Computer Automated Design of Systems

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COMPUTER AUTOMATED DESIGN OF SYSTEMS

Larry Paul Vines


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I EY KNOX LIBRARY, .__POSTGRADUATE SCHOOL

NTEREY, CALIF. 9394Q


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NAVAL POSTGRADUATE SCHOOL

Monterey, California

THESISComputer Automated Design of Sy stems

by

Larry Paul Vines

June 1976

The. sis Advisor: George J. Thaler

Approved for public release; distribution unlimited.

T175039


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REPORT DOCUMENTATION PAGE READ INSTRUCTIONSBEFORE COMPLETING FORM1. REPORT NUMBER 2. GOVT ACCESSION NO. 3. RECIPIENT'S CATALOG NUMBER

4. TITLE (and Subtttt*)

Computer Automated Design of Systems

5. TYPE OF REPORT ft PERIOD COVEREDMaster's ThesisJune 1976

PERFORMING ORG. REPORT NUMBER

7. AUTHORS

Larry Paul Vines

a. CONTRACT OR GRANT NUMBERfa;

9. PERFORMING ORGANIZATION NAME AND AOORESS

Naval Postgraduate School

Monterey, California 93940

10. PROGRAM ELEMENT. PROJECT, TASKAREA ft WORK UNIT NUMBERS

11. CONTROLLING OFFICE NAME ANO AOORESS



12. REPORT OATE

June 19761). NUMBER OF PAGES

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Approved for public release; distribution unlimited.

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IS. SUPPLEMENTARY NOTES

1*. KEY WORDS (Contlnua on rawaraa alda II nmcaaaarr and Identity or aloek numaar)

Automatic control, computer automated design, optimum design

feedback control, compensator, desired response,

20. ABSTRACT ; Continue on ravaraa aldm H naeaaaary and tdantlly by fcjoe* mama at)

An automated digital computer technique of control systemdesign is presented. The emphasis is on compensator designbut the method is applicable to the design of any systemwith free parameters. Signal representation and systemresponse are in the time domain.

The inputs required from the engineer are the systemconfiguration, the desired output response and the free

DD 1 JAN 73 1^73 EDITION OF < NOV 61 IS OBSOLETE(Page 1) S / N 0102-014-6601 I

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ftCUWITY CLASSIFICATION OF THIS P>GEfW..n Hrtm l.f..'

parameters. A parameter minimization routine is then used tominimize a specific cost function and to set the free parameter^A graphical output of the desired response and actual systemresponse is then produced for comparison by the engineer.

DD Form 14731 Jan 73

S/N 0102-014-6601 SECURITY CLASSIFICATION OF THIS F*OEfWin Data Enlmfd)


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Computer Automated Design

of

Systems

by

Larry Paul Vines

iant. United Stati

B.S.E.E., Purdue University, 1970

Lieutenant, United States Navy

Submitted in partial fulfillment of the

requirements for the degree cf

MAST5H OF SCIENCE IN ELECTRICAL ENGINEERING

from the

NAVAL POSTGRADUATE SCHOOL

June 1976


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DUDLEY KNOX LIBRARYNAVAL POSTGRADUATE SCHOOL'MONTEREY, CALIF. 93940

ABSTRACT

An automated digital computer technique of control

systea design is presented. The emphasis is on

compensator design but the method is applicable to thedesign cf any system with free parameters. Signal

representation and system response are in the time

domain

The inputs required from the engineer are the

systen configuration, the desired output response and

the free parameters. A parameter minimization routineis then used tc minimize a specific cost function and

to set the free parameters. A graphical output of the

desired response and actual system response is then

produced fcr comparison by the engineer.


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PECEIEM III. E

VIII. CMS EROGEAM LISTING80

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LIST OF FIGURES

1 Ijfical System to be Optimized 14

2 CADS Erogram Flow Chart 24

3 Hard-Leonard Speed Control System 26

4 lachcmeter Compensated System 28

5 CADS Elcck Diagram of Tachometer System 29

6 CADS Compensated Tachometer Feedback SystemSesfcnse 30

7 Type Three Block used for Pseudc Tachometer

Feedback 31

8 Eseudc Tachometer Feedback 32

9 Pseudc Tachometer Compensated System^Response 33

10 Third Order Plant to he Compensated 34

11 lag Compensated System 35

12 Occcirpensated, Compensated BODE PLOT 36

13 Conventionally Compensated System Response .... 37

14 CADS Lag Compensated System 33

15 CADS Compensated System Response 40

16 Plant for Lead Compensation , 41

17 Conventional Lead Compensated System Response . 42

18 CADS Elock Diagram for Lead Compensation 43

19 CADS lead Compensated System Response 44

20 CADS Compensated System Block Diagram 45

21 CADS Relaxed Boundary System Response 46

22 Serve Drive Mechanism, Original Compensation .. 48

23 CAES lagraa of Servo Drive Mechanism 49

24 Original Servo System Response 51

25 CADS Compensated C System Response 52


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26 CAES Compensated C -1 System Response 54N

27 C2ES Compensated C -2 System Response 55N

28 CfiDS Optimized System Response 57

29 Serve System Response, CADS Values with8 = 0.5 60

30 Elcck Reduced Ward-Leonard Drive 63


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ACKNOWLEDGEMENTS

The author wishes to thank Dr. George Thaler fcr his

guidance and encouragement during the course of this

investigation. Kris Butler, Ed Donnellan and M. Anderson of

the W. R. Church Computer Center deserve special recogcitioc

fcr their patient and professional assistance. Lastly, I

thank my ife, Patricia, without whom this thesis could

never have teen accoaplished.


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I. IHTBODUCTION

The past two decades have been witness to an ever

increasing use ci the digital computer. Engineering usage

and design are prctably the most important justif icaticn for

the large memory, high speed computers of tcday. Classes in

computer application to engineering problems have become

part of the established curricula at most universities.

Numerous papers have appeared on the application of the

computer to engineering problems [ 1 J which until recently

were unsclvatle or at best required long, tedious

procedures which gave only approximate solutions.

