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VLIEGTU[Ü30UWKUNDE ^ ^ ^ ^^^^^ ^ ^ " ' ^ ° Michiel ét Ruylarweg 10 - DELFT

\ W¥

Kh:

### by

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Page 20. 6th l i n e should read

### 0 ( 3 ; = - . ^ . Ê m Z j l d

M 3+Pjji M ( s ;

7th, 8th and 9th linea

Replace M ( 5 ) by ^ j j

Page 2. Equation 2.6. Left hand side should be ejtj^ and not e(tJ,

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'^mt^m^immmm

### V

-THE COLLEG-E OF AERONAUTICS

EERATA - C o . A . Note No. 128 DETERMINATION OF DYI-TAIvIICAL MODELS FOR

ADAPTIVE CONTROL SYSTEMS

Page 3 . E q u a t i o n (2.11 ) shoiild r e a d

oo

### •P(ju)) = / e~'^'*' ^ f ( t ) d t = i e"'^"^*f(t)dt

since f(t) = 0 for t<0. F(jw) is the Fourier Transform of f(t)

Page 5. After equation 3.12. add - "where N ( 3 ) comprises the factors in (3.10) with the exception of the first factor,"

Page 13. Para, (f) should read - "as s->cx), then y^s (s) etc."

Page 13. Para. 5.2. First sentence should read - "f/ith Networks A and B representing one port netw»rks, i.e. combinations

of resistors and capacitors between input and output terminals, the conventional analogue computing circuits are given."

Page 16. Equation (5.14) should read

^ ^ ^ V V2T(l+skT)/ s+a

Page 17. Delete 'orthonormal' from first sentence after equation (5.29)

Page 20. I^t jLine_ add - "where M(s) is the Laplace Transform of the input probing signal"

3rd line - Delete M ( S ) Zfth line - Delete M ( S ) 5th line - Delete M ( S )

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T H E C O L L E G E O F A E R O N A U T I C S C R A N F I E L D

Determination of Dynam^ical Models for Adaptive Control Systenas

b y

-R. J. A. Paul, B.So. (Eng.), A . M . I . E . E . , A. M.I. M e c h . E .

SUMMARY

A method is described for the synthesis of a dynamical niodel of a linear systena based on the use of orthonormal functions. It is shown that if the nominal va ues of all poles of a system a r e known, and if onjy one pole changes from its nominal value, then this change may be detected. It is also demonstrated that the numerator t e r m s of the transmission transfer function of the system may be found provided the denonainator is known. Active networks a r e described, for the simulation of the relevant orthonormal functions.

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CONTENTS

Page Summary

Introduction 1 Basic Method of Measurement 1

Orthogonal Functions 3 Synthesis of the dynamical model 7

4 . 1 . Determination of the denominator t e r m s 7 4. 2. Determination of the numerator t e r m s 10

Simulation of the Dynamical Model 12 5 . 1 . The Basic Arrangement 12 5.2. Use of one-port networks 13 5 . 3 . Use of two-port networks 14 5.4. Simulation of the relevant orthonormal

functions 15 5 . 5 . The Dynamical Model 17

P r a c t i c a l Considerations 21 6 . 1 . On-Line Operation 21 6 . 2 . The effect of system non-linearities 21

6 . 3 . The injection of test signals at several

points in the system 22

Conclusions 22

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1. Introduction

The approximate determination of the dynamics of p r o c e s s e s has been the subject of several recent p a p e r s .

One form of adaptive control systems , in which the controller is automatically adjusted to maintain a particular performance index in the presence of random

variations of the system p a r a m e t e r s , Is based on the realization of a dynamical model of the p r o c e s s . This model, under normal operating conditions of the p r o c e s s , must be capable of automatic adjustment to ensure that it r e m a i n s a reasonable

approximation of the p r o c e s s when the p a r a m e t e r s of the latter a r e subject to random variation.

The approximation of a linear system, based on the use of orthogonal functions, has been considered in many p a p e r s , including those by Gilbert* ' and Kitamori*^'. A spectrum analyser has been described by Braun et al* ' and uses a set of orthogonsil functions.

In these p a p e r s , very little knowledge, if any, is assumed of the process dynamics in the determination of the approximate model. However, in many applications the nominal values of some of the system p a r a m e t e r s a r e known, and this knowledge may be used to realize the most economical model of the system. This approach is

justified on the grounds that there is no virtue in constructing an adaptive system which does not make use of all available information concerning the p r o c e s s . This paper is concerned with such applications and a procedure is outlined, based on the use of orthonormal functions, for the determination of a dynamical model of the p r o c e s s . 2. Basic Method of Measurement

A null method is used and is similar to that described by Kitam.ori arrangement is shown in Fig. 2. 1.

(3) T h e INPUT PROBING ilGNAL m

### (t)

PROCESS DYNAMICS A DYNAMICAL MODEL B OUTPUT C ( t ) SQUARING AND AVERAGING CIRCUIT

### ;ro'

SIGNALS TO ADJUST PARAMETERS OF MODEL

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- 2 • •

Block A r e p r e s e n t s the dynamics of the process to be controlled.

Block B r e p r e s e n t s the dynamical model consisting of active RC networks whose p a r a m e t e r s a r e adjusted automatically to give a minimum value of mean square e r r o r e(t) .

Block D r e p r e s e n t s a squaring and averaging circuit, the output of which r e p r e s e n t s the mean square e r r o r .

