System identification by means of a digital computer method.

University of Windsor

University of Windsor

Scholarship at UWindsor

Scholarship at UWindsor

Electronic Theses and Dissertations

Theses, Dissertations, and Major Papers

1-1-1969

System identification by means of a digital computer method.

System identification by means of a digital computer method.

John D. McGee

Follow this and additional works at: https://scholar.uwindsor.ca/etd

Recommended Citation

Recommended Citation

McGee, John D., "System identification by means of a digital computer method." (1969). Electronic Theses and Dissertations. 6573.

https://scholar.uwindsor.ca/etd/6573

SYSTEM IDENTIFICATION BY MEANS OF A DIGITAL COMPUTER METHOD

by

J o h n D. McGee

A T h e s i s

S u b m i t t e d t o t h e F a c u l t y o f G r a d u a t e S t u d i e s t h r o u g h t h e D e p a r tm e n t E l e c t r i c a l E n g i n e e r i n g i n P a r t i a l F u l f i l l m e n t o f t h e R e q u i r e m e n t s

f o r t h e D e g r e e o f M a s t e r o f A p p l i e d S c i e n c e a t t h e U n i v e r s i t y o f W in d so r

U M I N u m b e r: E C 5 2 7 5 6

I N F O R M A T IO N T O U S E R S

T h e q u a lity o f th is re p ro d u c tio n is d e p e n d e n t u p o n th e q u a lity o f th e co py

s u b m itte d . B ro k e n o r in d istin ct print, c o lo re d o r p o o r q u a lity illu s tra tio n s a n d

p h o to g ra p h s , prin t b le e d -th ro u g h , s u b s ta n d a rd m a rg in s , a n d im p ro p e r

a lig n m e n t c a n a d v e rs e ly a ffe c t re p ro d u c tio n .

In th e u n lik e ly e v e n t th a t th e a u th o r did n o t s e n d a c o m p le te m a n u s c rip t

a n d th e re a re m is s in g p a g e s , th e s e will b e n o te d . A ls o , if u n a u th o riz e d

c o p y rig h t m a te ria l h a d to b e re m o v e d , a n o te w ill in d ic a te th e d e le tio n .

®

UMI

U M I M ic ro fo rm E C 5 2 7 5 6

C o p y rig h t 2 0 0 8 by P ro Q u e s t L L C .

All rights re s e rv e d . T h is m ic ro fo rm e d itio n is p ro te c te d a g a in s t

u n a u th o riz e d c o p y in g u n d e r T itle 1 7 , U n ite d S ta te s C o d e .

APPROVED BY:

j

/

ABSTRACT

S y s t e m i d e n t i f i c a t i o n , t h a t i s , a c c u r a t e l y o b t a i n i n g t h e v a l u e o f s y s t e m p a r a m e t e r s i s an i m p o r t a n t p r o b le m i n t h e s i m u l a t i o n o f many c o m p l e x s y s t e m s .

ACKNOWLEDGEMENTS

The a u t h o r w i s h e s t o e x p r e s s h i s t h a n k s t o D r . P . A . V . Thomas f o r h i s h e l p f u l s u g g e s t i o n s and g u i d a n c e .

TABLE OF CONTENTS

P a g e

ABSTRACT i i i

ACKNOWLEDGEMENTS i v

LIST OF TABLES v i

LIST OF FIGURES v i i

I . INTRODUCTION 1

I I . SAMPLED-DATA SYSTEM THEORY 2

2 . 1 Z - t r a n s f o r m T h e o ry 2

2 . 2 S y s t e m A l g e b r a 5

I I I . IDENTIFICATION 7

3 . 1 I d e n t i f i c a t i o n M ethod 7

3 . 2 F i r s t O rd er S y s t e m s 13

3 . 3 S e c o n d O rd er S y s t e m s 14

3 . 4 H i g h e r O rd er S y s t e m s 16

3 . 5 C om p u ter S o l u t i o n 16

I V . EXPERIMENTAL RESULTS 20

V. DISCUSSION OF RESULTS 23

V I . CONCLUSIONS 27

APPENDIX A. Z - t r a n s f o r m T a b l e 28

APPENDIX B. Com puter P ro g r a m s 29

BIBLIOGRAPHY 35

L IS T OF TABLES

P a g e

1 . F i r s t O rd er X — l.O O , 2 0 0 s a m p l e s 20

2 . F i r s t O rd er X = 1 . 0 0 , 150 s a m p l e s 20

3 . F i r s t O r d e r A, = 7 . 5 3 2 0 , 2 0 0 s a m p l e s 21

4 . F i r s t O rd er X = 2 0 . 1 5 0 , 200 s a m p l e s 21

5 . F i r s t O rd er X — 1 . 0 0 , 2 0 0 s a m p l e s 21

6 . S e c o n d O rd er S y s t e m s 22

L IS T OF FIGURES

P a g e

1 . I d e a l S a m p le r 2

2 . C o n t i n u o u s and S a m p le d I n p u t s 5

3 . True E r r o r Model 9

4 . L i n e a r R e g r e s s i o n E r r o r 10

5 . I t e r a t i v e Model E r r o r 13

6 . Program F lo w C h a r t 17

7 . ‘f E r r o r V e r s u s S a m p li n g P e r i o d 24

I . INTRODUCTION

The e n g i n e e r i s o f t e n f a c e d w i t h t h e p r o b le m o f o b t a i n i n g t h e c h a r a c t e r i s t i c s o f t h e d i f f e r e n t i a l e q u a t i o n s r e p r e s e n t i n g a c o m p l i c a t e d s y s t e m .

S i m u l a t i o n and d e s i g n p r o b l e m s o c c u r r i n g i n t h e d e t e r m i n a t i o n o f a e r o d y n a m i c c o e f f i c i e n t s , d ynam ic c h a r a c t e r i s t i c s o f t i r e s , a u t o ­ m o b i l e s u s p e n s i o n s y s t e m s and a i r c r a f t l a n d i n g g e a r s f o r e x a m p l e , r e q u i r e

an a c c u r a t e k n o w le d g e o f t h e p a r t i c u l a r s y s t e m t o p e r m i t a m e a n i n g f u l s i m u l a t i o n t o be m a d e. R e c e n t l y a t e c h n i q u e f o r ' S y s t e m I d e n t i f i c a t i o n Ey Means o f a I m p l i c i t S y n t h e s i s M e t h o d 1 w as p r e s e n t e d by C . L . S h e n g a.nd M.Y. Wu [ 7 j . The p a p e r i n v e s t i g a t e d t h e d e t e r m i n a t i o n o f t h e p a r a m e t e r s i n f i r s t o r d e r l i n e a r and n o n - l i n e a r s y s t e m s and p r o p o s e d a m e th o d f o r h i g h e r o r d e r s y s t e m s . A g e n e r a l p u r p o s e E A I-T R -48 a n a l o g c o m p u t e r w as u s e d f o r t h e i d e n t i f i c a t i o n .

