in which constant controllers fail to stabilize the systems. Extensions to non-homogeneous systems are given.

INTRODUCTION

In this paper we are concerned with the design of controllers for systems of the type

m 1: B. (x)u.

i=1 l l

x(t ) o

x o

( 1 )

where x E Rn is the state of the system at time t E [to,oo) and where the ui, i=1 , ... ,m represent the control variables at time t. The vector functions Bi(x), Bi : Rn + Rn are assumed to be continuous in x, i=1, ... ,m. We shall assume further that the controls are chosen in feedback form as ui(x) = gi(X), i=1 , ... ,m where, unless otherwise stated, gi : Rn + RI is a continuous function of x.

Bruni et al. [1] refer to (1) with Bi (x) = Bix, Bi E Rnxn , as a bilinear system, homogeneous-in-the-input. Following this terminology, we shall refer to (1) as a nonlinear system, homo­geneous-in-the-input.

The most famous and important (regulator) problem In optimal

389

390 D.H. JACOBSON

control has been the Linear-Quadratic-Problem (LQP} which was solved in 1960 by R.E. Kalman [2]. The advantage of a regulator formula­tion over direct stabilization of a system (say, by pole placement in the linear case) is that, in addition to yielding a controller which produces asymptotic stability, it permits the designer to choose a stabilizing controller which also minimizes a desired performance criterion.

Systems described by nonlinear ordinary differential equations such as (1) are perhaps not as common as those described by linear differential equations, but in any event the nonlinear character of (1) makes it a differential equation worthy of investigation. Indeed, we believe that the results presented in this paper relating to the stabilizability and optimal control of (1) are similar in generality and appeal to those known for linear systems. It should be noted, however, that a vast literature, see bibliography in [1], already exists for the case where Bi(x) is of the form Bix - the bilinear case, though the investigations have been confined largely to questions of controllability, observability and control on mani­folds. A recent paper [3] presents certain results on the stabilizability of bilinear systems which are related to ours. However, in that paper the close tie between stability and optimal control is not exploited.

We begin by applying to (1) certain (converse) theorems of Liapunov to yield conditions for asymptotic stability.

STABILIZABILITY

It turns out that a rather straightforward set of conditions lS obtainable which for all practical purposes answers the question of whether or not x = 0 can be made to be a (globally) asymptotically stable equilibrium point of (1). Note the trivial fact that x = 0

is a stable equilibrium point of (1) if the controls ui are identi­cally zero.

Theorem 1. Suppose that ui(x) = gi(x), i=l , ... ,m and that the gi and Bi are once continuously differentiable with respect to x. A necessary condition for x = 0 to be an asymptotically stable equilibrium point of (1) is that there exists a positive-definite function V(x), V : Rn + RI , which is continuously differentiable, such that the scalars

V (x)B. (x) x l

i= 1 , ... ,m (2 )

are not all zero for each non-zero x ERn.

Proof. Since the right-hand side of (1) with this choice of ui, i=l, ... ,m is independent of t and is continuously differentiable with respect to x, we can apply an inverse Liapunov theorem directly,

STABILIZATION AND OPTIMAL CONTROL OF NONLINEAR SYSTEMS 391

see [4]. This theorem guarantees the existence of a positive­definite function V(x), which is once continuously differentiable, such that v(x) = Vx(x)x is negative-definite. In our case, we have

m V(x) = L: V (x)B. (x)g. (x)

i=1 x l l

It follows immediately that this expression cannot be negative­definite unless the theorem is true.

Sufficient conditions for (global) asymptotic stability are provided by the following Liapunov Theorems which do not require differentiability of Bi(x) and gi(x).

(3)

Theorem 2. A sufficient condition for x = 0 to be made into an asymptotically stable equilibrium point of (1) is that there exists a positive-definite function V(x), V : Rn + RI , which is once continuously differentiable, such that the scalars

V (x)B. (x) x l

i=1, ... ,m (4)

are not all zero for each non-zero x ERn.

Proof. Let us set ui(x) = -Vx(x)Bi(x), i=1 , ... ,m. Then, the ui(x)~continuous and

V(x) m L:

i=1

2 [V (x)B.(x)] x l

which is negative definite because the Vx(x)Bi(x) do not vanish simultaneously for any non-zero x ERn.

Theorem 3. If in Theorem 2, V(x) is also radially unbounded then u.(x) = -Vx(x)Bi(x), i=1 , ... ,m causes x = 0 to be a globally asympt~tically stable equilibrium point of (1).

Proof. The proof follows from a standard Liapunov theorem [4].

Note that the gap between the necessary and the sufficient conditions, Theorems 2 and 3, is only the assumed differentiability of Bi(x) and ui(x).

Theorem 3 is useful especially in those cases where (1) cannot be stabilized by means of constant controls ui = ki' i=1 , ... ,m , ki E RI. We illustrate this by an example.

Example 1: Bilinear equation in R2.

Let

(6 )

392 D.H. JACOBSON

In this case it is easy to verify that there are no real numbers k1 and k2 such that

m L

i=1 B.k.

1 1

is a stability matrix. In other words, constant controls cannot asymptotically stabilize (7).

However, setting

we see that

and

(8)

2 V (x)B x = -x x 2 1

( 10)

which do not vanish simultaneously. As Vex) 1S radially unbounded, we have the result that setting

yields x = 0 as the globally asymptotically stable equilibrium point of (7).

( 11 )

Note that it is a trivial matter to prove that if there exist ki E Rl, i=1, ... ,m such that

m L

i=1 B.k.

1 1

is a stability matrix, then there exists Vex) = ~xTSx, S > 0, such that the scalars Vx(x)Bix = ~xT(SBi+BiTS)x, i=1 , ... ,m are not zero simultaneously for non-zero x ERn, and hence that the null solution of

m L

i=1 B.xu.

1 1

is asymptotically stable in the large under the action of u.(x) _~xT(SB.+B.TS)x, i=1, ... ,m. 1

1 1

STABILIZATION AND OPTIMAL CONTROL OF NONLINEAR SYSTEMS 393

OPTIMAL CONTROL

In the previous section we gave a condition which is sufficient to ensure that x = 0 can be made to be the unique asymptotically stable equilibrium point of (1). A set of controls which accomplishes this was shown to be ui(x) = -Vx(x)Bi(x), i=1 , ... ,m. It is clear, however, that 1

2p+1 u. (x) = - [V (x)B. (x)] l x l

i=1 , ... ,m ,

where p E {O,1 , ... } also has a similar effect on (1), which leads one naturally to the question of whether or not an "optimal" stabilizing control can be constructed. In this section we show that stabilizing controllers which minimize a wide variety of performance criteria can be constructed for (1). For completeness we first state and prove a general theorem.

Define the following cost function

00

J 1

f {q(x) + 2(p+1) E o i=1

m u. 2 (p+1)}dt,

l

where q(x) is positive-definite, q : Rn ~ Rl, and p E {O,1 ... }.

( 12)

Theorem 4. Suppose that there exists a radially unbounded, positive-definite function V(x), which is once continuously diffe­rentiable, which satisfies the algebraic Hamilton-Jacobi-Bellman equation

[q(x: 1 m 2(p+1) mln + 2 (p+ 1 ) E u.

i=1 l u. l

i=1 , ... ,m m

+ E V (xiB. (XiU] = 0 i=1 x l l

Then, the controls

u. = l

1 2p+1 -[V (x)B.(x)] , x l

i=1 , ... ,m ( 1 4 )

globally asymptotically stabilize (1) and minimize J in the class of control functions which causes x(t) ~ 0 as t ~ 00 •

Proof. It is easy to verify that (14) is implied by (13). Furthermore, substituting (14) into (13) yields

q(x) - 2p+1 2(p+1)

2 (p+ 1 ) m E [V (x)B. (x)] 2p+1 = 0

i=1 x l ( 1 5 )

394 D.H. JACOBSON

which implies that Vx(x)Bi(x), i=l, ... ,m , cannot all be zero for a non-zero x £ Rn.

Next, we note that 2(p+1)

m 2 +1 Vex) = - E [V (x)B. (x)] P

i=l x 1 = -2(p+1) q(x}

2p+1

which is negative-definite. Controls (14) therefore globally asymptotically stabilize (1).

Now, along a solution of (1) we have

00

Vex ) - v(x(oo)) + J V(x)dt = 0 o

o

which lS 00 m

Vex ) - v(x(oo)) + J o 0 i=l

E V (x)B.(x)u. dt = 0 x 1 1

Adding (18) to (12) and using ( 15) yields

00 m 2(p+1)

J = Vex ) - v(x(oo)) + J { E 2p+1 [V (x)B. (x)] 2p+1 0 i=l 2(p+1) x 1 0

m m 1 u. 2 (p+1)}dt + E V (x)B. (x)u. + E 2(p+l) i=l x 1 1 i=1 1

( 18)

Now we choose ui from the class of functions which causes x(t) + 0 as t + 00. For this class of controls it then follows that V(x(oo)) = o and, in particular, the controls (14) minimize the integral in (19) to give its minimum value of zero. Hence the minimum value of J is V(xo ).

