Computer Automated Design of Systems

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COMPUTER AUTOMATED DESIGN OF SYSTEMS

Larry Paul Vines

I

EY

.__ KNOX LIBRARY, POSTGRADUATE SCHOOL
1

iONTEREY, CALIF. 9394Q

NAVAL POSTGRADUATE SCHOOL
Monterey, California

THESIS
Computer Automated Design of Sy stems

by
Larry Paul Vines

June 1976

The.sis

Advisor:

George J. Thaler

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Computer Automated Design of Systems
7.

Master's Thesis June 1976
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Larry Paul Vines
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Naval Postgraduate School

Monterey, California 93940
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KEY WORDS

(Contlnua on rawaraa alda

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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 system design is presented. The emphasis is on compensator design but the method is applicable to the design of any system with free parameters. Signal representation and system response are in the time domain. The inputs required from the engineer are the system configuration, the desired output response and the free
DD
1

JAN 73 1^73
1)

(Page

S/ N

EDITION OF NOV 61 0102-014-6601
GEfW»..n Hrtm

£»l.f..'

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

DD

Form
1

1473
SECURITY CLASSIFICATION OF THIS F*OEfWi»n Data Enlmfd)

Jan 73 S/N 0102-014-6601

Computer Automated Design
of

Systems

by

Larry Paul Vines
Stati Lieutenant, iant. United States Navy

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

1970

Submitted in partial fulfillment of the

requirements for the degree cf

MAST5H OF SCIENCE IN ELECTRICAL ENGINEERING
from the NAVAL POSTGRADUATE SCHOOL June 1976

DUDLEY KNOX LIBRARY NAVAL POSTGRADUATE SCHOOL'" MONTEREY, CALIF. 93940

ABSTRACT

An automated digital computer technique of control

systea

design

is

presented.
with
free

The

emphasis

is

on

compensator design but the method is applicable to the
design cf any system

parameters.

Signal

representation
domain
The

and

system

response

are in the time

inputs

required
A

from

the

engineer are the
and

systen configuration, the desired output response
the free parameters.
is then used tc minimize a specific cost function

parameter minimization routine
and
A

to set the free parameters.

graphical output of the
response
is

desired response and actual system

then

produced fcr comparison by the engineer.

TABLE Cf CONTENTS

I.

IN1FCDUCTION
EROGEAM DEVELOPMENT AND IMPLEMENTATION
A
E

10

II.

12
12
15

GENERAL
MAIN PROGRAM

C

FLANT
1.

15
16

Block Data Card Block Connections

17
17

Drives

Standard Transfer Function Blccks

18

Parameters to be Optimized
D

19

DESIRED RESPONSE
CCST FUNCTION

(XDATA)

20 20

E
F

(PERFORMANCE INDEX, PI)
,

MNIMIZATICN ROUTINE
1.

22
22

General

2.

Explicit/Implicit Constraints and
Start Points
,

22

GRAPHICAL OUTPUT
III.
A B

23
..,

INVESTIGATION OF PROGRAM PERFORMANCE
IATA INPUT, AN EXAMPLE

25
25

1ACHOMETEF FEEDBACK
CASCADE COMPENSATION CASCADE LEAD COMPENSATION
SERVO SYSTEM COMPENSATION

28
34
41

C
D
E

47
...

IV
A E

DISCUSSION, CONCLUSIONS AND RECOMMENDATIONS
DISCUSSION

61

61 64

CONCLUSIONS

C

V.

RECCMMINDATIONS FOR FUTURE WORK APPENDIX A BLOCK DATA CARD FORMAT
APPENDIX APPENDIX
E

65

67

VI.

PLANT FLOW CHART
CADS MODIFICATIONS FOR

69

VII.

C

PECEIEM III.
VIII.

E

80
82

CMS

EROGEAM LISTING

LIST OF FIGURES

1

Ijfical System to be Optimized
CADS Erogram Flow Chart

14

2
3

24

Hard-Leonard Speed Control System
lachcmeter Compensated System
CADS Elcck Diagram of Tachometer System
CADS Compensated Tachometer Feedback System

26
28

4 5

29

6

Sesfcnse
7

30

Type Three Block used for Pseudc Tachometer

Feedback
8

31

Eseudc Tachometer Feedback

32

9

Pseudc Tachometer Compensated System^

Response
10

33
34 35

Third Order Plant to he Compensated
lag Compensated System

11

12
13

Occcirpensated, Compensated BODE PLOT

36
....
37 33

Conventionally Compensated System Response
CADS Lag Compensated System
CADS Compensated System Response

14
15
16

40
,

Plant for Lead Compensation

41
.

17
18

Conventional Lead Compensated System Response
CADS Elock Diagram for Lead Compensation

42

43 44
45 46
..

19

CADS lead Compensated System Response
CADS Compensated System Block Diagram
CADS Relaxed Boundary System Response

20
21

22

Serve Drive Mechanism, Original Compensation
CAES lagraa of Servo Drive Mechanism

48 49
51

23
24 25

Original Servo System Response
CADS Compensated C

System Response

52

26

CAES Compensated C -1 System Response
N

54

27 28

C2ES Compensated C -2 System Response
N CfiDS

55

Optimized System Response
0.5

57

29

Serve System Response, CADS Values with
8

=

60 63

30

Elcck Reduced Ward-Leonard Drive

ACKNOWLEDGEMENTS

The

author wishes to thank Dr. George Thaler fcr his
and

guidance

encouragement

during

the

course
M.

of

this

investigation.
fcr

Kris Butler, Ed Donnellan and

Anderson of
Lastly,
I

the W. R. Church Computer Center deserve special recogcitioc

their

patient

and professional assistance.

thank my «ife, Patricia,

without

whom

this

thesis

could

never have teen accoaplished.

I.

IHTBODUCTION

The past two

decades

have

been

witness

to

an

ever

increasing

use

ci the digital computer.

Engineering usage
Classes in
have
become
the

and design are prctably the most important justif icaticn for

the large memory, high speed computers of tcday.

computer application to
part
of

engineering
curricula
on

problems

the

established
engineering
or
at

at most universities.

Numerous papers have appeared

the
[ 1 J

application
long,

of

computer
were

to

problems
best

which until recently

unsclvatle

required

tedious

procedures which gave only approximate solutions.
Contrcl system engineering has
relied
met

increasingly
and
a

on

computer
design

sinulaticn

of

large scale systems to verify that
have
teen
to

specifications
a

make
any

mcdif icaticns to

system even before producing

prototype.

There are

programs
circuit,

available
or
of

to

simulate
first
order

virtually

electrical
eguations.

control system, either in transfer

function form or as a system

differential
design

ethers
which

draw the Bode, Nichols or Nyguist plots

of open and clcsed loop systems.

Some programs help

compensators

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 model for large order systems. the optimum reduced order MacNamara [5 1 went even further and used an iterative method
to

find the optimum compensation for an aircraft autopilot.
and
to

It would appear that these techniques could be extended

applied

the

direct

simulation

and

design of control

10

system circuits in the time dcmain.
The ictert of this thesis was to develop
a

user oriented
of

program which could
systems
means
of and

simulate
the
a

a

wide

variety

control poles and

determine

values of the gains,

zeros necessary to produce

desired respoDse.

