-----------------

INTERNATIONAL SYMPOSIUM

ON

OPERATOR THEORY OF NETWORKS

AND SYSTEMS

Volume I

AUGUST 12-14, 1975

MONTREAL

Copyright

~b ) 6 [t7)') M(J', y\

©1976 by Western Periodicals Company

13000 Raymer Street

North Hollywood, California 91605

j ....... .. '<~ '}.~

.f

PREFACE

TIle enclosed Dapers constitute ones discussed at the oDcrator Theory of

Networks and Systems International Symposium, OTNS, held at Concordia

University, Montreal, Canada, August 12-14, 1975. On behalf of the audience

wishing contact with these ideas I wish to thank the authors for ~~ing therrl

available and Western Periodicals for publishing them. The OTNS Symposium has been set up to bring together researchers Horkinr;

in the area for cross-fertilization and codification of results. By its nature

the field is one where both engineers and mathematicians have made extensive

contributions. Consequently, we have been happy to see that the OTNS Symposium

has positively contributed to fertile interactions between the two disciplines

and we hope such will continue within the OTNS structure now established.

The success of the Montreal OTNS Symposium is owed to a number of people.

Primary am:mg these is Professor N. Levan of UClA who as Program Chairman set

the tone of the meeting. Besides him I would like to publicly thank \v. Porter,

R. M. De Santis, V. Ramachandran, R. Saeks, M. N. S. SI.Jamv, ,T. Baras, 11. DeClaris,

as well as my Co-Chairman A. Zernanian, all ow whom contributed to the administra-

tion of the Symposium. We also appreciated the institutional support of our Co-Sponsors:

1. Department of Applied Mathematics and Statistics, SUNY, Stony Brook,

New York. 2. System Science Department, UCLA, Los Angeles, California.

3. Electrical Engineering Department, Concordia Universitv, Montreal,

Canada. I look forward to seeing a renewal of these activities at the next OTNS

Symposium presently being planned for summer 1977.

III

R. H. Newcomb 1975 OTNS Co-Chairman Electrical Engineering Department University of Marvland College Park, }1aryland

f

TABLE OF CO~TENTS

"Structure ne~ul t for ",anI inear Passive Svstems" .. B.D.D. :\nder~oJ\ and P . .1. ~foYlan', Department of I:lectrical Engineering, Univer~ity of Nel,castle, Australia

",\n Algebra of Operator :'etlwrks". II'illiam 0:. Anderson, Jr., Department of ~!athematics and George I:. Trapp, Department of Statistics and Computer Science, Il'est Virginia University, ~lorgantown, IVest \·irginia

"ContractiYe l'erturl'ation~ of Restricted Shifts" . ,Joseph A. Ball., Vi rgi nia polytechnic Institute and State University, Blacksburg, Virginia and Arthur Lubin,North­~estern University, Evanston, Illinois

"Frequency Response ~!ethods in ~!ultivariable Infinite Dimensional Linear S:'~tems"

,John S. Baras, Electrical Engineering Department, University of ~!aI'\'land, College Park, ~laryland

"On Simulataneous niagonalization of a Collection of Ilermitian !-Iatrices" S. Chakrabarti, B. B. Bhattacharyya and '1. N. S. Swamy, Department of Electrical Engineering, Concordia University, Montreal, Quebec, Canada

1

5

· 16

· 24

· 29

"J\ Wal~h Operational ~latrix for Solving Variational Problems". .41 C. F. Chen and C. II. Hsiao, Electrical Engineering Department, University of HGuston, Houston, Texas

"A Complex Form of the Generalized Fourier Series and Transforms" . 48 Dan A. Ciulin, Poly technical Institute of Bucharest

"Triangularization of Some Restricted Shifts". . 54 n. N. Clark, University of Georgia, Athens, Georgia and S. Sickler, Eastern Nazarene College, Quincy, Massachusetts

"Further Result~ on the Association of Variables". . . 57 James Conland and f. L. Koll, University of Regina, Regina, Saskatchewan

"The Feedback Interconnection of ~,!ul tivariab1e Systems: Simplifying Theorems for Stability" . .63

C. A. Desoer and W. S. Chan, Department of Electrical Engineering and Computer Sciences and the Electronics Research Laboratory, University of California, Berkelev, California

"The 'Fourier' Transform of a resolution Space and a Theorem of Masani" R. A. DeCarlo, R. Sacks and~!' Strauss, Texas Tech University, Lubbock, Texas

"Lumped-Distributed Networ]; S:'ntilesis Via Invariant Subspace Theory" . P. Dewilde, Departement Elektrotechniek, Katholieke Universiteit Leuven, \leverlee, Belgium and .T. S. Baras, Electrical Engineering Department, University of l'-laryland, College Park, Maryland

"Linear Ililbert Networks Containing Finitely ~lany Nonlinear Elements". Vaclav Dolezal

"Livsic's Chain Svnthesis" T. T. Ila and R. W. Newcomb, Electrical Engineering Department, Uni versi ty of ~laryland, College Park, l'-Iaryland

v

•. 69

· 75

· 80

• 82

I

"Radar Target Recognition--An Operator Theoretic Approach" ..... . D. E. Hammers and A. J. MacKinnon, ITT Gilfillan, Van Nuys, California

"Infinite Dimensional Realizahility Theory" . . . .. . ... , J. William Helton, University of California, San Diego, La Jolla, California

"Linear Network Synthesis Using Iteration Methods" . . . . . . . .. Y. ~. Jan and F. R. Chang, National Chiao-Tung University, Taiwan, Republic of China

"The Transform'ation Operator Approach to Multi-Subsystem Dynamics" . William Jerkovsky, The Aerospace Corporation, El Segundo, California

"A Note on the Nagy-Foias Lossy and Lossless Space" ..... . N. Levan, Department of System Science, 4532 Boel tcr lIall, University of California, Los Angeles, California

"An Output Control Problems Containing Input Derivatives". Victor Lovass-Nagy and David L. Powers, Clarkson College of Technology, Potsdam, New York

85

95

98

105

113

118

"An Explicit Treatment of Dilation Theory" . . . . . . . . . . . . 122 P. Hasani, University of Pittsburgh, Pittsburgh, Pennsylvania

"Characterizations of Operations Derived from Network Connections" . . . 133 Kaysuyoshi Nishio, Department of Information Engineering, Faculty of Engineering, Ibaraki University, Hitachi, Iharaki, ,Japan and Tsuyoshi Ando, Research Institute of Applied Electricity, Hokkaido University, Sapporo, Japan

"A Functional Analysis Approach to Minimum Sensitivity Control Design" J. Gary Reid, United States Air Force Avionics Lahoratorv , Wright Patterson AFB, Ohio .

"Passivity'and LP-Stability of Some Nonlinear Evolution Equations" . Dinu Wexler, with the Department of Hathematics, Facultcs Universitaires N.D. de la Paix, Namur, Belgium

"Contractive Transfer Ratios of Operator Networks" . . . . . . . . . . A. H. Zemanian, State University of New York at Stony Brook, Stony Brook, New York

VI

142

149

152

F

STRUCTURE RESULT FOR NONLINEAR

PASSIVE SYSTEMS

B.D.O. Anderson and P. J. Moylan Department of Electrical Engineering

University of Newcastle Australia

Abstract

A class of nonlinear, finite-dimensional, dynamic systems is studied for which the input and output vectors u and y satisfy a passivity condition. It is shown that such systems may be viewed a~ a cascade of a memory less passive nonlinear system and a dynamic lossless system. Th~ discussion is related to known results on linear network synthesis.

1. INTRODUCTION

Classical network synthesis, for linear, lumped,

finite, passive networks, is concerned with the

problem of passing from a port description of a

network in terms of, say, a positive real imped­

ance matrix Z(s), to a collection of (passive)

network elements and a scheme for interconnecting

them to produce a network with impedance matrix

equal to that prescribed, [1-3]. State-space

approaches to the same problem [4] commence by

assuming known a state-variable realization

{F, G, H, J} (generally minimal) of Z(s) - thus

Z(s) = J + H~ (sI-F) -lG (1)

and then attempt to construct from this an inter­

nally dissipative realization {F, G, a, J}, i.e.

one for which

[

J+J~

-(ll-G)

Equivalently, one needs to find a positive

definite P such that

[J+J~

-(H-PG)

-(H-PG)l >0

-PF-F~pJ

(2)

(3)

Once (2) or (3) is obtained, it is then possible

to define easily a nondynamic coupling network

Nc ' synthesisable merely with (passive) resistors,

transformers and gyrators, such that termination

of some of the ports of Nc in inductors leads to

an impedance Z(s) being observed at the r~main­

ing ports. For details, see [4].

Our purpose here is to describe hew some of these

results will carryover to a nonlinear situation.

Consider a system described by

x

y

f(x) + g(x)u

hex) + j(x)u (4)

where uCo) and yeo) are real m-vector functions

of time, x(o) is a real n-vector function of time

and f(o), g(o), h(o), j(o) are suitably smooth

re~l functions of appropriate dimension, with

f(O) = 0, h(O) O. We call such a system pas-

sive if for all u( 0) and tl, given x(t o) = 0,

one has

(I u~y dt > 0 (5) to

This definition is a natural extension to that

applying in the linear case; one can think of

u(o) and y(o) as current and voltage vectcrs

respectively, so that the integral in (5) consti­

tutes the energy input to a network with port

variables u(o) and y(O), computed ovpr

[to, tl]' and with the network initially unexcited.

Our task is to provide a no~inear internally

passive synthesis for (4)0 That is, we wish to

find a nonlinear, nondyna.!c or memoryle~s,

passive coupling network together with nonlinear

passive inductors so that the arrangement depicted

in Figure 1 (coupling network loaded at some ports

by inductors) has u related to y by (4). Note

that while (5) is a passivity condition on the

network of Figure I, it is an external one, direct­

ly putting constraints on the port behaviour alone

of the network, and not the behaviour of internal

variables nor the properties of components within

the network.

Note also that our specification of both the non­

linear inductor network and the nondynamic

coupling networ!, re:::ulting from the synthesis pro­

ceoure will be simply via port descriptions of

these networks - we shall not attempt to describe

how to undo any mutual coupling of the nonlinear

inductors for example. Accordingly, the contribu­

tion to practical network synthesis is virtually

nil; the result is more one concerning the theory

of passive systems, with electrical networks pro­

viding one means of visualizing the results.

In section 2, we present background results drawn

from [5] which allow reinterpretation of the con­

dition (5) in terms of the state-variable equations

(4). These results are used in section 3 to

present a passive synthesis. Section 4 contains

concluding remarks.

2. BACKGROUND

Returning to the linear problem for the moment, we

note that it is possible to associate with a passive

(or positive real) Z(s) in (1) a variational

problem, the solution to which defines a positive

definite P satisfying (3). It turns out that

the same sort of idea can be employed in studying

the passivity of (4).

2

Following [5], we shall assume that (4) is com­

pletely controllable in the sense that for any

finite states Xo and Xl' there exists a finite

time tl and a smooth control defined on [0, til

such that the state can be driven from x(O) = Xo

to x(t l ) = Xl' Further, we assume a form of local

controllability: for any Xo and any Xl in a

suitably small open neighbourhood of x o' there

exists a u(o) and tl as above with the

additional property that

J;I u'(t)y(t)dtl < p(llxl -xoll) (6)

for some continuous p(o) such that p(O) = O.

(This equation in effect demands that changes of

state must not use arbitrarily large amounts of

energy). The main theorem of [5l then states that

a necessary and sufficient condition for (5) to

hold is that there should exist real functions

P(o), ~(o) and w(o) with P(x) continuous and

with, for all x,

P(x) > 0

f' (x) 'iJ P (x)

~g' (x) 'iJ P(x)

j(x) + j '(x)

and P(O) o

-~' (x) ~(x)

h(x) - w'(x)~(x)

w' (x)w(x)

(7)

(8)

These equations generalize those applying in the

linear case, [4l. The results of the linear case

are recovered by setting f(x) = Fx, g(x) = G,

h(x) = H'x, j(x) = J and P(x) = x'Px. The

variational problem used in the linear case when

translated to the nonlinear case yields the follow­

ing characterization for one of the functions P(x)

satisfying (8): T

P(x) = - lim inf J 2u'(t)y(t)dt T-- u(o) 0

Let us observe for later use that (8) imply

[w(x) ~(x)l > 0

(9)

(10)

It is also possible to define a loss less system by

specializing (4) and (5) somewhat, and to obtain a

corresponding specialization of (8). Thus we say

(4) is loss less if (a) it is passive and (b) if

x(t o) = x(tj

) = 0, then

o (11)

for all u(·). In this case, the results of [5]

show that (7) and (8) hold with £(x) and w(x)

both zero, and the matrix on the left side of (10)

is accordingly zero.

3. SYNTHESIS PROCEDURE

For a single nonlinear inductor carrying current

and flux ¢, assumed to have a ¢-i charac­

teristic passing through the origin, the stored ¢

~nergy is JOi(¢)d¢. For n mutu~lly coupled

nonlinear inductors with current vector i and

flux vector ¢, the stored energy is J:i'(¢)d¢;

since this integral must be path-independent for

a lossless set of inductors, i(¢) must be of the

form VQ(¢) for some scalar function Q of ¢,

nonnegative on account of the passivity property.

In our problem, we identify the state variable x

witn the vector uf inductor fluxes, and the func­

tion ~(x) with the stored energy. Since

P(x) > 0 for all x, this means that the inductors

are certainly passive, indeed lossless. The

current corresponding to the flux vector x is

~VP(x). [In abstract terms, one may regard the

inductor simply as the map x H~VP(x)].

In Figure 1, we may evidently identify the vari­

ables v and i as x and -Vp(x) respectively. 2

l;ow observing (4) we see that the only way the

coupling network could provide the requisite rela­

tion between u and y is if it sustains precise­

ly the voltage-current pairs.

or

[h(x)+j(x)ul

~ (x) +g (x).d

These pairs in effect define the coupling network.

3

Note that in the event that the map x ~Vp(x)

is invertible, the coupling network will be current

controlled (i.e. any current can exist at its

ports) as will the coupled inductors. Further

the coupling network is plainly nondynamic. If

x~VP(x) is not invertible, the network is still

nondynamic, though controlled by something external

to it, viz. the vector x of inductor fluxes.

Let us now observe the passivity of the coupling

network. The instantaneous power flow into the

network is

u'g'(x)+f'(x)] I u l l:wP(xll

[u'j'(x)+h'(x)

[u 1], j (x)+j '(x) h(X)-~g'(X)Vp9

~(X)-~g'(X)VP(X)]' -f'(X)VP(x2J

> 0

using (10). Moreover, in case (4) is lossless,

we know that £(x) and w(x) in (8) are zero,

[:J

and use of (10) then implies here that the instant­

aneous power flow into the coupling network is

zero. Hence if (4) is passive, so is the coupling

network and if (4) is lossless, so is the coupling

network.

4. CONCLUSIONS

The main result of this paper is the demonstration

that a clatis of passive systems can be viewed as a

cascade of a memory less passive system (termed

earlier the coupling network) and a dynamic loss­

less system (termed earlier the coupled inductor

network). Some immediate variations on this theme

are clearly possible; for example, one could work

with admittances and capacitors, or one could

exhibit ~ synthesis starting from an analogue of

the scattering matrix. [Results akin to those of

[5] have been developed by one of the authors

which handle this problem].

Perhaps of more interest would be an examination

of the extent to which reciprocity ideas could be

incorporated into the study. Presumably one would

parallel some of the linear system ideas used in

[4], but the details remain unclear.

---.-------..... --.--,-,-----~-.--.. -- -------.--.-~---------~ ~~~----=-==x=

REFERENCES

[lJ E. A. Gui11emin, Synthesis of Passive Networks,

Wiley, New York, 1957.

[2J L. Weinberg, Network Analysis and Synthesis,

HcGraw Hill Book Co. Inc., New York, 1962.

[3] R. W. Newcomb, Linear Multiport Synthesis,

McGraw Hill Book Co. Inc., New York, 1966.

[4] B. D. O. Anderson and S. Vongpanitlerd,

Network Analysis and Synthesis - A Modern

Systems Theory Approach, Prentice Hall,

New Jersey, 1973, 548pp.

[5] P. J. Moylan, "Implications of Passivity in

a Class of Nonlinear Systems", IEEE

TFansactions on Automatic ControZ, Vol. AC-19.

No.4, pp. 373-381, August 1974.

u i.J ... ,.. . Passive

-? ,- Nonlinear :F y Nondynamic v - Coupling Network -....

J'\..

FIGURE 1

Passive Coupled

Nonlinear Inductors

Cascade Decomposition of Nonlinear Network

4

AN ALGEBRA OF OPEnATOR NETlV()I~KS

William N. Anderson, Jr. Department of Mathematics

and George E. Trapp

Department of Statistics and Computer Science

West Virginia University Morgantown, West Virginia

Abstract

An algebraic treatment of the foundation of the theory of networks whose elements are linear operators. The operation which determines the input impedance opera­tor from the branch impedance operator is analyzed. Special cases, such as pos­itively connected networks and resistive networks, are considered in detail.

1. INTRODUCTION

In this paper we present an algebraic treatment of

the foundations of the theory of networks whose

elements are linear operators. This treatment is

based on methods which we have earlier used to

study the interconnection of networks (4); ulti­

mately these methods derive from the ploneering

paper of Bott and Duffin (9). Related treatments

are given by Dolezal and Zemanian (13), (14), (30).

The purpose of this study is to formulate precise

mathematical statements of the physical assumptions

underlying the theory of operator networks, and to

show how certain aspects of the theory can be de­

rived from this comparatively simple set of assum­

ptions.

Our primary results concern the relationship be­

tween the branch impedance operator of an arbitrary

network and the input impedance operator of an

associated n-port network. Given a graph with

operators in the edges, the branch impedance oper­

ator determines branch voltages as a linear func­

tion of branch currents. By choosing n pairs of

nodes of the graph we may consider this network as

5

an n-port network and seek to determine the asso­

ciated input impedance operator. Simple graphical

conditions, similar to those given by Cederbaum

(11), guarantee the existence of the n-port imped­

ance operator. By defining a Kirchhoff subspace

to be the set of allowable current flows in the

branches and ports, we are able to show that a

branch impedance operator and a Kirchhoff subspace

give rise to the n-port impedance operator. Not

all Kirchhoff subspaces arise from graphs; our

theory holds for any such subspace, allowing us to

consider matroids and other non-electrical situa­

tions, see.(9), (15).

We begin in section 2 by reviewing the necessary

linear algebra. We then define almost right

definite operators. These operators are a natural

generalization of Hermitian positive semidefinite

operators and are closely related to positive real

operators. We discuss various properties of posi­

tive real operators, in particular we show that

the Moore-Penrose generalized inverse of a posi­

tive real operator is again positive real.

Motivated by Kirchhoff's current laws, we define a

Kirchhoff subspace in section 3. We show that a

natural dual space, the voltage space, exists and

obtain matrix representations of these spaces. If

A, the branch impedance operator, is a positive

real operator, and r is a Kirchhoff space, we show

that there exists a new positive real operator

• (A) (the n-port impedance operator). Using the

shorted operator (2), we obtain an explicit rep­

resentation for ;(A).

The formalism of section 3 is given network inter­

pretations in section 4. In particular, positive­

ly connected networks, and network duality results

are given.

Specializing to resistive networks in section 5,

we are able to demonstrate various inequalities

concerning" (A) by using the classical power mini­

mization principle. Moreover we determine when

two Kirchhoff spaces yield the same ~ function.

One application of this result is to the study of

the interconnection of networks with and without

transformers.

Finally, in the last section, we consider some

possible generalizations of this work.

2. POSITIVE REAL OPERATORS

Throughout this paper, we will consider finite

dimensional complex vector spaces, with inner pro­

duct .denoted by <','>. The book by Halmos is a

good reference for the linear algebra we use,

(19) •

If A is a linear operator, then by A* we mean the

linear operator defined by <Ax,y> - <x,A*y> for

all vectors x and y. We denote the range of A by

ran (A), and the null space of A by ker (A). If ..L

W is a subspace of V, then W is the orthogonal

complement of W. A Hermitian operator is a linear

operator such that A* • A. We say that a Hermitian

operator is positive Semidefinite if <Ax,x> ~ 0

for all vectors x; if moreover A is invertible,

we say that A is positive definite. The well

known Fredholm alternative theorem, in the form we .J. will use it, states that ran (A) * ker (A*) ; for

the special case that A is lIermitian or skew­

Hermitian, it follows that ran (A) • (ker (A)r.

6

It is well know that if A is positive semidefi­

nite, then <Ax,x> - 0 only if Ax = O. General­

izin~ this result to non-Hermitian operators, we

say that the operator A is almost right definite

if

i~<Ax,x> > 0 for all vectors x

ii)~<Ax,x> - 0 only if Ax = 0

Lewis and Newman (22) have used a similar defini­

tion for operators on real vector spaces; they use

the terminology almost positive definite.

LEMMA 2.1: Let A be a linear operator, with

A - ~ + As' where ~ is Hermitian and As is skew­

Hermitian. Then A is almost right definite if and

only if

iii) ~ is positive semidefinite

Iv) ran (As)<:ran (~).

Proof: It is easy to see that&t<Ax,x> - <~x,x>,

so that i) and iii) are equivalent.

Now suppose that iii) and Iv) hold, and that

~ <Ax,x> - O. Then <~x,x> - 0, and since ~ is

positive semidefinite, it follows that ~x • O.

Therefore Asx • 0, and thus Ax - O.

Conversely, suppose that i) and Ii) hold. Then if

ran (As>4- ran (~), there is a vector x such that

~x - 0 but Asx ". O. Then~<Ax,x> - <~x,x> • 0

but Ax ". 0, contradicting ii).

QED

LEMMA 2.2: Let A and B be almost right definite

operators, on vector spaces WI and W2, respective­

ly. Then the operator~ ~ - A ® B on WI (t) W2 is almost right definite.

Proof: For any vectors x£Wl and y£W2 , <A + B (x,y),

(x,y» - <Ax,x> + <By,y>. Since ~ <Ax,x> ~ 0

andP-a <By,y> ~ 0, it follows that ~ [<Ax,x> + <By,

y>] ~ O. Moreover, if~ [<Ax,x> + <By,y>] - 0,

then;}~ <Ax ,x> - 0 and ~<By ,y> - 0, so that Ax • 0

and By - O. Therefore A <t:' B (x ,y) - O.

QED

It is an easy consequence of lemma 1 that if A is

almost right definite, then ran (A) - ran (~) and .J...

ran (A) - ker (A). It follows that Al - AI ran

(A) maps ran (A) one-to-one onto ran (A). The op­

erator At, the generalized inverse of A, is defined

by

Ai I ran (A) - A-I 1

Ai" I ker (A) - 0

LEMMA 2.3: If A is an almost right definite oper­

ator, then At is almost right definite.

~: Let P be the Hermitian projection onto ran

(A). For any vector x, there is a vector yE ran

(A) such that Ay • Px. Then <Ax,x> - <J: PX,Px>

- <y,Ay>. Therefore~<A "x,x> -f'...<.<y,Ay> > O.

Horeover, ifR..<A~x,x> - 0, then 0 - Ay = Px.

Therefore XE ker (P) - ker (A), and thus At is al­

most right definite.

QED

The generalized inverse we have constructed above

is commonly known as the Moore-Penrose generalized

inverse. For many of our purposes it would suffice

to consider any operator A' such that AA'A - A, our

A' is such an operator, but there exist in general

many others which are not almost right definite.

The various types of generalized inverses are di­

scussed by Albert (I), and Rao and Hitra (24).

Let D denote the open right-half plane of the com­

plex plane. A positive ~ operator A (z) is an

operator valued function defined on D such that

i) A (z) is analytic on D

ii) A (z) is Hermitian on the real axis

iii) ~ <A (z) x,x> ?.. 0 for all vectors x and all ZED

LE~~ 2.4: Let A (z) be an analytic operator val­

ued function on D such that A (z) is Hermitian on

the real axis. Then A (z) is a positive real op­

erator if and only if A (z) is an almost right

definite operator for all ZED.

Proof: If A (z) is almost right definite for all

ZED, then iii) holds.

Conversely, suppose that~<A (zo) x,x> - 0 for

some z ED and vector x ~ 0 ronsider the scalar o

function f (z) - <A (z) x,x>; then~f (z ) - O. o

Since f (z) is a harmonic function, it follows

7

that either ~(f) - 0 on D or f is negative at po­

ints arbitrarily close to zoo The latter condition

violates the positive reality of A (z). Therefore

<A (z) x,x> - 0 on the real axis; since A (z) is

Hermitian there, it follows that A (z) x - 0 on the

real axis. Now let y be any vector. Then <A (z)

x,y> is an analytic function which is zero on the

real axis, and therefore throughout D by analytic

continuation. Therefore A (z) x - 0 throughout D;

in particular A (zo) x • O.

QED

LEMMA 2.5: Let A (z) be a positive real operator.

Then A' (z) is a positive real operator.

Proof: In view of lemmas 3 and 4, we need only show

that A'r is an analytic function of z. It follows

from the proof of lemma 4 that ker A (z) is a con­

stant subspace for ZED. Therefore A (z) - Al (z)

~O, where Al (z) is an invertible positive real

function acting on the subspace R· ran A (z), and

o is the zero operator on ker A (z). Then At (z)

• Al (z)Ef)o; since a true inverse is analytic, so

is Ai" (z).

QED

LEMMA 2.6: Let A (z) be a positive real operator

which is a rational function of z. Then A is a

rational function also.

The Eroof is a simple consequence of the fact that

the inverse of a rational function is again rational.

It is not true for general analytic operator func­

tions that the Moore-Penrose generalized inverse

is analytic; however, as Bart, Kaashoek, and Lay

(8) have shown, an analytic pseudo inverse A' sat­

isfying AA'A • A can always be constructed.

3. KIRCHHOFF SUBSPACES

We will let sand t denote respectively m and n

dimensional complex vector spaces, with inner pro­

ducts <., ·>s and <., ·>t respectively, let V - S

(f; T with inner product

«sl,t l ), (s2,t2»v = <sl,s2>s + <t l ,t2>v·

We assume that fixed orthonormal bases E - {el •••••

em} and F - {fl,···.fn } are given fo~ Sand T res­

pectively; then EuF is a basis for V. We call the

vectors of E branches, and the vectors of F ports.

All matrices will be written with respect to

these fixed bases.

A Kirchhoff Subspace, abbreviated KS, is a sub­

space r such that

i) ~ t£T, ]s£S '(s,t) £ r

ii) (O,t) £ r -> t· 0

In terms of networks i) states that any current t

may be put in the ports and a internal distribution

s exists in the branches; ii) requires that if all

currents in the branches are O,s • 0, then all port

currents must be 0, t • O.

We have previously considered this type of sub­

space when connecting two n-port resistive net­

works, see (4). A Kirchhoff subspace is just a

formal way to state Kirchhoff's current laws for

an arbitrary network.

If r is a KS, then we define the dual confluence

r' by

r' • {( a,T) £vl < a,s>s • <T,t>T for

all s,td}

We note the dual KS may be thought of as voltages

across ports and branches or, in terms of graphs,

the currents in the dual graph.

LEMMA 3.1: If r is a Kir~hhoff space in V, then

a.) r" - r b.) dim r + dim r' • dim V

c.) r' is a KS

The proof is an easy exercise in linear algebra,

and is omitted.

It will be convenient to represent r by 'certain

matrices; recall in this regard that fixed ortho­

normal bases have been chosen for Sand T.

LEMMA 3.2: There exists an m x m matrix K and an

n x m matrix L such that the columns of the matrix

J r -r:' :,] form a basis for r. Moreover, n ~ m.

Proof: By condition i) of the definition of a KS,

we can choose the first m columns of J as shown;

8

the remaining columns are chosen so as to be a bas­

is for the set of vectors in r which are of the form

(s,O). The inequality n < m follows from parts b

and c of lemma 3.1.

QED

In a similar manner, we let the columns of the

matrix be a basis for r'.

LEMMA 3.3: The vector (s,t) E r if and only if

l~ls - l~l. if and only if there is a vector ~ such

that [K* L*l [~] • s. The vector «(1 .T) E r' if

and only if [~1 (1 - l ~ 1. if and only if there is a

vector II such that l M* N*] l~] . (J

The proof follows by direct computation from the

definitions of J r and J r ,.

THEOREM 3.4: Let r be a KS and A an almost right

definite operator on S. Then given a vector c£T,

there exists a vector aES and a unique vector YET

such that (a,c) £ rand (Aa,y) E r'.

Proof: Using lemma 3.3, we wish to solve the eq­

uations

(3.1) l~,:J l: l~l We will employ the Fredholm Alt. Thm. The homo­

geneous adjoint system to (3.1) is

(3.2) r~ ~ :] \~~l f~l lM* N* -A* u3 lo

For any solution to (3.2), we have

(3 • 3) [ M* N* J r uu1

2

1 l - A*u3

Then A*u3 is orthogonal to all solutions to [:]

w - 0; since u3 is such a solution we have <u3 ,A*u3>

- O. Since A is almost right definite, it follows

that A*u3 • 0, so that the right hand side of (3.3)

is 0, and thus (3.3) is the homogeneous adjoint sy-

::::) 'O[:1h: :Y[:i[

By the definition of KS, system (3.4) has a solu­

tion for all vectors c, and thus <ul'c> ~ 0 for

all c. Thus ul

= :), so that (1) has a solution.

Moreover, since y = 0 for c - 0, it follows that

Y is uniquely determined by c.

Definition: Let r be a KS and A an almost right

definite operator.

fined by 4> (A) c = y.

;p (A) is the operator de-

THEOREM 3.5: Let A be an almost right definite

matrix and let r be a KS. Then;P (A) is an almost

right definite matrix.

Proof: Given a vector c, there exists an jI 50

(jI ,c) £ r and (AI> ,Y) £ r'. Therefore <Aa,a> -

<'r,c;. ., <4» (A)c,c>. Therefore~ (~(A)c,c) ~ o. Now if ~ (p (A)c,c) = 0, then~ (Aa ,a ) = 0 50 o 0

that Aao a O. Since r' is a confluence, it fol-

lows that Y ., 0; that is 4> (A)c ., o.

QED

We will derive an explicit expression for the

function ~ (A) in terma of a special function known

as the shorted operator. Let A be an almost right

definite matrix, partitioned A _fAll A121 where

l~l An

All and ~2 are square.

S(A), is defined by

Then the shorted operator,

(3.5) S(A) = All - A12

In previous papers we derived properties of the

shorted operator - for the special case of pos­

itive semidefinite A - using a variational problem

(2), (6). Here we use a KS. The formula (3.5)

itself is old, and finds application in many areas

not related to this paper (10), (12).

THEOREM 3.6: Let r be a confluence and A an al-

most right definite operator.

A [K* L*J)

Then, (A) .. S.( l~ 1

By theorem 3.4, given a vector cET, there is a

vector a£S and a unique vector y £1' such that (a,

c) E rand (Aa,Y) £ r'. Using lemma 3.3, we find

that there exists a vector). such that [K* L*J

l~1" a. Moreover [~l Aa ., [11' so that

(3.b) 6 ~ A[K* L*) c A

9

In particular, 0 = LAK*c + LAL*A. Since, is known

to exist for any c, we can solve for one possible

choice of X ~ A = - (1.0\1.*) LAK*c. Substitutinp; for

A in (3.6), we have y ~ (KAK* - KAL* (LAL*) L',"

which is the theorem.

QED

In the special case that K = L ., I, we have ~ (A) = S(A) •

If A (z) is a positive real operator, then LA (z)

L* is also a positive real operator. Lemma 2.5

yields that (LA (z) L*) is analytic and hence

~(A (z) ) is analytic. Then theorem 3.5 and lemma

2.4 imply that ~ (A (z) ) is positive real. We

have therefore demonstrated the followinh theorem.

THEOREM 3.7: Let A (z) be a positive real operator

and let r be a KS. Then;p (A (z) ) is a positive

real operator.

The preceding theorem deals with the .;ase where ,\

varys as a function of the scalar variable z; one

can also consider general matrix variations. In

this connection let us recall that a funet ion f is

said to be (Frechth) differentiable if there b

a linear function L such that f(x + h) = f(x ) + o 0

Lh + o(\\h\ I); for scalar or vector functions this

Frechet derivative is of course the usual deriva­

tive, and L is commonly expressed as the Jacobian

matrix. However, it is not necessary to express

L as a matrix, in writing the derivative of the

shorted operator it proves convenient not to do so.

The chain rule for derivatives is usually expressed

in terms of matrix multiplication; this convenience

is due to the fact that multiplication of matrices

is equivalent to composition of the corresponding

linear functions. When we use the chain rule later,

we will have to express this composition _,( fun.:­

tion directly, since we have not written the deri­

vative of the shorted operator as a Jacobian mat­

rix.

LEtIMA 3.8: \2] be an almost posi­A22

tive definite matrix such that A22 is invertible,

and let E -1·El1 El. 21 be a matrix such that A22

lE21 E22

+ E22 is invertible. Then

(3.7) S (A +E) • S (A) + [1 - A12A;~] \ Ell

LEn .' 1

\ I -1 I -1 -1

L-A22 An J-<A12A22E22 - E12) (A22 + E22 ) -1

(E22A22AZl - E2l). The proof is a direct compu-

tation.

THEOREM 3.9: Let A be an almost right definite

operator such that A22 is invertible. Then the

shorted operator SeA) is a differentiable function

at A.

~: For sufficiently small E, A22 + E22 will

be invertible, and the final term in (3.7) is

01 lEI I. Therefore the middle term in (3.7) is

the desired linear approximation.

QED

Theorem 3.9 of course imples that if A22 is in­

vertible, then SeA) is continous at A. In the

absence of invertibility, SeA) need not be con­

tinousi an example is given in (6).

If higher derivatives are desired, one may expand -1

(A22 + E22) in a Neumann series.

The concept of Quality is important in network

theory. In the present context, the duality is

expressed in terms of dual KS. That is, if r is

a KS and ~ is the corresponding matrix, we con­

sider the dual KS r' and the corresponding matrix

operator ~ '.

THEOR1:l! 3.10: Let:' be a KS and :' be the dual

is. Let. and ~' be the corresponding matrix

operations. Then if A is an invertible almost

right definite operator, ~ (A) _ ij' (A-I» -1 •

Proof: Let y - ~ (A)c. Then there exists a vec­

tor a such that (a,c) £ rand (Aa,y) £ r'. Let

a- Aai then (A-la,c) £ r" - rand (a,l) £ r'.

Therefore; '(A-I) - c, so that'. '(A-I) ~(A) - 1.

QED

4. OPERATOR NETWORKS

An operator network is a pair (r,A) where r is a

Kirchhoff space and A - A (z) is a positive real

operator acting on the vector space E, that is,

10

the space spanned by the branches. The operator A

is called the branch impedance operator.

Given a vector cET, called the port current vector,

we wish to find a ~r~n~~ current ~ a£S and a

port voltage vector YET such that (a,c) E rand

(Aa, y) £ r'. The correspondence between c and y is

given by the equation y - ~ (A)c; the operator. (A)

is called the input impedance operator.

In this section we will prove some basis theorems

about operator networks, and discuss some import­

ant examples. The fundamental theorem is:

THEOREH 4.1: Let (r,A) be an operator network.

Then for any port current vector c there is a un­

ique port voltage y, and a branch current vector

a such that (a,c) £ rand (Aa,y) £ r'. The opera­

tor ~ (A) defined by the equation y • , (A) c is a

positive real operator.

Proof: This is merely a restatement of theorems

3.4 and 3.7.

The operator network (r,A) is said to be positively

connected if LAL* is invertible, equivalently if

ker (AL*) - O. This definition would appear to

depend on the choice of Li the independence from

the choice of L follows from the fact that the co­

lumns of L* form a basis for a uniquely determined

subspace of S. In particular, if A is invertible,

then (r,A) is positively connected.

TIIEOREM 4.2: Let (r,A) be a positively connected

operator network. Then the branch current vector

b is uniquely determined by the port current vec­

tor c.

Proof: By assumption LAL* is invertible. Then we

can solve (3.6) for ~ obtaining ~ __ (LAL*)-l

LAK*c, and then

(4.1) a - [K* L*] l~l - (K* - L*(LAL*)-lLAK*)c

QED .-J

Following Cederbaum (11), we call K - (K* - L*(LA t

L*) LAK*)* the modified circuit ~atrix of the op-

erator network (r,A). An extensive study of the

properties of K has been made by Thulasiraman and

~urti (27).

THEOREM 4.3: Let (r,A) be a positively connected

operator network. Then t (A) is a differentiable

function of A.

Proof: Since LAL* is invertible. theorem 3.9

applies. Moreover. in view of (3.7). we see that

the linear approximation to ~ (A + E) is ~ (A) + UK;".

QED

For the special case of an n-port network. this de­

rivative formula has been given by Cederbaum (11).

One important example of a general variation in A

is given by letting the impedance in a single bra­

nch vary. A particularly simple formula is avail­

able in this case. Let the branch be branch i; we

will use the notation dA/daii for this derivative.

THEOREM 4.4: Let (r.A) be a positively connected

operator network. Let 0; be the current in branch

i corresponding to a port current of 1 at port j

and 0 at the other ports. Then

1...tJ.& .. [<;. ••••• aml* [<;. ••••• aml. Claii

Proof: For ease of notation. assume that i-I.

Then by theorem (4.3) and the chain rule, the de-

"vaUv. is< [~ g F" -K [f (l.'··.) ii., By

theorem (4.2). the vector [<;..··.aml is the first

row of K*. But this row is the product [lO"'OlK*.

QED

As mentioned previously. one familiar example of

an operator network is an ~-port network. Let G'

be an oriented graph with m + n edges, m of which

are called port edges. Let P denote the set of

port edges. and let G be the sub graph of G' con­

taining the edges of G which are not in P. We

assume that the set P contains neither a circuit

nor a cocircuit of G'. The branches e i then cor­

respond to the edges of G, and the ports fi to the

port edges.

The current space of G' is in fact a KS r. The

hypothesis that P contains no cocircuit yields

condition i of the KS definition. and the hypoth­

esis that P contains no circuit yields condition

ii. The voltage space of G' is also a KS r. The

11

KS r', the dual to r, is defined by (0,r) e: r' if .l..

and only if ~. ,-1) E r. The duality of rand r'

here follows from the fact that r and r~are ortho­

gonal complements.

The matrix J r may be found by imbedding P in a co­

tree of G' and then letting the columns of Jr

be

vectors corresponding to the fundamental basis of

circuits relative to the cotree. Similarly, form

J r , by imbedding P in a tree. This construction is

described in more detail by Cederbaum (11).

In the electrical interpetation, the edges of G are

inside a ''black box", and the port edges are order­

ed pairs of terminals which are accessible from the

outside. If (s,t) E r, then t is a vector of cur­

rent sources at the ports of the network, and s is

the vector of branch currents inside the "black

box". Similarly, if (o.t) then t is the vec-

tor of port voltages and a is the vector of branch

voltages. The duality of rand r' expresses the

physical principle that power measured at the ports

is the same as power measured at the branches. The

matrix A has diagonal entries with a ii • R > 0 for

a resistance; aii = Lz for an inductance; and aii -

l/Cz for a capacitance. Mutual inductance will re­

sult in off diagonal entries. but in any case A (z)

will be positive real.

The term operator network is more commonly applied

to the generalization of this example in which the

edge impedances are operators. Then each entry of

1 in the J r matrix is replaced by an identity ma­

trix of appropriate Size, and the O's of J r are re­

placed by zero matrices. The operator A is then

block diagonal, unless mutual inductances are all­

owed. Special cases of this generalization are the

series and parallel connection of networks (3);

the more general connections treated in (4) do not

necessarily admit such simple graphical models.

Instead of starting with a graph and obtaining a

KS. we now start with the KS, and reversing the

earlier procedure, we obtain a generalization of a

graph. knQol\l as a matroid. Let r be a confluence.

If x e: r, the support of x is the set of branches

and ports corresponding to the non-zero components

of x. A circuit is a minimal non-empty support

set; the set of circuits forms a matroid. This de­

finition is given by Tutte (28); there are many

well known equialent definitions (23), (29). If

instead of r we consider r', then we speak of co­

circuits and the dual matroid. (Tutte uses r~ in­

stead of r', but it is easy to see - and is a spe­

cial case of theorem 10 of (9) - that the two de­

finitions are equivalent)

THEOREM 4.5; Let (r ,A) be a resistive operator

network where A is a diagonal matrix. Then the

network is positively connected if and only if ev­

ery circuit containing only branches contains at

least one branch with non-zero resistance.

Proof: Let t£T be a non-zero vector such that

RL*t - O. Then every branch 1. n the support 0 f

L*t has zero resistance, so that any circuit con­

tained in the support of L*t will contain only

zero resistors.

Conversely, if some circuit contains only zero

resistors, then this circuit is the support of a

vector s£r such that Rs - O. Since the columns

of L* are a basis for the vectors in r whose

support is contained in the set of branches, the

vector s - L*t for some vector t; therefore

RL*t - O.

QED

The assumption that every circuit contains one

non-zero resistor is common in electrical net­

work literature (18), (23); the term "positively

connected" is due to Duffin (17), where an equi­

valent definition, in terms of cocircuits, is

given.

Minty has proved theorems 4.1 and 4.5 in the

special case that the matroid is regular (23); he

uses the term digraphoid instead.

5. RESISTIVE OPERATOR NETWORKS

Let (r,A) be an operator network. We say that the

network is resistive if the matrix A is a constant

matrix. In this case A must of course be Hermi­

tian and positive semidefinite.

A classical variation principle states that in a

resistive network the current always takes the

12

path of least resistance. In the case of resistive

operator networks we have an algebraic formulation

of this principle.

THEOREM 5.1: Let (r,A) be a resistive operator

network, and 41 the associated matrix operation.

Let c be an arbitrary vector and Y = P (A)c. Then

for any vector (a,c) E r.

(5.l) <Aa,a> ~ <'r,c>

Moreover, equality holds if and only if (Aa,n Er'.

Proof: fiy the definition of KS equality will hold

if (Aa, Y) E r'. In fact, by theorem 3.4 rr~"'"

exists an ao such that (ao'c) E rand (Aao,-r) E r'.

Since r is a vector space (t,o) E r, and thus <Aao

'

t> - o. Therefore <Aa,a> • <A(ao

+ t), ao

+ t> = <Aao,ao> + <At,t> ~ <Aao,ao> = <Y,c>. Thus the in­

equality is established. Moreover, if equality

holds, then <At,t> • O. Since A is positive semi­

definite, it follows that At • 0, that is Aa ~ Aao

'

so that <Aa,Y> • <Aao'·Y) e: r'.

QED

If A and B are positive semidefinite operators, we

say that A ~ B if A - B is positive semidefinte.

Another expression of this variational principle

can be given in terms of this partial order of op­

erator.

THEOREM 5.2: Let (r,A) and (r,B) be resistive op­

erator networks, with ~ the corresponding matrix

operation. Then> (A + B) ~; (A) + ~ (B) •

Proof: The special case for ~ being the shorted

operator is theorem 5.1 of (2). The result then

follows from theorem 3.6.

QED

An alternative proof, based on theorem 5.1, can be

constructed analogous to the proof of theorem J.j

Nishio and Ando have shown how some of these in­

equalities actually characterize certain matrix

operations as deriving from network models (25).

Corollary 1: The input impedance operator is a

monotone function of the branch impedance operator.

Proof: If the branch impedance operators are A

and C, with A ~ C, then C - A is also positive

semidefinite. Therefore; (C) .; (A + (C - A» >

<t (A) + <t (C - A) ~ ?(A) •

QED

Corollary 2: The input impedance operator is a

concave function of the branch impedance operator.

Proo f : If 0 <: a.5.. 1, t hen ~ ( 0 A + (1 - 0) B) ~

1> (0. A) + p (1 - 0) B) " o.,~ (A) + (1 - 0) P (B) ;

QED

Dolezal has proved theorems similar to these two

corollaries, see (13), (14).

In addition to defining a partial order of opera­

tors, we can define an order relation on the mat­

rix operations derived from Kirchhoff spaces.

Let r l and r2 be Kirchhoff spaces and PI' P2 the

corresponding matrix operations. We define f' 1 ::...

~ 2 if 1> 1 (A) .5.. p 2 (A) for all positive semidefinite

operators A.

)In the next theorem we give a condition on the

Kirchhoff spaces which is equivalent to the con­

dition 1> 1 .5.. P 2' In order to prove our theorem we

need two results on operator ranges. First, it is

well known that if A and B are positive semidefin­

ite, and A.5.. B, then ran (A) ran (B). Second,

theorem 1 of ( ::) states that if A is positive

semidefinite, and S is a subspace, then ran (S(A»

" ran (A) S.

THEOREM 5.3: Let p 1 and 1> 2 be matrix operations

derived from the Kirchhoff spaces r l and r Z res­

pectively. Then <p 1 .5.. P 2 if and only if there is a

constant k with Ikl ~ 1 such that for every vector

(a,kc) c r2 , the vector (alkc) £ r l •

Proof: Suppose that there is such a k. Let A be

a positive semidefinite operator. Then given an

arbitrary port current vector c, there exists a

vector a such that (a,c) £ f2' and (Aa,y) [ rio

By hypothesis, (k-la,c) £ rl' so that by theorem -1 - 1 1-2 5.1, <1> 1 (A)c,c" .5.. <Ak a,k a>" k <Aa,a>"

I 1-2 1 I ,k ~2(A)c,c>. Since kl > I, it follows that

PI (A) .5.. p 2 (A) for all A.

Conversely, suppose that 41 1 (A) ::... P 2 (A) for all

positive semidefinite operators A. Let US con­

sider an arbitrary vector (0 , Yl) £ ri with Yl '

O. Then aa* is a positive semidefinite operator,

13

and thus PI ( oc! *) ::... p 2 (aa *) by hypothesis. Com­

puting these operators, we have by theorem 3.6

and lemma 3.3 1> 1 (oa *) • S kl ao * [k1 * L] *] •

L1

k2

L2

[Y2* 82*] - S )')'2* ye 2* for some

fl 2y z* fl2B2*

We notice that if 32 ; f) then since Y2 h2* (;2*] e2

has rank 1 and because of aforementioned result on

the range of the shorted operator, we must have

p 2 (oa *) " O. However ~ 1 (00 *) is non-zero, and

PI ::...t2' thus f'Z = 0 and therefore ~2(oa*) =

'(2Y2*' !1oreover, since 1>l(a,,*) = '(1'(1*' we must

then have ran ('(l-Y l *) ran ('(2'(2*)' Since these

ranges are one-dimensional, it follows that y~ •

ky l ; furthermore Ikl ~ 1 since Yl'l* ~ '(2-(2*'

Concei vably k depends on a. However let Y and C:;

be two linearly independent vectors with (a ,r) (

ri and (B,O) (: r~. Then for some constants k1' k2

and k3 , (Il ,kl " [ ri, (S,k 20) [ ri, and (CL+ ~,k3

(y + 0» £ rio Since 'i is a subspace, it follows

that (O,k3(O + -y) - kly - k20) ( 'i' and since ri

is a Kirchhoff space, it follows that (k3

- k2

)0 +

(k3

- kI)y K O. But 0 and yare linearly indepen­

dent; thus kl - k2 " k3 ·

We now know that if (0 ,y) ( ri, and Y , 0, then

(a ,k'y) (: rio We need to show that if (0 ,0) £: ri, then also (0 ,0) (: rio To see this, consider some

(B,y) (: ri.with y ~ O. Then (S,ky)£fi' Since'i

is a subspace, (0+ B,y)£ri, so that (o+i3,ky}t"i"

Since r i is a subspace. we have (Q +2 ,ky) - C3 ,ky)

=(a,O)£fi.

We have proved that if (0 , y) £: ;' i then (u ,k-r> c

fi; the theorem however refers to r 1 and f 2 . To

complete the proof, consider a vector (a,e) ( '2"

Then for any vector (c,,-,) .~ ri, we have ('J ,ky) <:

ri. By the definition of Kirchhoff space, it fol­

lows that <a,o > = <c,ky> .. <k*c>(>. Therefore, by

the definition of Kirchhoff space (a,k*c) f ;1'

Since Ik*i = Ikl ~ 1, the theorem is proved.

QED

Cor,'llary: Let rl

and r2

be Kirchhoff spaces, and

~ 1 and ~ 2 the corresponding matrix operations.

Then ~l(A) • ~2(A) for all positive real operators

A if and only if there is a constant k with Ikl •

1 such that the vector (a,c) c r l if and only if

the vector (a,kc) £ r2•

Proof: If equality holds for all positive real A,

it holds for all positive semidefinite A. Then ~l

~ ~ 2 and ~ 2 ~ ~ 1 and the theorem applies. The

converse is proved as in the first part of the

proof of the theorem.