The p u r p o s e o f t h i s r e s e a r c h was t o t r y t o e q u a l o r im p ro v e u po n t h e a c c u r a c y o f t h e Sheng-Wu i d e n t i f i c a t i o n o f l i n e a r s y s t e m s

I I . SAMPLED-DATA SYSTEM THEORY

2 *1 Z - t r a n s f o r m T h e o ry

By t h e v e r y n a t u r e o f a d i g i t a l c o m p u t e r , t e c h n i q u e s f o r s y s t e m i d e n t i f i c a t i o n r e q u i r e d i s c r e t e i n f o r m a t i o n . M eth o d s o f a n a l y s i s w h i c h u s e d i s c r e t e d a t a a r e c a l l e d s a m p l e d - d a t a s y s t e m m e t h o d s . A s a m p l e d -d a t a s y s t e m c a n be -d e f i n e -d a s o n e i n w h i c h t h e f l o w o f c o n t i n u o u s i n f o r ­ m a t i o n i s t r a n s f o r m e d i n t o a s e r i e s o f p u l s e s o r n u m b e r s . T h i s s a m p l i n g p r o c e s s i s a n a l o g o u s t o i d e a l s w i t c h i n g . The d i s c r e t e p o i n t s o r p u l s e s c a n o n l y r e l a t e t o t h e c o n t i n u o u s d a t a a t t h e s a m p l i n g i n s t a n t s . T hat i s , f ( n T ) c a n be c o m p l e t e l y d e t e r m i n e d fro m f ( t ) b u t f ( t ) c a n be known o n l y a t t h e d i s c r e t e t i m e i n t e r v a l s T , 2 T , 3 T , e t c . , when f ( n T ) i s known. f ( t ) and f ( n T ) a r e t h e f u n c t i o n s shown i n F i g . 1 . T h i s t r a i n o f p u l s e s ,

r 2T 57- 4 T _{Ti m e}

t i m e

C D I S T l M v J O O S D A T A s a m p l e d DATA

r-J - f c ) ---— — --- f ( n T )

T

3

f ( n T ) , c a n be a n a l y s e d m a t h e m a t i c a l l y w i t h t h e i n t r o d u c t i o n o f i m p u l s e m o d u l a t i o n . F or a c o n t i n u o u s i n p u t f u n c t i o n , f ( t ) , t h e s a m p l e d - d a t a

*

o u t p u t f u n c t i o n f ( t ) c o n s i s t s o f a s e r i e s o f i m p u l s e s o f a r e a s f ( n T ) , i . e . m a t h e m a t i c a l l y ,

f * ( t ) = f ( t ) 8 * ( t ) ( 2 - 1 )

• k

w h e r e 8 ( t ) r e p r e s e n t s a u n i t i m p u l s e c a r r i e r t r a i n * E x p a n d in g ( 2 - 1 ) we o b t a i n

* 03

f ( t ) = f ( t ) E 5 ( t - nT) n=0

CO

= E f ( n T ) S ( t - nT) ( 2 - 2 )

n=0

T h i s b e c o m e s t h r o u g h s t a n d a r d L a p l a c i a n t r a n s f o r m a t i o n

F * ( s ) = E f ( n T ) e " nTS ( 2 - 3 )

n=0

N o w 5 i f f o r c o n v e n i e n c e we u s e t h e t r a n s f o r m a t i o n 2 n = e nTS and u s e F (Z ) t o r e p r e s e n t t h e i m p u l s e m o d u l a t e d f u n c t i o n , ( 2 - 3 ) b e c o m e s

F (Z ) = E f(n T ) Z n--0

- n

- f ( 0 ) + f ( T ) Z- 1 + ... + f ( n T ) z"n + . . . ( 2 - 4 )

Thus t h e Z - t r a n s f o r m c a n b e c o n s i d e r e d i n d e p e n d e n t l y o f a L a p l a c e t r a n s f o r m , w i t h t h e e x p o n e n t o f Z b e i n g an o r d e r i n g i n d i c a t o r . F o r an e x a m p le o f t h e c a l c u l a t i o n o f a Z - t r a n s f o r m , l e t f ( t ) = e a t . Then m o d u l a t i n g t h e u n i t i m p u l s e c a r r i e r w i t h f ( t ) we o b t a i n

4

i . e . f * ( t ) = 1 + e -aTZ-1 + e ' 2aV

2 + . . . + e ' 8"1 z ' n + . . .

W r i t i n g t h i s i n c l o s e d f o r m , i t b e c o m e s

F (Z ) = ---1--- ( 2 - 5 )

i _aT 7 _1

1 - e Z

A t a b l e o f s t a n d a r d Z - t r a n s f o r m s c a n be f o u n d i n A p p e n d ix A. Assume now t h a t F (Z ) o f ( 2 - 5 ) i s a t r a n s f e r f u n c t i o n o f t h e Q

fo rm ~ ( Z ) , w h e r e C( z ) i s t h e o u t p u t v a r i a b l e and R(Z) i s t h e i n p u t R

v a r i a b l e . The n u m e r i c a l e v a l u a t i o n o f C(Z) may be p e r f o r m e d b y s e v e r a l m e t h o d s , two o f w h i c h w i l l b e b r i e f l y d e s c r i b e d . One m e th o d u t i l i z e s c o n v e n t i o n a l l o n g d i v i s i o n , b u t i n t h i s c a s e t h e i n p u t R(Z) m u st be o f

a known fo rm s u c h a s s i n e , ramp, s t e p , e t c .

. .

, R( z )

c(z)

i _aT y ' 1

1 - e Z

I f R(Z) i s a ssu m ed t o be a u n i t s t e p f u n c t i o n , t h e n R(Z) — --- ~ 1 - Z C(Z ) t h e n b e c o m e s

C(Z) = _

( l - z " 1) ( l - e -a V 1)

1 - (l+ e' aT)z"1+e"aTz“2

E v a l u a t i n g

by

be

f U )

-1 - a V 1 1 - e Z

i. . e .

C(Z ) - e " aT C(Z ) Z_1 = R(Z)

C(Z) = R(Z) + e " aT C(Z) Z- 1 ( 2 - 6 )

w h i c h i s e q u i v a l e n t t o

C(nT) - R(nT) + e " aT C [ ( n - l ) T ] Now i f R(Z) i s a u n i t s t e p f u n c t i o n

t h e n

C ( l ) = R ( l ) + e " aT. C ( 0 ) = 1 as C ( 0 ) = 0 C( 2 ) = R ( 2 ) + e - a T . C ( l ) = 1 + e " aT C ( 3 ) = R ( 3 ) + e ~ a T . C ( 2)

- i + -a T -2 a T

= 1 + e " h e

Thus t h e o u t p u t C(Z) i s t h e same by e i t h e r m e t h o d .

2 . 2 S y s t e m A l g e b r a

I n a d d i t i o n t o t h e Z - t r a n s f o r m ^ a k n o w l e d g e o f t h e a n a l y s i s o f o p en l o o p t r a n s f e r f u n c t i o n s i s n e c e s s a r y . C o n s i d e r t h e two e x a m p l e s shown i n F i g . 2 .

Rfs)

G

y c*(s)

_UJ

(a)

6

G ( s ) = [ R ( s ) G (s) ] ( 2 - 7 )

w h i l e t h e o u t p u t i n 2 ( b ) i s

C * ( s ) = [ R * ( s ) G ( s ) ] *

= R ~ ( s ) G ~ ( s ) ( 2 - 8 )

1 a

Now i f , f o r e x a m p l e , R ( s ) = — and G ( s ) = t h e n ( 2 - 7 ) b e co m es

* , a *

C ( . ) =

t h a t i s

c <z > =

w h e r e ^ d e n o t e s t h e Z t r a n s f o r m o f t h e te r m i n p a r e n t h e s i s .

C(Z)

E q . ( 2 - 8 ) c a n be w r i t t e n

c <z > = ‘ 3 - 2 ( 1 - e ~ aT)

( Z - l ) ( Z - e " aT)

( Z- l ) ( Z - e " a^

k k k

T h u s , t h e o u t p u t s R ( s ) G ( s ) and [ R ( s ) G ( s ) J a r e n o t t h e sa m e . The a c t u a l c o n f i g u r a t i o n o f t h e s y s t e m i s t h e r e f o r e v e r y i m p o r t a n t .

I I I . IDENTIFICATION

3 . 1 M ethod

I t w o u l d be d e s i r a b l e t o be a b l e t o i d e n t i f y t h e c o e f f i c i e n t s o f a l i n e a r d i f f e r e n t i a l e q u a t i o n o f any o r d e r w h i c h i s o f t h e form

— + A f + + A Y = u ( t ) ( 3 - 1 )

d t " 1 d t " ’ 1

w h e r e Y ( t ) = s y s t e m o u t p u t u ( t ) = s y s t e m i n p u t

and t h r o u g h A^ a r e t h e unknown c o e f f i c i e n t s o f t h e s y s t e m and a l l i n i t i a l c o n d i t i o n s a r e z e r o .