Block F r e p r e s e n t s the control circuits which control the adjustment of the p a r a m e t e r s of the model. Only one p a r a m e t e r is adjusted at any time to give a minimum value of e ( t ) * .

m(t) r e p r e s e n t s the common input probing signal to both the process and the model. Although the process is subject to random variations, if it is assumed that these occur slowly compared with the response of the p r o c e s s , block A may be represented in mathematical form by its transmission transfer function G(s), where s is the complex variable cr + jw. In other words the process is assumed to be time invariant during the period required to estimate its c h a r a c t e r i s t i c s and to c o r r e c t the model. If the p r o c e s s is assumed to be linear, then

o r C(s) = G(s). M(s) c(t) = / g(X). m(t - X ) . dX o+ (2.1) (2.2)

where g(t) i s the unit impulse response of the system.

If G*(s) r e p r e s e n t s the t r a n s m i s s i o n t r a n s f e r function of the model, then, C*(s) = G*(8). M(s)

o r (t)

## r

^ o+

g*(T ) m(t - T )dT ,

where g (r) is the unit impulse response of the model. e(t) = c(t) - c*(t) Thus L e t Then f(t) = g(t) - g * ( t ) .

### i(t? = r CiCh.). f(T)

c(t) - c'(t) g(X). m(t - X)dX T L'o ' o

### rg*(T).m(t-T

(2.3) (2.4) (2.5) (2.6) - T)dT l i m T

### è ƒ '

-T m(t - X ) . m(t - T)dt d t . (2.7) (2.8) dX . dT . . . (2.9) i . e . ë ( t r = ƒ ƒ fW. iU) 0 ( X - T ) d X d T , o o

where ^ ( ^ is the auto correlation function of m(t).

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### F(ju) =. I i

J -to

Now F ( j u ) =. / e " ^ * ^ f ( t ) d t ( 2 . 1 1 ) w h e r e F(juj) i s t h e F o u r i e r T r a n s f e r of f(t).

• OP

A l s o » (u) » / e"^"* ^ju) d/Lt ( 2 . 1 2 )

### « ^ J.»

w h e r e * (u) i s t h e s p e c t r a l d e n s i t y of m ( t ) . m m T h u s f(t) = -è- I e^'^*F(Ju)dw ( 2 . 1 3 ) • — 00 and * ( M ) = ^ T e ^ " " • (w) du . ( 2 . 1 4 ) 2 ^ j m m » —CO * ( X - T ) = ^ f e^"^^"^^ * ( u ) d u . ( 2 . 1 5 ) 2fr I m m • - 0 0

### = ^ ƒ * (u) F(jw). F(-jaj)du , (2.18)

2tr ƒ m m ^ * - 0 0 i . e . Wl* = -^ / " ' | F ( j u ) l * * (w)dw , ( 2 . 1 9 ) J - 0 0 1 . 0 0 2 = T - / I G(ju)) - G*(jw) I * (w)dw . ( 2 . 2 0 ) 2ir / ' I m m i -oo T h u s f o r a given s p e c t r a l d e n s i t y of t h e input p r o b i n g s i g n a l m ( t ) , t h e m e a n s q u a r e e r r o r e(t) i s m i n i m i s e d a s G*(jw) b e c o m e s a c l o s e r a p p r o x i m a t i o n t o G(jw). T h e f r e q u e n c y r e s p o n s e function G (jw) i s s y n t h e s i z e d b y a conabination of a c t i v e RC n e t w o r k s w h i c h r e p r e s e n t a s e t of o r t h o g o n a l f u n c t i o n s . 3 . O r t h o g o n a l F u n c t i o n s An o r t h o g o n a l s e t of f u n c t i o n s 0 ^ ( s ) , Ojia) . . . O ( s ) , m a y b e d e r i v e d f r o m a n y s e t of l i n e a r l y i n d e p e n d e n t functions L ( s ) ; L (s) . . . L ( s ) , b y m.aking O ( s ) a s u i t a b l e l i n e a r conabination of L ( s ) , w i t h r

### ± / _ } q 0 " ) . o ^ ( - j u ) * ^ ( . ) . d . ' [ l ' 7 i ; i , (3.1)

(3 5) O (s) m a y b e r e p r e s e n t e d by t h e e q u a t i o n s '

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0^(8) = L ( s ) L (s) L (s) r r - 1 1 P P . . . P r - i , r r - 1 , r - 1 r - i , i P P . . . P r - a , r r - a , r - i r - 2 , i w h e r e q , r 2w ƒ " L^(JU ) L (-jw) * (u) dw . r •' m m p p . . . p r - 1 , r - 1 r-1 , r - a r - i , i P P . . . P r - a , r-1 r - a , r - a r - a , i P P P i , r - i i , r - 8 , , , P P P r r r , r - i r , i P P . . . P r - 1 , r r - 1 , r - i r - i , i P P, , P 1 , r < , r - i 1 , 1 (3.2) (3.3)

T h u s a n y s e t of o r t h o g o n a l functions i s d e p e n d e n t on the s p e c t r a l d e n s i t y of the input s i g n a l mi(t).

