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A special purpose analog computer for statistical system identification Fieguth, Werner 1965

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A SPECIAL PURPOSE ANALOG COMPUTER FOR STATISTICAL SYSTEM IDENTIFICATION by WERNER FIEGUTH B . S c * ( E . P . ) , U n i v e r s i t y of M a n i t o b a , 1963 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF APPLIED SCIENCE I n the Department of E l e c t r i c a l E n g i n e e r i n g We accept t h i s t h e s i s as conforming t o the standards r e q u i r e d from c a n d i d a t e s f o r the degree of M a s t e r of A p p l i e d S c i e n c e Members of the Department of E l e c t r i c a l E n g i n e e r i n g THE UNIVERSITY OF BRITISH COLUMBIA May 1965 In p r e s e n t i n g t h i s t h e s i s i n p a r t i a l f u l f i l m e n t of the requirements f o r an advanced degree at the U n i v e r s i t y of -B r i t i s h Columbia, I agree that the L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e f o r ref e r e n c e and study, I f u r t h e r agree that per-m i s s i o n f o r extensive copying of t h i s t h e s i s f o r s c h o l a r l y purposes may be granted by the Head of my Department or by h i s r e p r e s e n t a t i v e s . I t i s understood that, copying or p u b l i -c a t i o n of t h i s t h e s i s f o r f i n a n c i a l gain s h a l l not be allowed without my w r i t t e n permission* Department of £lECXRJCAL ENGINEERING The U n i v e r s i t y of B r i t i s h Columbia, Vancouver 8, Canada ABSTRACT An i t e r a t i v e method of system i d e n t i f i c a t i o n based on s o l v i n g the i n t e g r a l e q u a t i o n 0 y x ( r ) « J h(tr) 0 x x(t-(r) dc f o r h(cr') a t t e n e q u a l l y spaced p o i n t s (cr\, i = 1,2,..,10) i s d e s c r i b e d . R e p l a c i n g the above i n t e g r a l by a f i n i t e sum a t t e n d i f f e r e n t v a l u e s of t r e s u l t s i n a s e t of t e n e q u a t i o n s i n the t e n unknowns h ( t f \ ) . A s t a t i s t i c a l l y i d e n t i c a l and much more e a s i l y h a n d l e d s e t of e q u a t i o n s , o b t a i n e d by u s i n g one-sample p r o d u c t s i n p l a c e of the a c t u a l c o r r e l a t i o n f u n c t i o n s , i s th e n s o l v e d by a G a u s s - S e i d e l - l i k e i t e r a t i o n method, the convergence p r o p e r t i e s of which show t h i s approach to the i d e n t i f i c a t i o n problem t o be a u s e f u l one f o r a l a r g e c l a s s of system i n p u t s i g n a l s . A s i m p l e computer t o r e a l i z e the above i d e n t i f i c a t i o n method i s d e s c r i b e d i n some d e t a i l . The use of a simple q u a n t i z a t i o n form of c o r r e l a t i o n a l l o w s s h i f t r e g i s t e r s t o c a r r y out the r e q u i r e d d e l a y o p e r a t i o n s . Storage f o r the computer ,s e s t i m a t e s of the h(o^) i s i n the form of s t e p motor d r i v e n p o t e n t i o m e t e r s , which a l s o c a r r y out one of the m u l t i p l i c a t i o n o p e r a t i o n s . The v e r y e n c o u r a g i n g r e s u l t s of a number of r e l a t i v e l y r e a l i s t i c i d e n t i f i c a t i o n t e s t s u s i n g the computer are g i v e n . i i i TABLE OF CONTENTS Page A b s t r a c t i i L i s t of I l l u s t r a t i o n s i v Acknowledgement v i 1. I n t r o d u c t i o n 1 2. > Theory 4 2*1 G e n e r a l Background and Development of the I t e r a t i o n Procedure 4 2.2 Choice of the Method of C o r r e l a t i o n 8 2.3 The Convergence of G a u s s — S e i d e l - L i k e I t e r a t i o n s 11 3. The Computer 17 3.1 Ge n e r a l O u t l i n e of the Computation Scheme 17 3.2 B l o c k A of F i g . 3-3 21 3«3 B l o c k B of F i g . 3-3 23 3.4 B l o c k C i of F i g , 3-3 24 3«5 B l o c k D of F i g . 3-3 28 3«6 B l o c k E of F i g . 3-3 31 4. Test R e s u l t s 40 4.1 Test 1 40 4*2 Test 2 42 4.3 Test 3 44 4.4 D i s c u s s i o n of the Computer Test R e s u l t s 46 5. C o n c l u s i o n s 49 Ref e r e n c e s 51 Appendix I 52 i v LIST OF ILLUSTRATIONS F i g u r e Page 2-1 S u b d i v i s i o n of the I n t e r v a l [0»Tg"] 5 2- 2 Sampling P o i n t s of 0 (t)» h(cr)At, and 0_-.CC) 6 yx xx 3- 1 The Sampling Scheme f o r x ( t ) and y ( t ) 17 3-2 G e n e r a l Computer O u t l i n e 19 3-3 B l o c k Diagram of the Computer 20 3-4 G e n e r a t i o n of Sgn X Sgn y 21 3-5 R e s i s t i v e Network f o r W e i g h t i n g b^ 22 3—6 Step motor D r i v i n g L o g i c 25 3-7 Timing of the C o r r e c t i o n o f 26 3-8 Schematic f o r the P o t e n t i o m e t e r M u l t i p l i c a t i o n 27 3-9 Storage P o t e n t i o m e t e r B i a s i n g 28 3-10 G e n e r a t i o n of f, 30 k 3—11 B l o c k Diagram of the C o n t r o l C i r c u i t s 32 3—12 C i r c u i t Diagram f o r B l o c k A 33 3—13 C i r c u i t Diagram f o r B l o c k B 34 3-14 C i r c u i t Diagram f o r B l o c k C^ 35 3—15 C i r c u i t Diagram f o r B l o c k Dj P a r t 1 36 3—16 C i r c u i t Diagram f o r B l o c k D, P a r t 2 37 3-17 C i r c u i t Diagram f o r B l o c k E 38 3- 18 C o n t r o l S i g n a l Waveforms 39 4— 1 Computer I n p u t s f o r Te s t 1 40 4-2 Computer R e s u l t s f o r Te s t 1 42 4-3 A u t o - C o r r e l a t i o n of x ( t ) f o r Test 2 43 4»=4 Computer R e s u l t s f o r Test 2 43 F i g u r e Page 4 - 5 Comparison of 0 y x('C) and h ( t ) 4 4 4—6 Impulse Responses f o r Test 3 4 5 4 - 7 Computer R e s u l t s f o r Test 3 4 5 ACKNOWLEDGEMENT The author wishes t o express h i s thanks to h i s s u p e r v i s o r , Dr. E« V<» Bohn, 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 e n e r a l guidance i n the course of the r e s e a r c h d e s c r i b e d i n t h i s t h e s i s . Acknowledgements are due to the N a t i o n a l Research C o u n c i l f o r i t s awarding of a b u r s a r y i n 1963 and a s t u d e n t s h i p i n 1964, as w e l l as f o r i t s g e n e r a l support of the p r o j e c t under Grant Number BT-68. A SPECIAL PURPOSE ANALOG COMPUTER FOR STATISTICAL SYSTEM IDENTIFICATION 1 . INTRODUCTION Many p h y s i c a l systems are of such a c o m p l e x i t y t h a t i t i s not p o s s i b l e to determine system dynamics from a mathematical model.. B e f o r e the system can be c o n t r o l l e d i t i s then e s s e n t i a l to e s t i m a t e the dynamics from the o p e r a t i n g d a t a . I d e n t i f i c a t i o n i s an e s s e n t i a l , and p o s s i b l y the most d i f f i c u l t , s t e p i n any c o n t r o l scheme f o r a system w i t h unknown dynamics, the o t h e r s b e i n g the e v a l u a t i o n of the i d e n t i f i c a t i o n d a t a a c c o r d i n g to some p r e d e t e r m i n e d c r i t e r i o n , and the subsequent adjustment of c e r t a i n system parameters on the b a s i s of the outcome of the e v a l u a t i o n . I n o r d e r to account f o r p o s s i b l e parameter changes the method of i d e n t i f i c a t i o n must operate i n r e a l - t i m e , so t h a t the changes i n p l a n t dynamics are d e t e c t e d as soon as p o s s i b l e a f t e r they o c c u r . T h i s requirement a u t o m a t i c a l l y e l i m i n a t e s a number of methods proposed i n the p a s t , w h i c h depended on the e x t e r n a l p r o c e s s i n g of i n p u t and output d a t a . ^ ^ A method may u t i l i z e e i t h e r the normal i n p u t and output s i g n a l s of the system, or may r e l y on s p e c i a l t e s t s i g n a l s w hich modify the normal o p e r a t i n g i n p u t i n a p r e s c r i b e d manner. Where c o n t i n u o u s m o n i t o r i n g of the p l a n t i s d e s i r e d , as i t g e n e r a l l y w i l l be i n an a d a p t i v e system, t h e r e w i l