T h i s r e s e a r c h w i l l d e a l w i t h f i r s t and s e c o n d o r d e r s y s t e m s a l t h o u g h t h e m e th o d t o be d e s c r i b e d may be e x t e n d e d t o h i g h e r o r d e r s y s t e m s .

T h i s m e th o d d e p e n d s upon a l i n e a r s a m p l e d - d a t a m o d e l w h i c h a s s u m e s t h a t t h e i n p u t and o u t p u t s a m p l e s o f a g i v e n s y s t e m a r e r e l a t e d by a Z - t r a n s f o r m m o d el o f t h e f o l l o w i n g form

F( z ) = ^ f ( 3 - 2 )

w h e r e N ( z ) = a -f a Z ^ + a Z ^

' 0 1 n - 1

D(Z) = 1 + ^ Z ' 1 + p 2z ' 2 + . . . . j3n Z 'n

and n i n d i c a t e s t h e o r d e r o f t h e s y s t e m .

The i n p u t t o t h e s y s t e m t o b e i d e n t i f i e d c a n b e o f any f o r m , b u t i n m o s t o f t h e i d e n t i f i c a t i o n s , a s t e p f u n c t i o n o f c o n s t a n t a m p l i ­ t u d e and s t a r t i n g a t t i m e z e r o w as u s e d . T h e s e s t e p f u n c t i o n s w e r e s a m p le d and t h e s a m p le d v a l u e s w e r e u s e d t o d r i v e t h e ' b l a c k b o x '

s y s t e m . The o u t p u t o f t h e s y s t e m a t e a c h s a m p l i n g i n s t a n t w a s r e c o r d e d and t h e c o r r e s p o n d i n g v a l u e s o f i n p u t and o u t p u t w e r e l i s t e d t o g e t h e r . T h i s l i s t o f i n p u t and o u t p u t s a m p l e s i s r e f e r r e d t o a s t h e i n p u t - o u t p u t r e c o r d , and c o n t a i n s a l l o f t h e s a m p l e s a t t h e sampling i n s t a n t s u n t i l t h e i n p u t was r e m o v e d . The i n p u t r e c o r d c a n be e x p r e s s e d m a t h e m a t i c a l l y a s t h e sum o f t h e s a m p le d v a l u e s o f i n p u t a t t h e s p e c i f i c s a m p l i n g t i m e . Thus i f t h e i n p u t i s d e n o t e d b y X ( z ) t h e n t h e t o t a l r e c o r d o f t h e i n p u t u s i n g n o r m a l Z - t r a n s f o r m n o t a t i o n i s

m

X (Z ) = E X (n T )Z _n ( 3 - 3 )

n=0

w h e r e m = number o f s a m p l e s

X (n T ) i n d i c a t e s t h e v a l u e o f t h e i n p u t a t t h e s a m p l i n g i n s t a n t nT.

and Z n i s t h e t i m e o r d e r i n g i n d i c a t o r . S i m i l a r l y t h e o u t p u t W(z) c a n b e w r i t t e n as

m

W(Z) = E W(nT) Z~n

n=0

w h e r e W(nT) i n d i c a t e s t h e v a l u e o f t h e o u t p u t a t t h e s a m p l i n g t i m e nT.

T h e s e e x p r e s s i o n s f o r t h e i n p u t and o u t p u t r e c o r d s c a n a l s o b e w r i t t e n a s

m

9

m

W (Z ) = £ W. z"J j = 0 J

w h e r e j i n d i c a t e s t h e t i m e o f s a m p l i n g and and X^ a r e t h e v a l u e s o f W(Z) and X (Z ) a t t h e t i m e j , t h i s b e i n g t h e fo rm o f n o t a t i o n u s e d i n l a t e r e x p r e s s i o n s .

I f t h e same i n p u t i s a p p l i e d t o t h e unknown s y s t e m and t h e Z - t r a n s f o r m m o d e l and t h e o u t p u t s o f t h e two c o r r e s p o n d e x a c t l y a t e a c h s a m p l i n g i n s t a n t , t h e n t h e m o d e l i s a t r u e s y s t e m m o d e l and t h e c o e f f i ­ c i e n t s o f t h e a c t u a l s y s t e m a r e k n o w n . I f t h e two o u t p u t s a r e n o t e x a c t l y t h e same h o w e v e r , t h e e r r o r b e t w e e n them i n d i c a t e s how c l o s e t h e m odel c o e f f i c i e n t s a r e t o t h e t r u e v a l u e s . Thus i f we w e r e t o m i n i ­ m i z e t h i s e r r o r w i t h r e s p e c t t o t h e c o e f f i c i e n t s o f t h e m o d e l N ( Z ) / D ( Z ) we w o u ld o b t a i n t h e b e s t a p p r o x i m a t i o n t o t h e s y s t e m . The e r r o r d e t e r ­ m i n a t i o n i s shown d i a g r a m m a t i c a l l y i n F i g . 3 .

/

F i g . 3 T rue M odel E r r o r

The e r r o r a t e a c h s a m p l i n g i n s t a n t i s e ^ . The t o t a l e f f e c t i v e e r r o r i s f o u n d by summing t h e s q u a r e s o f t h e v a l u e s o f e^ o v e r t h e r e c o r d

10

E ( e ) 2 --- <j> IX ( Z) ‘ d("z} " W^Z) | 2 ^ = minimum ( 3 - 5 )

■* 2nj ' '

I f t h i s e x p r e s s i o n w e r e m i n i m i z e d w i t h r e s p e c t t o t h e c o e f f i c i e n t s and o f t h e m o d e l , t h e r e s u l t w o u l d g i v e t h e b e s t i d e n t i f i c a t i o n . T h i s m i n i m i z a t i o n h o w e v e r c a n n o t b e s o l v e d e x a c t l y [ 9 ] .

A m e th o d s u g g e s t e d by S t e i g l i t z and M cB ride [ 9 ] w i l l now be d e v e l o p e d t o s o l v e t h e a b o v e m i n i m i z a t i o n p r o b l e m . The m i n i m i z a t i o n i n v o l v e d i n F i g . 4 , w h i c h h a s no p h y s i c a l i n t e r p r e t a t i o n , c a n be e a s i l y s o l v e d .

F i g . 4 L i n e a r R e g r e s s i o n E r r o r

A g a in u s i n g t h e i n v e r s i o n i n t e g r a l t h e e x p r e s s i o n f o r t h e e r r o r i s

E ( e . ) 2 = — & ! X ( Z ) . N ( Z ) - W ( Z ) . D ( Z ) I 2 — = minimum ( 3 - 6 )

j 2 tcj u i ' L t

The e r r o r a t any t i m e j i n v o l v e s t h e v a l u e s o f i n p u t and o u t ­ p u t a t t i m e j p l u s s e v e r a l p r e v i o u s v a l u e s . The number o f p r e v i o u s v a l u e s i s d e p e n d e n t upon t h e o r d e r o f t h e s y s t e m . Thus t h e p r o d u c t a t X (Z ) and N(Z) c a n be w r i t t e n a s

The e f f e c t o f X ( Z ) . N ( Z ) a t t i m e j i s t h e r e f o r e n - 1

E a . X . . i = 0 1

n

S i m i l a r l y t h e p r o d u c t o f W(Z) and D(Z) a t t i m e j i s E (3, W. , i = 0 1 J ” 1 The e r r o r a t t i m e j i s t h e n

n - 1 n

e . = E a . X . . - E 8.W . ( 3 - 7 )

J i = 0 i J - i i = o 1 J ' 1

n - 1 n

o r s e t t i n g S o = 1 e . = E a . X . . - E B.W, . - W.

° J i = 0 1 J _ 1 i = l 1 J " 1 J

We c a n w r i t e t h i s i n m a t r i x fo rm i f we l e t

8 ' = C V V - " ...' V

q' . = r x . , x . x . , . , w . ---W. ] J L 3 * J - l j - n + 1 " j - 1 * J ~ n J

and 8 and q be t h e t r a n s p o s e o f 8 ' and q ' j * T h e r e f o r e

e . = q ' .8 - W.