Of somewhat more interest is the case where the

operator network is derived from a graph, with

scalar resistors .1f operators allowed in the ed­

ges. In terms of A, this means that A is res­

tricted to be a diagonal or block diagonal matrix.

I.e conjecture that in this case an analogue to

theorem 5.3 is true, except that there is a

separate constant for each edge. That the exist­

ence of such constants implies the inequality is

clear, but we have no proof for the converse.

Even the case of equality seems difficult; the

scalar case has been proved by Bott - Duffin (9),

theorems, (9) but their proof does not seem to

generalize.

An important special case of theorem 5.3 arises

when r2<= r l ; that is k a 1. This case arises,

for example, in treating the connection of n-port

networks with and without ideal transformers. For

any type of connection, series, parallel, hybrid,

cascade or even more general, the use of trans­

formers merely restricts the space of available

currents. Thus by the theorem the impedance of

the connection without transformers is always less

than or equal to the impedance with the use of

transformers. We have discussed the inequalities

for the parallel connection in (7); conditions for

equality are given there in terms of the topology

of the networks. Other authors have treated the

case of equality, without discussing the inequal­

ities (21), (24). The series connection appears

somewhat more difficult to treat, the difficulty

being that the series connection of n-port is not

14

commutative unless transformers are used.

6. CONCLUSION

In this paper we have presented the foundations of

an algebraic treatment of operator networks. Many

other properties of ordinary networks can be shown

to hold in this wide context. For example, one

can generalize Duffin's theory of extremal length

(16) and heat-flow networks (17); these generali­

zations depend primarily on theorems 4.4 and

5.1 of this paper.

Another subject that one may wish to treat is the

infinite dimensional generalization of operator

networks. For the case of invertible operators in

all the branches, Zem anian has developed an exten­

sive theory (30) without this assumption of inver­

tibility, theorem 4.1 need not hold. Counter

examples for the parallel connection of two oper­

ators are discussed in (6). However, the shorted

operator can be shown to exist in very general

circumstances (6), (20), and the formula of theo­

rem 3.6 will still yield useful results. For

example, one can consider infinite ladder networks

and prove fixed point theorems similar to the ones

in (5) without resort to the limiting arguments

used there.

1.

REFERENCES

A. Albert, Regression and the Moore-Penrose

Pseudoinverse, Academic Press, New York 1972.

2. W.N. Anderson, Jr., "Shorted Operators"

SIAM J. Appl. Math. 20(1971) 576-594.

3. W.N. Anderson, Jr. and R.J. Duffin, "Series

4.

5.

and Parallel Addition of Matrices", J. Math.

Anal. Appl. 26(1969) 576-594.

W.N. Anderson, Jr., R.J. Duffin and G.E.

Trapp, "rtatrix Operations Induced by Network

Connections", SIAM J. on Control 13(1975)

446-461.

W.N. Anderson, Jr., G.D. Kleindorfer, P.R.

Kleindorfer and M.B. Woodrodfe, "Consistent

Estimates of the Parameters of a Linear System'

Ann. Math. Stat. 40(1969) 2064-2075.

6. I .. N. Anderson, Jr. and G.E. Trapp, "Shorted

Operators II", SIAM J. Appl. ~fath. 28(1975)

160-171.

7. W.N. Anderson, Jr. and G.E. Trapp, "Inequ­

alities for the Parallel Connection of Resis­

tive n-Port Networks", J. Franklin lust 299

(1975) 305-313.

8. H. Bart, M.A. Kaashoek and D.C. Lay, "Re-

lative Inverses of Finite Meromorphic Operator

Functions", IndQg Math. to appear.

9. R. Bott and R.J. Duffin, "On the Algebra

of Networks", Trans. Amer. Hath. Soc. 74(l953}

99-109.

10. D. Carlson, E. Haymsworth, and T. Markham,

"A Generalization of the Schur Complement by

Means of the ~oore-Penrose Pseudoinverse",

SIAM J. App!. ~iath. 26(l974} 169-175.

11. I. Cederbaum, "On Equivalence of Resistive

n-Port Networks", IEEE Trans Circuit Theory

CT-12(1965} 338-344.

12. R.W. Cottle, "Hanifestation of the Schur

Complement", Linear Algebra Appl. 8(l974}

lB9-2ll.

13. V. Dolezal, "Hilbert Networks I", SIMi J.

14.

Control l2(1974}.

V. Dolezal and A.H. Zeman ian , "Hilbert

Networks II - Some Qualitative Properties",

SIAM J. Control 13(1975) to appear.

15. R.J. Duffin, "An analysis of the Wang alge-

bra of networks", Trans. Amer. Math. Soc.

93(1959} 114-131.

16. R.J. Duffin, "The External Lenght of a

Network", J. Math. Anal. App!. 5 (1962) 200-

215.

17. R.J. Duffin, "Optimum Heat Transfer and

Network Programming", J. Math. Mech. l7(196B}

759-76B.

lB. H. Flanders, "Infinite Networks I - Resis-

tive Networks", IEEE Trans Circuit Theory

CT-lB(197l} 326-331.

19. P.R. Halmos, ~ Dimensional Vector

Spaces, Van Nostrard, Princeton, 196B.

20. M.G. Krein, "The Theory of Selfadjoint Ex-

tensions of Semibounded Hermitian Operators

and Its Applications I", Mat. Sbornik N.S.

20(62} (1947) 431-495 (In Russian, with En­

glish summary).

21. A. Lempel and I Cederbaum, "Parallel Inter-

15

connection of n-Port N~tworks", IEEE Trans

Circuit Theory, CT-14(1967} 274-279.

22. T. Lewis and T. Newman, "Pseudoinverses of

Positive Semidefinite Matrices", SIAM J. Appl.

Math. l6(196B} 701-703.

23. G. Minty, "On the Axiomatic Foundations of

24.

25.

26.

the Theories of Directed Linear Graphs, Elec­

trical Networks, and Network Programming", J.

Math. Mech. l5(1966} 485-520.

V.C.K. Murti and K. Thulasiraman, "Parallel

Connections of n-Port Networks", Proc. IEEE

55(1967} 1216-1217.

K. Nishio and T. Ando, "Characterizations

of Operations Derived from Network Connections"

preprint.

C.R. Rao and S.K. Mitra, Generalized Inverse

of Matrices and Its Applications, Wiley, New

York 1971.

27. K. Thulasiraman and V.C.K. Murti, '~odified

Cut-Set Matrix of an n-Port Network", Proc

lEE ll5(1968}.

28. W.T. Tutte, "Lectures on Matroids", Nat.

Bur. of Standards J. Res. 69B(1965} 1-48.

29. H. v.'hitney, "On the Abstract Properties of

Linear Dependence", Amer. J. Math. 57(1935}

509-533.

30. A.H. Zemanian, "Infinite Networks of Posi-

tive Operators", Circuit Theory and App1. 2

(l974) 69-78.

William ~. Anderson, Jr. recieved the ES, !is and

Ph') from Carnegie-Hellon Uni'lersity in 1960, 1967

and 1968 respectively. From 1960 to 1964 he ser­

ved in the US Army Signal Corps. Since receiving

the Ph) he has been at The Rockefeller University,

the University of ~1aryland and West Virginia Univ­

ersity. He is a member "f SIAM, ACM and Sigma Xi.

George Trapp recieved the BS, }IS and PhD from

Carnegie-~Iellon University in 1966, 1967 and 1970

respectively. He has been at West Virginia Univ­

ersity and a consultant for Westinghouse Electric

Corporation since 1970. He is a member of ~lAA,

SIMI, ANS, Pi Hu Epsilon, Sigma Xi, Tau Beta Pi,

and Phi Kappa Phi.

CONTRACTIVE PERTURBATIONS

* OF RESTRICTED SHIFTS

by Joseph A. Ball Virginia Polytechnic Institute and State University

Blacksburg, Virginia 24061

and Arthur Lubin Northwestern University

Evanston, Illinois 60201

Abstract

The characteristic function of a certain type of contractive perturbation of a restricted shift operator is determined in terms of that of the unperturbed oper­ator. Also a spectral representation is computed explicity for a class of uni­tary perturbations. These results generalize some of the finite-dimensional re­sults of P.A. Fuhrman related to stability of linear control systems.

1. INTRODUCTION

In this paper, we study a class of pure contrac­

tive and unitary perturbations of (generalized)

restricted shifts acting in a Sz.-Nagy-Foias

space generated by an analytic contractive opera­

tor-valued function S(z), and we consider some re­

lations between the characteristic functions and

spectra of the original operators and the pertur­

bations. Restricted shifts (at least the case

where S(z) is unitary-valued) arise in the reali­

zation theory of discrete linear control systems,

in which case the analysis of the perturbations

studied here has applications to stability theory

for linear control systems. D.N. Clark [2J stud­

ied the one-dimensional unitary perturbations of 2 restricted shifts in H , i.e. S(z) a scalar inner

function. The general unitary perturbations are

implicit in work of de Branges and Rovnyak [lJ,

though in the context of the de Branges-Rovnyak

model theory rather than the Sz.-Nagy-Foias.

P.A. Fuhrman [4J considered a class of completely

nonunitary and unitary perturbations for the case

S(z) an inner function on a finite-dimensional

space. In this case, the maps considered are

always compact perturbations. Our purpose here i,

to generalize the results of [4J and [2J to the

context of a more general Sz.-Nagy-Foias space.

2. PRELIMINARY RESULTS

For notation, let C and C* be separable Hilbert 2 2 2 2 spaces, let L (C), L (C*), H (C), H (C*) denote tl

standard vector-valued Lebesgue and Hardy spaces

defined on the unit circle. (See [5J or [7J for

general references). We will use "t" to denote tl

argument of a function (vector or operator-valued:

defined on the unit circle, and for analytic func­

tions, we will freely identify h(t) on the circle

with its extension to the disc, denoted h(z). ThE

symbol S denotes a fixed purely contractive anal)

tic operator-valued function from C to C*' i.e.

S (z): C-tC*, II S (z)1I s; 1 for all I zl < 1 and

* Most of these results will appear elsewhere in revised form.

16

s

Ie

Ie

r-

\lS(O)cll < \lcll for all c ~ C, and let t.(t) = * \ 2 2 (I-S(t) Set»~ • Let H=H (C*) ~ t.L (C), where the

2 second summand denotes the L (C) closure of

[t.(t)g(t)\gE L2 (C)}, and M=[(S(z)f(z),t.(t)f(t»\

fE H2(C)}C H. Then M is invariant under the (uni­

lateral)~hift U+ in H defined by U+(f,g) = lt (zf(z),e get»~, so K = H0M is invariant under

* * U+, where U+ is the left-shift defined by * -1 -it U+(f,g) = (z (f(z) - f(O», e get»~. We call K

the Sz.-Nagy-Foias space generated by S. Let T

denote the restricted right shift in K, i.e. the

compression of U+ to K. Thus, for (f,g) ~ K,

T(f,g) = P(zf, eitg) , where P denotes projection

* * onto K, and T = U+\K' Note that if S is inner,

then t.(t) = 0 a.e. and K = H2

(C) (3 SH2

(C).

Let S(z) be the analytic operator-valued function - * - * defined by S(z) = S(z) , i.e. Set) = S(-t) :C*~.

Analogously to above, define 6(t): C*~by

6(t) = (I-S(t)* S(t»%, H = H2(C) ~ AL2

(C*), M =

USf, 6f)\ fE H2

(C*)}, K = H0M, and T = puli('

where u+ is the unilateral shift in Hand P is

projection onto K. Note that S is inner if and

only if S is inner. (We use "inner" in the sense

of [5], i.e. Set): C ~ C*is unitary a.e.; in the

terminology of [7], this is called "inner from

both sides".) The following result indicates how

a restlcted left shift may be represented as a

resticted right shift, and is basic for our

analysis. The inner case is proved in (3). The

proof is a direct computation and will be omitted.

2.1 THEOREM

(i) Let L=L2(C*) ~ t.L2(C) , i=L2

(C) ~ 6L2

(C*). _ -it *

Then T : ~L given by '1" (f,t.g)=e (S(-t) f(-t) + 2 e _ e

t. (-t)g(-t), t.(t)(f(-t)-S(-t)g(-t») is unitary.

(ii) '1"='1" \ K is a unitary map from K to K e

implementing a unitary equivalence between the -* -right shift T on K and the left shift TonK:

-* TT = T '1".

We can now derive an explicit formula for T which

will be useful later on.

2.2 COROLLARY . t

For (f,t.g)i K, T(f,t.g)=(zf(z)-S(z)Q(0),e1

t.(t)g(t)-

t.(t)Q(O) where Q(O) is the first component of

T(f,t.g). Proof: This is obtained by computing

*- * (T T T) (f,t.g).

If F=(f,g) E K and T(F) = (Q,h), denote by (TlF)(z)

the C-valued function Q(z). We state several tech­

nical lemmas needed later on. The proofs are rela­

tively straight-forward computations.

2.3 LEMMA.

For \ w\ < 1, X E C*' Y E 'I(

C, let

17

k :<!-S(z)S~w) w,x,y 1 - zw

x, -

+(S(z) - Sew) z - w y,

Then k E K and W,X,y

* t.(t)S(w)

1 - eit

1i'

t.(t) it

e - w

x)

P«x/(l - z~), 0» = k 0' w,x,

P« S~t2 t. {t 2 y»=k 0 • it y, it

- w w, ,y

e e - w

2.4 LEMMA

If (f ,g) = F E K, then

(i) (F, k O)K = w,x,

(ii) (F ,k 0 )K = w, ,y

(f(w),x)C *

«T IF)(W) ,y) c.

In particular

(iii) for X,y E C*' , and '11 in D,

* (k km ) = (I - S(TI2 S{C2 ',x,O'--1j ,y,O K 1-1],

(iv) for x,y E C, - * -

(k 1<- ) = (I - s(l) S(i:.) ) "O,x' 1I,0,y K I _ '11' x, y C

and

(v) for x E C, y,; C*' S(ll) - S {')

(k.. 1<- ) = (~w..L----=~ x, y)c -1.,O,x' lI,y,O K T) _, *

We note that if F = (f,g) E K is orthogonal to

K for all w ~ D, x,; C* and y E C, then f = 0 W,X,y

and '1" IF = O. From the formula for T l' it follows

that also g = 0, and henc(c F is the zero element of

K. This implies that [k \w E D, x E C*' y& cJ W,X,y spans a dense subset of K. This fact will make

the formulas iii) - v) useful for computations

later on.

The next Lemma follows from the Corollary to Theo­

rem 1.1 and direct computations.

2.5 ~

(i) Tk = W-l(k - k ), w 1 o. w,x,o w,x,o O,X,O

(ii) Tk = wk - k - • w,O,y w,O,y O,S(w)y,O

(iii) T*k = wk - k * w,x,O w,x,O O,O,S(w) x

(iv) T*k 0 = w-1(k 0 - kO 0 ), w 1 o. w, ,y w, ,y , ,y

We wish to distinguish two subspaces of K defined

by

k = the closure of {kO olx E C*} 0 ,x,

K = 0 the closure of (kO,o) y E C}.

Let us simplify the notation for this special case by writing

d for kO 0 and D for kO 0 • x ,x, y , ,y

2.6 LEMMA

Let F = (f,g) E K. Then * -1 -it (i) T F = (z f(z), e g(t» if and only if

F lko' it 1 (ii) TF = (zf(z), e g(t» if and only if F KO'

2.7~

Let PkO

and PKa

denote the orthogonal projection

onto kO and KO respectively. Then PkOF = dx

'

* -1 where x = (I - S(O)S(O» f(O) and PK

F = D , o y

where y = (I - S(0)*S(0»-1(T1F)(0). (Note since

S(z) is a pure contractive function, x and y

are well-defined for F in some dense subset of K.)

3. THE PERTURBATIONS .-

3.1 DEFINITION

Let A: C ~ C* be a bounded linear map. We define

Z(A) to be the unique bounded linear map on K such

that

Z(A)F = ITf

dAy if f = D Y

18

3.2 REMARK

* It follows that Z(A) is given by

Z(A) F= * IT * F if F L kO

* -1 * D if F=d ,where y=(I-S(O) S(O» A y x *

(I-S(O)S(O) )x

We note that T = Z(-S(O» (by lemma 2.4), and tha * Z(A) dx = DA*x

if and only if

* * (1) AS(O) S(O) = S(O)S(O) A

3.3 THEOREM

(i) Z(A) is a contraction if and only if * * * (2) A (I - S(O) S(O~ ~ (I - S(O)S(O) )

(ii) Z(A) is unitary if and only if

A=(I-S(O)S(O)*)\V(I-S(O)*S(O»% for some unitary

(iii) If A satisfies condition (1), then

Z (A) is a contraction if and only if II All ~ 1 and

Z(A) is unitary if and only if A is unitary.

Proof

(i) Since Z(A) maps KoL isometrically onto koL

and sends KO onto kO' Z (A) is a contraction if ani

only if it is contractive on kO' By lemma 2.4,

this holds precisely when

IIAYIl2-IIS(0)*AYII~lIyjI2_IIS(0)YIl2 for all y E C, but

this is clearly equivalent to (2).

(ii) As above, Z(A) is isometric precisely when

equality holds in (2). By [4, theorem 1.7(i)J,

this holds if and only if A = (I - S(O)S(O)*)-%

V(I - S(O)*S(O»~ for some isometry V. By

* lemma 2.4, Z(A) is isometric if and only if

(I - S(O)S(O)*) = (I-S(O)S(O)*)~VV*(I-S(O)S(O)*)%: * which holds if and only if VV = I, so V must be

unitary.

(iii) If (1) holds, then (2) reduces to * * *

(A A)(I-S(O) S(O»~(I-S(O) S(O», which, using (1)

again, holds if and only if A*A~ I, Le. IIAII ~ 1.

In the second case, (1) implies that A = V.

4. CHARACTERISTIC FUNCTIONS AND SPECTRA

V.

d

The Sz.-Nagy-Foias model theory for contractions

assigns to each contraction T on a Hilbert space

} * ). H the triple [19

T,19

T*,EilT(A ) where DT = (I-T T) 2,

D * = (I-TT*):\;; 19 = DR 19 * = D *H and Eil (A) = T '* 1:1 T' T T' T

[-T +ADT*(I..I.T) DT

J\19 is an analytic operator-T

valued functiun whose values are contractions from

19T

to ilT*' the defect spaces of T. (This holds

since TDT = DT*T.) We call this triple the char­

acteristic function of T, and if T is completely

non-unitary (c.n.u.), i.e. there is no reducing

subspace on which T is unitary. Then T is uni­

tarily equivalent to the adjoint of the restricted

shift on the Sz.-Nagy-Foias space generated by its

characteristic function [7, p. 248J. In most

cases, one is unable to get any "concrete" infor­

mation from this representation for a specific

operator because of computational difficulties

involved in simplifying the form of the charac­

teristic function. However, if A satisfies (1),

then we can apply Fuhrmann's proof [4, p. l69-l72J

verbatim to get the following two theorems.

4.1 THEOREM

If A is a strict contraction satisfying (1), then

Z(A) is a c.n.u. contraction on K with character­

istic function [KO' kO' SZ(A)(z)}, where EilZ(A)(z)

is given by

EilZ(A) (z)Dy=dG(z)y where

(I-S(O)S(O)*):\;;G(Z)(I-S(O)*S(O»-:\;;

(I-AA*)~(I-r (z)A*) -1([ (z)-A) (I-A*A) -:\;; and

r (z) =

(I-S(O)S(O)*):\;;(I-S(Z)S(O)*)-I(S(z)-S(O»(I-S(O)* S(O»-~

Note that the above are matrix fractional linear

transformations.

We call an open arc Y of the unit circle regular

for S(z) if S(z) has analytic continuation over

Y and for all AE y, S(A) is unitary. Let a(T)

and a(Z(A» denote the spectrum of T and Z(A)

respectively. Recall [7, theorem VI, 4.lJ that

a(T)=(1 zl < 11 S(z) is not boundedly inverible}

19

U £I A I = 1111. lies on no regular arc of S}.

4.2 THEOREM

Under the assumptions of 3.l,(i) a (Z(A»=l\A\=11

A lies on no regular arc of S} U Llz\ <l\(f(z)-A)

is not boundedly invertible}.

(ii) n *n Z(A) and Z(A) both converge to zero in

the strong operator topology if and only if S(z)

is inner. We note that (ii) is the condition for

asymptotic stability for a certain discrete linear

control system.

5. UNITARY PERTURBATIONS

Since the characteristic function of a unitary

map is zero, the above method fails totally when

Z(A) is unitary. However, when A satisfies (1) we

can still get spectral information about Z(A) by

adapting techniques of D.N. Clark [2J to a more

general setting. We begin with two technical

lemmas, and omit the proofs.

5.1 LEMMA

If A is unitary and satisfies (1), then a a = A (I + S(O)A*) (S(O)*+ A*)-l is unitary from C to C*'

5.2 LEMMA

For F = (f,g) E K,

(i) (Z(A)-!XF)=kO 0 where ,x, * * -1 x = -(a + S(O» ('rlF) (0)

* * (ii) (Z(A) -T )(F) = kO 0 where 1 ' ,y

y = -(a + S(O»- f(O).

For Iz\ < 1, defineCP(z): C* ... C* by

* * -1 cp(z) = (I-S(z)a )(1 + S(z)a) • Then straight-

forward calculation gives

* (3) cp(') + cp (1])

2(1 + s(')a*)-l(I-S(')S(I])*)(I + as(I])*)-l

and hence (let z = , = ~) cp(z) has non-negative

real part for \z\ < 1. By the opertor-valued

version of the Herglotz theorem, there exists a

non-negative operator-valued measure ~ on [O,zn]

2Ii:iS i9-1 such that cp (z) = J (e + z)(e -z) d ~ (9).

o

I

il' :1

Thus

(4) cP (')-+<P (11)* =2f(1-,Tj) (l-e -i9 ') -1(1_ei9 11) -ld)J.(9).

Comparing (3) and (4) yields

* * * I-S{C)S{]) = f I+s(')a d)J.(9) I+aS(n)

l-,Tj l_e- i9 , t_e i9li (5)

Similar computations give

(6)

and

(7) I-S{')S(U) _lTTa*I+aS{'t d)J.(9) I+S{li>a* ,-li 0 e -i9 _, e i9 _Tj

2 We define the Hilbert space L ()J.) as in Shulman

[6). For f-XtXEl

+ ••• + Xn~n a simple C*-valued

function where ~ ••••• ~ are characteristic 1 n

functions of disjoint Borel sets and xl ••••• xn are

corresponding elements of C* define

IIf\l2 = )J.

f(d)J.(t)f(t).f(t»=()J.(E1)xl·x1)+···+()J.(En)Xn·Xn)·

This does not depend on the representation of f(t)

in terms of characteristic functions. Let

a- (f(t): [0.2TT) .. c*lf is Borel measurable.

J \If(t)1I 2d()J.(t)x.x) < co for all x E C*. the range

of f(t) is contained in a finite dimensional

subspace of C*}. For f i a let e l .e2 ..... ek be a

basis for the smallest subspace which contains the

range of f(t). and define

a(f.t)-()J.(t)e l .e1) + ••• + ()J.(t)ek·ek)·

The definition is independent of the choice of

basis for this subspace. and IIfI1 2 s:Jllf(t)11 2da(f.t) )J. whenever f is a simple function. For f £ a. there

is a sequence of simple functions (f (t)} such n

that the range of fn(t) is contained in the range

of f(t) for n=1.2 ••••• and such that

20

J Ilf (t)-f(t)112da(f.t) .. 0 as n .. co. We can n 2

define Ilf(t)11 unambiguously as )J.

Ilfll2= 11m Ilf 112. )J. n-tx> n)J.

2 By L ()J.) is meant the Hilbert space completion of

the inner product space of equivalence classes of

functions with finite-dimensional range in )J.-norm. 2 The definition of L ()J.) is such that explicit for-

mulas can be written only for an element associated

with the equivalence class of an element of a. This. however, causes no difficulties for our pur­

poses. It is clear. for example. that the trans-it 2 formation h(t) .. e h(t) is unitary in L ()J.). with

spectrum equal to supp()J.) (the complement of the

largest open set on which )J. is zero).

We are now in position to define a unitary trans­

formation of K onto L2()J.) which transforms the

operator Z(A) on K to the operator of multiplica-it 2 tion by e on L ()J.).

5.3 THEOREM

Define V on elements in K of the form k, by .x.y

x -- * 1+ 5") a

it -e -,

ay.

Then V is well-defined and extends uniquely to a 2 unitary transformation (also V) of K onto L ()J.)

such that VZ(A) = ei~.

Proof:

We first check that V is an isometry on those

vectors where it is defined. Note. for x.y i C*.

* (k k ). (I-S(C)S(]) y.x)C 11.y.0· ,.x.O K l-Ti, *

* * - (JI+ S(C)a d (t) 1+ 05(1)) ) b -it)J. it= y.x C y l-e' l-e n *

(5)

* * .. (1+05{T]) y. 1+05{') x) 2 l_eitli l_e it , L ()J.)

= (Vk- O.Vk, O)L2(). 'I.Y. .x. )J.

Also. for x.y. £ C. - * -

(k .k ) .. (I-S{C) Sen) Y.x)c 11.0.y "O.x K 1-li,

- * - * S 1+ as(C) 1+ S(1))c:v )

(c:v -it d~(t) it _ c:v y,x C e -, e -~

(Vk-,O,y' Vk~ 0 ) 2( ) II .. ' ,x L ~

and finally, for x £ C* and y E C,

(VL ,Vk ) 2 . 1I,0,y "x,O L (~)

Hence V is isometric (and hence also well-defined)

on its domain. Since elements of the form k~ 'I,x,y span a dense set in K, V extends by linearity and

continuity to be an isometry of K into L2(~). Since the range of V contains all elements of the

i~ / it -form x/(l-e w) and x (e - w) for x £ C* and

Iwl < 1, it follows that V is onto L2(~).

It remains to show VZ(A) = ei~. By Lemmas 2.5

and 5.2, -1 -1

Z (A) (k 0) = w k -w k w,x, w,x,O O,x,O -1

+ w kO'(c:v* +S(ot)-l(S(O)*-S(w)*)x,O -1

= w (kw,x,O-kO' (c:v*+s (ot) -l(c:v* + S(wt)x,O)

and hence

VZ(A)k O· w,x, -1 -1

w (l_eitw)-l(I+oS(W)*)x-w (I+aS(wt)x

-1 it- -1 * = w [(1 - e w) - lJ(I+ oS(w) )x

Similarly

Z(A)k = Wk - k w,O,y w,O,y O,X(W)y,O

-k * * -1 * 0, (0' +S(O» (I - S(O) S(W» y,O

So

21

= eitVk • w,O,y

The theorem follows.

We note the following inversion formula for V.

5.4 THEOREM

* 2 Let V : L ~ K be defined, for F in G, by

* V F = (WlF, W2 F) where (WlF)(z)

(1+ S(z)O'*) S (l-e -itz)d~(t)F(t)

and (W2

F)(t) = lim(I - s(reitts(reit»-!:2. r~l

S it * it * it * i(t-e) -1 - (S(re ) -S(re ) S(re )0 )(l-re ) d~(e)F(e)

* Then V is the adjoint of V defined in Theorem

4.3

Proof:

To obtain WI' rewrite equation (5) substituting z

for , and noting that

* Vk- - I + as (1)) x to obtain lI,x,O- l_eitrj

* I-S(z)S(!) x = 1 - '11

Similarly, using equation (6),

- * S(z)-S(]) =SI+S(Z)C:V d (t)(VL )(t) - Y -it ~ . II 0 y

z-T)- l-e z "

This proves the correctness of the formula for

WI for all F of the form Vkm , and hence by -q,x,y approximation for all F £ G. To obtain the for-

mula for W2

, we first find a formula for

* (TlV F)(z). By an argument dual to that above, we

find

* (T IV F)(z)

* * S -it -1 -0 (I+aS(z» (e -z) d~(t)F(t).

The formula for W2

is then obtained by using the

* explicit formulas for T and T in Theorem 2.1.

5. 5 THEORE}l

Let A be unitary and satisfy (1). Then

cr(Z(A»=(I~1 - ll~ lies on no regular arc of

S} U (I~I - 11 ~ lies on a regular arc of S but

(1+ S(\)a*) is not boundedly invertible.}

Proof:

Since Z(A) has a representation as multiplication

by e ia on L2(1J.), we have cr(Z(A» = supp(IJ.), the

complement of the largest open set on which IJ. is

zero. By the integral representation of ~, we see

that the complement of supp(lJ.) is the set of ~ at

which ~(z) has analytic continuation with

Re~ (~) .. O. Since

* * -1 Cjl(z) = (I-S(z)a )(I+S(z)a) ,we have

(I+Cjl(z» - 2(I+S(z)a*)-1 and

S(z) = (I-~(z»(I+CP(z»-lQ'

Now, suppose cp(z) has continuation at ~ and

Re~ (~) .. O. Then (I+cp(\» is boundedly inver­

tible, and hence (I+Cjl(z»-l extends to an analy­

tic function in a neighborhood of~. Thus, S(z)

* has analytic continuation at ~ and (I+S(~)a ) is

boundedly invertible; since Reel' ~) = 0, S~) is

unitary. Conversely, suppose S(z) has analytic

* continuation at~, (I+S(~)a ) is boundedly inver-

tible, and S~) is unitary. Then (I+S(z)a*)-l

is analytic in some neighborhood of~, so ~(z) has

analytic continuation at \; since S(\) is unitary,

Re ~ ~) .. O. By taking complements, the theorem

now follows.

* * * * Since (I+S(\)a )-[(I+S(O)A) -S~)S(O) +A») * -1 * (I+S(O)A) ,we see that (I+S(\)a ) is

boundedly invertible if and only if B(\) -* * * -[(I+S(O)A ) - S(\)(S(O) +A ») is boundedly

invertible. With r as in Theorem 4.1, we have,

since A satisfies (1),

(r (\) -A) =

(I-S(O)S(O)*)%(I-S(\)S(O)*)-lB(\)A(I-S(O)*S(O»-%.

Thus, (r(~)-A) is invertible, but not necessarily

bounded1y, if and only if B(\) is invertible.

Since boundedness follows immediately in the finite­

dimensional case, we have the followins generaliza­

tion of [4, Theorem 3.6) to the case of general

analytic contractions S(z).

5.6 COROLLARY

If A is unitary on C, C finite-dimensional, and A

satisfies (1), then cr(Z(A»=(I\I=ll\ lies on no

regular arc of S} U (\ \ = 1\ \ lies on a regular arc

for S but (r(\) -A) is not invertible}.

In the finite-dimensional case, Z(A) is a compact

perturbation of T. Hence by the known spectral

behavior of T and Wey1' s theorem, q \ \ • 1] \ lies on

a regular arc for S but r(\) - A is not invertible}

must be eigenvalues for Z(A).

We can also adapt Fuhrmann's calculations [4, page

174) to determine eigenvalues in our more general

setting.

5.7 THEOREM

If A is unitary and satisfies (1), and \ lies on a

regular arc for S, then \ is an eigenvalue for Z(A)

if and only if the range of r ~) - A is not dense

in C*.

22

BIBLIOGRAPHY

(1) L. de Branges and J. Rovnyak, Canonical models

in quantum scattering theory, Perturbation

theory and its applications in quantum mecha­

nics, Wiley, New York, (1966), 295-391.

(2) D.N. Clark, One dimensional perturbations of

restricted shifts, J. Analyse Math. 25(1972),

169-191.

(3) P.A. Fuhrmann, On the corona theorem and its

application to spectral problems in Hilbert

space, Trans. Amer. Math Soc. 13291968), 55-66.

(4) P.A. Fuhrmann, On a class of finite dimensional

contractive perturbations of restricted shifts

of finite multiplicity, Is. J. Math 16(1973),

162-175.

(5) H. He1son, Lectures on invariant subspaces,

Academic Press, New York, 1964.

(6) L. Shulman, Perturbations of unitary trans­

formations, J. Math. Anal. and App1. 28 (1969),

231-254.

(7) B. Sz.-Nagy and C. Foias, Harmonic analysis of

operators in Hilbert space, North-Holland

Publishing Co., 1970.

i

23

FREQUENC Y RESPONSE METHODS IN

MULTIVARIABLE INFINITE DIMENSIONAL LINEAR SYSTEMS

John S. Baras Electrical Engineering Department

University of Maryland College Park, Maryland 20742

Abstract

Recent results on the analysis of models and structural properties of linear distributed systems are presented. The presentation emphasi7.es the role play­ed by harmonic analysis in these studies. The conclusions are that a careful selection of mathematical methods makes possible a satisfactory classification and detailed analysis of distributed systems models. These methods provide simple models that reflect input-output data of engineering importance.

SUMMARY

Modeling distributed parameter systems one finds

a number of intrinsic problems that do not appear

in lumped parameter systems modeling. Typically

a linear distributed system is modeled by a differ­

ential equation

dx(t) = A x(t) + Bu(t) dt

y(t) = C x(t) } (1)

Here for a great variety of problems it suffices to

assume that x(t) is in a Hilbert space X [1]. The

operator A arises from a formal partial differen­

tialor integrodifferential operator and it may in­

clude boundary conditions through the definition of

its domain t(A). In all situations A is as sumed to

generate a strongly continuous semigroup of bound­

ed ope rators on X. This last statement is an ab­

stract phrasing of the usual assumption that the

system of equations under study be well-posed.

The controls u are for us square integrable C[;n_

valued functions and the outputs yare square

24

integrable C[;m -valued functions. So u € L 2

and 2 n

y € L Certainly other input and output function m

spaces can be utilized. It turns out however that

2 the L topology gives rise to a particularly rich

theory. This does not state that other function

spaces can not provide theories with similarly rich

structures. The latter remains to be proved how­

ever. It is fair to say that to date other theories

(based on distributions for example (31) have not

produced detailed results like the ones we desc ribe

here.

Describing the properties of the operators Band

C in (1) above is more intricate. Indeed there are

various possibilities that are due to the following

facts: in distributed systems we can (a) apply

distributed control, that is control distributed in

the spatial domain of our partial differential op­

erator or, (b) apply boundary control, that is con­

trol through the boundary conditions of our p. d. e.

system; in distributed systems we can, (c) have

as outputs linear functionals of the whole solution

x, that is distributed observations (in the forITl of

a weighted average) or, (d) have as outputs linear

functionals of the boundary values of the solution

and (or) its derivatives, that is boundary observa­

tions. In (1) B: ([;n -+X and C: X -+ ([;ITl. In the case

of distributed control B is bounded, appears in (1)

directly froITl the physical description of the sys­

teITl and usually Range (B)~ lIl(A). SiITlilarly with

distributed observation C is bounded. In case of

boundary obervation C turns out to be typically

unbounded. The usual situation however is that

C is A-bounded [4]. That is its dOITlain ;I) (C)~

£(A) and Ilcxll([;rfi klllAXllx+k21lx\lX for SOITle

positive kl,k

2 and for all xe:lIl(A). The situation

with boundary control is a little ITlore subtle. In

such cases the physical description of the system

does not result directly in a model like (1). Typ-dx(t)

ically one has a p. d. e. ~ = cr x(t) and a bound-

ary partial differential operator 'I, which gives

the control via 'I x(t) = u(t), with 'I being cr -

bounded. One has to work further to bring the

original description into the forITl of (1). At the

end of this construction one ends with an operator

B that is "unbounded ", in the sense that B now

ITlaps ([; n into V '-:;) X -:;) V whe re Vi is the dual of

V (note here that V is included in X as a set and

not as a Hilbert space, the inner products in X

and V may be considerably different) (see [1] or

[2] for details). However as a ITlap froITl o:;n into

Vi B is clearly bounded. These ideas have been

used formally in engineering probleITls when re­

placing boundary controls with delta-function type

distributed controls.

We mainly analyze here ITlodels that have both

operators Band C bounded. We would like to

point out however that ITlost of the results can b~'

extended to the other cases with additional work

required by the more eOITlplex ITlatheITlatica1

technicalities. The basic ideas reITlain the sanH'.

The matrix valued function T(t) = CeAtB associated

with (1) is the weighting pattern of the system. The

Laplace transform of T is the transfer function A -1 T (s) = C (Is -A) B which is 0 riginally well defined

in SOITle right half plane. The triple (A, B, C) is a

regular realization for T or T, when Band Care At

bounded and T(t) = Ce B. This last equation is a

25

representation for the function T, and thus we

expect classical function theoretic representation

results to be quite useful here. We shall see that

this is indeed the case.

It is clear that spectral properties of the generator

A are crucial for the analysis of systems like (1\.

Utilization of spectral infornlation can provide

structural and qualitative analysis of great detail.

On physical grounds it is desirable that the spec­

tral properties of A "faithfully repres('nt input-

output ITleasureITlents ". Let us nlake the last stat e-A -1

ment ITlore precise. Clearly T(s) = e(ls-A) I;

can be analytically continued in ;) 0 (A), tIll' connect­

ed cOITlponent of the resolvent st"t of A that contain,;

+ 00. To siITlplify the discussion we aSSU111C that

p(A) is connected. Then if we lct ,~(T) denote the ~

set of nonanalyticity of T we have the spectral in-

clusion property [5]. J(T)~,' (i\\. A realization ~

(A, B,C) is spectrally minimal [ ,,] if ,'(T)=,' (A), ~

for sonle analytic continuation of T, and with

ITlultiplicities countl'd whenever l1waningful. Our

position is that spectrally mininlal rl'alizations are

very useful and natural nlOdels for linear distrib­

uted systems. Afterall physicists and engine'l'rs

usually nleasure things like natural frl'qu,'nei,'s.

spectral lines, radiation 111Odl's that arc' rt'flectl'd

in the singulal"ities of T. VI'<' \\'ant thl'n to inn'sti-

gate ('xistl'nce of such tl1odds, find sinlph' 1l1odds

of this type and study rdations lwtwel'n such 111odds.

What follows is a very bril'f SUllUllary of rt'sults in

this dirl'ction. For details and further rl'fl,rt'IKes

Wl' rde r to [5] [(>] 17].

A regular realization (A, B, C) is reachable \\'hl'n-

A':'t ever B':' e x"O for t"2 0 implies X" 0; is obser-

vable whenever C eAtx" 0 for t ~ 0 implies X" 0;

is canonical whenever it is reachable and obser-

vable; is exactly reachable whenever the limit

\ Ibn, eAt BB~' eA*t dt exists as a bounded and

oJ tl-+'" 0

boundedly invertible operator; is exactl~ observ­t 1 A':'t At

vable when eve r the limit lim ' e C ':'C e dt .; t"'''' 0 I

exits as a bounded and boundedly invertible opera­

tor. First notice that the existence of regular

realizations implies certain properties for T. In­

deed we have:

Theorem 1: Let T be an mxn matrix weighting

pattern. If T has a regular realization then T is

continuous and of exponential order. On the other

hand if T is locally absolutely continuous and its

derivative T is of expontial order, T has a reg­

ula r realization.

To proceed in the analysis we need to use the

theory of Hardy functions H~' H'" , H2 2 k mxn mxn

H (.t(C , N» (see [7] for notations). Then

" Theorem 2: Let T be analytic in Res> O. If

T(iUJ) =- C (i:l,)':' R(iUJ) a. e. with C E: H2(.t (Cm

, N) ),

2 n 13 E:H (.t(C., N» where N is an auxiliary Hilbert

" space, then T has a regular realization.

This latter realization is given by

X" H2

(N)

G" en ... X; (Gu)(i'JJ)" 13(iu.;)u

Ft c x P 2 M iwt x

H (N) e

'" I r' H:X"'e

m; Hx" znJ C'(i'JJ)x(iW)dW

-'" whc re M i'lit is the ope rato r I multiplication by

ei,),t I l~his is the translation realization.

(2)

It is interesting to ask when does the factorization

condition of the previous theorem become neces-

ZT5

sary? Then

" Theorem 3: Let T be a transfer function matrix.

If eithe r

(a) T has a dissipative (Le. for xE:£(A),

(Ax, x) + (x, Ax) ,;; 0) globally as symptoticall y

stable (i.e. lim \IeAt

x\\=O, VxE:X) regular t -+'"

realization, or

" 2 T £ H anc:l has a reachable and exactly mxn

(b)

observable regular realization,

then the factorization condition of Theorem 2 is

also necessary.

We would like to analyze case (b) a little further.

Note that the square integrability assumption is

inessential. The Hankel operator is then well

defined: a:>

(HTu)(t)" J T(t+ 0 )u(o )dO (3)

o

or in the frequency domain

H" U "P 2 M" Ou T H T

(4)

m

where ou(iW)" ~(-iW). Then the following is a well

26

defined regular realization:

X" Range (H,,) c H2 T m

" (Bu)(iw)" T(iw) u

1 r'"' C x " 2n J x(iw)dW

-a:>

(5)

(5) is the restricted translation realization. But

we know [6] [8] that if (A, n, C) and (F, G, H) are

two regular, reachable and exactly observable real­

lizations of the same weighting patte rn T, then

the re exists a bounded and boundedl y inve rtible

operator P so that PA" FP, PB" G, C" HP. So it

suffices to analyze the restricted translation real­

ization for this class of weighting patterns (and

thus systems). Note that this is an extremely

i

simple model and that the Fourier transform

(which is a classical function theoretic represen­

tation theorem) was utilized in its construction.

Now Range (H,,) is a left translation invariant sub-T Z~

space, and therefore Range (H,,)= (Q H ) ,ksm, T r k

where Q (itlJ) is isometric a. e. The important r

case is when k=m.

space of full range.

Then Q is inner and the sub­r

This fact must reflect some

" properties of T. The relevant property is that of

existence of a pseudonleromorphic continuation of

bounded type in the open left half plane. A transfer

" function matrix T analytic in Res> 0, has the above

mentioned property if there exists a matrix func­

tion G and a scalar function g, both bounded and " analytic in Res< 0 so that T(iW)=G(iW)/g(iW) a.e.

on the iW -axis. This is a generalization of the

concept of regular analytic continuation. Then the

following are equivalent:

" (a) T has a meromorphic pseudocontinuation of

bounded type in Res < O. Z

(b)(Range(H,,))~=QH ,Q inner. " T r m r

(c) T has a right coprime factorization T(iW) =

= Q (iW) P ':'(iw), with Q inner and P eH"" • r r r rmxn

Now Q determines the spectrum of A in the r

restricted translation realization (5) with multi-

plicities: a (A) = fl..! € OLP such that Q ':'(-W) has r

non null kernel} U r points on iW -axis through which

Q cannot be continued analytically}. Q also r r

determines the singularities of the pseudocontinu-

ation of T and 06) = a (A), multiplicities counted.

Note that, except for pathological cases, the

pseudocontinuation will be a true analytic continu-

" ation fo r T. Thus we have:

Theorem 4: Suppose T€HZ

nH"" T has a mxn mm meromorphic pseudo-continuation of bounded type

" in O.L.P., and T has a reachable and exactly

observable regular realization. Then i) the

restricted translation realization is spectrally

minimal, ii) any other reachable and exactly

observable realization is spectrally minimal.

Note also that Qr

gives a precise state space de­

composition for this class via the Jordan model"

theory of Nagy- Foias [10, ch. III]. Similar results

for discrete time systems can be found in [8]. [9].

and the references therein. We would like to re-

mark again that all of the above can be extended to

the other cases, i. e. B or C or both being un­

bounded. What is involved is a careful analysis of

the restricted translation realization (5) (which

can formally be written for any H"" function) in mxn

order to make the various operators well defined.

This is as far, invariant subspace theory and

Hardy spaces go. There are however inportant

classes of distributed systems that arise from

engineering and physics that are not included here.

To produce examples one needs only consider

transfer functions with branch points. For a

simple example consider heat transfer along a

long bar:

27

Ox (t, z)

at

x(O, z) = 0

x(t, 0) = u(t)

Z o x(t, z)

az Z

lim x(t, z) = 0

z-+ ""

- x(t, z)

y(t) = (temperature at z = 1) = x(t,1)

(6)

Then T(t) -t

e 1 -1/4t d T" () -ISTI e an s = e .

One can write a translation realization for this T

x = LZ[O,"")

At l' [0) e = left trans ahon an ,00 (7)

"" Cx = J g(t)x(t)dt, g(t)=e-

t

o This is a canonical regular realization. However

a (A) = closed L. P. while C1 cI') =

fbranch cut from -1 to ...0:>1. Thus no spectral

minimality. But certainly (7) is an unnatural Theorem 6 ([6]): Let (A, B,C) and (F,G,H) be

model for (6), because it ignores the great internal canonical regular realizations of T, with A = A''',

symmetry of (6). One has to use other means. In

particular many problems from mathematical

physics lead to models like ( 1) where A is selfadjoint

or normal. Then one can show by the use of the

spectral thoorem that if (A, B, C) is a canonical

regular realization for T, and A = A ':' this

realization is spectrally minimal [6]. The inter­

nal symmetry of the system results to additional

properties for T, which then can be utilized to

construct simple models. A simple example,

which illustrates the point, and also indicates how

classical function representation results can be

used here, is provided by the well known to elec­

trkal engineers completely monotonic and positive

definite functions. These arise naturally from

lumped distributed RC networks. A function cp is

completely monotonic, if it is C'"' on [ 0, co) and

(_l)n cp (n) (t) ~ {) for t> 0, and is positive definite

on (-'"', '"') if L: cp(t.-t.) a.a . ., 0, for every set of .• 1 J 1 J 103

real numbers ft.} and complex numbers fa.} . 1 1

Then we have [6]:

Theorem 5: A weighting pattern T is completely

monotonic if and only if it has a regular realization

(A, b, b) with A = A* and stable. T has a positive

definite extension on (-,"" ... ) if and only if it has a

regular realization (A, b, b) with A = -A':'.

One uses Bernstein's representation of completely

monotonic functions and Bochner's rep res entation

of positive definite functions to construct simple

models.

Under such symmetry the state space isomor­

phism theorem can be improved. It is important

to note that spectral minimality results from

assumptions on A alone (like A = A'~, A normal),

while the state space isomorphism theorem re­

quires additional symmetry:

28

B=C'~, F=F'::, G=H':'. Then they are similar via

a unitary map.

REFERENCES

[ 1] J. L. Lions, Optimal Control of Systems Gov­erned by Partial Differential Equations, Springer- Verlag, 1971.

[2] H. O. Fattorini, "Boundary Control Systems ", SIAM J. Control, Vol. 6, No.3, pp. 349-38S, 1968.

[3] A. Bensous san and J. P. Aubin, "Models of Representation of Linear Time Invariant Sys­tems in Continuous Time ", Univ. of Wisconsin­Madison, Math. Res. Ctr., Report MRC #1286, Sept. 1972.

[4] T. Kato, Perturbation Theory of Linear Oper­ators, Springe r- Ve rlag, 1966.

[S] J.S. Baras and R.W. Brockett, "H2 Functions and Infinite Dimensional Realization Theory", SIAM J. Control, Vol. 13, No.1, pp. 221-241, Jan. 1975.

[6] J. S. Baras, R. W. Brockett and P. A. Fuhrmann, "State-space Models for Infinite Dimensional Systems ", IEEE Trans. on Aut. Control, Vol. AC-19, No.6, pp. 693-700, Dec. 1974.

[7] J. S. Baras and P. Dewilde, "Invariant Sub­space Methods in Linear Multivariab1e Dis­tributed Systems and Lumped Distributed Network Synthesis ", to be published in IEEE Proceedings, Special Issue on Recent Trends in System Theory, Jan. 1976.

[8] J. W. Helton, "Discrete Time Systems, Oper­ator Models and Scatte ring Theory", J. Funct. Analysis, 16, 1974, pp. lS-38.

[9] P. A. Fuhrmann, "Realization Theory in Hilbert Space for a Class of Transfer Functions ", J. Funct. Analysis, 18, pp. 338-349, 1975.

PO] B.Sz-Nagy and C. Foias, Harmonic Analysis of Operators on Hilbert Space, North, Holland, Amsterdam, 1970.

L

ON SIMULTANEOUS DIAGONALIZATION

OF A COLLECTION OF HERMITIAN MATRICES

S. Chakrabarti, B.B. Bhattacharyya

and

M.N.S. Swamy

Department of Electrical Engineering Cor cordia University

Montreal, Quebec H3G 1MB, Canada.

Abstract

Existing results on simultaneous diagonalization of a pair of hermitian matrices have been extended for more than two matrices where the necessary and sufficient conditions are derived for simultaneous diagonalization through a single con­gruent transformation or through a pair of contragradient transformations under various conditions. It has been shown that the techniques preserited here are also applicable in a straightforward fashion for matrices with multivariate functional entries.