Contrcl system engineering has relied increasingly on

computer sinulaticn of large scale systems to verify that

design specifications have teen met and to make

mcdif icaticns to a system even before producing a prototype.

There are programs available to simulate virtually any

electrical circuit, or control system, either in transfer

function form or as a system of first order differential

eguations. ethers draw the Bode, Nichols or Nyguist plots

of open and clcsed loop systems. Some programs help design

compensators which use an iterative method to achieve the

desired frequency response [2] [3] Researchers continue to

better adapt the computer to engineering usage.

Cantalapiedra 4] has used an iterative method to find

the optimum reduced order model for large order systems.

MacNamara [5 1 went even further and used an iterative method

to find the optimum compensation for an aircraft autopilot.It would appear that these techniques could be extended and

applied to the direct simulation and design of control

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system circuits in the time dcmain.

The ictert of this thesis was to develop a user oriented

program which could simulate a wide variety of control

systems and determine the values of the gains, poles and

zeros necessary to produce a desired respoDse. A convenient

means of data card input was to be provided to specify (1)

the control system which was to be optimized and (2) the

desired response. fi locally available function minimization

routine (ECXE1X) was to be used to optimize the simulated

system's output.

To simulate the system which is to be optinized,.

transfer functions which are commonly encountered were

reduced to first order linear differential eguations. These

egcations were then programmed so. that the transfer function

blocks could be connected in an arbitrary fashion by data

card input. Several common nonlinear transfer blocks were

also provided. The program will simulate the system withthe known parameters and then allow all unknown or

adjustable parameters to be fixed by the computer

optimization routine to achieve the desired response.

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II. PROGRAM DEVELOPMENT AND IMPLEMENTATION

A. GENEEA1

Any program which is to be of maximum benifit tc the

user must fcave a simple means of data input and an outputwhich is- easy to interpret and apply to the problem at hand.

The input data should have a physical significance that does

net lose its relevance through the programming of numerous

eguatiens. The program should be a readily usable tool and

not a profclei in itself. The ictent of this thesis is to

present such a program, Computer Automated Design of Systems

(CADS),

which is readily used for simulation withoptimization of the output. Optimum in the sense that the

output response cf a given system configuration is as near

the desired response as possible.

Most ccntrol system design starts with a proposed

transfer function block schematic to achieve a desired time

domain response. CADS has the common transfer functions

built into the program and the input data can come directly

from the proposed system schematic. The transfer tlocks

which are available for system simulation are presented in

Table I. These blocks should be adequate, either seperately

or in various cascade combinations, to represent most

ccntrol systems.

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CP

EICCK

SYMBOL EQUATION SOLVED

9o= Q9 Vi . G 9,.

9 -P 9 * G 9;i , GS*P

9o

q -P 9o + G(9i + H9\)ei. G S+ 29

ft

9 = -25^6o-^9 +6eL_G - 9 o ^

S 2 +Z6W5 * co n2

6 = G 9 C 9 L ^ G9 i 9 U

9o -- 6 . 9 - * fie .

9 = wG6

^< 9 u

8i.

uVx~X 6l -9-0

Zj 9 c

9o * M < 9 69 = G 9- |9 e | > 9. 3

8;

|9. rS.0

)19 9;

t

TAELE I Program Transfer Functicr, Elocks

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To use the program the engineer must know the system

configuration in transfer block form and the desired second

order output response. If other than a second orderresponse is desired, this may be easily specified but

requires seme knowledge of how to input the desired

response. CALS will take the transfer block system, cennect

it, compare the given system response to the desired

response and set any free parameters to achieve the closest

match possible of the system output to the desired response.

Figure 1 is representative of the type of systeir the

program car simulate and optimize. In figure 1, the

parameters of the numbered transfer function blocks are

known or fixed by equipment limitations. Transfer blocks X,

I, and Z contain variable parameters which must be selected

by the prcgrair to make the system output reproduce a desired

output function as closely as possible.

x


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E. MAIK EBCGFAM

The KAIK program was developed to control the selection

of the sutf unctions used in the optimization process and to

compute the desired response cf the systems which were to be

optimized. A second order step response is calculated from

eguation (I) as the desired response and stored in the array

XEAIA.

Ge = u(t> (i)

S2 + 2 8 WtjS + W 2

The MAIN is essentially a bookkeeping routine which controls

the prccraa execution. The sufcfunction operations it

controls are defined in the following sections.

C. ELANT

The ccrnncn simulation routines LISA, DSL, ECAP, CSP and

INTEG were investigated in an attempt to adapt them tc the

general crotlem specified above. These programs reguired

either the incut cf the system's state variableeguaticns,

Laplace transform equations or the node-transfer function

pair for each system simulated. These programs are useful

for simulation if all of a system's parameters have been

specified. Eowever, each simulation is problem specific and

these technigues are not readily integrated with other

sutfuncticn ticgrams. A simulation subprogram which was

capable cf accepting data card input tc simulate a wide

variety cf ccntrcl systems and of working with other

existing subprograms had to be developed.

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cards used tc input the system configuration is shown telcw.

|1 ]11 | 20 | 40 |60 (Column Numker)

ELKCCE=EEVV G P Z

Hhere:

CC = Position number of the block

D - Type of tlocic (number)

EE = Input node number

VV = Cutput ncde number

G, P, and Z are parameters of the transfer function.

2. Ilcck Connections

The rumfcer of transfer blocks in the system and the

data cards associated with each of the blocks are read upon

initial entry into the simulation routine. The prograi then

connects the transfer function blocks in the proper orderby

comparing the input node number of a block to the cutput

ncde nunfcer cf every other block. Whenever these two

numbers are equal, the blocks are knowc to be connected and

a flag is set equal to 1.0. The flags are then used to

identify the input drives to each block.

3

.

Er i v es

The input guantity to each block is called the

EBIVE. The program was designed to allow for multiple

inputs to the system being simulated. This flexibility was

achieved by making the input to each block equal to the sum

of the outputs of all blocks connected to the input node

plus anj external forcing function (DBVIN) feeding the

transfer tlcck. The input DRIVE to a block is determined as

shewn in equation (II) .