Because of the special form of the H-J-B equation (15) it is obvious that we can, without loss of generality, replace q(x) in (12) by the form

2(p+1 )

2p+1 2(p+1 )

m E [V (x)B. (x)] 2p+1

i=l x 1

This tells us, in effect, that instead solving the H-J-B equation for Vex) we suitable vex) and hence specify q(x). statement of this approach.

of specifying q(x) and should rather choose a The following theorem is a

Theorem 5. Suppose that there exists a radially unbounded, positive-definite function Vex), V : Rn + Rl, which is once con-

STABILIZATION AND OPTIMAL CONTROL OF NONLINEAR SYSTEMS 395

tinuously differentiable. such that Vx(x)Bi(x). i=1 •...• m are not all zero for each non-zero x £ Rn. Then. the controls

1

u. (x) = - [V (x)B. (x)] 2p+1 i=1 •..•• m ~ x ~

(20)

globally asymptotically stabilize (1) and minimize

J = ""! { () 1 ! u. 2 (p+l )}dt o q x + 2(p+l) i~l ~ (21 )

~n the class of control functions which causes x(t) + 0 as t + "".

where

q(x) = 2p+l 2(p+l)

2(p+1) m l: [V (x)B. (x)] 2p+l

i=l x ~ (22)

By choosing an appropriate V(x) we can synthesize a desired q(x). In many cases the class of q(x) defined by (22) with V(x) a quadratic function. viz. V(x) = ~xTSx. S > O. is adequate. The following example illustrates this in the two-dimensional case.

Example 2: Consider the same equations as in Example 1 and let V(x) = ~xTSx. with S = I. Then. for p = O. we have

2, T 2, 2 222 q(x) = l: :2 [x B.x] =:2 [(2x1x2 ) + (Xl -x2 ) ]

i=l ~

='( 22)2 :2 Xl +x2

The control given by (11) therefore minimizes

"" =!{'( 22)2,( 2 2)} J :2 Xl +x2 +:2 u 1 +u2 dt o

We note that for large P. (21) is approximately

(23)

(24)

J ""! {! Iv ( )B ( )1 1 ! u. 2(P+1)}dt (25) ~ 0 i~l x x i x + 2(p+l) i~l ~

which yields an optimal controller (20) which tends to

u. (x) = -sign [V (x)B. (x)] ~ x ~

i=l •...• m (26)

as p + "". Indeed. this reasoning leads us to the following theorem,

396 D.H. JACOBSON

which 1S easy to verify.

Theorem 6. Suppose that there exists a radially unbounded, positive-definite function V(x), V : Rn ~ Rl, which is once con­tinuously differentiable, such that Vx(x)Bi(x), i=1, ••. ,m , are not all zero for each non-zero x £ Rn. Then, the controls

u. (x) = -sign [V (x)B. (x)] 1 x 1

i=1 , •.• ,m

globally asymptotically stabilize (1) and minimize 00

J = f q(x) dt (28) o

in the class of control functions which causes x(t) ~ 0 as t ~ 00

and which satisfies the control constraints

-1 ..;; u . ..;; 1 1

where, with no loss of generality

m

i=1 , .•. ,m

q(x) = L Iv (x)B. (x) I i=1 x 1

(30)

Theorem 6 is important in that it tells us how to design a bang-bang controller which asymptotically stabilizes (1) and mini­mizes the integral of a positive-definite function. Modulo the assumption of the existence of a solution V(x) to the H-J-B equation, there is no loss of generality in choosing q(x) to be of the form (30) •

Our next theorem is also of some interest.

Theorem 7. Suppose there exists a radially unbounded, positive­definite function V(x), V : Rn ~ Rl, which is continuously differen­tiable, such that Vx(x)Bi(x), i=1 , ••• ,m , are not all zero for each non-zero x £ Rn. Then

u.(x) = -[V (X)B.(x)]2p+1, i=1, .•. ,m, p £ {0,1, ... } (31)_ 1 x 1

globally asymptotically stabilizes (1) and minimizes

00

J = f {q(x) + 2p+1 o 2(p+1 )

2(p+1)

~ I u. I 2p+ 1 } dt i=1 1

STABILIZATION AND OPTIMAL CONTROL OF NONLINEAR SYSTEMS

in the class of control functions which causes x(t) + 0 as t where the positive-definite function q(x) is

q(x) = 2(P~1) ~ [V (x)B. (x)] 2(p+l) i=l x 1

Note that for large p we have

00 m J ~ f {q(x) + L iu.i} dt

o i=l 1

NON-HOMOGENEOUS SYSTEMS

+00 ,

397

(33)

(34)

Here we extend some of our results to systems of the form

m x = f(X) + L

i=l B. (x)u.

1 1 x(t ) = x

o 0

Theorem 8. Suppose that there exists a radially unbounded, positive-definite function V(x), V : Rn + RI, which is once con­tinuously differentiable, such that Vx(x)f(x) is negative semi­definite. Suppose further that Vx(x)f(x) and Vx(x)Bi(x), i=l, •.. ,m are not all zero for each non-zero x ERn.

Then,

u. (x) = -V (x)B. (x) 1 x 1

i=l , .•. ,m ,

globally asymptotically stabilizes (35) and minimizes

00 m J = f {q(x) + ~ L

o i=l

2 u. } dt 1

(36)

in the class of control functions which causes x(t2 + 0 as t + 00,

where q(x), positive-definite, is given by

m 2 q(x) = -V (x)f(x) + ~ L [V (x)B. (xU (382

x i=l x 1

There is considerable flexibility in choosing a V function to obtain a suitable q(x); in many cases, especially when f(x) is of the form Ax, a quadratic function 1S adequate.

Example 3: Suppose m

x=Ax+ L i=l

B.xu. 1 1

398 D.H. JACOBSON

Suppose V(x) = ~xTSx satisfies the requirements of Theorem 8. Then

i=l , ... ,m (40)

globally asymptotically stabilizes (35) and mlnlmlzes

J co T m T T 2 m 2 J {x Qx + ~ L [x (SB.+B. S)x] + ~ I u. } dt o i=l l l i=l l

(41)

where Q = -(SA+ATS) ;;;. o. Theorem 8 and the above example tell us how to design an

asymptotically stable controller which minimizes the performance criterion (37) in the case where the uncontrolled system x = r(x) or Ax is stable. If the uncontrolled system lS not stable this lS not always possible.

Theorem 9. Suppose that the real parts of the eigenvalues of A in (39) are not all non-positive. Then there does not exist a stabilizing controller for (39) which minimizes a criterion of the form (37) and which satisfies the H-J-B equation with continuously differentiable V(x).

Proof. Suppose the contrary, then the optimal control would be

u.(x) = -v (x)B.x l x l

and the controlled system would be

m x=Ax- L

i=l

i=l , ... ,m (42)

B.xV (xlB.x l x l

(43 )

However, for x sufficiently small, the stability properties of (43) are determined by A (recall that Vx(xI lS a continuous function of x) .

Regardless of the stability properties of A in (39), the following theorem shows that, under certain conditions, a bang-bang control exists which stabilizes (39) and which minimizes a perfor­mance criterion of the form

00

J J q(x) dt (44) o

Theorem 10. Suppose that -1 ~u. ~ 1, i=l , ... ,m. Suppose that there exists a radially unbounded, po§itive-definite function V(x), v : Rn 7 Rl, once continuously differentiable, such that

STABILIZATION AND OPTIMAL CONTROL OF NONLINEAR SYSTEMS 399

m V (x)Ax - E Iv (x)B.xl

x i=l x 1

is negative-definite. Then, the controls

u.(x) = -sign [V (x)B.X] 1 x 1

i=l , ... ,m (46)

globally asymptotically stabilize (39) and minimize (44) in the class of control functions which causes x(t) + 0 as t + 00, where, without loss of generality, q(x) is of the form

m q(x) = -V (x)Ax + E Iv (x)B.xl

x i=l x 1

CONCLUSION

In this paper we presented and proved theorems which allow us to choose controls which globally asymptotically stabilize (1) and which minimize performance criteria of the type (21), (28) and (32). We proved also that, with no loss of generality, the form of q(x) could be chosen according to (22), (30) and (33). OUr results are therefore very general and their applicability depends only upon the choice of V(x) which, even when restricted to quadratics, yields considerable flexibility in determining q(x). Some extensions to non-homogeneous systems were also given.

REFERENCES

[1] BRUNI, C., DI PILLO, G. & KOCH, G. Bilinear systems: an appealing class of "nearly linear" systems in theory and applications. IEEE Trans. Automatic Control, AC-19, No.4, 1974, pp. 334-348.

[2] KALMAN, R.E. Contributions to the theory of optimal control. Bolo Soc. Mat. 1\1exicana, 5, 1960, pp. 102-119.

IONESCU, T. & MONOPOLI, R.V. systems via hyperstability. AC-20, 1975, pp. 280-284.

On the stabilization of bilinear IEEE Trans. Automatic Control,

[4] HAHN, W. Stability of motion. Springer Verlag, Berlin, Heidelberg, New York, 1967.

STABILITY OF LARGE SCALE INTERCONNECTED SYSTEMS

Jan C. Willems

Mathematical Institute University of Groningen Groningen, The Netherlands

INTRODUCTION

A large scale system may be defined as an interconnection of a large number of individual subsystems. Usually the dynamic features of the system stem practically always from the component subsystems and the interconnection laws simply identify the signal flow graph between the various subsystems. The complexity of the system is in the first place due to these interconnections and makes it usually very hard to predict the overall behavior of th~ interconnected system. A de­sirable feature of an analysis or synthesis procedure for large scale systems is the separation of these two levels. One thus likes an analysis or synthesis procedure to be based, on the one hand, on the "local" subsystems (through the construction of, say, Lyapunov functions for the local subsystems or the design of dynamic con­trollers around these local subsystems) and, on the other hand, on the "global" interconnection laws (through the analysis of these interconnection laws, or, in the case of synthesis, through the de­sign of the signal flow graph which specifies how the physical in­teraction is and how information it to be transmitted between the subsystems) .