l be no o p p o r t u n i t y to use t e s t s i g n a l s a l o n e . Numerous p r o p o s a l s of 2 i d e n t i f i c a t i o n methods i n the l i t e r a t u r e have made use of some form of t e s t s i g n a l added to the normal i n p u t . ^ 2 ^ ' ^ ^ These s i g n a l s are o f t e n chosen to be n o i s e , e i t h e r G a u s s i a n or G a u s s i a n - l i k e i n the sense t h a t t h e i r a u t o - c o r r e l a t i o n f u n c t i o n s approximate the d e l t a f u n c t i o n . T h i s y i e l d s a v e r y e a s i l y i n s t r u m e n t e d method, s i n c e the c r o s s - c o r r e l a t i o n between the i n p u t p l u s n o i s e and the output i s j u s t e qual to the impulse response o f the system, assuming the n o i s e to be u n c o r r e l a t e d w i t h any of the normal system s i g n a l s . D e t e r m i n i s t i c t e s t s i g n a l s such as p u l s e s , s t e p s , s i n u s o i d s , e t c . have a l s o been used, but i n a p p l i c a t i o n s o t h e r than c r o s s - c o r r e l a t i o n . These methods have the advantage of b e i n g independent of the natu r e or l e v e l of the normal i n p u t s i g n a l ; i n d e e d , they work b e s t i n the complete absence of such i n p u t s which become, to the i d e n t i f i c a t i o n scheme, a form of " n o i s e " i n the t e s t s i g n a l . The c h i e f d i s a d v a n t a g e of t e s t s i g n a l methods i s t h a t the normal output of the system Id d i s t u r b e d , an e f f e c t e s p e c i a l l y u n d e s i r a b l e , or even i n t o l e r a b l e , i f the output of the system i s b e i n g o p t i m i z e d i n some sense, as i t u s u a l l y i s . There i s a l a r g e c l a s s of systems i n which the normal i n p u t i s of s u f f i c i e n t magnitude and v a r i a b i l i t y t o a l l o w an i d e n t i f i c a t i o n method u s i n g the normal s i g n a l s alone to f u n c t i o n p r o p e r l y . However, no s u c c e s s f u l such method has y e t been proposed, except i n the form of e x p e n s i v e , l a r g e - s c a l e , u n i v e r s a l i d e n t i f i e r s , w hich c o u l d not e c o n o m i c a l l y be used i n ( 4 ) ( 5 ) most a p p l i c a t i o n s . P a r k e r ^ d i d attempt the c o n s t r u c t i o n of an e c o n o m i c a l , r e a l - t i m e , o n - l i n e a n a l o g computer t o u t i l i z e o n l y normal o p e r a t i n g s i g n a l s , but f a i l e d to a r r i v e a t a 3 workable scheme because of i n h e r e n t shortcomings i n h i s c a p a c i t o r memory and time a v e r a g i n g approach. The work d e s c r i b e d i n t h i s t h e s i s i s an e x t e n s i o n of P a r k e r ' s work. H i s d i f f i c u l t i e s are overcome by u s i n g a s t e p — m o t o r - p o t e n t i o m e t e r form of s t o r a g e i n c o n j u n c t i o n w i t h a d i f f e r e n t type of i t e r a t i o n , b r i e f l y o u t l i n e d by P a r k e r i n h i s c o n c l u s i o n s . T h i s method, l i k e P a r k e r ' s o r i g i n a l i t e r a t i o n s , i a an o f f - s h o o t of the G a u s s - S e i d e l i t e r a t i o n method f o r the s o l u t i o n of a s e t of l i n e a r , a l g e b r a i c e q u a t i o n s , f i r s t a p p l i e d ( 6 ) to the i d e n t i f i c a t i o n problem by Bohn. 2. THEORY 2»1 Ge n e r a l Background and Development of the Method of I t e r a t i o n The c o n v o l u t i o n i n t e g r a l f o r l i n e a r t i m e - i n v a r i a n t systems g i v e s y ( t ) = J o h(er) x(t-er) dC (2-1.) where y ( t ) = system output h(o') = system impulse response x ( t ) = system i n p u t . F o r s t a b l e systems, w i t h bounded i n p u t s , JT h(tr) x ( t - c r ) d<r-^0 as T 9 0 . (2-2) L e t T = T be such t h a t the i n t e g r a l i n Eq. (2-2) i s n e g l i g i b l e w i t h r e s p e c t t o y ( t ) , except i f y « 0. That i s , T i s the s e f f e c t i v e s e t t l i n g time of the system. Then PT y(t) «* J s h(cr) x(t-o-) d<r (2-3) and the t i m e - i n v a r i a n c e r e q u i r e m e n t on the system may be r e l a x e d to e f f e c t i v e i n v a r i a n c e over any p e r i o d T g. The i n t e g r a l i n Eq* (2—3) i s now approximated by the sum f 10 y ( t ) & h((m-i)AT) x(t-(m-£)AT) AT. (2-4) m=l The s u b d i v i s i o n of the i n t e r v a l [ 0 , T g J and the a p p r o x i m a t i o n of h(o') x(t-(T) over t h i s i n t e r v a l are shown i n F i g . 2-1. U s i n g t e n s u b i n t e r v a l s i s c o n v e n i e n t because a simple decade c o u n t e r 5 can t h e n be used i n the c o n t r o l c i r c u i t r y . A l s o , t e n p o i n t s w i l l i n g e n e r a l s u f f i c e t o s p e c i f y an impulse response w i t h r e a s o n a b l e a c c u r a c y . or F i g u r e 2-1. S u b d i v i s i o n of the I n t e r v a l [0»T g]. For Eq. (2-4) to h o l d w i t h any r e a s o n a b l e a c c u r a c y , x ( t ) s h o u l d v a r y s l o w l y over any i n t e r v a l AT, but the a v e r a g i n g e f f e c t i n -h e r e n t i n the i t e r a t i o n s d e s c r i b e d below removes t h i s s t r i n g e n t c o n d i t i o n and r e q u i r e s m e r e l y t h a t the a u t o - c o r r e l a t i o n f u n c t i o n of x ( t ) be a p p r o x i m a t e l y c o n s t a n t over i n t e r v a l s AT. F o r reasons to be g i v e n l a t e r i t i s n e c e s s a r y to c o n v e r t Eq. ( 2 - l ) i n t o a form c o n t a i n i n g the a u t o — c o r r e l a t i o n and c r o s s - c o r r e l a t i o n f u n c t i o n s . M u l t i p l y i n g Eq* (2-4) by x ( t + f ) y i e l d s 10 (2-5) x(t+T) y ( t ) = 3>*. h((m-^)AT) x(t-rT) x(t-(m - ^ ) A T ) A T , m=l o r , r e w r i t i n g , 10 b ( r ) = h((m-4 )A r ) a ( r - ( m - i ) A r ) A t , (2-6) m=l where b(T) i s a one-sample e s t i m a t e of the c r o s s - c o r r e l a t i o n 6 m f u n c t i o n 0 ( f ) , and a ( r ) i s a one-sample e s t i m a t e of the a u t o - c o r r e l a t i o n f u n c t i o n 0 ( f ) . I f T = ( k - ^ ) A T . Eq* (2-6) becomes 10 b ( ( k - i ) A T ) = h((m-i ) A T ) a((k-m ) AT )A r . (2-7) m=l D e f i n i n g b ( ( k - £ ) A T ) £ b f c h((m-i ) A T ) A r £ h r a((k-m ) A t ) £ a ^ reduces Eq. (2-7) t o 10 \ ~ 2 g h m a k - m (2-8) The sample p o i r i t s f o r the f u n c t i o n s 0 ( T ) , 0 (T) and yx xx h(rr)AT" are shown i n Fig« 2—2, where the numbers o p p o s i t e the sampling times r e f e r t o ffche s u b s c r i p t s of the q u a n t i t i e s b^., a-j^jj and h m sampled a t those p o i n t s . S i n c e t h e r e are t e n unknowns h m to be f o u n d , t e n e q u a t i o n s of the form of Eq. (2—8) are r e q u i r e d ; t h a t i s , k as 192^ »..,10. "/ w2 "3 •4 '5 '6 "7 '8 '9 */0 " far) | B . . t . » » . » i ' *t % % V *6 r *8 9 fo * ' -& -8 -7 -6 -S -4- -3 -2 - / - I 0 A T < o ' "2*3 % V 6 7 8 -9 ' I 0 A T F i g u r e 2-2. Sampling P o i n t s of 0 ( T ) , h((T)AT, and 0 X X(T)< L e t f be the computer's e s t i m a t e of h a t any g i v e n time m r m 7 where f = h + e m , (2-9) m m m e b e i n g the e r r o r i n the i d e n t i f i c a t i o n of h . S u b s t i t u t i n g m ° m ° f f o r h i n Eq. (2-8) w i l l l e a v e the e q u a t i o n u n s a t i s f i e d , m m ^ ' r e q u i r i n g an e r r o r term, zt t o b a l a n c e i t . That i s , 10 m ~m 1 0 1 0 - b k ~ S ^-m hm " S ak-m em (2-10) m=l m=l U s i n g Eq. (2-8) reduces Eq, (2-10) to the form 10 e k = - S ^ e . (2-11) m=l On the average, the l a r g e s t c o e f f i c i e n t of the e m i n the sum of Eq. ( 2 - l l ) w i l l be a Q» because i t i s a g e n e r a l r e s u l t t h a t f o r a l l a u t o - c o r r e l a t i o n f u n c t i o n s 0 (0) ^ 0 ( t ) . T h e r e f o r e XX XX the term a^e^ may be r e g a r