I. INTRODUCTION

The problems of simultaneous diagonalization of a

finite number of matrices of finite order, through

a single transformation or through a pair of

specially related transformations, occur often in

statistics and engineering, particularly electri­

cal engineering. Initiated by Weierstrass' study

of strict equivalence and the canonical forms

for regular pencils of matrices and supplemented

by Kronecker's study of more general problems

involving singular pencils, similar problems have

been receiving continuing attention from the

mathematicians. (1-3,6) The necessary and suffi­

cient conditions for diagonalizing a finite set

of n x n nondefective* matrices through a single (1)

similarity transformation are well-known.

*An nxn matrix is called "nondefective" if and only if, it has n linearly independent eigenvalues,( 4) i.e. the matrix is similar to a diagonal matrix •

29

Similarly, the necessary and sufficient conditions

for diagonalizing a pair of hermitian matrices A

and B through a single congruent transformation,

i.e., through transformations QHAQ and QHBQ

where* Q is nonsingular, so that both QHAQ and

QHBQ are diagonal, are known. (3) Various special

cases of the last problem have been discussed in

(2) and (3), of which (3) includes the most detail­

ed study of this problem (and related problems)

known to the authors.

Simultaneous diagonalization of a pair of hermitian

matrices, A and B, through contragradient

transformations, i.e., through transformations of

the form Q-lAQ-lH and QHBQ has also been ob-

*Throughout this paper, symbols of the form AH will denote the transposed of the complex conjugate of a matrix A.

tained by Rao and Mitra. (3) The necessary and

sufficient conditions for simultaneous diagonaliz­

ation of a finite set of hermitian matrices {Ai}'

through a single unitary matrix transformation of H the form U AU ,where U is unitary, are also

known. (3,5) To the best knowledge of the authors,

the problems of simultaneous diagonalization of a

finite set of more than two hermitian matrices

through a single congruent transformation or

through a pair of contragradient transformations

are yet unsolved. The purpose of the present

paper is to extend the existing results on the

problems of the type mentioned above. In particu­

lar, the necessary and sufficient conditions for

the simultaneous diagonalization of a set of

hermitian matrices through a single co-gradient

transformation is presented.

It may be recognised that since the matrices

involved are hermitian whenever QHAQ or H

Q-lBQ-l are diagonal (where A and B are her-

mitian), these are real diagonals.

Finally, it has been shown that the problems of

simutlaneous diagonalization of hermitian matrices

the entries of which are complex-valued functions

of many complex variables can be converted into

the equivalent problems of simultaneous diagonal­

ization involving constant matrices.

2. SIMULTANEOUS DIAGONALIZATION

OF CONSTANT HERMITIAN MATRICES

In this section, we begin with a result quoted

from Rao and Mitra. (3)

Lemma 1 [Theorem 6.4.5] Let A and B be a pair

of hermitian matrices of the same order. Then a

necessary and sufficient condition that there

exists a nonsingular matrix H

Q such that

Q-lAQ-l and QHBQ are both diagonal is that:

rank (BAB) ; (ii) AB is nondef-(i) rank (BA)

ective with real eigenvalues.

However, it is easy to prove that for ~ two

matrices A and B, not necessarily hermitian,

such that BA is nondefective, rank (BA) = rank

(BAB) always holds. Note that rank (AB) < min

30

{rank(A) ,rank (B)}

«BA)B) < rank (BA)

Hence, rank (BAB) = rank

We also know that rank of a

matrix M is equal to the rank of its square, i.e.

rank (M) = rank (M2) if, and only if, the elemen­

tary divisors of M corresponding to zero eigen­

value are linear. Since BA is nondefective, all

elementary divisors of BA are linear and hence,

rank (BA) = rank (BABA) , i.e., rank (BA) = rank

«BAB)A) ~ rank (BAB) • This and the last inequal­

ity together yield rank (BA) = rank (BAB).

In that case, clearly, the condition (i) of Lemma

1 is superfluous in view of condition (ii).

Without using the condition (i) of Lemma I, we

shall now present an alternative proof for the

necessity and sufficiency of condition (ii) of the

same lemma. The proof is more straightforward

than the one presented in (3).

Lemma 2. Let A and B be hermitian matrices

of the same order. Then there exists a nonsingular H

Q-lAQ-l matrix Q such that and are

both diagonal if, and only if, AB is nondefect­

ive with real eigenvalues.

Proof. If anyone of the matrices A and B is

the null matrix, the lemma holds trivially. Let

us consider that both A and B are non-null.

In order to prove necessity we note that if Q

-1 _lH H exists as above so that Q AQ and Q BQ are

diagonal, and hence real diagonals, the product

Q-lABQ is then real diagonal. The necessity

follows.

In order to prove sufficiency, we shall consider

two separate cases, viz., AB = 0 and AB + 0 •

Since A, B are hermitian, AB o if, and only

if, BA = 0 ,i.e., A and B commute with each

other. Hence there exists a unitary matrix U H

such that UHAU = UHAU-l and UHBU are diagonal

and the sufficiency is proved.

Finally, let us consider the case AB + 0 Then

AB is nondefective with real eigenvalues means

that there exists a nonsingular T such that

1 -1 _lH H T- ABT = A is real diagonal. Hence, T AT T BT

H THBTT-lAT-l i ; .e., the matrices

H T-lAT- l and THBT, which are hermitian since

A, B are hermitian, commute. Hence, there exists

a unitary transformation U such that H

UHT-lAT-l U and UHTHBTU are simultaneously

diagonal. Thus Q = TU is the desired nonsin­

gular diagonalizing transformation.

Q.E.D.

Theorem 1. Let Al ,A2,···,Anl and Bl ,B2 , ••• , Bn2

be non-null hermitian matrices of the same order

and let anyone of these matrices, say Bl be

invertible. nonsingular

matrix Q

Then there exists a H

such that Q-lA Q-l i

and are

diagonal for i=1,2, ••• ,nl

and

if, and only if, (i) AiBl and

j = 1,2, ••• ,n2 -1

Bl Bj are non-

defective with real eigenvalues, V i and j = 2,

3, ••• ,n2 ' and (ii) the non-null elements of

the set -1 -1

{AlBl,A2Bl,···,Anl Bl,Bl B2 ,Bl B3"'"

pairwise commute.

Proof. Necessity: Given that

singular matrix Q such that

are real diagonal, V i,j

-1 = Q AiBjQ is real diagonal for

and j = 1,2, ••• ,n2

• Similarly,

= Q-lB-lB Q is real diagonal for 1 k

k = 2,3, .•• ,n2 •

The necessity of (i) follows. Since the matrices

and -1

Bl Bk are simultaneously diagonaliz-

able through a single similarity transformation,

the necessity of (ii) follows immediately.

Sufficiency: Note that the conditions

(i) and (ii) given above imply that there exists

a nonsingular matrix T such and

-1 -1 T Bl BjT = ~j are real diagonal for

and j '" 2,3, ... , n2 • Then clearly,

31

and

and

Vi and Vj , the matrices

pairwise commute and the matrices

pairwise commute. Since for any

two commuting matrices M and

exists, M-l also commutes with N, it follows

that the elements of the entire set of hermitian

matrices H -1 H

{(T BIT) , Al ,A2,···,Anl

' (T BIT) ,

pairwise commute. Hence there

exists a unitary matrix U such that

H H H U , U T Bl TU , U Ai U V i

real diagonal. Thus,

is real diagonal, Vi. Similarly,

= UHTHB TUUH~ U is real diagonal for 1 j

j =2,3, ... ,n2

' and hence, UHTHBkTU is diagonal

for k=1,2, ••• ,n2

• Identifying Q=TU as the

desired diagonalizing transformation. the

sufficiency follows.

Q.E.D.

Theorem 2. Let and

be non-null hermitian matrices of the same order

and let at least one of the matrices from each

collection, say Al and Bl , be invertible.

Then

that

there exists a H

Q-lA Q-l and i

nonsingular matrix Q such

QHB.Q J are diagonal for

i=1,2, ... ,nl

and j =1,2, ... ,n2 if, and only if,

(i) AB is nondefective with real eigenvalues

and (it) the non-null products {AiBj,i = 1,2, •••

,nl

; j = 1,2, ••• ,n2} pairwise commute.

Proof. Necessity follows in the same fashion as

in Theorem 1. For sufficiency we note that since

(i) and (ii) hold, there exists a nonsingular

matrix

diagonal,

V i,j

commute Vj , i = 1, 2 , ••• , n1

• Then

H H -1 T-1A T-1 = Ail (T B1 T) , Vi i

H H (T-1A1 T-1 )-1 A

1j Vj T BjT

i.e. , Ail commutes with H -1 (T B

1T) , Vi and

A1j

commutes with THA-1T Vj, because H 1 '

T-1A T-1 and THB T are hermitian, V i and i j

V j Then, Vi,

k = 1,2, ••• ,n1

Similarly, V j, R.; j,

R. = 1,2, ••• ,n2 • Hence, the hermitian matrices

H H -1 _lH H H T-1A

1T-1 ,T-1A

2T-1 , ••• ,T An1T ,T B

1T,T B

2T, ••• ,

THB T pairwise commute with one another. Hence, n2

there exists a unitary matrix U such that H

u~-lAiT-1 U, Vi and UHTHBjTU, Vj , are diag-

onal. Identifying Q = TU as the diagona1izing

transformation, the sufficiency follows.

Q.E.D.

Theorem 3. Let Al ,A2,···,An1 and B1

,B2

, ••. ,Bn2 be two collections of non-null hermitian matrices

of the same order such that the products

are non-null nonderogatory* with real eigenvalues

V i,j

such that

Then there exists H

Q-1A Q-1 and i

a nonsingu1ar matrix

QHB Q are diagonal, j

* A matrix is called derogatory(4) if the same eigenvalue of this matrix can occur in more than one elementary Jordan li10cks. O~~erwise, the matrix is called non-derogatory( •

Q

32

V i,j , if, and only if, (i) Ai B j is nondefect­

ive with real eigenvalues and (ii) the products

AiBj commute with one another, V i,j

Proof. The necessity is obvious. To prove suffi­

ciency, we note that (i) and (ii) imply that there -1

exists a nonsingu1ar matrix Q such that Q AiBjQ

= Aij is real diagonal, V i,j , and further,

for any fixed i and for any fixed j , the diago-

nal elements of Aij are different from one anoth-

er. In that case,

Similarly,

H (Q BjQ)Aij , V i,j

Using a well-known result (1; Theorem 3, p.223) it

follows immediately from the structure of Aij'S H

that Q-1AiQ-1 and QHBjQ are diagonal, Vi

and V j

Q.E.D.

In the next two theorems, the conditions are sev­

erely restricted so as to make the identification

of the diagona1izing matrices particularly simple.

Theorem 4. Let A1

,A2,.·.,An1 and B

1,B

2, ••• ,Bn2

be two collections of non-null hermitian matrices

of same order such that (i) the products AiBj

are non-null, nondefective with real eigenvalues

and pairwise commuting, V i,j ; and (ii) for all

Bj (resp. Ai)' there exists

a Bj , say B1), such that

nonderogatory*, V j (resp.

exists a nonsingu1ar matrix H and Q B. Q , j = 1,2, .•• , n2 J -1

only if Q A1

Bj Q, V j and

real diagonal.

* Ibid.

an Ai' say A1 (resp.

A1Bj (resp. Ai

B1

) is

Vi) • Then there

Q such that

are diagonal if, and -1

Q Ai B1 Q, V i are

!

~

Proof. Necessity is obvious. For sufficiency

we note that:

-1 Q AlBjQ Alj

-1 Q AiBlQ An

are real diagonal, V j and V i and further,

for each of these diagonal matrices the diagonal

mutually different. Then H H

Q-lA Q-l • QHB Q' Q-lA Q-l = A iIi n

H H-l and (Q B.Q)Al . = Q B.QQ Al

J J J

Vj Again, using the result

given in (1; Theroem H

Q-lAiQ-l , V i and

3, p.ZZ3), it follows that

QHBjQ, V j ,are diagonal.

Q.E.D. Theorem 5. Let Al,AZ""'~l and Bl,BZ, ••• ,BnZ

be two collections of non-null hermitian matrices

of same order such that (i) the products AiBj

are non-null, nondefective with real eigenvalues

and pairwise commuting; and (ii) let there exist

one matrix in each collection, say Al and Bl

,

such that Al and Bl are nonderogatory and

invertible. Then for any nonsingular matrix Q -1 such that Q AiB.Q is real diagonal, V i,j

J H the matrices Q-lAiQ-l, V i and QHBjQ, V j

are diagonal.

Proof. Condition (i) guarantees the existence of -1

a nonsingular matrix Q such that Q AiBjQ = Ai' H J

is real diagonal, V i,j Then (Q-lA Q-l ) = 1

H -1 H All(Q BlQ) is hermitian implies that Q BlQ iSH

diagonal (1; Theorem 3, p.ZZ3) and H

is diagonal. Then Q-lA Q-l i

hence Q-lA Q-l 1

H -1

diagonal, V i and QHB.Q J

is diagonal, V j

= An (Q BlQ) is -1 _lH_l

Aij(Q AlQ ) Aij

Q.E.D.

If a matrix A is both nondefective and nonderog­

atory, then the dimension of the eigenspace asso­

ciated to each eigenvalue of A is 1 and the

eigenvectors of A are linearly independent. Then

if B is another nondefective and nonderogatory

matrix such that AB = BA , it follows immediately

33

that any matrix Q which diagonalizes A through

a similarity transformation also diagonalizes B

through the same similarity transformation. In

the case of Theorem 3, Theorem 4 and Theorem 5, if

Q is a diagonalizing transformation, Then -1

Q AiBjQ is real diagonal, V i,j ,regardless of

whether the products AiBj are derogatory or non­

derogatory. However, as far as the nonderogatory

products are concerned, if these products commute,

then we need to identify only a modal matrix of

anyone of these products as the matrix which

simultaneously diagonalizes the nonderogatory

products AiBj through a similarity transform­

ation. Let us consider anyone of the derogatory

products, say

matrices Ai B j

values of AZBZ

AZBZ ' of the pairwise commuting

and let V be one of the eigen­

with a geometric multiplicity (4)

p ,where 1 < P < n n being the order of

the matrices. Then the dimension of the eigen-

values associated to V is p. This p-dimensiDn­

a1 eigenspace contains an appropriate set of, -1

exactly p column vectors of Q, since Q AiBjQ

is diagonal, Vi ,j Note that for AZB Z ' any

nontrivial linear combination of p linearly

idnependent eigenvectors associated to V is

also an eigenvector associated to

Q is a nonsingular matrix such that

Hence, if ·-1 ' Q AZBZQ is

.-1 • diagonal, Q A.B.Q

1. J will not necessarily be diago-

na1 for i" Z , j " Z Thus we conclude that for

Theorems 4 and 5, the diagonalizing matrix Q

can be identified as any modal matrix of anyone

of the nonderogatory products

The rest of this section will deal with the simul-

taneous diagonalization of hermitian matrices

through a single cogradient transformation. In

order to do so, the following result is needed,

which is quoted almost verbatim from Rao and

Mitra(3); the theorem number is quoted in paren­

thesis.

Lemma 3 (Theroem 6.4.Z). Let A and B be a

pair of n x n hermitian matrices. A necessary

and sufficient condition that there exists a non­

singular transformation Q such that QHAQ and

QHBQ are both diagonal is

.I.H .LH J. (i) rank (B A) = rank (B AB) ,

.L .LH .L _ .LH (ii) [A - AB (B AB) B AJ B - is nondefective with

.I. real eigenvalues; where B denotes the matrix

whose range space is the orthogonal complement of

the range space of Band B is a generalized

inverse of B where BB B = B; the superscript

"-" denotes this kind of generalized inverse.

In proving this result Rao and Mitra snowed that

the matrix Q, henceforth called a "diagonalizing

matrix" is of the form

where S is a nonsingular matrix such that

E and Fare hermetian matrices, D is real

diagonal, K is a nonsingular matrix such that

KHEK and KHDK are diagonal, U is a unitary

matrix such that UHFU is diagonal; E ZHLAZ

F = BJ.HAB J. , Z = yH(e~H)-l e is an nxr

matrix such that B = eDeH eHe I and I is r r

the r xr identity matrix where r = rank(B) ;

y is such that (B-)HBB-= yHy where y~ = I .I. .I.H .L _ .LH r

and A is diagonal; L = In - AB (B AB) B

and finally, S has the partitioned form as

follows:

Our next step is to generalize this result for

more than two hermitian matrices.

First we note that for any two n x n non-null

hermitian matrices A and B if A =!;B for

some non-zero complex number ~, then for any non­

singular matrix Q, QHAQ is diagonal if, and only

if, QHBQ is diagonal. Hence, without any loss of

generality,

Al ,A2 ,··· ,Am

complex number

Assumption I.

we can assume that

are such that

!; 'I i,j

Ai ; !;Aj for any

Let us call this

34

Let us now assume that the non-null hermitian mat­

rices A and B have common-position columns

which are the same in both A and B, i.e., the

ith column in A is identical to the ith column in

B and so on. These columns may be all-zero or may

contain non-zero matrices. Then we know that there

exists a real nonsingular transformation P ,

expressible as a product of standard elementary

transformations, such that

where the superscript "T" denotes matrix transposi-

tion, A and B are hermitian matrices of the

same order such that there are no common-position

columns which are the same in both A and B.

Let Al ,A2, ... ,Am denote the reduction of the non­

null hermitian matrices of same order, Al

,A2

, .•. ,

Am ' as performed in (2) such that Al ,A2

, ••• ,Am do not have any common position columns which are the

same in all these matrices. In that case it follows that

Lemma 4. Al

,A2, ... ,Am are simultaneously diagona­

lizable through a single cogradient transformation

if, and only if, Al

,A2 , ... ,Am are simultaneously

diagonalizable through a single cogradient trans­

formation.

Proof. Necessity: Let P be the real nonsingular

transformation such that

A = i

as in (2). Let Q be a nonsingular matrix such H

that Q AiQ = Ai

M = P-lQ. Then

is diagonal, 'I i Define

M is nonsingular and partition-

ing M as

where

that

MIl has the same order as H -

MllAiMll is diagonal, 'Ii

Ai ' it follows

Sufficiency: Given that there exists a H -

matrix, say MIl ' such that MllAiMll are diagona~

Vi. Then it is trivial to verify that for any

matrix M,

M

where MZl and MZZ are arbitrary submatrices

"uoh <h", "~i :]M a<e well-defined, • i ,

MH rAo- i °o~ M the matrices L J are diagonal. In that

case, if we define Q = PM , then is

diagonal, V i and sufficiency follows.

Q.E.D.

As a consequence of this lemma, it can be assumed

without any loss of generality that the non-null

hermitian matrices Al,AZ, ••• ,Am do not have any

common position columns which are the same in all

these matrices. Let us call this Assumption II.

Finally, let ai denote the a-th column of Ai' -a

1 < a < n , where the order of Ai is n x n for il3 il3 il3 il3

all i. Let ~a = ~l~a = ~ a = ••• = ~ ~ 1 Z-al PI3 apl3

10,1< P _< n - 1 , Z < 13 < m, ~l'~Z'···'~ - - PI3 are all non-zero complex numbers, where we also

assume that no other columns of Al,AZ,···,Am satisfy this relation (if there are other such

columns, the following procedure can be applied

successively and only the final result need be

considered). Then again we note that there exists

a nonsingular transformation E, expressible as a

product of standard elementary transformations,

such that

where Ai is a hermitian matrix of order

(n - m~n PI3) x (n - m~n PI3) , Vi, and there exists

at least one Ai such that the columns of this

matrix cannot be expressed as a non-zero complex

multiple of one another. It is easy to check

that Lemma 4 holds even when Ai is replaced by

Ai. Hence using the same kind of arguments as

before, it can be assumed without any loss of

generality that Al,AZ, ••. ,Am are such that there

exists at least one matrix Ai' 1 ..::. i ..::. m , such

that all columns of Ai are different from one

another. Let us call this Assumption III.

Definition 1. A set of non-null hermitian matrices

of the same order, containing at least two elements

of this set simultaneously satisfy the Assumptions

I, II and III, will be called an almost regular set

of matrices.

It is easy to see that every almost regular set of

matrices contains at least two elements, say Ml

and MZ

' such that there exists a non-zero real

number A where Ml + AMZ is nonsingular. This

is due to the fact that for any almost regular

set of matrices we can pick up two elements of the

set, Ml and MZ ' such that {Ml,MZ} is an almost

regular set of matrices and one of the matrices,

say Ml ' is such that its column cannot be express­

ed as a non-zero complex multiple of one another.

Then treating A as a formal parameter, it follows

that Ml + AMZ has full rank and det(Ml + AMZ) ,

expressed as a polynomial in A , has exactly

rank(Ml

+ AMZ

) roots. Then for any real A ,

35

A I 0 , which is not also a root of det(Ml + AMZ)

clearly Ml + AMZ is nonsingular. For such a A,

the triple (Ml,MZ,A) will be called a regular

triple for the almost regular set of matrices.

Note that the above argument also yields a proce­

dure for determining A for any regular triple.

We can now state and prove the following theorem

which incorporates the construction of a diagonal­

izing transformation whenever it exists.

Theorem 6. Let {AI ,AZ ' .•• ,Am} be an almost

regular set of matrices and let {Al,AZ,A} be a

regular triple for this set. Let B = Al + AAZ •

Then there exists a nonsingular matrix Q such H -1

that Q AiQ is diagonal if, and only if, (i) B Ai

are nondefective with real eigenvalues, Vi, and

(ii) The matrices B-lA. pairwise commute with 1.

one another, Vi.

Proof. Necessity: Given that there exists a non­

singular matrix Q such that QHA Q is diagonal, i

and hence real diagonal, V i Then QHBQ = A is

real diagonal and necessity follows.

Sufficiency. Given that the conditions

(i) and (ii) hold. Hence there exists a

lar matrix T such that T-lB-lA T = A i i

diagonal, Vi Then T-lB-lT-lHTHA T i

THAi T = (THnT)Ai

, V i Since THA T i

nonsingu-

is real

is hermi-

tian, (THBT)Ai Ai(THBT) i.e., THBT commutes

with Ai' Vi and since Ai's are diagonal, H the elements of the set {T BT ,AI ,A2,· •• ,Am}

commute pairwise. Thus there exists a unitary

matrix U such that UHTHBTU and UHAi U, Vi

are real diagonal. Hence

Le., UHTHAi TU is real diagonal, Vi. Then

Q = TU is the desired diagonalizing transform­

ation and the sufficiency follows.

Q.E.D.

The procedure for determining Q may be formal­

ized as follows:

Step 1: Reduce the given set of matrices to an

almost regular set of matrices by using standard

elementary transformations as given in (2) and

(3).

Step 2: Select any regular triple {Ai,Aj,A} for

this almost regular set of matrices and construct

B = Ai + AAj •

Step 3: Check whether (i) and (ii) of Theorem 3

are both satisfied for this B and the almost

regular set of matrices. If these conditions are

satisfied go to the next step. Otherwise, stop.

Step 4: Construct the diagonalizing matrix Qll

as given in the proof of sufficiency in Theorem 6.

Then the desired diagonalizing transformation will

be of the form PQ

vbe« Q - [:~: as in the proof of Lemma 4,

o J and P is the product Q22

of standard elementary transformations used to

reduce the given set of matrices to the almost

regular set of matrices as shown in (2) and (3);

Q2l

and Q22 are arbitrary matrices such that

36

Q is nonsingular and PQ is well-defined.

Before concluding the discussion on this problem we

shall present one more result which follows as a

direct corollary to Theorem 6.5.2 in Rao and

Mitra (3) •

Lemma 5. Let Ai

,A2

, ••• ,Am

be non-null hermitian

matrices of the same order and let one of these

matrices, say Al ' be positive semidefinite, and

)t(Ai ) c )t(Al

) for i = 2,3, •.• ,m , where J't(Ai

)

denotes the column space of Al and so on. Then

there exists a nonsingular matrix Q such that

QHAiQ is diagonal, Vi , if and only if, AiA~Aj = AjA~A.1 ,for i = 2,3, ••• ,m and j = 2,3, .•• ,m ,

where Al is any generalized inverse of Al such

that AiA~Al = Al

The proof is obtained in the same manner as for the

one given for Theorem 6.5.2. in Rao and Mitra(3).

3. SIMULTANEOUS DIAGONALIZATION OF MATRICES

WITH FUNCTIONAL ENTRIES THROUGH CONSTANT "

COMPLEX TRANSFORMATIONS.

In this section a brief discussion will be present­

ed on the simultaneous diagonalization of hermitian

matrices with functional entries through a single

constant congruent transformation or through a pair

of constant contragradient transformations. Consi­

der a finite collection of (nl

+ n2

) hermitian

matrices, denoted by Ml ,M2,···,Mnl and Nl

,N2

, ••

• ,Nn2 ' each of the same finite order n x n Let

the matrices be of the form as given below:

i I< j

i I< j

where i,j = 1,2, ..• ,n; the subscript and super­

script k corresponds to the kth matrix ~, and

similarly for l; x(k) £ ¢Yk , y<l) £ (,H; '\

and 0l are positive integers> l, Vk,l; C[

"denotes the field of complex numbers; the symbolsof

the form ~Yk denotes the yk-dimensional vector-

space over ([.

copies of ct ; obtained as the product of

(k) Yk Il ij £ [([ :([), II i,j ,k

II i,j,R, where the symbols of

denotes the linear space of all Yk

complex-valued functions on ~ with ~ as the

underlying field. Whenever possible, we shall

drop the characterizers (~(k» and (y(R,».

The problems of diagonalization may be formulated

as follows:

Problem I. Find the necessary and sufficient

conditions that a non-singular constant n x n

matrix Q exists such that

are diagonal for k = 1,Z, ••• ,nl

Problem II. Find the necessary and sufficient

conditions that a non-singular

matrix Q exists such that

nxn constant

(Sa)

(Sb)

and diagonal for k = 1,Z, ••• ,nl

and R, = 1,Z, ••• ,nZ

Let the symbol ~[fl,fZ, ••• ,fp] denote the finite­

dimensional linear space of functions generated by

the finite set of complex-valued functions

{fl,fZ, ••• ,fp} with ~ as the underlying field.

Similarly, let .-&g <f l' f Z ' ••• , f g > denote the

linear space of functions over ~ for which a

finite set of g linearly independent functions

{fl,fZ, ••. ,fg} form a basis. Whenever there is

no danger of confusion, the symbols of the form

~g <fl,fZ, ••• ,fg> will be abbreviated to ~g The "script" capital letters will be reserved for

representing such finite-dimensional linear spaces

of complex-valued functions with , as the under­

lying field.

and

Lemma 6. A necessary condition that ~ will be

diagonalized for all k in the form of equation

(4) through a complex constant invertible matrix (k)

Q is that every Il ij - entry of ~, II i,j,

belongs to a p(k)-dimensional subspace of

[~Yk :~], denoted by J(. (k) where 1 < p (k) _< n; p -

37

k = 1, Z , •.. , nl

• The subspace .ltp

(k) is the small­

est such subspace in the sense that for every other

subspace Jtp over d containing all Il~~)-entries of ~, we have p (k) 2. p, (k) = S'Jt and for

(k) p P P P = P , jt (k) = JI: .

p p

Proof: It is sufficient to prove this lemma for

only one k. The most general form of the n x n • (k)

diagonal matrix, ~(~ ), may be written as

follows:

(k), i-I Z • where all II. s , -" ••. ,n , are not necessar-,(l.

ily linearly independent of one another or not even

different. Let

f::. D (k) (k) (k) Yk ftp (k)=/.){Il l ,liZ ,···,Iln }C[4: :(.]

Evidently, p(k) 2. n. On the other hand, p(k)~ 0

corresponds to the trivial case of the subspace

{O}. Hence, 1 2. p(k) 2. n In that case,

denoting the largest subset of the linearly inde-

pendent elements of the set of generators (k) (k) (k)

{Ill ' liZ , ••• ,11 } of ~ (k) (k) (k)

by {Ill ' liZ ,. (k) n p

",Ilp (k)} , (after re-indexing, if necessary), it

is possible to write:

i.e., the rest of the functions belonging to the (k) (k)

set of generators, viz., II (k)+l,1l (k)+l"'" (k) p P

lin ' may be expressed as a non-trivial linear com-

bination of the basis function. Therefore, since

equation (1) holds, we can write:

which yields

II i,j ; (6)

the "tilde" denotes the complex conjugation. Thus (k) Il ij e: fl.p (k), II i,j. The last statement is

obvious from the definition of ~ (k) • p

Q.E.D.

From now on, our attention will primarily be

restricted to simultaneous diagonalization through

a single cogradient transformation;however, it will

be obvious from the subsequent discussion that the

same arguments can be used for the case of simul­

taneous diagonalization through a contragradient

transformation.

Except in the simplest case of p(k) equal to 1,

the problem is now to identify a basis for Ji (k) P

Since these spaces are finite-dimensional, the

most natural approach for determining the bases is

to follow the methods for doing the same in the

case of ~n as closely as possible. This means

that the spaces ~ (k) should be pre-Hilbert p

spaces with respect to suitably defined inner-

products. The inner product structure enables us

to employ Gram-Schmidt orthonormalization techni­

que for identifying an orthonormal set of basis

functions for each ~ (k) • P

The Gram-Schmidt orthonormalization is then per-(k) (k) (k) (k)

formed on the sets {~ll '~12 '···'~ij '···'~nn } using the standard procedure (including whether

the Gramian is zero or not at every step of ortho-(1) normalization) , for k=1,2, ••• ,n

l• Let

{A(k) A(k) A(k)} ~l '~2 " ••• ,~p (k) denote an orthonormal

basis of Jl (k) obtained in this fashion. Since

for p(k) >Pn the simultaneous diagonalization

is not possible as mentioned in Lemma 6, in order

to proceed further, we shall now assume p(k) < n

\/k

Let <·1·> denote the inner-product operation

corresponding to the appropriate space. Let

Then

(k) ~ij

Similarly,

(k) ~r

then,

(k) lID ,tij e:1I', \/i,j,k,m m

••• (7)

t(k) = <~ (k) A(k) (8) ~ >

ijm ij m

let

(k) =! 4I(k) A (k) , .~k) e: ct, \/ r,k,m; (9) ~m m=l rm m

41 (k) < (k) A (k) (10) ~r ~ > r m m

38

Let us now define n x n matrices

onal matrices ~(k) as follows: m

T(k) and diag­m

~(k) m

[t~~) ];i,j=1,2, .•. ,n 13m

. (k) (k) (k) d1ag{<I> 1 ' <1>2 , •.. ·<I>n }

m m m

with k=1,2, ••• ,nl

and m=1,2, .•• ,p (k)

(11)

(12)

Since

~ are hermitian, \/ k , obviously, T~k) mitian, \/ k,m •

are her-

Theorem 7. The collection nl

hermitian matrices

Ml

,M2

, ••• ,Mnl is simultaneously diagonalizable as

in (4) if, and only if, Q-lT(k)Q-lH ~(k) are m m

real diagonal matrices, \/ k,m •

Proof. Necessity: Substitution of (9) into (6)

yields:

(k) (k) ! Y _ (k) A (k)

~ij m=l r=lqirqjr<l>rm ~m (13)

Comparing (13) with (17) we have:

t (k) n _ (k)

ijm r~lqirqjr<l> r m (14)

Le.

T(k) Q ~(k)QH Vk,m m m ' (15)

where ~(k) is a diagonal matrix as defined in (12~ m H

Thus Q-lT(k)Q-l = ~(k) are diagonal, \/k,m m H m H

Since (Q-lT(k)Q-l)H = Q-lT(k)Q-l ~(k) must m m m

havr only real diagonal entries.

Sufficiency. Given that the matrices H

Q -IT (k) Q-l ~ (k) are diagonal, \/ k m There-m m ' •

fore, T(k) Q~(k)QH. Note that m m

(k)

~ = Jl T~k)~~k) (k)

I! Q~(k)QHA(k) ~ m ~m

m=l

(k)

Q( I ~(k)~(k»QH (16) m=l m m

(k)

= ! ~ (k)~ (k) m=l m m

(17)

Since ~~k) are diagonal, V k,m ~ are diagonal,

Vk •

Q.E.D.

(.q (q Let U e ' l = 1,2, .•• , n2 and e '" 1,2, .•• , d

denote the matrices derived from

where U(l) are the counterparts (l) e

d are the counterparts of the

Nl

,N2 , .. ·,Nl ,

of T(k) (where m

dimensionalities

p(k». Similarly, let ~(l) denote the real e

diagonal matrices which are counterparts of ~(k) m

The next theorem follows as

of Theorem 7.

an obvious corollary

Theorem 8. The collection of (nl

+ n2) herme­

tian matrices Ml

,M2

, ••• ,Mnl and Nl

,N2 , ••• ,Nn2 is simultaneously diagonalized through contra­

gradient transformations as shown in (5a) and (5b) if d 1 if Q-lT (k)Q-lH = ~ (k) V k m , an on y, m m"

and Q~(l}Q ~(l) V l,e, are real diagonal e e

matrices.

Depending on how the Gram-Schmidt orthonormaliz­

ation is initiated, different orthonormal bases

may be obtained for the same space. It will now

be shown that Theorem 7 and Theorem 8 hold regard­

less of the choice of the orthonormal bases; the

diagonalizing transformation Q remains invariant.

~(k) .(k) .(k) Let {~l '~2 ""'~p (k)} be

mal basis for the space fi (k) • P

be the counterparts of T(k) ~(k) m ' m

determined using this new basis.

another orthonor-. Let f(k) ~(k)

m ' m

, respectively,

Theorem 9. The matrices

diagonal if, and only if,

Q-lf(k)Q-lH are real

Q~lT(k)Q-lH are real m

diagona1.

The proof is simple and hence omitted.

The following corollary is obvious.

Corollary. The matrices

The implications of Theorem 8 and Theorem 9 are

that the problems of simultaneous diagonalization

of matrices with functional entries can be convert­

ed to equivalent problems of constant matrices.

The rest of the problems then· become identical to

those discussed in the section II.

This concludes our discussion on the simultaneous

diagonalization of functional matrices.

39

V. CONCLUSION

In this paper, existing results on simultaneous

diagonalization of hermitian matrices through a

single congruent transformation and through a pair

of contragradient transformations have been gener­

alized for an arbitrary but finite number of

matrices. The results obtained here are shown to

be applicable to the matrices with multivariate

functional entries which satisfy appropriate con­

ditions. An example of simultaneous diagonaliz­

ation through contragradient transformations is

also included.

ACKNOWLEDGEMENT

This work was supported by the National Research

Council of Canada under Grant Nos. A-7739 and

A-7740.

The authors wish to thank Prof. N.K. Bose of the

University of Pittsburgh, Pa., and Professor M.

Vidyasagar of Concordia University, Montreal, for

useful discussions and suggestions.

REFERENCES

1. F.R. Gantmacher, Matrix Theory, Vol.I, Chelsea Publishing Co., New York, N.Y., 1960.

2. A.C. Aitken, Determinants and Matrices, 9th Ed., Oliver and Boyd, Edinburgh, U.K., 1956.

3. C.R. Rao and S.K. Mitra, Generalized Inverse of Matrices and Its Applications, John Wiley and Sons, Inc., New York, N.Y., 1971.

4. D.M. Young and R.T. Gregory, A Survey of Numerical Mathematics, Vol.II, Addison-Wesley Publishing Co., Reading, Mass., 1973.

5. P. Bhimasankaram. "On Generalized Inverses of Partitioned Matrices", Sankhya, Series A, Vo1.33, 1971.

6. A Ben-Israel and T.N.E. Greville, Generalized Inverses: Theory and Applications, Wiley­Interscience, New York, 1974.

* * * S. Chakrabarti received the B.Sc.(Hons) degree in Physics, and the B. Tech., M.Tech. and D.Sc. degrees in Radiophysics and Electronics from the University of Calcutta, India, in 1965, 1967, 1968 and 1973, respectively.

From 1972 to 1974 he was with the Department of Electrical Engineering, University of California, Davis, supported by a National Scholarship awarded by the Govt. of India. Since October 1974, he has been a Post-doctoral Fellow at Concordia University Montreal. His current research interests are in the areas of Digital Signal Processing, Optimiz-

-

i

'I'

:11'

III

ation Techniques and Systems Theory.

Dr. Chakrabarti is a member of the IEEE, SIAM and MAA.

B.B. Bhattacharyya received the B.Tech. (Honors) and the M.Tech. degrees from the Indian Institute of Technology, Kharagpur, in 1958 and 1959,respective1y, and the Ph.D. degree in elect­rical engineering in 1968 from Nova Scotia Techni­cal College, Halifax, Canada.

From 1959 to 1965 he held appointment as a Technical Teacher trainee and a Lecturer at, respectively, the Indian Institute of Technology, Kharagpur, and the Indian Institute of Technology, Madras. He joined the Electrical Engineering Department of the Novia Scotia Technical College, Halifax, Nova Scotia, Canada, in 1965 and, in 1968, moved to the University of Calgary to become Assistant Professor in electrical engineering. He joined Sir George Williams University (now known as Concordia Univ­ersity), Montreal, Canada, in 1970 as an Associate Professor of electrical engineering. He became a Professor in 1973. He has published a number of articles in the area of network theory.

M.N.S. Swamy was born on April 7, 1935. He received the B.Sc. (Honors) degree in mathematics from Mysore, India, in 1954, the Diploma in electrical communication engineering from the Indian Institute of Science, Banga10re, in 1957, and the M.Sc. and Ph.D. degrees in electrical engineering from the University of Saskatchewan, Saskatoon, Saskatchewan, Canada, in 1960 and 1963, respectively.

He worked as a Senior Research Assistant at the Indian Institute of Science until 1959, when he began graduate study at the University of Saskatchewan. In 1963, he returned to India to work at the Indian Institute of Technology, Madras. From 1964 to 1965, he was an Assistant Professor of Mathematics at the University of Saskatchewan. He has also taught as Professor of Electrical Engineering at Nova Scotia Technical College, Halifax, and the University of Calgary, Calgary, Alberta, Canada. He is now Chairman of the Department of Electrical Engineering, Concordia University, Montreal, Canada. He has published a number of papers on number theory, semiconductor circuits, control systems, and network theory. He is also Associate Editor of the Fibonacci Quarterly. During 1976, he is the vice-president of the IEEE Circuits and Systems Society.

A WALSH OPERATIONAL MATRIX FOR SOLVING VARIATIONAL PROBLEMS

C. F. Chen and C. H. Hsiao

E.lectrical Engineering Department, University of Houston

Houston, Texas 77004

Abstract

The Walsh function was initiated by Rademacher [1] and independently developed by \~alsh [2] in the early nineteen twenties. In recent years, the Walsh theory has been innovated and applied to various fields [3-8] in engineering and scibnce. To the authors' knowledge, however, this powerful tool has not been used for solving variational problems.

This paper establishes a clear procedure for the variational problem solution via the Walsh functions techniques. In the beginning part of this paper, we will introduce Walsh functions and briefly summarize the properties. Therefore, it is tutotial in nature. Then we will derive an operational matrix for the in­tegration use. The variational problems will be solved by means of the direct method with Walsh series.

I ntroduct ion

The basic idea of direct method for solving vari­ational problem is to convert the problem of extremi­zation of a functional into a problem of extremization of a function which involves a finite number of vari­ables. Ritz's method is a well known one in this field. This paper introduces Walsh functions first which is tutotial in nature; then presents a direct method for solving variational problems via Walsh functions. The procedure involves (1) assuming the admissible functions by Walsh series with coefficients to be determined; (2) establishing an operational matrix for performing integration; and (3) finding the necessary condition for extremization and (4) solving for the algebraic equation obtained from the previous steps to evaluate Waolsh Coefficients. Because of the orthonormal property of the powerful Walsh series, the new direct method is simpler in reasoning as well as in calculation. An illustrative example and a practical application to heat conduction problem are included.

Rademacher and Walsh Functions

Rademacher's function is a set of square waves of unit height with periods equal to 1,1/2,1/4,1/8, ...

2(I-k) respectively. The first five square waves are shown in Fig. 1. Alternatively, we state that thek_l number of cycles of the square waves of rk(t) is 2 It is noted that the set is not complete since, except for ro(t), the set involves only functions which are odd aflout t=I/2.

In 1923, Walsh independently developed a complete set which is known as Walsh functions. The set of

4)

Walsh functions and the set of Radamacher functions have the following relations.

<P (t) r (t) 0 0

<p (t) r (t) I I

I 0

<p (t) (r (t» (rl (t)) 2 2

I I <p (t) (r (t» ( r (t»

3 2 I

<p (t) n

where

q

(1 )

in which ['J Therefore,

means taking the greatest integer of II II

n = b q

q-2 0 2 + ... bl

. 2 (4)

where bqbq_l ... bl is the binary expression of n.

Therefore if a particular Walsh function ¢ (t) is given and its Rademacher function components arg re­quired, we simply change n into binary form and then substitute into (2).

Rademacher functions are easy to draw, so are Walsh functions.

Walsh Coefficient Determination

A function f(t) which is absolutely integrable in [0,1] may be expanded into Walsh series.

where cn are coefficients of Walsh series of f(t).

It is desirable to determine the coefficients c such that the integral square error n

N [f(t} - L

n=O c ~ (t)J2dt = £

n n (6)

is minimized. Taking the partial derivative of € with

respect to cn yields

Jl

~ = 2c - 2 dCn n 0

fIt). (t)dt and then setting it n

equal to zero, we have:

c n J

l

o .n (t) fit) dt

This simple result is due to the orthornormal property of Walsh functions.

Let us illustrate the Walsh series expansion by the following simple ramp function example.

fIt) = t

Substituting fIt) into (7) and taking only 4 terms yield,

o

1 II

After substituting these obtained values of coefficient into (5), we have

1 1 1 t = 2.0(t) - 2 .l(t) - 8" .2(t) + 0 .• 3(t)

Which is the four-term Walsh series expansion of the ramp function.

Discrete Formula

If the given function is not in its analytic form but in tabulated data or in graphical form and its Walsh series expansion is derived, we would modify (5) and (7) into discrete forms:

k=0,1,2, ••. (m-I) (5-a)

n=O, 1 ,2 , ••. (m- I ) (7-a)

where fk is the average value of the function in question in the k-th subinterval and. k the value of n, the n-th Walsh function in the k-th sub-interval; and m is the total number of subintervals.

For illustrating the use of discrete formula, let us evaluate the Walsh series again for the ramp func­tion in its tabulated form. Given

k o 2 3 1/8 3/8 5/8 7/8

The corresponding graphical form is shown in Fig. 3.

-------~~-

Equation (7-a) in its expansion form for m=4 is as follows,

--1

r col

1'00

'-1

·01 ·02 ·03 1 fO

Cl ! ·10 ·11 ·12 ·13 fl

=1 1 (8)

c2 i ! ·20 ·21 ·22 ·23 f2 11

I

I I I c3 i ~- ~ L·30 ·31 ·32 ·33

-.-l f3

L -

42

Substituting the tabulated data of the ramp function into (8), we have

.-Col f1 11 I-

i I : 1/8

I -1 -I 3/8 c1 I I i Ii I

5/8 ! c i -1 -1 2. I I

c3 ! -1 -I I 7/8 I (9-a) _I ~-

__ J L... -'

11/2 I

(9-b)

The square matrix defined in (8) and the numerical values shown in (9-a) are easily recognized from the definition of Walsh functions.

It is seen that the Walsh series of a unit ramp function obtained from discrete formula, or (9), and that obtained from the analytic formula are, of course, the same, since fk is exactly equal to the average value of fIt) in the k-th interval.

Eq. (8) can be written into a general compact form.

c = Wf . 1. m

Where W is called the Walsh matrix.

Operational Matrix

(10)

In the previous section, we showed that the ramp function can be expressed by a Walsh series, or

t ; 1.. (t) - 1.. (t) - ~ ~2(t) 2 0 4 1 0 'I'

( 11)

However, a ramp function can be considered as the first integration of a unit step function, or .o(t). Therefore, we write the following

s:

The first integration of • (t) is a triangular function, and if we expand the !riangular function

(12)

into Walsh series by using discrete formula with' m=4.

we have

Jt I I

<1>1 (t)dx = [If' 0, 0, - 8] o

Similarly, we can evaluate the Walsh series coefficients of the first integration of <l>2(t) and <l>3(t) and easily obtain

r o

and

I <1>2 (x)dx ;;; [ 8' 0, 0, 0]

I [ 0, 8' 0, 0]

>-<1>01 I

<I> 1 i <1>2

<1>3

Combining (12) through (15), we have

<1>1 (x)dx I I , 4" I

i'" 1 1=

i I <1>2 (x)dx 8

<l>3(x)dx 0 i .....; +-

or in compact form

I - If

0

0

I a-

j: <I>(x)dx = P (4x4) ~(4) (t)

( 14)

( 16)

~(llx4) is called the operational matrix of dimension

4 which relates Walsh functions and their integra­tions. It is chosenas a square matrix for the reason of convenient calculation.

By use of (17), integration becomes multiplica­tion, therefore, we consider P as an operational matrix. -

If we divide the unit [0,1) into 8 sub-intervals

instead of 4, and evaluate J<I>Odt, J<I>ldt, ...

J <l>7dt by either analytic method or discrete formula,

we would obtain a group of triangular waves as shown in Fig. 4. Then expand the triangular waves into Walsh functions. Consequently, we arrive at the

43

following formula.

WO"lii 1 I 0 I I 0 0 0 r<l>o 1 -4" -a- !-IT

j<l>l dt It 0 0 1 ! 0

I 0 0 1$1 -a- -16

If ill 0 0 0 0 0 I 0 $2 I $2dt I i a- -IT

J ,I I I 1 I 0 0 ! 0 0 I $3dt i . 0 8 0 -16 $3 ! ~: -- - --

, j'$4dt -I :6

---------- - - ---0 0 0 0 0 0 0 $4

, j'$5dt 10 I 0 0 0 0 0 0 $5 16

0 I 0 0 0 0 0 $6 16 ("1. 0

f $7dt I 0 0 0 I 0 0 0 0 $7_ 16 .-J -- --which is

J: P(8)(x)dx;;; ~(8x8)P(8)(t) (18-a)

It is interesting to note that the left up corner

of ~ (8x8) is exactly ~(4x4) in (17), the right up

corner and the left down corner are unit matrices multiplied by -1- and ~ respectively, and the right

d • 16. 116 11 .. own corner IS simp y a nu matrix. Following a similar reasoning line, we can write

a general expression for the operational matrix P of

order m (which is a positive integer power of 2).

P ",(mxm)

I

1 I 2 - 1- - I m I 2 1 m-(,:-) , - " , ___ L _______ ,

~I 0 :, (~\

1 , 1 1

- - I (m\, m ::: 11',

m ;:: ({) :: 0', ' I ' 1

- - - - '- - - - - - - ~ - - - - - - - ~ - 2m !. (r) 1 1 ,-

- I (~\ I 0 (m) 1 m:: 41 '" 41 , ' -------------------,------

1 2m ::: o (~\

::: 21

This operational matrix will play an important role in the direct method for solving variational problems.

Direct Method for Simplest Variational Problem

The regular method for solving problem of a functional:

Jl .

J = 0 F(t,x(t),x(t»dt

the extremization

(20)

Ii I

is through the Euler equation

F _...E.. F. = 0 x dt x (21)

However, the differential equation so obtained can be integrated easily only in exceptional cases. There­fore, many direct methods have been developed. Ritz's and Gelerkin's methods are well known [9,10]. This paper mainly uses Walsh's functions to establish the direct method for variational problems.

Not like other direct methods starting with the assumption of the variable itself, the method we developed starts with the rate variable. In other words, first assume the rate variable x(t) as a Walsh series whose coefficients are to be determined.

00 • x( t) E c i4>i

i=O (22)

Taking a finite terms as an approximation, we have

~ (tl ;. cOiflO + c l 4>1 + ... + cm- l 4>m-1

~ :' ~ (23)

From ( 17) we know that

ft p (>")d A = P ~ (t) (24) ::::

0

Then the variable x( t) can be expressed as

.. jt

x( tl o

x(A)dA + x(o) c' ~ ~(t) + x(o)

(25)

The other terms In the functional of (20) are known function of the independent variable t and can be expanded Into Walsh series with known coefficients. Expressing x, x and t in terms of Walsh functions through substitution, we finally have

(26)

The original extremization of a functional problem shown in (20) becomes the extrimization of a function of a finite set of variables In (26).

Taking partial derivatives of J with respect to c i ' and setting them equal to zero, we obtain,

~=o dC. '

I

(1=0,1, ..• m-I)

Solving for c l ' and substituting into (25), we will have the answer.

(27)

We note that the above proposed method implies Euler's direct method of finite difference and is similar to Ritz's method using power series and Fourier series; but considering (i) the orthonormal property of Walsh series and (ii) the product property of P shown

in (24) and the operational property of P itself, we can claim that the new direct method via~Walsh func­tions is simpler and more powerful than any previous methods.

Let us establish the detailed procedure via several classical problems.