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DBIVE(i) = THA (j) *FLAG (j f i) DRVIN (i) (II)IHA(j) is the output of block j. FLAG(j,i) is 1.0 if block

j is connected to block i and zero if it is cot connected.

EEVIN (i) is any external forcing function specified by the

user whicfc drives block i. DRVINs must be inserted in

subroutine IIANT as Fortran IV statements. The standard

program has CSVIN(1) = 1.0 specified as a unit step incut to

the system. Problem III-E in the section Investigation of

Prcgram Eerfcrmance demonstrates how multiple, time varying

inputs are tc be inserted in the program.

4. Standard Transf er Function Blocks

The transfer function blocks available for system

simulation are shown in Table I. These blccks were selected

because cf tteir ccmmcn usage in the modeling process. They

were alsc found to be adequate either separately cr invarious cascade combinations to represent most control

systems.

The transfer function equations for each tjpe of

block were written in state variable format and stored in an

array named THA. The program reads a number from a block's

data card which specifies the type of transfer function

associated with the block. The system equations ar then

solved seguentially by selecting the type of equation

assccitaed with block number one, solving for its output and

then sequencing tc block number two, etc. The integral

eguations are solved by a modified RKL fourth order method

in the subfunction routines. Three integration routines

were necessary tc store the intermediate results obtained

for those equations involving a double integration. The

four sutfunctions, RKLDE2, RKLDE3, CCPIX and RKIDI4 are

called by IIAKI. BKLDE2 is used for the integration of a

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type twc blcck. BKLDE3 is used for integration of a type

three blcck and to store intermediate quantities for

sutfuncticn CCPLX. CCPLX calls RKLDE3 and RKLDE4 for

integraticn cf a type four block.

Nc prevision has been made for the input of initial

cenditiers cr the integrators. Therefore, the integrations

must start at time equal zero. The user must specify the

step size tc fc used for integration (DT) and the problem

run time (IF) . To conserve computer time ET should be made

as large as possible. DT = TF/1000 is suggested as

appropriate in most cases. Equally important is that TF beas small as pcssible.. Use of the program has shewn that

letting TF te greater than the transient response time of

the systea is rarely justified. For preliminary analysis TF

was kept tc enly slightly longer than the time of the first

undershoot fcr second order dominant systems. Because of

the complexity of subroutine PLANT a flew chart of PLAKT has

teen included as Appendix B.

5 . Pa^am eter s to be Optimized

The parameters of the system which are to be

optimized ty the minimization routine EOXPLX must be

specified in PLAN1. The minimization routine returns the

trial values cf the variable parameters to PLANT in an array

labeled C l . fiegular Fortran IV statements equate the

variable parameters to the values in this array. If a

pcle-zerc pair were to be optimized/ the following

statements wculd be inserted in PLANT: P(i) C(1)

and Z (i) = C (2) . The location fcr the preceding statements

is clearly iEdicated in the program listing for PLANT.

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D. DESIBEI* BESPONSE (XDATA)

The ciitciia against which the system response will be

compared will vary according to the application cf the

system. Since most design work on control systems is done

en the tasis of second order dominance, the program was

written tc provide a second order step response with

adjustable 5,8 and gain as the basic criteria againstn

which the sinulated system will be compared. The desired 6,

H and gain are read in as input data. The step response isn

then computed in the main program by subfunction RKLEEC and

stored in the array called XDATA. Any ether time domain

response nay be specified by the user by removing the second

order step response equations and replacing them with the

equations of the desired response. An example is provided

by problem III-E in Investigation of Program Performance.

If the program is being used for simulation only and nc data

curve is desired, setting the input variable LEAP = 1 will

cause the program to bypass the computation cf rhe standard

secend order data equation.

E. C0S1 FOSCTION (PERFORMANCE INEEX, PI)

The achieved system response is compared witt the

desired response and the difference is the error. The

program searches for the parameter settings which will

ainimize this error. The cost function may be specified bythe user tc weight the system outputs as desired in

subfuncticn FE. The default cost function cf the program is

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rr

tte integral errcr sguared. J = / (Err) 2 dt. An example

of a weighted cost function is presented in Section III.

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F. MINIMIZATION RCOTINE

1

.

G ene r al

lh free parameters are optimized to reduce the cost

function hy the complex method of M. J. Box. [6] Box's

constrained optimization method has been programmed at the

Naval Postgraduate School by B. B. Hilleary as a subroutinecalled ECXFLX. This subroutine will find the minimum cf an

arbitrary function (cost function) subject to arbitrary

explicit constraints and for implicit constraints. Explicit

ccnstraints are defined as upper and lower bounds on the

free parameters. Implicit constraints may be arbitrary

functions of the free parameters (e.g. P P < 178).1 2

Two function subprograms are used to evaluate the

objective function and implicit constraints, FE and KE

respectively. The method EOXPLX uses to search for the

values of the free parameters which minimize the cost

function is explained in the computer program listing.

2. Ex elicit/ Implicit Ccnstraints and Start Points

ECXFLX searches a feasibility region (n-dimensicnal

space, where n = number of free parameters) defined ty upper

and lower bcunds on the free parameters for a minimun cost

function. The sialler the region defined by these

boundaries, the nore rapidly the program will convergeto

the optinum parameter settings. Good engineering judgement

will be necessary to keep the feasibility region as small as

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possible. The boundaries of the search regicn are read from

data cards by the main program as the upper bound (XG) and

the lower bcurd (XL) for each free parameter.

Implicit constraints may be any arbitrary furcticn

of the free parameter desired. If an implicit constraint

such as the product of a pole-zero pair must be less than

seme number is to be evaluated, it must be supplied bj the

user to the subfunction KE. No implicit constraints were

used in this thesis.