C o n s i d e r the p a r t i c u l a r c a s e when * (w) i s a c o n s t a n t which m a y be

n o r m a l i z e d to unity. Such a s p e c t r a l d e n s i t y r e p r e s e n t s t h a t of a white n o i s e s i g n a l . A l t h o u g h the l a t t e r i s not p h y s i c a l l y r e a l i z a b l e , it m a y be shown, t h a t , p r o v i d e d

* (w) i s c o n s t a n t o v e r a bandwidth w h i c h i s nauch g r e a t e r t h a n t h a t of the p r o c e s s u n d e r i n v e s t i g a t i o n , it i s a r e a s o n a b l e a p p r o x i m a t i o n to a s s u m e t h a t * (w) i s c o n s t a n t o v e r a n infinite r a n g e of f r e q u e n c y . If, t h e r e f o r e , \$ (w) i s a s s u m e d t o b e u n i t y then e q u a t i o n 3.3 m a y be , m m e x p r e s s e d a s qr

### ^ /ls<^"^- V"^'^^^"

(3.4) 1 a s w •• 00 , t h e n If L (jw). L (-jw) a p p r o a c h e s z e r o m o r e r a p i d l y t h a n e q u a t i o n 3 . 4 m a y be e x p r e s s e d a s a c o n t o u r i n t e g r a l w h e r e the c o n t o u r e n c i r c l e s t h e e n t i r e r i g h t half s - p l a n e , T h u s ,

### 2-^i L

L ( s ) . L ( - s ) d s q r 3~ L ( s ) . L ( - s ) d s q r r e s i d u e s of L ( s ) . L ( - s ) in r i g h t q r half s - p l a n e . ( 3 . 5 ) If a l i n e a r , s t a b l e , d i s c r e t e p a r a m e t e r s y s t e m i s now c o n s i d e r e d , i t s t r a n s m i s s i o n t r a n s f e r function i s c h a r a c t e r i z e d by i t s p o l e s and z e r o s , which m a y be r e a l o r

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F o r such a system the set of linearly independent functions may be expressed in the f o r m : -L, (s) ' - ~ k s + p,, for k = 1 . . . . m (3.6) and ^ m ^ 2 r - l <«) 2 2 „ * s + 2a s+a +/3 r r r L ^„ (s) = s L ^_ , (s) ra+2r m+2r-l where p, , a and 6 a r e positive constants.

'^k r r '^

for r = m + l . . . . n

( 3 . 7 )

If this set of functions is substituted in equation 3.1 the following orthonorm.al functions m.ay be derived.

0,(8) ^ . ^ , s - p ,

s + Pi

s-r^

### s + p,

(3.8) and in general for k < m

/ 2 p s - p k s + p, 8 + p k - 1 s - p s - p k - 1 8 + Pg S + p^ (3.9) If r > m V i a Vo* + ^ »

### Hn+2r-i«)= -.

s" + 2a s + a* + fi r r r s* - 2a s + o* + fi* 1 1 1 s + 2a^s + a* + j3* s^ - 2a s + a» + ^^ r - 1 r - 1 r - l s + 2a s + a* + j3* r - 1 r - 1 r - 1 s - p m s + p "^m -/ïa Vo'' + Ö r r r s + 2 a s + a + ^ r r r . N(s) " - P i s + p^ (3.10) (3.11) and O , „ (s) = — m+2r » s V4a r s + 2 a s + a +fl r r r N(s) (3.12)

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The transmission transfer function of the model may be expressed as N i = 1 ^ where and G*(8) = l_j \a, 0.(s) , (3.13) k / 2 p s - p s - p k o , ( 8 ) = - ^ — — - . ^ ^"^ . . . f o r i < m (3.14) i i 8 + p ^ + Pi 1 ^ + Pi 2 / 0 : [k] B + k' ' V ^ + /3f 1 s* - 2a s + a + ^' s-p i L i i 1 '^iJ 1 1 "^1 '^m k.O,(8) = • ^ ^ ^ s*+2a^ s + a^'+ ^» s''+20^ s + a ^ ^^ ^^^m s-p^ . . . for i > m . s+p, (3.15) F r o m equation 2. 20 e(t) ra 1 2*r / |G(JU) - G*(ja»)| \$ («) du . (3.16) J ' • mm é ^ 0 0 With « («) = 1 (3,17) mm ^ 0 0 i Ö T = ~ / |G<3<^) - G*(ju)) f d« (3.18) • - e o it 0 0 = ^ ƒ [G(i«) - G * ( j « ) ] |G(-JW) -G*(-jw)] du. . (3.19) » - 0 0

Substituting equation 3.13 into equation 3.19

CO JJ ü \'

-[G(J«)

### N,

+ \ k* . (3.20) i =1

F o r the minimum value of e(t) by adjustment of k

~ ' 'hr I ' ^ [G(J«)0^(-JW) Idu + 2k. = 0 1 • - 0 0

^, = 4- / G ( j « ) 0 . ( - j « ) d « (3.22) 1 ^ir I 1

• - 0 0

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Since G(ju) O (-ju) a p p r o a c h e s z e r o m o r e r a p i d l y than •:— a s u ^ „ 1 J w ( 3 . 21) m a y be e x p r e s s e d a s k. = \ R e s i d u e s in r i g h t half s - p l a n e of G(s) O ( - s ) ) \ - ) = \ R e s i d u e s in r i g h t half s - p l a n e of G ( - s ) 0 . ( s ) ) for i = 1 , 2 N . 4, S y n t h e s i s of the d y n a m i c a l m o d e l T h e t r a n s m i s s i o n t r a n s f e r function of a l i n e a r , d i s c r e t e p a r a m e t e r s y s t e m m a y be e x p r e s s e d by the p a r t i a l f r a c t i o n e x p r e s s i o n : -a -a -a, -a b s + c

=

+

+ . . .