Illustrative Example

It is required to find the extremal of the fol­lowing functional.

J = J~

and boundary

x(O)

x (1)

(~2 + t ~)dt

conditions

0

1/4

(28)

(29-a)

(29-b)

This is the exercise #7, Ch.1 of Elsgole's book [II]. For solving this problem by the Walsh direct

method, we assume that

14>ol • I 4>1 ; x (t) = [c , c l ' c2 ' c

3] p

4>2 4> '- 3_

(30)

Here we let m=4 for clarity in presentation; more accurate results can be obtained by using a larger m.

There is a variable t involved in (28) explicitly, we need its Walsh series expansion also. Using (II) cirectly yields

I 4'

~ h' pet)

I B"' 0] Ij> (t) (II-a)

Substituting (30) and (II-a) into (28), we have

fl, , , , J = [~ pet) p (t) : +: pet) p (t)

o h]dt (31)

However, the vector function 4>(t) has a particular

property due to the orthonormality of Walsh function,

I

J ~(t) p' (t}dt o

After integration, (31) simply becomes

J = c'c + c'h

(32)

So far, we have not introduced the boundary con­ditions into consideration. For the initial one, we easily see that,

t It x(t) f ~(A)d>" + x(o) ,

4>(A)dA + 0 c

0 0 (34-a)

;;; c' P pet} (34-b) .. since x(O) = O. For the final boundary condition, substituting (29-b) into (34-a) yields

X(J)=C,!I </>(\)d\ - 0-

It is Interesting to note that the definite integral of </>0 from 0 to I is equal to I, while the definite inte-

grals of </>1' </>2 and </>3 are all equal to zero; or

(36)

i=1 ,2, .•.

Substituting (36) into (35) simply gives

o (37)

o

Co is found without too much effort at all. This information should be substituted into (33) also, we then have

I 2 2 2 I I I J = 16 + c I + c2 + c3 + ~ - ~I - SC2

For extremization, we take the partial derivatives of J with respect to c i ' i=I,2,3, and set it equal to zero:

dJ/ClC I 0: 2c

I -I 0 4=

dJ/dC 2 0: 2c2 - ~ = 0

dJ/dC3

= 0: 2c = 0 3

Therefore,

And x(t) is obtained from (34-b) ,

x(t) = c' P </>(t) - --

I

/,' = "8

or I (38) c2 =16

c = 0 3

I I I 16 </>1 (t) - 32 </>2(t) -128 </>3(t)

(40)

If Euler equation is used for the analytic solu­tion, the answers should be

x(t) = t t(1 - t t)

~(t) = t (I - t)

(39-a)

(40-a)

respectively. The graphical comparison of the solu­tions via Euler's analytic method and via Walsh's direct method is shown in Fig. 5. It is seen that even m=4, the Walsh direct method is quite satisfac­tory.

Application to a Heat Conduction Problem

Consider the following functional extremization

45

problem:

J = J: [t y 2 - y g(x)]dx = r: F(x,y,y )dx

(41)

where x is the independent variable; and y the depen­dent variable. The function g(x) is defined as

I _ { -I for 0 2. x 2. 4 and

g (x) - I I 3 for - < x < -4 - 2

(42)

The values of y(O) and y(l) are unspecified. This is a functional extremization problem with moving bound­aries at both ends. Using classical variational analysis, we can find the two conditions from F(x,y,y )

F. I 0 y x=O y (0) = 0 impl ies

(43)

Fy Ix=1 = 0 impl ies y (I) = 0

Schechter [12] gave a physical meaning to this problem by assigning y(x) to be the temperature in a solid slab; g(x) the rate of internal heat generation per unit volume, and x the space variable as shown in Fig. 6. It is noted that g(x) satisfies the follow­ing:

,1 J g(x)dt = 0 o

( If'/- )

which means that there is a volume weighted equality of sources and sinks of thermal energy.

What we are interested in is to apply Walsh direct method-to solve this functional problem.

First of all, we assume y (x) as a Walsh series with 8 terms.

00

y (x) = L c',</>', i=O

= cO</>0+cI</>I+c2</>2+c3</>3+c4</>4+c5</>5+c6¢6+c7¢7

~ :' ¢ (x)

Integrating Y (x) and using operational matrix P, we obtain

y(x) y (\)d\ + y(O) ~ ~' (x </>(\)dA + y(O) .. 0

_ ~ c' P ¢(x) + y(O)

The Walsh series expansion of g(x) can be shown to be,

(47)

Substituting (45), (46) and (47) into (41) and apply­ing orthonormal property of ¢(x), we have,

J = f>h'~(x)f (x):-:'~~(x)2' (x)~-y(O)g(x)]dx = 1 c' c - c' P h (48)

2 - - - :::

The boundary conditions of (43) can be expressed in terms of Walsh functions

In minimizing J with respect to c subject to the con­straint (49), we apply Lagr~nge'; techniques with two multipliers Al and A2 and define,

J* ~ J + Al~'~(O) + A2~'!(I)

= t ~'~ = ~'P~ + Al~'!(O) + A2:'!(I) (50)

where "*" means the auxiliary function. Setting the partial derivatives of J'~ with respect to c equal to zero, we have

3J'~ ~ = ~ - ~~ + Al!(O) + A2!(I) = 0 (51 )

Equation (49 and (51) may be combined

I ~(O) p(l)l r - ~ l -:~l ::;8x8 I

I I I

.p' (0) 0 0 ! : Al i= I I

I .p' (1) 0 0 .-J A2 I

~ - j

Solving (52), we have

(53)

Therefore,

y(x) = x P p(x) + y(O) -I = ~ [-39~0+52~1+12~2-34~3+8~4-7~5+~6-8~7]

(54)

The Walsh series solution curve obtained from the above equation and the actual solution are drawn in Fi g. 7.

Conclusions

After briefly reviewing the Walsh series tech­niques, we form an operational matrix for performing integrations in Walsh functions analysis. A direct method of variation is established by using Walsh series. A simplest extremization problem and a vari­ational problem concerning heat conduction are completely solved step by step by the new proposed procedure. It is believed that the approach is more powerful than Ritz's or Euler's direct methods for solving variational problems.

References

1. H. Rademacher, Einige Salze uber Reihen von

46

allgemeinen Orthogona-functlonen, Math. Ann., Vol. 87, 1922, 712-738.

2. J. L. Walsh, A Closed Set of Orthogonal Functions, Am. J. Math., Vol. 45, 1923, 5-24.

3. J. D. Lee, Review of Recent Work on Applications of Walsh Functions in Communications, Proc. Walsh Function Symp., Nav. Res. Labs., Washington, D.C., 1970, 26-35.

4. H. F. Harmuth, Application of Walsh Function in Communications, IEEE Spectrum, Nov. 1969, 82-91.

5. J. E. Gibbs and H. A. Gebbie, Application of Walsh Function to Transform Spectroscopy, Nature, Vol. 224, Dec. 1969, 1012-1013.

6. C. W. Thomas and A. J. Welch, Heart Rate Represen­tation Using Walsh Functions, Proc. Walsh Function Symp., Nav. Res. Labs., Washington, D.C., 1972, 154 154-158.

7. F. Picher, Walsh Function and Optimal Linear Sys­tems, Proc. Walsh Function Symp., Nav. Res. Labs., Washington, D.C., 1970, 17-22.

8. M. S. Corrington, Solution of Differential and Integral Equations with Walsh Functions, IEEE Trans. on Circuit Theory, CT-20, No.5, Sept. 1973, 470-475.

9. Gelfand, I. M. and S. V. Fomin, Calculus of Varia­tions, Prentice-Hall, 1963.

10. C. D. Brewster, Approximate Methods of Higher Analysis, Interscience Publishers, Inc. New York, 1958.

II. Elsgolls, E. L., Calculus of Variation, London, Pergamon Press, 1961.

12. Schechter, R. B., The Variation Method in Engin­eering, New York: McGraw Hill Co. 1967, pp. 23-24.

13. Neuman, C. P. and A. Sen, "A Suboptimal Control Algorithm for Constrained Problems Using Cubic Splines," Automatlca, 9, Sept. 1973, pp. 67-69.

14. Mang, J. H., "A Sequency-Ordered Fast Walsh Trans­form," IEEE Trans. on Audio and Electroacoustics, Vol. AU-20, No.3, Aug. 1972, pp. 204-205.

~o (f ~lt------

Fig. I Rademacher Functions

Fig. 2 Walsh Functions

1 ----------------

Fig. 5 Fig. 3 Ramp Function

Fig. 6

Fig. 4 Walsh Functions and their First Integrations Fig. 7

47

Solutions of Extremal Problem with Fixed Boundary Conditions

3W=br-o 05 1 X

-I

>-=5,,-,IO:..:bc-+--_X

Heat Conduction In a Solid Slab

Solutions of Heat Conduction Example

A COMPLEX .FOlilil OF THE GENERALIZED FOUAIlli SilUES

AND TJiANSlOliIIS

Dan A. oi u.l1n Polyteohnioal Instjt~te of Euchar!st

Abstract

Starting from are'll fllnction whioh satisfies the Dir:l.chlet oon di tions for develoving in a Fourier ser1.es r.tn 1 transform", and uti­lising the properues of the analytic'":!.l signals an al go r1 thm to generate an orthogonal set of functions is prlsented. Relation which represent in a complex form the orthogo:18.1 sets of funotions is obtained directely. A partic'~ll'tr C'lse of tlis relation is the well-known set ejnwt. By means of these rel~tions the ~eneralized frequenc3 (sequency) is defined and a complex form for LSner'llized Fol1rier/Laplace series and tr.ansforms is dedl~l)ed. As applications relations for generalized DirlC pulse, periodioal generalized Dirac pl~lse and a gener'llized form of convolution theorem in which the natllre of the Fourier trmsforrn applears evi­dent are deduced. The application of the rel"ttions presented i~ the paper leads, through other ways, to the rllsl11 ts known in the spec1al1 ty field.

1.INTdOD{JCTION

Relations for represen tlltion of the s1 g_

nala,[l] ,[1.4] ,by means of gener':l.lized .1<'011-

riel' ser1es and transforms are known.

These relations are sill!1.1<tr to the real forms of' the Fourier series 'md tr3.nsforms. Sometimes thi s repre eent·'ttion C'ln 'lssu.me

2. SETS 01 ORTHOGOlI'AL PUlI'C!IOlfS

The well-known orthogon'll set hnctions ejnwo t C'ln be ','Iritten in the form

ein.,.,.t _ [e it] 11~ r t " tJ nw .. - = L CO'3 T J !lL11.

=[crnt -3 d( (C03t) ] Il"il

By replacing the 'lm.lytj cal sig:nStI ejt

(1 )

a oomplex form too,{2]. The util:lz::ttion by <.Ulother ~Ul.:llytic"tl si611al zT(t) whose of suoh representation is ·:ldv r:tnt C1.geolls J.'e::tl pS1.rt only when the orthogonal set of functions regarding whi oi the developing t1ike pl·.oe

is apriori known (as for eX'd.mple the 7'-:llsl). f 'mction, Legendre polynomi!11s etc.) If

the se~ of ianct:i ons 1 s to be deduced

Btart1~g from ~ 6i van re:d fi.mction throClgh

an ort~ogon';llized iflethod,{3'~{14} ,[16J ' then, Uec:.t.!.se of rec:~rp.nces proceed'lce :l sed, rot I 1 riSe ·'3Jlloun t 0 f c'J.l c'~l3. ti on s take s plaoe.

48

(2 )

S'l.tisfies the Dil.'ichlet conditione J.nd

whi eli is ,..i ther :;1 j?Clriod.i e'.u f.J.tlction wi th

tho p?riod T, or ,~ f.l.nction defin1 te on ':l

fini te into rv"l.l [C ,T)"I.nother set of fl1no­tions i 0 obt·j,1r.ed;[~]:

[Zr{t)j"/'o = br (t)-iJf[1r(t J)j n/,o . Zr-3rltJ] np.,

=[ V~:(t)+ lt~(tJ] e-JM3~ ~(t) J =einPo['f{t)-i'f(tJ] (3)

rel .. tion it~ \\hich v,e not~d

'1m = - alta tl 'K [MO] d (J 1r{t) (4)

t(i)== exp[jln(1;(f)+'J{2[1r{tJj)] (~)

i::Uld n= 1,' 2, 3, •••

p = ':l, real n:lmOer o np

The orthoe;on',ll set 01' f,mctions [ZT(t)] 0

obt''lir:.ed 'is1bove, s'l.t1 Rt'i es the condi tiona

01 ortHOGoYuli ty in accord'l.nce to thE' re-

lations: T

f fRe[z;~t)] Jm[r;p-(t)]dt =0 o

r

.if]m[~np'(tJlJm[i!T"'P.{ilJdi =[~rI. for n=m T T 'J 0 ';:0 I' nRn

o

(6)

(7)

( 8)

,Froof: npo 11

Since o3i tLer the i'..mctA~n "(e [ZT (tu

01' ";hC' r;Ulction Jm[ZT 0 (t)] s'itisfies

tl:E' -:11 rich]."'t r.oYJd1 tions by developine, in

'j, ~'ol.,d<2.c 8e1'ie s, [5Jwe c'm wri te:

" IT T Re[lTn'~{t)J]m.[zrnp~{t)]di;. .. 0, T - jk~t .... J"H4Jt (9)

= -1.1 LC(kw.}e !.$(Jw,,)1r iv/oe ]tit T 0 At: • .,. _ L:-oO

in which w.=2JI 'lnd T is the P2L'iod or T

the dc>fjr.j!1;~ interV':l.l of the sign':il Re,

1m [z~po (il)

Into,rc!J;;U1[SL!b the order of tile Sel:ll 'md

S:.1;'1 3.Ild integ.c'ition oper''ttior:.s we obt:dn:

where ((oj = A(·),. i B(.)

To prove t,l"le relation (8) the same method

as for the proof of the relation (7) can

be Ll.tllized. Con8eqLl.ently, the relation

(3)can be considered as a relation repre­

senting any orthogonal set of fQnct10ns

which s3.tisfies the Dirichlet conditions

for developing in a ]'oQrier series. Choo­

sing a partic'J.lar form for functions 'f(.)

'3.lld r (.) from the m<:l.in rel·1.tion (3) a re­

lation to represente a given orthogon,'ll

set of fQnctions will be obtained. Thus,

~or "'(t): 0 and 'fIt) : t the set of l'u.nction; ejnwo t will be obtained.

The number p has a meaning simil'ir to , 0

the number wo' If the an'lli ti c signal

ZIt) is periodic'll with the pertod T we

can wri te : '[ . J Po JPo ftl) -I 't(t)

l.r (tl = e POf

J~['f{t .. tT)-i't{t40nrJ] == z., t+nT): e

(13 )

n=o,1,2, •••

According to the rel'':I.tion (13) there re-; j;e[zT"P-(fJ] Jm[z/P"(t)dt ]dt=

o .... 2 :: -/.L /({iw.J! ~~rziwo :: 0

8-1.1 ts th'it either the f;.lnction of modulus (10) ePo''-f (t) and the fi.l.nction of phase

ejPo'f(t) 'ire periodical fll.nctions (some­

an'lly-times tl'ley cau be const.'illts) Because fmc­

:E'OLl.- tions e j (.) is periodicoU too there re­sill ts t

L =·00

~\,c two: ,Jrodlf of the r;~l':ition (7) the

ti c ,1 c1J.l zT (t) is det'eloped in :3.

rjec se.:.':'.es

(ll ) or

49

po 'f(T) " 27i 2Ji po = 'fiT} (14 )

a rel'ltion which represents ~ generaliza­

tion of the rel<:ltion

(15 )

for 'f(t) I t

By <:In'llogy with the meqning of the fre-w, Wo

quency given for t~e number 21 (n;; fo)

the number s;; ----2-2

"" has bee'Ul c9.11ed in o JI-

[1] sequency (gener"lli zed freque~cy). an.p. [f(t)-j 'f'{tO The orthogonal set of functions e

can be nonnalised. Because, [4] ,[5J

n~r(t} 1/ :H'K[~{t)JII:: +{r~2(t}dt o

(17)

The ;malitic nOl'm'ilised signal z.(t) will

be gi ven by reI'). ti on :

Z,.,{i}::: ~(tJ _ ' 'R [M+J) 1/ '3r (t)U J 1/ dl [~7(t)JQ (18)

=~(t)-aK[1r{t)J _ ldtl 1117 (t}f - 1/1rft}Q

If function sT(t) is known the'Ul'1l1 tic

dgnal zT(t) oan be c'llculated. ]\mction

t[sT(t)J is calcJ.bted either by utilj ~dng t,he rel'ltion -}( [~(t)] = II! f ~(t) dr;

// t - (; - 0-

or, for the periodi c'il flJ.nction

utilising the relation,[6] T

'J([~r{t)] :: ; J 1r(r;}df[J,(t-~>]dr; o

T

:: 2~ J :1T tr;)cta JiG d~ o d T

(19 )

s.r (t j, by

(20 )

or by '-ltiliSing the FouMer series. If

~(t) = ZAn co;, 27i nt 1" Bn din?!! nt (21 ) I'l T T

then

50

3.GEI'ir;iiALIZED RlJ.Q~ ... t SiLtI~S

:; .1.HEP~SEI; TATION C]1 G'!;N3!t."iLI Z3D ~'t J.U t;;t SE.tIES

The rel~:itions for representation of 3. pe­

riodlC'.l or defined on :m interV'tl [0,'1] f1lnctlon g(t) which s'.ttist'ies tIle Dirich­

let condttions, by me9.ns of a Gener~liz9j

louirer 8erie8,[1],[7], 'n'<\:

(23)

where

T

G" =f J9r{t){'(t}dt o

(?4 )

and

97 (tJ = ~r (t+lT) (25)

In tllese rel'ttions the complex conjll,;'1.te

of tlie fmction fk (t) has been noted f: (t) ~1.l: d tne 0 L'tho bornl se t 0 f Llll C ti on s

reg~trding whi eh the d"'velopint; in tile Ge­

nerqlized Fourier seriAS is done W':J.S no-te d fk (t ). rl.!'i ting

fl( (the iKpo['f(i)- i'f'(tJ] (26)

we obtqin rel~tions

n:-()I>

and respectively

u' The coefficients G(nppi 'tre complex mlmbers

b) In the reLt,tiona (27' md (28) the valu.e for n is to be tOtk"n b t

,'- e lVeen :!.:. 00. By non si derj ng th:? r,? l:t ti on s (9) (10) 'md (12) we e'll1 ob"'f!'" -/-, , J

'-'--ve uil it +'l" o rtl10 gon ''t11 ty j s kent fa t' th'" ": -

• ' w nee~tlve

v'll~e of n too.

c) ... tel:ition (27) represent tile p.ign3.1s

gT(t) eitller if ePo *(t),. 0 or if ePol\'(t)

V''tr.ifl':(~S only for '1 n'l'nb"r nf points of

"zero me:'ts'lre".

d) ;{el!tion (27) Cin be dedloed from re­

Ll.tion (29) by developil1b in i /oiuier

series of tl:~ sign'-ll gT(t)

~ lkWpt: Q (t) = L- G., (KuI,,) e ( ?9 ) tfT" l(

by tHe re':1rl"tne;ing ':l.nd regro;']'ping of the

ter'its. In tLi s w"lY if one cansi ders the

develoojng in '1. FO,lorier series oJ each

ter'lJ of the orthogo'1<tl 8Pt of f'moti ons e info [ f(i) - it(t}] %

- 'f27it (30) ;: L Cn {r2Ji}e J T r." T

'Uld by replacing in rel'ltion (27) tLere

res;llts:

a reL'.Ltion from whi ch we c'm dedlce rp-lCl­

t ion (?9) thrOll.gh ''1 ra'lrr'mging 'md re­

gro:J.pin,~ of the t~rms.

e) l'he rel'itions (28) 'md (31) fihow th'3.t , . anp.['f(t) -i.,(tJ]

tLeortr;ouo1n1 set of j,mctlonse

i s compl e te •

f) rlelqtion (27) c'm be rew,J:'itten in

'~fiotHer form, n'lmely,

:r jnp.C'fttJ-jtMJ -f" jnp .. ['f/tJ-jtllll ar{t) = LG(np.}e tGlo) + Li{np,Je

11:-00' n: 4

;p np.tM -nf. t{l;) = Ao'" 11[6{J1p,)e + 6{-np,)e ] c.tr.)rlfo itt) +

+ i[G{np .. )enp.tlt~ 6l_np,)enpot{-t14irz nt. i(t J] 00'

= Ao'" I A{t,nfi)cron/'Di{t) T'5{~rtfD)~"rznp.'f{I:)

(31 )

in whioh

51

T

AD = f 1 ~T{t)dt (32) o

... {elation (31) is tLe ger.er':l.liz3d form of

relation:

0"

Q ttl: AD T LAn wn,v.t T 8n.1mnUlot (33) dT n:"

TIle '-1p'~ri();Lic'il fmotions C'H' b8 conside­

red to proceed iro:n the 8~~.i odi.c,:tl func­

tions by "l.); 'Ul1:im.;t~d :incr'~'jse of the pe­

riod? B'lt,:in -I,hi" 0,\8', ':iccorrting to

r~l'ltjon (14) po n'lst be 0 djfferenti:'tl

'(ll;o'~nt dp c--:.nd til>? d1 f3C:"3t2 variable

will be considerecl'.lSi cOI!tinous one.

ilnd8r tbe Sf' condi ti on s, by '~tili zing re­

I ':I. ti on (;;' 8) we will 0 b bin:

&;m'"8 'f{~i)6IP}:: 7-.00 = f3(t)e-if['f{t)+i'rCtJk-

-011 relqtion in which G (p) repr8sents

seq,lency 0;):;otr:il d"lnsity f:,motion

(35 )

(36 )

the

:irg

f(') represButs tLe inverse of fllllction

'f( • )

(37 )

.!!lCi g(t) the aperiodic,:tl 81..)1::.1.1.

Under tue s",:ne ccndi ti ons flU 011 rel3.ti on

(?7) :h>;>re r'3S'llts:

cw in.,. ['f(t)- j't(tll . 3{t) = linn L G(n.~)e _

T~OI> n:-....

0- - j,,~. ['({t)-tNt)] 21

:t.m. L ~ .,(~ )6(nr-)e Zi,,~/;(g)(38) T..,,,. ft.·eo It

.... t ,;; / ip['f(t)·i'l'ltJ.!h. F 1('1) :: z7i J .. l.7td'Je -r

if the linli t

t . _2.;;..'ii~ ___ "Pl. • ::"

T""" p.""''f{~) (39) :: (~m. ,'f(T) - I,m..!!!!.) T.... Art 'ft'ffTJ] - T-,OI' T

exi st s rmd is eq'.nl wi th k. It :i s known

th~t for ~t)::. t 'md Y;(t) = 0 the limit

exi sts '~'ld is equal wi th 1.

Considering the relation (38) 'lnd repla­

cing Gl (p) by Gl

(p) tfp (p) ("p (p) repre­

sents the periodical D~rac pulsg with Po

3.S period) there resJ.lts that,[8j

[

0- ip['f(~)-i't(t}J hrltJ ::2~ __ 4 (PJI,. (/') e dp

- 000.., il' ['f{t)-i'l'(t)] ::"2; /4{p)L/(p-np.)e df (10)

_ De 71=-000

Changing the order of sum and integration

we will obtain

ZOO' in,. r'f{tJ-i'l'(tJ)

Ir.,{t):: 64 1",.1 e (41 ) ". -01'

a relation of the sa.me form as relation (27) • In a formal way, the fact tllQt.t the li:n:i t

T'3.king in to ""ccoun t reI 'i tion (27) the pe­

riodi c'il gener'lli zed Dir:~c pulse C'U'l be

defined

('f) [00 info ('f(t)- i'l'ltJ) b (t) = e (45) p.

n: -01'

Tne relations (36),(38) ~d (44) &llow a

gener!.tlized theorem of convolution to be

dedClce<i. By considering the sic;nals

i I'";. ;p['f{tJ-tNt)] ~(t) ;' 27 J{I'} e c¥

-00>

(46 )

fO" ,,['f{tJ-j't(t)]

A{t) = ~ H(f} e rlf 2n

(47)

-00

we c~n ded'~ce

I ... 4, [,,{t)- i'l'{t)j 3(t): 2jj J '5'(p)J.I(pJe (J rip

jro; ~P['f(tJ-j'H.JJ -i~["{r)tJ'f'{r)lip[yI4A.)"''''MJ :2~jjJ~{~)h{u)e e e. J'."P .

-..,.

or exists has been j.tstifii:ld in relltion (3~).

In the following, as in the case of the

1ol1rier transform, we will consider k=l.

By means of the fo.£'m;"l change of variable

in relation (38):

31'::S : v+a· w

we can obtain reVil.tion for tile general1zed

Lapl~ce transfona

I f :1 ['1ft) -i'l-{t)) lIt). iF G4(~J e tid

III (421

and

5.APPLICA!IO.

By means of r.l,~,n (38) the generalized D:l.rao pul.e S (t) we can define

SC'I) , r- JP [,!,tj-i'f{tJj (t).- 2;; __ e iJ.p. (44 )

52,

-g{t): ff#~}h(u.)J("itl r,u)d~ tkt (48) --

For !f(t)=. t ~d 'f(t)=.O there results

Ji'f}tru)J =. J(t-,,-u.l (49) I' '({.,.t /

'f'{o) so .

The rel~tion (48) represents the generali­

ze1.f~nVOll1tion theorem. The funotion

& (t, ?:,I1) depends only on functions

'fl. )~d '/'(.) which define the genera­lized Fol1rier tr'Ulsform.

The relations ('6),('8),(27) an4 (28) allow

the dedu.cing ot a sampling theor.lls ot a

signal wi th liIl1t1n, time dl1r&tion and li­

miting sequen07 speotrum, a sampling theo­

rem wi th "V'lrt able step" and a Q,l1an t1 sin,

theorem ,[9],.1 th reell h It mllar to tho •• presented in[l) ,flo].

The same relations allowe4 I1S to dedl10e •

't

i

In''itL(~:!Jitic~il rel'ltions fOl' dp fin1ng'i pro- Use of urthogonarlllalized dC fJ.nctions,

b.ibUity dcr;sity f.mction for 't [;iven phy- Telecommunication Conference, Bucharest,

sic.tl syste,ll [n]. A 'll'lin ap91iciition con- 1972

sjsts in 'tn'11ysis 'tnd synthesis of systems [3]D.C.doss, (}rhtonorm9.1 Exponenti':l.ls, IEEE

with "time v·:tri'1.ble" p':trameters,[l] ,[2],[4J ,Transactions on Communic"l.t1.on Electronics,

[}. 2J • rihrch, 1964.

[4]D. Ciulin, Tlle ,\nalysi s 0 f Time-Varying

6. CONCLUSION Systems, Polytechnic'll Institute of Bucha-

In tile p3.per ::m 'j,lgOrithm for direct gene- rest, December 1973

r:J.ti.on of:m orthoe;on'11 set of functions, [5]V.Cizek, An'llitic SiGn'll '~nd some of

st':lrting f'ro:n'~ re'tl fclnction which s"ttisi.' its Applic~tion. Symposium Summer School

fjes tfle Dirichlet conditions, is presented. on Circuit Theory, Prag 1968

it !,'1rticllu'e C"ise of this "tlcsorithm is the [6]V.Cizek, Discrete Hilbert 1ricJ.!lsform,

t:;e!:er'J.tion of the well-known set ejnwo t IEEE Tr"tlls'lction on A.ldio and lHectroacu.s-

st'..irting from fu.nction cos wot. l'ne rela- t.ics,' \01. l\.U-18,no.4,1970

tionp obt'lined per,ni t tile genGr'ill s:J.tion [7]A'.Zygmund, Trigonometric Series. Carn-

of tbe freq·tency me'l!ling, by lntrod i.lcing brige at the University Press, 1959.

the :ae.:lnir..u 01' seq:lency for the (3ener"il1- [8]D.Oiulin, Sl.gnal aepresentationsby Me-

sed freq'.lency,[l]. By u.tilising the re- thods of P..mction:l.l Analysis. Telecornmu-

l'.tions t111lS obtained rel:'l.tio!ls 1'"r gene- n1c~tion Ltev1ew, Buch'lrest ,no.5 ,1972.

r':l.lized Fourier seri es :md trH.nsform and [9]D.Ciulin, Description of the ::;'ilTlpling even for gener3.1ized L::J.pl3.ce tr!:l.Ilsfonn are

ded.lced. The se rel''l t'i ons h"lve been \1 til1-sed flJr defining t.il'3 gene r''ll t ;jed Di r"l.C

ptlse .md ti,e llerjcdic"ll gener'lllz<:ld 'D1r'lC

plll se. A ger~er'J.li zed form of the convolu­tion theorem in which appear in a direct

way the nature of the IltUised Fourier

trCJ.nsform is dedllced too.

The reldtiono dedllced have been atilised

in p;)8c1 ':tli zed li teL'~t.u:e .tvr appli c'itions,

[9] ,[ll] , [12]. !tesul ts like those pl'esented

in[l] ,[10] h3.ve been obhined, [9].

TIle :nOl.in applic'J.tion of the [resented re­

l:.1tion consists of the 'iD'llysis and syn­

tbesis of the systems with time v:3.r!':l.ole p"tra:llete r s [4] ,[12].

rlEFE.dENCBS

[1] H. i.harIDllth, A G<>1'1er<il1 zed Concept of

FreqHmCj s.hd some tI,pplications. 13EE Tr"l.r,~

s"J.ctions cn Inlormation Theory, vOl.II-Il' ,

no';, 1968. [2]D.iV • .mPilre cnt, ,v. ~chmi d t, Appro ximation

of Butterworth ~nd ~auer Filters by the

53

Signals Using Function"!l Analysis. '.rele­

communication rlev1ew, no.12,Bucharest, 1972.

[10] I .Kluvanec, Sampling Theorem in Abs-

tract H'irmonic Analysis. Mathematicko fyzk"l,l­

ny Casopis Sloven, Aka.d,Vied 15,1965.

[llJ V.M. Catuneanu, Un the FU,nctional Spc1.ce

'ind SOill(;l Implic~tions in the ltea11bility

Theory. Proceeding of the 3-th SymposLlm

on Real1b111ty,Budapest,1974.

[12] D. Ciulin ,.te son~iD t T1me-Varian t C1 rcui t.

Proceeding of the Second International Sym­

posium on Network Theory, Yugoslavia,1972

[13] Lt. 'N.Newcomb, Operator Theory of Net­

works. C1rC!lits 'lnd Systems IEEE,1974

[14] H.F.Harm!lth,Tr'1nsmission of Info rma­

tion by Crthogon'l.l Functions. Springer-Ver­

lr:l.g ,Berlin ,Heidel berg, New-York ,1970

[15] C.BooBwetter, Slgn'llanaly:o: smd Syn­

these ' it Hllfe Crthogonaler ~"il ter,

Frequenz,no.l,1971

[16J H.L.Armstrong, On the iUtpredenhtion

01' Transients by Series of Orthobonal Func­

tions, I!~ Tr'ms':lctions on Circ:"u t '.rheory n~.4 ,1959.

TRIANGULARIZATION OF SOME RESTRICTED SHIFTS·

D. N. Clark University of Georgia

Athens, Georgia

and S. Sickler Eastern Nazarene College

Quincy, Mass.

Abstract

Triangularization of the res~icted shift along a chain of vector regular factorizations is obtained, in case the characteristic function is scalar-valued and pseudo-meromorphic.

1. INTRODUCTION

The purpose of this note is to point out how the Darlington,synthesis methods of

Arov (2) can be used to obtain a new ~i­

angularization of restricted shifts in the pseudo-meromorphic case. Our results take place in the scalar case, but perhaps

they point to matrix-valued generaliza­tions. Also, results peculiar to the pseudo-meromorphic case may be of interest because of the physical differences in that case.

2. DEFINITIONS

We say we have a triangularization of a bounded operator T if we have a

(discrete) representation of T as a triangular matrix or a (continuous) repre­sentation of T as a multiplication

operator and a Volterra integral operator on some L2 space. In either case, we need a chain (discrete or continuous) of invariant subs paces of T.

3. HISTORY

For a contraction operator T having a

scalar (Sz.-Nagy-Foia~) characteristic

·communicated in written form only

54

function ~, triangularizations have been given by Ahern and Clark (1) for the dis­crete case, and in (I), Kriete (4) and

Clark (3) for the continuous case. All

these papers use chains of invariant sub­spaces arising from factorizations

~= fP:J. ~ of the characteristic function into two scalar factors.

But there are other factorizations of ~

which give rise to invariant subspaces. Indeed, according to Sickler (6), if

I~(eit) 1< 1 a.e., there are chains corresponding to factorizations of the form

t ~= (~1~2) (t,Oll~2)

where t,Oij E Hoo

with I t,Ou l2 + It,Oi2 12 = 1 , i = 1,2 We pose the problem of finding

triangularizations corresponding to such chains, and we show how to do it in a special case: the case in which ~ is "noncyclic" or "pseudo-meromorphic". We use a Darlington Synthesis method of Arov (2).

4. AROV I S METHOD

The pseudo-meromorphic assumption on tp

amounts to the existence of an inner func­

tion B such that B~ E H2 , or, by

Arov (2), the existence of two invariant

subspaces Ml and M2 of T such that

Ml::l~, TIM 2

is isometric, and

T*I~~ is isometric. Let

(1- Ic,o(eit ) 12)1/2 and let

~(t) =

e be the

outer function with Ie I = ~. The sub-

spaces Ml and

factorizations of

M2 correspond to the

c,o:

c,o 1 c,o= (1 0) (!Jx'!) , c,o= (c,o ~e) (0)

respectively, where Ii> , ~ are inner.

The result of Arov (2) also asserts that

the characteristic function of the com­

pression of T on Ml - M2 is given by

A-C A concrete version of Arov's result is

given as follows.

Lemma. The operator

u- C maps each of the three spaces in the

decomposition

[0 (i) (~) ~]

Ei' [(~ EBH2)eA(~ (+\H2)]

('fl A(O (+)~)

unitarily onto the corresponding space in

the decomposition

(K e Ml

) EB (Ml e M2 ) (i) M2

of the model space K of T

5. CONCLUSION

Of course triangularization of U*TU can

be obtained, at least if one assumes a

chain of factorizations A = AkAk of A.

For example, one may pick the orthonormal

basis

Ht Zt =A(O,e )

for U*K, and get

55

1 (TUxj , UYk) = 0lj «0,1) , Ak (0»

o (TUXj , UZ k ) = OJ 1 0kO «0,1) , A (0) (1»

(TUYk' UX t ) = 0

(T~:~ (:~:(-1 ~ a::1 t::] ,A. ((1- a~ .hrJl if t.,k

(TUYk' Uz t )

_ '«,cAk' ( .it ~1 -ak,it) -1), (~))

(TUz t ' UXj ) = (TUz t ' Uy j) = 0

(TUZ t , UZ j ) = °jt+l •

More satisfactory versions of the above

can be obtained by using a triangulariza­

tion of the Sz.-Nagy-Foia~ model of A;

see Lubin (5).

(1)

REFERENCES

P. R. Ahern and D. N. Clark, On func­

tions orthogonal to invariant sub­

spaces. Acta Math. 124 (1970),

191-204.

(2) D. Z. Arov, Darlington's method for

dissipative systems. Soviet Physics­

Doklady 16 (1972), 954-956.

(3) D. N. Clark, Concrete model theory

for a class of operators. J. Func­

tional Analysis 14 (1973), 269-280.

2

(4) T. L. Kriete, Fourier transforms and

chains of inner functions. Duke

Math. J. 40 (1973), 131-143.

(5) A. Lubin, Representations of vector­

valued invariant subspaces and con­

crete model theory, to appear.

(6) S. Sickler, The invariant subspaces

of almost unitary operators.

Indiana U. Math. J. 24 (1975),

635-650.

Douglas N. Clark is Associate Professor

at the University of Georgia. He

received the Ph.D. (Mathematics) from

Johns Hopkins in 1967, and was

Assistant Professor at UCLA.

Sheldon Sickler is Assistant Professor

at Eastern Nazarene College. He

received the Ph.D. (Mathematics) at

UCLA in 1973.

56

FURTHER RESULTS ON THE

ASSOCIATION OF VARIABLES

James Conlan and E.L. Koh University of Regina Regina, Saskatchewan

Abstract

This paper summarizes and extends some results on association of variables obtained by us earlier. Specifically, the fractional calculus is used to extend some classical results for Laplace trans­forms. These results are then applied to some problems in systems analysis.

1. INTRODUCTION

If ~(t) is the input to a non-linear time

invariant system, then the output can be

represented as a Volterra series of the

form Lgn(t), where 00 00

g (t)=f ··.f h ('Il,···,'I )~(t-'Il)'" n n n -00 _00

see references (1) ,(2) ,(12). Here hn is

the impulse response of an n-th order

system.

If we let 00 00

then upon taking the n-dimensional Laplace

transform, L , of f , we have n n

Lnfn=Fn(Sl,···,Sn)=Hn(sl,···,sn)~(sl)···

~(s ), where Hn=L h , and ~=L~, where L=L l • n n n

From now on we omit the subscript "n" from

the functions fn,Fn' and gn' To obtain the

output as a function of t, we need to find

57

-1 Ln F=f(tl,···,tn )· Upon setting t l =···=

tn=t, we obtain g(t)=f(t, ... ,t). If we let

G(s) be the Laplace transform of g(t) we

have the following diagram, see reference

(3) •

G (s) -------4) 9 (t)

The function G=A F is called the associated n

transform of the function F.

2. DERIVATIVE THEOREMS

2.1 FRACTIONAL OPERATORS

In what follows we will make essential use

of the concept of fractional integration

and differentiation. There are various

ways to define these operations, not all of

which are equivalent, see reference (11).

The particular integrals we will need are

the following, see reference (6). The

Riemann-Liouville fractional integral:

A 1 JX A-l I {fix) }=f(If (x-y) f(y)dy. o

The Weyl fractional integral:

A 1 Joo A-l K {f(x)}=f(If (y-x) f(y)dy. x

Here, and throughout the paper, we assume

that A is a complex number with positive

real part, R(A»O. The corresponding

fractional derivatives are

and

, k k-A{ } D".f(x)=-(d/dx) K f(x) , respectively,

where k is the integer satisfying

k-l~R(A)<k. Note that in the limiting case

A A where A=k-l, we have DOf(x)=Doof(x)=

(d/dx) k-l f (x) .

2.2 APPLICATIONS TO SYSTEMS

In what follows it will be convenient to

use the following notation. We let

S=(sl,···,sn)' and ~m=(sl,···,sm_l,sm+l' ... s). In reference (8), Koh extended

n a result of Chen and Chiu, refenmc:e (3),

to obtain the following result.

Theorem 1. If F(i)=(s -a)-(k+l)Fl(~ ), m m

58

Proof. -1 -g (t) = f ( t , . . . ,t) = L F ( s) I -t= (t t ) n , ••• ,

Upon taking the Laplace transform of both

sides, the theorem is proved.

In references (4) and (9), Conlan and Koh

extended the two classical formulas for

Laplace transforms

t L{t-nf{t)}=(J ds)nf(x) , and

o L{tnf(t) }=(-d/ds)nf(s),

where n is a positive integer, to the case

for general n. Specifically, the

following results were obtained.

Theorem 2. Assume: (l) v is a complex

number with R{v) >0; (2) f (x) is "uch that

f(x)=O for x<O, and f(x)e- cx is absolutely

integrable on [0,00) for some c; (3)

x-vf(x) is absolutely integrable on [0,1].

Then L{t-Vf(t) }=KVL{f(t)}.

Theorem 3. Under the hypotheses (l) ,(2)

of theorem 2,

One can prove theorem 2 by setting

KVL{f}=r(~)J:(Y-S)V-l{J:f(t)e-ytdt}dY'

interchanging the order of integration,

and using the definition of the gamma

function. The proof of theorem 3 follows

in a similar fashion from

Using theorem 3, one can obtain the

following theorem, the proof of which is

similar to that of theorem 1, see refer­

ence (4) for details.

Theorem 4. If F(S)=(Sr-a)-(V+l)Fl(~r)'

(_l)k-l v then G(s)- r(v+l) D~Gl(s-a).

3. DISTRIBUTIONS AND

ASSOCIATION OF VARIABLES

3.1 FRACTIONAL OPERATORS AND DISTRIBUTIONS

The question remains as to possible exten­

sions of theorem 4 to the case where the

exponent of the monomial factor of F has

positive real part. To answer this

question it is useful to apply fractional

operators to distributions. One can

follow Erdelyi and McBride, reference (7),

and define IAf by

A A <I f,CP>=<f,K cP> (1)

where cP is a test function, and f is a

distribution with support in [O,~): see

reference (13) for definitions of these

concepts. Note that if f is a function,

this definition reduces to the formula

for interchanging the order of integration.

Using (1), the Laplace transform of

eatIA[o(t)] can be written as

L{eatIA[o(t»)}=<IA[O(t»), eate-st>

=<o(t) 'KAe-(s-a)t>=r(~)JOOyA-le-(S-a)YdY. o

Putting y=v/(s-a), this becomes 00

1 -AJ A-l-v r (A) (s-a) v e dv. o

Since the integral is just r(A), we have

We can find the Laplace transform of

eatD~O(t) in a similar manner:

(2)

L{eatD~O(t)}=«d/dt)kIk-Ao(t) ,e-(s-a)t>

and as in the proof of (2),

<oCt), Kk-Ae-(s-a)t>=(s_a)-(k-A). Hence

at A A L{e DOo (t) = (s-a) , s>a. (3)

3.2 A PRODUCT THEORID!

The answer to the question raised at the

beginning of paragraph 3.1: is embodied in

the following theorem.

_ A A

Theorem 5. If F(s)=(s -a) Fl(S ), then m m

G(S)=Kk-A{e-(s-a)t[(S-a)-d/dtlk9l(t)}lt=0'

s>a.

-1 A -1 A

=L {(s -a) }L l{Fl(S )},

59

m n- m at A and so by (3), g(t)=e DOO(t)gl(t). Hence

{ at A A G(s)=L e DOo(t)gl(t)}=<DOO(t),gl(t) x

x e-(s-a)t>

=(-l)k<o(t) ,Kk-A{(d/dt)k[gl(t)e-(s-a)t)}>

=(_l)k<o (t) ,Kk- A{ ~ (~) [-(s-a) )k-jgi j ) (t) x j=O J

=<o(t) ,Kk-A{e-(s-a)t[(S_a)_d/dt)k9l(t)}>

=Kk-A{e-(s-a)t[(S-a)-d/dtlk9l(t)}lt=0' q.e.d.

Example.

Therefore by theorem 5,

1/2{ -st }j G{s)=K e [s-d/dt]gl{t) t=O

_ 2 1 Joo -1/2 -sy 3/2 - lIT r(1/2) oy e [s-d/dy]y dy

2Joo -sy 1 =- {sy-3/2}e dy=---TI 0 TIS

One can verify the above result by direct

computation as follows:

=L-l{st/2}~ IE2 t 3- ~ D~/20(t)t~/2t3·

Therefore g{t)=~ D~/20{t)t3/2.

any test function ~(t),

<g,~> ~ <D01/ 20{t) ,t3/2~(t»

In

=-~ <0 (t) ,Kl / 2 { (d/dt) [t3/2~ (t)] >

Now for

=- ~ ____ 1 __ Joo

(y_t)-1/2(d/dy) [y3/2~(y)]X vTI r (1/2) t

x dY!t=O

where H(t)=l for t~O, and zero otherwise.

Thus, g(t)=-~H(t), and G(s)=-~L{H}=-~ " TI TIS

4. SERIES EXPANSIONS

4.1 A GENERAL METHOD

In this section we present a general

method for deriving operational formulas

f0r the association of variables. We

consider the function F of the form

F{S)=Fl(sm)F2{~m)' where Fl(Sm) can be ex­

panded in an absolutely convergent series,

for ISml>R, of the type

00 A. Fl(S ) = L a./s J

m j=O J m (4)

with {A.} an arbitrary increasing sequence J

of positive numbers which approach 00.

Under these conditions, one can take the

inverse transform of Fl(Sm) term by term

to yield

00 1..-1 fl{tm) = L a.t J /r(A.)

j=O J m J (5)

which converges for all real and complex

tm~O~ see reference (5). This fact enables

us to prove the following.

Theorem 6. If F(S)=F l (Sm)F2 (Sm)' and

00 A. Fl(Sm) L a./s J where O<AO<Al< ... ~oo,

j=O J m then

G(s) a. k. ",-1

\ ---1- (-l) J D J G (s) l. r(A.) 00 2 '

j=O J

where k.-l<A.<k .. J - J J

Proof. From (5), we have

00 1..-1 L a.t J

j=O J m

Hence get)

f2(t )/r{A.)

m J

",-1 L a.t J g2(t)/~(AJ')' and so

j=O J

00 1..-1 t G(s)=J { L aJ.t J g2(t)/r(A

J.)}e-

s dt.

o j=O

Since g2(t) is Laplace transformable, and

the a. are such that the series converges J

absolutely, we can interchange the orders

of summation and integration. 'rhus

00 00 1..-1 G(s)= L [a./r(A.)]J t J g2(t)e-stdt.

j=O J J 0

By theorem 3, we obtain the conclusion of

the theorem.

4.2 APPLICATIONS

Example 1. _ 2 2 -v A

Let F(s)=(s +a) Fl(S), m m

R(v»O. It is easly shown that

00 j 2J' L (-1) r(v+j)a j=O j!r(v)s~(j+V)

Thus

G(s) 00 ' 2' k \ (-1) J r (v+j) a J (-1) j o 2j+2v-l L J'!r(vjr(2J'+2v) 00 x

j=O

x

where k,-1<2v<k,. From Legendre's duplica-J J

'- 2z-1 1 tion formula, rnr(2z)=2 r(z)r(z+Z)'

we get

L j=O

00

G(s)= L j=O j! rn r (j+l/2)

where k>j+l/2, i.e., k=j+l. Therefore,

G(s)=-,I Ia j /(j!rn»)o!-1/2 Gl(s). J=O

Since o-1/2=_Kl / 2 , and OV=Ov=(d/ds)v for 00 00

, -1/2 aO 1/2 ~nteger v, G(s)=n e K Gl(s), or

lfoo -1/2 G(s)=n (y-s-a) Gl(y)dy s+a

(7) •

This formula is certainly well defined for

the Laplace transform Gl{s). As an

application consider F{sl,s2)=[1/ls l +l) x

x [1/{s2+1»). Here Gl{s)=l/{s+l), and the

associated transform of F is

G{s) ! f [l/Iy-s-l) [l/{y+l)]dy n s+l

! 1 f dv, on letting v 1T Is+2 0 (v+l) IV

Hence G{s)=1/ls+2, since the last integral

(6), is simply 1T. This result can be verified

where J is the Bessel f.'nction of order ~. ~

Strictly speaking, this formula (6) is

symbolic in nature. It becomes more tract­

able for specific i~tegral values of v. For instance, when v=l, the formula reduces

=-!sin(aO )Gl(S)=-!sin(aO)Gl{s), where a 00 a

O=d/ds, which was obtained by Koh in ref­

erence (8).

Example 2. Let F(S)=(S +a)-1/2Fl{~ ). m m

Again, it can be shown that

(s +a)-1/2 m

Thus

61

by direct inversion.

(l)

(2)

(3)

REFERENCES

Barrett, J. F.: "The use of function-'

als in the analysis of non-linear

physical systems". J. Electron

Control, 15, 1963, 567.

Brilliant, M.B.: "Theory of the

analysis of non-linear systems".

Report 345, Research Lab. of Electron,

M.LT., 1958.

Chen, C.F., and Chiu, R.F.: "New

theorems of association of variables

in multiple dimensional Laplace trans­

form". Int. J. Systems ScL, 1973,

4, 647.

(4)

(5)

Conlan J. and Koh, E. L.: "A fract­

ional differentiation theorem for

the Laplace transform". Canadian

Math. Bull., to appear.

Doetsch, G. :"Guide to the applica-

Professor James Conlan received the

bachelors degree and the masters degree

from the University of California,

Berkeley Campus. He received the Ph.D.

in mathematics from the University of

Maryland in 195B. He has worked as a

tion of the Laplace and z-transform", research mathematician for the U.S. Navy,

Van Nostrand, 1971. and has taught at the University of

Western Australia, and at Howrard

(6)

(7)

(B)

Erdelyi, A., et.al.:" Tables of

integral transforms", McGraw-Hill,

1954.

Erdelyi, A., and McBride, A.C.:

"Fractional integrals of distribu­

tions", Siam J. Math. Anal., 1,

1970, 547.

Koh, E.L.: "Association of variables

in n-dimensional Laplace transform",

Int. J. Systems Sci., 1975, 6, 127.