J J J

The t o t a l e r r o r c a n b e o b t a i n e d by s q u a r i n g t h e s a m p l e d i n s t a n t e r r o r s and summing o v e r t h e r e c o r d l e n g t h .

2

E ( e ) i s t h e t o t a l e r r o r w h i c h we w o u l d l i k e t o m i n i m i z e J

w i t h r e s p e c t t o t h e c o e f f i c i e n t s a and i . e . w i t h r e s p e c t t o t h e 2

m a t r i x S . I f we t a k e t h e g r a d i e n t o f E ( e ^ ) w i t h r e s p e c t t o 8 we w o u l d o b t a i n a m i n i m i z a t i o n c r i t e r i o n

7 9e .

i . e . g r a d ( E ( e ^ ) ) = 2 (E e^.

= 2 E q . e . = 0 J J

But e . — q ' j 8 - W.

12

T h e r e f o r e £ q . q ' . 8 - £ q.W. = 0

J J ] J

l e t Q = £ q . q '. and c = £ q.W.

J J J

Then t h e c o e f f i c i e n t s i n 8 a r e

8 = Q_ 1 . c ( 3 - 8 )

f o r minimum e r r o r .

B ut t h i s c r i t e r i o n d o e s n o t mean a n y t h i n g a s f a r a s t h e o r i g i n a l p r o b le m i s c o n c e r n e d .

I f t h e v a l u e s o f and p^ and t h e c o r r e s p o n d i n g v a l u e s o f N(Z ) and D ( z ) d e n o t e d a s N ^(Z) and D^(Z) w h i c h w e r e f o u n d u s i n g t h e

a b o v e t e c h n i q u e a r e u s e d t o a d j u s t t h e v a l u e s o f i n p u t and o u t p u t s u c h t h a t

W (Z )

new _________ 1______________ _ 1

and

^ o r i g i n a l ^ ^ 1 + (B^Z + . . . p^Z n D^(Z)

x _new ( z ) _{_} -j _{1 _} ,₁

^ o r i g i n a l ^ ^

1 + P^Z ^ + . . . p^Z n

D^(z)

t h e n new v a l u e s o f a and p a r e o b t a i n e d . The new v a l u e s o f N ( z ) and I )(z ) t h o a b e c o m e N^(Z) and D ^ Z ) r e s p e c t i v e l y . T h i s p r o c e d u r e i s c o n ­ t i n u e d s e v e r a l t i m e s . I f we l e t i e q u a l t h e number o f i t e r a t i o n s , t h a t i s , t h e number o f t i m e s t h e o r i g i n a l e q u a t i o n 8 = Q ^ . c i s s o l v e d

t h

t h e n on t h e i i t e r a t i o n we s o l v e f o r N (Z ) and D^(Z) w i t h t h e o r i g i n a l i n p u t and o u t p u t r e c o r d s b e i n g f i l t e r e d by ^ ( Z ) , t h e v a l u e o f

D(z)

e,-13

F i g . 5 I t e r a t i v e Model E r r o r

2 dZ

9 1 N . ( Z) D . ( Z )

S ' . j ) = — f | * < * ) - W<Z ) - D ^ | 2 i

. t h .

I f o n t h e i i t e r a t i o n t h e v a l u e s f o u n d f o r a and (3 a r e t h e same t h e v a l u e s f o u n d o n t h e p r e v i o u s i t e r a t i o n , t h e n D ^ (z ) = D^, ^ (Z ) t h e e r r o r m i n i m z a t i o n e q u a t i o n b e c o m e s

V Z> ? V Z> 5 d ?

§ lx <z >- M z T - w ( z ) l 1 - 5 ^ - C z f l

1 ' 1 - 1 '

- min

( 3 - 9 )

a s and

— m m .

l . e .

N.

f | X ( Z ) . ~ K Z ) - W(z) | 2 ~ = m i n . ( 3 - 1 0 )

T h u s , t h e i t e r a t i v e p r o c e d u r e p r o v i d e s a s o l u t i o n t o t h e o r i g i n a l m o d e l e r r o r m in im iz a t io n a t c o n v e r g e n c e o f t h e c o e f f i c i e n t s .

t r u e

3 . 2 F i r s t O rd er L i n e a r S y s t e m

H a v in g f o u n d (3^ we m u st now r e l a t e t h e Z - t r a n s f o r m t o t h e t i m e domain c o e f f i c i e n t s o f t h e e q u a t i o n

Y + \ Y = u ( t )

T h i s e q u a t i o n may b e w r i t t e n i n a t r a n s f e r f u n c t i o n fo r m u s i n g L a p l a c e t r a n s f o r m a t i o n a s

Y , . ____1 _

u ^ s + \

T h i s e q u a t i o n c a n now b e w r i t t e n i n t e r m s o f Z - t r a n s f o r m s a s

}K z ) = - 1

-T h e r e f o r e (3^ = - e

i . e .

\ = ~ ln (^~^l^ ( 3 - 1 2 )

T

w h e r e I n = n a t u r a l l o g a r i t h m T = s a m p l i n g p e r i o d (3^ = c o e f f i c i e n t o f D ( z ) .

3 . 3 S e c o n d O rd er S y s t e m s

The Z - t r a n s f o r m m o d e l f o r s e c o n d o r d e r s y s t e m s i s “ 0 + “ i Z’ 1

| ( z ) --- — z i

1 + p xz L + P2Z

15

s2Y (s) + S A Y (s) + B Y ( s ) = u ( s ) o r

^ ( s ) = — 1 = ( s + a ) ( s + a ') ( 3 " 1 5 )

u S +AS+B lS

w h e r e s = a r e t h e r o o t s o f t h e c h a r a c t e r i s t i c 2

e q u a t i o n s + As + B.

( 3 - 1 4 ) b e c o m e s u po n a p p l i c a t i o n o f Z - t r a n s f o r m a t i o n - a T - a T

Y 1 ( e 1 - e 2 )Z 1

“

“ V l ' l - t e - a i V - ^ V 1 + e-<‘l+a

> V 2

T h e r e f o r e e q u a t i n g c o e f f i c i e n t s o f ( 3 - 1 3 ) and ( 3 - 1 6 ) - a T - a T

^ = - ( e + e ) - ( a l + a 2 ) T

P2 - e

We c a n s o l v e f o r a^ and a 2

= " I n r l v l ?.>

a l T 2

' h - ^ 2 - 4 ! i

-( 3 - 1 7 )

- "In /

v 1

2

x

a 2 T 2 '

rhen t h e c o e f f i c i e n t s o f (3-14) becom e

3 . 4 H i g h e r O rd er S y s t e m s

S y s t e m s o f a h i g h e r o r d e r t h a n two may a l s o b e i d e n t i f i e d . I f a l o o k up t a b l e i s d e v e l o p e d t o r e l a t e t h e t i m e domain c o e f f i c i ­ e n t s o f s u c h s y s t e m s t o t h e e q u i v a l e n t Z - t r a n s f o r m c o e f f i c i e n t s , t h i s t a b l e c a n t h e n be s t o r e d i n c o m p u t e r memory and t h e a p p r o p r i a t e r e l a ­ t i o n s h i p s c a n b e r e t r i e v e d when a p a r t i c u l a r o r d e r o f s y s t e m i s t o be i d e n t i f i e d .

A f e w e x a m p l e s w i l l b e shown t o i l l u s t r a t e t h e u s e f u l n e s s o f t h i s t e c h n i q u e .