+ . . .

### —22-

+ (3.23) S + D S + p , 8 + p , S + p a , „ j ^ 2 ^ o « ^ ^» *^k "^m 8 + 2a^ s + a, + p ^ " " .

### 8 ' + 2 a ^ s + a^ + ^J

w h e r e t h e p a r a m e t e r s p^ . . . . p , a . . . . a , ^ . . . . ^ a r e p o s i t i v e c o n s t a n t s . a , . . . a , b . . . . b__, c . . . . c „ a r e unknown c o n s t a n t s which m a y have a p o s i t i v e 1 m 1 N 1 N J r or n e g a t i v e s i g n .

4 , 1 . D e t e r m i n a t i o n of the d e n o m i n a t o r t e r m s

If, a t a n y t i m e , a l l but o n e , s a y p. , of the d e n o m i n a t o r coefficients a r e known, a m o d e l m a y be c o n s t r u c t e d in a way s u c h that the m i n i m i z a t i o n of the m e a n s q u a r e e r r o r , b y v a r i a t i o n of one pole p o s i t i o n , g i v e s the v a l u e of the unknown p a r a m e t e r . C o n s i d e r the m o d e l w h o s e t r a n s m i s s i o n t r a n s f e r function i s given b y

nr- s - p s - p, ^ s - p, , , 6 - P s* - 2 a s + a" + ^ ' /-.*/ \ \Iy. 1 ^ - 1 *^k+l *^m 1 1 ^1 u \ s ; = S + y s + p B + p, , s + p, , , s + p a , n , a , „a ' ^1 "^k-l *^k+l '^m s + 2a^ 8 + flj + ^ , (4.2) s* - 2 a s + a* + 8 * » ^«N N ^N 8* + 2a-, s + a ' + p' N N N U n d e r t h e s e c o n d i t i o n s , for e(t)* to be a m i n i m u m by a d j u s t m e n t of y , t h e n f r o m ( 3 . 1 9 )

### W~ ^ 'h I ' h rG(jü.)G*(-jw) + G(-j«)G*(+jw)"[ du = O (4.3)

= - 2 g ^ \ R e s i d u e s in the r i g h t half s - p l a n e of G ( - s ) G*(s) . ( 4 . 4 )

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8

-Hence

9 y \ p ^ + y y = 0 . (4.5)

i . e . ( P u + y ) ^ - ^ ^ - V ^ = 0 . ( 4 . 6 )

i . e . Pk " y • ( 4 . 7 )

T h u s if e(t) i s m i n i m i z e d , the v a l u e of y r e q u i r e d i s e q u a l to the unknown p a r a m e t e r y .

If now the n o m i n a l v a l u e s of a l l the p o l e s of the s y s t e m a r e known but in a p e r i o d , which i s long c o m p a r e d with the m e a s u r e m e n t t i m e , one pole can change f r o m i t s nom^inal v a l u e , t h i s change m a y be d e t e c t e d by a slight r e a r r a n g e m e n t of the m o d e l , i . e . •j-Ti— s - p s - p , , s - p , s - p , , s - p s ' - 2 a s + q ' + ö f _,*, . V 2y *^i "^k-l '^k *^k+l '^m i T ' ' i S + y s + p. s + p, , s + p, s + p. , , s + p a, _ , a, oa "ï *^k-l '^k ^k+1 '^m s + 2a^s+a^+p^ s ' - 2 a ^ s + a ^ + ^'^

### s% 2aj^8+a; + 4

(4.8)

T h e m o d e l h a s now a n e x c e s s r e a l pole c o m p a r e d with the s y s t e m . If any one of the r e a l p o l e s ( s a y p ) Is subject t o v a r i a t i o n , then the m i n i m i z a t i o n of

a g '^

|— e(t) will a g a i n give the condition that p = y . In t h i s c a s e , h o w e v e r , the

p a r t i c u l a r pole which h a s changed f r o m i t s n o m i n a l v a l u e i s not known and a p r o c e d u r e i s n e c e s s a r y to give t h i s i n f o r m a t i o n . T h i s m a y be a c h i e v e d in the following m a n n e r . After the m i n i m i z a t i o n of e(t) the f a c t o r / s - p \ i s r e m o v e d f r o m the t r a n s f e r

\ s + p, /

function of the m o d e l . If no change in the m i n i m u m condition for e(t) o c c u r s a s a r e s u l t of t h i s o p e r a t i o n t h e n it i s obvious that p i s the v a r i a b l e p o l e . If h o w e v e r ,

/ s - p \

a change o c c u r s , the f a c t o r I ) i s r e - i n s e r t e d and the p r o c e s s r e p e a t e d with I—7— J. T h i s s e q u e n c e of o p e r a t i o n s Is r e p e a t e d until the r e m o v a l of a p a r t i c u l a r

### ' ' V " " PrA

f a c t o r I — 7 — j c a u s e s no c h a n g e . T h e pole p then r e p r e s e n t s the n o m i n a l