(9) Koh, E.L. and Conlan, J.: "Fract­

ional derivatives, Laplace trans­

forms, and association of variables",

Int. J. Systems Sci., to appear.

(10) Lubbock, J.K. and Bansal, V.S.:

"Multidimensional Laplace trans­

forms for solution of nonlinear

equations", Proc. I.E.E., 1969,

166, 2075.

(11) Oldham, K.B. and Spanier, J.: "The

fractional calculus", Academic

Press, 1974.

(12) Volterra, V.: "Thecry of function-­

als", Blackie, 1930.

(13) Zemanian, A.H.: "Distribution

theory and transform analysis",

McGraw-Hill, 1965.

62

University.

Professor Eusebio L. Koh received his

Ph.D. in 1967 at the State University of

New York, Stony Brook, working under

A.H. Zemanian. His dissertation was on

the Hankel transformation of generalized

functions. He holds an honors degree

from the University of the Philippines

and masters degrees from Purdue and

Birmingham. He taught engineering in the

Philippines until 1964 and mathematics

in South Carolina for a year before going

to Regina in 196B. He is on Sabbatical

Leave at the Technischen Hochschule

Darmstadt during 1975-76. His research

interests are distribution theory,

integral transformation, and differential

equations. Professor Koh is married and

has four children.

TilE FEEDBACK INTERCONNECTION OF MULTI VARIABLE

SYSTEMS: SIMPLIFYING THEOREMS FOR STABILITY

C. A. Desoer and W. S. Chan

Department of Electrical Engineering and Computer Sciences and the Electronics Research Laboratory

University of California, Berkeley, California 947Z0

Abstract

We consider the stability of the feedback interconnection of possibly unstable, n-input n-output subsystems whose interconnection is described by el = ul - yZ' eZ = uz + Yl and Yi = Gi(ei), i = 1,Z. We give three theorems which simplify the stability tests. Theorem 1 deals with nonlinear time-varying subsystems. It gives conditions on GZ so that the stability of ul ~ Yl guarantees that of the feedback system. The other two theorems consider continuous-time linear time­linvariant subsystems. It is noted that in the multivariable case, the stability of ui ~ Yi, i = 1,Z are not sufficient to guarantee the stability of the feedback system and Theorem Z specifies some additional required conditions. Theorem 3 Shows that if GZ and Gl(I+GZGl)-l are in some special stable classes, so is the transfer function of the feedback system. In both theorems, corollaries specialize the results to lumped and single-input single-output cases. The paper ends by showing how these results can be translated for the discrete-time case.

INTRODUCTION

This paper may be viewed as a first step toward a general input-output theory for arbitrary inter­connections of multi-input multi-output subsystems. In contrast to [1] it does allow, in several results, unstable subsystems. It is closely related to [Z] which gives necessary and sufficient conditions for stability allowing for unstable subsystems. The thrust of the paper is towards finding conditions under which stability tests are greatly simplified. The results below constitute an extension of results presented at the 1974 Allerton Conference [18]. The discrete-time extension is described in section IV.

The point of view adopted in the paper is that pioneered by Sandberg and Zames [3,4]. This approach to stability problems has been developed in many papers [5-9] and books [lO-lZ]. A slightly different but closely related approach is to be found in [13-16].

In the first section of the paper we describe the system under consideration and review the pertinent definitions and facts needed to state our results. The second section presents two basic examples which are needed to understand some basic points related to the new results. The third section states precisely the three basic theorems and

63

tries to describe the nature and interrelationships of the results. A more complete version of this paper will appear in the Proc. IEEE, Dec., 1975.

Notations. lR, 0:, lR(s)".)\: denote, respectively, the fields of real numbers, complex numbers, rational functions with real coefficients, and the convolution algebra defined in [5], [6] and [lZ]. The elements of the convolution algebra,.)\: are generalized functions described in (10), below; it is easy to show that if f,g EI.)\: then f+g and cf (for all real number c) are in~, furthermore if as "product" one takes the convolution of f and g, then f*g E,~; for this reason,.)\: is called a convolution algebra. Superscripts nand nxn are used to denote th~ co~resgonding classes of ordered n-tuples (e.g. lR , (/; ".J\') and nxn arrays (e.g. lR(s)nxn), respectively. Laplace transforms are denoted by a '. Operators and matrix-trans fer­functions are denoted by capitals (e.g. Gl' Gl)· Scalar transfer functions are denoted by lower case letters, (e.g. g(s». The abbreviations MIMO and SISO denote "multiple-input miltiple-output" and "single-input single-output," respectively. (/;+ and &+ denote the closed and the open right half-plane.

I. SYSTEM DESCRIPTION AND PRELIMINARY DEFINITIONS

We consider a feedback system S whose inputs,

outputs, etc. are defined on 0' c 1R: typically 0' = lR+, for continuous-time systems, and 0' = l+, (the nonnegative integers) for discrete-time systems. Let',J = {f: jJ +CV} where CV is a normed space with norm 11.11. For any T E 0', fT(t) = f(t) if t < T, and zero for t > T. Using the usual definitions of addition and scalar multi­plication, we define the vector space

To avoid long concatenations of subscripts, we shall write

The feedback system S is made up of two subsystems as shown in Fig. I. If CV = lRn, then the two subsystems are n-input n-output subsystems.~ )The inputs ui, errors ei' outputs Yi belong to Sfe . We define for i = 1,2

(1)

Note that it is not required that the Gi be linear. The equations are then

(2)

(3)

We make a general existence assumption ~~ich w~ll hold througho~t thl:!laper: lI(ul,u2) E Sf e x Sf e , 3 (el> e2) E ~e x Sf e which satisfy the equations (2), (3) of the system. For general existence criteria see [4,11,12]. Note that uniqueness is not required. If uniqueness holds, there is a map, denoted by He such that

If uniqueness does not hold, He becomes a relation [17].

:0 Gi is said to be ~-stable iff (4)

. 3 I( < '" :3 IIx E S£, liT E g e

II Gixll < kllxD T - T

The gain of Gi is defined to be the infimum of all such k; it is denoted by y(Gi). Calculations of the gain for SISO and MIMO systems can be found in [3,4,11,12]. The incremental gain of Gi, Y(Gi),(t) is defined as

(t)The superscript - used to distinguished the incremental gain from the gain, has nothing to do with the II-II sometimes used to denote Z-transforms.

64

IIG x - G x II < yllXl

- x II }. il i2T- 2T

(5)

For linear system y(Gi ) = y(Gi ). Let u, e, and y denote the order pairs (ul,u2), (el,e2). and (Yl,Y2)' respectively. We also have the map H : u ~ y. It is important to note that if we def~ne J : ~e x Sfe + 'le x Sf e by J(u)

J(ul.u2) = (u2.-ul). then

(6)

where I denotes the identity. LoLl"

If both G1 and G2 are linear maps, the map He : u ~e is obtained as follows: (a) operate with G10n (2). use linearity, eliminate Glel using (3) and solve for e2; (b) operate with G2 on (3), use linearity, eliminate G2e2 using (2) and solve for el; (c) the linearity of G2 imllies (I+G2Gl)G2 = G2(I+GlG2) hence G2(I+GlG2)- = (I+G2G1)-lG2; (d) similarly, the linearity of Gl implies Gl(I+G2Gl)-1

(I+GlG2)-IGI' The final result is

e = [elJ [(I+G2Gl )-1 e

2 = G

I(I+G

2G

l)-1

where G1G2 denotes the composition of Gl with G2'

1:h)e TIJ1f' (or the relation). He is said to be .) Sf x~ -stable iff 3k < 00 such that lIul,u2 E: ~e' liT E ~,.for i = 1,2

(8)

In other words, if in the product space we choose the norm lIuli = Ilulll + IIU211. then we see that (8) is equivalent to y(He ) < "'. From (6), y(He ) < 00 if and only if y(Hy) < ~.

For the continuous-time. linear, time-invariant case, for i = 1,2, we define Gi by a convolution: to alleviate notation. we also use Gi to denote the kernel of the convolution operator, thus Gi : 1R+ + lR nxn and

(9)

Using A to denote Laplace transformed quantities • we have

In the linear, time-invariant, distributed case, we introduce the Banach algebras ,J\ and,.) as follows (see [5],[6],[12])

,J~ {f : 1R+ + lR I f(t) L fio(t-t i ) + fa(t) i=o

where (10)

L Ifil < "', t > 0 IIi, fa ELl} i=o i-

A Eu~nxn reans t;, that each element of the matrix A ErJt. 1,.7' nxn ~ lA I A Ej\ nxn}. It is well kpown that if Gl>~2 Ej\:nxn, then (;1 + (;2, (;1(;2 Ey\nxn and Gil Ej\:nxn * inf Idet Gl(s) I > O.

sEG:+

IIhn t;, 2: Ih.1 + loo Ih (t) Idt, (11)

a i=o 1. a

0

Ifh ( ) E fin hl ,h2,··· ,hn -

Ilhll t;, = maxllh II ,

a CY. a CY.

and if H E rJ't nXn

t;, n IIHII = max ~ Dh .. U (12)

a i j=l 1.J a

Then if 1 u E Ln ~nxn < p ..:: 00 , and H Er then

p

HH*u U < IIHII lIu Y (13) p- a p

where II. lip denotes the .£th norm [12]. It can also be shown that if H ErJnxn and if u : 1R+ -+ :R n is continuous and bounded (or almost periodic, or periodic) then H*u has the same properties, resp. Property (13) is often expressed by saying that u ~ H*u is L~stable for all p E [1,00].

Ii rA lnxn Two elements : . ./\J, ~u of rJ\ are said to be pseudo right coprime, abbr. p.r.c., (resp. pseudo left coprime, abbr. p.l.c.) [12,19] iff

(i) det CUJ (s) ~ 0 Vs E t+

and (ii) q)Ji +C\jct) = q,t; (resp ... Klcfl-fl--m~iti) Gixen a function G : ~+ -+ ~nxn, the ordered pair (;\I,Cf)) is said to be a p.r.c. factorization, abbr. p.r.c.f. (resp. p.l.c. factorization, abbr. p.l.c.f.) of G iff

(i) G =JGc{)-l (resp. G = cO-S~)

In ~he linear, time-i~variant. lumped case, Gl,G2 E 1R (s)nxn and Gi is said to be proper iff all its elements are bounded at infinity, and

Gi is said to be exponential stable (abbr. expo s~) iff it is proper and bas all its poles in ~_, (the open left half plane).

If G E 1R (s) nxn and (; is proper, then G has both a left- and a right-coprime factorization, [12].

The following observation will greatly simplify the analysis below: The map J defined above in (6) is a linear isometric map, therefore the two equations (6) lead to the l~vJ

Lemma: a) In the general nonlinear case: He is --g: x S£ stable if and only if Hy is :1 x -:J: stable;

b) in the linear time-invariant case:

He E ,,2nx2n 1.·f d 1 'f Jt an on y 1

H E .fi2nx2n. y •

also for the lumped case, . He is expo st. if and only if Hy is

expo st.

It is this lemma that allows us to restrict our attention to He exclusively. We choose He because it is more convenient to work with.

II. INSTRUCTIVE EXAMPLES

In the linear case, He is given by (7): He splits into four partial maps: ui» ej. i,j =)1,2. Each one of these four partial maps may be ~-stable or not: this gives 16 = 24 possible patterns of instability; this number is further reduced to 10 by interchanging subscripts 1 and 2. In view of the fact that each of the four partial maps depends on the same two functions Gl and G2, one might expect that not all possible patterns of instability might ?ccur and hence that one might prove the sr x Sf -stability of He by studying only a proper subset of the four partial maps. This is, in fact, not so. Consider the following two linear time­invariant examples.

(ii)~,cf) are p.r.c. (resp. il,q) are p.l.c) Example 1. If gl(s~ = lis, g2(s) = s/(s+l), then all submatrices of He are expo stable except 81(1+8281)-1 which has a pole at s = O.

" lim inf IdetC[)(s.)I > 0 • i-+oo 1.

TheAfoll~wing fact ha~been established in [121. If G E ,...An n A and v(1,q ,) is a p. r. c. f. or a p.l.c.f. of G then p E t+ is a pole of G

* P E (;+ is a zero of det cf).

65

Example 2. If al(s) 1/ s]

l/s

G 2 (s) = [1/ (s+ 1) 1/ s l o l/(S+ld

then all submatrices of He are expo stable except (1+(;2(;1)-1 which has a pole at s = O. A detailed

study of all 10 possibilities is reported in [18].

In conclusion, even in the lumped, linear, time­inyarj~nt case, in order to prove that He is ~ x :::r stable, one must in the general case investigate the stability of each of the four partial maps ui ~ ej' i,j = l,Z.

III. THE SIMPLIFYING THEOREMS

In most design procedures and stability considerations one assumes uz = 0 and studies the stability of the map ul ~ Yl' namely, Gl(I+GZGl)-l. An interesting question is then: under what general conditions does_nhe(V-stability of Gl(I+GZGl)-l imply the Sf x~ -stability of He? The following theorem answers the question for a broad class of nonlinear systems:

Theorem 1. (Nonlinear time-varying MlMO)

Let Gi be defined_~s in (1). If GZ and Gl(I+GZGl)-l are ~-stable, and if the incremental gatn .of Gz, y(GZ)' is finite, then He and Hy are ~x ~ stable. -

Comments: (a) It s~ould be stressed that Gl is not required to be ~-stable. (b) In particular, if as in most practical cases the feedback sub­system, GZ' is linear, then the c~ndition Y(GZ) < 00 is equivalent to that GZ be Sf-stable. Therefore for the linear time-varying MIMO case, the ~-stability of)Gztnd that of Gl(I+GZGl)-l imply that He is ~ x ~ stable. (c) If GZ is unbiased (i.e. GZO = 0), choosing xZ = 0 in (5) and comparing with (4), we see that y(GZ) ~ Y(GZ)· Hence, we have the following

Corollary 1.1. (Nonlinear time-varying MIMO)

If Gl(I+GZGl)-l is ~-stable, if GZ is unbiased and if GZ has a finitF i~Tremental gain, y(GZ), then He and Hy are '.:f x ~ stable. -

In order to bring to bear analytical tools, we restrict ourselves to linear time-invariant distributed systems. An important feature of Theorem Z and its corollaries, is that they do not impose any stability conditions on either Gl or GZ. This is in contrast to Theorem 1 which requires that y(GZ) < 00.

Theorem Z. (Linear time-invariant distributed MIMO)

Let Gl and GZ be represented by con~olution oper~tors as in (9). S~ppose that Gl has p:l.c.f. and GZ has p.r.c.f. or Gl has p.r.c.f. and GZ has p.l.c.f. Suppose that V sequences (Si)1=1 C '+ and I si I + 00

lim infldet[I + Gl(si)GZ(si)]I > 0 (14) i-->oo

Under these conditions (abbr. U. Lc.) if (a) 9l(I~GZGl)-l, GZ(I+GlGZ)-l are in,~nxn, and (b) Gl, GZ have no common ~ pole, then He and Hy E ')ZnxZn. -

66

Comments: (a) this conclusion implies that He is Ln-stable for all p E [1,00], see (13), and that t~e system takes inputs u that are continuous and bounded, or almost periodic, or periodic into errors e and outputs y of the same class, resp. (b) Note that neither Gl nor GZ are assumed to be L~-stable.

Corollary Z.l. (Linear time-invariant lumped MIMO)

~et, for i = 1,Z, Gi be a convolution opera~o:, Gi(s) E 1R (s)nxn and be pr~per. Let det(I+GlG2) (00) # O. U.t.c., ifAGl(I+G2Gl)-1, GZ(I+GlGZ)-I are expo st~ and it Gl and GZ have no common ~ pole, then He and Hy are expo st. -

The condition det(I+G1GZ) (00) # 0 is related to well-posedness [11,15]: with the Gi(S) E 1R(s)nxn and proper, this determinantal condition is violated if and only if (I+GlGZ)-l and (I+GZGl)-l have a pole at infinity, i.e. the closed-loop system transfer function He includes differentiators!

Corollary Z.Z. (Linear time-invariant lumped SISO)

~et, for i = 1,Z, gi gi(s} E ~(~) and be and gZ(l+glgZ)-l are expo st.

be a convolution operator, A A A 1 proper. U.t.c·Aif gl(!+gZgl} expo st., then He and Hy are -

Observe that if the assumption of Corollary Z.Z were replaced by "(1+8Z8l)-1 is expo st." then the conclusion does not follow. See counterexample in [18]: gl(s) = s/(;:l); 8Z(s) = (s-l)/s.

Corollary Z.3. (Linear time-invariant distributed SISO)

Let, for i = 1,Z, Gi be SISO, hence denoted by gi and let it be a convolu~ion ~p~rator. Let 8l,8Z ~ave ~.~.f. U.t.c. i¥ gl(l+~Zgl)-lAand ~ gZ(1+gl gZ)-l are in J\- then He and Hy EcJtZxZ -

Theorem 3 and its corollary are more restrictive: tyey exploit the properties of the algebras ,finxn and 1R (s)nxn, resp. and impose some stability requirement on GZ'

Theorem 3. (Linear time-invariant distributed MIMO)

If GZ and Gl(I+GZGl)-l are in,~nxn, then He and Hy are in,~ZnxZn. -

Since the proof of Theorem 3 is purely algebraic, it obviously extends almost verbatim to the lumped case.

Corollary 3.1. (Linear time-invariant lumped MIMO)

If GZ and Gl~I+GZGl}-l are exponentially stable, then so are He and Hy . -

Note that it is this corollary which justifies the common design procedures and the elementary discussions of MIMO feedback systems.

Comments: (a) in,TheoremJ3 and Corollary 3.1 it is

not assumed that Gl is in, nxn or exponentially stable resp., (b) Theorem 3 and Corollary 3.1 bear

I :\

a striking resemblence with Theorem 1. The nature of the results are different. Theorem 1 deals with nonlinear operators and the conclusion that some operators have finite gain is reached on the basis of similar assumptions concerning G2 and Gl (I+G2Gl )-l, together with the auxiliary assumption y(G2) < 00. Theorem 3 deals with a s~ecific algebra of linear operators, namely. ~n~n, the conclusion to be reached is that He ~ ~2nx2n. The proof is purely calculational relying on properties of linear maps and on the various closure properties of the algebra. The same comments hold for Corollary 3.1. (c) One might think that Theorem 3 follows from Theorem 1 by replacing "'::i-stable" with "Ll}-stable for all p E [I,"']": this, however, is incorrect because the class of convolution operators that are L~-stable for all p E [1,00] is larger than,~nxn. Indeed the latter excludes the possibility of singular measures in the convolution kernel.

IV. THE DISCRETE-TIME CASE

The results above except for Theorem 1 and its corollary are stated for the continuous-time case. A study of the proofs would easily show that they extend easily to the discrete-time case. The required changes are listed in the Table I: B(O,l) and B(O,l)C denote the open unit ball centered on 0 in ~ and its complement, resp.; tl denotes the convolution algebra of absolutely

convergent sequences: £1 = {(z.)'" c c:1 ~lz.I<,"'}, (for details see [12]). 1 0 0 1

Table I

Laplace transform -> Z-transform

.A -> '~l

,;nxn nXn -> 2.1

0

I: -> B(O,l)

c:+ -> B(O,l)C

s -> -> z -> 00

1R(s)nxn -> 1R (z) nxn

Research sponsored by the National Science Foundation Grant GK-43024X.

[3] 1. W. Sandberg, "Some Results on the Theory of Physical Systems Governed by Nonlinear Functional Equations," Bell Syst. Tech. Jour., 44, p. 871-898 (May-June 1965).

[4] G. Zames, "On the Input-Output Stability of Nonlinear Time-Varying Feedback Systems," IEEE Trans. AC-ll, 2, 228-238; 3, 465-467, (1966).

[5] C. A. Desoer and M. Y. Wu, "Stability of Linear Time-Invariant Systems," IEEE Trans. CT-15, p. 245-250 (1968).

[6] , "Stability of Multiloop Feedback Linear Time-Invariant Systems," J. Math. Anal. Appl, 23, p. 121-130 (1968).

[7] F. M. Callier and C. A. Desoer, "Necessary and Sufficient Conditions for Stability of n-input n-output Convolution Feedback Systems," IEEE Trans. AC-18, 3, p. 295-298, June 1973.

[8] F. M. Callier and C. A. Desoer, "LP-stability, (12 p 2 00 ), of Multivariable Nonlinear Time-Varying Feedback Systems that are Open Loop Stable," Int. Jour. of Control, 19, I, 65-72, 1974. --

[9] M. Vidyasagar, "Some Applications of the Spectral Radius Concept to Nonlinear Feedback Stability," IEEE Trans. CT-19, p. 608-615, Nov. 1972.

[10] J. M. Holtzman, "Nonlinear System Theory," Prentice-Hall, Englewood Cliffs, New Jersey, 1970.

[11] J. C. Willems, "The Analysis of Feedback Systems," MIT Press, Cambridge, Mass, 1971.

[12] C. A. Desoer and M. Vidyasagar, "Feedback Systems: Input Output Properties," Academic Press, New York, 1975.

[13] W. A. Porter and C. L. Zahm, "Basic Concepts in System Theory," Tech. Report 33, Systems Engg. Lab., Univ. of Michigan, Ann Arbor, 1969.

[14] M. J. Damborg and A. Naylor, "Fundamental Structure of Input-Output Stability of Feedback Systems," IEEE Trans. Vol. SSC-6, p. 92-96, 1970.

References [15] R. Saeks, "Resolution Space, Operators and Systems," Lecture notes 82, Springer-Verlag, 1973. [I] D. W. Porter and A. N. Michel, "Input-Output

Stability of Time-Varying Nonlinear Multiloop Feedback Systems," IEEE Trans. AC-19, 4, [16] p. 422-427, Aug. 1974.

[2] F. M. Callier and C. A. Desoer,"A Stability Theorem Concerning Open-loop Unstable Con-volution Feedback Systems with Dynamical [17] Feedbacks," to be presented at the IFAC Congress, 1975.

67

R. De Santis, "Causality, Strict Causality and Invertibility for Systems in Hilbert Resolution Spaces," SIAM J. Control, g, 3, p. 536-554, Aug. 1974.

S. MacLane and G. Birkhoff, "Algebra," The MacMillan Co., New York, 1967.

[18] C. A. Desoer and W. S. Chan, "Interconnection of Unstable Linear Systems," Proc. Twelfth Allerton Conference 1974, U. of Ill., Urbana, Illinois.

[20] F. M. Callier and C. A. Desoer, "Open-loop Unstable Convolution Feedback Systems with Dynamical Feedbacks," to appear in Automatica (Abbreviated Version Proc. IFAC '75).

68

[19] M. Vidyasagar, "Coprime Factorization and Stability of Multivariable Distributed Feedback Systems," (to appear in SIAM J. Control).

F

THE "FOURIER" TRANSFORM OF A RESOLUTION SPACE AND A THEOREM OF MASANI*

R. A. DeCarlo, R. Saeks, and M. Strauss Texas Tech University

Lubbock, Texas

1. ABSTRACT

Using two classic theorems (one of.Mackey an~ another of Stone) and ~ recent result of Masani and Rosenberg, thlS paper pleces together a genera11zed frequency response theory for an abstract Uniform Resolution Space. The present theory assimilates past work as done by Fa1b, Freedman, Anton, Masani and Rosenberg, and one of the authors. The results of this paper are not new, but are merely a rearrangement of subtleties uncovered by the aforementioned aut~ors. An interesting consequence of this work was that an abstract Uniform Reso1utlon Space has both a "time transform" and a "frequency tranSform". Such a duality is not readily identifiable in an L2 function space since the time transform, there, is the identity.

2. INTRODUCTION

Fourier analysis is basic to the design and under­standing of physical systems. The property that convolution in the time domain maps into a product in the frequency domain, yields a theory both prac­tical and aesthetically pleasing. This note pro­vides what is hoped to be a generalized frequency response theory for arbitrary, closed, linear, time invariant operators on a uniform resolution space. Previous attempts at providing a general frequency theory have illuminated numerous subtleties, yet still appear inadequate for one reason or another. Interestingly enough, the mathematics necessary for such a synthesis is well entrenched in the 1itera-ture. This paper merely pieces these results to­gether and reinterprets them in light of the work

two fundamental ideas--the idea of a "transform" from time to frequency and the property of a time­

invariant mapping to a product of functions in fre­quency. We desire a Fourier representation for time invariant operators defined on an appropriate space. Two avenues arise. A traditional approach uses a Fourier-like integral to obtain the repre­sentation. In an abstract approach, the Fourier representation is a spectral representation of the abstract operator relative to an appropriate spec­tral measure. This road is both more general and eliminates the need for a specific representation of the operator.

Fa1b, Freedman and Anton (3), (5) developed a gen-era1ization closely paralleling the classical

of Fa1b, Freedman, Masani, Rosenberg and Saeks. (3), theory. The formulation considers Hilbert space­(9), (12), (21)

Classical Fourier analysis consists essentially of valued L2 functions (square integrable relative to the Haar measure), defined over a locally compact

*This research supported in part by Air Force Office of Scientific Research Grant AFOSR 74-2631.

69

abelian (LCA) group, G, and operators which are characterized by an Ll convolutional weighting function. The theory is highly representation-de­pendent and fits awkwardly into the setting of an abstract resolution space. In fact, the identity and unit delay are not admissable to the theory. The major advantage is that one obtains an opera­tor-valued Fourier representation.

Masani and Rosenberg (11), (7) use a spectral theoretic vehicle to alleviate the difficulty of a specific representation of the operator. Moreover, the theory settles nicely into an abstract setting. Yet, the frequency response is always scalar-valued, even in the multivariable case, and the concept of a "transform" is absent.

Finally, Saeks (21) has a Masani-like development whose Fourier representation assumes values in a suitably restricted class of operators. The advan­tages are the compatibility with abstract spaces and an operator-valued frequency response. Yet, still, the concept of a transform is missing and major existence questions are still present.

The structure of the present theory rests on the classic theorems of Mackey (7) and Stone (4) and a recent theorem of Rosenberg and Masani. (10) With this comment, we define the setting.

2. UNIFORM RESOLUTION SPACE

A resolution space is a pair, (H,E), where H is a Hilbert space and E is a spectral measure on an or­dered LCA topological group, G. On an ordered LCA group, a spectral measure determines a resolution of the identity, and conversely. Thus, it is ad­vantageous to work with the resolution of the iden­tity Et = E([_oo, t]), rather than with the spectral measure E, as illustrated at the end of this sec­tion.

As an example, consider the Banach space, L2, to­gether with the truncation operator, Et , defined as

(EtX)(q) = {x(q) q ~ t t 0 q > t

or equivalently, the spectral measure, defined via

(E(B)x)(q) = [Xo(q) q £ B

q ¢ B

for all Borel sets B.

In addition, L2 admits a group U of shift operators ut , defined as

(Utx)(q) = x(q - t).

Thus, the concept of time invariance is well de­fined in a classical L2 setting. In general, a re­solution space lacks the concept of time invari­ance. Such a property requires an extension of the concept of the L2 "time-shift". A group of such operators, in general, fails to exist in an arbi­trary resolution space.

In particular, we seek a strongly continuous group of unitary operators (i.e., Ut -s = Ut(Us)-l for all

t and s in G), such that

UtE(B) = E(B + t)Ut

for all t in G and Borel sets B. A resolution space, together with such a group U of shift oper­

ators, ut , is a Untform Resolution Space (URS) , de­noted by the triple (H,E,U).

Underlying each URS is an ordered LCA group, G, which, for our purposes, i~ time. Associated with G is a "character group", G, which is the group of continuous homomorphisms from G into the multi­plicative group ~f complex numbers of magnitude one. Note that G is, in general, not ordered.

In like manner, attached to each URS (e.g., (H,E, U)), defined over G, is a "dual" character space (H,U,E)*, defined over G. E and U are a spectral measure and group of shift operators, respectively, defined via the two equalities

t £ G

and A

Y £ G.

Here, (y,-t) denotes the complex number of magni­tude one, resulting from the operation of the

A

*We have adopted the ordering (H,U,E) bekause via Stone'~ theorem, E and U cQntain the same information. Moreover, U and E do. Thus, (H,U,E) rather than (H,E,U).

70

r " character y in G acting on -t in G, and where the

integral is the Lebesgue integral. Stone's theor-" em (~) assures the existence and uniqueness of E

and U. " " Oddly, the charact~r space (H,U,E) is not a resol-

~t~on space since G is not ordered. However, (H, U,E) displays all the resolution space properties which do not depend on the ordering of G. In fact, by Stone's theorem (~), (7), (12), U is a group of shift operators for E, satisfying the imprimitivity equality over G--i.e.,

UYE(B) = E{B + y)UY•

For our purposes, the character group plays the role of frequency.

Now, the physical properties of causality, memory­lessness, time invariance, etcetra, have precise descriptions in the uniform resolution space struc­ture. In particular, for bounded operators, T, on (H,E,U), causality is equivalent to EtT = EtTEt (1), (2), (20); anticausality, to EtT = EtTEt *; memorylessness, to EtT = TEt which, in turn, is equivalent to T, being both causal and anticausal. Since memorylessness is a symmetric conce~t~ it has an analog in the character space, (H,U,E), whereas causality does not. Because of this, we say a bounded operator, T, is time invariant if E{B)T = TE{B) for all Borel sets B of G. Via Stone's theorem, this is equivalent to UtT = TUt

for all t in G. Clearly, we emphasize the charac­ter space in the definition of time invariance.

In the case of unbounded operators, T (e.g., the derivative operator), T is causal if EtT ~ EtTEt **; T is anticausal if EtT ~EtTEt; T is memoryless if EtT = TEt; and, finally, although ~omewhat ~on-in­tuitively, T is time invariant if E{B)T ~TE(B)

A -

for all Borel sets B in G, where, again, we empha-size the definition in the character space. For unbounded operators, Stone's theorem, in general, does not yield an equivalent statement (such as

UtT = TUt ) in the original resolution space. How­ever, for the case of linear, single-valued, closed operators with domain dense in H, then UtT = TUt

if and only if EtT '= TEt. (9), (10) The funda­mental role of the character space becomes more clear in the following section.

3. EQUIVALENT SPACES

In this section, Mackey's theorem verifies an equivalence between an abstract URS, (H,E,U), and a function space, (L2{G,K), XB' crt). Now, the rele­vant information contained in (H,E,U) is also con-

" " tained in (H,U,E). Thus, applying Mackey's theo-" " rem to (H,E,U){under the guise of (H,U,E», another

equivalence to (L2{G,K), crY, xs)"exists. Further­more, (L2{G,K), XB' crt) and (L2{G,K), crY, Xs) have an affinity via Stone's theorem.

After numerous references throughout this develop­ment to the above two authors, we, at last, pre­cisely state their results. Hopefully, this will facilitate understanding of the maps between the various spaces, hinted to in the above paragraph. The following is a statement of Stone's theorem for

LCA groups. (4), (14)

Suppose G is an LCA group and G, its "dual" char­acter group; let (y, -t) be the complex number of magnitude one, resulting from the operation of Y in G on -t in G; define E{L) as the cr-algebra of Borel ~ets of G{G); finally, let {utlt in G} ({UYly in G}) be a strongly continuous group of unitary op­erators on a complex Hilbert space, H, onto H." Then, there exists a unique spectral measure, E{·) (E{.», for H on E(L), such that for all t in G (y

" in G),

or

UY = IG{y, -t)dE{t).

The initial task, now, is to construct an equiva-

t *Et = E«t, 00]) = I - E •

**For an unbounded operator, T, onta resolution space, (H,E,U), the domain of EtT is smaller than the domain of TE. As such, the containments indicate that, where the domains of EtT and TEt coincide, then EtT = TEt.

71

I'

II ii, :ii:

lence between twa L2 spaces via this thearem.

Cansider the URS'A(L2(G,K), XB' at), and the char­acter s£ace, (L2(G,K), aY, Xs)' where G is an LCA graup; G, its character graup; XB' the character­istic functian af the Barel set B in L; at, the classical shift aperatar (i.e., (atf)(q) = f(q -t)*, and, lastly, the measure an the space will be the Haar measure, m.

Now, the Faurier transfarm maps L2

(G,K) to. L2(G,K)

in such a mann~r that XB is taken to. the spectral measure an L2(G,K), whase integral is aY. Simi-

t A

larly, a maps to. the unitary graup an L2(G,K), whas~ assaciated spectral measure is XS' As such, (L2(G,K), aY, Xs) is the Faurier transfarm af the character space far (L2(G,K), XB' at).

Interpreting this, we have XBcampletely determin­ing aY and XS' completely specifying at. The link between these twa spaces and an abstract resalutian space is Mackey's thearem. The statement af his thearem far LCA graups fallaws. (19)

Let E be a spectral measure an the Barel sets af an LCA graup, J, and let U be a strangly cantinuaus, unitary representatian af J, such that

E(B + t)Ut = UtE(B)**

far all Barel sets B af J and all t in J; then, there exists a unique Hilbert space, K, and a uni­tary transfarmatian, M,

such that

(1) ME(B)M- l = XB far all Barel sets af J; and

(2) MUtM- l = at far all t in J,

where K is a complex Hilbert space and m, the Haar measure.

Far an arbitrary URS, (H,E,U), defined aver an LCA graup, G, by design, E and U satisfy the imprimi­tivity equality. Mareaver, the character space,

A A

(H,U,E), passesses the praperty that E and U satis-fy the imprimi ti vi ty equality. (20) Fi na lly, it is clear that (L2(G,K), XB' at) and (L2(G,K), aY, Xs) satisfy the hypatheses af the thearem. Therefare, by blending Mackey's and Stane's thea rem , the fal­lawing cammutative diagram results:

A A

(H. E (U), U (E))

V ~M (L2(G,K), XB' at )( F )(L2(G,K), aY, Xs).

Figure

Remarkably, the diagram reveals that an arbitrary URS is equivalent, under a memaryless, time invari­ant, unitary transfarmatian--a unifarm resalutian space isomarphism, to. an L2 space. Distinctians among the spaces far a fixed G, therefare, depend anly an the cardinality af the space, K. (19) Mix­ing this equivalence with a recent result af Masani and Rasenberg (10), gives the desired structure--i. e., "time invariance" maps to. multiplicatian.

4. APPLICATION OF THE MASANI-ROSENBERG RESULT

This sectian begins with the result af the abave mentioned authars. The thea rem is nat stated in its general farm (10), but is restricted to. a graup J, a Hilbert space, K, and the Haar measure, m.

Let T be a closed, single-valued, linear aperatar with dense domain an L2(J,K,m), such that T cam­mutes with the aperatian of multiplication by the characteristic functian af a Borel set--i.e.,

far all B in L (the a-a1gebra of all Barel sets af J).

A

Then, there exists a measurable functian, T an J, whase values are aperatars an K, such that

(Tf)(j) = T(j)f(j) j in J.

This thearem applies to. functian space. Thus, to. verify the saught after praperties an the abstract URS, we first apply the Mackey Transforms as in

*Here! {atlt in G} and ~~IB in the set af Barel sets af G} serve as the strangly cantlnuaus graup af Shlft aperators and the spectral measure, respectively.

**This is the imprimitivity equality.

n

Figure 1, redrawn below for simplicity. " " (H, E (U), U (E))

V, F ~y (L2(G,K), XS' a )( ) (L2(G,K), a , xB)·

Our discussion dwells upon two types of operators in the abstract URS, memoryless operators and time invariant. First, we consider the time invariant case.

Let T be a closed, linear, single-valued, time in­variant operator on (H,E(U), U(E)). Recall that T is time invariant if E(B)T C TE(8). Under the

" -Mackey Transform, M (which we term the Mackey fre-quency transform), we have an equivalent statement in (L2(G,K), aY, Xs), as follows:

XSTM ~ TMXS' " where TM is the image of T under the M transforma-

tion. Clearly, the conditions of the Masani-Rosen­berg theo~e~ are satisfied. Thus, there exists a mapping, T:G+K, such that

" (Th)(Y) = T(y)h(y) " " for all Y in G and h in L2(G,K). This says that

time invariant, closed, linear, single-valued oper­ators on an abstract URS are, as hoped, multiplica­tions in the "frequency domain"--i .e., in (L2 ,(G,K), Y ") a , Xs •

Now, let T be a memoryless, linear, closed, single­valued operator on (H,E,U). Recall that T is mem oryless if EtT = TEt. Thus, by reasoning similar to the time invariant case, the image of T under the Mackey-time transform, M, commutes with Xs in (L2(G,K), XS' at). Thus, it is equivalent to a multiplication by the Masani-Rosenberg theorem.

This structure, then, shows that certain operators on an abstract URS have the "right" properties. It is interesting to note that there is a duality in­herent in this formulation. The presence of a Mackey "time-transform" and a corresponding "fre­quency-transform" is apparently necessary for the cohesiveness of the theory.

5. CONCLUSIONS

The above ideas assimilate past theories in a num­ber of ways. The theory generalizes the Falb-

Freedman-Anton work because of the abstract set­ting and because it is valid for a larger class of operators. Clearly, there is no restriction to scalar-valued frequency responses as in (9) and (13). Moreover, it circumvents the existence questions associated with Saeks' work. (21) In fact, as in (21), given appropriate conditions, a multiplication on a function space can be viewed as an integral over the spectral measure, defined via multiplication by X, i.e.,

T = f T(w)dX(w).

Hence, the pre-image of T under the Mackey frequen­cy transform in (H,E,U) takes the form

" " T = f T(y)dE(y),

as was specified in (21).

In a private conversation with one of the authors, Desoer raised a question about the fact that dif­ferential operators satisfied the definition of memorylessness in an abstract setting. The ques­tion caused some doubt in our minds as to the appro­priateness of the definition. This note gives a partial answer, in that closed, memoryless operators are multiplications. Hence, the apparent pathology noted by Desoer can only arise in the case of non­closed operators.

5. REFERENCES

1. R. DeSantis, Causality and Stability in Resolu­tion Space, Proc. 14th Midwest Symposium on Circuit Theory, Univ. of Denver, 1971.

2. , Causality Structure of Engineering Systems, Doctoral Thesis, Univ. of Michigan, Ann Arbor, 1971.

3. P. L. Falb and M. '1. Freedman, A Generalized Transform Theory for Causal Operators, SIAM J. Control, 7 (1969), pp.452-471.

4. P. A. Filmore, Notes on Operator Theory, Van Nostrand-Reinhold, New York, 1970.

73

5. M. I. Freedman, P. L. Falb and J. Anton, A Note on C usality and Analyticity, SIAM J. Control, 7 (1969), pp. 472-47S.

6. G. W. Mackey, Induced Representation of Gro~pSk and Quantum Mechanics, w. A. Benjamin:-kew or, T90s.

I I!I! Iii

7. , A Theorem of Stone and von Neu-mann, Duke Math J., 16 (1949), pp. 313-326.

8. , Group Representations, Lecture Notes, Oxford Univ., 1966-1967.

9. P. Masani and M. Rosenberg, When is an Opera­tor the Integral of a Spectral Measure, (to appear).

10. , The Multiplication Operator in L? over a Localizable Space and Bochner I s TheOrem, (to appear). _

11. P. Masani, The Normality of Time-invariant Subordinate Operators in a Hilbert Space, Bull. Amer. Math Soc., 71 (1965), pp.546-550.

12. Orthogonally Scattered Measures, Advannces in Math .• 2 (1968). pp. 61-117.

13. • Quasi-Isometric Measures and their Appllcatlons. Bull. AMer. Math Soc •• 76 (1970). pp. 472·528.

14. F. Riesz and B. SZ-Nagy. Functional Analysis. Fredrick Ungar. New York, 1953.

15. R. Saeks. Causality in Hilbert Space. SIAM Review. 12 (1973)'. pp. 283-308.

16.

17.

18.

19.

20.

21.

f State in Hilbert Space. SIAM Review. 15 (1973}. pp. 283-308.

• On the Existence of Shift Operators. ·T-ec~h-.~Memor- EE-707. Univ. of Notre Dame. fndi­ana. 1970.

• Causal Factorization. Shift Opera­Tt~or~s-a~nC7d the Spectral Multiplicity Function. Vector and Operator Valued Measures and ~­catrons:--Academic Press. Inc •• 1973.-

• Resolution Space Operators and ~­....... te-m-s-...... S-pringer-Verlag. New York. 1973-.-

• Fourier Analysis in Hilbert Space. ~SI~A~M~R~e-view. 15 (1973). pp. 604-637.

Segal. I. E •• Equivalence of Measure Spaces. Amer. J. Math. 73 (1951). pp. 275-313.

74

LUNPED-DIS'I'RIBUTED NE:rWORK SYNTHESIS

VIA HJVAIHANT SUBSPACE 'I'HEORY

by

P.Dewilde

Departement Elektratechniek

Kathalieke Universiteit Leuven

Heverlee, Belgium

and

J.S.Baras

Electrlcal Engineering Department

Universit.y of Haryland

College Park, Maryland 20742

75

SUHMARY

'l'he goal of this paper is to provide cascade synthesis methods for

general nonrational impedance/scattering or transfer/scattering

matrices. It is shown that the invariant subspace theory of

Helson [1], Lax [2]and the factorization theorems of V.p.Potapov

[3] provide the means for synthesis techniques, which generalize

those used by Belevitch [4,5] and Oono-Yasuura [6] in the rational

case. In contrast with the standard multivariable approach to

lumped distributed network synthesis [7-9 ] our methods are single

variable. It should be emphasized that in the Russian literature

many examples can be found, where similar single variable tech­

niques f~om operator theory are applied not only to network syn­

thesis problems but to more general systems synthesis as well.

See for example ~O-ll] and the interesting book by Livsic [ 12]

We analyse both lossles s and lossy synthesis .. rIe assume that

we are given a scattering matrix S(jw), a transfer scattering

matrix L21 (jw) or an input impedance Z(jw), which are non-ratio­

nal and we discuss methods to synthesize a cascade lumped -dis­

tributed network realizing the function at hand. Starting from

an analysis of the system properties of an (eventually lossy)

scattering matrix, we deduce the crucial property that a lossy

(ordinary or transfer) scattering matrix can be synthesized in

a lossless cascade if and only if its nullspace is big enough,

or if and only if it has a pseudo continuation of bounded type

in the left half plane [13] or if and only if it can be embedded

in a lossless scattering matrix. Such matrices are called

·"roomy". Any factorization technique for a 10ss1e~s scat'::ering

matrix ego Potapov [11], w~ll then yield a cascade~synthesi~:

Such a factorization has the form :

(1") S I = B.P

where B is a Blashke product and P a singular part. Bdsed on

the ordering of matrix inner functions by divisibility a degree

theory is developed for (eventually nonrational) scattering

matrices \·;hich generalizes that of the ratioi1al case.

The notion of reduction of degree in this generalized sen~c is defined. Based on that we shov' how one can pullout of

76

2.

(1) elementary fnctors that arc directly synthcsizablc and thu~

pJ::ouuce a cascadc synthesis. For preliminary results i1'. this

direction we refer to 14-15-16-17-18

Next., technical and computational problcms are discussed.

First there is the problem of obtaining a spectral factorizdtion

and next the problem of computing a lossler,s embedding.

A method to do both at once based on a Belevitch synthesis type

procedure is presented and its applicability is discussed.

\~e give various restrictions that should be imposed on S, so

that the resulting networks will be realistic and we illustrate

these results by particular electrical networks.

At the end vIe present a discussion that conn~cts the networks

synthesis problems discussed here with infinite dimension

realization theory.

..

77

REFERENCES -------

[11 H.nelson, L~cturcs on Invariant Suhspaces, Academic Press, N.Y.

1964.

[2J p.D.Lax, "Translation Invariant Spaces," Acta Math, Vol. 101,

1959

[3] V.P. Potapov, "'fhe Multiplicative Structure of J-Contrative

Hatrix Functions," A.M.S.Transl.Ser.2, Vol. 15, 131-243.

[4] V.Belevitch "Factorization of scattering matrices with

application to passive net\'lOrk synthesis" Phil.Res. Repts

18 (4) 275-317 (Aug 1963)

[5] V.Belevitch, Class·ical NetvTOrk Theory, Holden Day, San

Francisco, 1968.

[6] Y.Oono and K. Yasuura, "Synthesis of finite passive 2n-ter­

minal Networks with prescribed Scattering Matrices,"

Mem. Fac. Eng. Kyushu Univ. 14 (2) ,125 -177 (May 1954~

[7] D.C.Youla, "A Revieyl of Some Recent Developments in the

Synthesis of Rational Multivariable Positive-Real Matrices,

"SIAM-Al'lS Proceedings, Vol.III, Mathematical Aspects of

Electrical Network Analysis, H.S .tH1f and F .Harary (edts.)

1971, 161-190.

[8] T. Koga, "Syn thes is of a Resistively Terminated Cascade of.

Uniform Lossless Transmission Lines and Lumped Pa~>f;ive

Lossless Two-Ports," IEEE Trans .Cir~it Theor~, Vol. CT-18,

1971, 444-455. - -

[9] D.C.You1a, J.D.Rhodes and P.C.Marston,"Oriving-Point Synthesis

of Resistor-Terminated Cascades Composed of Lumped Lossless

Passive Two-Ports and Commensurate TEM Lines,"IEEE Trans.

on Circuit Theory. CT-l9, No 6,1972, 648-664.

78

[10] A.V. Efimov , "Realization of Reactive J-expanding

Hatrix Functions" (in Russian) I Akad .. Nauk. JI.rmjansk ,

SSSR Isvcst V, No.1 (1970) I 54-63.

[l1J V.P. Potapov , "Ccneral Theorems on the Structure and

Detaching of Elemen t.ary Factors of Analytical Hatrix

Functions,"{in Russian) Dokl. Akad. Nauk.Armjansk,

SSSR, XLVIII, No.5, 1969, 257-263.

[12J H.S. Livsic, Operators, Oscillations, Waves, Open

Systems, A.H.S. Transl. of l-!3.th. Monographs, Vol. 34,

1973.

[13J R.G. Douglas and J.\v. Helton, "Inner Dilations of

Analytic Matrix Functions and Darlington Synthesis,"

to appear.

[14J P. Dewilde, "Roomy Scattering Matrix Synthesis,"

Technical Report, Dept. of Hathematics, Univ. of Cali­

fornia, Univ. of California, Berkeley, 1971.

[15] _______ , "On the Finite Unitary Embedding Theorem

for Lossy Scattering Matrices," Proc. 1974 European

Conference on Circuit Theory and Design lEE, London.

[16] , "Cascade Scattering Matrix Synthesis"

[17]

Techn. Rapt. No. 6560-21, Info. Systems Lab. Stan.ford

Univ. 1970.

V. Belevitch and R.W. Newcomb, "On the

Problem of Degree Reduction of a Scattering Matrix

by Factorization", Journal of the Franklin Ins.t.

Vol. 291, No.5, May 1971, 387-401.

[18J J.S. Baras, "Lumped-Distributed Network Synthesis and

Infinite Dimensional Realization Theory," Proc. 1974

International Symposium on Circuits and Systems, San

Francisco, 76-80. ,

79

Linear Hilbert Networks Containing Finitely Many Nonlinear Elements

Vaclav Dolezal

Abstract

This paper establishes conditions for the existence and uniqueness

of a regime in a linear (finite or infinite) Hilbert network which con­

tains finitely many nonlinear multivalued elements. These conditions are

given in terms of the driving point-set impedance of the linear network

and the operator describing the nonlinear elements.

80

Linear Hilbert Networks Containing Finitely NJany Nonlinear Elements

Vaclav Dolezal

Technical Summary

The objective of the paper is to give relatively simple conditions which guarantee the existence and uniqueness of a regime in a linear (finite or infinite) Hilbert network containing finitely many nonlinear multivalued elements.

Using the theory developed in papers [lJ - [3J, we first prove a theorem on the existence 0·' the driving point-set impedance of a (non­linear, in general) Hilbert network.

Then we establish the main theorem on existence and uniqueness of a regime in a network under consideration. It turns out that the nec­essary and sufficient conditions in question are given by the behavior of the mapping R + Z+, where R is the driving point-set impedance of the linear part of the network, and Z+ is an operator describing the nonlineQr element.

Moreover, it will be shown that the main theorem holds also for finite networks whose variables are in a linear space that need not be a Hilbert space.

As examples illustrating the application of the results, we consiciel' a finite R, L, C network with constant elements, which contains either a nonlinear resistor or a nonlinear inductance, and a DC~urrent network containing several nonlinear resistors.

References

[lJ V. Dolezal, Hilbert Networi{s I, SIArIJ J. Control, Vol. 12, No.4, Nov. 1974.

[2J V. Dolezal and A. H. Zemani~, Ililbert Networks II - Some ,!mlit:1t ive Properties, SIAM J. Control, Vol. 13, No.1, Feb. 1975.

[3J V. Dolezal, Generalized Hilbert Networks, to appear in SIAM J. l'ontl\'l.