A

convenient
(1)

data card input was to be provided to specify

the control system which was to be

optimized
optimize

and

(2)

the

desired response. routine
(ECXE1X)

fi

locally available function minimization
the

was to be used to

simulated

system's output.
To

simulate

the

system
which

which

is

to

be

optinized,.

transfer

functions

are

commonly
so.

encountered

were

reduced to first order linear differential eguations.

These
data

egcations were then programmed
card
the

that the transfer function

blocks could be connected in an arbitrary
input.

fashion
the all the

by

Several common nonlinear transfer blocks were
The program will simulate

also provided.
known

system
unknown

with
or

parameters
parameters
to

and
be

then

allow
by

adjustable

fixed

computer

optimization routine to achieve the desired response.

11

II.

PROGRAM DEVELOPMENT AND IMPLEMENTATION

A.

GENEEA1

Any program which is to be of

maximum

benifit

tc

the

user
which

must
is-

fcave

a

simple means of data input and an output

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
is

eguatiens.
"present such
(CADS)
,

The program should be a readily usable tool and

not a profclei in itself.
a

The ictent of this
used

thesis

to

program, Computer Automated Design of Systems
is
of

which

readily
the output.

for

simulation
as

with

optimization

Optimum in the sense that the
near

output response cf a given system configuration is
the desired response as possible.

Most

ccntrol

system
CADS has

design
the

starts
common

with
a

a

proposed

transfer
built
which
or

function block schematic to achieve

desired time
functions
tlocks

domain response.

transfer
The

into the program and the input data can come directly

from the proposed system
are

schematic.

transfer

available for system simulation are presented in
These blocks should be adequate, either seperately

Table I.
in

various

cascade

combinations,

to

represent

most

ccntrol systems.

12

CP

SYMBOL

EQUATION SOLVED

EICCK

9i

.

G

9,.

9o= Q9 V

9i
,

G

9o

S*P

9

»

-P

9

*

G

9;

ei.

G

S+ 2
•

9
ft

q

»

-P

9o +

G(9i + H9\)

0L_

G
S 2 +Z6W5
2
*• co

-

9

o

^

9

=

n

-25^6o-^9

+6e

«uVx~ X
8i.

6

=

G 9C
6
«.

9L ^

G9

i
fie
9
"«.

9U

X

6l
9c

-9-0—

9o
9

--

9

-

*

—Zj
8;

=

°w

G6 ^ 

9.

3

)
t

TAELE I Program Transfer Functicr, Elocks

13

-

To

use

the

program

the engineer must know the system

configuration in transfer block form and the desired second second order If other than a order output response. response is desired, this may be easily specified but
requires
response.
it,

seme
the

knowledge
given

of

how

to

input
to

the the

desired
desired

CALS will take the transfer block system, cennect

compare

system

response

response and set any free parameters to achieve the

closest

match possible of the system output to the desired response.

Figure
program

1

is representative of the

type
In

of

systeir
1,

the
the are
X,

car

simulate

and

optimize.

figure

parameters of the
I,

numbered

transfer

function

blocks

known or fixed by equipment limitations.
and
Z

Transfer blocks

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

<y

i

o-

3

O

+ -6- * -Q

—

IIGQBE

1

Typical System to be Optimized

14

E.

MAIK EBCGFAM

The

KAIK program was developed to control the selection
to

of the sutf unctions used in the optimization process and

compute the desired response cf the systems which were to be

optimized.

A

second order step response is calculated

from

eguation
XEAIA.

(I)

as the desired response and stored in the array

G

e

=

u(t> S2 + 2
8

(i)

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, CS£P and

INTEG were investigated in an attempt to adapt them

tc

the

general

crotlem

specified
equations
all

above.

These programs reguired

either the incut cf the system's state

variable

eguaticns,
are
useful

Laplace
for

transform
if

or the node-transfer function

pair for each system simulated.

These programs

simulation

of a system's parameters have been

specified.
these

Eowever, each simulation is problem specific and
are
A

technigues
cf
cf

not

readily
card

integrated

with other

sutfuncticn ticgrams.
capable
variety

simulation
data

subprogram
of

which
a

was
wide

accepting

input tc simulate

ccntrcl

systems

and

working

with

other

existing subprograms had to be developed.

15

Subfurcticn PLANT was developed to simulate the systems It provides the versatility tc he optimized. which were
necessary to simulate numerous system configurations and is capable cf working with a minimization routine to optimize
the variable parameters.
T^e

program

reads

the

transfer
in

fucction
state
the

blcck

connections
format.

from data cards.

This data is

then used to automatically set up the

system

eguatiocs

variable

These state variable eguatiocs are

solved as first order, ordinary

differential

eguatiocs
are
in

by

Bunge-Kutta-Gill
The

forth

order method.
of

The programming

was done in Jortran IV and all calculations

double

precision.

capability
in

connecting

the

transfer
forward,

fucction
f eedback

blccks
,etc
.)

any

configuration
having
each

(feed

is

possible

with this program.

Simulation
withic
the

flexibility is provided by

blcck

system capable of accepting an external forcing function.

1 -

JiccJS Data

Card

Several

possible
means

ways

of

inputting
a

the

data

necessary tc simulate arbitrary system
investigated.
directly fica
a
A

configurations
chosen
as

were
the

of defining

system configuration

blcck diagram schematic was

most preferable method.
of the system.

Using this approach, each data card

was designed to be directly related to a

transfer

fucction
to

This resulted in having direct access

the

transfer fucction parameters for optimization.
1c

simulate the system,
a

a

data card is prepared for
The data card

each transfer blcck in the schematic diagram.

ccctains
the

field cf cumbers which specify the block number,

the type cf transfer functioo ccctained

within

the

block,

input

cede

and

the output node to which the blcck is

connected and the values of any parameters associated with general fcrmat of the data the transfer function. The

16

.

cards used tc input the system configuration is shown telcw.

|1

]11

|

20
G

|

40
P

|60
Z

(Column Numker)

ELKCCE=EEVV
Hhere:

CC
D

=
-

Position number of the block
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

The prograi then initial entry into the simulation routine. connects the transfer function blocks in the proper order by

comparing the input node number of a
ncde
nunfcer

block

to

the

cutput

cf

every

other

block.

Whenever

these two

numbers are equal, the blocks are knowc to be connected and The flags are then used to is set equal to 1.0. a flag

identify the input drives to each block.

3

Er i v es

The input guantity

to

each

block
to

is

called

the

EBIVE.

The

program

was

designed

allow for multiple
was

inputs to the system being simulated.

This flexibility
to
the

achieved
plus
anj

by making the input to each block equal to the sum

of the outputs of all blocks connected

input

node

external

forcing

function

(DBVIN)

feeding the

transfer tlcck.
shewn in equation

The input DRIVE to a block is determined as
(II)
.

17

DBIVE(i) =
IHA(j)
j

£

THA

(j)

*FLAG

(j

f

i)

DRVIN

(i)

(II)

is the output of block j.

FLAG(j,i)

is 1.0 if block

is connected to block i and zero if it is
(i)

cot

connected.