The starting values cf the free parameters (XS) areread in by the MAIN frcm data cards. A good choice of

starting values will dramatically reduce the time required

for optiii2aticn. A preliminary root locus or Bode plot

methed of estimating the best values of the free variables

should be accomplished whenever possible.

G. GRAPHICAL OOIPUT

Two subroutines were written to provide for the

graphical output of the desired response and the best system

response achieved by the optimization process. The user may

select, by data card input, either subroutine PPLI which

provides a high speed printer plot or subroutine PIC which

provides a calcomp graph. Every fifth integration point

stored in the arrays XDATA and THAODT is plotted.

Subroutines PPIT and PIC call the subroutines PLOTP and CRAW

respectively. PLOTP and DRAM are standard plotting routines

at the NP computer facility and are not a part of the

simulation program. Figure 2 diagrams the information flow

and data input to the program.

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No . Runs XDATA No. Trials

No. Var 6 > wn XU, XLdt, TF Type Graph

r

XS

i

, t

Plot MAIN . . m w output

1

1

response1 L1

1

1

i

t

I

1

1

1i

t

1

1

1

1

BOXPLX ~ * ] s.E1

1

1n i

>v I /^Implicit

BLOCKDataInput

1

J

I

t

'

NO / xTli Constraints< K>| / MetFE

YES ^

Drives1

1

[

ir l

i *4t

Si

PLANTSystem

.mulation

1

i

i ... RKLDEs

Optimization

Simulation

FIGG3E 2 CADS Program Flow Chart

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III. INVESTIGATION OF PROGRAM PERFORMANCE

The example problems presented below were used to aid in

tie development cf the optimization program. The order of

difficulty cf the problems progresses frcm a simple text

bock single variable, single input system to a multivariafcle

operatioDal servo drive system which has multiple inputs and

discrete level feedback. An example of how a schematic

diagram representation of a system is prepared for input to

the progran is presented prior to considering the example

problems.

A- EATJ INECT, AN EXAMPLE

Ihe Rard-Ieonard drive system [7] shown schematically in

Figure 3 (a) has two variable parameters. The gain cf the

amplifier and the tachometer feedback are available to

adjust tie system's response. To simulate the system a

blcck diacraa representation of the system is drawn as shown

in Figure 3 (c) using the transfer blocks frcm Table I.

Cortjt

fTAc*


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t\-

URvjnCj)

GO)S*P GCOGO)S+PW -($

nn

G(4)

Gfs)

EC

(c)

fIGCBE 3 Ward - Leonard Speed Ccntrcl Systei

tsing Feedback. (A) Schematic Diagram

(B) Elock Diagram (C) Blcck DiagramUsing CADS Blocks.

P(1) = 1/T

P(3) =0

Hfcere G(1) = K/t

G(2) = Km/Ba

G(3) = 1/J

G


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a type two transfer block connected between nodes 1 and 2.

The data care input for this block would be

BIKC 12=0 102 K/ T 1/ T

The program reads the data card input and connects the

blocks by setting a FLAG = 1. whenever the input node

nunber to a block is the same as the output ncde number of

any other block. The input to a transfer block is then

determined tc be the sum of the outputs of all blocks

connected tc the input node plus any external forcing

function driving the input node. The program has a unit

step specified for DEVIN (1). If this is the only input tothe system, no action is necessary on the part of the user.

If ether than a unit step input to the system is desired or

if there are other external forcing functions such as

EEVIN (3) , they must be specified and placed within the body

cf the subfurction PLANT as Fortran IV statements. For the

example shc%n in Figure 3 (c) , a card with the equation

DEVIN (3) = f (T )

wculd have tc be inserted preceding the drive equations. An

example cf hew multiple, time varying drives are specified

is given in section III. E.

When cptinizing a system, seme of the input quantities

willbe

unkcewnor variables. These variables must also be

assigned *itfcin subroutine PLANT. To optimize the variables

K and K cf Figure 3 the following two statements would be1 t

inserted in IIANT:

G(1) = C(1)

G(5) = C(2)

where C(1) and C (2) are the variables which will beoptimized by subroutine BOXPLX.

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. IflCBCMJlIE JEIDBACK

The first optimization problem attempted was one which

had an exact solution that can be found by algehraic

methods. Ac instrument servo [8] with unity feedback and

forward transfer function

1000

(III)G(S) =

S(S+10)

was to be ccopensated with tachometer feedback as shewn inFigure 4. The enly specification fcr the system's

performance cf this single variable, secend order system was

the simple requirement that the closed leep roots have a

5 = 0.7.

Df O \OOa

i \ f- SCs+io)

K4 S

.

IIGUEE 4 Tachometer Compensated System

The systeii shewn in Figure 4 was redrawn as Figure 5 in

order tc achieve the tachometer feedback. The

characteristic equation of the system shown in figure 5 is

S 2 (10 + 103 k ) S + 103 =

t

(IV)

The required K = 0.0343 may be calculated from equationt

IV.

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>Q) loeo /C S +10 s

m

___

t

1

FIGDBE 5 CADS Block Diagram of Tachometer System

CADS was programned to optimize the system with the

variable, F , specified to be between the limits

0.C1 < K < 1.0. The desired response was specified tc bet

the standard seccnd order step response for 5 = 0.7,

Wn=

100C. CADS determined the optimum value for Kto

be

K - 0.03*43. Figure 6 shows the system's step response andt

the desired response are virtually superimposed.

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The tyce one system, in the preceding example, allowed

derivative feedback from the forward path without requiring

a block which was capable of differentiation. One may

encounter a type zero system or a system which cannot beconfigured to provide derivative feedback from the forward

path. Tbe possibility of providing a pseudo derivative

feedback using a type three block was investigated for these

cases.

A type three block with Z = and G = P is shewn in

Figure 7. If the pole is placed far out or the real axis,

this block actroximates derivative feedback.

FIGUFE 7 Type Three Block used for

Pseudo Tachometer Feedback

The effect of using this pseudo tachometer feedback on

the complex roots of the closed loop system is negligible.

It does add a real root at P = -964.