81

Livsic's Chain Synthesis

T. T. Ha and R. W. Newcanb Electrical Engineering Department

University of Maryland College Park, Maryland 20742

ABSTFACI'

The factorization method of M. S. Livsic for J-lossless matrices is presented

in concise form.

"on the other bank, the palms beckoned us". [lJ

I. INTROOOCI'ION

In the book of M. S. Livsic [2J a factoriza­

tion theory for J-lossless matrices is presented

which is useful for ,the cascade synthesis of

lossless networks. Unfortunately this is spread

throughout a large segment of the book and in a

language not too familiar to western engineers.

Consequently we here attempt to concisely

present the ideas and in a form more accessible

to western engineers. Al th:::>ugh we hope this

paper will be self-contained we point out the

canpanion paper [3] where the background ideas

are presented.

II. PRELIMINARIES

We consider as given an n x n matrix

transmission operator S (p) of the canplex

frequency variable p = 0 + jw. associated with

S(p) is a constant matrix J satisfying J = J a,

2 where superscript a ~ans adjoint, and J = I,

with I the identity. Physically a transmission

onerator can be interpreted as a chain, scattering

or transfer scattering matrix (or various canbina­

tions of these), the interpretation depending upon

the associated J. Thus, for example, J = I corresponds to the scattering matrix while

J =[~~] is associated with tre chain matrix

[3].

We assume, as per [3], that the transmission

operator can be written in the form

(1a)

82

where T, the interior operator, and r, the input

to state space operator, are consequently taken as

known. The interior operator is also taken to

satisfy the relationship [3J

(lb)

Knowing the transmission operator in this form we

clearly know the input to interior operator

[ -1 R = T-pIJ r (2a)

fran which

S(p) = I + Jr~ (2b)

As a final preliminary we ccmnent aht if Q

is an isanetry, Q~ = I, [4, p.1SJ, then the

replac~nt

(3a)

f = Qr (3b)

leaves S(p) invariant, that is

(4a)

though we find

R = Q~ (4b)

III. FACI'ORIZATION - RATIONAL CASE

At this point we assume that S(p) is rational

in p in which case T in (la) can be taken to

be an m x m matrix operator with m finite.

By putting T in lower triangular form, (S), we

will show that a decomposition of the interior

operator R, (10), results to yield a factoriza­

tion, (12).

By standard techniques [5, p. 75J there exists

a W1i tary transfornation Q, which is a special

case of an isanetry, for which T can be put in

lONer triangular form through (3a). Thus ~

assume that T is in lower triangular form and,

hence, by partition, written as

(5)

where Tn and T22 are also lONe!' triangular,

with Tll of any desired number of rows (smaller

than m, assuming m.:.. 1).

Assuming nON that T is in lONer triangular

form and to go along with the partition of (5)

we define projection matrix operators

(6)

where 11 and 12 are identities of appropriate

dimension; here £1 + 22 = 1. By (5)

from which our definitions for Tl ,T2, r l ,r2 should be clear. Further, as £1I'£2 = 0,

(7f)

(7g)

With these points in mind we can invert T-pI

83

(-1 -1 -,22 T2-pI) £2~1 (Tl-pI) El ( Bb)

= El(Tl-pI)-~1+E2(T2-PI)-122

(Bc)

consequently we obtain, on directly using (Bc) in

(2),

(9a)

(9b)

(9c)

(10)

(9d)

(ge)

(9f)

(9g)

Equation (10) is the main one which allONS the

factorization, for

S=I+Jra[T-pI)-lr

=I+J(r~+r;)[~+~Sl]

=sl+Jr~2S1 = (I+Jr~)Sl

=S2S1

(lla)

(llb)

(llc)

(lId)

(12)

Having obtained the desired factorization, (12),

we comment that if the degree 8[S], [6, p. 195]

is m, that is the state-space realization of S(p)

for (1a) is minimal, then

as seen from (9) using 1:1 + .£2 = I. [1] G. Nam, "Night Crossing", in ''We Promise one

Another", The Indochina l"bbile Education

IV. FACIDRIZATION - IRRATIONAL CASE Project, Washington, OC, 1971, pp. 57-59.

If S(p) is irrational in p then the

above procedure still works whenever T can be

put into the lower triangular form (5) by an

isometry Q. The disucssion in [7, p. 73]

indicates that this will always be the case if

S is a contraction. See also [2, Chap. 6].

V. DISCUSSION

Given an S(p) in the form of (1a) we have

presented the ideas of Li vsic which lead to its

factorization, as we have seen always in the

rational case. This has the advantage that

optimal degree reduction occurs. Too, if one

desires degree one factors then such can be obtain­

ed by the use of a partition at (5) in which

Tll is 1 x 1. However, it should be pointed out

that the triangularization of T required may

lead to oomplex valued T and r. The S(p) under consideration all satisfy

(14)

in which case they are often called J-lossless[8].

Such correspond physically to lossless network

structures when S(p) has appropriate analyticity

properties to guarantee possi vi ty.

"A few Irore strokes! The bank was now closed"

[1] .

84

[2] M. S. Livsic , "Operators, Oscillations,

Waves (open systems)", Translations of

Mathematical Monographs, Vol. 34, American

Mathematical Society, Providence, RI, 1973.

[3] T. T. Ha and R. W. Newcanb, "Fundamentals

for Livaics Chain Synthesis", Proceedings of

the 18th Midwest Symposium on Circuits and

Systems, August 11-12, 1975, pp. 337-340.

[4] P. A. FillIrore, "Notes on Operator Theory",

Van Nostran Reinhold Co. , New York, 1970.

[5] C. C. Mac Duffee, "The Theory of Matrices",

Chelsea, New York, 1956.

[6] R. W. Newcanb, "Linear Multiport Synthesis",

McGraw-Hill, New York, 1966.

[7] B. Sz-Nagy and C. Foias, "Harmonic Analysis

of Operators on Hilbert Sj)aces", North­

Holland, Amsterdam, 1970.

[8] N. Levan, "Theory and Applications of J.

Lossless Scattering Systems", Journal of

the Franklin Institute, Vol. 294, No.5,

November, 1972, pp. 312-321.

RADAR TARGET RECOGNITION -­

AN OPERATOR THEORETIC APPROACH

D. E. Hammers and A. J. MacKinnon

ITT Gilfillan

Van Nuys, Cal.

Abstract

The problem of signal design for radar target recognition is studied from an operator theoretic point of view. Target operators are structured from meas­ured signatures so that recognition becomes a matter of identifying a target operator as a member of a priori known set of operators. A signal design criteria is established and shown to be optimum in relation to the statistical reliability of recognizing a target in the presence of White Gaussian Noise.

1. INTRODUCTION

Prior to 1970 radar systems were designed on the basis of classical analyses as established in various significant Radar Texts. (1)(2) These methods, for the most part, are based on impor­tant fundamental work by many individuals (to many to name here) towards developing target and background models which are used to develop the important radar subsystems--antenna, trans­mitter, receiver, and signal processor. The goals stressed are primarily those of detecting and tracking targets in a variety of corruptive backgrounds. In the late sixties digital methods began to be heavily applied for the realization of signal processors needed to extract the impor­tant target parameters for these goals.

The emphasis on digital proces sing coupled with the emergence of modern methods of systems optimization theory has influenced radar systems analysts to begin treating new design problems from an overall systems viewpoint. In this con­text, it has become apparent that the major item in any radar system which integrates the entire system together is the transmitted waveform. Each subsystem, including the target, should be viewed as a transformation acting on the desired waveform producing a response which ultimately leads toward the primary goals. The fundamental waveform design approach usually taken by Radar systems analysts is that of deriving a signal which provides the peak response on the time-

85

frequency domain of the Radar Ambiguity Func­tion. (3) The latter is a correlation function which characterizes the degree to which a signal can be distinguished or resolved from another signal in the time and frequency domain. The waveforms associated with certain additional radar functions cannot be easily treated by this method, in par­ticular, distinguishing or recognition of multiple targets as well as unwanted or bogus targets. In reality, what is needed, is an extension of the ambiguity function approach to cover this prob­lem by linking waveform design to the statistical likelihood of identifying the target as a known operator. That is, an operatortheoretic approach might provide a more unifying mathem~tical treatment of waveform design for target recog­nition than the classical method stressed by the Radar Ambiguity Function.

In this paper we study the recognition problem by an operator theoretic approach such as that applied by Balakrishnan to systems identifica­tion(4) or by Weber to signal detection(S) to pro­vide a rigorous foundation for developing the above concept. In fact, the techniques developed here are similar to modern methods of designing probing signals of identifying linear systems; e. g. , Esposito and Schumer(6) solve for a signal which discriminates between two linear filters in colored Gaussian noise. Mosca(7) generalizes this result by using a functional analytic method for deriving the probing signals for indentification of one of M linear channels in colored Gaussian·

noise. Our approach is to derive a signal design criteria which is nlore dependent on the proper­ties of the target operator rather than noise. This is due to the fact that we assume that the target has already been detected, located and that colored noise (clutter) can be removed by stan­dard radar filtering methods. (8)

The concept of a Radar Target Operator is intro­duced in relation to measurements made on a tar­get by a test waveform. An Impulse Response Operator is then characterized from these mea­surements with respect to the cor relation time assigned to the target. (Correlation time here refers to changes occurring while the target is fixed in space.) These same measurements and operator generation are assumed possible for all ta rgets in the set to be identified.

A probing signal design criteria is derived for identifying any of the Radar Target Operators of the known set. The criteria is based on maxi­mizing the mean square difference (separation distance) between the responses of the target set. The problem of achieving a signal which is equally likely matched to all operators is discussed and as a result the requirement of a constrained signal design criteria evolves. It is shown that the desired signal is one which maximizes a con­strained bilinear functional created from the operato rs of the target set.

Finally, the probability of correctly recognizing a member of the a priori set is investigated. It is shown that the mean square difference criteria for the probing signal is optimum in the sense of achieving the optimum Bayes decision in the presence of White Gaussian Noise.

2. STRUCTURE OF THE RADAR TARGET OPERATOR

The concept of the Radar Target Operator as developed in this paper depends on the preserva­tion of linearity of the subspace of all waveforms which are transmitted and received from the tar­get. That is, a real vector space is preserved from the signal domain through the electromag­netic field domain and back to the signal do­main. (9) Now, referring to Figure 1, a target, denoted by.T, can be regarded as a scattering system which is characterized by a linear target operator, S, which maps incident waves, wi, into reflected waves, w r :

wr = Sw

i (2-1)

The particular target ope rator which we are con­cerned with is the Target Impulse Resp?nse Operator. Consider Figure 1 and let Wi be the

86

<l>1Il 0--- =-=0= '---:----' - S - L....-A-R

- ......

p' AR S AT<I>

COMPOSITE OPERATOR MODEL

<l>1t) ~L._~~_~_~_:;'_ ... ~ P III

T pit); T <1>(1)

WHERE T ; AR SAT

SINGLE OPERATOR MODEL

pit)

Figure 1. Modeling of Target Operators

Dirac 5(t-r) function; also let h:r be the corre­sponding reflected wave, i. e. ,

(2-2)

consequently:

wr

(t) = f h.r(t, C) wi (C) d C (2-3)

_CD

i for any w in the signal domain. Note that in (2-3)

we have assumed ~ to be non-stationary. This is necessary to accommodate changes caused by the targets motion relative to the spacial location of the radar. To this end, a time cor relation interval, tJ, is defined over which these changes will be considered. We exclude the target's trajectory so that tJ is chosen small enough to only observe motion occurring about a fixed point in space. Now letting AT and AR denote the trans­mitting and receiving antenna functions we have

and

i w = ATD(t)

r p(t) = AR w

(2-4)

(2-5)

then, substituting (2-1) and (2-4) into (2-5) we have

Now let aT and aR

be the antenna impulse responses so that

(2-6)

(2 -7)

where ". " denotes the convolution (or composi­tion) operator. But in Figure 1, we define T to be the Radar Target System Operator which maps (tJ (t) into p(t) or

p(t) = T(tJ(t) (2-8)

So finally we have a target characterized by

T = (A SA) = ((a • h -). a ) R T R':/ T

(2 -9)

We note that p(t) would be a random variable if for instance the target orientation (aspect angle of the target) is unknown relative to the direction of the transmitted wave caused by (~(t). In this case p(t) is mapped 1 to 1 into the a priori range space of T with probability < 1 so that T can be considered as a stochastic operator. If the orientation is known or if p(t) can be mapped 1 to 1 into the a priori known range space of T with probability 1; then T would be classified as a deterministic operator. If wi and w r are hor­izontally or vertically polarized electric field intensity vectors, then by choosing

we have the impulse response matrix operator:

(2-10)

In our discussions we will only consider one polarization vector.

In general, tractable mathematical representa­tions for T do not exist for complex targets such as aircraft so we assume that a priori measure­ments (signatures) under ideal conditions can be made to approximate it. For example, if the target were suspended in free space and (tJ (t) ~ ~ (t) with a

R ~ aT ~ ~ (t) then

p(t) = T (t) f, (t) ~ I:y(t) (2-11)

This is reasonable since under controlled condi­tions, a target could be held fixed in the far field * of the antenna and measured by a test signal, (tJ(t), consisting of a sequence of very narrow pulses for a length ot time equivalent to 6. These pulses would be transmitted at a rate which is high enough to prevent aliasing (violation of Shannon's sampling theory) the measurement of the frequen­cies defining motion within a bandwidth of 1 16.

The pulsewidth l' would be selected less than dlc (ratio of target line-of-sight length to the speed of light). An estimate of the power spectral den­sity associated with p(t) might appear as in Fig­ure 2. Note that the transmitted RF frequency would also be selected such that the wavelength is much less than d, creating the situation of the

5 twl p

r-- } . i::~~~0\~6~lO'E OF

-- ~~- L11----- -. '. , ___ ',,_ " _ ~\'i~;~ ENVELOPE

---........ ",- ""

t~J SptW' T POWER SFECTRAL DENS lTY OF pw

Figure 2. Example of Target Frequency Response

so-called "extended target." For example, if the target dimension along the line-of-sight of the radar was 25' then a reasonable value of l' and A might be 10 ns and 3 cm. On the basis of the above, the te st signal !P (t) will be approxi­mated by a sequence of pulse functions modulating the RF carrier,

jw t c

e

Considering a finite set of N complex valued points (T

S apart) on the time interval [0, 6], an

N -dimensional vector space, <1>, can be defined which is spanned by an orthogonal basis generated from the test signal !p(t); vis.

Then each !Pi is an Nxl vector with component, such that

CP11 0 0

0 (tJ22

(tJ 0 . !PZ 1 0 ···!PN =

(2-12)

0 0 !PNN

with

jw t {) c

!Po • .e 11 1

(2-13)

2 ':'The antenna "far field" begins at a distance which is usually defined as 2D IX, where D is antenna aperture and A is wavelength.

87

6. _ .!. [u(t-iT )-u(t-i(TS+T))], T« d (2-14) 1 T S c

u(t) - unit step function

Now let a system 0 function be a signal vector in ~ in the sense that it is formed by unity projec­tions on each cp.. Thus the system impulse response can b~ represented in vector form as

(2-15)

where h~is an Nxl vector, T j7 a NxN matrix and ,~ ,J

o a Nxl unity vector. The components of h,fare hi such that

h. (2-16) 1

o

But hywill be assumed stationary for t( [0, TS] so that

(2-17)

o

3. SIGNAL DESIGN CRITERIA

After establishing the definition of the Radar Target Operator, we proceed to develop a wave­form design criteria which will lead to an optimal probing signal other than cp (t). cp (t) could be used, but it is our purpose to find a signal of lower dimensionality and bandwidth which can be used to identify the Radar Target Operator observed by the radar. As in Section 2, the components of cp are the basis vectors which span a finite dimensional waveform space ~. Then for W, a subspace of ~, another signal, w, can be gen­erated by operating on the most basic element of ~, i. e., 0 such that

w = Uo, wE' W (3 -1)

*. .... where U is a degenerate operator actmg on..,. Further, we assume that the target has initially been detected (i. e., in the sense that its position is known) and its response is that of a determin­istic operator. As such, it is thought to be a member of an a priori known target set, ,r, with

where ~(k) is the target operator of the kth ta~­get, .~-( ). In the radar recognition problem, it is equally likely that anyone or none of these operators exist. A design criteria is needed to systematically derive a probing signal, w, which generates a response in the range space of T(k) and which, in some sense, is ideally separated from corresponding responses of the range spaces of the remaining operators of .~-. This must be accomplished under the liITlitations of acceptable bandwidths, power levels and duty cycles of the radar transITlitter. U will provide this requireITlent. Note that the above goal is also reasonable when considering the addition of noise to the response, since then respective range spaces are enlarged and can overlap. As seen later, the probability of recognizing.1-(k), is related to the design criteria of w.

Since T(k) is the iITlpulse response operator of the kth tar~et, it follows froITl (2-15) that the response p k) can be written in vector forITl as

(k) P

Then

(k) = T w, (3-2)

(3 -3)

whe re II x II denotes the no rITl of an e leITlent x in a Hilbert space of cOITlplex valued functions defined on a set of N poInts on the tiITle interval [0, 1\] II p(k) II can be considered as a ITleasure of the voltage received froITl the kth target when w is transmitted. SiITlilarly,

II p(i)II_11 pU)11 =11 T~11-11 T~1L i",j, lsisK, (3-4)

is a ITleasure ot the ditference voltage received froITl ta rgets i and j. Applying the triangle inequality, we have

(3-5)

This implies that greater separation is possible between targets when coherent rather than non­coherent inforITlation is considered.

3.1 REMARK 1

Regarding the ability to identify the presence of one of the target operators in 3", let us consider the following situation. Let R

l, R

2, R

3, be the

range spaces of the set

9"= (T(l), T(2), T(3))

':'A degenerate operator maps an entire space into a finite diITlensional sub-space [10].

88

and suppose these are subspaces of a space P such that .T:W- P. Then the operators are completely identifiable on W under the following logic:

(3-6)

, where 11 and U imply the operations of inter­section and union and ¢ is the null set (see Figure 3).

Figure 3. Completely Identifiable Targets

With the above preliminaries, we now develop a distance function d(w) on P, relative to the set ,T. Let dij denote the norm square of the difference between the ith and jth targets or from (3-3) and (3-5) we have

d .. = II (T(i) _ TO)) wl\ 2 1J

(3-7)

or since the adjoint of T(i) - T(j) exists, we have

the quadratic form

(3-8)

(3-9)

3.2 REMARK 2

Again considering the three target set, it follows from (3-6), that it is desirous to have d12, d13 and d 23 as large as possible for some w £ W. Letting p(1), p(2), and p(3) be vectors on P, we

seek to maximize the norms of the difference vectors as shown in Figure 4.

d • II (2) _ ml12 12 p P ~

p (2) /' /:

2 A / d

23 • II p(2) _ pmll ~ / /

II

Figure 4. Distance Functions for the Three Target Set

89

Each d .. cannot be maximized separately since w which {~ appropriate for d12 may not be for dl3 or d 23 . Suppose instead the average va lue of dij is considered; i. e.,

(3 -1 0)

then the triangle inequality implies that

max a.(w) :5: max dlZ

+ max dl3 + max dZ3 w w

Maximizing (3-10) as it is written may still have the tendency to separate the range spaces of one pair of operators while causing the range spaces of another pair to overlap. Thus we need a con­straint on the maximation of a.(w). a(w) can be generalized to K targets by considering d ij for all target pairs, with i "j, and letting a(w) be the sum of the distance between all targets, it follows that

L

a.(w) ~ d .. L ij 1J

or letting L

D L r D ..

ij 1J

the generalized distance function becomes a quadratic form

(3-11)

(3-11a)

where

L K(K - 1)

2 pairings.

Recall that the dimensionality of <I> is N

Also from the definition of U it follows that

and also that

cl .. (w):5: d .. (Ii) 1J 1J

"

(3-12)

Considering another subspace Y where U: - y

such that YeW then

(3-13)

and for y E: Y

d .. (y) "d .. (w) 1J 1J

(3-141

Thus ordered subspaces can be created by U on [O,~] and the cor responding distance functions monotonically increase as the subspace dimension increases.

3.3 REMARK 3

Considering the three target example, a plot of the pairwise distance functions versus subspace dimension can now be structured as shown in Figure 5. Then relative to the dimension of the desired waveform space, W, the following bounds are apparent:

d max

and

d . mln

AVERAGE

max d .. (w), i, j lJ

min d .. (w) i, j lJ

PAIR-WISE --t-----:r------:;...-F-----­SEPARATION FOR Nw

u z

~ '"

~ NW

SUBSPACE DIMENSION

(3-15)

(3-16)

Figure 5. Target Separation in Relation to the Dimension of the Waveform Space

The desired constraint is formulated by requiring that the difference between d and d . be bounded, or for ( > 0 we forlRax mln

(3-17)

or equivalently, letting fJ (w) = w':'Bw we have

{3 (w) = 1 (3-18)

where

B (3-19)

and

D .. from d lJ max

D D .. from d . lJ mln

This ( constraint insures that the paired mean square differences between all targets will be

90

maximized uniformly within ( so that maximum separation waveform will not favor targets farther apart than others.

The technique of constraining one quadratic form by another can be expressed more succinctly in the form of a Rayleigh Quotient, d(w), of D rela­tive to B(lO). That is, under all conditions pre­vious ly established, the probing signal des ign criteria d(w) is well defined for w ~ 0 and can be simply written from (3-11) and (3-18) as

d( ) = a..t& w (3 (w) (3-20)

Thus the derivation of a maximum separation waveform becomes a problem of finding that w which maximizes d(w). It can be shown that the desired w is the eigenvector which corresponds to the maximum eigenvalue of d(w)(ll).

4. PROBABILITY OF TARGET RECOGNITION

We begin by assuming that the received signal can be the response of the target and its local background to the transmitted waveform, w. This signal is defined to be an element in an NW-dimensional response space, P, since it is in the range space of the linear operator defined for the target and background. We further assume that White Gaussian Noise is added to the target by the receiver. Referring to Figure 6, let x be the received signal, p(k) the response of the kth target, c the background response and 11

the noise, we have for K possible targets:

(k) x=p +c+17 (4-1)

or equivalently

(k) x = T w + Cw + 17 ls;ks; (4-2)

TRANSM ITTER ANTENNA

w

~ y (kl - RECEIVER - ANTENNA x -

1)

Figure 6. Definition of Target, Background and Noise

(k) . IIere T and C are the hnear operators repre-senting finite approximations to the impulse responses of the kth target and associated back­ground. We note that the background return can appear as the localized environment in which the target exists, or as some unknown distortion to the target operator. For simplicity, all opera­tors are considered deterministic; i. e., the target exists at a known aspect angle while the background is unable to change during the dwell time of the N-dimensional waveform, w. In the following treatment the background is non-exis­tent, c = 0 and the variance of the noise, 0

2 , is known or can be measured. The problem for non-deterministic taryets and for c * 0 is treated in reference ( 2).

Under the above conditions, the conditional probability density for the presence of the jth target can be written as

e

2 ('J 2 x (4-3)

2 Here, x denotes norm of x on [0, 6] and 0"

the variance of x assuming p(j) was observe~

Now assuming that each target is equally likely to occur, it follows that the optimal Bayes decision logic for selecting the kth target is

exp -~II x - p (k) 112 (4-4)

x

max K

exp (- 2 :: Ilx -pof) 1 Sj SK

Taking In of both sides and since In is mono­tonic it fonows that the desired decisfon logic is

(4-5)

lsj S K

4.1 REMARK I

Considering once again the space P containing the ranges of the three target set, it is evident from (4-5) that the ideal hyperplane decision

91

surface is achieved when the response vectors form a plane defining an equilateral triangle (see Figure 7).

Here it is clear that the distance between responses is the same for each pair and the unconstrained signal design criteria (l (w) can be applied to derive the optimum w.

Figure 7.

4.2 REMARK 2

HYPERPLANE ::.-- DECISION SURFACE

Symmetric Target Responses Plus Noise

A more likely response geometry is depicted in Figure 8. For this case it can be shown that maximization of the average difference distance, (l (w), will not necessarily result in the optimum w since it would tend to favor responses p(l) and p(3). Consider the following:

Suppose x = p(2) + 1/ (4-6)

and assume that the noise, TI, is such that

(4-7)

p

Figure 8. Non-Symmetric Target Responses Plus Noise

Now define d .. (x) to be the shortest projection of x on (p._p.), 1J ls(i, j)s:3.

1 J

,-

I

Iii

Then for the geometry shown in Figure 8

select lover 2 (4-8a)

select 3 over 2 (4-8b)

select lover 3 (4-8c)

The above logic fails to select response 2 even though we assumed it present. Further, it follows that

max a. (w) ¢:::::;> max d .. , 1~ i, j"; 3 1J

(4-9)

w

so that conditions (4-8) a, b, c are only worsened by maximizing a. (w). The decision logic can be balanced by deriving w under the constraining quadradic form fJ (w) established in Equation (3-18). That is, letting

(4-10)

an acceptable value of E" can be found by estab­lishing confidence limits around the required Probability of Recognition, Pc' and then com­puting the corresponding values of dmax and dmin' To do this we need to further investigate the method of computing the pair-wise Pc as a function of dij and RMS noise level.

Without loss of generality we limit the target set to two targets, say

so that

g G w

h Hw

From (4-5) the two target hypothesis test becomes:

If II x - g II 2 -II x - h II 2 sO, then the

target is 'f)

If II x - g II 2 -II x - h II 2 > 0, then the

target is .1('

(4-11)

(4-12)

Since G and H, the operaf~rs of '!J and .:If', are bounded linear operators 3), their adjoints G':' and H':' exist so that (4-12) can be written as:

92

I (x, Aw) I 2 (w,:i w) ~ 0 ~ <!j

< 0 ~ .Yt'

(4-13)

where (x, y) denotes the inner product of x and y on [0, t,] and

Aw

l:!w

(G-H)w

(G':'G-H':'H)w

(4-14)

(4-15)

The recognition of a particular target, say 'lJ , implies that the following inequality has been satisified,

2( (Gw + TI), Aw) - (w, l:!w) ~ ° (4-16)

Since ~(w) = 1/2 (w, A':'Aw), we get

Q! (w) + (TI, Aw) ~ ° (4-17)

Recall that from (3-7) that max a.(w) produced that waveform which separat'fs the targets in the mean square sense. Thus (4-17) shows this criteria to be ideal for the two target case in the sense of Bayes Decision Logic since a greater a. (w) strengthens the inequality. Note that for the two target set, D+ = D_ and therefore B=O. Thus a. (w) = d(w) as used in the following equations.

The success of recognizing target ,1 can be com­puted from the conditional density fTl (x/g). 1£

we define P c(g) as the probability of cor rectly recognizing 'fl, then from (4-13) we have for a threshold Q(w),

Pc(g) = PR

[ I (x, Aw) I ~O (w) I x = g + TIl (4-18)

Now let z = (x, Aw), then since x is characterized by a Gaussian density so is z, or letting f (z/x)= f (x/ g) the probability density function of z given target :1 is

f (z/ g)

2 2 -(z-j./ ) /2 a

g z (4-19) e

. (12) where It can be shown that

(4-20)

a 2 = a 2 (2d) z (4-21)

and P (g) is computed from c

Pc(g) = fa> f (zig) dz

O(w)

or equivalently

(4-2l)

2 2 -(z -I-' ) la (2d)

g e dz

(4-23)

4.3 REMARK 3

The required signal design constraint, £, for the three target system can be determined from (4-23). Assuming a fixed, plots of Pc (p(2)) versus d l2 and d

23 can be made relative to a common d

axis for a given waveform dimensionality NW:S; N (e. g., see Figure 9).

The £ criteria can be determined by selecting dmin and dmax for the required Probability of Recognition. Note that if an acceptable £ cannot be determined for the required P" (i. e., it does not exist for NW :s; N), then an unreliable target signatures have been obtained from the N sam­pled test signal, "o(t), on (O,~).

REQUIRED Pc-_~----"""'..c:._---:::;....r

Figure 9. Probability at Recognition Versus Pair-Wise Distance

5. RELA TlONSHIP TO THE RADAR AMBIG UITY FUNCTION

The measure of separability (or distance) between target responses for the required recognition performance was shown to be a constrained quad­ratic form which depended on the paired differ­ences between target operators of the a priori known set. This approach can be viewed as an

extension of the concepts implied by the Radar Ambi~'1i4r Function. For example, it can be shown that for two targets the Ambiguity Function can be used to derive the best trans­mitted signal to discriminate between responses due to the Doppler shift of two targets. Here identification or recognition is not the goal, but rather it is the ability to indicate that two targets exist. More specifically, refer back to Equation (3-7) and suppose we define our distance function, d .. (w), on a continuous time interval (-a>, 00)

i~Jstead of the sample interval (0, 0. Then assuming that p.(t) and p.(t) are square inte-g rable on (-00, 00) we can Jdefine

(5-1 )

Now if p.(t) and p.(t) are modeled as similar functionJ only wifh time and frequency differences

such that

p. (t) ( ) jw.t ( 5-2a) - wt+T.e 1 1 1

p.(t) w(t + T.) jW. t (5-2b) - e J

J J

then letting T = T. - T. and W = W. - W. it follows that the cross pr~ducf term fro~ EqJation (5-1)

becomes

93

00

Ij!(T, W) ~[ T T

w(t + '2) w(t - Z-lcos wt dt (5-3)

which is the Ambiguity Function form. By mini­mizing Equation (5-3) with respect to w(t), we in effect maximize the distance between target responses and can optimally separate the targets. However, in the recognition waveform design problem, the simple modeling used in Equations (5-2a) and (5-2b) cannot be used, rather Equa­tion (3-2) holds, so that Equation (5-3) becomes

00 • Ij!(T, Iol) =[ T.(t,T) w(T) (T.(t,Iol)W(\-l))··'dt (5-4)

1 J '"

which can be defined as the target operator form of a cross-ambiguity function. Thus minimizing Equation (5-4) with respect to w(t) is equivalent to maximizing Equation (5 -1). The multiple tar­get constrained form of the pairwise cross-ambi­guity function could be similarly derived. Note that if the signal space domain was restricted to N points on (0, 6), then finite dimensional target operato rs could be used and a Rayleigh Quotient form for the constrained minimization problem could also be applied.

to

r !

6. SUMMARY

By considering the radar target as a finite dimen­sional operator on a Hilbert Space, we have attempted to develop an operator theoretic meth­odology for designing an optimum target recogni­tion system relative to the realizable as pect of m.easuring the target signatures. The approach has been directed at seeking a balanced solution depending upon the association of the target observation space to the transmitted signal space through the radar target ope rator. The context of "optimum" has been taken to mean the best

References

(1) Skolnik, M. W., Radar Handbook, McGraw­Hill, 1970.

(2) Barton, D. K., Radar Systems Analysis, Prentice-Hall, 1964.

(3) Rihaczek, A. W., Principles of High Resolution Radar, Chap. 5, McGraw-Hill, 1969.

(4) Balakrishnan, A. V., "Stochastic System Identification Techniques," "Stochastic Optimization and Control," Wiley & Sons,

(5) Weber, C. L., Elements of Detection and Signal Design, Chap. 14, McGraw-Hill, 1968.

(6) Esposito, R. and M. A. Schumer, "Probing Linear Filters - Signal Design for the Detection Problem," IEEE IT, Vol. IT-16, No.2, pp. 167-171, March 1970.

(7) Mosca, E., "Probing Signal Design for Linear Channel Identification, " IEEE IT, Vol. IT-18, No.4, pp. 481-487, July 1972.

94

transmitted waveform which would provide the required probability or recognition. As noted in Section 3, this meant a degenerate (lower dimen­sional) form of the original measurement signal when Bayes Decision Logic is applied to the pos­sible responses in the presence of White Gaussian N ois e. The characteristics of the signal we re derived relative to the discrimination properties necessary to identify (recognize) a target as a particular linear system in an a priori known set of linear systems. Thus a system identification approach was taken.

(8) Nathanson, F. E., Radar Design Principles, Chap. 9, McGraw-Hill, 1969.

(9) Levan, N., "Target Dis crimination Studies, Report No.1, " Unpublished International Telephone and Telegraph (Gilfillan) Report, Van Nuys, California, August 1972.

(10) Bellman, R., Introduction to Matrix Analysis, pg. 110, McGraw-Hill, 1960.

(11) Hammers, D. E., "Generation of a Finite Dimensional Matched Filter," Unpublished ITTG Report, Dec. 1970.

(12) Hammers, D. E., A System Theoretic Approach to Radar Target Recognition, Phd. Dissertation, UCLA, 1973.

(13) Akhiezer, N. 1. and 1. M. Glazman, Theory of Linear Operators in Hilbert Space, Ungar Publishing Co., 1966, pg. 43.

(14) Muddle, R. P., "An Eigenfunction Problem Concerning the Ability of a Radar to Distinguish Between Two Targets, " Royal Aircraft Establishment, Technical Report No. 66348, Nov. 1966.

INFINITE DIMENSIONAL REALIZABILITY THEORY

J. Hilliam Helton University of California, San Diego

La Jolla, California

This talk is on realizability theory for systems with infinite dimensional state space. As is generally known the study of [A,B,C,D] type linear systems was dis­covered to be closely related to Lax­Phillips scattering theory or equivalently to operator model theory. This was dis­cussed by several people at the first con­ference in this series 2 years ago. ~y

purpose here is to give a general descrip­

tion of what has happened and of the state of the art now. The talk will dwell on generalities and downplay technical details. For more detail one is referred to a fair­ly complete article which will appear in the forthcoming issue of the I.E.E.E. Pro­ceedings devoted to 'Recent Advances in Systems Theory.' The work described in this talk is largely due to Baras, Brockett, Fuhrmann, DeHi1de, and myself.

To speak generally the main accomplishment of the last two years has been to estab­lish analogs of the basic facts from finite dimensional systems theory and also to give what seems to be a reasonable de­finition of infinite dimensional system. The last point sounds strange so lets con­sider it first. r'lhat reqUirements would one like for an [A,B,C,D] linear system? One would like for the theory to be gen­eral enough to contain many examples but be specific enough so that , .. e can prove

95

substantial things about it. To be more specific some desirable properties are

(1) A,B,C,D act on some vector spaces.

(2) A,B,C,D can be associated with most distributed devices.

(3) Any natural notion of energy as­sociated with the physical situa­tion corresponds to a topology on the vector spaces which is 'compatible' with the system.

(4) The basic general properties of finite dimensional systems have valid analogs.

Maybe more should be said about energy. In finite dimensions one can ignore such considerations but in infinite dimensions one must have a topology on the space. Traditionally in mathematical physics the topology is painstakingly selected to have a direct physical interpretation. Usually it is an inner product and gives a Hilbert space structure to the vector space in­volved. This seems like a very good tradition to maintain in the case of infinite dimensional systems. For dis­crete time systems this proves not to be difficult although it must be mentioned that energy considerations are a little

" ~

garbled in discrete time. In continuous ,1"

time one runs into a lot of technical td

:'

i

" II: I

II: I , ,

i

problems. Certainly more than I originally

anticipated. Baras and Brockett showed that if one picks A,B,C,D in the mathe­matically nicest way then one could not

even do a transmission line. This sug­gests one use a set up like Balakrisna's

where B,C are distributions. That's plenty general but there is not a natural

Hilbert space structure and so energy con­siderations are ignored. Aubin and

Bensoussan of the Lions school pursued the

Balakrisna distribution approach and gave

an abstract analysis of it. They then im­

posed a Hilbert space structure in a cer­

tain way and found the same difficulties

which Baras and Brockett found. The first point to be emphasized in this talk is

that there is a reasonable way of defining

infinite dimensional systems which meet conditions (1) through (4). The set up is

complicated enough that it requires more discussion than can be given in a short

talk and so we only list the definition

and give no further explan~tion: A system consists of

X,U,Y are Hilbert spaces.

A is a closed operator on X defined

on 19(A) a dense sub domain of X and the semigroup eAt is bounded.

This gives rise to a rigged Hilbert space structure ~(A*)- , X , ~(A*).

The operators B,C,D are continuous linear

B:U -- 19(A*)- C:19 (A) -- Y

D:U -- Y

A compatible system satisfies

(ZOI-A)-lB c ~(C) (all systems in the

authors experience are compatible). In addition there are straightforward

notions of continuous, approximate, and

~ controllability and observability.

The set up just described has very good properties; it satisfies (1) through (4)

in fact. Let's expand on (4): Our infinite

96

dimensional system has the properties-­

(a) An operator function F analytic

near infinity can always be real­ized with a compatible system.

(b) Given two canonical systems with

the same frequency response func­

tion there is an M such that

(c)

MA = Al1 ME = B CM= C D = D

If (A-A)-l and (A-A) -1 are meromorphic in (b), then they have

the same poles. If the frequency response function F in (a) is

'pseudomeromorphic' then the

singularities of (A-A)-l equal

those of F. If F is not

'pseudomeromorphic' everything breaks down.

It should be mentioned that (c) is a

little delicate and one cannot expect more to hold. In fact the first part is a little surprising and seems to be the

physically correct and useful result. It

was first done in the Lax-Phillips theory.

That brings us to the next main property of our systems.

(5) Every infinite dimensional system

corresponds to an abstract Lax­Phillips scattering situation.

The correspondence is very ex­plicit.

Thus the study of this type of system is equivalent to Lax-Phillips scattering

theory ("lith D+ and D_ orthogonal) . Advantages of this are: TheoreMs from the Lax-Phillips theory (both present and

future) can be converted to theorems about

systems and vice versa. We are guaranteed that all of the many examples they treat can be fit into this systems set up, and lastly the Lax-Phillips avoids all compli­cations involving distributions and is

couched quite simply in terms of Hilbert

space. Thus an alternative formulation of continuously approximately controllable and observable compatible system is The

Lax-Phillips Model.

T(t) is a uniformly bounded semigroup on a Hilbert space H containing two

distinguished closed orthogonal sub­

spaces D+ and D_ such that

(i) T(t)D+ c D+ and T(t)*D_ c D_;

(ii) n T(t)D+ = {a} and n T(t)*D_ = {a};

(iii) st lim PD +D T(t) 0 and o -

st lim PD +D T(t)* 0; o +

(iv) T(t) is isometric on D+ and

T(t)* is isometric on D

(v) T(t)D+ is 1 to T(t)(D_ ~X)

and T(t)*D_ is 1 to

T(t)*(D+ $X).

It seems very likely this set up will be easier to use in some circumstances than the half distributional-half operator theoretic [A,B,C,D] formulation. One project which might be informative would be to see if one could translate 'least squares quadratic control' and'estima­

tion' into the Lax-Phillips set up. Pos­sibly it would lead to some interesting constructions. One might also determine how some of the parabolic and hyperbolic

partial differential systems in Lious framework fit into the Lax-Phillips model.

A final property which should be empha­sized is that of 'compactness.' It is not included in the definition of system, however, many systems which carefully model a physical situation will satisfy a compactness condition. Experience with

the Lax-Phillips theory indicates it

should enter as

(6) 3 T > 0 so that .eAT (lA-A) -1 is a compact operator (that is, it is

approximable by a finite rank

operator).

For discussion of the physical signifi­cance of (6) in several situations see §S

of the longer paper.

This brief sketch hopefully gives some idea of the present state of the art. To my view the general realizability theory is now complete enough that we must begin to go to more specific levels. TtTe must

impose physically reasonable additional assumptions on the systems structure which lead to more refined theories. T·Thereas,

the restriction of finite dimensional state space is so strong that in tradi­tional systems theory no classification of

additional structure has developed; in infinite dimensional situations such a

classification is necessary.

97

BIOG~HV

BILL HELTON was born in Jacksonville, Texas on November 21, 1944. He received the bachelors degree in mathematics froM the University of Texas in 1964, the masters degree and the Ph.D. degree in mathematics from Stanford University in 1968, respec­

tively.

From 1968 to 1973 he was an Assistant Pro- .f

fessor at Stony Brook, New vork. During the beginning of 1974 he was a visiting Associate Professor at UCLA and currently is an Associate Professor in the Department of Mathematics at the University of Cali­

fornia, San Diego in La Jolla.

LINEAR NETWORK SYNTHESIS USING ITERATION XETHODS

Y.G.Jan and F.R.Chane National Chiao-Tunc UniTersity

Taiyan, Republic of CAina

Abstract

We employ two iteration methods, Tiz. maximum eicenTalue iteration and Gauss-Siedel iteration, te rind the rational appreximation te t .. truncated La«uerre transform of the impulse response of some linear networks. Usin~ the deriTed alcoritaa, the executinc procedures fer synthesizing taree simple networks are discussed in detail. Taeir results are plotted and tabulated.

I. SIGNAL REPRE3El.nl'IO;; .-.NO LAGUlSJ.1RE

POL YJ( O}lIALS

I-I. Si~al aepresentation

There are many different meanin«a of si~alG,

we consider the representation of a sienal as a

real function ret) on (O,OoJ. Furthermore, ye

assulle f( t J to be intepoable in tlle mean squared

sense, i.e. 5;2(t) dt (00. We can express tllo eienal f(t) by a set of com­

-plete aHtIaftGt..t b .. b { In (t); noaO, 1,2, ... } [1} Of course. t .. enersy ef In(t) is &Ssuaed finite.

Jllen f(t) is reprosented by the basis fUnctien

discussed abeTe. tllen ye kaTe: C)o

f(-t; .. 1:; C.ln (t) ?I._O

YHre C;a JOOf(t) 1 (t) dt non

HeyeTer in practieal ease. ye ean u.e only a

finite number ef basi. functions in eur repro­

-.entation. Suppeso {In(t), noaO",2, •••.•• K} be

tlle finite s.t ef ortllonoraal b .. i. functions

yhicll are used to represent the sisnal f(t).Tllen

tlle apprexi.ation f (t) can be expres.ed as: " a f (t) .. T C I (t) (3)

a L..J n n n .. o

98

wllere the ceefficient. {cnl are cllesan in

sucll a Ya::! tllat the mea squared error

f "1~(t)-fa(t»2dt (4) is ainillum.

After impl.mentation of equation (4) by sub­

-stitutinc equatien (3) into equation (4),

tae ceefficients { Cn\ in equation l3) are

still ciTen as equation (2). Tlle mean squared

errer t can be reduced by increasinc tlle nu.­

-ber K since ye have assumed tllat tlle basis

functions form a complete set.

1-2. La«Uerre Pelynoaials

A particular important set of cOllplete ort .. -

-normal basis is the Lacuerre polynomials

In (t). noaO,1.2 •.••• whicll appear as: [2]

{

~2.tt e-o.t ~ ('tt) (-2oJ:f 1 (t)_ 1(=0 f( k r t~o ~ . (5)

o t<o Yhicll can be obtained by applyinC Graa-

Scllmidt precedure te {( at )ne -at .naO,1, 2 ..... }

Yllere a is 8 positive real nu.ber.

l'he corresponding Laplace transforll of equation

(5) is: ../20... s-a. n

Ln(s)=( .51-a. ) ( sta. ) (6)

It appears that tile nth LSbUerre pol;rnomial

Ln\S) is the impulse response of the cascade

cOllbination of a single pole network and naIl

pass networks each with a pole at S=-a (1].

~upp.se we have a system response h(t), taen it

can be approximated by Lagurre polynomials as: f(.

]l(t)~h(t) ='Z:Cl(t) (7) a ?\=o n n

where Cn=~oa h(tjln(t) = (00 n(f) L *(f) df .J-60 n

B.r droppi~ off signals at the network junctions

and summing with proper veighting factors { Cn\

then tae overall network can approximate the

desired function, as shown in figure 1. (1J.

1-3. Lacuerre Transform

b'rom previous section, we conclude that ~

finite energy sigDal can be represented by 00

• Laguerre polynollials as f(t)=Z"C 1 (t).:rhe

coefficients { Cni in the rep~;;e:t:tion are naaed as Laguerre speotrull wAioh was adopted , from Steiglitz s ~per (3] , and the associated

polynollial F (z)"~Cnzn is defined as the L8BI1erre e ?Ie"

transform of f(t), wllere the dumlQ' variable z is

a complex variable inside the unit circle.

Let us make the following transformation:

S-(l z= sta.

... a.l~ ~

(8)

i.e. it transforms the unit circle Izr -1 into

s-jw, tae outside of the unit circle in z domain

into tlul left half plane Gf the s domain, and

the inside of the unit circle into the right half , plane of the s dOllain. Then froll Parsevals relatie.

and equation (6) vo can han the !Gllowing

relationship: (4) ~ S~

F(a)" (~)p e( sta.) and aillil arly we havo:

..fit( a (ltZ» Fe(Z)--;=Z- p( I-Z,

(9)

99

The above property, i.e. the relationShip between

equations (9) and (10) offers a simple .ethGd to

compute the Lacuerre spectrum of a fUnction with

rational Laplace transform.

II. APPROXHIATION OF TRUllCATED LAGUl!!RRE

POLYBOMIALS BY RATIONAL FUliCTIONS

In the si~al representation as equation(3J, ve

should increase the nu.ber of coefficients to

reduce the approximation error, in other words,

we should increase the cOllplexity of the network.

It turns out that the syntheais of the resulted

network would be rather complicate and seriGusly

uneconomical. In order to avoid aboTe drawbacks, it is possible

to employ a rational function R ~.) to appro xi-e mate the truncated Lacuerre transform (5) . That

is we use the following approximation:

B,y equation(9), the associated Laplace transferm

of R (z) is: e

The realization of equation(12)is shown in fi,2,

which we have one stage Of~, and P or Q (de­

-pend on which one is larger) stages of( ~~ ).

It is to be noted that. the number of unknown

coefficients is P+Q+1. Howenr in the polynOmial

representation, as in fi«ure 2, we need one atac­

of ~ and K staces of (~+& ).Its corre&­

-ponding number of coefficienta is K+1. Suppo ••

K is an even number, and also if P=Q=K/2, th. ( S-Cl \

we can saTe half stages of ~ I by the

rational approximation as equation (11). In the

following, we will firat review tAe Pade'.etbod

( 5 J for finding the required rational

expression.

:, i;I'

II-1.Tke Pade' Approximation

If we choose K=P+Q, tken equation (,1) can be

rewritten as: ( 2 ( 2 Q. cO+c , z + c2z +"" 1+b, z+b2z + ••••.••• +bQz )

Then equation (13) can be rewritten into the

following matrix form: a., Co at (, Co .,. c b,

-1:~ C~tl

o . . (pt ~)x' Go~ c" bQ. (!:I·t' II\X I or equivalently ~ .. 'J

-~ · --f::l b [J [ ] <piQit)Y(Q ,)

where r c 1 Jand (c 21 are corresponding upper and

lover parts of the matrix in equation (,4).

Zquation (,5) can be solved in the follov1n«

wa;r:

The vectors i, $, b

"

and the matrix [C41are the

correspondinc vectors and matrix in equation (14).

If the iDvers. of [041 exists, then from equation

(17) we have:

-1

$, -- (041 c3

and also we uve the solution for a &8:

( 19)

hen the rational approximation 1l (z) is expanded e

100

in power series by long diVision, then by section

1-3, the coefficients of tAls series will be

" identical to those of H (z), the truncated 1a-e

-t;I1erre transform, up to degree P+Q.

In the above approximation method we only con­

-sider P+Q.+1 coefficiants ot the Laguerre trans-

" -torm H (z). iith the saae tecknolo~, Salomon-e -sson [51 modified above method and extend the

coetficiants of c to K+1 terms with K~P+Q. In n this situation, we have more linear equnt~ons

than the number ot unknown variables {ail .

and {bi)Q • And consequently equation (t~ is i=O no longer valid, and therefore an error vector

sAould be introduced, i.e. we have: Q" "c..,

(, Co (c

_~t /"

~ = '1' - - --0 CPt-! " " .. '" ... C~, c 0 b~

{' I

C . I ") CI' " , ci< Q • (j( "T"I )I tit' ~')>(I"t I) ( tl~ t f ....

or alternative y we have tae tolloY1n« matrix .~~

torm:

-cd- _\~l~~Y~f~ ~ rc,l =e [1 t (kti)X~ _ u (~_~»)( (I<-p); - b

Or - ~ +[A) ~ == e (22) (2f) where [ I 1 is the identity matrix of order

(P+1) X (P+1 ), Tectors i and Ii 1 are defiaed in

equation (15). Vectors c6' i, e and matrix 05

are defined as:

'Co - /a 0 ~

(, u, €I

Cb = -x,- €"-Cl. ap -~I (.23)

Ck -b61. €"

(q~l~; 0

0 0 Co

C,,_I C~-2 ..... c,.~~

Hence ( A) is a matrix of order( K+1)X (P+Q+1 J •

From aboye consideration, we for.ulate the

problem as to find the unknown vector etc

- ~ x = SUCA that tae mean squared errer c

-~ '2 -bQ

- - -T ( J - -T- -T = e e - x B _ x -2x v + c6 c,

is minillUm. where [B )= (A)T [A] is a real sy_etric matrix

( ) ( , - AC" 1 T -of order P+Q+1 X P+Q+1) and y = A c6 , an

(P+Q+1) tuples yector. From the gradient method in calculus of vari-

-'II -ation,tae optimal solution % ll.appens to be:

end its associated error is:

-T _ _*T_ E ... c6 c6 - x v

Salomonsson eaployed above procedures, i.e.

involying a direct matrix inYersion. to find

tae required coefficients for the si&nal repre­

-sentation for some simple networks. (5)

11-2. Tae First Kind of Iteration .etud Per

Searching Rational Coefficients

From equation (26). we know taat we sllould

inverse the real symmetric matrix (BJ to find tao -'II

tae optimal solution x • It is ,enerally a

tedious and time consuming 1I9rk. lIence we will

approach this problea from another point ef view.