EEVIN

is

any external forcing function specified by the
i.

user whicfc drives block

DRVINs

must

be

inserted

in

subroutine
the

IIANT

as
=

Fortran

IV statements.

The standard

program has CSVIN(1)
system.

1.0 specified as a unit step incut to

Problem III-E in the section Investigation of
varying

Prcgram Eerfcrmance demonstrates how multiple, time

inputs are tc be inserted in the program.

4.

Standard Transf er Function Blocks

The

transfer

function
I.

blocks available for system

simulation are shown in Table
were alsc found to

These blccks were selected
They
cr

because cf tteir ccmmcn usage in the modeling process.
be

adequate
to

either

separately
most

in

various
systems.

cascade

combinations

represent

control

The

transfer

function

equations

for each tjpe of

block were written in state variable format and stored in an

array

named THA.
with

The program reads a number from a block's the
type
of

data card which specifies

transfer
type
of

function
equation

associated
solved

the

block.
by

The system equations ar€ then
the

seguentially

selecting

assccitaed with block number one, solving for its output and integral etc. The then sequencing tc block number two,
eguations
were
are

solved by a modified RKL fourth order method
routines.
a

in the subfunction

Three

integration

routines
The

necessary

tc

store the intermediate results obtained

for those equations involving

double

integration.

four

sutfunctions,

RKLDE2,

RKLDE3,

CCPIX

and RKIDI4 are
of
a

called by IIAKI.

BKLDE2 is used for the

integration

18

type

twc

blcck.
and

BKLDE3 is used for integration of
to

a

type
for

three

blcck

store

intermediate
calls

quantities
and

sutfuncticn

CCPLX.

CCPLX

RKLDE3

RKLDE4

for

integraticn cf a type four block.
Nc

prevision has been made for the input of initial

cenditiers cr the integrators.
must
run
as

Therefore, the
(DT)

integrations
the

start
time

at
fc€
.

time equal zero.

The user must specify the
and

step size tc

used for integration
possible.
DT
=

problem
as

(IF)

To conserve computer time ET should be made

large

as

TF/1000

is

suggested
shewn

appropriate
letting
was
TF

in most cases.

Equally important is that TF be

as small as pcssible..
te

Use of the

program

has

that

greater than the transient response time of
For preliminary analysis TF

the systea is rarely justified.

kept tc enly slightly longer than the time of the first

undershoot fcr second order dominant
teen included as Appendix B.

systems.

Because

of

the complexity of subroutine PLANT a flew chart of PLAKT has

5

.

Pa^am eter s to be Optimized

The

parameters
ty

of

the

system

which

are

to

be
be

optimized

the

minimization

routine

EOXPLX

must

specified in PLAN1.
labeled
variable
•C
l

The minimization

routine

returns
equate

the

trial values cf the variable parameters to PLANT in an array
.

fiegular to

Fortran
the
be
to

IV

statements
in
the

the
If a

parameters
pair
=

values
in

this array.
P(i)

pcle-zerc
statements
and
Z (i)

were
be

optimized/
PLANT:

following
C(1)

wculd
C (2)
.

inserted

The location fcr the preceding

statements

is clearly

iEdicated in the program listing for PLANT.

19

D.

DESIBEI* BESPONSE

(XDATA)

The ciitciia against which the system response

will

be

compared
system.
en

will

vary
of

according
second
a

to

the

application cf the
is

Since most design work on control systems tasis
tc

done

the

order dominance, the program was
order

written

provide

second

step

response

with

adjustable

5,8
n

and

gain as the basic criteria against
The desired 6,

which the sinulated system will be compared.
H

and gain are read in as input data.
n

The step response is
and

then computed in the main program by subfunction RKLEEC

stored
order
by

in

the

array

called XDATA.

Any ether time domain

response nay be specified by the user by removing the second
step

response
III-E

equations and replacing them with the
An example
is

equations of the desired response.
problem
is

provided

in Investigation of Program Performance.

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

curve

desired, setting the input variable LEAP =

1

will

cause the program to bypass the computation cf rhe
secend order data equation.

standard

E.

C0S1 FOSCTION

(PERFORMANCE INEEX, PI)

The

achieved
response
this
tc

system
and
for
the

response

is

compared
is

witt

the
The

desired
ainimize
the

difference
parameter
system

the error.

program searches
user

the

settings
as

which

will

error. The cost function may be specified by

weight

the

outputs

desired

in

subfuncticn FE.

The default cost function cf the program is

20

rr

tte integral errcr sguared.

J =

/

(Err) 2 dt.

An

example

of a weighted cost function is presented in Section III.

21

.

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
called

optimization

method has been programmed at the

Naval Postgraduate School by B. B. Hilleary as a

subroutine

ECXFLX.

This subroutine will find the minimum cf an
(cost

arbitrary function

function)

subject
lower

to

arbitrary
Explicit
on

explicit constraints and for implicit constraints.

ccnstraints are defined as upper and
free

bounds
may
178).

the

parameters.

Implicit

constraints
P
1

be arbitrary

functions of the free parameters (e.g.

P
2

<

Two

function
and
The

subprograms
implicit
EOXPLX
method

are used to evaluate the

objective function

constraints,
uses
which

FE

and

KE

respectively.
values of
the

to search for the

free

parameters

minimize

the

cost

function is explained in the computer program listing.

2.

Ex elicit/ Implicit Ccnstraints and Start Points

ECXFLX
space, where
and
n

searches a feasibility region
=

(n-dimensicnal

number of free parameters)

defined ty upper
a

lower

bcunds on the free parameters for
The
the

minimun cost
by

function.

sialler
nore

the

region

defined

these

rapidly the program will converge to judgement Good engineering the optinum parameter settings. will be necessary to keep the feasibility region as small as

boundaries,

22

possible.
data

The boundaries of the search regicn are read from
by the main program as the upper bound
(XL)
(XG)

cards

and

the lower bcurd

for each free parameter.
may be any arbitrary furcticn
If an

Implicit
of the free

constraints

parameter desired.

implicit

constraint
bj

such

as

the

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

seme number is to be evaluated, it must be supplied
user
to

the

subfunction KE.

No implicit constraints were

used in this thesis.
The

starting values cf the free parameters
data
cards.
A

(XS)

are
of

read in by the MAIN frcm

good
or

choice
Bode

starting methed
of

values

will dramatically reduce the time required
A

for optiii2aticn.

preliminary root

locus

plot

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.
select, by data card input,

The user may
PPLI
which

either
Every

subroutine
fifth
and

provides

a
a

high speed printer plot or subroutine PIC which

provides

calcomp graph.
the

integration
THAODT
is

point

stored

in

arrays

XDATA

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.

23

1

*

*

No

.

Runs
r

XDATA
6
>

No. Var dt, TF
• •
i

wn

Type Graph

No. Trials XU, XL XS

•
•
,

«

t

•»«
1
1 1 1

MAIN
i

—.

.»

—

m

w»

—

—

Plot output response

L

1

1
t

I
1

1

1
1

i

t

1 1

BOXPLX
n
i

~"

*

]

s.E

1

1
1

1

BLOCK Data Input Drives
[

1

NO
'

J

I

t

FE
1

YES

/K>| < ^
l

>v

I

/^Implicit
/ Met

xTli Constraints

i
i

r

*4»

t

1
i

PLANT System
Si .mulation

i

—

«...