d e d as a s o r t of dominant term i n the sum, so t h a t the e r r o r can be a t t r i b u t e d m a i n l y to e^, the i d e n t i f i c a t i o n e r r o r i n h^. I t i s p r e c i s e l y because the c o r r e l a t i o n f u n c t i o n p r o p e r t i e s g i v e r i s e t o t h i s c o n v e n i e n t dominant term t h a t these f u n c t i o n s were i n t r o d u c e d a t a l l , i n s t e a d of w o r k i n g d i r e c t l y w i t h the s i g n a l s x ( t ) and y ( t ) as i n Eq. ( 2 - 4 ) , i n w h i c h case no convergent i t e r a t i o n scheme c o u l d have been found. I n k e e p i n g w i t h the above assumption t h a t i s due to e^ i t i s p o s s i b l e t o c a l c u l a t e "improved" v a l u e s of f^. by u s i n g 8 a g r a d i e n t type method wh i c h makes the change Af^. i n f p r o p o r t i o n a l to e, s A f k (=Ae k) = ae k » (2-12) where a i s a p r o p o r t i o n a l i t y c o n s t a n t . I t w i l l be noted t h a t o n l y i f sgn = -sgn e^ does the c o r r e c t i o n d e f i n e d by Eq«, (2-12) serve to reduce \ e^\. S i n c e t h i s c o n d i t i o n cannot be guaranteed to h o l d , e s p e c i a l l y f o r low v a l u e s of | | , the i d e n t i f i e d parameters f w i l l always be " n o i s y " . I f the i n d e x k i n Eqo's ( 2 - l l ) and (2-12) i s made to change a c c o r d i n g to k = l,2 s,3y «». ,10,1,2, »•>;,». an e n d l e s s i t e r a t i o n i s d e f i n e d i n which each of the computer e s t i m a t e s f o r the t e n v a l u e s h m i s a d j u s t e d i n a r e p e t i t i v e sequence. Some r e s u l t s c o n c e r n i n g the convergence of the i t e r a t i v e scheme d e f i n e d above are d i s c u s s e d i n s e c t i o n 3 of t h i s c h a p t e r , making use of the known convergence p r o p e r t i e s of the Gauss— S e i d e l method of i t e r a t i o n , t o which the above i t e r a t i o n s bear a s t r o n g resemblance. T h i s i t e r a t i o n method e s s e n t i a l l y g i v e s an approximate s o l u t i o n of h ( t ) from the i n t e g r a l e q u a t i o n the impulse response found by s o l v i n g Eq. (2-13) i s t h a t of the b e s t l i n e a r model of the system i n the mean-square e r r o r sense. (2-13) w h i c h i s the e x a c t form of the time average of Eq. ( 2 - 6 ) . (7) Woodrow has shown t h a t i n the case of n o n - l i n e a r systems 9 2.2 Choice of the Method of C o r r e l a t i o n A number of d i f f e r e n t t ypes of c o r r e l a t i o n are a v a i l a b l e f o r use i n i n s t r u m e n t i n g the computation scheme o u t l i n e d i n the p r e v i o u s s e c t i o n , each p o s s e s s i n g v a r i o u s advantages and d i s a d -vantages i n r e g a r d to ease of i n s t r u m e n t a t i o n , a c c u r a c y , and speed of convergence. To determine the e f f e c t on the proposed i t e r a t i o n scheme of u s i n g d i f f e r e n t methods of c o r r e l a t i o n , a b r i e f d i g i t a l computer s i m u l a t i o n of the i d e n t i f i c a t i o n computer was c a r r i e d out» On the b a s i s of t h i s s i m u l a t i o n , the p o l a r i t y c o i n c i d e n c e (8) method used w i t h c o n s i d e r a b l e success by Turner i n the c o n s t r u c t i o n of a c o r r e l a t o r , w h i c h would have been the e a s i e s t method t o i n s t r u m e n t , had t o be r e j e c t e d because i t gave c o m p l e t e l y u n s a t i s f a c t o r y r e s u l t s . A l t h o u g h the c o r r e l a t i o n e s t i m a t e s i n t h e o r y average out t o the t r u e v a l u e , i t seems t h a t i n d i v i d u a l l y they are so e r r a t i c as to cause the i t e r a t i o n t o become u n s t a b l e . Three o t h e r methods^ l i s t e d i n o r d e r of m e r i t , gave s a t i s f a c t o r y r e s u l t s . They ares (a) The e x a c t form, u s i n g a n a l o g p r o d u c t s , = x ( t ) x(t+kAT) (2-14) (b) A simple form of S t i e l t j e s c o r r e l a t i o n , ^ ^ u s i n g a f o u r - l e v e l q u a n t i z a t i o n , a k = x ( t ) Sgn x(t+kAT) Mag x(t+kAT) (2-15)' (c) The r e l a y method of c o r r e l a t i o n , a^ . = x ( t ) Sgn x(t+kAT) , (2-16) 1 0 where Sgn x = / l i f x > 0 l - l i f x < 0 Mag x = | 1 i f |x| > V R [ 1 i f |x| < V R , V R being some suitable, f i x e d reference l e v e l . A l l three methods gave very similar r e s u l t s . Method (a) i s very d i f f i c u l t to re a l i z e and was used c h i e f l y as a control on the other methods, to a l l o v d i f f e r e n t i a t i o n between errors due to shortcomings inherent i n the computation scheme and errors due to correlation function approximations. (9) A study by Watts has shown that even a moderate increase i n the number of quantization levels i n a S t i e l t j e s correlation causes very considerable improvements i n the accuracy of the f i n a l correlation function. Since the computer simulation was limited to rather i d e a l i z e d cases, and since i t i s l i k e l y that for more r e a l i s t i c signals the advantages of four-level quantization over two-level quantization would become much more apparant, i t was decided to use method (b) i n the computer instrumentation. Using method (c) would have resulted i n only, minor reductions i n the c i r c u i t complexity. It i s now necessary* i n Eq. (2r-10), to account for the fact that the individual computer correlation estimates are only approximate. Substituting the computer estimates b^ and a ^ - m i n place of the exact analog products b^ and m y i e l d s 1 0 e k = e k + S k - b k - S <-m fm > < 2- 1 7) m«=I where 6, i s the error due to the approximations made i n the c o r r e l a t i o n f u n c t i o n e s t i m a t e s * Eq.'s (2-10) and (2-17) g i v e 10 *k = <b'k " V ~ S ( - t ^ , " a k _ m ) f m (2-18) m=l w h i c h , when time averaged* becomes 10 *k = <b'k - V " ^ ^k-m " <W fm • < 2~ 1 9> m=l I f f remains a p p r o x i m a t e l y c o n s t a n t over the a v e r a g i n g i n t e r v a l * Ta? = r l u ITfl Ta? ^"1 J f ~ , (2-20) v k-m k-nr m v^t-m k-nr m 1 s ' and i f the c o r r e l a t i o n e s t i m a t e s , a l t h o u g h i n d i v i d u a l l y o n l y a p p roximate, average out t o the c o r r e c t v a l u e , 6^ 0 . (2-21) T h e r e f o r e the net c o r r e c t i o n t o f ^ over an a v e r a g i n g i n t e r v a l of N i t e r a t i o n s , < A fk>net » N £ k « N a ( e ^ + ^ ) « N a e k , (2-22) remains u n a f f e c t e d by i n a c c u r a c i e s i n the i n d i v i d u a l c o r r e l a t i o n e s t i m a t e s . 2*3 The Convergence of G a u s s — S e i d e l - l i k e I t e r a t i o n s F o r the s e t o f s i m u l t a n e o u s l i n e a r a l g e b r a i c e q u a t i o n s ^ ak,m xm - ^ = 0 r k = X ' 2 ' - ' N ' ( 2 ~ 2 3 ) m=l * 12 a Gauss-Seidel-like i t e r a t i o n scheme for solving for the N unknowns i s defined as follows: i complete it e r a t i o n s y i e l d x.^^, j = 1,2, ,N t 3 "fch the i estimates of a l l the unknowns. A further k-1 steps of the ( i + l ) * * 1 i t e r a t i o n r e s u l t s i n x.