In this paper we will outline a procedure for investigating the stability of large scale interconnected system based on this principle.

This paper was prepared while the author was visiting the Seminar fur Angewandte Mathematik of the ETH in Zurich.

401

402 J.C. WI LLEMS

The procedure consists out of several steps. The first step is called "tearing" which simply means that the large scale system is viewed as an interconnection of a number of sUbsystems. Since we are interested in stability analysis it is reasonable to assume the system with which we start to be an autonomous one. However the subsystems obtained in the tearing process will have inputs and outputs. These "internal" inputs and outputs cause the subsystems to interact. In order to obtain an analysis procedure which investigates the stabili­ty it is thus useful to obtain a concept which generalizes to input/ output systems the concept of Lyapunov function which comes up and is very effective in the stability analysis of autonomous systems. This generalization leads to a special class of dynamical systems which we have called dissipative systems. The second step of the procedure thus consists of investigating whether or not the component subsystems are dissipative. The third step investigates the proper­ties of the interconnection laws. These, in a sense, must be "matched" to the dissipativeness of the subsystems. If this is in­deed the case, then we obtain from this procedure a Lyapunov function for the original large scale interconnected system. Its stability properties may then, as a fourth and last step, be tested easily with this Lyapunov function at hand.

~le hasten to state that the procedure thus outlined is not par­ticularly subtle. However, it makes the stability analysis systema­tic and easy to oversee. The technique has, in a sense, some of the same drawbacks which any Lyapunov method has: some functions are in­troduced, the success of the method depends on their existence, but no constructive procedures for their computation is given. The situ­ation is, however, a bit more favorable here than in the general Lyapunov context since, whereas in the general case we are always bound .to have to investigate the total system in all its complexity, here we may always concentrate on subsystems which, evidently, will usually be much simpler to investigate.

The ideas of this paper have appeared on several places in the literature before. In particular, in [lJ this stability analysis procedure is introduced in general, whereas in [2J these ideas are used to show how one could view essentially all of the numerous re­sults in the area of feedback stability from this point of view. We will therefore omit detailed proofs and examples.

STABILITY OF LARGE SCALE INTERCONNECTED SYSTEMS 403

BASIC CONCEPTS

Let 2: ={u, V, Y, Y, X, </>, r} be a (continuous time) dynamical system with

U the input aZphabet, V the input space, Y the output alphabet, Y the output space, X the state space, </> the state transition function, r = the read-out function.

As usual, 2: is assumed to satisfy a standard series of axioms, which are well-documented in the literature [3, 4J. If U (and thus V) con­sists of a single element only then 2: is called an autonomous dyna­mical system. A very special but very important class of autonomous dynamical systems are called flows (classical mechanics and dynamical systems theory as it is understood in mathematics is basically a study of flows). A flow is an autonomous system with r(x, u, t) = x; it is thus defined by 2:* = {X, </> } but in a specific context some topological assumptions are usually added in the axiomatic structure. (In the notation 2:* = {X, </>} for a flow, one can suppress the depen­dence on u in </>, thus writing x(t) = </>(tl , t , x ». We will hence

o 0 assume that X is a metric space and that </>(., to' .) is continuous on [to' 00) x X. The flow 2:* is then called a smooth flow.

Lyapunov stability studies the asymptotic behavior of flows. Assume therefore that x* is an equilibrium state (i.e., </>(t, to' x*) = x* for all t, t ; t ~ t ) of the smooth flow 2:*. The concepts of boundedness of 2:*~ stabil~ty of x*, asymptotic stability of x*, and asymptotic stability in the large of x* may then be defined in the usual way.

A particularly effective method for studying the stability pro­perties of an equilibrium state of a flow is by considering a Lyapunov function. Thus the function V : G x R + R is said to be a Lyapunov function for 2:* on the set GcX if V(</>(t,to,x),t>':V(x,to ) for all x e G and t,to; t ~ to (most authors also require for a Lyapunov function that V ~ 0, but we will not do so here). An im­portant advantage is that if 2:* is described by a differential equation ~ = f(x,t) (f is then said to be the generator of </» and if V is sufficiently smooth then the required inequality in the de­finition of a Lyapunov function may be checked by verifying that "0'12:*::: 0 on G x R. Herevh::*: X x R + R is given by

404 J.e. WI LLEMS

av av V Il:* = at + ax • f, and its sign definiteness is

immediately verifiable from V and the generator f of cpo It is well­known that in order to conclude stability or instability on the basis of a Lyapunov function one must add certain other assumptions. These have to do with the positive definiteness, radial unbounded­ness, and/or decrescence of v. We will not dwell on these well-docu­mented ramifications ISI.The disadvantage of Lyapunov methods is that it is usually very hard to postulate a reasonable candidate for V. It is for this problem that the paper offers a systematic pro­cedure.

The usefulness of L~e concept of a Lyapunov function is thus limited to flows. However, it is possible to generalize this con­cept in a natural way to systems with inputs and outputs. This is one way of viewing the recently introduced concept of dissipative dynamical system. Assume thus that in addition to the dynamical system L we are given functions S: X x R + Rand w : U x Y x R+R such that w(u(·), y(.), .) is locally integrable for all u E U and y E Y. The function S is called the storage function and w is called the supply rate. The crucial element in the definition of a dissipative system is an inequality, called the dissipation inequa­lity. The inequality is analogous to the inequality required in the definition of a Lyapunov function (with the storage function playing the role of Lyapunov function and the supply rate being absent in the case of flows). However, not unalike the situation in the case of Lyapunov functions for flows,it turns out that the behavior of L is qualitative different de'pending on whether the storage function is nonnegative or not (as in Lyapunov theory, it would be more logical to distinguish between storage functions with or without lower bounds since the whole theory is invariant under addition of a constant to S). We will therefore introduce two classes of dynamical systems: the first are called dissipative dynamical systems and the second are called (for reasons which will become clear later on) cyclo-dissipative dynamical systems.

Thus {L, w, S} is called dissipative if (i) S ~ 0, and (ii) the

dissipati~n inequality:

S (x ,t ) + o 0

STABILITY OF LARGE SCALE INTERCONNECTED SYSTEMS 405

is satisfied for all x o ' xl E X'to,~tl?: to and UE U. In here xl = <P(tl , to' x o ' u) and w(t) (abusively) denotes w(u(:) ,~(t) ,.t) with y(t) = r(<p(t, to' x o ' u), u(t),t). It is called cycZo-d~ss~pat~ve if the dissipation inequality (ii) is satisfied.

INTERCONNECTED SYSTEMS &~D TEARING

Assume now that we are given a family of dynamical systems LP' = { Ua , yo., yo. , Xa ,<po., ra} where a ranges over some index set A and an interconnection constraint N which specifies how the systems EO. are to be interconnected together. In general this will not com­pletely specify all the signals in the system since certain inputs will remain free and act as external inputs to the interconnected system. In the first place N could be defined as a subset of

n (Ua x yo.) (in general, one could even allow N to be timevarying al:A also). In specific situations it turns out that N has usually a great deal more structure. Setting U = IT Ua and Y = ITAya then one

aeA ae typically has that N is defined as the graph of a mapping from Y into

Ui where Ui x Ue = U is a partition of U into its internal and ex­ternal components. Moreover, in fact, this map of which N is the graph is usually one of a very special kind since it identifies components of Ui with components of Y. Thus setting Ui= S~B uS and Y = Y~C uY this map is then defined by a map P : B + C. The inter­connection constraint then states that uS = Yp(S) ,thus

«S~BuS' u e ), ~cYY) N if and only if uS = yp(S) for all SeB.

We introduce these ideas here in detail eventhough they will not be used in the sequel since we believe the concept of an inter­connection law to be an important one in the understanding of system structure and the vague and lor overly general definitions which are now being used cannot be expected to be very fruitful. Note, finally, that the usual interconnections (series, parallel, feedback, electrical network interconnections) are of the latter kind.

Given EO., aeA, and N one may now examine the interconnected system a~A EftIN. If the interconnection is what is called weZZ­posed then aeA EalN defines an dynamical system in its own right. For formal definitions to this effect, the reader is referred to [2J. In general there will be "free" inputs in a~A Eo. IN. If N is defined through a map from all of U into Y (i.e., if U = Ui) then

one expects the interconnected system a~A EalN to be an autonomous one. In stability studies one is primarily interested in this case.

Assume thus that a~A EalN = E*~ a~AXa, <P} is the resulting flow

406 J.e. WILLEMS

a (the ~ is to pe computed from the ~ 's appearing in the definition of ~a by solving for the inputs ua ).

1 A typical interconnected system is a feedback system. Thus let ~=[ul, vi, yl, yl, xl, ~l, rl} and ~2:(u2, V2, y2, y2, x2,~2,r2} be interconnected according to the feedback interconnection con­straint u l = Y2; u2 = YI' The resulting system ~* = ~l x ~21 feed­back will thus be a flow (assuming well-posedness); it has state space xl x x2 and its state transition function is to be obtained by solving some implicit equations. In the case that ~l and ~2 are described by differential equations and one of them is strictly causal (i.e., r(x, u, t) is independent of u) then these implicit equations may in fact be solved explicitely.