In stead of inversing II&trix (B). we use the

following iteration aethod to find the required

solution x :

i =i_A( (BJin-i (z]) n+1 n n

where in denotes the value of x at the nth itera-

-tion stage and A is the step size at the nth n

iteration process.

WUn O<.An <: 2/ A.. max ' where .A. mu is the largest eigenvalue of matrix (B J. then as the

iteration stages is large enough in+1 will

approach tae optill&l. solution i*. (B) -1 T. Tlle selection of the iteration step size and its

associate computation procedures can be referred

to reference (41 .

11-3. The Second Kind Of Iteration Ketaod For

linding Rational Coefficients

Tao approximation procedure in tae preYious itera

-tion algori tllm is designed in such a manner that

not only the convergence rate should converge

fast but also tu step size An be no greater taan

the value 2 /.A. • In other words. the i tora-max -tion method involves maxillUm eigenyalue cal-

-culations yhich in certain situations m87 become

an annoying step. In order to overcome above difficulty. we intro­

-duce another iteration algorithm .. :

in+1 ... in - 1/2 C (D1 + (LJ}-1. V£ (28)

where aatrices (D).(UJ .(L) , are diagonal, upper

triaJlClllar. and loYer triant=Ular matrix of CBl respectively. Froa the definition of gradient. equation (28)

is equivalent to the following expression:

%n+1'" i n- 1/2 [[D) + [L)r1

X 2 ( LBl ~ - i

=xn- (CD) + (Llr1 «(BJ in -- v

(29)

Let rlb1 + [L J]-1 ((B) iii TJ -Ain• then ye have

another form for equation (29). 1. e.

i 1C1i -~i n+ n n

Since IIatrices (D J and (L 1 are the diagonal

and lower triangular part of the matrix (D) • then by the back substitution method it is Dot

difficult to calculate the value A in.

Equation ,29) can be further simplified as:

101

xn +1= - (lD J + lL1)-1

+ [CD) + lLl)-1 -v

Equation (31) is just a Gauss-Siedel iteration

form in numericnl analysis for solving simul­

-taneous equations. Since (B 1 is a real, sym-

-metric, positive- definite matrix, then from

Jauss-~iedel iteration property we conclude

that the iteration process in equation (31)

will conver~e no mAtter the initial guess

of the va:ue Xo (6" The convergent rate of

this iteration process can be referred to (4) •

I II. t!XAKPLE.3

, . t· In order to il:ustrate the Pade approxkma 10n

and iteration methods as illustrated in the

previous sections • .ie have calculated the appro­

-xi;aations of the following impulse responses:

~ t < 1

~ t < 2

t? 2

o~t ~2

t> 2

In theee examples, the parameter a is chosen

as 1, and for comparison we have calculated

approximations of these desired impulse responseS

to the truncated La...",1.lerre series ( equation (11».

For truncated 1,aguerre seriee, the degree for

the numerator and denominator are both 5, and A

therefore the degree of Laguerre series B (z) I e

is 10, however in modified Pade approximation

the degree of the power series is selected as 12.

·.!.'he results are plotted on figure 3 to figure 5.

and also tabulated on table 1.

IV. COll~LUJII,)HS

:Ie have presented two iteration methods for

finding the coefficients of Laguerre polynomials

in the synthesis of some linear networks. In­

-stead of original matrix inversion method, these

two methods used iterative adjusted algorithms

to find the approximation parameters and the

performance of these methods are based on the

minimization of mean squared error. The first

kind of algorithm converges under the condition

that the iteration step size is less than some

certain constant value while the second algoritha

will converge under all possible circumstances.

This analysis us been illustrated by the synthe­

-sis of three simple networks and their results

have been plo tted and tabulated and cOllpared.

V.REFERENCES

1. :c.ewis Franks," Si8Ilal Theory," Prentice -Hall,

1969.

2.Nobert Wiener," Extrapolation, Interpolation

and Jmoothing of Stationary Time Series,"

John '.iiley and 30ns, 1949.

3.K. 3teigli tz, n Rational Transform Approximation

via the Laguerre Spectrum," J.of the Franklin

Institute, Vol.280,No.5,pp.387-393.

4. Y G.Jan, F.R. Chang, "Signal Representation with

~aguerre Polynomials Using Iteration Net.od, "

Technical Report, Telecommunication Labs.

TeL-B No. 143, Oct. 1974.

5.Goran Salomonsson, "Linear Network 3ynthesis

with Laguerre Polynomials," Ericsson Technics,

No.2, pp. 84-109, 1971.

6.J.H. Wilkinson," Rounding Errors in Algebraic

Processes," Prentice-Hall, 1963.

102

Truncated ,

Pade ,

Modified Pade

** Pirat iteration

** :3econd iteration

Table 1 * Sum of :3quared Irror

A, (t) 112(t)

1.0982 0.33789

0.53206 0.21829 10-1

0.52967 0.20740 10-1

0.81507 0.83576 10-1

0.78846 0.79277 10-1

* Tho samp~ing interTal i8 set equal to 0.1 fro. t=O to t-5.

** Tke initial point ~ i8 ckosen arbitrail7 .. (-1,-1,-1, ••••.• -1)

Tlle TalUO of E 18 choson as 10-4

h3{t)

0.56918

0.29672 10-1

0.21769 10-1

0.13344

0.12558

lI'il'llro 1. J.pprozi.aUon of desired impulae reaponse b7 a

linoar combination of Laruorre pol7noaials

Ficure 2. aoali.ation of Re{z) wit. M=N.

103

~f!Oe 3 lx..ple I

'il .." \

1

0.5

I

0.5

1 2

\

\ \

\ --- -. .. _},' 4

Approxiaation bv Dane wthod

r,..e4-lx"pip 2:

h2(t)

1 2 4

ADproximation by ~ad~ method

Example 3:

h3(t)

1

fi,.~

""---2 3 4

n")r~)(imat1011 by pa IE' method

1 ' - -." r 0.5

0.5

5

1

::l.5

\ , ,

1

\ \ \

2

\ \

... ~~ , ...

ADprox iro"t i"" by modifi~d "a ,e method J~: r.t' !"'\bt rix i nvprse

h2(t)

1 :2 3 4

Approximaticr, r.Y l'1o(Jlfipd ::>ale method using ffibtrix ir vrrsp

h3 (t)

1

I , 2

---_ ....

A;Jprnxim&t :n!. t., 'r:" .... :If'O os f

mf't.t,..., as:' T1t ,..- .• r l..X invPTf r

1 r',

0.: ,

I

"

I I

\

-•. - - - 4

Ar·':)!"'oJ<imbti:"ltl ~ y r.v- i~ !'ir.·~ "'f:~: , .... ~s:,.nt I~· . ::~.' r·· ... •

h;,:t)

').5

') .

5

2 1.

A"::>roximat ion ty "l0uifip(i metnoc .1siI!£;, lEt i!ltJrbt iell

hJ (t)

!, ;~\ " , .1

\ , I ,I \ , \

j \ .. -......

,,")'.)]':-y : tr..:'lt . '! ...i.S -;.; " !".:.I i'e. ~ .

).5

5

0.5

1

~ ,

, \

I I \

I \

\

--~--

A;nrux~rr.&.i:i -II !-': ... lJ.S i !'".r LY""": 1 t,' ..... ~ TJ

r,~ (t:

... ----" ...... :"'. ---5

Aoproximat ien by mo·:ifi. n ~ l!"ptho': using 2nd itpr&tion

h) (t:·

~ ~ \ " ~

J I' ~ \ . ~

:. :' ~~'(

• o

THE TRANSFORMATION OPERA'lOR APPROACH*

TO MULTI-SUBSYS'mM DYNAMICS

William Jerkovsky The Aerospace Corporation El Segundo, California

Abstract

A new approach to formulating nonlinear dynamics equations is presented. This new approach is a unified approach which abstracts the essence inherent in the Newton, Euler, Lagrange, and Boltzmann-Hamel methods of formulating dynamics equations. The key ingredient is the linear velocity transformation operator which transforms "Old" velocity variables to "new" velocity variables. The new approach is illustrated by a simple example of interconnecting two rigid bodies.

1. INTRODUCTION

Gabriel Kron 1-3 has developed a tensorial approach

to generating equations of motion for a system

when the equations for the subsystems and the

interconnection equations are given. The most

significant feature of this tensorial approach is

that the resulting equations of motion are the

exact nonlinear ordinary differential equations

for the system. Currently there are many workers

in the field of network and system theory who use

approaches that are similar to Kron's, but most of

these methods are only applicable to linear systems

(because, in effect, they are linearized versions

of Kron's method).

The purpose of this paper is to cast Kron's work

into a modern mathematical framework, and to

illustrate the method with a multi-body spacecraft

dynamics example.

The discussion of the method given herein differs

from previous discussions of Kron's work because

Kron and his followers did not separate the method

into its algebraic and geometric aspects. The

algebraic aspects are e~sentia1ly the same whether

the system is linear or not; the geometric aspects

of the system contain all the nonlinearity of the dynamics.

2. PRIMITIVE EQUATIONS

In any finite dimensional physical system we can

always specify a kinetic energy T whi~h is

quadratic in the system velocity vector . .J:

T=~-Jtl.J.cr (1 )

where at is the transpose of ~. and u is the

system metric tensor. In general, I.J. is positive

definite symmetric, and therefore it has a positive

definite symmetric inverse v

-1 ) v = I.J. (2

The product I.J. 0 is called the sO's tern momentum r.:

G = I.J. .) (3)

Even though in general I.J. depends on the coordi­

nates, nevertheless G is a linear function of the

veloc i ty o. The kinetic energy is now t i VE'n by

(4 )

The equations of motion for any dynamical system

can always be written as

G - C G = K

where G is the time derivative of G. K is the

sys tem force including forces derivable from a

potential or dissipation function), and C is tb~ 4 system Cartan operator In general C depends op

the velocity a and on the

*This work was supported by USAF SAMSO Contract No. F04701-74-C-0075.

lOS

\

;i

(5) is a nonlinear ordinary differential equation

for G. If we introduce the notation

c • G = G - C G (6)

then (5) becomes D

G = K

" G is the covariant time derivative of G.

If we have a network of n subsystems, then the

equations for the ith (i = 1,2, ... ,n) s~bsystem iii ai ·i i i

can be written as G = ~ 0, G = G - C G, and • i i G = K (note that the superscript i is a counting

index, not a vector component index). Let 0 be

the direct sum of oi for i = 1,2, ... n; similarly, i

let G be the direct sum of the G ; etc. Thus, the

composite system consisting of n subsystems can be

described by G = ~ 0, G = G - C G, and G = K. Note

that this composite system has as many degrees of

freedom as the sum of the number of degrees of

freedom for all the subsystems.

3. TRANSFORMED EQUATIONS

If it is desired to reduce the number of degrees of

freedom of the composite system-because of the

constraints due to the interconnections- we

introduce a new velocity vector a which has smaller

dimension than o. We write the transformation in

the form

(8)

where A is a linear transformation operator (note

that in general A depends on the coordinates but

not on the velocity). We can actually write the

transformation equation (8) even if we do not want

to reduce the number of degrees of freedom of the

( compos j te ) system. If cr and o have the same

dimens ion (i.e. , if we do not reduce the number of

degrees of freedom) then A is invertible. If (f

has smaller dimension than 0 (because of the

constraints) then the operator A still has rank

equal to the dimension of 5; hence, A is an injec­

tive operator. When we convert from:; to j we

s imul taneously convert from G to G, where

G = At G (9)

where At is the transpose of A. The kinetic energy

can now be written as

106

where

1 -t­:2 G 0

t i:1=A ~A

(10)

(11)

~ is the transformed metric tensor. Since ~ is

positive definite symmetric and A is injective, ~

is also positive definite symmetric. We can now

also write

G ~ 0 (12)

and

o = v G (13)

where _-1 v = ~ (14)

Evidently ii is also pos i tive definite symmetric.

We now define the linear transformation operator B

by

(15)

Since ~ and v are bijective and A is injective, it

follows that B is surjective. In fact, we easily

see that B is a left inverse of A, and hence (8)

can be inverted:

a Bo (16) We can also write

The transformed equations of motion are now given

by " G if (18)

where

.. . G=G-CG

C=AtBt+AtCBt

Since K = At K we also have

~ = At G (20)

" and hence G is obtained from 8 (via At) in the

same way as G is obtained from G.

Thus the equations of motion for a dynamical system

are always given by the statement that the covar­

iant time derivative of the momentum is equal to

the force. The equations G - C G = K and

B - C G = K are equivalent to (but simpler than!)

the Boltzmann-Hamel equations 5,6. If the system

velocity is actually a holonomic velocity then

these equations reduce to the Lagrange equations

(or equivalently, to the Hamilton equations).

In principle, the amount of effort required to

formulate the equations of motion for a dynamical

system is the same no matter what approach is used.

The advantage of formulating the equations as

discussed herein is that all of the nonlinear

dynamics is buried in the Cartan operators (C or C). It may take quite a bit of effort to get an

explicit expression for the Cartan operator, but

after it is found the equations of motion are very

simple when expressed in terms of this operator.

Furthermore, the equations of motion always have

the same form, and all the terms required in the

equations are always obtained in the sare way. In

a sense, the approach described herein is a non­

linear generalization of similar ideas in linear

largescale system theory 7

4 • TENSORS AND VEC'IDR SPACES

It is interesting to note that we can consider 0

and cr to be contravariant vectors, and we can

consider G, G, and K, K to be covariant vectors.

The relationships G = ~ 0 and IT = ~ (j then show

that~, ~ are covariant of degree 2. Hence v,V

are contravariant of degree 2. The relationship

o = A cr shows that A (and B) is covariant of degree

1 and contravariant of degree 1. The kinetic 1 t -t -

energy is, of course, a scalar; T= 2 G 0= G 0, or

G 1 t 1 _t - - It' t h 8 = 2" 0 ~ 0 = 2 0 ~ O. l.S easy 0 s ow

that the tire derivative of T is T = Kt 0 = ft cr, and this is, of course also a scalar.

We can consider cr to be an element in the velocity

space /7 (denoted by cr ~. Similarly a .. 9, G .. Jf ,

Gt~Ke.~ Ke.j(. These spaces are related as shown

in the commutative diagram in Figure 1.

'§ A .9'

i! I ~.

At

a

:K •• -----:K

Figure 1: Commutative Diagram for Transformation Operator A

9'and '§ are tangent spaces to the configuration

manifold Which describes the dynamical system;

.it andJ" are cotangent spaces which are the duals

of the tangent spaces (and vice versa). The

tangent and cotangent spaces are ~ vector

spaces even though the configuration manifold is

in general nonlinear. The metric tensor relates

these dual spaces to each other. Since the

effect of A on 9'and of At on .it is invertible, we

also have the commutative diagram in Figure 2.

107

Figure 2: Commutative Diagram for Transformation Operator B

If we introduce basis vectors in each of the above

vector spaces, then cr, G, K, C, ~, are represented

by the matrices E., g, K, Q, ~, respectively; here

Q, g, and.K. are colwnn matrices of scalars, and

.c. and .l:. are square matrices of scalars. The

elerent of !!. Which is in the i t.h row and j Yl

column can be expressed as

.Qij (21)

where Qk

is the k:t.b elerent of Q., and Where the

coefficients rik are the components of the affiDe

connection (with respect to the chosen basis). It

o is a holonomic velocity, then the rik are the

Christoffel symbols of the second kind for the

metric matrix 1!:.: The coefficients rik are usually

very tedious to evaluate; however, we don't need

to evaluate them when we use the transformation

operator approach because we only need the Cartan

operator C (or the Cartan matrix Q.), and we can get

an expression for C directly without first evalua­

ting the rik.

It should be noted that we really don't even need

C, because only the product C G appears in the

equation of motion. Often we can write C = Cl +C2 where C

2G = o. Hence, C G reduces to CIG.

In practice it is usually simplest to introduce

X = - C G so that (5) and (6) become

G+X K (22)

G+X

The transformed equations then become

(23)

where

(24)

The significance of the above development is that

we have expressed the equations of motion for an

arbitrary dynamical system in terms of the

algebraic and geometric aspects. The whole devel­

opment is centered around the transformation

operator A. If we make another transformation, say - - - = -t- = -"b = .!.t-t cr = A ?J, then we get G = A G, K = A K, C = A B + -t - -t ~ .!. - = ~ = A C B , G = G - C G, and G = K; alternatively,

~ == = = -t - !..t-we have G = G + X where X = A X - A G. Since

cr = A cr we also have cr =../0 where A = A A. Since

A and A are covariant of degree 1 and contravariant

of degree 1, so isA; since A and A are injective,

so is .H.

5. EXAMPLE

The transformation operator approach to dynamics is

being used at the Aerospace Corporation to generate

the dynamics equations of motion for various 8 spacecraft • Its utility stems from the fact

that the same systematic technique can be used for

108

spacecraft of arbitrary configuration. After the

equations of motion are expressed in the form . . G = C G + K or G = - X + K these equations are

numerically integrated for G, and then cr is found

by solving G = ~ cr for a.

We will now give an example of deriving the equa­

tions of motion for a system consisting of two

rigid bodies, given the equations of motion for a

single rigid body. This system of two rigid

bodies might represent a spacecraft in which one

of the bodies represents the "main body" of the

spacecraft and the other rigid body represents an

"attached body" such as a gimballed antenna. For . --i l = 1,2, let H c. denote the angular momentum of

and let !i body i about itg center of mass c i ' c. l

denote the torque (or moment) on body i about the -i

center of mass c i ; also, let ~ denote the linear

momentum of body i, and let Fl be the force on

body i. The equations of motion for body i are

now given by

, . j{l

c. l

(25)

Let wi denote the angular velocity of body i with

respect to inertial space, and let v denote the c.

translational velocity of the center lof mass c. l

with respect to inertial space. The angular -+i --i

momentum H and linear momentum P are now given c. by l

(26)

where Ii is the inertia of body i about the center c i i

of mass c i ' and M

have also introduced

the property that

for any vector V.

is the mass of body i. We

the unit dyadic E which has

(27)

The primitive momentum and force for this two­

body system can now be denoted by

HI Ll cl

cl

H2 L2

G c2 K

c2 (28) pI ffL p2 ¥

From (25) we now have G + X = K where X O. The

primitive velocity is

_1 (jJ

--2 w CY= v c

l

v c2

and the primitive mass/inertia is

(30)

where the elements of ~ not shown are zero dyadics

(Cj). From (26) we now have G = I.lo a.

Next we assume that the two-body configuration is

such that there are no relative translational

degrees of freedom between the two bodies. Thus,

the composite system has only 9 degrees of freedom

(3 translational and 3 rotational degrees of

freedom for body 1, plus 3 rotational degrees of

freedom for body 2). We incorporate the con­

straints ( of no relative translational degrees of

freedom) by expressing the 12-dimensional vector

CY in terms of a 9-dimensional vector~. In order

to find the required transformation operator A

we examine Figure 3 and introduce the following

notation.

Body 2: -2

= Ui + n 1: (jJ

Inertial Reference

Figure 3: Diagram of Two-Body Configuration

If P and q are points in space let 1 and r p q denote the position vectors to points p and q,

respectively, from the origin of an inertial

reference; also, let R denote the position pq vector to point p from point q. Evidently

Rpq = Tp - rq • Let a be an arbitrary reference

point fixed in body 1, and let h be the "hinge

point" where bodies 1 and 2 are interconnected.

We can now write

109

r + R + R h a >ba c2

Taking the inertial time derivative of these - ~ equations and introducing the notation v = r p p (i.e., v is the translational velocity of point

p p with respect to inertial space) yields

Since R is fixed in body 1, we have CIa

where we have introduced the following notation: if .• - -t A and B an~ vectors, then A and A are defined by

We can now write

A'B = AxE -t -A =-A

is also fixed in body 1 we get

(34)

..1 S"l 1 ~ -Rt .2 N • W lml ar y, n h = h • w, ow c2

c2

ndenote the angular velocity of body 2 with

respect to body 1, and for convenience write

u..~ as;. Then

From this we get

. .1 W

.2 w w+O

where It It h + ~ • (35)to (37) can now be c2a c2 a

written as

t] it 0 0 -l ;] (38) -2 E E 0 w - R't 0 if

::~ cla

R't ~t E R h c2

a c2

We now define a by

The transformation operator A is now given by

comparing::; = A·awith (38).

Now that we have A, we get the transformed momentum

and force from IT = At.G and K = At.K. Thus we get

E 0 (40)

OE

Ifl +W + R .T +R ~ .p c

l c2 cla c

2a

If + R .p2 c

2 c2

h

If a

where If is the angular momentum of bodies 1 and 2 a

about a, ~ is the angular momentum of body 2

about h, and P is the linear momentum of bodies

1 and 2. Similarly we get

L a

K= ~ h

(41)

F

where r is the torque on bodies 1 and 2 about a, a

~ is the torque on body 2 about h, and F is the

110

force on bodies 1 and 2. Since X = 0 we get

_·t [va x pJ X = -A ·G = .,. = ..... ~ vh x P

0"

The transformed mass/inertia is t

jj; = A .~.A =

y 12 a ah

= ~a ~ M'Rt ~ 'Rt

ca c2h

where

MR ca

M2'R c2

h

ME

(42)

(43)

- ="T + Ml 'R .R't I a cl cla cla

+7 +~ R' • 1ft C2

c2a c2a

'":!21

'":!2 -2 - 0;>( t h = I + M- R h·.11 h

C2 C2 C2 (44 )

M=Ml+M2

MR =MlR' +~R ca cla c2a

Note that Ta is the inertia of bodies 1 and 2

about a;~ is the inertia of body 2 about h, Mis

the mass of bodies 1 and 2, and point c is the

center of mass of bodies 1 and 2 (so that If is ca the position vector to the composite center of

mass from the arbitrary reference point a fixed

in body 1).

The final transformed equations of motion are now

given by

[~J and

H - - 'fah I M1f W' a a ca

~ ~a -y2 ~1f h • "0 h c2 (G=;1".O) (46)

-- M~ M2'Rt ME --P v ca c2h

t-

In a spacecraft dynamics application, F is the

total external force on the spacecraft, L is the a total external torque on the spacecraft about the

point a, and if body 2 is an antenna, ~ is the

torque on the antenna about the hinge point h.

This example can easily be generalized to a

system consisting of an arbitrary number of rigid

bodies interconnected in a fashion so that the

graph of the configuration forms a tree (no

closed lOOps). If the configuration does not

form a tree, the equations are considerably more

involved; however, even this more complicated

case can be handled by cutting the graph so that

it becomes a tree and then enforcing the cut

constraints by appropriate constraint forces

(i.e., Lagrange multipliers). If the bodies are

flexible rather than rigid, the same approach

still applies, but the equations get much more

complicated (primarily because the equations for

a single flexible body are much more complicated

than for a single rigid body).

III

m:FERENCES

1. G. Kron, IINon-Riemannian Dynamics of Rotating

Electrical Machinery", Journal of Mathematics

and Physics, Vol. XIII, No.2, pp 103-194,

May 1934.

2. B. Hoffman, "Kron's Method of Subspaces",

Quaterly of Applied Mathematics, Vol. II; No.

3, pp 218-231, 1944.

3. L.V. Bewley, Tensor Analysis of Electric

Circuits and Machines, Ronald Press, 1961.

4. H.W. Guggenheimer, Differential Geometry,

McGraW-Hill, 1963.

5. E.T. Whittaker, A Treatise on the Analytical

pynamics of Particles and Rigid Bodies,

Cambridge University Press, 4th Edition, 1937·

6. D.C. White and H.H. Woodson, Electromechanical

Energy Conversion, Wiley, 1959·

7. D.M. Himmelblau (ed.), Decomposition of

Large-Scale Problems, North-Holland Publishing

Co. (American Elsevier Publishing Co.), 1973·

8. W. Jerkovsky, "The Transformation Operator

Approach to Multi-Body Spacecraft Dynamics",

Aerospace Corporation Report (rough draft),

(3 volumes), dated October 1974.

BIOGRAPHY

William Jerkovsky was born in 3atschka Palanka,

Yugoslavia, in 1940. He received the B.S. degree

from Loyola University, Los Angeles, in 1962 and

the M.S. degree from the University of California,

Los Angeles, in 1965; both degrees are in physics.

He joined the Aerospace Corporation in 1972 where

he is currently involved in various aspects of

spacecraft dynamics and control. He was previously

in the controls department at TRW Systems,

Redondo Beach, and in the preliminary design

department at Garrett AiResearch, Torrance.

112

A NaIE ON THE NAGY-FDIAS

LOSSY AND J1)SSIESS SPACE*

N. Levan Departnent of System Science

4532 Boelter Hall University of California

Los Angeles, California 90024

Abstract

We discuss in this paper the construction and properties of a Nagy-Foias soace. The neanings and applications of such a space in systerm and networks will be studied in sone details.

1. IN'OODUCI'ION

A Nagy-Foias space is an abstract Hilbert space on

which a model of a contraction Hilbert space

operator can be constructed. Plainly speaking, a

ITDdel of an operator is another operator (or

operators) which is better understood, but at the

sarre tine, has a lot more structures.

In what follows, we shall investigate the· system

neanings and applications of a Nagy-Foias space,

as well as the applications of the Nagy-Fbi as m:xiel

theory in system and network realization.

2. THE NAGY-FDIAS SPACE

Let H be a (separable) Hilbert space, the space

of analytic functions f'ran the unit disc I zl < 1

to the vectors in H is denoted by H2 (H) • Thus ,

H2(H) consists of power series

00 f(z) = L f zn, for Izl < 1, f in H

n=O n n

and ~llfI12<00 (2-1) n=O n

2 '!he irmer product and norm in H (H) are defined

in the usual way 00

lif'his work was supported by National Science Foundation under Grant number ENG 75-11876.

(2-2)

and

IIfl12 = jollfnll~ (2-3)

The space H2(H) is actually a Hilbert space, and

furthermore it can be identified with the space

L;(H) [lJ, which consists of Fourier series with non-negative powers of eit . For a function f(z)

in H2(H) •. there exists a boundary function f(eit

)

in L;(H). We shall need the space L2

(H) \ which

consists of Fourier series of all powers of eit

Clearly L2(H) = L~(H) 0 L;(H), where L~(H)

113

is the space of Fourier series of ne.~ti ve powers it of e .

If we consider the t;(H) sequence {fo ' f l ,···, fn

, ... } as a discrete sip::nal, startin~ from the

present tine 0, then f(z) is just its discrete

Fourier transform. In what follows we shall

regard L 2(H) as the space of allowed sip.nals over

all tine, while H2(H) (or L;(H)) is the sub­

space of present-future sip::nals and L2

(H) is the

subspace of past signals.

Given two Hilbert spaces HI and H2, a funCtioo

e(z) from Izl < 1 to the operators (linear

bounded) from HI to H2 is denoted by {e(z), •

HI' H2}. Such a function is bounded when

and is analytic when it has the power series .

expansion 00

e(z) = I e z, 1 zl < 1 n=O n n

(2-5)

where en are operators from Hl to H2 [lJ.

To each analytic e(z) we can have the boundary

function e(eit ) defined almost everywhere. It

then follows that the multiplication by e(z) is

a map from H2(Hl ) into H2(H2) while the

multiplication by e(eit ) is a map from- L2(Hl )

into L 2 (H 2). Clearly, e (z) can be regarded as

a system transfer function, and e(eit ) is a

system frequency function. In this paper we shall

consider those e(z) which has a causal [2,3J

boundary function e(eit ). This can be best

explained as follows. If we represent e(eit )

by a matrix operator with respect to the

decomposition L2(H.) = L2(H.) ® L+2(H.), 1 - 1 1 i = 1, 2 , then

(2-6)

then clearly e12 is a ~ap from the present­

future input subspace L~(Hl) into the past out-2 it put space L_(H2), therefore, for e(e ) to be

ca~al e12 must be 0, that is the function

e(e1t ) is lower triangular [2,3J. Note that e22 can actually be identified with the multiplica­

tion by e(z). In what follows we shall need

the adjoint e(eit )* which is clearly given by

e(eit )* = I e-inte* n=O n

* where en is the adjoint map of en.

A bounded analytic fUnction {8(z), Hl , H2}

is said to be contractive if 1 le(z)hll IH

.:: IlhlllH for any hl in Hl . For a 2 1

(2-7)

contractive e, we can define the positive self-

adjoint operator

(2-8)

It is clear that l'l(t) is bounded between ° and

1, and the multiplication by ~(t) is a map from 2 2 L (Hl ) into L (Hl ). Let u(t) be an element of

114

L2(Hl ), then

11~(t)u(t)112= Ilu(t)11 2-lle(eit )u(t)II Z (2-9)

Thus if u(t) is the incident voltage of a linear

network whose scattering operator [4J is e(z) then

1 1~(t)u(t)1 12 is just the net amount of energy

absorbed by the network. We corrment that linear

passive networks are characterized by contractive

analytic functions e.

A contractive analytic function e is said to be

inner if the map e(eit ) is an isometry,

consequently for an inner e, ~ (t) = ° - and this

clearly corresponds to the case of a lossless

network.

The Output - Energy Dissipation Space

Given a contractive analytic function {e(z),

Hl , H2} we associate with it the following input

and output spaces: L2(Hl), H2(Hl ), L2(H2) and 2 H (H

2). We now define the space

(2-10)

where (!) denotes the direct sum, and stands

for the closure. Since ~(t) is bounded below

by 0, elements of H are of the form (v(z),

~(t)u(t)) with v(z) in H2(H2) and u(t) in 2 L (Hl ). We shall refer to H as the output -

energy dissipation space. H is a Hilbert space

[1J whose inner product and norm are defined

in the usual way

[(Vl'~~)' (v2,~u2)J [vl,vlJ + [~ul,~u2J and

Note that for a pair (v ,~u), the output v needs

not come from the input u, infact both v and u

are quite arbitrary. In H consider the sub­

space of all present - future outputs which come

from present - future inputs, i.e., the subspace

"t 2 M = {(e(z)w(z),~(t)w(el )),w in H (Hl )

We have

II(ew,~w)112 Ilewl12

+ Ilw112-llewl12

IIwl12

( 2-11)

Hence M is a closed subspace of H, as a

consequence, we can consider the orthogonal

complement in H of M

W- = H2(H2

) 0 f>(t) L2

(Hl )

<3 {(e(z)w(z) ,t;(t)w(eit »), w

(2-12)

W- is called the Nagy - Foias space generated by

{e(z), Hl , H2}

We now explore further to see what are the

meaning;s and applications of this space to system

and network theory. Clearly W- is basically a

subspace of the output - "input" space H, and

its elements must be such that

(v,f>u)cW-<~(v,f>u)JL(ew,f>w),

w in H2(Hl ) (2-13)

or

(v,f>u)cw-<=>(e*v + f>2u ,w) = 0,

2 w in H (Hl ) (2-14)

2 2 Thus as a function in L (Hl ), (e*v + f> u) must

be orthogonal to the subspace H2(Hl ), that is

it can only be expanded into a Fourier series of it +-

negati ve powers of e Letting Pi and Pi'

i = 1, 2, be the projection operators onto 2 L+(H

i), we can express (2-13) and (2-14) simply

as

Given an element (v,f>u) of W-, we can row

construct the element

(2-15)

(2-16)

where u + u+ = u, and it follows readily from

(2-15) that (e21u_,f>u_) is in Mf while

(~2u+,f>u+) is in M. The element v is just the

response due to u. Using (v,f>u) we can write,

for ~ (v,f>u) in W-

Consequently

(v,f>u) = (e21

u ,f>u )+P (v-v,O) - - M"'-

(2-rn

In what follows we shall concentrate on two

special subs paces of W-, which are denoted by

~ and ~ and are defined by

~ = closure {(e2lu_,f>U_), U_cL:(Hl )}

and

~ = closure {Pw-(Y, 0), y in H2

(H2)}

(2-18)

(2-19)

Plainly speaking ~ is just the subspace of

present-future outputs which result entirely P'om

past inputs, while ~ is the subspace of outputs

whose corresponding inputs are not prescribed

before hand.

3. 'IHE IDSSLESS NAGY-FOIAS SPACE

For the loss less case t;(t) = 0, and therefore

W- = H2(H2

) e {e(z)w(z), w c H2(Hl )} (3-1)

The subspace ~ in this case can be simply

described as follows. Setting

2 n e(z) = e

O+el z+e2z + •.• + enz + ..•

We have

+( it ) e12u_ = P2 e(e )u_

(ela_l + e2a_2 + •.. ) 1

+ .....

(3-3)

'lhus the matrix of e12 with respect to the it 2it

orthonormal basis {l, e ,e ," • } is

115

(3-4)

This infinite matrix is called the Hankel operator

H8 generated by 8. We conclude that

(3-5)

The subspace ~ in this case is all of W since, by definition ~ is the closure of

2 Pw{Y} for any y in H (H2).

2 Let Y be any element of H (H2), then from

(2-15), it is in W if and only if P~(8*y) = 0

or equivalently

* 822 Y = 0 ( 3-6)

Furthernore, since 8 is inner, it can be

* readily verified that 822 821 = O. Thus, *

those y of W are in the null space of 822 ,

and in turn, are also in the range space of 821 ,

We therefore have, for an inner 8:

0-7)

The subspace W in this case is the family of

all present-future outputs due entirely to past

inputs - it can therefore be taken to be a

state space [5], and due to 0-7), the system in

this case is both controllable and observable.

4. '!HE LOSSY NAGY-FOIAS SPACE

We now consider the case in which 8 is not inner,

this corresponds to a lossy passive network.

We shall briefly describe here the basic struc­

tures of ~ and ~. For details, we refer to

[ 6 ].

In what follows we will need the definition of the

restricted shift operator. Plainly speaking the

shift operator on a space ~ is the multiplica­

tion by z. Fbr the space H, the shift operator

116

s is just the multiplication by z and by it e

Furthermore, it can be shown that [1 ], the sub­

space M is invariant under S -- and hence W is invariant under S*. The restricted shift T

is a linear bounded operator from W to W, defined by

·t T(v,Au) = P w(zv,e

l ~u),(v,~u) in W (4-1)

It then follows that its adjoint is

( 4-2)

We now set

4>o(z) 8(z)

4>1 (z) 4>O(z) - 4>0(0)

z

<P (z) - 4> (0) 4>n+l(z)

n n z

then, clearly 4>n(O) = 8n , the coefficient of zn

in the power series expansion of 8(z). Next,

define

Then clearly [6] for Ci in HI

K Ci = [4> ,~e-int]Ci n n

(4-3)

-int -int [82le Ci,~e Ci] (4-4)

Thus ~ is in ~ for n > 1. Furthernore

it can be readily verified that

Kn+lCi = T*nxlCi, n > 1

We have

Theorem 1

Mi = span {KlCi, T*KlCi, •.. , T~lCi, ... }

(4-5)

--- *1 and therefore KlCi is a cyclic subspace of T ~'

* ~ ~ the restriction of T to ~.

Similar results can be gotten for ~ [6 ]. Indeed

we have

Theorem 2

M2 = span {kOS, klS, k2S, .•• , knB}

where

and therefore kOS is a cyclic subspace of

For the lossy case, the space W can again

be taken to be a state space, except that, in this

case, we also have the dissipated energy along

with the state. We note that for an element in

~, we have

which is just the energy stored in the system -

due to input u _ in the past. Thus, in some

sense, this shoos same connection between state ani

energy. The space ~ can be thought of as the

space of "error states" - since originally, for

a pair (V,tlU) of H, the output v and input

u are quite arbitrary.

We conclude with the comment that, since

Kla and kuB are cyclic for T*IJ~ and Tl~

respectively, they will result in canonical

realization of e For a full treatment of this,

we refer to [6].

REFERENCES

1. B. Sz-Nagy and C. Foias, "Harm::mic Analysis of Operators on Hilbert Space", North Holland­American Elsevier, Amsterdam, NevI York, 1970.

2. W. A. Porter, "Some Circuit Theory Concepts Revisited", Int. J. on Control, Vol. 12,- pp. 433-448, 1970.

3. R. Saeks, "Causality in Hilbert Space", Siam Review, Vol. 12, pp. 357-383, 1970.

4. R. W. Newcanb, "Linear Multiport Synthesis", NcGraw-Hill, New York, 1966.

5. A. V. Balal<rishnan, "State Space Theory of Linear Time-Varying Systems" pp. 95-125

117

of "System Theory", L. A. Zadeh and E. Pollack Editors, McGraw-Hill, New York, 1969.

6. N. Levan, "Canonical Realizations of Transfer Operators': Proceedings 7th LF.LP. Conference on Optimization Techniques, NICE, September 8-13, 1975; Springer-Verlag 1975.

On output Control Problems Containing Input Derivatives

Victor Lovass-Nagy David L. Powers

Clarkson College of Technology Potsdam, New York 13676

Abstract

Given plant and control equations, either or both containing the derivative of the input, the problem is to find a control which drives the output along a prescribed path. In this article, a method is developed which avoids the Laplace transform and uses the concept of a matrix generalized inverse. Some criteria are found for existence of a solution, and techniques are given for simplifying the computation of the solution.

Consider the time-invariant control all functions are as differentiable as

systCr.1

dx dt (1)

(2)

where x is the (nxl) state vector, u is

the (pxl) input vector and y is the (qxl)

output vector. One important problem

in control theory is to determine an

input or control vector u which forces

the output to be a prescribed function

of time. This problem (often called

output function observability or

functional reproducibility of output)

can be attacked by means of Laplace

transform. The case where Bl=O and

01=0 has been treated by sevGral authors

[Brockett, 1970, p. 81; Wolovich, 1974,

p. 163]. In this note a method will be

developed which avoids the Laplace

transform and uses matrix generalized

inverses to determine inputs which drive

the output of the system (1), (2) along

a prescribed path. We will assume that

118

necessary.

Let y be a given function of time.

It is desired to find a control u(t)

satisfying equations (1) and (2)

together with some initial conditions,

x(O) = x O' u(O) = u O•

First we rewrite the original problem

(1), (2) in the form

or dw P dt Qw + f

(3)

(4)

(5)

(6)

-B ] 1 Q =

Dl r:] f =r:]· Now we treat equation (6) as an

1 b · ., h . h dw. t b a ge ra~c equat~on ~n w ~c dt ~s 0 e

found. A symbolic way to solve (6) is

by multiplying through by a conditional

inverse, pc, of P, that is, any matrix

satisfying the equation ppcp = p

[Graybill, 1969, p. 162]. pC is also

called a g-inverse [Rao and Mitra, 1971,

p. 21], or a {l}-inverse [Ben-Israel

and Grevi11e, 1974, p. 8].

If equation (6) does indeed have an

algebraic solution, its general form is

where h is an arbitrary function of t.

In order to survey the legitimacy of the

passage from the equation (6) to equation

(7) it is convenient to specify the con­

ditional inverse

(8 )

where o~ is a conditional inverse of 01

(i.e., any matrix satisfying the equation c 010 101). Hence we calculate

It is clear that if ppcQ = Q and

ppcf = f, then equation (7) implies

equation (6). In terms of the blocks

of the partitioned matrices, these con­

ditions are equivalent to requiring that

Certainly these conditions are fulfilled

if 01 (and hence P) is of full row rank,

for then 010~ = I. However, even when

these conditions are not fulfilled, it

may happen that

(I_PPC) (Qw + f) = 0

for an appropriate choice of the function

h. In terms of smaller matrices, this

means that

o. (9)

This condition, of course, can be checked

only ~ posteriori.

The arbitrary function h appearing

in equation (7) deserves some additional

comment. If we partition h in the same

way as w,

119

then we may calculate

c lB1 (1-01 O~) h2 (I-P P)h =

(I-010~)h2

Evidently, the components of h1 do not

even enter into equation (7) nor its

solution. Of the components of h 2 ,

some may be needed to satisfy the con­

dition (9). Any remaining components

may be used to minimize a cost functional

if desired.

Let us now concentrate on the

solution of the initial value problem

Setting M = pCQ, the solution of the

initial value problem (10) is

(10)

w(t) = exp(Mt) w(O) + J:eXP(M(t-T».

pCf(T)dT + J:eXP(M(t-T» (Z-pcp)h(T)dT

(11)

Since P and Q usually have more

columns than rows, the matrix M = pCQ

will have rank less than its order. It

is then convenient to introduce the matrix

N = Qpc. If we use the power-series

definition of the exponential, we find

11'11'" I

'I

2 exp{Mt) = I + pCQt + pCQpcQ ;1 + ••.

QPCt2 I + pC{t + --2-1-- + ..• )Q

I + pC F{Nt)Q

where the function

Nt2

N2

t3

ft F{Nt) = t + ~ + ~ + •.. = oeXP{NT)dT

Moreover, by a similar manipulation,

exp{Mt)Pc = pC exp{Nt).

By these means, we may recast the

terms of the solution (11), as follows.

C ft = p oeXP{N{t-T» f{T)dT

exp{Mt)w{O) = (I + pCF{Nt)Q)w{O)

Finally we arrive at the solution

wet) (I + pCF{Nt)Q)w{O)

+ pC f:eXP{N{t-T»f{T)dT

+ f:C1 + pCF{N{t-T»Q) l~lJp{T)dT C

where p (t) = (I-01 01) h2 (t) •

Here, everything is expressed in terms

of functions of N only. We make note

of two additional facts: first that

exp(Nt) = I +NF(Nt);

and secondly, if N is nonsingular,

-1 F(Nt) = N (exp(Nt)-I).

(12)

(13)

Thus, only one of the two functions F(Nt)

and exp(Nt) need be calculated.

120

Example. Since the derivative of the

input vector u can be removed from one

(not both) of the equations (1), (2),

[Balabanian and Bickart, 1969, p. 245]

[a,l[ [:j That is,

Then

p

Q

[-1 O~-

A = 2 -2

c = [0,1]

we find that

~ 0 0

U 1 0 0 1

o -2 -1

1 1 0

°0 0 °1

[~ 0

pC 1 0 0

-~

0 0

a J 0 0 0 0 -1/2 1/2 0 1/2 -1/2

0 1 -2 1 -1/2 0 -1/2 0

N 0

~/2J -2 -1

The equation that one solves is

where

dw at

pC f = ~/2YJ and

li/2Y

y.

[1,1] •

0 0 1/2 1/2_

-u

o

or (I-pCP1h ~ rn. where r ~[_~] and n i,

an arbitrary scalar function of t.

Note also these features: P is of full row

rank (rank P=3) so that ppcQ = Q and

Ppcf f for any Q and f; also the

matrix N = QPC is nonsingular, 3x3, while

M = pCQ is singular 4x4.

Our solution, in terms of N, is

w(t) (I + pcN-l(exp(Nt)-I)Q)W(O)

+ pc J:eXP(N(t-T))f(T)dT

It c -1 + 0 (I + P N (exp(N(t-T))-I)Q)rn(T)dT.

Since N in this example is a diagonal i­

zable matrix, the exponentials are

relatively easy to compute. The

arbitrary function n may be used to

minimize some performance index, if

desired.

References

1. Balabanian, N., and T.A. Bickart,

Electrical Network Theory (New York:

wiley) 1969.

2. Ben-Israel, A., and T.N.E. Greville,

Generalized Inverses, Theory and

Applications (New York: Wiley) 1974.

3. Brockett, R.W., Finite Dimensional

Linear Systems (New York: wiley) 1970.

4. Graybill, F.A., Introduction to

Matrices with Applications in

Statistics (Belmont, California:

Wadsworth) 1969.

5. Rao, C.R., and Mitra, S.K.,

Generalized Inverse of Matrices

and its Applications (New York:

Wiley) 1971.

121

6. Wolovich, W.A., Linear Multivariable

Systems (New York: Springer) 1974.

Victor Lovass-Nagy received B.S.

and M.S. degrees in Electrical Engineering

and the Ph.D. in Mathematics from the

University of Technical Sciences in

Budapest and later taught there in the

Faculty of Mathematics. After four years

as an engineer in the Ganz Electrical

Works, Budapest, and two years as Reader

in Engineering Mathematics at the

University of Khartoum, he came to

Clarkson College of Technology, where he

is now Professor of Mathematics. His

research interests are: theory and

applications of matrices, systems science,

network theory and control theory.

David L. Powers received B.S. and M.S. degrees in Mechanical Engineering

from Carnegie Institute of Technology

and Ph.D. in Mathematics from the

University of Pittsburgh. After two

years at Universidad Santa Maria in

Valparaiso Chile, he joined the

Department of Mathematics at Clarkson

College of Technology, where he now is an

Associate Professor. His research

interests include: control theory, matrix

theory and numerical analysis.

Cfl"I""'"""

, .' . ,

I

il;

,

AN EXPLICIT TREATMENT OF DILATION THEORY

by

P. Masani

University of Pittsburgh, Pittsburgh, Pa. 15260

Abstract

In this paper w-to-W* operator-valued positive definite

kernels K(··) are defined, where W is a Banach space, and the

Moore-Aronszajn Reproducing Kernel Theorem extended to such K(··).

Congruent Hilbertian varieties X(·) whose covariance kernel is

K(·o) are thereby obtained. The general notion of a propagator

or controller of X(o) is introduced, and necessary and sufficient

conditions established for its existence. It is shown that if

dilations are redefined in terms of isometries rather than pro­

jections (as seems more natural), the dilation R(o) of a given

operator-valued function R(o), is precisely the propagator of a

Hilbertian variety whose covariance kernel K(·o) is that ob­

tained from R(o) by the methods of Halmos and Nagy. Dilation

Theorems are thus rendered explicit, and their method of proof

routinizedo

122

Characterizations or operations derived

rrom network connections

"

Katsuyoshi NISHIO

Department of Information Enginee~ing, Faculty. of Engineering

Ibaraki University, Hitachi, Ibaraki, Japan

Tsuyoshi ANDO

Research Institute of Applied Electricity,

Hokkaido University, Sapporo, Japan

123

1. Introduction

Through study of electrical network connections, Anderson

and Duffin (2] introduced the concept of parallel ~ of two . Hermi tian semi -defini te matrices, an,d subsequently Anderson (1]

defined a matrix operation, called shorted operation to a subspace,

for each Hermi~ian semi-definite matrix. If A and Bare im-

pedance matrices of two resistive n-port networks then their para-

llel sum A : B is the impedance matrix of the parallel connection.

If ports are partitioned to a group of s ports and to the re-

maining group of n-s ports,then the shorted matrix AM to the

subspace M spanned by the former. group is the impedance matrix

of the network obtained by shorting the last n-s ports.

~ . Parallel addition and shorted operation can be defined on the

class of all bounded positive linear operators on a Hilbert space

and are of great interest from the point of view of operator theory. "

In fact, Anderson and Trapp (3J pushed through this program and

terconnections.

124

Our purpose in this paper is to give some characterizations,

of parallel addition and shorted operation among operations on the

c1a. ss of all positive ope'rators or. a Hilbert space. Theorem 1

will show that the series-parallel inequality aTl~ the transformer ...

inequality are, in some sense, char~cteristic for parallel addition.

In Theorem 2 shorted operation is recovered through c"~mutativity

wi th parallel addition, while Theorem 3 will charac,terize ,sho::ted

operation in terms of some inequalities of con~ve type. In the

final section we make some comment on range inclusion relations

associated with shorted operation and paxallel addition.

2. Preliminaries

In this paper we shall be concerned with (bounded linear)

operators on a complex Hilbert spaceH. The range of an. operator T

will be denoted by ran(T). A Her.itian operator A will be called .,

posi tive if the quadratic form (Ax.x)~ 0 for all x. H. For, two

Hermi t ian opera tors A and B, we denote A ~ B if A~ - B is

positive. The unique positive square root of a positive operat~r

'.

125

A will be denoted by A projection will always mean a

Hermitian projection. Convergence is understood as strong conver-

gence unless convergence in ~ is mentioned explicitly.