^ +^|- j = 1,2, ,k—1» J The k t h step of the ( i + l ) t h i t e r a t i o n uses the lat e s t th available estimates of the unknowns i n the k of the N equations to give the error term k J k ^ -} k,m m 1--(i) v. i l 5* ^ " ^ ak,m Xm N (i) m=k+l which i s used to form x, ^ + " ^ according to (2-24) x k ~ x k + c k e k (2-25) or. (i+1) _ „ (i+1) 'k k A x k = c ^ e ^ (2-26) where c^ i s a constant* Under suitable conditions x, as i -In matrix notation Eq» (2-23) becomes AX - T = 0 . (2-27) Defining A = L + D + U (2-28) 13 a l l o w s Eq.'s ( 2 - 2 l ) and (2-23) t o be r e w r i t t e n as E ( i + 1 ) = I - L X U + 1 ) - D X ( i ) - U X ( i ) (2-29) and A X ( 1 + 1 ) = C E ( i + 1 ) . (2-30) Case (a) ; For the s t a n d a r d G a u s s — S e i d e l i t e r a t i o n C = D"1 . (2-31) Then Eq.'s (2-26) and (2-27) g i v e D A X ( i + l ) = T — L X ( i + 1 ) - D X ( i ) - U X ( i ) , (2-32) or 0 = I - L X ( i + 1 ) - D X ( i + 1 ) - U X ( i ) . (2-33) That i s , the step of any i t e r a t i o n c o r r e c t s x k ^ ^ s o a s t h e l i m i n a t e e n t i r e l y the e r r o r i n the k e q u a t i o n . Eq. (2-33) can be w r i t t e n as 0 = T - (L + D) X ^ i + 1 ) - U X ( i ) . (2-34) R e i c h ^ ^ has shown a n e c e s s a r y and s u f f i c i e n t c o n d i t i o n f o r the convergence of the i t e r a t i o n s t o be the p o s i t i v e d e f i n i t e n e s s of the m a t r i x (L + D) + U = A* Case (b) % For a s l i g h t l y more g e n e r a l i t e r a t i o n l e t C = p D"1 , (2-35) 14 where p i s any real constant. Then Eq, (2-29) gives E ( i + 1 ) - D A X ( i + l ) = T - L X ( i + l ) . - D X ( i ) - U X ( i ) - D A X ( i + l ) (2-36) which, using Eq.'s (2-30) and (2—35), becomes (1-p) E ( i + 1 ) = Y — L X ( i + 1 ) - D X ( i + 1 ) - U X ( i ) . (2-37) Using Eq. (2-29) to substitute for E ^ 1 + 1 ^ i n Eq. (2-37) gives (l-p)( Y - L X ( i + 1 ) - D X ( i ) - U X ( i ) ) = T - L X ( i + 1 ) - D X ( i + 1 ) - U X ( i ) , (2-38) or> c o l l e c t i n g terms, 0 = p I - (pL + D) X ( i + 1 ) - (pU - (1-p) D ) X ( i ) . (2-39) Comparison with Eq. (2-3l) shows that the condition for conver-gence i n th i s more general type of i t e r a t i o n becomes the positive definiteness of (pL + D) + (pU - (l-p)D) = pA. That i s , the condition i s unchanged* The i t e r a t i o n used by the computer of this thesis can be shown to approach as closely as desired, i n an average sensej the more general type of Gauss—Seidel i t e r a t i o n discussed above. The set of equations for the computer, * k « b k - i f c V-m fm m=i A f k a a e k f k -1,2, i&i ,10 , (2-40) corresponds to Eq.'s (2-24) and (2-27), since the f^ are always the l a t e s t available estimates of h^ As they stand, Eq.'s (2-40) have time varying co e f f i c i e n t s and do not constitute a simple s e t of sim u l t a n e o u s e q u a t i o n s . However, l e t them be averaged over N i t e r a t i o n s . Then 10 t j* -k-m "m I f a i s s u f f i c i e n t l y s m a l l t o re n d e r a l l f a p p r o x i m a t e l y c o n s t a n t over the N i t e r a t i o n s * IP. i s ¥ith obvious changes i n n o t a t i o n , 10 \ = B k - S F m . (2-43) m— The n et change i n f ^ d u r i n g the N i t e r a t i o n s i s = ^  « = N a ( A f k ) n e t S a e f c ^ E k , (2-44) or, A F k = N a E f c (2-45) F o r r e a s o n a b l y l a r g e N and time i n v a r i a n t s t a t i s t i c a l p r o p e r t i e s of the s i g n a l s x ( t ) and y ( t ) the v a l u e s of and j[, w i l l be v e r y n e a r l y c o n s t a n t . Thus, w i t h s u f f i c i e n t l y x s m a l l a, the computer's i t e r a t i v e scheme reduces t o the s o l u t i o n o f a s e t of c o n s t a n t — c o e f f i c i e n t s i m u l t a n e o u s e q u a t i o n s by a G a u s s - S e i d e l - l i k e i t e r a t i o n * a l l o w i n g use t o be made of the convergence c o n d i t i o n s above. These c o n d i t i o n s may or may not be v a l i d f o r l a r g e v a l u e s of ou Th e r e f o r e a n e c e s s a r y ( and s u f f i c i e n t , i f a i s s m a l l ) c o n d i t i o n f o r convergence of the 16 computer's i t e r a t i o n s i s the p o s i t i v e d e f i n i t e n e s s of the m a t r i x 0 _ ( O ) 0 (-A*) 0 (-2ATT) xx 0 (-10AT) ^ A t > > ^ x x ^ 0 ) 0 x x ( l O A t ) » • • • * 0 (0) xx 7 3. THE COMPUTER 3.1 Ge n e r a l O u t l i n e of the Computation Scheme I t i s d e s i r e d t o compute 10 k k *^ J k-m m m=l (3-1) and t h e n to c o r r e c t f k a c c o r d i n g to A f k = a e k , k = l , 2 , ... ,9,10,1,2, ... (3-2) C o n s i d e r the sampling scheme i n F i g . 3-1. |l S | 6 , 7 , ^ k %>w, k-4, t-o F i g u r e 3-1. The Sampling Scheme f o r x ( t ) and y ( t ) . 18 The f u n c t i o n y ( t ) i s sampled ^Af l a t e r t h a n x ( t ) i n o r d e r to g i v e the r e q u i r e d s h i f t to the sample p o i n t s of the c r o s s - c o r r e l a t i o n c u r v e * (See P i g . 2-2 ). The q u a n t i t y X i s the a c t u a l a n a l o g v a l u e of x ( t ) j sampled every lOA't and h e l d f o r a p e r i o d l O A f , w h i l e the x^ , i = 1,2, .» ,10, are d i g i t a l a p p r o x i m a t i o n s of x ( t ) 9 Sgn x ( t ) Mag x ( t ) , w h i c h can be s t o r e d i n two s h i f t r e g i s t e r s . , S i m i l a r l y y i s the d i g i t a l a p p r o x i m a t i o n Sgn y ( t ) Mag y ( t ) of y ( t ) a t the l a t e s t s a m p l i n g p o i n t f o r t h a t f u n c t i o n . Forming the p r o d u c t s of X and each of the s h i f t r e g i s t e r elements i n t u r n y i e l d s which are the c o e f f i c i e n t s r e q u i r e d t o form the sum i n Eq. ( 3 — l ) . The p r o d u c t of X and the l a t e s t sample of y ( t ) g i v e s w h i c h , t o g e t h e r w i t h the sum r e s u l t s i n . F i g . 3-2 shows i n s i m p l i f i e d schematic form a method f o r r e a l i z i n g the sampling scheme of F i g . 3-1 and the subsequent c a l c u l a t i o n s f o r f o r m i n g A f ^ , and i s a v e r y rough o u t l i n e of the a c t u a l computer. A more r e a l i s t i c b l o c k diagram of the computer, g i v i n g the f l o w of i n f o r m a t i o n among the v a r i o u s main l o g i c b l o c k s , appears i n F i g * 3-3. The b l o c k s , whose d e t a i l e d c i r c u i t diagrams are shown i n F i g . ' s 3-12 to 3-17, are i n d i v i d u a l l y d i s c u s s e d i n the f o l l o w i n g s e c t i o n s of t h i s c h a p t e r . X x (3-3) X y « x(-kATT) Sgn y(-l£X) Mag y ( - * A t ) = b k , (3-4) 19 The n o t a t i o n used i s as f o l l o w s : Fo r G= x,y,X, I f 0>Cy Sgn e 3^ 1 ( -14 ) Sgn e = -1 ( o • • ) I f e<o* Sgn e = -1 ( o V. ) Sgn G 1 ( -14 V. ) F o r 0= x,y, I f |e|>vR , Mag 9 = 1 ( -14 V. ) Mag 6 = 1 2 ( o V. ) I f i e l < v B , Mag 0 = 1 2 ( o V, ) Mag 6 1 ( -14 V, ). Vp i s some p o s i t i v e r e f e r e n c e l e v e l . F i g u r e 3-2. G e n e r a l Computer O u t l i n e . A +1X1 •IXI SqnX B v7 c CO CD <-0 V v X Ho Cr-> E fc-counter-I I I ' I I 1= l+l 1 X £ m to -H i x r cn cn a v v 3^ I I I I I I c en to c to t i l t CD 5 i l l c Co aK-L f[ D F i g u r e 3-3. B l o c k Diagram of the Computer. 21 3.2 B l o c k A of F i g . 3-3 The c i r c u i t diagram f o r b l o c k A i s shown i n F i g . 3-12* T h i s b l o c k uses the system output y ( t ) , sampled at the a p p r o p r i a t e t i m e , t o g e t h e r w i t h the s i g n a l X from b l o c k B to c a l c u l a t e b^. , b k = X Sgn y Mag y = |x|Sgn X Sgn y Mag y . (3-5) T r a n s i s t o r s T-^  and T^ of F i g . 3-12 form a h i g h g a i n d i f f e r e n t i a l a m p l i f i e r , the output of w h i c h , Sgn y ( t ) , s w i t c h e s between 0 v. and -14 v . as the i n p u t y ( t ) moves from - o 0 5 v. to •05 v . . Sgn y ( t ) and Sgn y ( t ) , which has been o b t a i n e d from the i n v e r t e r T^, form the i n p u t s to the b i n a r y s t o r a g e c i r c u i t c o n t a i n i n g T^ and T,-, w h i c h samples and s t o r e s the i n p u t s on command from the c o n t r o l p u l s e V^ .. The p r o d u c t Sgn X Sgn y i s o b t a i n e d , as shown schema-t i c a l l y i n F i g . 