3 . 5 Com puter S o l u t i o n

The i n p u t and o u t p u t r e c o r d s u s e d a s d a t a f o r t h e c o m p u t e r i d e n t i f i c a t i o n w e r e g e n e r a t e d by m eans o f a r e c u r s i o n f o r u m l a o f t h e a p p r o p r i a t e o r d e r . T h i s d i g i t a l g e n e r a t i o n p e r m i t t e d many s y s t e m s t o be t e s t e d w i t h o u t any d i f f i c u l t y o b t a i n i n g d a t a . The number o f s a m p le o b t a i n e d and t h e s a m p l i n g p e r i o d a r e known and a r e a l s o i n p u t t o t h e p r o g r a m . The i d e n t i f i c a t i o n o f f i r s t and s e c o n d o r d e r s y s t e m s w as a c c o m p l i s h e d by u s i n g two p r o g r a m s . T h e s e p r o g r a m s w e r e w r i t t e n i n F o r t r a n and w e r e run o n an I . B . M . 1 6 2 0 . They a r e now d e s c r i b e d w i t h r e f e r e n c e t o F i g . 6 .

The v a l u e s o f i n p u t and o u t p u t o b t a i n e d fro m t h e r e c u r s i o n f o r m u l a a r e f i r s t r e a d i n t o t h e c o m p u t e r a l o n g w i t h T - t h e s a m p l i n g p e r i o d and N - t h e o r d e r o f t h e s y s t e m . E.ach v a l u e o f X (Z ) t h e i n p u t and W(Z) t h e o u t p u t , i s s t o r e d a s an e l e m e n t o f v e c t o r XJ and WJ

17

i; 0 . 0 0 0 1

< 0.0 0 0 /

T h e st e p s u s e d

H - F R E A R E T H E

S A M E A S T H O S E

S H O W N A T F A R

Le f t , t h e o n l t

D i F F E f i E N C E

o c c u r s / a / t h e

A c t u a l f o r m u l a s

U s e d .

< o

E N D

S T A R T

I F

I I - )

So l v e Fo r

c a l c u l a t e

m o d e l

Ou t p u t

J e t u p

i n i t/a l/s f

M A T R I C E S

Re a d o r i g i n a l

X T £ W T

S u r v \ e i = ^ S3" * W 3

saue =

2-

cct-Qtt

R E A D

X J W J N T

Me a n s q u a r e

E R R O R £ V A R I A N C E P R E F I L T E R X T ^ W T

s t o r e c o e f f i c i e n t i n Al a p

i n I T I A L I 2 E P A R A M E T E R S

I I , N P , N Z A L A P

c a l c u l a t e c o e f r

Co m p a r e W i t h

A l a P

18

t h e i t e r a t i o n c o u n t e r , N2 and NP, and ALAP w h i c h c o n t a i n s t h e p r e v i o u s v a l u e s o f t h e t i m e dom ain c o e f f i c i e n t s a r e s e t t o t h e i r i n i t i a l v a l u e s . SQUE and SUMCI a r e i n i t i a l l y s e t t o z e r o . The a p p r o p r i a t e v a l u e s o f XJ and WJ a r e s t o r e d i n t h e m a t r i c e s QJ and QJT. QJ i s t h e n u s e d t o

fo r m t h e m a t r i x C l = QJ.WJ and Cl i s summed o v e r t h e r e c o r d l e n g t h and t h e r e s u l t a n t s t o r e d i n SUMCI. A l s o t h e p r o d u c t o f QJ and QJT i s summed and s t o r e d i n SQUE w h e r e SQUE = £ QJ.QJT. The m a t r i x SQUE i s i n v e r t e d u s i n g a l i b r a r y s u b r o u t i n e and t h e r e s u l t a n t p r e m u l t i p l i e s SUMCI. T h i s p r o d u c t i s t h e c o e f f i c i e n t m a t r i x S . Up t o t h i s p o i n t i n t h e c a l c u l a t i o n s t h e o r d e r o f t h e s y s t e m i s i m m a t e r i a l a s l o n g a s i t i s s p e c i f i e d . H ere t h e two p r o g r a m s b e g a i n t o d i f f e r . The c a l c u l a t i o n s h a v e e s s e n t i a l l y t h e same fo rm e x c e p t t h a t t h e a c t u a l f o r m u l a s f o r m o d e l o u t p u t , e t c . a r e s p e c i f i c a l l y f o r f i r s t o r s e c o n d o r d e r s y s t e m s . The number o f i t e r a t i o n s a l r e a d y p e r f o r m e d i s c h e c k e d b y t e s t i n g I I . I f I I i s o t h e r t h a n z e r o t h e o r i g i n a l i n p u t o u t p u t r e c o r d i s r e r e a d . U s i n g t h e c o e f f i c i e n t s o f t h e 8 m a t r i x and t h e i n p u t r e c o r d XJ t h e

o u t p u t o f t h e m o d e l N ( Z ) / D ( Z ) i s c a l c u l a t e d , and t h i s o u t p u t i s co m p ar ed w i t h t h e a c t u a l o u t p u t WJ o f t h e unknown s y s t e m . The mean s q u a r e e r r o r

f i l t e r l / D j j a s d e s c r i b e d i n S e c t i o n 3 . 1 . The new c o e f f i c i e n t s r e p l a c e t h e p r e v i o u s o n e s i n A L A P , t h e c o u n t e r I I i s i n c r e a s e d by o n e , and t h e

I V . EXPERIMENTAL KESELTS

S e v e r a l s y s t e m s w e r e i d e n t i f i e d u s i n g t h e m e th o d d e s c r i b e d . V a r i o u s s a m p l i n g p e r i o d s and n um bers o f s a m p l e s w e r e u s e d f o r some s y s t e m s . A l s o v a r i o u s i n p u t s w e r e u s e d . The r e s u l t s o f t h e s e t e s t s a r e l i s t e d i n t h e f o l l o w i n g t a b l e s .