AM Given a closed subspace M of H, the shorted operator of a

"

positive operator A to M is defined as the maximum of all

posi tive aperators B such that B' A and ran(8)~ M. The ex-·

istence of such maximum is guaranteed by Anderson and Trapp Ca. Theorem 1) , but it was pointed out far earlier by Krein (7,

Theorem 1 J. The operation A ~ AM will be called the shorted

operation to M. Anderson and Trapp (3, Theorem 6J showed that

shorted operation admits the following variational description

(minimum power principle):

= inf (A(x+y),x+y) YEw'-

for

where M~ is the ortho-comp1ement of M.

x E. H,

Given two positive operators A and B, we obse~ve the

operator on direct ,sum H ® H with (operator) matrix (:

(1)

Now the parallel sum A : B is defined as the restriction of the

shortGd operator (A A,

A) . t6 the iJubspao. "H e {o} , A+B H era}

identified with H itself. Then we have by (1)

126

(A:Bx,x) = inft(AY,'y) + (Bz,z). x=y+z} for XCi~. (2)

The binary operation (A,B) r---+ A : B will be called parallel

addition. The following formula, derived from (2), is otten of use:

IA:B = (A- l + B-l)-l if A and B have bounded inverse. (3) -. We can see immediately trom definition that shorted operation

has the following properties (cf. [3] ):

(i) AM' A,

(H) (c(.A)M = 0(. AM

(Hi) (AM)M = AM'

for 01. > 0, , •

(iv) AM+BM,,(A+B)M'

Correspendingly, we can derive immediately fro~ (2) that parallel

addition satisfies the followino:conditions(cf. C3J):

(I) A:B = B:A,

(II) (A:B):C = A: (B:C),

(III) (o(.A):(c(.B) = cL(A:B) tor oC. > 0,

(IV) 1 A:A = '2A,

(V) A:B + c:o ~ (A+C~: (B+O), "

(VI) T*(A:B)T ~ T*AT:T*BT, for every operator T,

(VII) If An(resp. Bn) conY.~9 •• d.c~ ••• inoly to A (~ •• p. 8)

then so does A :B n n

127

to A:B.

, ,

Condition (V) means that the impedance of series-parallel

connection is greater than that of parallel-series connection.

T*AT is the impedance when a transformer is connected. Therefore

(VI) means that the impedance of parallel connection with trans-

"

former first is greater than that w~th transformer last •.

The important interconnections between parallel additi~n and

shorted operation were established by Anderson and Trapp (3,

Theorem 12)

A: c(. P converges to AM in norm as do ~ 00 , (4 )

where P is the projection to the subspace M. In particular,

since AH = A for all A by definition,

(VIII) A:o(. converges to.A in norm as tl ~ 00

An important consequence of (4) is the commutativity of parallel

addition and shorted operation:

J • (5 )

128

3. Parallel addition

In the preceding section we pointed out that parallel addition

satisries condition (I) to (VIII). Our main result in this section .. is to show that these conditions si~gle out parallel addition. To

this end, ~he rollowing Lemma on functional equation plays a basic

role.

Lemma (Bohnenblust (4, Theorem 4.1.) . .!::!.!! be .! function'

defined on [O,OO)X (0,00) with value in [0,00). If i satisfies

the rollowing conditions:

(A) ~(r{, r) = f( r ' ct ), .

(B) ~( ~ (c{ , r ), D) = !( d., i ( r ' -r) ) , (C) ~("rct , 'J'r) = 1!c('f) for .,.:> 0,

(0 ) ~(ol ,p ) ~ ¥(c{"r' ) I for 0( ~ oi. .!.!!2 r "(5' J

(E) ~(l,O) = 1

then either ~ has ~he form .,

~ ( d. , r ) = max ( 0( , ~ ) ror o(,p~o,

~ there is .! constant 0 < ~ <ac)such that

0;

129

In particular, if ~(l,l) = 2 in addition, then

for

Theorem 1. A binary operation (A,B) ~ AoB on the

. cl~ss of all positive operators ~ out to be parallel addition,

i.e. AOB ~ A:B for all positive operators A and B,!! and

only if it satisfies the following conditions:

J

(I) AOB = BOA,

(II) (AOB )oC = Ao (BoC) ,

(III) (otA)o(oC.B) = d(AoB) for d > 0,

(IV) 1

AoA = '2A,

(V) AoB + CoD ~(A+c)o(B+O),

(VI) T*(AoB)T~ T*AToT*BT for every operator T,

(VII)

(VIII)

If A (~p.. B ) converges decreasingly to n n

then so does A 0 B n n

A0eL converges to

to

A (~p. B)

Proof. Suppose that a binary operation (A,B) ~ AoB

satisfies conditions (I) to (VIII); Let us show that if a pro-

jection P commutes with A and. B then P commutes with AoB,

and more precisely

(6)

130

In fact, since commutativity implies

A = PAP + (l-P)A(l-P) and B =PBP + (l-P)B(l-P),

it follows from (V) and (VI) that

AoB > PAPoPBP + (l-P)A(l-P)o(l-P)B(l-P)

~ P(A.B)P + (l-P)(AOB)p..-~).

Therefore ~he operator

C = AoB - P(A.B)P - (l-P)(AoB)(l-P)

is positive with

pcp = (l-P)C(l-P) = o.

Then by positivity of C we conclude that

CP = C(l-P) = 0 hence C = 0,

which ovbiously implies the expec~ed commutativity of P and AoB

Since scalars commute with all projections, it follows from

the above that for each pair 01., r the operator o(.p cOlIIIDutes

~i th all projections, consequently oeop . must be a scalar. we

consider the scalar function i of two positive variables,

defined by

for 0( , ~ > 0, (7)

which has the meaning because by· (IV) and (V)

-1 -1 1 -1-1 ct 0 P ~ 2m i n (ol , ~ ) > o.

131

Now it follows from (1),(11),(111) and (V) that

(A) ~ ( ol , r ) = ~ (r ,c(. ),

(B ) ~ ( ~ (cl 'r ), 'f) = I (a( , ~ ( ~ , l' » , (C) ~(t'ol, Q~) = l' ~(~ ,~ ) -for 1" ~ 0,

(D) ~ ( c( '~ ) ~ ~ ( cI. " r") '. for 0(, 0(' and r ~ pl. On the basis of (A) and (D) we can extend q; over [0, ot) )( [0, r ) by the formula

~ (0<. ,0) = ~ (0, c{ )~lim I (0( ,(3 ) f -) 0 ,

and

f(O,O) = o.

The extended function satisfies also conditions (A) and (D), and

(E) 2(1,0) = I,

because by (VIII)

!(l,O) = lim ~(,l/n) = lim(l.n)-l = 1. n-)_ n+ao

'; Now since ~ satisfies conditions (A) to (E) and ~(1,1) = 2

by (IV), it follow,s from Lemma of Bohnenblust that

for ,~ 'r ~ O. (8)

Since by (3)

for 0( , r > 0,

(7) and (8) implies

132

and consequently by continuity

for «,~~ o. (9)

For a positive operator A, in view of the spectral theorem '.

(cf. (a, § 107]), there exists a sequence {An 1 such that AI> A2 > ... and An Converges to A and such that each An has the form

N A = Lei .p .

n i=l n,1 n,1

where ol . ~ 0 (i=1,2, ... , N=N(n» and P • are projections n,1 n,1 N

such that p .' p . = 0 n,1 n,] (iFj) and ,EP . = l. i=i n,1

commutes with each p . n,1 we have by (6) and (9)

N N A o 1=.L.A P .OP .=2:.0(. .P •• P.

n i=l n n,1 n,1 i=l n,1 n,1 ,n,1

N N =.2:(0{ .el1)P . =2:.(oC .:l)P . ,

i=l n.1 n,1 i=l n,1 n,1 • J

and correspondingly N

A : 1 = 2:.( 0£. . : l)P . n . i=l n,1 n,1

hence

A .1 = A : 1 n n (n=1,2; ... ) .

Now it follows from (10) and (VII) that

A-I = lim A .1 n-too n

= lim A :1 = A:l. n-t_ n

133

Since

(10)

(11)

';.11,' I,

!II, ' ;i:

To get the final goal, we re~tk that ;.r a po~ i Uve °PQr·ator

C has bounded inverse then

C(AeS)C = (CAC).«('.C). ( l~)

In fact, (VI) implies

C (AeS)C .:EO (CAC)e (CSC) ,

and

Now for atbi trary positive operators A and B, since (B +.~ ~ has bounded inverse for each n')o, we have by (11),(12) and (VII)

AoS = lim Ao(S+l/n) n~""

= lim(S+1/n)~{(S+l/n)-~A(B+1/n)-~.11 (B+1/n)~ ..... n~oo

= lim A n~GII

(S+l/n) = A

This comp1ietes the proof.

B.

Remark, In the proof of Theorem 1 conditions (I), (II), (III),' "

(IV) and (VIII) are used only for scalars, and condition (VI)

only for positive T.

4. Shorted operation

In § ~ parallel addition is defined in terms of shorted

operation, and the latter is recaptured from the former by (4).

We shall first characterize sport~d operation through commutativity

with parallel addition.

Theorem 2. An operation JC(.) £!! the class of· posi tive

operators coincides with shorted operation to ~ closed subspace

M, Le. ~(A) = AM for every positive operator A,·g ~ only

i! it satisfies the following conditions:

• J

(a)

(b)

x( d. A) = OC> :n:.(A)

X(A:B) =:n.(A):B = A: X(B).

Proof. In' 2 we pointed out that shorted operation satisfies

(a) and (b).

Suppose that an operation ~(.) satisfies (a) and (b).

Then we have

1: n :rt ( 1) = 1: :It ( n) = :n: (1: n) = X( 1 ) : n. (13)

Since by (VI~I) the sequence ;e(l):n converges to ~(1) in

norm as n ~oo l:n:7t(l) converges increasingly to 'to(l).

135

, I , 1

, ' 'I I I

I :i!i:'

, '1:'1',:1'",: : 'I, Iii

In view of the spect ra1 theorem (cf. (8, i 107) ) :7C( l) admits

spectral representation 00

"(1) = ~ )"dP(A.) o

where P().) are projections such that for A ~ ,...,

lim P(r) = P()oJ r -+ ~+

and lim P(~) -- 1. ~7OO

Tben we have by the spectral

representation

j[(1) ~ ~ (l-P()..» for A. > O. (14)

It follows from (4), (13) 8fia (14) that

J[;(1) = lim l:nJt(l) ~ lim l:n:l(l-P().» = 1-P(l,..) n~oo n+oo

and consequently

X(l) ~ 1 - P(O). (15 )

On the other hand, since ran(~(l» is contained in M = ran(l-P(O»,

we have

)&(1) ~ 1\'3r(1) 1I·(l-p(O». (16)

Now for each positive operator A it follows from (a) and (b)

conbined with (4), (15) and (16) that

3C(A) = lim :Jt(A):n = 11m A::nX(1) n~oo n-too

~ lim A: n( l-P(O» = A n+oo M

• J

and similarly

x,(A) ~ lim A:n UX(l)ft(l-P(O» = AM' n-t ..

Therefore the operation" coincides with the shorted operation

to M. '.

The following theorem correspoods to Theore. 1 tor parallel

addition.

Theorem ·3. An operation x( .) 2.!! the cU~.s of all positive

operators ~ out to be ~ shorted operation ~ ~ closed

subspace M, i.e. ~(A) = AM ~ every positive operator A,

if and only if it satisfies the -following ,conditions:

{i) x(A) ~ A,

(ii) X(c( A) = «.;,t(A) f-or 0,

(iii) X(x.(A» = 'teA),

(iv) X(A) -: X(B) ~ X(A+B),

(v) 't (A+ 't(B)C 'l:.(B» ~ :n.: (A) + X(B)C le(B),

(vi) 't(A2) ~ lC(A)2.

Proof. In § 2 we pointed out that the shorted operation to

a closed subspace M sa-tisfies (i) to (iv). Also (v) tollows

137

! I:

:",1 I '"

1','

11": ,

ill"

II il':I.1

immediately from (1) because 8 My = 0 for OVC!r), V E MJ. To

see (vi), let

A = A + lin and 8 = P + lin n n

where P is the projection to M. Sinc" A n and 8n have

bounded inverse, it follow£! fr~m (3) th.t fat

-2 (A -1+ m -18 .')2 (An:~8n) = n n

Since order relation between two positive operators i. reversed

by forming respective inverses, we have

(1 ) -l(A 2 mB 2) m +m n: n

8y (VII) the sequences A 2: mB 2. and n n converge to

2 2 A : mP and (A:mP) respectively as n --t 00 , hence

-1 2 2 m (1 +.) (A: mP) ~ (A: mP) . (17)

8y (4) the left side of (17) converges to (A2)M while the right

side does to T~is proves (vi) for shorted operation.

Suppose conversely that an operation ~(.) satisfies conditions

(i) to (vi). It follows from (i) and (vi) that

o~ X(l) ~ 1 and . 2 2

~(l) = X(l) ~ XCI) ,

which implies immediately that 'tel) is a projection. Let M

138

denote the range of 'tel). Now in view of (ii) we have only to

show that

for 0' A ~ 1.

Let O~A' 1. Then since lelA) ~ :eel) by (iv), ran(JC(A» .. is contained in M, so that we conclude from (i) and the definition

of shorted operation that

(10)

Then it follows from (10), (iii) and (iv) that

:1C(AM- X(A» + X(A)

.. X(AM-X(A» ... X(X(A» ~ "CAM) ~ aCA)

hence

" (11)

Now since (10) implies

'Japplying (vi) we have by (11)

~(l) = X[(AM- X(A» + [:IC(l)

"

hence

This together with (10) conchdes' the proof. '.,

139

5. Range inclusion

Remarkable properties of parallel addition and shorted operation

from the point of view of range inclusion have been discussed by

Anderson and Duffin (2) and Anderson (11 in finite dimensional

• J

case and 'by Fillmore and William (6) and Anderson and Trapp [31

in general case: for positive operators A, B and a closed sub-·~

space M

(cl )

I (cl )

(,~ )

( ~' )

ran(AM~) = ran(A ~)n M,' (3, Theorem 1]

ran(AM) 2 ran(A )("\M,

ran«A:B)~) = ran(A~)(\ran(B~),

ran(A:B) "2 ran(A)" ran(B).

t 3, Theorem 11]

On the other hand, we showed in the course of the proof of Theorem 3

We remark that (~") (resp. (rU» can be cosidered to give quanti-

, I tative expression to (c(.) (resp. (~». In fact, for instance,

by (0£) applied to A2, relation "(ct.') is equivalent to

ran(AM) :2ran( (A2)M~);

which says, by a result of O.ouglas [5 J , that ·there is • constant

1'> 0 such that

we can take

140

Referenc ••

'1 W.N. Anderson, Jr., Shorted operators, SIAM J. Appl. Math. 20(1971), 520~525.

. 2 W.N. Anderson, Jr. and R.J. Duffin, Series and parallel

addition of matrice., J.'Math. Anal. Ap,l. 26(1969), ,576-594.

3 W.N. Anderson, Jr. and G.B. Trapp, Shorted operator. II, (preprint) •

4 F. Bohnenblust, On axiomatic characterization of Lp space, Duke Math. J, 6(1940), 627-640.

5 R.S. Douglas,

" inclusion, On majorization, factorization and range

Proc, Amer. Math. Soc. 17(1966), 413-416.

6 P.A. Fillmore and J,P, Willia •• , On operator ranges, Advances in Math. 7(1971), 254-281. \

7 M.G. Krein, Theory of selfadjoint extensions of ... i-1 bounded operators and its application, Mat. Sb.

20(62)(1947), 431-495 (Russian).

8 F. Riesz and B'. Sz. -Nagy, Functional ana1y.ia, Ungar, New York, 1955.

"

141

Iii! I I

'iii ~ II

'II

A FUNCTIONAL ANALYSIS APPROACH TO

MINIMUM SENSITIVITY CONTROL DESIGN

J. Gary Reid United States Air Force Avionics Laboratory

Wright Patterson AFB, Ohio 45433

Abstract

This paper treats the problem of open-loop minimum sensitivity control design from a gradient iteration method of solution rather than from previously well­known Riccati equation techniques. By using standard methods from functional analys~s~ the quadratic cost functional of the generally high dimension sensit~v~ty system is transformed into a low-dimension minimum norm problem on the control space. A recently obtained matrix-operator form of the para­meter sensitivities in linear time-invariant ordinary differential equation systems then enables one to compute the gradient function with a very small number of integrals. Finally, extensions are suggested to more general linear systems defined on an arbitrary Hilbert space, and some of the compu­tational considerations of computing the gradient are discussed.

1. INTRODUCTION

A classic problem in control theory is the

design of control laws which are "forgiving" of

one's inaccurate knowledge of system parameter

values. One method to approach this problem is

to treat the nominal trajectory parameter sen­

sitivities as additional "states" of the system,

and then use the formalism of the Maximum Prin­

ciple (Pontryagin, et al (7) to determine the

optimal control which minimizes a cost func­

tional of both the natural system states and

the sensitivity "states". (See, eg, Guardabassi,

et al [I), Holtzman and Horing (3), Kahne (4).

If the system is described by a set of

linear ordinary differential equations, then it

is well-known that the parameter sensitivities

also satisfy a linear set of differential equa­

tions termed the "sensitivity system". There­

fore, if the sensitivity cost functional is

selected to be quadratic in the control, system

states, and sensitivity "states", then minimum

sensitivity open-loop control may be determined

explicitly via solution of a matrix Riccati

differential equation (eg, Kahne (4). We

142

emphasize that the control law so obtained

is merely open-loop, and it cannot be made

closed-loop (as is the case when the quadratic

cost functional does not contain sensitivity

constraints) except by approximation (See, eg,

Lamont and Kahne (5). This is a well-known

dilemma (eg, Deyst and Price (9), and basically

stems from the fact that the optimal control is

computed as a linear combination of the nominal

trajectory parameter sensitivities which, in

turn, are computed under the assumption that

the control is open-loop. Since the parameter

sensitivities are not "physical" variables of

the system, but are merely mathematical opera­

tors (partial derivatives) which are dependent

upon the mathematical nominal trajectory to

compute their values, they lose their meaning

in an on-line, feedback control law.

One problem of such Riccati equation

approaches is often the extremely large number

of differential equations to be solved. If n

is the state dimension and p is the parameter

dimension, then there are potentially n(p+l)

"states" in the sensitivity system and the

symmetric Riccati matrix differential equa­

tion has dimension n(p+l) x n(p+l). If the

system is time-invariant, then this number of

differential equations may be reduced sub­

stantially by using low-order sensitivity models

(eg, Wilkie and Perkins [13]) or taking con­

trollability properties into account (eg, Gupta

and Mehra [2] or Reid, et al [10] [11]), but

regardless there still can be a great amount of

computation if either the state or parameter

dimensions are high.

This paper treats this same open-loop,

linear system, minimum sensitivity control pro­

blem, but it approaches the solution in a funda­

mentally different way. Using well-known

methods of functional analysis, the quadratic

cost functional is transformed into a minimum

norm on the space of controls. (See, eg, Porter

[8]). This problem may then be solved by

standard gradient iteration techniques. For

simplicity, the case of linear time invariant

ordinary differential equation systems is

treated first. Using recently reported (Reid,

et al [10] [11]) algebraic representations of

the sensitivities in such systems, a highly

efficient method of computing the gradient

function is thereby obtained. Following this

development, extensions to more general linear

systems described by linear operators on a

Hilbert space are discussed.

2. PROBLEM FORMULATION

We consider the n-dimensional time­

invariant system

~(t) • A(v)x(t) + B(v)u(t) x(O)mxo

with observable output

yet) - C(v)x(t) yet) £ Rm

(1)

(2)

where v is a constant, unknown parameter vector

which parametrizes A, B, and C. The nominal

value of v £ RP is designated Vo and all system

quantities will henceforth be evaluated at vo;

therefore, the explicit reference to v will be

143

deleted from the notation. The components of

v are designated Vi' i • 1, 2, ••• p, and

partial derivatives with respect to Vi are

designated with a subscript "(i)".

For a given u £ L2 (0, t f ; Rr) and a given

Xo £ Rn

, the nominal output of the system (1)

is uniquely specified by

yet) - T(t)xo + Wet) u (3)

where

T(t) :: CeAt t Wet) ::! CeA(t-s)B(o) ds (4)

o Then the output parameter sensitivities are

given by

z(i)(t) :: Y(i}(t) - T(i)(t)xo + W(i) (t)u (5)

where it is fairly easy to show that [10]

(6)

The computation of these partial derivatives will

be discussed in the next section.

Now it is assumed that we wish to minimize

the quadratic cost functional

tf

J(u) - <Y(t f ), SfY(tf » + ! <yet). S(t)Y(t»dt o

tf (7)

+ ! <u(t}, u(t» dt o

where we define the m(p+l) dimensioned augmented

vector

yet) T(t) Wet)

z (1) (t) T(l)(t) W(l)(t)

yet):: x + u 0

z(p)(t) T (p) (t) W(p)(t)

:: T(t)x + W(t)u (8) 0

and it is assumed that Sf and Set) are non­

negative and symmetric. This is a very gereral

sensitivity cost functional as it weights the

parameter sensitivities at both the terminal

time as well as along the nominal trajectory.

and various modifications of this cost func-

tiona1 have been used by a number of previous

researchers (eg, [1] [3] [4]).

The problem, then, is to calculate the

minimum sensitivity, open-loop control law

which minimizes the functional (7). As dis­

cussed in Section 1. this problem may be solved

explicitly via a Riccati equation technique;

however, we will take an alternate approach by

using standard tools of functional analysis to

transform the cost functional (7) into the form

tf

J(u) k + 2 J <h(t), u(t»dt o

where

tf

+ J <u(t), P(t)u>dt o

k = <Y z • i • (t f ), Sf Yz • i . (tf »

tf

+ J <Yz • i . (t), S(t)Yz . i • (t»dt o

-* h(t) = W (t f - t) Sf Yz •i . (t f )

tf -* + J W (s-t) S(s) Yz • i . (s) ds

o

r.* -P(t)u - u(t) + w (tf-t)SfW(tf)u

(9)

(10)

(11)

tf

+ J w* (s-t)S (s)W(s)u ds (12) t

Y i (t) = T(t)x z. . 0 (14)

Since Sf and S(·) are assumed positive, the

optimal control exists and is uniquely specified

by

u * (t).. _P- 1(t)h (15)

However, it is not practical to invert P(t)

directly except by Riccati equation techniques

and so iterative method of solution is appro­

priate.

The gradient of J(u) is easily shown to be

VJ(t; u) = 2(h(t) + P(t)u) (16)

and so a gradient or conjugate gradient

algorithm may be conveniently utilized (eg,

Luenberger [6]). The key to such an approach

is, of course, obtaining an efficient means

to compute the gradient function. This is

the topic discussed in the next section.

3. COMPUTATION OF THE GRADIENT

Recently Reid, et a1 [10] [11] have shown

that the zero-input and zero-state response

of the system (8) may be put into the form

Yz • i • (t) = T(t)xo = F f(t) (17)

t Y (t) z.s.

W(t)u = J H f(t-s)u(s) ds (18) o

where f(') is

with elements

a 2n-dimenslona1 vector function j A t of the form t e k where Ak ,

144

k = 1, 2, .•• q, are the destinct eigenvalues of

A (which, if complex, would also introduce

factors of sines and cosines), and j = 0, 1,

.•. 2nk

where nk

is the eigenvalue multi­

plicity of Ak in the characteristic polynomial

of A. The m(p+1) x 2n and m(p+1) x 2nr

dimensional matrices F and H are obtained

directly from the quantities (CAxo)' (CAxo) (i)'

(CAB), CAB) (i)' i = 1, 2, ••. p, and by the in­

version of a 2n x 2n dimeri-s'ioned Vandermonde

matrix. (See Appendix)

Now if we assume, for simp1icit!, a single

control input (ie, r=l) then we may, put h(t) and

P(t)u, equations (10) - (11), into t.he form

T T h(t) = f (t f - t)H Sf Yz.~.(tf)

(19) o

tf T T·

+ J f (s-t)H S (s)Y (s) ds z.s.

(2G)

t

where

F f(t) (21)

t Y (t) = H J f(t-s)u(s)ds z.s.

(22) o

For more than one control input, th~ computa­

tions are mere1yrepea,ted Jor each iIl-put and

corresponding column vec~or of B. A~so note

that the indicated 2n convolutions in equations

(18), (19) and (2l) may be transformed into an

equal number of quadrature integrals through

use of relations such as

~(t-s) \t -~s ~t -A s (t-s}e z tee - e se k (23)

Therefore, in examining equation (18) -

(2l) we see that the gradient function

VJ(t; u) - 2(h(t} + P(t)u} (15)

may be computed with only 2nr quadrature inte­

grals if S(o} = 0 (terminal sensitivity only)

or 4nr quadrature if S(o} ; 0 (trajectory

sensitivity). Thus the computations for each

iteration of a gradient minimization algorithm

are quite low and increase only linearly with

the state dimension, n, and are independent of

the parameter dimension, p. This is in contrast

to the Riccati equation approach in which, at

worst case, the computations may increase with

the square of nand p. Since the cost func­

tional is quadratic, we are assured that either

a gradient or conjugate gradient algorithm will

converge ~o the unique minimizing control (eg,

Luenberger [6]). Additionally, if the system

(I) is output controllable and if it is desired

to meet the terminal constraint, y(tf } • Yf'

exactly, then the gradient projection .ehtod

of Rosen [12] may be conveniently utilized

to calculate the minimizing control.

4. EXTENSIONS TO GENERAL LINEAR SYSTEMS

The problem formulation of Section 2. is

easily extended to the more general linear

system with output given by the operator equa­

tion

yet} a T(t; v)xo + Wet; v}u (24)

yet} £ Rm

v £ RP u £ U Xo £ X

where the control space U and state space X are

assumed to be arbitrary real Hilbert spaces, and

the zero-input and the causal zero-state system

operators, T(t; 8) and Wet; 8), are each assumed

to be continuous in t and continuously differ-

145

entiable with respect each vi at the nominal

Vo £ RP• Then the output parameter sensitivities

are once again given by the operator equations

(25)

and the cost functional J(u), equation (7), may

be for.ed in the same manner as before. Defining

the augaented operators T(t) and Wet) in a -* siai1ar manner to Section 2. and letting W (t f )

denote the adjoint operator of W(t f } £

II (p+l) -* Lc (U, R ) and W denote the adjoint operator

of W{o} £ Lc(U, L2(0, tf

; Rm(P+l»), then we may

once again transform the cost functional J(u)

into the form

J{u} - k + 2 <u, h> + <u, Pu>

wbere

(26)

(27)

(28)

sod k and Yz•i • (t) are defined as in Section 2.

Rote that each term of the above expressions

correlate with the corresponding terms for h(t)

aDd P{t}u in equations (18) - (19).

The gradient of J(u) is once again given

by VJ(u} - 2(h + Pu), and so a gradient or con­

jugate gradient algoritm. .ay be utilized in

co.puting the optimal control. However, once

again the key to such an approach is how easily - ~. it is to compute W(o)u and the adjoints w (t f ) -* aDd V. For linear ti.e-invariant ordinary

differential equation systellS we saw thdt tbese

quantities could be co.puted quite efficiently;

the pri.ary reason for this ease of coaputation

st~ frca the fact that 801M! nor.ally very

tt.e coosuaing convolutions in coaputing W{ o)u -" aod V could be transfotw!d into the far .ore

coavenient quadrature integral fora. However.

for geoeral linear systeas tbis conversion of

the convolutions would generally not be feasible.

Beoce. it is a significant observation that only ... the -trajectory sensitivity" adjoint, W ,

~l"es a convolution, while inherently the -* -teraiDa1 sensitivity" adjoint. W (tf ) does DOt

(see equation (18) - (19». If one only visbea

to weight the sensitivities at the terminal time

t f and not along the entire nominal trajectory,

then this gradient method of solution may still

be a computationally practical method of solu­

tion. It is also of interest to note that this

substantial difference in computational burden

bct'Jeen the "terminal" and "trajectory" cost

functional problems is not a feature of the

Riccati equation type approaches, for there the

number of differential equations remains in­

variant regardless of such a change in the cost

functional. Therefore, in the terminal sensi­

tivity problem, particularly, the gradient type

approach might be an attractive alternative to

such explicit methods of solution.

5. EXAMPLE

To illustrate the theory of Section 3. we

consider the simple second order linear system

d dt

[lOJ ~l (tJ _ 61 (tJ o (t) - x (t) t E: [0,1]

2 2 (30)

with nominal parameter vector v = [-2 -3 l]T. o

Then the nominal eigenvalues are Al = -1 and

A2 = -2. From the Appendix the Vandermonde

matrix V and its inverse are

-1

1

-2

1

1

-2

4

-4

-~l V-I = [_1: -8 - 9

12 - 2

Then it is fairly easy to show that

F

2 o -2 0

-3 2

5 -2

4 -2

-6 2

o o

o o

-1

2

3

-5

-4

6

o o

o o

21

-2

-2

4

o o

H

1

-1

-2

3

3

-4

1

-1

4

8

5

1

o o 1

-1

-1

1

o o

5

12

9

2

-1

2

2

-3

-3

4

(31)

o o 1

-2

-2

4

-1 0

2 0 (32)

146

and

f(t) = [e-t te-t e-2t te-2t ]T

Then for the sensitivity cost functional

+ 1 2 J u (t)dt o

the gradient function

1 J byT(t)Y(t)dt o

VJ(t; u) = 2(h(t) + P(t)u)

(33)

(34)

(35)

may be computed from expressions (19) - (22)

where the matrices Sf and S are replaced by the

scalars a and b, respectively. For each new

guess of the control there are four integrals

required to compute Y (0) and four to com-z.s.

pute the third term of P(o)u (the trajectory

sensitivity term).

On the other hand, since there are two

system states and three parameters there are a

total of eight "states" in the sensitivity

system. It may be shown from controllability

considerations that the complete sensitivity

system for this example may be generated with

only six differential equations [10], but

regardless, the Riccati equation method of sol­

ution would require solution of either an 8x8

or 6x6 nonlinear matrix Riccati equation.

Using a steepest descent algorithm

(Luenberger [6]) and the initial guess uo(t)

-h(t), the gradient method of solution was

mechanized on the CDC 6600 digital computer.

The results for two different sets of

weighting constants, a and b, are shown in

Tables 1 and 2. We see that in either case the

gradient method converges quite rapidly.

ITERATION

START

1

2

COST

.717

.715

.715

NORM2 GRADIENT

.180

. 385xlO-3

.1l0xlO-5

Table 1. a = 1, b = .1

ITERATION COST NORM2 GRADIENT

START --4

.108x10 • 180x104

1 • 632xl03 • 413x102

2 • 612x103 • 338x10

3 • 61lx103 • 777xlO -1

4 • 61lxl03 • 642x10-2

Table 2. a ~ 10, b = 1

6. SUMMARY

By using standard methods of functional

analysis, a gradient method of solution has been

developed as a computational alternative to

Riccati equation approaches to the minimum sen­

sitivity control problem in linear systems. For

an n-dimensional linear time-invariant ordinary

differential equation system with r control in­

puts, a recently obtained algebraic representa­

tion of the parameter sensitivities [10] [11]

allows us to compute the gradient function with

merely 2nr or 4nr quadrature integrals for the

terminal or trajectory sensitivity problems,

respectively. This method of solution may then

be a practical alternative to Riccati equation

techniques where the computations increase at

a rate greater than the square of the state

dimension n.

Finally, extensions have been suggested for

more general linear systems, and it was observed

that the computations of the "terminal" cost

functional problem are inherently far less than

the "trajectory" cost functional problem due to

the elimination of time-consuming convolutions.

APPENDIX

In this appendix we briefly describe how

the matrices F and H used in equations (20) and

(21) are computed. Far more details and a

derivation of this matrix-operator form for the

parameter sensitivities may be found in [10]

[11] •

For simplicity. assume that the nominal A

matrix has des tinct real eigenvalues, Xk ,

k - 1, 2, ••• n, and assume that there is only

one control input so that B is a column vector •

Define the 2nx2n generalized Vandermonde matrix

147

1 Xl

0 1

1 X2 V -

o 1

).2 1

2Xl X2

2

2A n

X2n- 1 1

(2n-l}Xin- 2

).2n-l 2

(2n_1}X2n- 2 n

(A-I)

and the m(p+1)x2n dimensioned matrices

E -

G

Then

and

CB

(CB) (1)

(CB)(p)

CAx o

H _ GV-1

2n-1 CA x o

(CA2n- l x ) o (D)

(A-2)

CA2n- l B

(A-3

(A-4)

The extension to the case for multi-inputs,

non-destinct and complex eigenvalues is easily

made. (See [10] for details).

BIBLIOGRAPHY

1. Guardabassi, G., A. Locatelli, and S.

Rinaldi, "On the Optimization of Continuous

Linear Systems with Sensitivity Constraints."

Preprints Second IFAC Symposium on System Sensi­

tivity and Adaptivity. Dubrovnik, Yugoslavia,

August 1968.

2. Gupta, N. K. and R. K. Mehra, "Computational

Aspects of Maximum Likelihood Estimation and

Reduction in Sensitivity Function Calculations."

IEEE Trans on Automatic Control, AC-19: 774-783,

Dec. 1974.

3. Holtzman, J. M. and S. Horing, "The Sensi­

tivity of Terminal Conditions of Optimal Control

Systems to Parameter Variations," IEEE Trans on

Automatic Control, AC-10: 420-426, Oct 1965.

4. Kahne, S., "Low-Sensitivity Design of Opti­

mal Linear Control Systems," IEEE Trans on Aero­

space and Elect. Systems, AES-4: 374-397, May

1968.

5. Lamont, G. and S. Kahne, "Comparison of

Sensitivity Improvement Techniques for Linear

Optimal Control Systems," IEEE Trans. Aerospace

and Elect. Systems, AES-5: 142-151, March 1969.

6. Luenberger, D. G. Optimization by Vector

Space Methods, New York: John Wiley & Sons, Inc.

1969.

7. Pontryagin, L. S., V. G. Bo1tyznsku, R. V.

Gamkre1idze, and E. F. Mischenka, (trans by K. N.

Trirogoff, ed by L. W. Neustadt) The Mathematical

Theory of Optimal Processes, New York: John

Wiley, 1962.

8. Porter, W. A., Modern Foundation of Systems

Engineering, New York: Macmillan, 1966.

9. Price, C. and J. Deyst, "A Method for Ob­

taining Desired Sensitivity Characteristics

with Optimal Controls", Proceedings Joint

Automatic Control Conference, University of

Michigan, 1968.

10. Reid, J. G., "Sensitivity Operators and

Associated System Concepts for Linear Dynamic

Systems", PhD Dissertation, Air Force Institute

of Technology, 1975.

148

11. Reid, J. G., P. S. Maybeck, R. B. Asher, and

J. D. Dillow, ,"An Algebraic Representation of

Parameter Sensitivities in Linear Time-Invariant

Systems," Journal of the Franklin Institute,

January 1976.

12. Rosen, J., ·"The Gradient Proj ection Method

of Nonlinear Programming, Part I, Linear Con­

straints", J. S·oc. Industrial Applied Math.

8: 181-217, 1960.

13. Wilkie, D. F. and W.R. Perkins, "Genera­

tion of Sensitivity Functions for Linear Systems

Using Low-Order Models", IEEE Transactions on

Automatic Control.

BIOGRAPHY

J. Gary Reid was born in Newark, New Jersey,

on 12 Novermber 1945. In 1967 he re~~ived a n:s. in Aeronautics from the United Stat~~'Air Force

Academy and was commissioned in the Air Force.

In 1968 he received an S.M. from M. 1. T. in Aero­

nautics and Astronautics, and he is currently"

working to complete a Ph.D. degree at the Air

Force Institute of Technology with a'specia1ity'

in estimation and control theory. :'

Captain Reid is a student member,of the IEEE

societies on automatic control, computers, and , .

aerospace and e1ectr?nic systems. His current

research interests include linear system theory,

identification, adaptive nonlinear estimation,

and pattern recognition.

PASSIVITY AND LP-STABILITY OF SOME NONLINEAR EVOLUTION EQUATIONS

Dinu Wexler, with the Department of Mathematics, Facultes

Universitaires N.D. de la Paix, Namur, Belgium.

Abstract One states some results on passivity and LP-stability of the input-output ope­rator associated with nonlinear evolution equations involving a monotone opera­tor.

1. INTRODUCTION

We discuss passivity and LP-stability as defined in Systems Theory (see for instance (3), (5)) for the input-output operator associated with the dif­ferential equation

du dt + Au :3 f, (1)

where A is a maximal monotone operator (possibly nonlinear, unbounded and multivalued) of a real Hilbert space H. A significant class of ordina­ry and partial differential equations may be written in form (1). The recently developed ge­neral theory of these equations is closely rela­ted to nonlinear contraction semi groups. From this point of view it represents a nonlinear ver­sion of the well-known Hille Yosida Phillips theory. It is also related to variational ine­qualities for partial differential equations. A systematic exposition may be found in the mo­nograph of H. Brezis (2).

Recall first some basic definitions and results in this theory. A (multivalued) operator A H",CP(H) with domain

D(A) = {x E H : Ax F 0}, is said to be maximal monotone if : (i) A is monotone, i.e. (x1-x2'Y1-Y2) ;> 0, Vxl'x2 E D(A), Y1 E A xl'

Y2 E A x2;

and (ii) A does not possess proper monotone exten­sions. An important class of maximal monotone ope­rators consists of subdifferentials of convex lo­wer semi continuous functions ~ : H~) - 00, + 00),

~ =! + 00.

Recall the following existence, uniqueness and re­gularity results: let f E L~oc(to' + 00, H). Then for any Uo E ~ there exists a unique weak solu­ti on on [ to' + oo[ of the Cauchy problem

~ + Au :3 f, u(O) = uo. (2)

This solution is a strong one whenever (i) A is a subdifferential and f E L~oc(tot + 00, H); or (ii) Uo E D(A) and fEB Vloc (! to' + 00[, H). The solution u depends continuously on Uo and f in the following sense: if u and u are the solutions of

(2) and

149

~ + Au :3 f, u(O) = Uo (3)

respectively, then

lu(t) - "(t)1 ~ luo - "01 + )t If - fl do, 't ~ to' to

In this setting one may consider various problems for systems with input f and output u. We state here our main results on passivity, strong passi­vity and LP-stability. Proofs will be published

: I !

elsewhere.

2. PASSIVITY

Assume 0 E AO. One says that eq. (1) is passive if to any f E L;(R, H) (the causal extension of L2(R, H)) one can assign a weak solution u of (1) on R such that u E L;(R, H) and such that

J:.(f - i, u - u)db > 0, Vt E R,

for any f, f E L;(R, H) and any assigned solu­tions u, U respectively.

We first state the properties which are implied by passivity: when eq. (1) is passive, then the solution u assigned to f is unique, limt~_oo u(t) o and the operator f~ u is causal and commutes with any translation. Moreover, the operator f~ u possesses the following continuity proper­ty : if un and u are the solutions assigned to fn and f respectively and if f ~ f in L2(_ 00, T,

2 n H), then un ~ u weakly in L (- 00, T, H) and strongly in Lr (_ 00, T, H), for any rE12, +001 (hence uniformly onl - 00, TI).

We state a sufficient passivity condition : assume o E AO; if there exists a > 0 and p > 0 such that

(x, y) ~ alxl 2, V x E D(A), Ixl < p, y E Ax,

then eq. (1) is passive. Some nonlinear diffusion equation satisfy the above conditions. Another simple example is furnished by the subdifferential of the norm of H .(so that Ax = Ixl-1 x, if x f 0

and AO is the closed unit ball of H with center 0).

3. STRONG PASSIVITY

One says that eq. (1) is a-strongly passive (a > 0) if it is passive and if for any f and f in L;(R, H)) the assigned solutions u and u respec­tively satisfy

J:;: - i, u - u)db >" J:~u - ul' d , V t E R.

The maximal monotone operator A is said to be a-strongly monotone if A - a I is still monotone (I being the identity on H). Recall that for any A > 0, the (nonlinear) resolvent JA = (I + A A)-l

of A is a single-valued operator defined on the whole of H. We now state our main result.

Theorem 1 : Assume 0 E AO and let a > O. The fol­lowing three conditions are equivalent: (i) eq. (1) is a-strongly passive; (ii) A is a-strongly monotone; and (iii) the resolvent of A satisfies

(JA Y1 - JA Y2' Y1 - Y2) ~ A a IJA Yl - JA Y212,

V Y1' Y2 in H and A > O.

This extends to our framework a result established previously for linear evolution equations by Bel­trami and Buianouckas [11 by arguments which use in an essential way the linearity. The proof we give is based on certain results in the theory of nonlinear contraction semi groups.

Note that when A is a-strongly monotone, the ope­rator fl~ u possesses the following Lipschitz pro-

• 2 perty : for any f, f in Le(R, H), the assigned so-lutions u and u respectively satisfy

IIXt(u-u) II 2 .0; a-I II~t(f-f) II 2 ' Vt E R, L (R,H) L (R,H)

where;tt is the characteristic function of 1- 00, tl·

4. LP-STABILITY

Some results on LP-stability with p E [1, + oo[ may be obtained by using the ideas we applied to dis­cuss passivity. We mention here only a result on LOO-stability. One says that eq. (1) is LOO-stable if for any [uo' fl in ~ x Loo(R+, H) the weak solution u of (2) on R+ belongs to L (R+, H). The maximal monotone operator A is said to be coercive

150

if there exists Xo E H such that

(x - xo' y) Ixl =+00. lim

Ixl~ yEAx

Theorem 2 : If A is coercive, then eq. (1) is Loo

_ stable and the operator [uo' fll+ u from D(A) x Loo(R+, H) into ~(R+, H) carries bounded sets into bounded sets.

The converse of Theorem 2 holds at least when A is a subdifferential. When dim H < 00 and A is a subdifferential, then eq. (1) is LOO-stable if and only if A is surjective, i.e.

U Ax = H. ~D(A)

REFERENCES

[1) E.J. Beltrami and F. Buianouckas. A Note on Passive Evolution Equations, J. of Math. Analysis and Applications 37 (1972), 227-230.

(2) H. Brezis. Operateurs maximaux monotones et semi-groupes de contractions dans les espaces de Hilbert. North-Holland/American Elsevier. 1973.

(3) C.A. Desoer and M. Vidyasagar. Feedback Sys­terns: Input-output properties, Academic Press, 1975.

(4) D. Wexler. Operateurs fortement monotones et equations d'evolution fortement passives. C.R. Acad. Sc. Paris 280 (1975). 201-204.

(5) A.H. Zemanian. Realizability Theory for Con­tinuous Linear Systems. Academic Press. 1972.

Dinu WEXLER was born in Bucharest. Rumania. on August 28, 1931. He received the Doctor degree in Mathematics from the University of Bucharest in 1966. From 1955 to 1971 he was with the De­partment of Mathematics of the Institute of Pe­troleum. Gas and Geology, Bucharest and the Ins­titute of Mathematics of the Rumanian Academy of Sciences. In 1972 he was with the Department of Mathematics. Universite de Paris VI (Pierre et Marie Curie). France and in 1973 he joined the Department of Mathematics. Facultes Universitai­res N.D. de la Paix. Namur. Belgium. where he holds the rank of Associate Professor. His re­search interests are in Differential Equations and Nonlinear Analysis.

151

CONTRACTIVF. TRANSFER RATIOS OF OPF:RATOR NF'lWORKS A. H. Zemanian

State University of New York at Stony Brook Stony Brook, New York 11794

Abstract

By an "operator" we mean a bounded linear operator in a complex Hilbert space. An operator network without mutual coupling, whose self-impedances are commut­ing operator-valued analytic functions corresponding to passive elements, have voltage and current transfer ratios that are contractions on certain cones in the right-half complex plane.

Summary. The concept of an electrical network whose parameters are operators in a complex Hil­bert space H has been discussed in a number of prior works (1) - (8). The purpose of the pre­sent paper is to extend to operator networks the fact that RIC scalar networks having no mutual coupling have voltage and current transfer ratios whose absolute values are bounded by one on the real positive axis. We show that networks whose parameters are invertible commuting positive bounded linear operators in H have operator-valued voltage and current transfer ratios that are strict contractions on certain cones in the right­half plane. The apex of every such cone is at the origin and its bisector is the real axis. Moreover, the angle of the cone depends in a cer­tain way on the number of nodes of the network.

We need some definitions: The numerical range W(f) of an operator is the bounded set of complex numbers

W(f) = [(£a,a) : a €H, 1\ a \\ = l}.

C denotes the complex plane in the following ex­pressions. For any positive integer m, C(m) is the open cone

C(m) = [)"'sC: larg)"'\ < TT/2m}.

For any fixed C € C (m), we define the closed cone

O(m,O = [)." € C: larg )"'1 ~ m larg CI}.

152

Let m be a maximal abelian self-adjoint algebra of operators containing the parameters of the networ~ Given any set of positive commuting operators, such an m can always be constructed. Q(m) will denote the set of all analytic operator-valued functions F em C(m) such that, for each C€C(m), we have F(C) €"n, W[F(C)]C O(m,O, and w[F(O] is bounded away from the origin.

We aSSume that the network N under consideration is a connected three-terminal or two-terminal-pair network having no mutual inductance and no inter­nal sources. Every self-impedance is assumed to be a member of Q,(l). (Note that every resistance, inductance, or capacitance that is a positive in­vertible operator in m has its impedance in Q(l).) Finally, we assume that there does not exist in N any short circuits and that all the external ter­minal nodes are distinct nodes.

We now generate a certain network N' equivalent to N as follows. We first replace all series connec­tions inside N by equivalent single branches and then replace all parallel connections by equiva­lent single branches. We continue repeating these two steps until a network N' with no internal series or parallel connections is obtained. N' is uni~uely determined by N and has the same behavior at its external terminals as does N. Throughout the following k will denote the number of internal nodes (not counting the external terminal nodes) of N'.

A cut-n~de ~f a c~nnected network is a node wh~se deletion coupled with the deletion of all the branches incident at that node results in a dis­connected network. An external terminal node of N is a cut-node if and only if it is a cut-node in N'.

Let T(,) and J(C) denote respectively the open­circuit voltage transfer ratio and the short-cir­cuit current transfer ratio of either a three­terminal or two-terminal-pair network. We are now ready to state our conclusions.

Theorem 1. Let N be a three-terminal network such that the input node not common to the output is not a cut-node. Then, for all ,eC(2k + 2), IIT(,)II < 1 and T(O has the form

T ( ,) = [I + A ( ,) r \ where A E: Q(2k + 2) and I denotes the identity op­erator in H.

Theorem 2. Let N be a three-terminal network such that the output node not common to the input is not a cut-n~de. Then, for all 'eC(2k + 2), IIJ( 011 < 1 and J( (;) has the form

J(') = [I + B(,)]-l,

where Be Q,(2k + 2).

Theorem 3. Let N be a two-terminal-pair network such that at least one of the input (respectively, output) nodes is not a cut-node. Then, IIT(,)II < 1 (respectively, IIJ(,)II < 1) f~r every ,CC(2k + 4).

These theorems are proven by extending Kirchhoff's third and fourth laws to operator networks and manipulating the numerical ranges ~f the operator­valued impedances in an appropriate way.

The values of k in these theorems cannot be de­creased; that is, given any, not in the cone in­dicated in the c~nclusion of anyone of the theorems, there exists a network that satisfies the stated assumptions and whose voltage or cur­rent transfer ratio is not a contraction at that ,. The assumption that all branch impedance operators commute is a severe one. However, it can be shown by example that it is necessary. Indeed, let r

l and r

2 be positive invertible noncommuting

operators connected in s~ries. Let T be the transfer v'Jltage ratio that maps the voltage drop across r l + r 2 into the v~ltage drop across r 2 • Then, by choosing r

l and r

2 appropriately, we can

make IITII greater than one.

Finally, we note that if s~me of the external ter­minal n~des are cut-nodes or if there exist paths of short circuits inside N c~nnecting the terminal nodes, then it is possible f~r T(e) and J(C) to have norms equal to but not larger than one.

153

RF,FERENCFS

(1) V. Dolezal, "Hilbert netw~rks: I", SIAM J. Contr~l, t~ appear

(2) V. D~lezal and A. H. Zemanian, "Hilbert net­works: II - Some qualitative pr~perties", SIAM J. Control, to appear

(3) V. Dolezal, "Generalized Hilbert networks", to appear

(4) V. Dolezal, "Networks", to appear

(5) A. H. Zemanian, "Passive operat~r netw~rks", I~E~ Trans. Circuits and Systems, v~l. CAS-21 (1974), pp. 184 - 193

(6) A. H. Zemanian, "rnfini te networks of positive ~perat~rs", Circuit Theory and Applicati~ns, vol. 2 (1974), pp. 69 - 78

(7) A. H. Zemanian, "Continued fractions of op­erator-valued analytic functions", Journal ~f Approximation Theory, t~ appear

(8) A. H. Zemanian, "Infinite electrical networks", Proc. I~EF" to appear

. ,~' .