RKLDEs

Optimization
Simulation

FIGG3E

2

CADS Program Flow Chart

24

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
a

bock single variable, single input system to

multivariafcle

operatioDal servo drive system which has multiple inputs and An example of how a schematic discrete level feedback.
representation of a system is prepared for input to example the progran is presented prior to considering the
diagram

problems.

A-

EATJ INECT, AN EXAMPLE

Ihe Rard-Ieonard drive system [7] shown schematically in The gain cf the (a) has two variable parameters. Figure 3

amplifier and
adjust
tie

the

tachometer
response.

feedback
To

are

available

to
a

system's

simulate the system
I.

blcck diacraa representation of the system is drawn as shown
in Figure 3
(c)

using the transfer blocks frcm Table

Cortjt

fTAc*

<a)

25

URvjnCj)

GO)
S*P<0
t\-

-5G>

GCO

GO)

S+PW

-($

nn
G(4)

EC
Gfs)

(c)

fIGCBE

3

Ward - Leonard Speed Ccntrcl Systei
(A)

tsing Feedback.
(B)

Schematic Diagram
(C)

Elock Diagram

Blcck Diagram

Using CADS Blocks.
= K/t
=

Hfcere

G(1)
G(2)

P(1)

=

1/T

Km/Ba
1/J
P(3)

G(3)
G <4)

=
=
=

=0

-Km
-Kt
= Er =

G (5)

DEVIN DEVIN
The

(1)

(3)

-T t
are
then
as

nodes

cf

the

block

diagram
The

nunbered
shcwc
in

sequentially
also

1,2, ...,n.

blocks between the nodes are
1,2, ...N
(N)

nunbered
3

sequentially
Data

Figure
input
within

(c)

.

cards

are then prepared for each

blcck which specify the block number,
ncde,
tfce

type

cf

blcck,

the

the

output node, and the parameters contained
In Figure
3

blcck.

(c)

,

for example, block

1

is

26

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
number

the

blocks
any

by
a

setting
block.

a

FLAG

=

1.

whenever the input node
of

nunber to
other

block is the same as the output ncde
The

input
of

to a transfer block is then

determined tc be the
connected
step
tc

sum

the

outputs
any
The

of

all
has
a

blocks
unit

the

input

node

plus

external forcing

function driving the input node.
specified for DEVIN
(1).

program

If this is the only input to

the system, no action is necessary on the part of the
If if there

user.

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

other

external

forcing

functions

such
For

as

EEVIN

(3)

they must be specified and placed within the body
the
in

cf the subfurction PLANT as Fortran IV statements.

example

shc%n

Figure
DEVIN
(3)

3

(c)

,

a
)

card with the equation

=

f (T

wculd have tc be inserted preceding the drive equations.

An

example cf hew multiple, time varying drives
is

are

specified

given in section III. E.
When cptinizing a system, seme of the

input

quantities

will

be unkcewn or variables.

These variables must also be
To optimize the variables

assigned *itfcin subroutine PLANT.
K
1

and

K
t

cf Figure

3

the following two statements would be

inserted in IIANT:
G(1)
= C(1)

G(5)

= C(2)

where

C(1)

and

C (2)

are

the

variables

which

will

be

optimized by subroutine BOXPLX.

27

£.

IflCBCMJlIE JEIDBACK

The
had an

first
exact
Ac

optimization problem attempted was one which
solution
that
can
be

found

by

algehraic

methods.

instrument

servo [8] with unity feedback and
1000

forward transfer function
G(S)
=

S(S+10)
was to be ccopensated with tachometer feedback as

(III)

shewn

in

Figure
the
5

4.

The

enly

specification
that

fcr

the

system's
a

performance cf this single variable, secend order system was
simple
=

requirement

the closed leep roots have

0.7.

D

f

i

O
\ f-

\OOa
SCs+io)

K4 S
.

IIGUEE

4

Tachometer Compensated System

The

systeii shewn in Figure 4
tc

was redrawn as Figure

5

in

order

achieve

the

tachometer

feedback.
5

The
is

characteristic
S2

equation
(10 +
= t

of the system shown in figure
103 k
t
)

S

+

103 =

(IV)

The
IV.

required

K

0.0343

may be calculated from equation

28

-) >Q

loeo
S +10

/

«C

m

s

___

«t
1

FIGDBE

5

CADS Block Diagram of Tachometer System

CADS was programned to

optimize
to

the

system

with

the

variable,
0.C1
the
Wn =
K
t

F

,

specified

be

between

the

limits
tc
5

<

K t

<

1.0.

The desired response was specified

be
0.7,
be

standard
100C.
-

seccnd

order

step

response

for
K

=

CADS determined the optimum value for

to

0.03*43.

Figure

6

shows the system's step response and

the desired response are virtually superimposed.

29

CM

00

o

Ho D O < S
Eh

o

O

0.0-

0.05

0.10

0'.15

0.20

0.25

Time

(sec)

E1GUBI

6

CADS Compensated Tachometer Feedback

System Response

30

The

tyce one system, in the preceding example, allowed

derivative feedback from the forward path without
a

requiring
One may
be

block

which
a

was

capable

of

differentiation.
system
a

encounter
configured
path.

type zero system or a
to
of

which

cannot

provide derivative feedback from the forward

Tbe possibility
a

providing

pseudo

derivative

feedback using
cases.
type
7.

type three block was investigated for these

A

three

block

with

Z

=

and

G =

P

is shewn in

Figure

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
the
on

complex
a

roots of the closed loop system is negligible.
real root at
P =

It does add

-964.

The system of Figure

5

was redrawn as shown in Figure

8

using the pseudo tachometer feedback.

31

+

1

S

—rv
-(oV+S

iO

J
,

r "\

S+10

\J

SHO 3

-i

FIGURE
The

8

Pseudo Tachometer Feedback
calculated
a
5

optimization

program

K
t =

=

0.0352 fcr the

abcve confignration.

This gave

0.715.

The
in

system response and the desired response are shown
9.
.

figure

The

two

responses

are

again
step

nearly
tc

sup erimpcsed

This method of providing derivative feedback
the

has the disadvantage of adding an integration

problem

solution

with

the concomitant increase in problem

sclution time.

32

0.0-

0.05

0.10 Time

0.15
(sec)

0.20

0.25

fIGURE

9

Eseudo Tachometer Compensated System's Response

33

c.

CASCADE CCMEEJJSATION

Having
a

jrcven the feasibility of the program optiaizing

single

variable
third

system
order

where

an

exact
two

solution

was

available, the next problem considered extending the problem
scope to
a

plant
to

with
be

variables.

The

unstable
Figure 10.

plant

that

was

compensated

is shewn in

o
II

—

(i«»i)(j«+062jft»+i)

EIGUEE

10

Third Order Plant to be Compensated

The
have

riant
<2,

was

to be stabilized using a single section

cascade cempensatcr.
H

The compensated plant was required

tc

without reducing the error coefficient. [6] The

PW
Tc keep the error compensated system is shown in Pigure 11. coefficient constant, the compensator used was a simple lag

network

(

t,< t%

)

.