In the case we are interested in here, we actually start with a flow ~* and we identify the subsystems ~a and the interconnection constraint N such that ~*ilYA~aiN is satisfied. This reverse process is called teaPing. Of course when we are using this method for ana­lysing a system, the systems ~a will have inputs. These are "inter­nal" variables in ~* but which act as true inputs to ~a: This notion is quite old in fact: it corresponds to the concept of virtuaZ dispZacements, etc •• Indeed the input to ~a which now could, in theory, be any ua~ua. will in reality in ~* be one specific UaEua.

STABILITY ANALYSIS

The method for constructing a Lyapunov function for an interconnected system is based on the following proposition:

Proposition: Assume that A is a finite set. Let ~*=cf~A ~al N be a flow and { ~a,w<l, Sa} be cyclo-dissipative for all a eA. Assume moreover that a~A w<l(ua , ya., t) = 0 whenevera~ A(ua , yo. ) E N(such interconnection constraints are said to be neutraZ with respect to the supply rates wa ). Then

S ( II xa , t) = ~ so. (xa , t) atA a6A

is a Lyapunov function for ~* on all of II Xa aeA

This proposition now leads to the following systematic proce­dure for constructing a Lyapunov function for a large scale inter­connected system ~*:

STABILITY OF LARGE SCALE INTERCONNECTED SYSTEMS 407

Step 1: (TeaPing) Identify subsystems E(l, (l E A, and an inter­connection constraint N such that E*= TT E (ll N

(leA

Step 2: (Local- Anal-ysis) Verify that { E(l ,w(l, s(l} is dissipative (resp. cyclo-dissipative) for all (l E A;

Step 3: (Gl-obal- Anal-ysis) Verify tha"1x.tAw(l 0 along N, Le., that TI (u(l , y(l ) e N implies E wa (u(l , y(l ) = 0

(l6A (leA

* Step 4: (Stabil-ity) Examine the stability properties of E using S = (l~A s(l as a Lyapunov function.

COMMENTS

1. Very roughly speaking one can state that E* should be stable if all of the {E(l, wa, S(l} are dissipative and unstable if one of them is cyclo-dissipative (with a storage function S(l without a lower bound). The exact stability properties depend, of course, on more detailed properties of EA S(land its derivative along E*. The

(lE point here is that we aPe sure in this way to obtain Lyapunov func-tion. Conceptually, this is the main difficulty. To derive stabili­ty conclusions from there is a matter of "technical details".

2.In the case of feedback systems Step 3 corresponds to verifying that wl(u, y, t) = -w2 (y, u, t). If this is satisfied, and if

{ El, wl , sl} and { E2, w2, S2} are both dissipative then E*= El xE21 feedback can be expected to be stable. Thus E* inherits the quali­tative behavior of El and E2. This is probably a (first) formali­sation of the widespread belief to the effect that "negative feed­back (wl = - w2) wiU not cause instabiUty".

3. The procedure outlined above may be modified such that it admits an analog for systems defined in input/output form.

4. A question which comes up in the above procedure and which is of obvious importance is the following: Given E and W3 does there exist a function S such that I.E, w, s} is dissipative? Much is known about this, both abstractly and for particular classes of systems (e.g., linear systems with quadratic supply rates). See, for example [2] or [6]. For time-invariant cyclo-dissipative systems there exists a function S such that by given E and w,{E , w, S} is cyclo-dissipative if and only iHf "" (t) dt ~ 0 where gi indicates that

408 J.e. WI LLEMS

the integration may be performed along an arbitrary closed path in the state space.

5. Much is also known about the relationship between input/output and Lyapunov stability. Very roughly speaking a system which is asymptotically stable in the large will be input/output stable and a system which is input/output stable and whose state space is reachable and irreducible will be asymptotically stable in the large. However, there is much more to this than meets the eye. Perhaps [7] contains the most general results in this direction.

6. Two general classes of supply rates occur very frequently in applications. The first one is w(u, y, t) =< ti, Y > and the second one is w(u,y,t) = 11u112 - II yll 2, or, more generally, w(u,y,t) =

f( Ilull ) - f( Ilyll ). If now the interconnection constraintis of the form N : II y -+ N ( II y ) then N will be neutral in the former case

aEA a aeA a if N Y is orthogonal to y (in electrical interconnections this is implied by Tellegen's theorem). It is also satisfied for the feed­

back interconnection ul=-Y2' u2 = Yl' In the second case (assuming all of the fls to be the same) N will certainly be neutral if N is a permutation of the components of aP Ya' Since interconnection constraints are usually of this type ~his situation also occurs quite frequently.

7. Note, finally, that stability will still follow if the con­di tion l: wa= 0 in Step 3 is changed to l: w'" ~ O.

a to A aEA VI.

CONNECTIVE STABILITY

The concept of connective stabiZity refers to the preservation of the stability properties of a system under a class of structural perturbations. This type of stability analysis would appear to be particularly useful and relevant in the context of large scale systems theory since it touches upon the problem of reliability and robustness which surely ought to be an important consideration in this area. A number of recent papers have appeared on this topic. The papers of Siljak [8] and of Cook [91 are examples of the kind of results which may be obtained (see also references in these papers). The Lyapunov functions and norms are usually .R-l-(or .R-oo-)

type norms and positivity conditions play also an important role.

It would take us too far to survey this work here. It is pos­sible,however, to interpret these results along the lines of the

STABILITY OF LARGE SCALE INTERCONNECTED SYSTEMS 409

stability mechanism explained above. The main point is this: assume that the interconnection constraint which was originally N is modified, as a result of structural perturbations, to N'.If it happens that (see Step 3) I wa < 0 along the new interconnection constraint N' ,then this stru~iftral ~erturbations will not influence the stability property. Similarly, if a structural perturbation takes place in one of the subsystems, but if this does not influence the dissipativeness of the subsystem involved then this will likewise not influence the stability. This procedure is particularly effec­tive in the feedback case I1 x I2 I feedback. Assuming {Il, II ulll - II ylll, Sl} and {I2, II u211 - II Y2 11 , S2} to be dissipative and that the structural perturbations do not increase the the gain of Il and I2(in a sense which can be made precise) then the dissipativeness will be preserved and the desired connective stability will result. This situation occurs frequently since structural perturbations (which usually have to do with disconnec­tions) have often the property that the gain is not increased by this perturbation. Siljak's result may indeed be interpreted in this way with the norms in question a properly choosen tl-type norm.

REFERENCES

III Willems, J.C., "Qualitative Behavior of Interconnected Systems", AnnaZs of Systems Researeh, Vol. 3, pp. 61-80, 1973

121 Willems, J.C., "Mechanisms for the Stability and Instability in Feedback Systems", Proeeedings IEEE, Vol. 64, No.1, 1976

131 Kalman, R.E., Falb, P.L., and Arbib, M.A., Topies in Mathema­tieaZ System Theopy, McGraw-Hill, 1969

141 Desoer, C.A., Notes fop a Seeond Coupse on Linear Systems, Van Nostrand-Reinhold, 1970

151 Willems, J.L., StabiZity Theopy of DynamieaZ Systems, Nelson, 1970

161 Willems, J .C., "Dissipative Dynamical Systems, Part 1: General Theory; Part II: Linear Systems with Quadratic Supply Rates", Apeh. Rat. Meeh. and AnaZysis, Vol. 45, pp. 321-351 and 352-392, 1972

171 Willems, J.C., "The Construction of Lyapunov Functions for In­put/Output Stable Systems", SIAM J. Contpot, Vol. 9, pp.105-134, 1971

410 J.e. WILLEMS

8 Siljak, D.D., "Connective Stability of Competitive Equilibrium", Automatica, Vol. 11, 1975

9 Cook, P.A., "The Stability of Interconnected Systems", Int. J. Control, Vol. 20, pp. 407-415, 1974

LARGE-SCALE SYSTEMS:

OPTIMALITY VS. RELIABILITY

D. D. Siljak and S. K. Sundareshan Department of Electrical Engineering & Computer Science University of Santa Clara Santa Clara, California

ABSTRACT

Optimal control systems may become unstable if subject to structural perturbations whereby parts of the system are disconnec­ted (and again connected) during operation. Multi-level schemes for controlling large-scale linear systems are proposed in this work, which provide a trade-off between reliability and optimality. By treating the interconnections among subsystems as perturbation terms, a two-level control strategy is developed. Local controls are used to optimize the decoupled subsystems with respect to quad­ratic costs, and global controls are applied to minimize the effects of interconnections. Although this control scheme results in a sub­optimal performance, it is inherently reliable.

1. INTRODUCTION

Intuitively, the higher the degree of cooperation among the subsystems, the higher the efficiency of the overall system. In­creased cooperation, however, means increased interdependence which, in turn, may jeopardize the functioning of the system when a number of subsystems cease to participate. Since for a proper operation of large-scale systems, it is essential that structural changes do not cause a break-down, a trade-off between optimality and reliabi­lity should be established o Therefore, the control schemes for large systems should be designed to ensure reliability despite a possible deterioration of the optimal performance 0

In this work, we propose a multilevel control scheme which in­herently incorporates the desired trade-off. Local controllers are chosen to optimize the performance of the subsystems, disregarding the interconnections among the subsystems. This strategy is shown

411

412 D.O. SILJAK AND S.K. SUNOARESHAN

to result in a reliable system when the interconnections are suita­bly limited. However, the interconnections cause a degradation of the system performance from the optimum proportional to their size. A suboptimality index is defined which measures the performance de­gradation and represents the trade-off between optimality and reli­ability. By introducing additional global controls to minimize the effect of interconnections, the performance degradation is reduced without upsetting reliability of the system. These controls are implemented by a controller on a higher hierarchial level that re­ceives the information about the subsystem states. Even if some states are not available to the global controller, a partial neutra­lization of the interconnections can still be achieved.