TABLE 1 F i r s t O rd er S y s t e m s

Y + AY = u ( t ) w h e r e A. - 1 . 0 0 0 and 20 0 s a m p l e s

T s e c . u ( t ) A. <j0 E r r o r

0 . 5 1 . 0 1 . 0 0 0 1 3 4 + 0 . 0 1 3 4

0 . 2 1 . 0 0 . 9 9 9 9 9 2 3 - 0 . 0 0 0 7 7

0 . 1 1 . 0 1 . 0 0 0 0 0 4 5 + 0 . 0 0 0 4 5

0 . 0 5 1 . 0 1 . 0 0 0 0 0 8 8 + 0 . 0 0 0 8 8

0 . 0 2 1 . 0 0 . 9 9 9 8 8 - 0 . 0 1 2

TABLE 2 Y + A.Y — u ( t ) w h e r e A. — 1 . 0 0 0

and 1 50 s a m p l e s

T s e c u ( t ) A. ^ E r r o r

0 . 5 1 . 0 1 . 0 0 0 0 6 8 + 0 . 0 0 6 8

0 . 2 1 . 0 0 . 9 9 9 9 9 2 3 - 0 . 0 0 0 7 7

0 . 1 1 . 0 0 . 9 9 9 9 4 9 - 0 . 0 0 5 1

0 . 0 5 1 . 0 1 . 0 0 0 0 2 9 + 0 . 0 0 2 9

TABLE 3 Y + AY = u ( t ) w h e r e A = 7 . 5 3 2 0 and 200 s a m p l e s

T s e c u ( t ) A $ E r r o r

0 . 5 1 . 0 7 . 5 3 6 7 1 + 0 . 0 6

0 . 1 1 . 0 7 . 5 3 3 6 4 + 0 . 0 2

0 . 0 2 1 . 0 7 . 5 3 3 0 0 + 0 . 0 1 3

0 . 0 1 1 . 0 7 . 5 3 2 0 7 + 0 . 0 0 1

0 . 1 5 . 0 7 . 5 3 1 7 3

• • •

- 0 . 0 0 4

TABLE 4 Y + AY = u ( t ) A = 2 0 . 1 5 0 and 2 00 s a m p l e s

T s e c u ( t ) A < jo E r r o r

0 . 1 1 . 0 2 0 . 1 1 1 1 - 0 . 2 0

TABLE 5 Y + AY = u ( t ) A = 1 . 0 0 0

and 2 0 0 s a m p l e s

T s e c u ( t ) A f E r r o r

0 . 1 3 . 0 0 . 9 9 9 9 2 7 - 0 . 0 0 7 3

0 . 1 5 . 0 1 . 0 0 0 0 0 4 5 + 0 . 0 0 0 4 5

2 2

SECOND ORDER SYSTEMS

TABLE 6 Y + AY + BY = u ( t ) and 20 0 s a m p l e s

T s e c ACTUAL VALUE IDENTIFIED VALUE

A B A B

0 . 1 3 5 . 0 3 0 0 . 0 3 5 . 0 8 5 6 3 0 1 . 0 6 2

1 . 0 6 . 0 5 . 0 5 . 9 9 1 4 4 . 9 9 3 4

1 . 0 5 . 0 6 . 0 4 . 9 9 0 8 5 . 9 8 4 4

0 . 5 5 . 0 6 . 0 4 . 9 8 5 0 5 . 9 7 7 7

TABLE 7 SECOND ORDER IDENTIFICATION ERRORS

ACTUAL COEFFICIENTS PERCENTAGE ERROR

1

A B A B

0 . 1 3 5 . 0 3 0 0 . 0 + 0 . 2 4 6 0 . 3 5 4

1 . 0 6 . 0 5 . 0 - Q . 1 4 3 - 0 . 1 3 3

1 . 0 5 . 0 6 . 0 - 0 . 1 8 4 - 0 . 2 6 0

V . DISCUSSION OF RESULTS

The a n a l o g i d e n t i f i c a t i o n t e c h n i q u e o f S h e n g and Wu [ 7 ] i s b a s e d on t h e p r i n c i p l e o f s t e e p e s t d e s c e n t and u t i l i z e s an i m p l i c i t s y n t h e s i s m e t h o d . I t i s u s e d f o r f i r s t and s e c o n d o r d e r l i n e a r s y s t e m s and a l s o n o n l i n e a r f i r s t o r d e r s y s t e m s . A m e th o d t o i d e n t i f y h i g h e r o r d e r l i n e a r s y s t e m s u s i n g t h e same i m p l i c i t s y n t h e s i s c i r c u i t s i s a l s o p r o p o s e d . The s y s t e m s t o be i d e n t i f i e d w e r e s i m u l a t e d u s i n g an a n a l o g c o m p u t e r c i r c u i t . The r e s u l t s o f t h e i r m e th o d w e r e c o m p a r e d t o t h o s e o f t h e d i g i t a l m e th o d d e s c r i b e d .

I n t h e d i g i t a l m e t h o d , t h e f a c t t h a t t h e i n p u t w as sam pled^ t h a t i s , i n t e r r u p t e d , i m p o s e d a m i n o r r e s t r i c t i o n . The s y s t e m t o be i d e n t i f i e d c o u l d n o t b e run u n d e r n o r m a l o p e r a t i n g c o n d i t i o n s . H o w ev e r, s i n c e t h e m a j o r a p p l i c a t i o n o f any i d e n t i f i c a t i o n t e c h n i q u e i s i n t h e r e s e a r c h f i e l d , t h i s r e s t r i c t i o n i s n o t f e l t t o be o f any g r e a t

s i g n i f i c a n c e .

The i n p u t - o u t p u t r e c o r d s w e r e c h e c k e d t o make s u r e t h a t t h e number o f i n i t i a l i z i n g z e r o s e q u a l l e d t h e o r d e r o f t h e s y s t e m . I f t h i s c o n d i t i o n was n o t t r u e t h e d a t a w as m i s l e a d i n g and c a u s e d e r r o n e o u s i d e n t i f i c a t i o n .

2 4

0 - t

X=

2 0 0 S A M P L E S

0-0/

X = l . o o o

150 s a m p l e s

0.001

"X= i.o o o 2 0 0 SA/APLES

0.5T

0 .0 5

0.01 _0.02 0 - 1 0.2

25

The p a r a m e t e r A. w as f o u n d t o be 7 . 5 3 6 7 i n t h e w o r s t c a s e and 7 . 5 3 2 0 7 i n t h e b e s t c a s e . A t h i r d s y s t e m Y + 2 0 . 1 5 0 Y = u ( t ) w as a l s o i d e n t i f i e d t o w i t h i n 0 . 2< j b u t t h i s v a l u e c o u l d b e im p r o v e d w i t h t h e p r o p e r s e l e c t i o n

o f s a m p l i n g p e r i o d . T a b l e s 1 , 2 , 3 l i s t t h e i d e n t i f i e d v a l u e s and a s s o c i ­ a t e d e r r o r s f o r two f i r s t o r d e r s y s t e m s w i t h s e v e r a l d i f f e r e n t s a m p l i n g p e r i o d s . F i g u r e 7 sh o w s t h a t t h e b e s t r e s u l t s a r e o b t a i n e d when t h e t o t a l s a m p l i n g t i m e i s e q u a l t o a p p r o x i m a t e l y 2 0 - 3 0 t i m e s t h e t i m e c o n s t a n t o f t h e s y s t e m .

The s e l e c t i o n o f a s a m p l i n g p e r i o d i s d e t e r m i n e d by two c r i t e r i a . I f t h e s y s t e m r e s p o n s e i s a s shown i n F i g . 8 and t h e

Ti m e

F i g . 8 Time R e s p o n s e o f F i r s t O rd er S y s t e m

s a m p l i n g p e r i o d T i s l a r g e w i t h r e s p e c t t o t h e t i m e c o n s t a n t l/A. t h e s a m p l i n g i s p o o r and t h e r e f o r e t h e e r r o r i s h i g h e r . When n T , t h e t o t a l s a m p l i n g t i m e i s s m a l l w i t h r e s p e c t t o 5/A., o n l y p a r t o f t h e s y s t e m r e s p o n s e i s s a m p l e d and t h e e r r o r i s h i g h e r . T h e re i s t h e r e f o r e an optim um s a m p l i n g p e r i o d w h i c h i s s m a l l co m p a r ed t o 1/A. p r o v i d i n g t h e

26

p e r i o d i s s m a l l co m p a r ed w i t h t h e t i m e c o n s t a n t w h i c h s a t i s f i e s t h e two c r i t e r i a . F i g . 7 sh o w s t h a t a s t h e s a m p l i n g r a t e i n c r e a s e s t h e a b s o l u t e v a l u e o f e r r o r i n c r e a s e s and a s t h e s a m p l i n g p e r i o d b e c o m e s q u i t e l a r g e

( t h e t o t a l s a m p l i n g t i m e i s lo n g ) t h e a b s o l u t e e r r o r a g a i n i n c r e a s e s . S i m i l a r r e s u l t s w e r e o b t a i n e d f o r A. = 1 . 0 0 w i t h 150 s a m p l e s and A. = 7 . 5 3 2 0 w i t h 200 s a m p l e s . T h e r e f o r e o n c e an a p p r o x i m a t i o n o f t h e s y s t e m i s

o b t a i n e d , a p r o p e r v a l u e o f T c a n b e c h o s e n t o o b t a i n t h e b e s t i d e n t i f i ­ c a t i o n .