3—4, by u s i n g two r e s i s t i v e "and" gates f o l l o w e d by an a c t i v e " o r " g a t e , T, . F i g u r e 3-4* G e n e r a t i o n of Sgn X Sgn y . The " o r " gate output determines whether T^ or Tg w i l l c onduct, t h a t i s whether + j x | or - | x | w i l l be s w i t c h e d i n t o the r e s i s t i v e network of F i g * 3-5» The r e a s o n f o r u s i n g - |x| 22 i n s t e a d of -14 v. f o r the " o r " gate s u p p l y v o l t a g e i s to make the gate output c o m p a t i b l e w i t h the b i a s i n g of t r a n s i s t o r s Ty and T Q which i s d e r i v e d from X k R, - V W W \ A -R2 vwww-33 K «3 V W W A -F i g u r e 3-5. R e s i s t i v e Network f o r W e i g h t i n g b^» I n F i g , 3-5, R^> B '.and R^ are chosen so t h a t tjt^- ( s w i t c h c l o s e d ) V. 2V, — ( s w i t c h open). (3—6) The s w i t c h , t r a n s i s t o r T ^ i n F i g , 3-12, i s o p e r a t e d by Mag y and c a r r i e s out the w e i g h t i n g o p e r a t i o n r e q u i r e d f o r the c o r r e l a t i o n method b e i n g used* Mag y i s computed and s t o r e d by t r a n s i s t o r s T^ t o T ^ • T^Q, because of the e q u a l r e s i s t a n c e s i n i t s e m i t t e r and c o l l e c t o r b r a n c h e s , has an output - y ( t ) as l o n g as the i n p u t y ( t ) i s n e g a t i v e . The e m i t t e r f o l l o w e r T.^ f o l l o w s the l a r g e r of the two i n p u t s y ( t ) and — y ( t ) g i v i n g | y ( t ) | which i s compared w i t h the r e f e r e n c e l e v e l V R by the h i g h g a i n d i f f e r e n t i a l a m p l i -f i e r ( T j ^ a n < l T ^ ) whose o u t p u t , Mag y ( t ) , i s 0 v. or -14 v» depending on the r e l a t i v e magnitudes of | y ( t ) | and V R . Mag y ( t ) and Mag y ( t ) , from the i n v e r t e r T^^, are sampled by the same p u l s e Yj t h a t sampled Sgn y ( t ) and s t o r e d i n the b i n a r y made up 23 of T 1 5 and T l 6 . The purpose of the p o t e n t i o m e t e r P.^  i n F i g . 3-5 i s to a l l o w c o n t r o l of the maximum a m p l i t u d e , i n terms of m e c h a n i c a l r o t a t i o n of the s t o r a g e p o t e n t i o m e t e r s , of the f i n a l i d e n t i f i e d h m v a l u e s , i n o r d e r to a v o i d d r i v i n g the p o t e n t i o m e t e r s p a s t t h e i r m e c h a n i c a l s t o p s . C l e a r l y , i f 1?1 i s s e t c l o s e to ground l e v e l , t>k i s always s m a l l , and Eq. ( 2 - 8 ) , 10 1 " u b, = ^> ' a, h k ^ i ' K—m ii r ^-^ m m=l w i l l be s a t i s f i e d f o r s m a l l v a l u e s of h^ 3«3 B l o c k B of F i g . 3-3 The c i r c u i t diagram f o r b l o c k B i s shown i n F i g . 3—13« T h i s b l o c k uses the system i n p u t x ( t ) t o compute x ( t ) | , , Sgn x ( t ) and Mag x ( t ) . The f i r s t two of these are sampled every 10AT and t h e n s t o r e d t o g i v e |x| and Sgn X. The l a t t e r , two,- and t h e i r l o g i c a l complements, form the i n p u t s to the s h i f t r e g i s t e r s i n the b l o c k s C^ $ where they are sampled and s h i f t e d a l o n g one p o s i t i o n every AT « T r a n s i s t o r s T^- t^, T l g , T i g , T 2 Q i n F i g . 3-13 perform the same o p e r a t i o n on x ( t ) t h a t T 9 to T 1 4 d i d on y ( t ) i n F i g . 3-12, namely c a l c u l a t i n g |x.("t")| and t h e n Mag x ( t ) and Mag x ( t ) , which form the Mag x s h i f t r e g i s t e r i n p u t s . S i m i l a r l y T.^ "fc^  T-^ y p e r f o r m the same f u n c t i o n as T^ to T,. i n F i g . 3-12, computing the Sgn x s h i f t r e g i s t e r i n p u t s Sgn x ( t ) and Sgn x ( t ) j and sampling them w i t h the p u l s e to r e s u l t i n the s t o r a g e of Sgn X i n the b i n a r y s t o r a g e c i r c u i t T ^ , T^^„. 24 The sampling of | x ( t ) | i s done by a r e e d r e l a y and r e q u i r e s about one m i l l i s e c o n d . Two d r i v i n g s t a g e s , T^ and T^, were found n e c e s s a r y to ensure t h a t b o t h c h a r g i n g and d i s c h a r g i n g of the 2(xf« s t o r a g e c a p a c i t o r , depending on whether the new |x| i s s m a l l e r or l a r g e r t h a n the o l d one, would take p l a c e s u f f i c i e n t l y q u i c k l y . The s t o r a g e c a p a c i t o r feeds a two stage d.c. a m p l i f i e r , T^ and T^, whose g a i n i s arranged such t h a t i t s o utput i s a p p r o x i m a t e l y t w i c e the c a p a c i t o r v o l t a g e , a l l o w i n g t h i s o u tput t o be f e d back p o s i t i v e l y to the s t o r a g e c a p a c i t o r through an a d j u s t a b l e , l a r g e r e s i s t a n c e to c a n c e l the c u r r e n t drawn by the 5.6 megohm a m p l i f i e r i n p u t r e s i s t o r . T h i s g i v e s r e l a t i v e l y l o n g s t o r a g e t i m e s (about 10 seconds) w i t h no a p p r e c i a b l e sag i n |x| . T r a n s i s t o r T^ s e r v e s to s t a b i l i z e the a m p l i f i e r a g a i n s t temperature i n d u c e d d r i f t s i n i t s o u t p u t . An i n c r e a s e i n temperature w i l l lower the b i a s v o l t a g e Vg and thus compensate f o r the i n c r e a s e d a m p l i f i e r g a i n 0 The double e m i t t e r f o l l o w e r s , T^, T^Q and T ^ , T were found n e c e s s a r y because +|x| and - | x | f e e d a i l e f f e c t i v e l o a d of o n l y 500 ohms. A s i n g l e e m i t t e r f o l l o w e r would cause l o a d i n g e f f e c t s i n the e a r l i e r , r e l a t i v e l y high-impedance c i r c u i t s . 3.4 B l o c k C^ ^ of F i g . 3-3 The c i r c u i t diagram f o r b l o c k i s shown i n F i g . 3-14. T h i s b l o c k performs two f u n c t i o n s ? (a) I t uses Sgn and the output of the i+1 p i n of the k - c o u n t e r to d e c i d e whether a c o r r e c t i o n to f. i s to be I 25 made, and i f so, i n wh i c h d i r e c t i o n i t i s to be. (b) I t uses X, x^ and f ^ to form the c o n t r i b u t i o n a ^ ^ - f ? towards the sum ^> a, m f . m=l F u n c t i o n (a) i T h i s i s c a r r i e d out by two g a t e s , T-^  and i n F i g , 3-14, which perf o r m the l o g i c o p e r a t i o n i l l u s t r a t e d i n F i g , 3-6. Mot of Driving Circuit F i g u r e 3—6* Motor D r i v i n g L o g i c P i n i + l of the k — c o u n t e r of b l o c k E i s used i n s t e a d of p i n i to gate the c o r r e c t i o n of f ^ i n o r d e r to a l l o w most of the computer to s t a r t c a l c u l a t i n g e. , w h i l e the r e s t c a r r i e s 1 T-L out the c o r r e c t i o n u s i n g the s t o r e d v a l u e s of |e^| and Sgn • T h i s arrangement, shown i n F i g * 3—7, p r e v e n t s the c o r r e c t i o n p r o c e s s from i n t e r f e r i n g w i t h the next sampling i f AT i s s m a l l * The motor d r i v i n g c i r c u i t i n F i g . 3-6 i s a commercial u n i t , bought w i t h the ste p motors, which a c c e p t s 26 Correct f, Correct f 2 Correct f 3 -« *- •« -< ~V V V Sample )£,| Sample l£2| Sample |£3I and Sgn £, and Sgn £ z and Sgn S-3 F i g u r e 3-7, Timing of the C o r r e c t i o n of f ^ • p o s i t i v e g o i n g p u l s e s of a t l e a s t 5 v o l t s and l e s s t h a n 2.5 microseconds r i s e time a t e i t h e r of two i n p u t s , a p u l s e a t one i n p u t c a u s i n g a 15° c l o c k w i s e s t e p i n the motor, a p u l s e a t the o t h e r i n p u t c a u s i n g an e q u a l s t e p i n the o p p o s i t e d i r e c t i o n . The purpose of the .02 [ i f . c a p a c i t o r s i n the i n p u t c i r c u i t r y to the gates T^ and T^ i s to p r e v e n t s p u r i o u s s t e p s of the motor i n response t o s t e p changes i n the Sgn or k — c o u n t e r i n p u t s to the g a t e s , by c a u s i n g the gate o u t p u t s i n response to such steps t o have r i s e times c o n s i d e r a b l y i n excess of 2,5 m i c r o s e c o n d s . F u n c t i o n (b) s The p r o d u c t a^.^ f± = |x| (Sgn X ) ( S g n x ± ) ( M a g x ± ) f ± i s o b t a i n e d by s i m p l y a p p l y i n g +|x| or -|x| to the t e r m i n a l s of the p o t e n t i o m e t e r s t o r i n g f ^ , the s i g n depending on the s i g n o f Sgn X Sgn x^, as shown s c h e m a t i c a l l y i n F i g . 