34

r,sn
S (S + i)

(.2Sf)

FIGOBE

11

Lag Compensated System

A

compensator
or
the

which

wculd

meet

the

required
as
a

specif icaticrs was calculated by conventional methods
check

program's performance.
The values of
=

The Bode plot cf the

uncompensated and conventionally compensated system is shewn
in Figure
12.
tfce
t
r,

and

r.

for Figure

11

which
tc

would meet
t,

design requirements
2

were

determined
shewn
in

be

-

1C,

100.

The compensated system's response
figure

using these values for the compensator is
15.

35

'a

«d <y CO 4-J <D
CO

ac e o a
a>

DU

o g c o

2 H U

jQ

o

O
CO

o
CN

36

0.0

2.0
13

4.0

6.0
(sec)

8.0

10.

Time

FIGCBZ

Cor ventionally Compensated System Response

37

.

The

conpensated
14

system

was

redrawn
program

as

shewn

in

Figure

using

the

standard

transfer

blocks

available for system simulation.

Q^Z^loJjL -O s+i (SfPc)
-I

O

S*5

-o

FIGURE 14 CADS Lag Compensated System

A
H

stardard
was

second
as

order
the
match.
<
P

response

of

&

=

.3,

=
n

0.75

chesen
program

desired

response

for

the

optimization

to

The limits placed en the
< C
.1

optimization program were .001
Arhitrarj
E

and .01

<

Z

<

1.

C

values
Z

to
.1

begin
in the

optimization were specified as
logarithmic
T,

=

.01 anc

centers
and
15

of
=

the

Co

Co

search

2cnes.

CAES determined

=

8.078

f2

67.47
the

as the optin.cn parameter

settings.
the

Figure

shows

desired
response

response

and

program

compensated

system

response cf the system compensated by CADS is more nearly the desired response than is the conventionally compensated sjstem. One should remember that the specified
The

response is for

a

true
is

second
forth

order
order.

systen

whereas,
a

the

ccnpensat€d

system

Therefore,

perfect

38

match cf

desired

versus

actual

responses

could

net

be

cttained.

39

T
•

rH

/f^\
CN
•

/ /

//

\

\^

r-\

II
o
O S B

\
•

HH D

•

/

/

//

\v ^
\
"'"^

\

,

R

//

\

//
//

\

^\

//

00

^

D

o

•

11

/
o
•

/
<*
•

o

/

/

/

CN

o

•

y?

/

0.0

y

2.0

4.0

6.0

8.0

10.0

Time

(sec)

FIGDBE 15

CADS Compensated Systeir Response

40

r.

CASCADE 1EAE COMPENSATION

The complexity of the next problem to be solved by
was

CADS

extended
was

to to

five free parameters.
be

The plant shewn in
a

figure 16

used

to

follow

unit

amplitude
anc
the Two

sice-wave input of 200 rad/sec.
output
cculd
not

The output amplitude was to
by

be almost exactly the same as the input amplitude,

lag

the

input

mere than 10°.

sections cf cascade compensation were to be used. [8]

SiH

ZOOi

*

K
S
x

G*

j

EIGURE 16

Plant for Lead Compensation

The specified requirements may be interpreted as an open
leep gain
a
> 15

db and a phase angle
A

^

90° at

W

=
n

200

from

Nichols
ttat

jlct.
a

cut
of
3

and
X

try solution on a Ecde plot
10 6
Z

shewed

gain

and
=

two

phase

lead

compensators
requirements.
of

with a double zero at

70 and a double pole

at P = 7CC will satisfy the closed loop magnitude and

phase

figure

17

shows

the

system

response and
The

desired response obtained for these values.
the

magnitude

compensated system's response is 33% of the desired

response and lags by 8.12°.

U1

V)

a o W
u>

C3

a

M
W
>i
UJ

i>

P
0)

x>

Ui

a

u

«

C

o
•H

C
> a o

u

-'

03 =3

1*1

J-nOVHi

42

The

prchlem

was

then run on the optimization program
The
free

*ith the tlcck connections as shown in Figure 18.

parameters
the

were the poles and zeros of the compensators and
of

gain
=

the

plant.
was

The

desired

data

curve

of

generated from the standard second order step response by setting 5 = Q, w = 200 and using
XDAIA
sir(.200t)
n

X(2)

as

tre

desired

response.
,

<y
-J

o^ 0-J&-

FIGUBI

18

CAES Block Diagram for Lead Compensation

The optinizaticn program solution

went

to

the

lower
were

lioits

fcr

tcth
500
<

poles

and

the upper limits for roth the
the
80

zeros and the gain.

The limits of
<

search
<
Z

zones
< i

relaxed
3.2 X

tc

P
i

<600,

9C

and
the and

106 < g(3)

3.4 X 10*.
the

The program again placed
of P
i

free
G (3)

variables
= 3.4
X

ce
.

limits

=

500,

Z

= i

9C,

1C 6

The system and desired response for these
in

values

is

shewn

Figure

19.

The system magnitude and

phase are much closer to the desired

cutput

response

than

the respcese cf the conventionally designed compensator.

43

o

•ji

a o

—

a>

Q3

<v

-M

W
>i
en

M

<y (0

w s
(i/

a.

u
n3

a o

a u
o\

en

w m U H
Cm

44

The phase difference is only 4.33°. the
fcouncaries

produced

the

Continued relaxation of compensated system shewn in

Figure 2C.
these
lags
the

figure 21 is a plot of the system's response for
The system's response is improved in that it
ty

values.
irput

only

3.46°

and

the

magnitude

is

essentially 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
the

program

was

activated
were

at

T

=

although
prccess

problem

specifications

for the steady

state resicnse.

The problem was rerun with the optimization The same
effctained.

started after the transient had died out.
fcr the
the

values

optimum

compensator

Mere

Apparently

snail

initial transient did not effect the

prctlea scluticn.

45

<v VI

c

o
Oi
VI i

M

(d

n
x
(0 .-»

s 3 o

03

a «s u

to

W
05

H
&4

T15

iftOYHi

{TT-

46

E.

SEBVC S1S1EM CO KPENSATICN

The

foregoing examples of the simulaticn - optimization
were

program
guickly

simple
with

examples
the

which
of

could

obviously
usace

be
A

sclved

standard cut and try methods.
program

mere complex and challenging example

is

presented

fcy

the sjstem shown in Figure 22.

The system is an operational servo drive mechanism

with

multiple
as

inputs
and

and

discrete level feedback.
(t> 75ms)

This bighly

nonlinear system's output was to follow the input as closely
possitle
the
in steady state

there was to be
and the

very little ccise ripple.
are

The free parameters of the system
of the compensator, C
Li

pcles

and

zeros

,

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
I

simulaticr

it was found that the current limiter for
|I
D
|

was
D

net needed because to

< 25 A and the

limiter

was

removed
(XDATA)

decrease program run time.
three
then
in

The desired response

was written as a set of

equations.

These

eguations
and

were

used

to

replace the second order step response

eguations
were

the standard program.