The i d e n t i f i c a t i o n o f s e c o n d o r d e r s y s t e m s a l t h o u g h n o t a s a c c u r a t e a s t h e f i r s t o r d e r s y s t e m s w a s w i t h i n 0 . 5 ^ i n e v e n t h e w o r s t c a s e . F o r s e c o n d o r d e r s y s t e m t h e s a m p l i n g p e r i o d m u s t b e c h o s e n t o

s a t i s f y two c r i t e r i a . A c c o r d i n g t o t h e s a m p l i n g t h e o r e m t h e p e r i o d o f s a m p l i n g m u s t b e a t l e a s t T = 1/2W w h e r e W i s t h e h i g h e s t f r e q u e n c y p r e s e n t . A l s o t h e t o t a l s a m p l i n g t i m e m u st b e s u f f i c i e n t t o a l l o w t h e s y s t e m t o s e t t l e w h i c h s e t s a l o w e r l i m i t o n t h e p e r i o d T . I f a p e r i o d o f a p p r o x i m a t e l y 1 / 2 o r 1 / 3 t h e u p p e r l i m i t i s u s e d , an a p p r o x i m a t i o n o f t h e s y s t e m c a n b e f o u n d . T h i s i s p o s s i b l e i f t h e b a n d w i d t h o f t h e s y s t e m c a n b e a p p r o x i m a t e d e v e n r o u g h l y . The p e r i o d c a n t h e n be r e ­ d u c e d u n t i l t h e e r r o r s t a r t s t o r i s e a g a i n . F o r e x a m p l e t h e s y s t e m Y + 5Y + 6Y = u ( t ) c a n be i d e n t i f i e d a t T = 1 . 0 s e c s , w i t h g r e a t e r a c c u r a c y t h a n a t 0 . 5 s e c s . The maximum v a l u e o f T i n t h i s c a s e w o u l d b e a p p r o x i m a t e l y T = 2 . 0 s e c s .

By a p p l y i n g t h e m e th o d s u g g e s t e d b y M cB ride and S t e i g l i t z w i t h a s a m p le d i n p u t i n s t e a d o f a c o n t i n u o u s o n e a g r e a t e r a c c u r a c y o f i d e n t i ­ f i c a t i o n h a s b e e n a c h i e v e d . S i m i l a r r e s u l t s c o u l d be e x p e c t e d f o r

V I . CONCLUSIONS

The d i g i t a l m e th o d d e s c r i b e d i n t h i s r e s e a r c h o f f e r s an a l t e r n a t i v e t o a n a l o g i d e n t i f i c a t i o n f o r l i n e a r s y s t e m s o f any o r d e r . The l o o k - u p t a b l e f o r h i g h e r o r d e r s y s t e m s w o u l d c o n s i s t o f r e l a t i o n ­ s h i p s w h i c h w o u l d be o f v a l u e t o t h e p a r t i c u l a r u s e r .

B o th s y s t e m s , t h e d i g i t a l and a n a l o g , g i v e a c c e p t a b l e r e s u l t s and t h e c h o i c e o f o n e o v e r t h e o t h e r w o u l d d e p e n d upon a v a i l a b l e e q u i p ­ m e n t and t h e p e r s o n n e l i n v o l v e d .

APPENDIX A

TABLE A . l Z T r a n s f o r m s

f ( t ) F ( s ) F (Z )

1₁ 1 Z

s Z - l

1 T Z

t

2

s ( Z - l ) 2

1 z

- a t

e s + a Z - e " aT

- a T -bT b - a Z (e " aT- e " bT')

e - e

( s + a ) ( s + b )

( Z - e - aT) ( Z - e ' bT)

a Z s i n a T

s i n a t 2, 2

s + a Z2 - 2 Z c o s a I + l

s Z( Z - c o s a T -)

c o s a t _{2+ 2}

29

I o o t

"j I r ' O N ' j I O N — w c I ( o » 1 ; « •«,• j ' o ) » o J T ( o ) » ~ - l ( o » 1 ) » ^ n L T ( o » i )

o i :i: ;o i o n x j .... j c o - - - ;

J I N 0 N 0 I ON , . ( NO i O n / i : ( ■» Or1 ) i 0 f. 0 0 ) i i u i O ( 6 i 0 !

O I . 1 6 ^ 6 I ON . - ( 0 . - )

0 0 0 ' r o ~ N T t r 1 2 . 7 - ! " ' - . . .. . . .

O n u * O N i J ! < ., ^

N o = O “N0

- - ■ •" — j •

- i :

o ~ o j o< Jl ( 1 i i ) — ■ » ^ ■' -6 0 7 C I ( In I ) - . . J O 6 0 0 N - 1 » >NO

J O 6 0 0 u = 1 »: -.n

J -■ N ^ L * N ) ^ ' • '

-T 0 0 'JiOC I ( K i 0. ) - ■- • ^ ^ v ~

J = ~ N P - N f I

j o l -j ; j =

---J 0 0 7 ’" T = ! 11

-; — I — 1

0 J ! I ) - X J < i_ 5 0 7 0 J T C I ) - X J C u )

_;o , . o i =r-j, - — r z j - - i - + t t

-J C I ) N v J ( J ) ON n J 7 ( I ) - l:; J ( : : )

0 I ( i » 1 ) = n .

J J- 7 - -0 :: 1 i N O

L - 1

y - o J r ( x )

0 IN r N L ) = X - N J ( J )

o J i - ' . O I ( . < » ! _ > = J ~ . ’. 0 1 ( O t D + O . O . t L I

A ( \ t L ) = X 7 J i _ i i \ ) - V

- NO = 7 0 ---

-NA = i 1

N _ = N 0

3 0

a o —> A — I * N — D U "O L - ! i HA

o 3 0 J._ ( A * L. ) — v , i L ) + .-s 5 - ^ L. i i ^ C C N T i TvUi_

.

jCj 3 K - 1 i . D ATT- "3" L'— _.- A ( ;■A t L )

F A - 0 2 ; p a-h o

C A L L P i j ( JaTk:-A PA A s = i , PAP

■

L._ ,

---■ A P A L ; = A A H - * p a l )

.-•. _ ?, : -p

■

m- mL

Fp = NA

PAP - 1

OALL---i-*ArA>-P J OALL---i-*ArA>-P C i i <_ • ! i .-1 ( 1 t J ) i J : i i i ) * I - 1 *. P <_ ) A L A - “ L 0 P r ( r-. ( » 1 ; ) / 1

f ' i « -N A r->

I F ( I I - i

A -r F.vJUi P.

i^trAipr • *jS" 1' • l ( X y ( v i h -,v _• ( J ) « vj= i v D s.- -j > j j i f ‘ s> i s i ■ i /-a i V I X 1 ■" - -- • / ^ i_ — - i_ # i )

A 15 A A A A 1 A = 1 • i 5

A J ( A / - • .

S ; i t .\ ; - •

-J .. A — - •

a- p '

-AI s —

/ i a ■

.

• _ •

AO 3 A T 5 A P P . l v .

P. P J = A ( 1 , 1 ) -p X J ( P ) + , , ( A , 1 ) P r_ p. A A L_ ( : ' ) - { A A A “ .v A ( .-. ) )

A ;A A A A = r A : 1A ■ ■ + ■. A ( In ) 0 3 7 rj'AA = «PAA

i \ i s P — A ^ n — ! \

'r-A 'r-A i < 0 i < — A A A A — / s i ; , --1 F A A A A d d P i ^ - A S j i ,

A u ‘-A F A A5P A T ( AHA 5-sHOrl I A i l. 1 6 i P !

y > N i •

■

- T Ak- T -A -. --A --A 3-5- 7 ; ; = : --A '

V Pis I ( I P P P P N I ( 5 ) 5 P A -A, A v A A T - v A P T -(- P A - . I

A A r - P u y - A P

V ■ s ini — v i n s i / ' i -"

"" ” P j ^ t F ' j A t i i V.-'<T - - ■

3 1

x o ( x ; + f - ( . . s i i ) p - w >

-■ 0 '..'J (;< ) j < m - t _ * i ; ... j < i )

i I = I I + 1

o x i . o r o o x « i i

■ ~ U 3 ' , i f ■

-X L - .< L + i A l_ A P - A L ■'

o J T O 0 J -

. y i ^ U i \ ‘ — A ^ J ( / i — / \

0 0 P O A . ' i A T ( 1 0 i : TA- _. O ^ r A I .. 1 _ A T i o i P 1 . 7 )

0 O P ' P 0 A A A T~( ~ n i i - ' i — )- - -

-£_ r y r \ . - : ; , T ! 4 l P • v O

0 0 .0 F p . P . - i A T t I 10

^ 0 - i"‘ j : \ t A 7 ( i X i r A i 7 i i i ... • / ) O A L L O X I T

B u d

3 2

O R O i

_2L3-I_0-12 . L J . w I C _ 0 i . l ) i j J ( i ) t

J o [ Oi'j X ( 3 — w ) , *'j C o = — )