This paper introduces the concept of structural perturbations [1,2] in the multilevel optimal control. This necessitates treat­ing the interconnections as perturbation terms, thus, ignoring their possible beneficial effects. Such an approach, however, opens a real possibility for treating nonlinearities in interconnections as well as in subsystems [3]. The suboptimality analysis conducted here uses the results of Popov [4] and Rissanen Is] obtained in the context of perturbed optimal systems. A similar approach has also been used by Bayley and Ramapriyan I6], and Weissenberger [7]. Neu­tralization of interconnection effects by the global control is mo­tivated by the work of Johnson I8] concerning disturbance rejection in linear systems.

Reliability of optimal control can be of concern even when the systems are not composed of interconnected subsystems, as pointed out by Rosenbrock [9]. However, the reliability aspects of control with regard to structural perturbations [1, 2] which are of interest in this work, are different from those considered in [9], which arise from such considerations as changes in loop gains.

The proposed approach to reliability of optimal systems is a part of the general theory of multilevel control for large-scale systems. In addition to the hierarchic optimization [3], the theory is concerned with the problems of decentralization, stabilization and estimation of large-scale dynamic systems [10-13].

2. STRUCTURAL PERTURBATIONS: AN EXAMPLE

A motivation for the subsequent analysis is provided by the following simple example: We are given a second-order linear sys­tem

SXl - 3x2 + u l -4xl + 6x2 + u2

with the quadratic performance index 00

J = f o

The optimal control

(1 )

(2)

OPTIMALITY VS. RELIABILITY 413

o u l = -lO.49xl + 7.l4xZ

o Uz = 7.l4xl - 11.18xZ (3)

which minimizes the index J is easily found using the quadratic regulator theory [14]. The resulting optimal closed-loop system is obtained as

xl = -5.49xl + 4.l4xZ

Xz = 3.l4xl - 5.l8x2 (4)

which is asymptotically stable. Now, the equations (1) may describe two distinct parts of the

system governed by equations . xl = 5xl + ul

x2 = 6x2 + Uz (5)

and -3x2 and -4xl may be considered as interconnections. If

during the operation of the overall system (1), disconnection of the subsystems is an expected possibility, then the application of the computed optimal control is not appropriate. When the inter­connections are removed, the use of control (3) for the disjoint subsystems (5) results into a closed-loop system

xl = -5 0 49xl + 7.l4x2

(6)

which is unstable. Therefore, if a system is expected to undergo structural perturbations, the standard optimization techniques do not provide a satisfactory solution to the control problem. In this case, we propose an alternative approach which produces a re­liable overall control system that is stable under arbitrary struc­tural perturbations. The cost involved in the approach is that the resulting system is suboptimal with respect to the specified per­formance index.

To incorporate the structural perturbations, we rewrite the original system (1) as

xl 5xl - 3e12x2 + ul

(7)

where e12 and eZl are interconnection parameters which take on

values 1 or 0 depending on whether the interconnections are pre­sent or not. When e12 = e2l = 0 , the subsystems in (7) are de-

coupled and described by (5). With each decoupled subsystem, we associate a performance index,

414 D.O. SILJAK AND S.K. SUNOARESHAN

00 00

(8)

o o We proceed to compute local optimal control for each decoupled sub­system in (5) with respect to performance indicies in (8), using the standard linear regulator solution. This gives the local opti-mal controls as

(9)

* * resulting in the optimal costs J l and J 2 • * * Controls ul ' u2 when substituted in (7), produce the system

(10)

which is stable for all (binary) values of the interconnection para­meters e12 , e2l • However, this reliability of the system (10) is

achieved at the price of deterioration in the system performance. * * With con~rols ul and u2 ' the system (7) cannot attain the opti-

* * * mal value J = J l + J 2 due to presence of the interconnections. The number

* E =

J - J *

= 0.85 (ll) J

can serve as a suboptimality index for the system under structural perturbations, where j is the actual value of the performance in­dex for the overall system (10) when e12 = e2l = 1 •

It is possible to improve the obtained result and decrease the suboptimality index E by introducing the global control functions. The role of these functions is to minimize the interconnection ef­fect among the subsystems. Such a choice of global control produces a value of the performance index that is closer to the optimum J*, without upsetting the reliability of the control. In the simple case under consideration, it is actually possible to choose global control so as to neutralize the interconnections and obtain exactly J* .

We consider the control functions posed of two components,

U~ + ug U - u~ + ug u l 1 l' 2 - 2 2

and

~ ~ ~* * where u l ' u2 are local controls chosen as u l u l in (9). The global control components we select as

ug = 4x 2 1

in (7) as com-

(12)

(13)

OPTIMALITY VS. RELIABILITY 415

Therefore, the control functions in (7) are

ul = -lO.lOxl + 3e12x2 , u2 = 4e2lxl - l2.0Sx2 (14)

and the closed-loop system is

(15)

From (15), we see that the subsystems are decoupled and~ therefore, the system is optimal with the value J* and the suboptimality in­dex E = 0 •

The above example has provided a motivation to develop a mul­tilevel optimization scheme for large systems composed of intercon­nected subsystems. The proposed procedures produce control func­tions which result in systems with a satisfactory suboptimal per­formance and. at the same time. guarantee reliable operation under structural perturbations.

3. CONNECTIVE SUBOPTIMALITY

Let us consider a continuous linear dynamic system

x = Ax + Bu • (16)

where x(t) e Rn is the state of the system and u(t) 6 Rm is the control. In (16), A = A(t) and B = B(t) are n x nand n x m matrices continuous in t on the interval T = (T, + 00) where T is a number or the symbol _00

We assume that the system (16) can be decomposed into s inter­connected subsystems described by

s Xl = A.x. + B.u. + I e .. A .. x.

1 1 1 1 j=l 1J 1J J i = 1, 2, •.. , s (17)

n. where x.(t) e R 1

1

is the control, and

matrices of proper

uous functions in

is the state

A. = A. (t), 1 1

dimensions.

t such that

m. of the i-th subsystem, u. (t) e R 1

1

B. = B. (t) • and A .. = A .. (t) are 1 1 1J 1J

In (17) • e .. = e .. (t) are contin-1J 1J

e .. e [0, 1] for all t e T . The 1J

functions e.. are 1J

elements of the s x s interconnection matrix

E = (e .. ) 1J

[1, 2].

In the proposed multilevel optimization scheme, u. (t) is considered as consisting of two parts, the

1

u~(t) and the global control u~(t) , 1 1

u. (t) = u~(t) + u~(t) . 111

the control local control

(18)

416

The local control

R. R. u. = -K.x. l. l. l.

R. u. (t) l.

D.O. SILJAK AND S.K. SUNDARESHAN

is chosen as a linear control law

(19)

to optimize isolated subsystems, and the global control law s

u? = -l. I

j=l e .. K?x.

l.J l.J J (20)

to minimize degradation of performance due to interconnections among the subsystems.

When E = 0 , the interconnected system (17) breaks down into s decoupled subsystems

R. x. = A.x. + B.u. , i = 1, 2, , s . (21) l. l. l. l. l.

We assume that all s pairs (A., B.) are completely controllable, l. l.

and that with each isolated subsystem (21) a quadratic performance index

00

J i (rO' xiO ' u1) = f to

(22)

is associated. In (22), Q. = Q.(t) is an n. x n. symmetric, l. l. l. l.

nonnegative definite matrix,

positive definite matrix.

The local control u~(t) l.

R. = R.(t) is an m. x m. symmetric, l. l. l. l.

in (19), can now be chosen to mini­R. mize the performance index Ji(tO' xiO ' ui ) in (22). From linear

R.* regulator theory [14] the optimal control u. is given by (19) where l.

K~ -1 T (23) = R. B.P. l. l. l. l.

In (23) , P. = P. (t) is an n. x n. symmetric, positive definite l. l. l. l.

matrix which is the solution of the Riccati equation

P. + P.A. T -1 T = 0 (24) + A.P. P.B.R. B.P. + Q. . l. l. l. l. l. l. l. l. l. l. l.

The optimal cost is

J: (to' xiO) = IlxiOII;. (25) l. T

Under the assumption that Q. can be factored as Q. = C.C. , l. l. l. l.

where C. is an n. x n. constant matrix, so that the pair CA., l. l. l. l.