DRVIN(1),

(4)

(5)

alsc

written

as

a

set

of

equations and placed in

subroutine E1ANT. The discrete level feedback to block two These was achieved by making DRVIN(2) = INTGEB (TBA (12) 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

%
<
6 o
II

o

^
tx

a o
•H
-P
(0
UJ

a
Pi

W°

a;
tti
t=

u

o

c
>0

o
it

•H en •H

u

O
>
ui

M
5

a

2 o

•H

C

«3

X!
<J <y

^^
§
*
/-».

3
+
•4

3?

+
*4

>
•H
c

C

V)

>?
-t-

3

3
<^>

v^

-t

tf
"7

n
6
«M

Q

M
O > M
<D

CO

a

o

c

h £

u

(N

W

a

•

» •
c

a
n;

*
>
i.

si
,c

-H
ii ii

m|«o
ii

•
"

48

•H

a w S
(TJ

JO,

o

•H

Q
O >

M

u
VI
M-l

O
(0

M
en

a
Vi

a u

««»

w « M

49

The
parts.
to

system

optimization was initially broken intc two

This sas tc reduce the number of variables that were
The above steps were

be optimized per run and to obtain near optimum starting

values fci the free parameters.
in

taken

an

effort

to decrease the computer time required for a

solution.
C
i

The first run was tc
of

optimize

the

compensator,

independent
run
C
8
.

the
to

noise

suppressor.
values
for

The
the

second
noise

cptimizaticn
suppressor,

was

select

The

rationale behind this separation was

that the two circuits perform different functions and should

therefore
system.
free

be

initially

separable

in their effects en the

The final optimization run was to be made with

all

parameters

available to the program fcr optimization.

The search zene centered on the values found above.

Figure values
Z
2

24

shows

the
in

response
Z
*

for

the
run
to
P
l

system
set
=

as

originally compensated.
fcr
£
2

The optimization
=

the

C

resulted
L
=

43.5,

21.0,
of

=

47.5,

5S2.C.

Figure 25 shows the
values.
The

simulation

the

system

using

these

initial

velocity
after

overshoot has teen reduced and the
the

average

velocity

transient

appears

more

equally distributed above and

below the desired velocity.

50

W

c o
Q4
V)

0)

u-

a
<v

*j

>i
</)

o > M
en

H
(0

a
•H o> •H

O
CM

M

-

H

O'Ofr

o*oe

cot o*03 (39S/UT) A^TOOXSA

51

W G O
Ui

o H

<y

=3

w
W

W

>i

o

00

o
Q) CO

u>

<0

6

w a
a>

O

E-t

u Q U
en

a o

o

m
CM

W «
H

O

O'Ofr

o'oe

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

52

The

iiiitial optimization

run for setting the values of

the noise suppressor used the same cost function

which

had

been
J
W

used
/

for
(upper

all

previous
<^

optimization
0.2

runs
and

=

Err 2 dt.
15C0

This resulted in
bound)
.

=

(lower bound),

=
n

Figure 26 shews the result of
was
to

using

this

cost

function
it

further

decrease

the

overshoot.
mere

Ecwever,
the

allowed larger switching cr noise
transient
Tc
J

transients as a result of weighting large
than

errors
this

lesser

noise

jitter.

overcome
=
/

prcrlem, the ccst function was changed to
the transient errors.

|Err|*t*dt.

This was tc weight the steady state errors mere heavilj than

Using this cost function
<f

the
H

values
=

set

by

the optimization program were
,

=

.

2

and

1430.

n

The damping factor, S The

was again placed on the lower limit.

initial

overshoot was still improved ever the original
the

system's response tut
iaproved
27.

switching

transients

were

not

ever the previous optimization trial.

The results
rigure

of the siiulction run using these values is shown in

53

%
V

H o



o M
«N
PM

0*0fr

0*02 (TOT Oe (D3S/UT) Aq.TOOI3A

o o

54

'

%
V

CM

o

•

<y

W a o
cu
CO

o H o

«
a
(V

cu

•

-M
CO

PH
CO

CN
1

o—
O
•

CO

o

u'
'O

CD

—

CO

P
s
CD

(V

CD

<d CO

e
•H KO Eh

O o

a,

•

u
a < u
CO

O

B

T
o o
•

r* CN

W 03 a
C3 tn

O

CN

O'Ofr

o*oe

o*oz

o*oi

o o
«

55

A

final

optimization
The

run

was

made

with

all

six

performance index was changed tc the weighted cost function shown in eguation V. The weighting factor was computed to equally weight the start up transient
variables
free.

and the steacy state responses in an effort

to

reduce

the

switching transients.
J

=/*|Err|dt
-

0.0

<

T

<

0.045
TF
(V)

J

In. 14*|Err|dt

0.045
= 521.,

< t

<

The optimuff compensator
P
2
fi

values
P
3

established
Z
3

by
6

CADS
=

were

=

27.1,
1266.

Z
2

= 52.5,

=

48.1,

0.32 and

=
o

figure 28 shows the fully
Ihe

compensated

system's

response.

initial

overshoot has been reduced acd the
tc

average velocity in steady state is clcser
value
than
the
the

the

desired
II.

criginal

system's response.
still
the

However, the
Table

transient summarizes
system.
The parts.

encr

en shut off is

present.

results

of

optimization of the servo

CPD time reguired for the optimization pr-ccess was

net decreased by splitting the
Ihe

problem

into

two

separate

time reguired for each optimization run en the

individual compensators was the same as the time reguired tc
optimize tie complete system with six variables.

56

'

<N
iH

O

•

0> to

e o

O
rH
•

w
CW

Oi

O

ua

a
u>

-P

W
>i
CO

00

o *-» o O
•
"

T3

—

<D CO

V N
•«1

a
•H
4->

CD

g
•H
vo En

O

U4

O o

•

u <J u
CO CN

CO

•^r

o o

•

w 03 o
CS

CM

O'Ofr

cot o*03 o*oe (oas/ui) At}TOO-[aA

57

o o m

o o en

o o
IT)

O m
^r

oo KO CM

CM in
«o

in

CM

CM

f* iT
eg

r* j»
riT

r^

^

o
in

r» cr

r»
^3"

00
3-

a o
•H
(d

N
•H

S
n
Ck

o o
in

•

t

•rt

CM en Ul

CM
cr>

CM

m

O

in

^> CM in

V
'-it

>
•H in
01

N

o
in

•

^ m
•

in cr

in 3-

m
i

a

M
O > M
Q)

m
zr

•

n

•

S3"

5T

IN in

CO

*—

m
o

N
04

o CM

r™ CN

«~ CM

r-

CM

p* CM

P W
r•

M

U

M (0 a s 9

•^

£3

4J

O
•H
4J
4->

4J

P
*

<3

H H W
f-J

U a
4J

T3
(M

T3
(M

—
W
M M

3"

CQ «C

u

u

^^^
W

M M

W

M M

+
-P

H

—

T3

m

M M

H
CO
1-1

M O e

Ui
(V

(0

c
•H
cr
••-»

Q4

u

o

a

o

M

58

This problem has presented an excellent example of

the

necessity
index

carefully select the cost function which will neasure the sjsten's performance. Defining a performance
tc

which

will

weight

the

more

objectionable
that
thej
2

characteristics of a system's response so reduced becomes difficult eliainatec or
ccuplexity ircreases.

are

as the system's
dt,

The performance index, J = /(Err)

was not adegtate in its treatment of the noise and switching

tracsients fcr the above
indexes
used

problem.