The system of Figure 5 was redrawn as shown in Figure 8

using the pseudo tachometer feedback.

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+1

S rv iOJ

, r \S+10 \ J

-(oV+S

SHO 3

-i

FIGURE 8 Pseudo Tachometer Feedback

The optimization program calculated K = 0.0352 fcr thet

abcve confignration. This gave a 5 = 0.715.

The system response and the desired response are shown

in figure 9. The two responses are again nearly

sup erimpcsed . This method of providing derivative feedback

has the disadvantage of adding an integration step tc theproblem solution with the concomitant increase in problem

sclution time.

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0.0- 0.05 0.10 0.15Time (sec)

0.20 0.25

fIGURE 9 Eseudo Tachometer Compensated System's Response

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c. CASCADE CCMEEJJSATION

Having jrcven the feasibility of the program optiaizing

a single variable system where an exact solution was

available, the next problem considered extending the problem

scope to a third order plant with two variables. The

unstable plant that was to be compensated is shewn in

Figure 10.

oI (ii)(j+062jft+i)

EIGUEE 10 Third Order Plant to be Compensated

The riant was to be stabilized using a single section

cascade cempensatcr. The compensated plant was required tc

have H


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r,snS (S + i) (.2Sf)

FIGOBE 11 Lag Compensated System

A compensator which wculd meet the required

specif icaticrs was calculated by conventional methods as a

check or the program's performance. The Bode plot cf the

uncompensated and conventionally compensated system is shewn

in Figure 12. The values of r, and r. for Figure 11 which

would meet tfce design requirements were determined tc be

t, - 1C, t 2 = 100. The compensated system's responseusing these values for the compensator is shewn in figure

15.

35


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'a

d


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0.0 2.0 4.0 6.0 8.0 10.Time (sec)

FIGCBZ 13 Cor ventionally Compensated System Response

37


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The conpensated system was redrawn as shewn in

Figure 14 using the standard program transfer blocks

available for system simulation.

Q^Z^loJjL(SfPc) -O s+i O S*5 -o

-I

FIGURE 14 CADS Lag Compensated System

A stardard second order response of & = .3,

H = 0.75 was chesen as the desired response for then

optimization program to match. The limits placed en the

optimization program were .001 < P < .1 and .01 < Z < 1.C C

Arhitrarj values to begin optimization were specified as

.1 in the logarithmic centers of the

search 2cnes. CAES determined T, = 8.078 and f2

= 67.47

E = .01 anc ZCo Co

as the optin.cn parameter settings. Figure 15 shows the

desired response and the program compensated system

response

.

The response cf the system compensated by CADS is more

nearly the desired response than is the conventionally

compensated sjstem. One should remember that the specifiedresponse is for a true second order systen whereas, the

ccnpensatd system is forth order. Therefore, a perfect

38


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match cf desired versus actual responses could net be

cttained.

39


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T

rH

/f^\CN // \

/ / \^r-\

II\

o / \v/ \ ' ^

HH // \ ^, RD

O // \S // \B // ^\

// ^ D00

11

o

/o/


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V)

ao

Wu>

C3

ai>

MW>iUJ

x>

P

0)

Uiau

Co

H

C

>aou

-'

03=3

1*1

J-nOVHi

42


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The phase difference is only 4.33. Continued relaxation of

the fcouncaries produced the compensated system shewn in

Figure 2C. figure 21 is a plot of the system's response for

these values. The system's response is improved in that it

lags the irput ty only 3.46 and the magnitude isessentially the sane as the input magnitude.

Si N LQQl -* Cs+a&Si-SOO O ifo+'^Q.S+*too 3-*fo/o O

-I

FIGUEE 20 CAES Compensated System Elcck Diagram

The OEtinizaticn program was activated at T =

although the problem specifications were for the steady

state resicnse. The problem was rerun with the optimization

prccess started after the transient had died out. The same

values fcr the optimum compensator Mere effctained.

Apparently the snail initial transient did not effect the

prctlea scluticn.

45


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E. SEBVC S1S1EM CO KPENSATICN

The foregoing examples of the simulaticn - optimization

program were simple examples which could obviously be

guickly sclved with the standard cut and try methods. A

mere complex and challenging example of program usace is

presented fcy the sjstem shown in Figure 22.

The system is an operational servo drive mechanism withmultiple inputs and discrete level feedback. This bighly

nonlinear system's output was to follow the input as closely

as possitle and in steady state (t> 75ms) there was to be

very little ccise ripple. The free parameters of the system

are the pcles and zeros of the compensator, C , and theLi

pcles cf the ncise suppressor C .n

To siaulcte and optimize the system it was redrawn using

the available program blocks as shown in Figure 23. Euring

simulaticr it was found that the current limiter for I wasD

net needed because |I | < 25 A and the limiter was removedD

to decrease program run time. The desired response (XDATA)

was written as a set of three equations. These eguations

were then used to replace the second order step response

eguations in the standard program. DRVIN(1), (4) and (5)

were alsc written as a set of equations and placed in

subroutine E1ANT. The discrete level feedback to block two

was achieved by making DRVIN(2) = INTGEB (TBA (12) ) . These

changes tc tte mair program and PLANT were all that were

neccessarj tc sinulate this system. The inplementation of

these changes to the program is shewn on pages 80 and 81.

U7


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%6 o

< o ^

II

tx

Pi W

u

a

oit

2o

>0

>

^^

* 3?/-. + 4

*4 C

-t-

-t

>?

tf

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c

3

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a

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,csi

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aw

HS(TJ

JO,

o

HMQO>uVI

M-l

O

(0

Men

aVia

u

w

M

49


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103/244

The system optimization was initially broken intc two

parts. This sas tc reduce the number of variables that were

to be optimized per run and to obtain near optimum starting

values fci the free parameters. The above steps were taken

in an effort to decrease the computer time required for a

solution. The first run was tc optimize the compensator,

C i independent of the noise suppressor. The second

cptimizaticn run was to select values for the noise

suppressor, C . The rationale behind this separation was8

that the two circuits perform different functions and should

therefore be initially separable in their effects en the

system. The final optimization run was to be made with all

free parameters available to the program fcr optimization.