^ " P / ^ s - p r

v a l u e of the v a r i a b l e pole of the s y s t e m and the r e d u n d a n t f a c t o r — m a y be r

r e m o v e d f r o m the naodel. T h e v a l i d i t y of t h i s p r o c e d u r e i s b a s e d on the a s s u m p t i o n , a s a l r e a d y s t a t e d , that the change in the pole p o s i t i o n d u r i n g the t i m e of m e a s u r e m e n t i s n e g l i g i b l e ,

An a l t e r n a t i v e s i t u a t i o n m a y a r i s e , h o w e v e r , in that the v a r i a t i o n m a y o c c u r in e i t h e r of the two n o m i n a l p a r a m e t e r v a l u e s of a p a r t i c u l a r p a i r of c o n j u g a t e c o m p l e x

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p o l e s . If it i s known that a particular pair of conjugate complex p o l e s has a damping factor which i s subject to variation, this variation m a y be detected by a slight rearrangement of the model with a transfer function of the form

2V/i w (s +w ) s ' - 2 4 , « s + u* s*- 24..W s + u ' n*l ^ ' ' ° ' ^ ° N ° N G (s) = . . . . 8*+ 2>i,w s + u ' s'+ 2Sg W 8 + w' s'+ 2^,W 8 + w* Pa °> ^ " ° N ° N S - p s - p . . . . . . . . - 2 2 - ( 4 , 9 ) S + p s + p • ^ ' 1 m

w h e r e (i r e p r e s e n t s the variable p a r a m e t e r and all other p a r a m e t e r s are known.

1

Note 5 u - a and w • Va* + fi* . ( 4 . 1 0 ) r o r o r r

r r

Under t h e s e conditions for eftr to be a minimum by adjustment of ii then

2 - ^ ^ =• - 2 5 " ^ y R e s i d u e s in right half s-plane of G ( - s ) G*(s) - 0 . ( 4 . 1 1 )

y Residues in right half s-plane of G(-s) ^ G * ( s ) = 0 ( 4 . 1 2 )

^ . a

, - , G ( - s ) . - I L . ( S + W ^ ) ( ^ - 2 M | U ^ S + W ^ )

i . e . ) R e s i d u e s in right half s-plane of ^j i < x 0 a .a (s* + 2/i, w 8 +w ) ' o, o, ' 1 s ' - 2 « « 8 + « • < * • " > ' o, o, i . e . ) R e s i d u e s of '• . = 0 ( 4 , 1 4 ) (s* + 2 R u S + «* )* 8* - 2 5 . « 8 + w ' o, o, ' o, «. This r e s u l t s in the condition that,

5 = M . ( 4 - 1 5 )

1

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10

-4. 2 Determination of the numerator t e r m s

The transmission transfer function of the system as expressed by 4,1 may be re-written as :-A :-A s-p :-A, s-p, , s-p, :-A s-p , s-p. _ . . i_ a | j k ^k-1 '^i m m - 1 J • s+p, s+p, • s+p, • • • s+pj^ • s+pj^_j ' s + p , • • s + p ^ • 8 + p ^ _ j ' • • s+p, B s + C s - p s - p , s - p 1 i _ ^m ^m-1 2^ * L O V J _ * S + P S + P / • • s + p. a o w . a o, (^ B s + C ( s ' ' - s 2 S , (J +u' ) (s''-82S w_+Uo ) s - p s - p , r r r - 1 O r - l O j . - ! * °i " i '^m '^m-l •*'s*+82S u + « • (s*+s24 u TT' ) (8*+82S w +w" ) ^"^Pm ^"^Pm-l r o o r - 1 o , o , i o, o, r r r - 1 r - 1 ' ' s - p , s+p, B__8+C-T (s*-s2JS u +0J* ) (8*-s25 « +u' ) s - p s - p , N N fj_i o „ , o „ , * o. o . m "^m-l N - 1 N - 1 s*+s2S„co +CJ* (s'+s2iS„ , u +(j« ) ( 8 ' + s 2 5 w + « M ^'^Pm ^"^Pm-1 N o^ o^ N - 1 o^_^ o^_^ s - p i_ • • • s+p, • (4.16) where S w * « and u = .f a* + ^ ' , (4.17) r o r o v r r r r N

I . e . G(8) = \~' I, 0,(s) where I, is a constant for i = 1 . . . .N . (4.18)

### 2 . " ' " ' " ' " " ' '

1

With the transmission transfer function of the model in this case given by :-N

G*(s) = ) K 0.(s) (4.19) the object is to determine the coefficients A . . . . A , B . . . . B „ and C.. . . . C „ .

'' 1 m ' N < N All these coefficients may be subject to variation but it is assumed that only one

coefficient at any time changes significantly from its nominal value, in the period required for measuring the change.

Note O-j(s) contains all the poles of the system which are assumed to be known.

### aë(tT*

F r o m equation 3. 22, it is obvious that k. = *. for the condition -^. = 0.