The
in

ether

performance
effects upon
functicc
was

were

also

marginal

their

parameter op tinization.

Although

the

cost
be

reduced

to

a

mathematically correct minimum, the system's
achieved. usinc
The

performance was not the best that could

system's

performance was only optimum due tc the definition
The system was

of the cost furcticn.

simulated

the

compensator values determined from the last optimization run 0.5. with the exception of 5 which was increased to «f =
This

siaulaticn was made based upon engineering judgment of Figure 29 shows that this change the effects of varying S . the damping factor produced a system response which was in
nearer the desired response than any of the previous runs.

59

o

•H

o O

W
a;

(0

a

in

o— O u
•

00

<D

a;

0)

o
W
0)

e
•H
<£>

O

Eh

w
>i

o

o

> u
a>

en

O'Ofr

coe

cox o*oz (D9S/UT) Aq.TOOX9A

60

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
time

developed

during
but

the
cf

investigation
control system
due
tc

proves

that

dcmain

optimization

responses is net only feasible
readily

desirable

the

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) (2)

the order of the system being simulated, the area cf the search zone determined by

the

upper
and
(3)

and lower bounds en the variable parameters

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. = number of the T Rut1 The prograi run time is therefore, trials X system simulation time. A high crder system may

61

require twenty seconds of CPU time for simulation.
trials
are

If

300

required

to

determine

the

optimum parameter

values, the total CPU time would be 100 minutes.

Example

Lower

Starting
Guess
XS

Upper Bound
XU

Number
of

CPU Time

Problem
III
b
(pseudo)

Bound
XL

Required

Trials

-300

-34.

3

-3

55

7min03sec

.001
c

.01
.1

.01

0.1 1.0

225 225

24min53sec 24min5 3sec

400

450 450
95 95

500 500

400 d
90

100
100

2,000
•

68min

90

3.4xl0 6 3.5xl0 6

3.5xl0 6

10
40
e

20

30

50

60

400
.3

500
.5

600
.55

230

4hr

1150

1300

1400

Table III
Table
III

presents
a

a

tises reguired to obtain

summary of the search zones and solution for seme of the problems
The amount of CPU time reguired for

considered
excessive.

in this thesis.

for a solution,

especially

problem

III-E,

may

seem

However, there are several considerations which should be mace prior to arriving at this conclusion. The equations of the system do not have to be (1)

written, programmed nor debugged if

the

system's

62

-

component transfer functions are known. lime domain requirements do not have (2) translated
A

to

be

irto

frequency

domain specifications

for system simulation and design.

systematic search is carried cut to obtain the cptinum parameter settiEgs. This assures that
(3)

with

valid

bounds on the variables an acceptable
be

solution

will

obtained

with

the

first
the
cesign
the

optimization attempt.
The time required to perform the above steps in

prccess

by conventional means may result ic many more hours

of CPU tine than if the program

CADS

were

used

from

beginning of the design process.

Several means of reducing the
fcr

computer
by

time

required
the
a

optimization
of

were

previously

outlined in section II. keepinc
should

The most significant reduction is obtained

number
nirimum.

integrations required for system simulation to
Elcck diagram reduction of the system

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.

tt
J

*s-rr\.

^o
flGUEE 30

K«

rs+i

^6

Z,SH

o—

Elock Reduced Ward-Leonard Drive

63

Section III.E.
cost often
will

showed the effect

of

using

different
is

functions

to

measure

the

quality

cf the optimized

response.

Although the integral error squared
a

criteria

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

carefullj ccrsider how to best define
properly

cost function
(err <

which

penalize

deviations
A

of the system response
1)

from the desired response.

small error

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
be used.

a

weighted cost function will have to

One nethcd cf arriving at a properly weighted cost
is to sinulate the system using first estimates of

function
over

tte variable parameters and recording the sum of the the different portions of the response.
be used
to
A

errors

ratio cf the
weighting

errors can then

arrive

at

proper

factors for each time section cf the response.
working BOXEIX will continue the optimization process, until it can no longer seventh significant digit the

in

reduce the ccst function.
the

Often an acceptable solution

for

system

parameters

has been found long before th€ ccst

judicious use A function has teen reduced to its minimum. 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

cptinization using the CAES program is a straight forward process which dees not require an simple, Economic in-depth analysis cf the system being optimized. times does dictate that intelligent starting use of CPU
lime donain
values and bounds be placed on the variable parameters which
are to b€ optinized.

64

The program is

a

readily
It
is

usable

tool

for

simulation
if

without

cptinization.
to,

competitive
simulation

with,

not

superior
simulating

other

common

routines

when

typical

control systems.

This feature alcne is

expected to bring the program into common usage.

C.

RECCKMOIATIOKS FOR FDTDBE WOHK
All
of

(1)

integral

calculations

performed

during
for

this

investigation nere done in double
accuracy
the system response.

precision
be

increased
to

The possibility of using

single precision

calculations

should

investigated

decrease cere reguirements.
(2)

The ability to begin the optimization

process

at

some

time

greater

than

zero

should

be

provided.

This will
and

necessitate ncdif icaticn of the data input cards

block

eguations so that initial conditions can be entered.
(3)

The program presently
(EBVINs)

reguires
the
by

that

external
which
are

fcrcing
to

functions
within
the

and

variables
Fortran
A

be

optimized be

specified

IV

statements

placed

bedy cf the program.

method of reading these

specifications from data card input should be developed so have to "shuffle" cards in the that the user will not
program deck.
(4)

The craphical output of CADS was all that was necessary

for the

investigations

conducted

in

this

thesis

tut

a

prevision
be
(5)

fci numerical output of selected responses should

provided for detailed analysis.
The

feasibility

of reducing the number of significant
a

digits ECXFLX uses should be studied as

means of

reducing
if
it

optimization time.
reject a system'
s

Also some criteria might be developed to

response before TF is

reached

is

determined tc te unacceptable.

65

(6)

A method of automatically relaxing the boundaries on the

variables

being

optimized

when

they

go

to their limits

should be developed.
The

(7)

standard

cost

function

provided

and

all

user

developed cost functions should be normalized.
permit
a

This

would

more

direct

and easier comparison of a system's

"goodness" when several different integration

step sizes or

run times have been used in the optimization process.

66

APPENDIX A
Block Data Card Format

|1

(11

1

20

|40
P

|60
Z

ELKCCD=EEVV
CC
E

G

EE VV
G
P Z
*

= = = =
=

= =

PC5ITICN NUMBER OF THE BLOCK TYPE OE BLOCK (NUMBEB) INEC1 NCDE NOMBER OUTPUT NODE NUMBER VAIUE OF GAIN VAIUE CF THE POLE* VAIIE CF THE ZERO*
8

Fcr block type 4,

is read into the P location and

Hn
*

is read into the Z position.

Per

block

type 5,
U

is read into the P position
Z

and
*

9

is read into the

position.
P

Bcr block type 6,

6
C

is read into the

position

and

Qe

is read into the Z position.

EXAKPIES
G

EIKC11=0102

10.
G

P
5.
P
Z

BIRCi2=0304

1.