Ci ) is completely observable, each closed-loop subsystem

OPTIMALITY VS. RELIABILITY 417

• -1 T x. = (A.- B.R. B.P.)x., i-I, 2, .•. , S 1 111 111

(26)

is globally asymptotically stable. R.* The control u. (t) , i = I, 2, ••• , s , will not generally 1

be optimal for the composite system (17), and it would not result in the optimal cost

(27)

R.* unless E = O. When E f 0 , the controls u. (t) produce the 1

value of the performance index for the composite system given as

where

s j(to' xO) = L Ji(to' xiO) ,

i=l (28)

(29)

It is obvious that

j(to' xo) ~ J*(to' xo) , Veto' xo) e T x Rn (30)

R.* . and the local control law ui ' 1 = 1, 2, •.• , s , can only be sub-

optimal for the composite system (17), with an index of suboptimal­ity E > 0 defined by

J(to ' xO) ~.(l+E) J*(to' xO), Veto' xo) e T x Rn (31)

From (30) and (31) , it follows that the index E for the com-posite system

R. s . (A. - L e .. (A .. - B.K~.)x. x. = B.K.)x. +

1 1 111 j=l 1J 1J 1 1J J

i = 1, 2, ... , s (32)

s depends on the size of the effective interconnections L e .. (A .. -

j=l 1J 1J

Motivated by the example given in the preceeding sec-B. K? .)x. 1 1J J

tion, we would like to design a system for which the suboptimality index is invariant under structural perturbations. That is, we are interested in the following:

Definitiono The system (32) is said to be connectively suboptimal with index E if there exists a number E > 0 for which inequal­ity (31) is satisfied for all interconnection matrices E(t) .

418 D.O. SILJAK AND S.K. SUNOARESHAN

To establish the property of the system (32) as specified by

the above Definition, we can choose the gain matrices K~ and K? R- 1 1J

of the local and global control. The gain matrices K. are selec-1

ted by the standard quadratic optimization scheme applied to the decoupled subsystems as in (26). The gain matrices K? are cho-

1J sen to minimize the effect of interconnections among the subsystems and, thus, produce the lowest value of the suboptimality index E. In doing so, we arrive at the closed-loop composite system which is connectively stable with invariant suboptimality index under struc­tural perturbations.

4. A CRITERION FOR CONNECTIVE SUBOPTIMALITY

Let us recall the meaning of the fundamental interconnection s x s matrix E = (e .. ) [1, 2], which has constant binary elements

1J e.. equal to 1 when the j-th subsystem acts on the i-th subsys-

1J tern, and it is 0 when the j-th subsystem does not act on the i-th subsystem. Therefore, any interconnection matrix E(t) can be generated from the fundamental interconnection matrix E by re­placing the unit elements e .. of E by the elements e .. (t) of

1J 1J E(t) • The suboptimality conditions are derived for the fundamen­tal interconnection matrix E, but are valid for all interconnec­tion matrices E(t) , thus establishing the connective suboptimal­ity of the system.

Now, we prove the following:

Theorem. The system (32) is connectively suboptimal with index E if

min A [W. (t)] s s _ I i m 1

I I E Vt € T (33) e .. ~ .. < '2 I+E AM[P i (t)] ,

i=l j=l 1J 1J - max i

where ~ T H .. (t)] and the matrices H .. (t) and ~ .. = AM[H .. (t) , 1J 1J 1J 1J

W. (t) 1

are given as

g H .. = A .. - B.K .. ,

1J 1J 1 1J

-I T W. = P.B.R. B.P. + Q. 1 111 111

(34)

and Am' AM are the minimum and the maximum eigenvalues of the re­

spective matrices at time t

P:oof. Since the decou~l~d s~bsystems hons V . (t, x.) = II x. I I P , 1 = 1, 2,

1 1 1.

ally the Hamilton-Jacobi equations

(26) are optimal, the func­.•. , s , satisfy individu-

OPTIMALITY VS. RELIABILITY

aV.(t, x.) 1 1

at T R, + [grad V.(t, x.)] [(A.- B.K.)x.]

1 1 1 1 1 1

IIK~X.IIR2 , 11.

1 n.

419

Vet, Xi) € T x R 1, i = 1, 2, ••• , s (35)

Now, the total time derivative V.(t, x.) can be calculated n. 1 1

along the trajectories x.(t) e R 1 of the composite system (32) 1

for the fundamental interconnection matrix E as

avo (t, x.) T R, V.(t, x.) = 1 at 1 + [grad V.(t, x.)] [(A.- B.K.)x.]

1 1 1 1 1111

S

+ L e .. H .. x. j=l 1) 1) )

(36)

By substituting (36) into (35) and rearranging the terms, we get

Ilx.ll w2 = - (l+e:) V. (t, x.) 1 . 1 1

1

s + (l+e:)[gradV.(t, x.)]T L e .. H .. x.- e:llx1.ll w2.(37)

1 1 j =1 1) 1) ) 1

T T T T where x = (Xl' x2' •.• ,xs) .

By summing equations (37) and integrating from to to ~,we obtain

* j(to' xO) = (l+e:) J (to' xO) + ~

s s L {[grad V. (t, x.)]T L e .. H .. x.- -Ie: Ilx.llw2 }

i=l 1 1 j =1 1) 1)) +e: 1 i

* where j and J are defined in (38)

(28) and (27). We use the inequalities

IIH .. i.11 < ~··lli·11 , 1)) - 1) )

(39)

and s

II [grad V. (t, i.)]T L ~ .. H .. i.11 < 1 1 j=l 1) 1) )

s 2AM (P1' ) L e .. ~ .. II X. 112 ,

j=l 1) 1) ) (40)

together with the inequality (33) of the Theorem, to establish in-

420 D.O. SILJAK AND S.K. SUNDARESHAN

equality (31) from (38). Since E(t) < E element-by-element for all t e T , inequality (31) holds for-all interconnection matrices E(t) and the system (32) is connectively suboptimal with index E •

The proof of the Theorem is complete.

The proof of the above Theorem is based upon a result obtained by Rissanen [5] in the context of perturbations of optimal systems.

As demonstrated by the example of Section 2, a system optim­ized by local controls in a decentralized scheme, remains stable despite the structural perturbations whereby certain parts of the system may be disconnected (or connected again) during the opera­tion of the system. That this is indeed true for the general case of system (32), when the main inequality (33) of the theorem is satisfied, we can show by choosing the function

s s v (t, x) I V.(t, x.) = I

i=l 1 1 i=l (41)

as a Liapunov function. Since each P. is a positive definite solution of the Riccati equation (24),1 Vet, x) is a positive defi­nite quadratic form. Taking the total time derivative of Vet, x) along the trajectories of the system (32) for the case E(t) = E , and using (33) and (39), we get

s Ilxill~.

s -T

s V (t, x) I + 2 I x.P. I e .. H .. x. < 0

1 1 1J 1J J i=l 1 i=l j=l

Vet, x) 6 T x Rn (42)

Therefore, the equilibrium x = 0 of the system (32) for Eet) = E is asymptotically stable in the large. Since E(t) < E element-by­element, the equilibrium x = 0 is asymptotically stable in the large for all interconnection matrices E(t) , that is, it is con­nectively asymptotically stable in the large [1, 2].

So far, we did not comment on the role of the global controls u?(t) except to modify the original interconnections A .. x. to

1 1J J obtain the effective interconnections (A .. - B.K?)x. It is ob-

1J 1 1J J vious that when the original interconnections A .. x. satisfy the

1J J inequality (33), then the connective suboptimality with index E is achieved by local controls u~(t) only (corresponding to the triv-

1

ial case K? = 0 ; i, j = 1, 2, 1J

The role of the global controls

, s ).

u~(t) 1

is to reduce the effect

of interconnections o This is useful when the size of the existing interconnections is such that no value of E can be found to satis­fy inequality (33). Even in such cases when an appropriate value of E could be found, the global control can be implemented to re­duce the value of E and, hence, the deterioration of the optimal performance.

OPTIMALITY VS. RELIABILITY

From inequality (33), we see that the index of the norm ~.. of the matrix H .. = A .. - B.K¥.

lJ lJ t lJ 1 lJ To minimize ~ .. , we choose K~. = B.A.. where

lJ lJ 1 lJ

421

E is a function defined in (34).

B: is the Moore-1

Penrose generalized inverse of B. [15]. When rank 1

(B~B.)-lB~ and

B. 1

1 1 1

In the IA .. ]

lJ choice

Kg .. = T )-1 T (B. B. B.A .. lJ 1 1 1 lJ

( 43)

particular case, when the rank of the composite matrix [B. is equal to the rank of the matrix B. itself, then the 1

1

(43) of K~. produces H .. = O. That means, the effective lJ lJ

interconnection from the j-th subsystem to the i-th subsystem in the closed-loop optimized system, is nullified. A similar idea for choos­ing control functions of the form (43) has been used by Johnson [8] in a different context of disturbance rejection in linear systems.

Apparently, our choice of K~. in (43) has been made because in the framework of our optimizatt6n approach, the interactions among the subsystems are regarded as perturbations which only de­teriorate the performance of the overall system. Such a choice of the global control excludes the possibility of using interactions as beneficial in improving the performance of the system. However, this sacrifice is made to retain the trade-off between optimality and reliability, and ensure a degree of suboptimal performance that is invariant under structural perturbations.

We should note that the use of global control presumes the availability of the states of the interacting subsystems at the corresponding subsystem. This may seem to be an unjustifiable re­quirement since availability of the subsystem states may encourage a centralized controller to optimize directly the overall system (16). However, as demonstrated by the example in Section 2, the centralized optimization scheme does not guarantee a reliable con­trol which is of principal importance in the control problem treated here. Furthermore, despite the high efficiency of modern computing machinery, the complexity of systems with large number of variables makes a centralized optimization unattractive even with the most re­fined computational techniques.

To illustrate the structural aspect of the Theorem outlined in this section, let us consider a system of equations

( 44)

which describe the motion of two identical pendulums coupled by a spring [14] subject to two distinct inputs as shown in the figure.