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Alternatively from 3, 23,

i,e.

or

\

### K

-\ Residues in right half s-plane of G(s) 0 , ( - s )

### ^ ^-^Pk-1 f;^

• • • s - p ,

\^' Residues in right half s-plane of G(8)

^ k \ "-Pk "-Pk-1 2P„ ^ ^ 1 k \ Also k' = Residues of r B ) B s + C r r 2 V g a T (-8) r o r 8 + s2S u + u^ r o o r r s" - s 2 S <•> + w r o o J r i . e . k ' - ^ r o or k ' 2VT~ü~ = B r r o r r and k* = Residues of r B 8 + C r r s + 8 25 " +w r o o r I

### 2VT~ü* . u)

r o o r r s' - s 2S u +«' r o o r r i.e. k" =- —

### or 2^/r~i^

o r o r r . CO k * r o o r (4.20) (4.21) (4.22) (4.23) (4.24) (4.25) (4.26)

The above analysis indicates that all numerator coefficients may be found provided that the poles of G(s) are identical to G*(s). Thus the search procedure for the model adaptation should be that the pole variation is first determined before

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12

-5. Simulation of the Dynamical Model 5 . 1 . The Basic Arrangement

It is required to generate the orthonormal functions so far described.

Use is made of a computing or operational amplifier associated with passive linear RC two port networks. The use of one-port networks may be considered a s a special case.

The general arrangement is shown in Fig. 5 . 1 .

t | O m^

### re.

o -o PASSIVE o NETWORK A 1-2

### 1

%2 -G PASSIVE NETWORK B o •

t 3

### e,

Fig. 5 . 1 .

Network B is a two port network placed In parallel with the computing amplifier i . e . active network C.

It may be shown ' that,

where and y^(s) 1 2 Ks) a

rS, . 2

### - Ëjs)

E ( s ) 3 y (s) 2 1

forward short circuit transfer admittance of E2(s) = 0 network A E,(s)=0 — r e v e r s e short circuit transfer admittance of network B

provided that the following assumptions a r e valid.

(5.1)

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(a) Networks A and B are initially relaxed ( i . e . zero charge on all capacitors) .

(b) The forward open circuit voltage transfer function of the amplifier -K G (s) •• - •*> over the frequency range of interest. (K is a positive constant and G (s) is a rational function of the complex frequency s) . (c) y (s) = O.i.e. amplifier input Impedance is infinite.

Q

(d) y (s) =-«»,i.e. amplifier output impedance is zero . B V (s) ~ v^(s)

(e) y (s) > •'^' -^n for the frequency range of interest . K

A B B (f) If y (s) or y (s) •• " as s •• «> that y (s) should approach infinity

2 2 M 1 1 a

as s •• OO .

(y2j(s) •= - short circuit admittance of output port of network A) . (y (s) a short circuit admittance of input port of network B) . 5.2. Use of one-port networks

With networks A and B representing combinations of r e s i s t o r s and capacitors between input and output p o r t s , the conventional analogue computing circuits are given.

i . e . if A r e p r e s e n t s a resistance R and B r e p r e s e n t s a capacitance C

### '"- if! = - sk • < » • "

E,(s)

which is the transfer function of an integrator.

AT ^3(3) R^

Also _ i _ ^ __f_ (5 4)

E,(s) R, • ^°'^' if network A r e p r e s e n t s a resistance R, and B a resistance R^^.

Network A may also represent several one port networks to each of which is connected an input voltage.

F o r this case, „

E,(8) = - y E^ ^ (5.5)

1 K

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14

-5 . 3 . Use of two-port networks

The parallel combination of standard T-networks to represent networks id B result

adjustment*''.

A and B results in transfer functions where coefficients are capable of independent

The forms of T networks together with their respective short circuit transfer admittance functions are given in Table 5 . 1 .

T A B L E 5 . 1 . NETWORK

### i

Cl 1 2 L| „ c R L j ei <»»o ) i t L| R R L2

+ 62 \

### +

+ 62 ^ 2 . ( 0 1 2 R 1 2R 1 2R ( =

### -.*cV'

1 + S C R 2»CR 1 + i C R 1 1+ sCR - y , 2 ( 0 )

Network A or network B may be arranged to comprise each of these networks multiplied by a scaling constant k.

Thus for example if

k S ^ C ^ R S k 2sCR + k A. , _1_ _i 2 3^ ^ z / ^ ' " 2R • (1 + sCR) k s*T*+ k 2sT + k 1 1 2 3 2R (1 + sT) and B, _!_ k / T % k^2sT + k, ^12^^' " ' 2R • (1 + sT) (5.6) (5.7)

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t h e n E , ( s ) E , ( s ) k s*T*+ k 2sT + k, 1 2 3 k y T ' ' + kj^sT + k^ w h e r e T = RC and the k ' s a r e <; 1 5 . 4 , S i m u l a t i o n of the r e l e v a n t o r t h o n o r m a l functions (i) G*(s) = ^ s+a w h e r e a i s to be c a p a b l e of a d j u s t m e n t and m a y be g r e a t e r o r l e s s t h a n unity 1 If a = kT w h e r e T i s a p o s i t i v e p r e - d e t e r m i n e d i n t e g e r and k i s a v a r i a b l e coefficient and < 1, t h e n r i * / \ ^ 2 k T ( 5 . 8 ) ( 5 . 9 ) ( 5 . 1 0 ) ( 5 . 1 1 ) ( 5 . 1 2 ) T h e a c t i v e n e t w o r k a r r a n g e m e n t which s a t i s f i e s equation 5 . 1 2 i s shown in F i g , 5 , 2 in which

### O

C . F . r e p r e s e n t s a c o m p u t i n g a m p l i f i e r r e p r e s e n t s a cathode follower and RC ,

Note: T h e m a g n i t u d e only of T i s used in specifying r e s i s t o r v a l u e s and 0.5 </2T, e t c .