3-8, and then w e i g h t i n g the p o t e n t i o m e t e r output w i t h Mag x^. T h i s w e i g h t i n g o p e r a t i o n i s c a r r i e d out by a r e s i s t i v e network i d e n t i c a l to the one shown i n F i g . 3-5, except t h a t the p o t e n t i o m e t e r P^ i s 27 t h e l i m i n a t e d , Sgn x^ and Mag x^ are o b t a i n e d from the i elements of t h e i r r e s p e c t i v e s h i f t r e g i s t e r s , T^tTj and T^,T^ . I n F i g . 3-8 e q u a l v o l t a g e s of o p p o s i t e p o l a r i t y are needed on the s t o r a g e p o t e n t i o m e t e r t e r m i n a l s because f ^ may be p o s i t i v e o r n e g a t i v e , n e c e s s i t a t i n g a f o u r - q u a d r a n t m u l t i p l i c a t i o n . I n the a c t u a l c i r c u i t i n F i g , 3-14 o n l y a p o r t i o n , about t h r e e q u a r t e r s , of the v o l t a g e |X| i s a p p l i e d to the p o t e n t i o m e t e r i t s e l f , the remainder of the v o l t a g e drop b e i n g r e s e r v e d f o r b i a s i n g . t h e t r a n s i s t o r gates T-^Q, T ^ , T.^ A N (^ • And) (And Gate G-ate T;3 '-HXI 1 M a g y.{ Weigkt 1 .p. -IXI F i g u r e 3-8. Schematic f o r the P o t e n t i o m e t e r M u l t i p l i c a t i o n . 28 S i n c e i n almost a l l p r a c t i c a l cases the most n e g a t i v e v a l u e of i s f a r s m a l l e r i n magnitude t h a n the most p o s i t i v e v a l u e , b e t t e r use can be made of the a v a i l a b l e t r a v e l d i s t a n c e s i n the p o t e n t i o m e t e r s by b i a s i n g the zero p o i n t f o r f ^ away from the mec h a n i c a l p o t e n t i o m e t e r c e n t r e by means of a f i x e d s e r i e s r e s i s t o r . The e n t i r e r e s i s t i v e network a c r o s s w h i c h +|X| and - X| are a c t u a l l y a p p l i e d i s shown i n P i g . 3-9 . ? 2.2 K I M / W -Gate 6,1 K--V 3.9 K—*- 2.2K 2.2K VWVVAfVWyVAAAAAAAA W s / V V W V W I A A A M Oir. Zero G-ate Bias B i a s W P + 'XI If SjnXSgnxt-l K ? J V=[ -|X| if SgnX5ghXL=-l F i g u r e 3-9. Storage P o t e n t i o m e t e r B i a s i n g . 3.5 B l o c k D of F i g . 3-3 The c i r c u i t diagrams f o r b l o c k D are shown i n F i g , 1 s 3-15 and 3-16. T h i s b l o c k t a k e s the ou t p u t s from b l o c k s B and C., j = 1 , 2 , ,10, forms and s t o r e s e, <? h. - *^ >* a- k_ m f m , and then performs the a p p r o p r i a t e c o r r e c t i o n on f ^ . F i g . 3-15 c o n t a i n s the p o r t i o n of b l o c k D connected w i t h the computation and s t o r a g e of . The summing c i r c u i t i s a one stage d.c. a m p l i f i e r ( T 2 — temperature s t a b i l i z e d by t r a n s i s t o r T^) whose normal o u t p u t of 6 v. f o r no i n p u t i s b i a s e d t o 0 v . by a l a r g e n e g a t i v e v o l t a g e b e f o r e b e i n g i s o l a t e d by the e m i t t e r f o l l o w e r T^ from T^ and T,_, which f i n d 29 the a b s o l u t e v a l u e of e^ .* As i n the case of | x ( t ) | , two stages and T^, are needed t o f e e d the Ijxf« s t o r a g e c a p a c i t o r through the r e e d r e l a y sampler* The s t o r a g e c a p a c i t o r i s f o l l o w e d by a d.c. a m p l i f i e r e s s e n t i a l l y i d e n t i c a l t o the one f o l l o w i n g the 2u.f. | x | s t o r a g e c a p a c i t o r i n F i g . 3-13. The c i r c u i t r y f o r c a l c u l a t i n g and s t o r i n g Sgn.e^j T ^ to T-^, i s i d e n t i c a l t o t h a t used f o r the same purpose f o r x ( t ) and y(t)« The two e m i t t e r f o l l o w e r s T^^ and T ^ p r e v e n t l o a d i n g of the st o r a g e c i r c u i t by the low impedance seen by Sgn . F i g . 3-16 c o n t a i n s the c i r c u i t r y t h a t c a r r i e s out the c o r r e c t i o n A f ^ = tte^ • T h i s c o r r e c t i o n cannot be r e a l i z e d e x a c t l y because the s t e p motors which p o s i t i o n the s t o r a g e p o t e n t i o m e t e r s are d i g i t a l d e v i c e s . I n s t e a d , A f ^ i s q u a n t i z e d . I n the computer a decade c o u n t e r i s used t o generate an a p p r o p r i a t e l y s c a l e d s t a i r c a s e f u n c t i o n , one s t e p per motor s t e p , to r e p r e s e n t the c o r r e c t i o n Af^. • As soon as the s t a i r c a s e exceeds i n magnitude, the motor i n p u t s i g n a l s , a t r a i n of p u l s e s from a f r e e r u n n i n g m u l t i v i b r a t o r , are b l o c k e d . See F i g * 3-10. There are two o t h e r c o n t r o l s on the motor i n p u t s i g n a l s . One a s s u r e s t h a t the p u l s e t r a i n i s b l o c k e d d u r i n g the t r a n s i e n t s t a t e w h i l e a new i s b e i n g sampled and s t o r e d . The o t h e r b l o c k s the s i g n a l s i n case exceeds t h e t e n t h l e v e l of the s t a i r c a s e f u n c t i o n ^ as soon as the decade co u n t e r reaches a count of t e n , the c o r r e c t i o n i s stopped. Both of these c o n t r o l s can be c a r r i e d out by one f l i p - f l o p , as i n F i g * 3-10. The monostable c i r c u i t i n the p u l s e t r a i n p a t h 30 s e r v e s merely as a p u l s e shaper. Astabk Muin Gate. Block Gate If F.F. Is in Reset State Block Gate if |VI > |£ kl V <,V = Af k Ftn*lO of Counter to motors F i g u r e 3—10. G e n e r a t i o n of A f ^ • The v a l u e of t e n as the maximum p o s s i b l e number of ste p s i n Af^., a s i d e from i t s convenience i n g e n e r a t i n g the s t a i r c a s e f u n c t i o n , i s a l s o a compromise between two c o n f l i c t i n g r e q u i r e m e n t s - the speed of c o r r e c t i o n , l i m i t e d by the motor's t o p s t e p p i n g speed, and the a c c u r a c y of A f ^ , l i m i t e d by the number of st e p s i n the c o r r e c t i o n . I t i s f e l t t h a t , s i n c e A f ^ i s o n l y a c o r r e c t i o n term anyway* r a t h e r a r b i t r a r i l y chosen and depending on the a v e r a g i n g e f f e c t of numerous i t e r a t i o n s , a q u a n t i z a t i o n i n t o t e n l e v i e s s h o u l d s u f f i c e . The motors are capable of s t e p p i n g about 500 times p e r second w i t h the p a r t i c u l a r m e c h a n i c a l l o a d b e i n g used. S i n c e 31 one e n t i r e c o r r e c t i o n must be c a r r i e d out i n a time AT or l e s s , a t e n s t e p c o r r e c t i o n would r e s t r i c t AT t o v a l u e s g r e a t e r t h a n 20 m i l l i s e c o n d s . T h i s g i v e s a lower l i m i t of .2 seconds f o r the s e t t l i n g time of a system t h a t may be i d e n t i f i e d w i t h t h i s computer. The upper l i m i t i s p r e s e n t l y d i c t a t e d by the c a p a c i t o r s t o r a g e c i r c u i t f o r | x | , but c o u l d e a s i l y be made a r b i t r a r i l y l a r g e by r e p l a c i n g the c a p a c i t o r system w i t h a simple a n a l o g -d i g i t a l s t o r a g e d e v i c e . The c a p a c i t o r s t o r a g e f o r |e k| p r e s e n t s no problem, s i n c e je^j i s not r e q u i r e d t o be known a c c u r a t e l y f o r l o n g e r t h a n 20 m i l l i s e c o n d s * 3«6 B l o c k E of F i g . 3-3 The c i r c u i t diagram f o r b l o c k E i s shown i n F i g . 3-17« T h i s b l o c k c o n t a i n s a l l ' the c o n t r o l and t i m i n g c i r c u i t r y of the computer. B e s i d e s p r o p e r l y c h a n n e l i n g the c o r r e c t i o n A f ^ w i t h the t e n k - c o u n t e r o u t p u t s , i t generates se'ven c o n t r o l v o l t a g e s , V., j = 1,2, ,7, whose waveshapes, a l o n g w i t h a statement of J t h e i r p urpose, are shown i n F i g . 