The search zene centered on the values found above.

Figure 24 shows the response for the system as

originally compensated. The optimization run to set the

values fcr C resulted in Z = 43.5, P = 21.0,L * l

Z = 47.5, = 5S2.C. Figure 25 shows the simulation of2 2

the system using these values. The initial velocity

overshoot has teen reduced and the average velocity after

the transient appears more equally distributed above and

below the desired velocity.

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107/244

oH

00o

O

oQ)

CO

6

E-t

o

WGOUi

iW

u>

aouenQU

mCM

W

H

O

O'Ofr o'oe o*o3 cot(09S/UT) Aq.TOO-[SA

52


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111/244

%V

Ho


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%V

CM


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116/244


117/244

to

eoOi

O wrH CW

ua

O au>

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CO00o *- T3

o VO

g U4H O

vo En

OCO

uo


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o o o O ooo o o m KOm en IT) ^r CM

in in CM CM

CM

o

eg

nCk

01

N

04N

3

OH4J

Ua

4J

uu

f* r* r^iT j ^

ao r r- r 00 oin cr iT ^3 3- H

(d

NHS

t rt

o CM CM CM ^> Vo en cr> O CM '-itin Ul m in in

>H

in in in M ^ cr 3- m ao i

in m m n IN OS3 zr 5T in

*

>MQ)

CO

mo

o r ~ r- p*CM CN CM CM

^

CM

P

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M(0

as9

4JHH

P r- 4J * f-JT3 T3 3 CQ(M (M M CM M M + HM M W -PW W T3^ ^ ^ MMm

MOHCO 1-1

Ui (0

e c(V HQ4 cr

a -o Mu o

58


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121/244


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123/244

o

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00o

uO i

o>ua>en

O'Ofr coe o*oz cox(D9S/UT) Aq.TOOX9A

60


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IV. DISCUSSION^ CONCIUJICNS AND RJCCMMEND A 110 *S

A. LISCDSS1CN

The objective of this thesis was to investigate the

feasibility cf developing a computer program which would

optimize a variety of control systems' respcnses in the time

domain. Ihe program developed during the investigation

proves that time dcmain optimization cf control system

responses is net only feasible but desirable due tc the

readily interpretable results of the optimization process.

CADS requires approximately 200K bytes cf computer core when

the high speed printer plot is used for graphical display of

the output arc 230K when outputting calcomp graphs. These

cere reguirenents are a maximum and could be reduced by more

careful, professional programming.

The cenputer time reguired for CAES to arrive at a

solution is dependent upon

(1) the order of the system being simulated,

(2) the area cf the search zone determined by the

upper and lower bounds en the variable parameters

and

(3) the nearness of the starting guess to the

optinuo parameter values.

Every trial ?alue cf a parameter selected by EOXPLX reguires

a complete simulation of the system in order to evaluate the

system's response and compare it with the desired response.The prograi run time is therefore, T Rut1 = the number of

trials X system simulation time. A high crder system may

61


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require twenty seconds of CPU time for simulation. If 300

trials are required to determine the optimum parameter

values, the total CPU time would be 100 minutes.

Example Lower Starting UpperProblem Bound Guess Bound

III XL XS XU

Number CPU Timeof Required

Trials

b -300 -34. 3 -3 55 7min03sec(pseudo)

.001 .01 0.1 225 24min53secc .01 .1 1.0 225 24min5 3sec

400 450 500

400 450 500

d 90 95 100 2,000 68min

90 95 100

3.4xl0 6 3.5xl0 6 3.5xl0 6

10 20 30

40 50 60

e 400

.3

1150

500

.5

1300

600

.55

1400

230 4hr

Table III

Table III presents a summary of the search zones and

tises reguired to obtain a solution for seme of the problems

considered in this thesis. The amount of CPU time reguired

for a solution, especially for problem III-E, may seem

excessive. However, there are several considerations which

should be mace prior to arriving at this conclusion.

(1) The equations of the system do not have to be

written, programmed nor debugged if the system's

62


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component transfer functions are known.

(2) lime domain requirements do not have to be

translated irto frequency domain specifications

for system simulation and design.

(3) A systematic search is carried cut to obtain

the cptinum parameter settiEgs. This assures that

with valid bounds on the variables an acceptable

solution will be obtained with the first

optimization attempt.

The time required to perform the above steps in the cesign

prccess by conventional means may result ic many more hours

of CPU tine than if the program CADS were used from the

beginning of the design process.

Several means of reducing the computer time required

fcr optimization were previously outlined in section II.

The most significant reduction is obtained by keepinc the

number of integrations required for system simulation to a

nirimum. Elcck diagram reduction of the system should be

accomplished whenever possible. The example of the Ward

Lecnard drive system shewn on page 26 can be reduced tc the

simple system shown in Figure 30 by block diagram reduction.

^o Krs+it *s-t -rr\.

^6Z,SH

o

flGUEE 30 ElockReduced Ward-Leonard Drive

63


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131/244

Section III.E. showed the effect of using different

cost functions to measure the quality cf the optimized

response. Although the integral error squared criteria is

often used to judge a system's performance, the user should

carefullj ccrsider how to best define a cost function which

will properly penalize deviations of the system response

from the desired response. A small error (err < 1) souared

becomes even smaller. If all the errors are small, the IES

is a valid cost function but if the system response involves

laroe and small errors a weighted cost function will have to

be used. One nethcd cf arriving at a properly weighted costfunction is to sinulate the system using first estimates of

tte variable parameters and recording the sum of the errors

over the different portions of the response. A ratio cf the

errors can then be used to arrive at proper weighting

factors for each time section cf the response.

BOXEIX will continue the optimization process, workingin the seventh significant digit until it can no longer

reduce the ccst function. Often an acceptable solution for

the system parameters has been found long before th ccst

function has teen reduced to its minimum. A judicious use

cf CADS nay be made by evaluating the system response after

ten to fifteen minutes of run time to see if an acceptable

solution has been found.