R >/2T

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18 -F r o m -F i g . 5 . 2 . A l s o ( i i ) E^(s)

### Eli)

E , ( s ) / 2 k T 1 + s k T / 2 E s + a E , ( s ) G*(s) - - 2 / ^ 2 > ^ \ _ (s - a) 2T(1 + s k t ) / "^ " (s + a) 8* + s2Sw + 0)'' s^ + s2Sw + cu* o o J ( 5 . 1 3 ) ( 5 , 1 4 ) ( 5 . 1 5 )

w h e r e w m a y be g r e a t e r o r l e s s than unity and g i s a p o s i t i v e coefficient which cannot e x c e e d unity. It i s r e q u i r e d t h a t ^ m a y be capable of independent a d j u s t m e n t .

If t h e n " o - T • G*(s) = 2VT v X - s T 1 + S 2 S T + s'^T*

### JL

1 + s25T + s'^T* w h e r e and G,(s) G,(s) = 2VT" T G / S ) + G^(s)3 Vg s T 1 + s2gT + s ^ T * 1 + s 2 ^ T + s^T^ ( 5 . 1 8 ) ( 5 . 1 7 ) ( 5 . 1 8 ) ( 5 , 1 9 ) T h e i n s t r u m e n t a t i o n of G (s) and G (s) i s shown in F i g . 5 . 3 . 1 2 If F ( s ) = S ' T ' - s2ST + 1 B ^ T " + s 2 ^ + 1 T h i s m a y be e x p r e s s e d a s F ( s ) = - ( 1 - s4CT 1 + S 2 4 T + S ' T ( 1 + 4 ^ . G (s) ) (5,20) ( 5 . 2 1 ) ( 5 . 2 2 )

T h e i n s t r u m e n t a t i o n of equation 5. 21 i s a l s o shown in F i g . 5 . 3 . With r e f e r e n c e t o the l a t t e r , T == RC , at output A V ? E ^ ( s ) - VZ- s T 1 + s 2 ^ T + s^T* . E , ( s ) . ( 5 . 2 3 ) ( 5 . 2 4 )

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E (s) At output B, E j s ) = V4 E (s) = - ^ E (s) . (5.26) R * 2 2 * 1 + 82CT + s T E (s) At output C, £3(3) = - (4S E,(3)+ E , ( s ) ) , (5.28) i e ""'^'^ - (1 4^;sT • i . e . ' , . - - (1 ) , ^i<^^ l + s 2 e r + s ^ T ' ' = / l - B 2 ^ T + s ' T ' N ^ ^ ^ g j \ 1 + s2ST + s^T* / '1 + s2ST + s

If the voltages E . ( s ) and E (s), respectively, are multiplied by the coefficient 2^ the required orthonormal functions are obtained. It should be noted that a four gang potentiometer is required to give the coefficients dependent on 5 •

Alternative network arrangements may be derived in which independent adjustment of <i) is also available as described in Ref. 7.

o

5 . 5 . The Dynamical Model

The dynamical model of the system may be constructed by connecting, in tandem, the networks described in 5 . 3 .

One possible arrangement is shown in Fig. 5.4. With reference to Fig. 5.4. , networks N . , . N are of the form shown in Fig. 5. 2 and networks N , • • . N.,

1 m m+1 N a r e of the form shown in Fig. 5 . 3 .

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18

-S CHANNEL

(24)

ö > t—I

>

ö

### r

Mt«)0|C«) •o \ o-O M(») O 2 U ) MCS)OK|1«) r V 'i r r " MU) OnC») , 1 M(.s) On («) '^ 2v/f^

### 4r

ADJUSTMENT FOR Kfc FACTOR

M ( s ) S KiOi^CO

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20 -f,(s) f (s) m ^m+l<«>

### ys)

0,(s) O (s) m «N<«) 8 - P . 0 - ( s ) S + p , m = M(s) k=l M(s) 11 s + p. s - 2^ u s + 0) ° m + l ° m + l s ' + 2S , w s + w^ "^^^ ° m + l ° m + l . f (s) . M(s) m » M(s) . f (s) m N

### n

r = m + l M(s) s + p, — 2 1 . f (s) . M(s) s + p m - 1 s" - 24 w s + w* r o o r r s* + 24 w s + w * r o o r r m M(s) M(s) M(s) 2 ^ ^ V N - ^^N ^

V N ! i l )

### 2 ^ ^ V ^ ^

i l 11 V + s24-,ej + cj* '' s' + s2S. ^N-l<^) ^N-l<^> N °N °N 2 >^?N " O N • " O N s + s24^^cj ^ ° N °N

### - T ^ ) • ^N-i<^)

+ Cü ' ( 5 . 3 0 )

The t r a n s m i s s i o n t r a n s f e r function of the m o d e l i s given by N

G*(s) ( 5 . 3 1 )

i=l

w h e r e K. i s a s c a l i n g coefficient which m a y be p o s i t i v e o r n e g a t i v e . In F i g . 5 . 4 . t h i s s c a l i n g i s achieved by m e a n s of the v a r i a b l e r e s i s t o r s and in the c a s e of

._ t h i s s c a l i n g i n c l u d e s the f a c t o r 2^'T^. Switches S,. . . S . . N " 1 < N n e t w o r k s N

m+1 N

connect the outputs of the n e t w o r k s to the input of a m p l i f i e r 1 o r a m p l i f i e r 2 to t a k e acount of the sign of K. r e q u i r e d .