G

BIKC23=0405

1.

10.

5.

67

G

d

UJ„

E1F

1Ci*=

1C06

200.
G
©u

.2

A
-5.

BIK115=1207

10.
G

20.
&c

%
10.

EIK126=0108
*£ee Table I.

1.

3.

68

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

CM

,

,

REAL *87FACL7

REAL *8XCATA

REAL *87 FA, TFA COT,
REAL *8F1,F2

T

f

OT ,CRI Vc »CI*G,CyGDOT,DRVIN,TF

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 =C

I\CLT=C
Fl

=CT

Ml
ICK

1-2

=C.5CG*F1
=C =C

N55

=c = 2*N

69

FEAC

1NFLT CAT/*
4
4

4444444444
4
4 4 4 4
.

CC
I

2

4

4

=lTN
I

***RE/>C

(5,23)
=

IC(I ),IC(I ),IE(I ),IV(I ),G(I)

tP(I)tZd)

4

4 4
4

SET

TMCIT
•
.

CITF17 CF LAST ELGCK

4

4
4 4 4
*

*

.

*

IF

*

.

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

*
I

4
4 4 4
4

2

4 4 4 4 4 4 4 4
4
4

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

)

2

4
4 4 4
•
=»

icm.sc.i

IF

*

.

*

4 4 4 4 4
4

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

F

4
4

4

4
4 4 4 4 4
4

IF

IC

(

I

J.EC.5
I

f^55 =

N554l

4
4 4

4 4 4 4
4

IC(I).EC6
•

IF

4
4

•

|

K66=N£64l

4

4 4
4

:

4
2

44444*4*fcR]TE

(6,26)

ICC

I

)

, I

C

(

I

)

,

IE

(

I

),IV(I),GC ),P(I ),Z(I)

70

SET

ThACIT

=

SPECIFIED

ThETA,

IF

ANY

•

*

IF

*

ISET.NE.C
*
.
I

ICIT=ISET

*

•

*

***WRITE (6,27) ICUT

Ml
^ 5 5

=^Nf s
=4*K(:t
=

=4*N11
N-1

N66 NEC

SCAN

FCR

INFITS
E\

CONNECT SYSTEN
I

SETTING FLAG=1. FGF CONNECTEC ELKS
+
+

4
4

4444444444
4
4

CC
4

4

4 4

J=ltN
4
4

+

4
4 4

4

4

4444444444
4
4 4 4 4 4
4

CC
4
K=

1,N

4

|

FLAGCKtJ) *C.O
I

4
4 4 4

4 4 4

I

.

4

*

.

4444444*4
4 4
*

„
I

* V

IF
(

*
(

*

K .

)

. E C . I E

J

)

*
•

.

*
j

*

.

*

FLAG(K,J)=1„C

I

4 + 4 4
4

*

,

*

F

TAKE eLK TYPE ANC SCIVE ECNS ONE EY CNE

71

CLE^ PING CLT FEGISTERS ANC
I

INITIALIZING CCLNTEPS

+
4i 4

444444444
I

tQ
£

4 4 4 4

4

ICLP=1,N

6

4 4 4 4 4 4 4 4 4 4 4444444J

7HACGTUCLF) =0.C0 CNG(ICLP) =C.CO = G«CO CNGCCT UCLF CFVINUCLF) =O.C0
)

TKA(ICIF)

=

C.CO

>

2(

ICLP

)

=C.CO

>2CCT(KLP) =O.CC
7

=C.QCC

NR«S CCNTFCL ENTPV
NF2
N

FCIM

IN

INTEGRATION SUBROUTINES

P2

b?4

=1 =1 =1

K2 CCNTFQLS

H-

1

C

t-

BLCCK EQUATION IS BEING SOLVEC

'V
ICCNT 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=0
IT

=C

ILAST'S CCNTFCL FFCGRAN CATA AND PROGRAM EXIT

ILAST=C I5LAST =C I6LAST =0

72

)

SPECIFY ANY F*F^ETEFS TC BE OPTIMIZED HERE
i

EG.

F(1)=C<2),
FCF

Z(1)=C(1),
If^FlTS

SET CPIVES
4 7
4

4+4 4 444 4+ 4
4

CC
9

4 4

MCRV =

1,/S

4
4 4 4

4 4 4 4
4

I

CFVIM

1

)

«1.
)

1

CFIVE (NCRV
4
4

=

C.CC
4

4
4

444
4 4
4

4

444444
4

CC
4
8 M = l,?\

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115

LIST OF BEFEBENCES

!§-DC_kj2arJ?

Papers in Electrical En^iiieering and Cogputer Scierce , v. K, edited ty Director, W., S. Dcwden, HutchinscE Boss, Inc., 1973.
fi

McDaniel,

H.

L.

Jr.

and
tc

Mitchell,

J.

B.,

"An

Inncvative

Approach

Compensator

Design",

NASA

CB-2248, Kay 1973.
Office
cf

Naval Besearcb CB 215-238-1, Optimal linear
to

Control
Sp.ecs_.J_

(Formulation
by
G.
L.

meet
C.

Conventional
A.

resign

Hartman,

Barvey and C. E.

Muller, 2S March 1976.

Cantalap iedra,

j.,

low

Order
Naval

Mcdles

for

Dynamic
School,

Systems M

M-S.

Thesis,

Postgraduate

Mcnterey, 1972.

MacNamaia,

M.

A.

S.

,

A

New Optimization Method A££lied
S.

%£ flutcpilot De sig n, M. Schccl, Mcnterey, 1975.
Eox
and
8
a

Thesis, Naval Postgraduate

M.

J.,

"A New Method of Constrained Optimization

Ccmparison With Other Methods", Computer Journal,
p.

April 1965,

45 - 52.
C.

Fitzgerald, A. B. and Kingsley,

Jr.,

Electronic
New York,

Machinery,
1961.

p-

186,

Fig. 4-22, McGraw Hill,

Thaler,
1973.

G.
S

J.,

Desi gn of Feedback Systems,
Inc.,

Dcwden,

Hutchinson

Boss,

Stroudburg,

Pennsylvania,

116

INITIAL DISTRIBUTION LIST

No.
1.

Copies
2

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Canercn Station
Alexandria, Virginia 2S314
2.

Litrary, Code 0212
Naval Postgraduate School

2

McDterey, California 93940
3.

Department Chairman, Code 52
Naval Postgraduate School

2

Monterey, California 93940
4.

Professor Gecrge J. Thaler, Code 52TE
Naval Postgraduate School

5

Monterey, California 93940
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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Mr. Peter Gramata

IBB

Monterey and Cottle Roads
San Jcse, California 95114
9.

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Bcfcert

Mcintosh

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IBM

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Mr. Ren Palmer

IEM

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12.

Mr. B. M.

Syn

IEC

Monterey and Cottle Roads
San Jcse, California 95114
13.

Professor
Dect.

A.

G. J.

MacEarlane

cf Electrical Engineering

University of Manchester
Inst, cf Science and Technology

Manchester England 067609
14.

Dr. Bui-lien Rung

Dept.

cf Applied Sciences
a

Oniwersite du Quebec
Chicoutimi, Quebec
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Chicoutimi

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118

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