B. CONCICSICKS

lime donain cptinization using the CAES program is a

simple, straight forward process which dees not require an

in-depth analysis cf the system being optimized. Economic

use of CPU times does dictate that intelligent starting

values and bounds be placed on the variable parameters which

are to b optinized.

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APPENDIX A

Block Data Card Format

|1 (11 1 20 |40 |60

ELKCCD=EEVV G P Z

CC = PC5ITICN NUMBER OF THE BLOCKE = TYPE OE BLOCK (NUMBEB)EE = INEC1 NCDE NOMBERVV = OUTPUT NODE NUMBERG = VAIUE OF GAINP = VAIUE CF THE POLE*Z = VAIIE CF THE ZERO*

* Fcr block type 4, 8 is read into the P location and

Hn is read into the Z position.

* Per block type 5, is read into the P positionU

and 9 is read into the Z position.

* Bcr block type 6, 6 is read into the P positionC

and Qe is read into the Z position.

EXAKPIES

G

EIKC11=0102 10.

G P

BIRCi2=0304 1. 5.

G P Z

BIKC23=0405 1. 10. 5.

67


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E1F 1Ci*= 1C06

BIK115=1207

EIK126=0108

G d

200. .2

G u A10. 20. -5.

G &c %1. 3. 10.

UJ

*ee Table I.

68


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141/244

APPENDIX B

Plant Flow Chart

FUNCTION FLAM (C)CI>ENSICN G < 25 ) , P(25) Z(25), FLAG(25,25),G(25), TFACCT(25), CNGCGK25), DPVIN(25),NF(25), IM25), IV(25), X2(25), X2DOTC25),

CM

CRI V E (25 1 , TFA(25), IC(25), IC(25), IE(25),TFACIT(2CC1) , XCATAOGGl), C(25)

REAL *87FACL7REAL *8XCATA

REAL *87 FA, TFA COT, T f OT ,CRI Vc CI*G,CyGDOT,DRVIN,TF

REAL *8F1,F2

RE*L *SX2,X2i:C7

CCMNCN 7,CT,TF,7FACUT,XDATA,M3,ICCNT,NEC, I5KIF,I7F

FCP OFT RUNS, INFIEIT REAC STATEMENTS AFTER REACING

* IF * .ISKIF-1 *

+

i 1 I 5 |

I ISKIF=2

I

***FEAC (5,24) f^,I3ET

INITIALIZE CCLNTEFS

ICLT =CI\CLT=CFl =CT1-2

MlN55ICK

=C.5CG*F1= C= C=c= 2*N

69


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143/244

FEAC 1NFLT CAT/*

444444444444444

. 44444444444

4444444444444

2 4444444444444444

444444444444444444

4

4

4 CC2

4 I =lTN4

I

***RE/>C (5,23) IC(I ),IC(I ),IE(I ),IV(I ),G(I) tP(I)tZd)

SET TMCIT = CITF17 CF LAST ELGCK

* * .. * IF * .

*(IV (I)-IVCCT J.LE.Q *

I\CLT=IV(I )KIT =IC(I)

= IF * .

icm.sc.i *

F

IFIC ( I J.EC.5

IFIC(I).EC6

:

I 2

I Ml = Nll4l...........

I f^55 = N554l

| K66=N64l

2 44444*4*fcR]TE (6,26) ICC I ) , I C ( I ) , IE ( I ),IV(I),GC ),P(I ),Z(I)

70


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145/244

SET ThACIT = SPECIFIED ThETA, IF ANY

* IF *

* .ISET.NE.C

* *

***WRITE (6,27) ICUT

I ICIT=ISET

Ml =4*N11^ 5 5 =^Nf sN66 =4*K(:tNEC = N-1

SCAN FCR INFITS

CONNECT SYSTEN E\ SETTING FLAG=1. FGF CONNECTEC ELKS

I

4 +

4 CC +4444444444 4 44 4 J=ltN +4 4 4444 44 4 CC4444444444 44 4 K = 1,N4 44444 | FLAGCKtJ) *C.O44 I44 I

4 4 . 4 * .4 * IF * *4444444*4 I V ( K ) . E C . I E ( J ) *4 * . *4 * . . *4 * , *+4 F44

TAKE eLK TYPE ANC SCIVE ECNS ONE EY CNE

j FLAG(K,J)=1C I

71


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147/244

CLE^ PING CLT FEGISTERS ANC INITIALIZING CCLNTEPS

I

i

4

4444444444444

6 4

+4- tQ

444444444 I ICLP=1,N

4

TKA(ICIF) = C.CO7HACGTUCLF) =0.C0CNG(ICLP) =C.COCNGCCT UCLF ) = GCOCFVINUCLF) =O.C0> 2( ICLP ) =C.CO

4444444J >2CCT(KLP) =O.CC7 =C.QCC

NRS CCNTFCL ENTPV FCIM IN INTEGRATION SUBROUTINES

NF2 =1N P2 =1b?4 =1

K2 CCNTFQLS H- 1 C t- BLCCK EQUATION IS BEING SOLVEC

'VICCNT IS ISEC TC CCNTPCL FRCGRAM FLOW

T ICCNT=C _T

IWAIT IS LSEC TC CCNTFCL TIME. TIME IS STEPPEC EVERY

FIFST *NC THIPC FASS THRU INTEGRATICN PCUTINES

IWAIT=0IT =C

ILAST'S CCNTFCL FFCGRAN CATA AND PROGRAM EXIT

ILAST=CI5LAST =CI6LAST =0

72


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149/244

SPECIFY ANY F*F^ETEFS TC BE OPTIMIZED HEREi

EG. F(1)=CFE ECN TC SCLVE

a * IF * .