3-18. The c i r c u i t r y i t s e l f i s v e r y simple and s h o u l d r e q u i r e no e x p l a n a t i o n , but may be e a s i e r t o f o l l o w w i t h the h e l p of the b l o c k diagram i n F i g . 3—11* 32 V a Set AT M 3 V4 6 ])= Differentiator M " Monostable k-counter '"10 Relay *L (Sample |£k|) to to to _ - »' • • Relay^l (5ampie |xfl)|) F i g u r e 3-11. B l o c k Diagram of the C o n t r o l C i r c u i t s * 33 (frow B) ~T$ ~TJ0 T2. \ Figure 3-12* C i r c u i t Diagram for Block A. 34 rxCt) %1 \0K$ 10 K$ g>< -14-v. Relay # / 2N404 - i x i + 14 v. Xto C,) (to A and G j ) Sgnx(t) Sgnx(t) SgnX SgnX (to Q Magx(t) Magx(t) Figure 3-13« C i r c u i t Diagram of Block B« 35 K-Counter (from E>) F i g u r e 3-14. C i r c u i t Diagram of Block C. „ F i g u r e 3-15. C i r c u i t Diagram of B l o c k D, P a r t 1 Correction. Pulses (to CO F i g u r e 3-l6« C i r c u i t Diagram of B l o c k D, P a r t 2 • F i g u r e 3-17. C i r c u i t Diagram of Block E 39 -1.5 msec. Store JXI (next pulse in lOAe .sec.) V9 S t o r e S g n X (next pulse in lOAr sec .) Sjn £ ft" Pulse for S g n x and M a g x S h i f t R e g i s t e r s Store "*-/• msec. " Count k, Set Control F.F. v 7 f S a m p l e and Store I" ^ n  r I Mac y a g y F i g u r e 3-18• C o n t r o l S i g n a l Waveforms . 4* TEST RESULTS A s e r i e s of t e s t s > d e s c r i b e d below, were performed on the computer to e v a l u a t e i t s performance. 4.1 T e s t 1 T h i s i s a h i g h l y a r t i f i c i a l example, the c h i e f u s e f u l n e s s of which l i e s i n the f a c t t h a t f o r the g i v e n i n p u t s i g n a l s a l l i n t e r n a l computer waveforms, a,s w e l l as the d e s i r e d f i n a l answer, are known. Thus t h i s t e s t , o r something s i m i l a r , c o u l d be of c o n t i n u i n g use as a t e s t program t o check out the computer i n g e n e r a l o r to t r a c k down the source of a c i r c u i t m a l f u n c t i o n . The computer i n p u t f u n c t i o n s are shown i n F i g . 4—1. xtt), A, F i g u r e 4-1. Computer I n p u t s f o r Test 1. 41 The period of the inputs i s synchronized with the sampling period of the computer,, If the signal magnitudes are chosen such that Mag A^ = 1 and Mag A 2 — 2 ? then by inspection, b l ~ A l M a g B = b b k = - A X Mag B = -b (k £ l ) a Q = A-^  Mag A-^  = 2a ak-m ~ A l M a g A2 = a (k ^  m) . (4-1) Therefore the computer solves the following set of equations for h. 1 1 b = 2a h^ + a h 2 + a h^ + .... + a h^^ -b = a h^ + 2a h 2 + a h^ + •••• + a h^Q -b = a h^ + a h 2 + a h^ + .... + 2a h^Q (4—2) The true solutions to the set (4—2) are n l 11 a h m 3 l H <4-3> or, h m = -.158 h± (m £ l ) . (4-4) Only this l a t t e r r a t i o i s meaningful i n checking the computer solution,.because the actual magnitude of the solution i s continuously adjustable to any desired l e v e l . The computer solutions are shown i n F i g . 4=2, where h^ has been normalized to 1,0 i n a l l cases. It can be seen that, under admittedly ideal conditions, the computer's accuracy i s quite good. The l a s t set of data 42 p o i n t s , a f t e r 650 i t e r a t i o n s , g i v e s the f i n a l s t a b l e s t a t e of the computer. Wt) X + o loo iterations Z50 iterations 400 iterations 650 iterations (stable) F i g u r e 4-2. Computer R e s u l t s f o r Test 1. 4.2 Test 2 Th i s i s a f a i r l y r e a l i s t i c t e s t of the computer, the soTt of t h i n g i t was d e s i g n e d t o s o l v e . The impulse response to be i d e n t i f i e d i s t h a t of an underdamped second-order system, as shown i n F i g . 4-4. The i n p u t used t o t e s t t h i s system i s Gau s s i a n n o i s e which has been low—pass f i l t e r e d t o g i v e i t a wide a u t o - c o r r e l a t i o n f u n c t i o n , as shown i n F i g . 4-3. T h i s a u t o — c o r r e l a t i o n curve was determined u s i n g the computer. ( Se Appendix I ) . The computer r e s u l t s a f t e r 4000 i t e r a t i o n s ( i n i t i a l l y f, w 0, a l l k) are shown i n F i g . 4—4, a l o n g w i t h the t r u e impul Figure 4-4. Computer Results for Test 2 44 response of the t e s t system, w h i c h has been s c a l e d v e r t i c a l l y t o f i t the d a t a p o i n t s . A l t h o u g h i d e n t i f i c a t i o n i s somewhat slower w i t h the f i l t e r e d n o i s e used i n t h i s t e s t t h a n w i t h the wider band G a u s s i a n n o i s e t a k e n d i r e c t l y from the n o i s e g e n e r a t o r , the a c c u r a c y of the f i n a l computer r e s u l t i s much the same i n b o t h c a s e s , showing t h a t the computer does not depend f o r e f f e c t i v e o p e r a t i o n on a t e s t system i n p u t s i g n a l whose a u t o — c o r r e l a t i o n f u n c t i o n i s narrow compared t o the system's impulse r e s p o n s e . To show the improvement of t h i s computer over a si m p l e c r o s s - c o r r e l a t i o n type of i d e n t i f i c a t i o n f o r the b a n d - l i m i t e d random i n p u t used i n t h i s example, F i g . 4-5 compares the c r o s s - c o r r e l a t i o n f u n c t i o n w i t h the known impulse r e s p o n s e . I t i s e v i d e n t t h a t i n . t h i s case 0 ( f ) g i v e s a poor e s t i m a t e of h ( t ) . « M O M F i g u r e 4-5. Comparison o f 0 (t) and h ( t ) f o r Test 2. 45 h(t) U Figure 4-6, Impulse Responses for Test 3. ° Impulse Response A X SOD Iterations after Change I m p u l s e x | _ i Response 8 £ + looo Iterations, 1 o after Change o +• + £ I0AT X 5? Figure 4-7o Computer Results for Test 3» 46 4.3 Test 3 T h i s t e s t checks the i d e n t i f i c a t i o n of a change i n the impulse response b e i n g measured. The i n p u t x ( t ) i s a g a i n G a u s s i a n n o i s e , but t h i s time u n f i l t e r e d , g i v i n g i t a v e r y narrow a u t o c o r r e l a t i o n f u n c t i o n . The computer i d e n t i f i e s curve A i n P i g . 4-6 u n t i l s teady s t a t e c o n d i t i o n s are r e a c h e d , a t which time h ( t ) i s changed t o curve B. F i g . 4-7 shows the computer i d e n t i f i c a t i o n of the impulse response A and the computer's e s t i m a t e s of impulse response B 500 and 1000 i t e r a t i o n s a f t e r the change-over. 4.4 D i s c u s s i o n of the Computer Test R e s u l t s A l t h o u g h the computer can be seen to work f a i r l y w e l l , a few shortcomings emerged i n the above and o t h e r t e s t s . The i d e n t i f i c a t i o n time i s q u i t e l o n g , but t h i s i s t o be expected i n a s t a t i s t i c a l method of s o l u t i o n . S t a r t i n g w i t h a l l f = 0 i n i t i a l l y , a t y p i c a l i d e n t i f i c a t i o n w i l l take of the o r d e r of 3000 i t e r a t i o n s , or system s e t t l i n g t i m e s . T h i s , of c o u r s e , i s not a v e r y r e a l i s t i c c r i t e r i o n because i n normal use, when c o n t i n u o u s l y m o n i t o r i n g an impulse response, o n l y the i d e n t i f i c a t i o n of a change i n h ( t ) i s of i n t e r e s t . T h i s can be a c c o m p l i s h e d i n o n l y about 1000 i t e r a t i o n s , as i n t e s t 3, and would appear to be adequate f o r t r a c k i n g the dynamics of a system w i t h s l o w l y v a r y i n g parameters. The computer was found to work not a t a l l w e l l w i t h s t r i c t l y p e r i o d i c i n p u t s . F or pure s i n u s o i d s i t can be shown t h a t the computer w i l l not work; nor w i l l any o t h e r i d e n t i f i c a t i o n 47 scheme. The r e a s o n i s t h a t f o r a s i n u s o i d a l i n p u t to a l i n e a r system, the o u t p u t i s a l s o s i n u s o i d a l w i t h the same f r e q u e n c y b u t , i n g e n e r a l , changed phase and a m p l i t u d e . However, t h e r e i s no unique way of i n d u c i n