We consider the case when the position aCt) of the spring can change along the full length of the pendulums, that is 0 ~ aCt) < t

422 D.O. SILJAK AND S.K. SUNOARESHAN

for all t. The problem of interest is to optimally regulate the system (44) for any unspecified function a(t). In view of the un­specified nature of the function

Interconnected Pendulums

aCt) , the standard "one shot" optimization of the system is diffi­cult to apply. Furthermore, if optimization is conducted for any particular position of the spring, the computed optimal control may be inappropriate for another position. This problem can be resolved by using the proposed multilevel control scheme.

Choosing the state vector as x = e81, 91, 82, e / the equa-tions (44) can be rewritten as 2

0 1 0 0 0 0

- ,[ - ka2 0

ka2 0

1 0 R, mR,2 mR,2 - mR,2 .

x = x + u , 0 0 0 1 0 0

ka2 0

g ka2 0 0

1 -2 - "[ - -2 - -2 mR, mR, mR, (45)

where = (ul , T System (45) can be decomposed into sub-u u2) two

systems described by

[: :] ~ 0 l l: :] [: : ] x, . xl = xl + ul + eU xl + e12 _-s _

[: :J + [-:] [: :l [: 0] *2 = x2 u2 + e21 xl + e22 X2

o (46)

OPTIMALITY VS. RELIABILITY 423

• T • T 2· where xl = (61 , 61) , x2 = (6 2, 62) , a = g/i, ~ = l/mi , y = kim,

2 2 and ell(t) = e12 (t) = e2l (t) = e 22 (t) = a (t)/i •

We note that 0 < e .. (t) < 1 for all t and i, j = I, 2, - 1J -

and the disconnection of the two subsystems occurs when aCt) = 0 , that is, e .. (t) = O. In that case, the spring is moved all the

1J way to the support, and two pendulums shown on the figure, are de-coupled from each other.

The decoupled pendulums (subsystems) are optimized with respect to the costs specified by Ql = Q2 = I 2x2 ' Rl = R2 = 1 , by the lo­cal controls

i i ul = -~[P12 P22]xl , u2 = -~[P12 P22]x2 (47)

1 2 2 !.: I!.: where p = - [(a + Q ) 2 _ a.] , and (1 2 ) 2 12 62 I-' P22 = S + P12 ' are elements of the 2 x 2 symmetric positive definite solutions PI' P2(PI = P2) , of the Riccati equations.

Due to the presence of interconnections, the computed local controls are suboptimal with index g given by the inequality (33) as

Vt € T (48)

where WI (WI = W2) is calculated from (34).

Now, by choosing the global controls

ug 1 O]x 1 S [-yell 0 ye12

ug 1 [ ye2l 0 O]x (49) 2 - S -ye22

We can reduce the suboptimality index to zero. The global controls (49) together with the local controls (47), result in a two-level control strategy that optimizes the system with respect to the cho­sen performance indices, and at the same time preserves reliability under structural perturbations caused by the changes in the position of the spring that couples the two pendulums.

That the decentralized optimization produces systems with high tolerance to nonlinearities in the interactions among the subsystems, was first observed by Weissenberger [7]. In the context of the pro­posed suboptimal design, that property of decentrally optimal sys­tems was established in [3]. To see this, let us consider an ob­vious modification of system (17).

s x. = A.x. + B.u. + I e .. h .. (t, x), i = I, 2,

1 1 1 1 1 j=l 1J 1J . .. , s

(50) n.

where the interconnection functions h .. : T x Rn + R 1 are defined, 1J

424 D.O. SILJAK AND S.K. SUNDARESHAN

d b d d T X Rn, d f h . continuous, an oun e on an satis y t e inequalit1es

II h. . (t , x) II < t;... II x. II, V ( t , x) 6 T x Rn 1J - 1J J

i, j = 1, 2, •.• , s

where t;... are nonnegative numbers. 1J

(51)

By replacing inequality (39) by (51), the following Corollary to the above Theorem is automatic:

Corollary: The system (50) is connectively suboptimal with index £ if the numbers t;. .• defined by (51) satisfy inequality (33).

1J

In the context of the two pendulums of the figure, we can in­clude nonlinearity of the spring and consider a modification of equations (46) according to (50), where k is replaced by k(t, x). Then, the condition

Ik(t, x)1 ~ my, Vet, x) € T x Rn (52)

and inequality (48) guarantee suboptimality of the system with the same suboptimality index £ as in the case of the linear model. It should be noted that we do not need to know the actual shape of the function k(t, x) • but only the constraint on its size. This is another important property of the proposed suboptimal control which should be added to reliability when it is contrasted to the conventional optimal control.

5. CONCLUSIONS

When a system is expected to undergo structural perturbations, the control policies obtained by standard optimization procedures, may not result in a satisfactory performance. In this situation, a trade-off between reliability and optimality is a meaningful proposition, and an acceptable control strategy should ensure re­liability even at the cost of a suboptimal performance. A multi­level control scheme has been developed, which accomplishes this trade-off.

REFERENCES

v 1. Siljak, D. D., "Stability of Large-Scale Systems", Proc. Fifth

IFAC Congr., Paris, 4(1972), C-32: 1-11 . ., 2. Siljak, D. D., "On Stability of Large-Scale Systems Under Struc­

tural Perturbations:, IEEE Trans o , SMC-3(1973), 415-417. y

3. Siljak, Do D., and M. K. Sundareshan, "On Hierarchic Optimal Control of Large-Scale Systems", Proc. Eighth Asilomar Conf., Pacific Grove, Calif., (1974), 495-502.

4. Popov, V. M., "Criterion of Quality for Non-Linear Controlled

OPTIMALITY VS. RELIABILITY 425

Systems", Proc. First IFAC Congr., Moscow, 1(1960), 173-176.

5. Rissanen, J. J., "Performance Deterioration of Optimum Systems", IEEE Trans., AC-ll(1966), 530-532.

6. Bailey, F. N., and H. K. Ramapriyan, "Bounds on Suboptimality in the Control of Linear Dynamic Systems", IEEE Trans., AC-18 (1973), 532-534.

7. Weissenberger, S., "Tolerance of Decentrally Optimal Control­lers to Nonl ineari ty and Coupling", Twelfth Allerton Conf., Monticello, IlL, (1974), 87-95.

8. Johnson, C. D., "Accommodation of External Disturbances in Linear Regulator and Servomechanism Problems", IEEE Trans., AC-16(197l), 635-644 0

9. Rosenbrock, H. H., "Good, Bad, or Optimal", IEEE Trans., AC-16 (1971), 552-554.

10. Siljak, D. D., and M. B. Vukcevic, "On Hierarchic Stabilization of Large-Scale Linear Systems", Proc. Eighth Asilomar Conf. Circuits, Systems, Computers, Pacific Grove, Calif., 1974, 503-507.

11. Silj ak, D. D., "Stabi lization of Large-Scale Systems: A Spin­ning Flexible Spacecraft", Proc. Sixth IFAC Congr., Boston, Mass., 1975, 35-1: 1-10.

12. Siljak, D. D., and M. B. Vukcevi~, "Large-Scale Systems: Stabi­lity, Complexity, Reliability", IEEE Proc o, Special Issue on Recent Advances in System Theory, Edited by W. A. Porter, 1975 (to appear) . .,

13. Siljak, D. D., and M. B. Vukcevic, "Multilevel Control of Large­Scale Systems: Decentralization, Stabilization, Estimation, and Reliability", Recent Advances in Large-Scale Systems, Edited by R. Saeks, Point Lobos Press, 1975 (to appear).

14. Bryson, A. E., and Y. C. Ho, "Applied Optimal Control", Blais­dell, Waltham, Masso, 1969.

15. Langenhop, C. E., "On Generalized Inverses of Matrices", SIAM J o Appl. Math., 15(1967), 1239-1246.

LIST OF PARTICIPANTS

Masanao AOKI Department of System Science University of California Los Angeles, California

K. J. ASTROM Department of Automatic Control Lund Institute of Technology S-220 07 Lund 7, Sweden

Jean-Pierre AUBIN Universite Paris - IX Dauphine 75775 - Paris Cedex 16, France

A. BLAQUIERE Laboratoire d'Automatique Theorique Tour 14, Universite de Paris 7 2 Place Jussieu, 75005 Paris, France

Kai-ching CHU IBM Thomas J. Watson Research Center Post Office Box 218 Yorktown Heights, New York 10598

S. CLEMHOUT, G. LEITMANN, and H. Y. WAN, Jr. Department of Mechanical Engineering University of California Berkeley, California 94720

J. B. CRUZ, Jr. Coordinated Science Laboratory University of Illinois Urbana, Illinois 61801

427

428

Edward J. DAVISON Department of Electrical Engineering University of Toronto Toronto, Canada

R. F. DRENICK Polytechnic Institute of New York 333 Jay Street Brooklyn, New York 11201

Theodore GROVES

PARTICIPANTS

Center for Mathematical Studies in Economics and Management Sciences

Northwestern University Evanston, Illinois 61201

G. GUARDABASSI and N. SCHIAVONI Istituto di Elettrotecnica ed Elettronica Politecnico di Milano Milan, Italy

D. H. JACOBSON National Research Institute for Mathematical Sciences Pretoria, South Africa

Roman KULIKOWSKI Institute of Organization and Management Polish Academy of Sciences Warszawa Palac Kultury, Poland