* IfcMT.EC.C*

* . *

. * IF * .* I C ( M 3 ) . E C . 1 *4 . *

F

I 7=T4H2

11

73


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151/244


152/244


153/244

5T4RT SCLITICN

type cf*E Ecimcf^s

SCLVES 7I-AC17 = G*THAIN

11 THMV3 ) = G (N3)*CFIVE(M3)IWAIT'IhAIT+lUAST=UAST + 1

. * IF * .*(ILAST,EC.N11 ).ANC.(ICGNT.EC.NEQ)* . . *

* . * T | 21

F

I 18 I

TYPE TfcC ECNS

SOLVES Tl-ACLT/TI-AIN = G/IS + P)

12 7HADG7 (N3) *-P(M3 )*ThA(M2 ) +G(M3*CPIVE


154/244


155/244

I

S=FKLDE2(CNG,CNGDC7,NP2)

7HA(M2) =XMn+l

* IF* S-l.C

*

I 17 I I 16 I

7YFE FCIP ECISS.

SCLVES ThACLT/TI-MN = G/(S**2 + 2*DEITA*WMS + kN**2>

14 V =CCFLXF,Zf C-,CRIVE,X2,CMG,NR4)

XkAiTxkHi+x

a * IF * .* V-l. ** *

I 17 I 16 I 21 I

7YFE FIVE ECNS

SCLVES ThAGIT = G*ThAIN FGR /G*7MIN/ < SA7 LIM7S

I 7HMN3) = GCN2)*0FIVE


156/244


157/244

. * IF * .* TM


158/244


159/244

* IF * .*(I6U$T.EC.N6).ANC5$ MS EEElv NACE FCP EACH ECN

.]-NP2 - ^P2-UI^P2 =NF2*1fvp4 =m;^ + iH =CICCNT=C

INCREMENT

* * .. * IF * .

* IUAIT.EC.ICK ** . . *-

F

I 7 i

I IWAIT=C

2C ***WRITE (6,20)

STCF

78


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161/244

21

22

CATA CCLLECTICN FCINT. CATA IS R5CQRCED AT 5NC IF

CCI>FLE7E TIME STEF(CT)

IT = 17 + 1

TFACLTUT ) =THA( ICIT)

TEST FCF ENC CF FIN

. * IF * .* T-TF ** . *

+

cc 23 |

I

RESET CCLNTEFS FCF NEXT PASS THRU EQUATIONS

NF2 =1f>P3 =1NF4 =1N3 =CICCNT=CIfcAIT=CILAST=CI5LAST =0ULAST =0

23 I FLANT=1.

24

25

26

27

28

29

20

FETIPN

FCFN AT (I22> f I2)

F C F y A T < 3 > , I 2 , 1 1 , 1 > t 2 I 2 , 8 X , 3 E 2 C , 7 )

FCFNAT (/t5h ELK f I2tIlt2H = , 21 2 ,6X, 3E20.7

FCFM7 ,'TFETA CUT IS THA ( , 12, ' ) '

FCFNAT (4CH *** EGN SWITCH CCNTRCL DID NCT fcCr


162/244


163/244


164/244


165/244

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LIST OF BEFEBENCES

-DC_kj2arJ? Papers in Electrical En^iiieering and Cogputer

Scierce , v. K, edited ty Director, S. W., Dcwden,

HutchinscE fi Boss, Inc., 1973.

McDaniel, H. L. Jr. and Mitchell, J. B., An

Inncvative Approach tc Compensator Design , NASACB-2248, Kay 1973.

Office cf Naval Besearcb CB 215-238-1, Optimal linear

Control (Formulation to meet Conventional resign

Sp.ecs_.J_ by G. L. Hartman, C. A. Barvey and C. E.

Muller, 2S March 1976.

Cantalap iedra, j., low Order Mcdles for DynamicSystems M M-S. Thesis, Naval Postgraduate School,

Mcnterey, 1972.

MacNamaia, M. A. S. , A New Optimization Method Alied

% flutcpilot De sig n, M. S. Thesis, Naval Postgraduate

Schccl, Mcnterey, 1975.

Eox M. J., A New Method of Constrained Optimization

and a Ccmparison With Other Methods , Computer Journal,

8 April 1965, p. 45 - 52.

Fitzgerald, A. B. and Kingsley, C. Jr., Electronic

Machinery, p- 186, Fig. 4-22, McGraw Hill, New York,

1961.

Thaler, G. J., Desi gn of Feedback Systems, Dcwden,

Hutchinson S Boss, Inc., Stroudburg, Pennsylvania,

1973.

116


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INITIAL DISTRIBUTION LIST

No. Copies

1. Defense Documentation Center 2

Canercn Station

Alexandria, Virginia 2S314

2. Litrary, Code 0212 2


McDterey, California 93940

3. Department Chairman, Code 52 2



4. Professor Gecrge J. Thaler, Code 52TE 5



5. Lt. Larry P. Vines 3

119 Indian Lane

Oak Ridge, Tennessee 37830

6. Dr. Jerrel R. Mitchell 1

Department cf Electrical Engineering

Mississippi State University

Mississippi State, Mississippi 39762

7. Mr. Jay Cameron 1

IB*

Monterey and Cottle Roads

San Jose, California 95114

117


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8. Mr. Peter Gramata

IBB


San Jcse, California 95114

9. Mr. Bcfcert Mcintosh

IBM



10. Mr. Martin Dost

IBMMonterey and Cottle Roads


11. Mr. Ren Palmer

IEM



12. Mr. B. M. Syn

IEC



13. Professor A. G. J. MacEarlane

Dect. cf Electrical Engineering

University of ManchesterInst, cf Science and Technology

Manchester England 067609

14. Dr. Bui-lien Rung

Dept. cf Applied Sciences

Oniwersite du Quebec a Chicoutimi

930 est, rue Jacgues-Cartier

Chicoutimi, Quebec

G7H iEI

118


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15. Mr. Dave NobleIBMMonterey and Cottle RoadsSan Jose, California 95114

119


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19 AUG79 260 11

Thesi s166W3

ff7 ^Computer automated

design of systems.

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1661*1*3V6d97 Vinesc - 1 Computer automated

design of systems.


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Computer automated design of systems.

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Computer Automated Design of Systems

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