g a g i v e n g a i n and phase change i n a s i n u s o i d of a g i v e n f r e q u e n c y . I n o t h e r words, a pure s i n u s o i d a l i n p u t s i g n a l does not c a r r y enough i n f o r m a t i o n to p e r m i t the r e c o n s t r u c t i o n of a " b l a c k box" from o n l y a knowledge of i t s i n p u t and o u t p u t . I t a l s o f o l l o w s from the n e c e s s a r y c o n d i t i o n f o r convergence d e r i v e d i n s e c t i o n 2.2 of t h i s t h e s i s t h a t the computer w i l l not work f o r s i n u s o i d a l i n p u t s . For pure s i n u s o i d s the m a t r i x 0 X X ( 2 A T ) i s o n l y p o s i t i v e s e m i - d e f i n i t e , whereas p o s i t i v e d e f i n i t e n e s s i s n e c e s s a r y f o r convergence of the computer's i t e r a t i o n s . V e r y poor r e s u l t s were a l s o o b t a i n e d f o r n o n - s i n u s o i d a l p e r i o d i c i n p u t s i f t h e i r time p e r i o d T was l e s s than the system's s e t t l i n g time T ( = 10AT") • Here a g a i n t h e r e i s a m a t h ematical s e x p l a n a t i o n . F o r p e r i o d i c i n p u t s , b o t h 0 Of) and 0 (T) are xx yx p e r i o d i c as w e l l , w i t h the same p e r i o d T. I f T < T i t t u r n s s out t h a t two or more of the t e n e q u a t i o n s s o l v e d by the computer are i d e n t i c a l or v e r y n e a r l y so* Thus the computer attempts to s o l v e n i n e or l e s s independent e q u a t i o n s f o r t e n unknowns. For 48 T > T the r e s u l t s , although s t i l l not e n t i r e l y satisfactory, s are at least q u a l i t a t i v e l y correct. Some d i f f i c u l t y was encountered i n controlling the zero l e v e l of the i d e n t i f i e d impulse response, but this i s probably an effect of the external testing c i r c u i t s rather than of the computer i t s e l f . A d.c. l e v e l i n the Gaussian noise signal, which i s adjustable but d i f f i c u l t to set accurately, would have this e f f e c t , as would imbalances or d r i f t i n the d.c. amplifiers i t was found necessary to introduce into the test system. 5. CONCLUSIONS A s t a t i s t i c a l method of system i d e n t i f i c a t i o n has been d e s c r i b e d and i t s l i m i t a t i o n s have been d i s c u s s e d . A simple computer t o c a r r y out t h i s method of i d e n t i f i c a t i o n has been d e s i g n e d , b u i l t and t e s t e d . The t e s t r e s u l t s are q u i t e s a t i s f a c t o r y and i t can be s a i d t h a t , w i t h i n the l i m i t a t i o n s of the g e n e r a l m a t h e m a t i c a l approach to the problem, the computer works. A few d e t a i l s remain to be r e s o l v e d b e f o r e the computer can s e r v e a u s e f u l f u n c t i o n i n a p r a c t i c a l s i t u a t i o n such as an a d a p t i v e l o o p . A t p r e s e n t the i d e n t i f i c a t i o n time i s e x c e s s i v e l y l o n g , b u t an a l t e r a t i o n i n the g e a r i n g r a t i o between the ste p motors and the p o t e n t i o m e t e r s , o r an i n c r e a s e i n the number of p o s s i b l e motor s t e p s p e r c o r r e c t i o n over the p r e s e n t l i m i t of t e n , s h o u l d r e s u l t i n some improvement. A l t h o u g h the l a c k of c o n t r o l of the zer o l e v e l of the i d e n t i f i e d i m p u l s e response may have been caused o n l y by the p a r t i c u l a r t e s t c i r c u i t r y used and thus pose no problem i n a p r a c t i c a l a p p l i c a t i o n of the computer, i t would lbe r a t h e r s e r i o u s i f i t d i d o c c u r , s i n c e almost a l l f i g u r e s of m e r i t d e r i v e d from an impulse response w i l l depend on the a c c u r a c y of i t s z e r o l e v e l . I t s h o u l d be p o s s i b l e , however, to e l i m i n a t e the problem e n t i r e l y . I f AX i s p r o p e r l y chosen, h ( l O A t ) 0 • U s i n g feedback t o f o r c e a l l f ^ up or down i n u n i s o n so as to keep f , n = 0 s h o u l d g i v e adequate c o n t r o l . T h i s would r e q u i r e 50 the use of a double potentiometer with the tenth motor, since the present one i s used f o r one of the computer's multiplying operations and does not have a constant reference gupply across i t . An alternative method of s t a b i l i z i n g the zero l e v e l would be to use the result tha.t for a normalized system h(t) dt a 1 , For the computer this would mean using feedback to maintain f = 1 . i a l Normalization would be obtained by a proper scaling of the signals x(t) and y(t.) or$ equivalently, by a suitable, fixed setting of potentiometer PI i n F i g . 3-12. I 51 REFERENCES 1, Goodman, T.P*, and Reswick, J*B., " D e t e r m i n a t i o n of System C h a r a c t e r i s t i c s from Normal O p e r a t i n g Records''^ A.S.M.E. T r a n s a c t i o n s , V o l . 78, Fe b r u a r y 1956, pp. 259-271• 2* L i c h t e n b e r g e r , W.W., "A Technique of System I d e n t i f i c a t i o n U s i n g C o r r e l a t i n g F i l t e r s " , I.R.E. T r a n s - a c t i o n s PGAC. V o l . AC-6, May 1961, pp. 183-199* 3* Anderson, G», A s e l t i n e , J * , M a n c i n i , A., and S a r t u r e , C., "A S e l f - A d j u s t i n g System f o r Optimum Dynamic Response", I.R.E. N a t i o n a l C o n v e n t i o n R e c o r d , 1958, P a r t 4 j pp. 182-190. 4* Gabor, D., V i l b y , ¥., and Woodcock, R., "A U n i v e r s a l Non-L i n e a r F i l t e r , P r e d i c t o r and S i m u l a t o r Which O p t i m i z e s I t s e l f by a L e a r n i n g P r o c e s s " , I.E.E.  J o u r n a l . J u l y 1960, p. 444. 5* P a r k e r , L.E.G., "A Real—Time Analogue Computer f o r the E s t i - m a t ion of System Dynamics", M.A.Sc. T h e s i s , U n i v e r s i t y of B r i t i s h Columbia, 1962. 6* Bonn, E.V*, "A C o n t i n u o u s l y A c t i n g A d a p t i v e Analog Computer f o r d e t e r m i n i n g the Impulse Response of C o n t r o l ! Systems w i t h G a u s s i a n S i g n a l s " , Trans. E n g i - n e e r i n g I n s t i t u t e of Canada, V o l . 5, No.3, 1961* 7* Woodrow, R.A., "Data F i t t i n g W i t h L i n e a r T r a n s f e r F u n c t i o n s " , J o u r n a l Q£ E l e c t r o n i c s and C o n t r o l , V o l . 6, No.5, 1959, pp. 454-480. 8* T u r n e r , R.M., 9* Watts, D.G., "A D i g i t a l C o r r e l a t o r f o r Low-Frequency S i g n a l s " , M.A.Sc* T h e s i s , U n i v e r s i t y of B r i t i s h C o lumbia, 1964. "A G e n e r a l Theory of Amplitude Q u a n t i z a t i o n w i t h A p p l i c a t i o n t o C o r r e l a t i o n D e t e r m i n a t i o n " , P r o c . I»E*E*» P a r t C, March 1962, pp. 209-218* 10» R e i c h , E., "On the Convergence of the C l a s s i c a l I t e r a t i v e Method of S o l v i n g L i n e a r Simultaneous E q u a t i o n s " , Ann. Math* S t a t i s t i c s . V o l . 20, 1949, pp. 448-451. APPENDIX I CORRELATION FUNCTION MEASUREMENT The computer, w i t h a v e r y minor m o d i f i c a t i o n , can be used to measure t e n e q u a l l y spaced p o i n t s of any a u t o - o r c r o s s -c o r r e l a t i o n f u n c t i o n . I f s w i t c h SWl i n F i g . 3-15 i s opened and the summing a m p l i f i e r i s r e b a l a n c e d , the normal computer e q u a t i o n 10 f •m m m=l i s changed t o e k b k * Thus, a f t e r N i t e r a t i o n s , where care must now be t a k e n t o ensure t h a t a l l f ^ are i n i t i a l l y z e r o , i = l = N b, k m N 0 y x ( ( k - i ) A r ) That i s , the computer o u t p u t s f ^ are d i r e c t l y p r o p o r t i o n a l o r r e l a t i o n 0 . yx y ( t ) i s s e t e q u a l t o x(t)« t o the c t i o n F o r an a u t o - c o r r e l a t i o n measurement 

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