Jiguan G. LIN DepartITlent of Electrical Engineering and

Computer Science ColuITlbia University New York, New York 10027

A. G. J. MACFARLANE Engineering Department University of CaITlbridge CaITlbridge, England

D.Q. MAYNE DepartITlent of COITlputing and Control IITlpe rial College of Science and Technology London SW7 2BZ, England

PARTICIPANTS

Geert Jan OLSDER Twente University of Technology Post Office Bos 217 Enschede, The Netherlands

E. POLAK and A. N. PAYNE Department of Electrical Engineering and Computer

Sciences and the Electronics Research Laboratory University of California Berkeley, California 94720

Nils R. SANDELL, Jr. Electronics Systems Laboratory Room 35-213 M.1. T. Cambridge, Massachusetts 02139

D. D, SILJAK and S. K. SUNDARESHAN Department of Electrical Engineering and

Computer Science University of Santa Clara Santa Clara, California 95053

J. H. WESTCOTT Department of Computing and Control Imperial College London SW7, England

Jan C. WILLEMS Mathematical Institute University of Groningen Groningen, The Netherlands

H. S. WITSENHAUSEN Bell Telephone Laboratories Murray Hill, New Jersey 07974

L.A. ZADEH Division of Cornputer Science Department of Electrical Engineering and

Computer Sciences Univers ity of California Berkeley, California 94720

429

INDEX

Advertising models, 187 Agents, interacting, 41 Aggregate;

demand, 254 dis-, 261 supply, 254 supply hypothesis, 43

Aggregation, 87, 259, 261 Allocation, 266 Analysis;

local, 407 global, 407

Asymptotically stable, 413, 390

Balance of payments, 257 equilibrium, 253, 254

Bankrupt-policy, 192 Bargaining process, 207 Bilinear system;

homogeneous -in-the­input, 389

Bochner, theorem of, 61 Canonical form;

block observable, 257 Channel, discrete noisy, 282 Channel capacity, 37 Classical single-variable

feedback theory, 325 Coalitions, 95, 97, 98, 100,

187, 371 fuzzy, 111

Competitive economy, 18 Constant, quasi-, 235 Constraint;

interconnection, proper equality, proper inequality,

405 118

118

Consumption, 254 Control;

centralized, 313 chattering, 235 decentralized structure, 11,

313, 303 hierarchical, 139, 279 hierarchy, 145 organizational, 149

Controller, bang-bang, 396 Coordinating signal, 141 Coordination, 140 Co-ordinator, 19 "Core", 118,198 Coupling, weak, 3 Cross-over point, 343 Data networks, packet switching,

9 Decentralization, 304, 412 De ce ntrali zed;

organizations, 149 regulation problem, 26

Decomposition, 18 Desoer, C. A., 257 Development, 265

decentralized management of, 266

Differential games, 108, 187 Leitmann-Liu model, 211 n-person, general sum, 207 permanent, 235

Diplomacy, 95, 105 fuzzy, 106

Directional extremum, 118 Directional shadow, 118 Displacements, virtual, 406 Dissipation inequality, 404

431

432

Disturbances; global, 43 local, 43

Dynamic dominance, 235 Economic growth, 253, 254 Economics;

macro-, micro-,

Eigenanalysis, Equilibrium;

Nash, Pareto, playable,

98

42 42

10

112 207

198 Essential game, Estimation;

simultaneous, 253, 255 Astrtlm-Bohlin technique,

255 Fixed modes, 304 Flows, 403 Freeway traffic corridor, 7 Function;

aggregation, 86 algebraic, 326 characteristic, 326 characteristic branch of,

reward, storage, transfer,

Fuzzy;

328 281 404

325

core, 371 logic, 341 Pa reto -optimal set, 356 restrictions, 342 set, 339

Games, 371 coalition, 232 fuzzy, 372 history-dependent, 232 multi-state, 232 sequential, 232 theory of, 71

Grammar, context-free, 350 Hamilton-Jacobi-Bellman

Equation, 393, 418 Hierarchial level, 412 Hierarchial optimization, 412 Hierarchial organization, 19

INDEX

"Imputations" , 118 Incentive compatible control, 149 Information, 71

complete, 28 imperfect, 42 null, 28 optimal-sufficient-, 33 partial, decentralization, 29 -pools, 29 structure, 25 value of transmis s ion, 28

Information pattern, 2 Information theory, 282 Input-Output;

conversion, 281 description, 69 mapping, 70

Interactions; dynamic, 30 indices, 307 performance, 30

Interconnected systems, stability of, 21

Interconnection matrices; E .. , 320

Interc~nnections, 411 Interface design, 1 Inverse, Moore-Penrose gener-

alized, 421 Investment, 254 Labor exchanges, 44 Leader-follower, 139 Lexicog raphic orde ring of the

criteria, 78 Liapunov function, 420 Liapunov theorem, 390 Linear-Quadratic-Gaussin, 25

253, 390 Linear regulator theory, 416 Linguistic approach, 339 Lyapunov function, 10, 403 Management-labor negotiations,

207 Markets, local, 51 Marshallian model, 42 Maximum likelihood state, 46 Mayne, D. Q., 257 Membership function, 343 Microeconomic, most probable

state, 46

INDEX

Mixing, ideal, 58 Periodicity constraint, 235 Model; Perturbations;

"Zeuthen-Hicks", 210 singular, 143 "Zeuthen-Nash", 210 structural, 411

Multicriteria decision problem; Playable strategies, 213 infinite alternatives, 84 Pole assignment; large, but finite, 80 decentralized, 304, 305 "small", 78 Polynomial;

Multicriteria optimization, 77, pole, 326, 325 18, 354, 117 zero, 326

algorithm, 81 Price stability, 253, 254 Multivariable; Processes, industrial and

feedback theory, 325 biological, 57 frequency-response design Processing times, 282

methods, 325 input-, 283 poles and zeros, 325 output-, 283,

Nash equilibrium; Product, max-min, non-cooperative, 207, Production factors,

174, 188 Proposition;

284 353 266

Nerode, 70 disjunctive, 346 Nonlinear system, homogeneous - conjunctive, 346

in-the-input, 389 ljJ-vector, 284

433

Normal structural form, 257 Psychology, experimental, 283 Nyquist-Bode design; Pure transport or plug flow, 58

generalized, 334 q-directionally convex, 122 Nyquist diagrams, 325 Quadratic regulator theory, 413 Observable, completely, 416 Realization; "Observer", 261 minimal state-space, 257 Off-line design problems, 35 observable pair, 257 Optimal control, 188, 393 Regulation, 303

multilevel, 412 Relational assignment equation, Optimal decision; 343

linear rules, 18 Reliability, 19, 411 Optimality, 411 Residence time distribution, 60

for non-linear systems, 19 Return-difference matrix, 333 fuzzy, 102, 104, 105 Riccati equation, 416

Ordering problem, 72 Riemann surface, 328 Organization; Root locus methods, 325

centralized, 288 Rules; chart, 285 enforcement, 149 decentralized, 288, 296 operating, 149 mathematical theory of, Scalarization, linear, 118

279 Scheme; member, 281 centralized optimization,

p-directionally convex, 121 421 Parameter, imbedding, 143 multi-level, 411 Pareto-optimal, 117,208,210 multi-level control, 411

fuzzy set, 339 Search model, 44 Partitions, coalitive, 97, 98, Self-fulfilling expectation, 42

100, 101, 108 Self-organizing, 36 Performance-relevant, 33 Self-regulating, 36

434

Separation theorem on convex sets, 121

Sequential, 72 Series connection, 61 Series expansion;

Laurent, 328 Servomechanism problem, 304

decentralized robust, 306 Sets, fuzzy, 102, 103 Smith-McMillan form, 332 Stabilization, decentralized, 303,

305 Stability, 410, 407

connective, 408 input/ output, 408 matrix, 392

Stabilization, 389, 412 Stabilizing controlle rs, 393 State, 69 State -space models, 325 Steady-state, 235 Stewart-Hamilton equation, 58,

66 Strategy;

collective, 114 feedback Stackelberg, 142 hierachical M-level, 143 S tackelbe rg, 139 Stackelberg equilibrium,

142 Strike, 207 Suboptimality;

connective, 415 index, 412

Subs ystems ; identical, 57 interconnecting identical,

57 "neighboring", 31

Supply rate, 404 System, decentralized dynamic,

25 Systems;

cyclo-diss ipative dynami-cal, 404

dissipative, 402 dissipative dynamical, 404 flow. 58, 59 inertial navigation, 3 interconnected, 401 large scale, 303 multi variable, 17

INDEX

Systems (cont. ) non-homogeneous, 397 open flow, 58, 59 power, 5

Tax; -able income, 261 personal income, 261 rates, 261 yield, 266

Team theory, 18, 25, 279 "Tearing", 402, 407 Trade, 253 Unemployment, 253, 254 Values;

acceptable performance, 77 decreasing marginal, 32 ideal, 90 noninferior, 77

Variables; decision, 77 intervening, 282 linquistic, 341 macroeconomic, 41 microeconomic, 41 primary-performance, 282 ps ychological, 284 secondary, 282

Vector; eN -linearly maximal, 124 eN -linearly supremal, 124 eN -sectionally maximal,

126 eN -sectionally supremal,

126 maximal, 118

Vector valued performance index, 236

Wages. 44 Worker, 44 World trade index, 255 Zero memory, 27 Zeros, 325