With s u c h a m o d e l e a c h switch and the a s s o c i a t e d v a r i a b l e r e s i s t o r would be o p e r a t e d s e q u e n t i a l l y to give a m i n i m u m value of e(t) P r o v i s i o n would be m a d e for one s e r v o - d r i v e to o p e r a t e a l l s e c t i o n s .

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6. Practical Considerations 6 . 1 . On-Line Operation

For on-line operation the adaptation of the model,as already described, is achieved by injecting the low level test signal into the input of the system in

addition to the actuating signal. The system output is then correlated with the test signal to give the desired signal for comparison with the output of the model.

The test signal in this application has a flat power density spectrum over a finite bandwidth, which with the latter, say, ten times that of the system, is suggested as a reasonable approximation to white noise,

The requirement for the cross-correlation, however, is a distinct disadvantage since it involves the instrumentation of a time delay, multiplier and integrator. Moreover, the assumption is implied that the process is ergodic which is not strictly true with slowly-varying p a r a m e t e r s . A compromise must also be made in choosing a finite time of integration instead of an infinite period, and this

results in functions of the actuating signals, which in this case are noise components, to appear at the output of the correlator. To achieve a reasonable signal to noise ratio the integration period must be maintained at some value T, with the result that changes in the system parameters can only be detected after a finite number of periods T. This is due to the fact that several coefficients may have to be adjusted and each adjustment occupies a number of periods T. Again, in order to obtain sufficient information regarding the system response, several correlations having different delays are required.

These considerations suggest the need for sampling and quantizing the input and output signals. In this way the correlator can be considerably simplified^ ' ' '. Alternatively, in some applications, the actuating signal itself nnay be used as

the test signal. However, as discussed in section 3, this means that the functions forming the model are no longer orthogonal and the adaptation of the model occupies a much longer time. Moreover, unless the signal approximates to white noise, there is no guarantee that the model, which gives a minimum e r r o r function, is a good approximation to the system.

Another factor i s , that in a system having a large number of poles, there will be difficulty in (detecting a minimum if say a low-pass filter is used as the averaging device. This is due to the fact that the output of the filter will be subject to random variation due to inadequate filtering,

The above considerations are severe restrictions on the use of the methods described in this paper. However, it appears that many of these restrictions are imposed on any method of model adaptation based on the use of test signals, 6.2. The effect of system non-linearities

Another disadvantage of any method based on test input signals is the fact that their presence may result in the system being operated in a non-linear mocie over

certain periods. For the case of simple non-linearities, such as well defined saturation effects, a possible cure might be the incorporation of identical non-linearities in the model. The success of this technique would depend on the

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22

-proportion of time the system was operating linearly to the time of non-linear operation. This technique has not been investigated by the author,

6 . 3 . The injection of test signals at several points in the system

If access is available to several points in the system, it would seem to be desirable to construct models for parts of the system by using several test signals and correlating the available output with the respective input signal in each case.

In this way, each model is simplified and the time of adaptation is reduced a s all the models may be adapted simultaneously. The disadvantage of this approach is the duplication of test equipment such as c o r r e l a t o r s . However, it may even be advantageous to adapt each component model sequentially with the same test equipment, if the time variation of the system parameter is sufficiently slow compared with the measurement time.

7. Conclusions

The synthesis of a model based on the use of orthonormal functions seems to have sontie m e r i t ,

The effect of non-linearities and inaccurate measurement techniques, however, warrants closer investigation to ensure that a reasonable approximation to system performance is achieved,

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8. References 1, Aseltine. J . A . 2. Gilbert, E . C . 3. Kitamori, T. 5. Braun, L . , Mishkin, E. , Truxal, J . G . Courant, R. , Hilbert, D. 6. Mathews, M. V. , Seifert, W.W. 7. Paul. R . J . A . 8. Suskind (Editor) 9. Watts, D . C . 10, Paul. R . J , A .

A Survey of Adaptive Control Systems.

T r a n s . Inst. Radio Engineers on Automatic Control, December, 1958.

Linear System Approximation by Differential Analyzer Simulation of Orthonormal Approximation Functions. T r a n s . Inst. Radio Engineers on Electronic Computers, June, 1959.

Applications of Orthogonal Functions to the Determination of P r o c e s s Dynamic Characteristics and to the Construction of Self-optimizing Control Systems.

Automatic and Remote Control, ( I . F . A . C . Proceedings Moscow, 1960), Vol.2, Butterworth Ltd.

Approximate Identification of P r o c e s s Dynamics in Computer Controlled Adaptive Systems.

Ibid, Vol.2.

Methods of Mathematical Physics.

V o l . 1 , 1953. (Text Book), Interscience Publishers L t d . , London.

Transfer Function Synthesis with Computer Amplifiers and Passive Networks.

1955 Western Joint Computer Conference, I . R . E . pp 7- 12. Simulation of Rational Transfer Functions with Adjustable Coefficients.

College of Aeronautics, Tech. Note No. 126, Feb. 1962. Notes on Analog Digital Conversion Techniques,

Chapter 1, (Wiley, 1957). Quantization.

Lecture presented at an Exposition of Adaptive Control. Imperial College of Science and Technology, April 1961. Hybrid Methods for Function Generation.

Lecture presented at the Third International Analogue Computation Meeting, Opatija, Yugoslavia, Sept. 1961.

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