--i 4 a l Ui 1 C 2 —- ‘ _ ( 2 , - )

) , : 1 2 2 , 2 :

i J T ( Q j , _ i

X ( 2 3 ) , J j 2 3 »

J E <

1 }_, 3 _ 3 , 3 )

L T -1 (

i’: _r-\vj O O O ( lJ

K i_ A 3 9 j o , T

_ i \ _ A 3 — 9 9 3 , ( X 3 t 3 2 ,

,

2 2 — 2 ’<■ !\l

l l - u

-S Q U E ( I , l ) - v . » ^ C < 3 ^ S U M C - L C l , I ) = 3 #

DO 6 0 0 K = l , N 2 D O - 6 0 0 L = l , N 2

S Q U E < L , K ) = 3 . o 3 3 U

DO- 7 3 0 K = l , N 2

L = l

-SU O IC -i- C - K , 3 - ) — 3 » u w u 0 ^ J = 3

n p=n+ i

D O 1 3 0 J = X E , 4 : 3

D O i . ( 1 — 1 , 2

i< - I — 1

0 J ( I ) = X J ( L )

Q J T T L ) _ - X J C o )

DO 2 o 1 = 2 2 , 2 0

1-1= J * ~ I 2 N

q j ( I ) = 3 J ( )

4-4 4/ r C oQo-2 ,J._ (!• ) _ __

0 I C 1 , 1 ) - 0 •

^ 3 7 K = x , i 1C

3 3

i i n u

■C -a i_ i_ P _ v

r p. . » C-( C I ! J ) l J - 1 l I } » i — 1 » N p ) I F ( I I — 1 ) 4 3 i ‘+ 4 t ‘i ‘ *

4 P - P A J 3 p --- .. .. .

P C A C 5 5 1 » ! X J ( J ) « p ( J ) i J - l i 2 - - ) ; j 1 r O C 5 1 A T C I X i r 1 - • . ' 3 X i 5 1 _ • , )

4 3 3 5 J •- 1 .< — 1 * i'i

5 5 L P ) - * •

j . i X J ( \ ) = 0 .

. . . i aU l = 3_»_o. . .

5 5 J = - .

. ; 1 1 - .. . ..

C P P O P - J .

-X Y - r t ( 1 , 1 ) 5 5 C 5 ) + , - , ( * _ ♦ ! ) ?5 - ; ^ ( 1 )

aaP .J = 5 0 5 t - t A L J » 4 ) ' - J ~ ■: ( C_». 1 ) 7;r > - 5 0 . 1 3 ',-j J i = B W J

/Af i K ] — B 4 ^ ■'.> A. ( X / )

1 [A P O I v — i A , A -5 A r ~:— \ Aa )

,... < . ... 5 — .. 5

P U N C H 5 3 -

, 5 i a 4. .5 T- £ . _ l o . 5

'5 v • £ T — _ • _

5 0 5 3 7 N = N P « ^ ~

-5 ^-5-5I-=XC:A113i-A-fA-5 (-5 1 1 -3-50. . . ... . .. . .

. . I VaaNT — V A P T + V . i

- - -

J —

P ... . —LAh'

VO'TA 1 - v 3 ;\T / M V,-'

P Ulas. rj _ 3 j y r V . - ■... T

l A o r ( i . 1; iV 5 - a 5 ^ . -5 _ r r 5 w K-! 1^. * 1 o • 5 )

_____ 5 3 5 ^ C = 3 3 P T F ( ( ( 3 v.U. ) l l p r - p . * 1 ~ L a ’J J 1.1...

M 1 -; “ L J 4 r C ( ( 4 < i ) Ti Ok^AV ) / <_ • ) u i . - - L . 0 G F !. ( ~ ( _■ , 1 ) - r C U ^ w ) / C . )

- ( A 1 + ^ 1 ) P P = CA 1 -X-i.5 1

3 4

. : . .0 A r ( - > O A.r I ‘ - ) “ ~ • ‘ I ’ * -- • * -' ■ d

t,-.. / -

l_

^h-a z/ t

w d - o i / i T ': <_ )

J J r-!U ^ C i ^ ,' J t n l - t L . i

— . : . . . . _• _ . . . _ . . .

0 3 F O X . "AT ( 3 1: F F ; _ A O o F F I ... I A. A T T - X - F " 1 3 . 7 i . - X * F 1 3 • 7 )

F ^ A ,iP -A d -

3 , ; P

-i-’J A ^ i : ' l u u j t - i - I

o o o r J k i - i A [ ( i' 1 ; * .1. . . i r 1 i_ • T )

V. J ( X ) J ( X ! r , , ( , 1 ) * . J ( ; - I ) i , -. ( £ . , 1 J ( X ~ d )

a - j x j c x j. -x J c ; ; ; + - - ( x , i ) c . - i ) + a ( 4 , n » x j c x - x >

i i - i i + i

d . v i . j - L - j u w . ) I A P J > F in d o o « i i

-... 4tG XjO d J d .. _.. .. . ...

o d d F O X ' A T ( Oi-i 1 I - I d )

,* ^ o C ~ T d • / )

A F O F F A T ! 4 Z I O . o 5 A T T F O X X X T C M )

0 0 ■ - r Oi- XmT C 1 X ». 1 - • i d 3a » i’ i ^ # A )

— s ■ o^Fi-TXL4-4AJ-..n. . . . . . .

F A L L XX I T

BIBLIOGRAPHY

1 . A.W. L a n g i l l J r . , A u t o m a t i c C o n t r o l S y s t e m s E n g i n e e r i n g , V o l . I I ( P r e n t i c e - H a l l , New J e r s e y , 1 9 6 5 ) .

2 . J . T . T o u, D i g i t a l and S a m p l e d - D a t a C o n t r o l S y s t e m s , ( M c G r a w - H i l l , New Y o r k , 1 9 5 9 ) .

3 . D . P . L i n d o r f f , T h e o r y o f S a m p l e d - D a t a C o n t r o l S y s t e m s , ( J . W i l e y and S o n s , New Y o r k , 1 9 6 5 ) .

4.. B . Kuo, A n a l y s i s and S y n t h e s i s o f S a m p l e d - D a t a C o n t r o l S y s t e m s , ( P r e n t i c e - H a l l , New J e r s e y , 1 9 6 3 ) .

5 . J . G i b s o n , N o n l i n e a r A u t o m a t i c C o n t r o l , ( M c G r a w - H i l l , New Y o r k , 1 9 6 3 ) . 6 . W. S e i f e r t and C . S t e e g , C o n t r o l S y s t e m s E n g i n e e r i n g , ( M c G r a w - H i l l ,

New Y o r k , I 9 6 0 ) .

7 . C . L . S h e n g and M.Y. Wu, ' S y s t e m I d e n t i f i c a t i o n by Means o f I m p l i c i t S y n t h e s i s ' , N a t i o n a l C o n f e r e n c e on A u t o m a t ic C o n t r o l , C a r l e t o n U n i v e r s i t y , O t t a w a , 1 9 6 5 .

8 . R .E . K alm an, ' D e s i g n o f a S e l f - O p t i m i z i n g C o n t r o l S y s t e m ' , T r a n s . ASME, V o l . 8 0 , p p . 4 6 8 - 4 7 8 , F e b r u a r y 1 9 5 8 .

VITA AUCTORIS

1 9 4 5 1963

1967

1969

B orn i n T o r o n t o , O n t a r i o

C o m p l e t e d Grade X I I I a t W a l k e r v i l l e C o l l e g i a t e I n s t i t u t e , W i n d s o r , O n t a r i o .

B . A . S c . i n E l e c t r i c a l E n g i n e e r i n g a t t h e U n i v e r s i t y o f W i n d s o r , W i n d s o r , O n t a r i o .