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UBC Theses and Dissertations

State estimation and optimization with application to adaptive control of Linear Distributed Parameter… Wong, John Kin 1974

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STATE. ESTIMATION AND OPTIMIZATION WITH APPLICATION TO ADAPTIVE CONTROL OF LINEAR DISTRIBUTED PARAMETER SYSTEMS by JOHN KIN WONG B.Sc.CEng.) Hons., Univ e r s i t y College of Swansea, United Kingdom, 1968 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY i n the Department of E l e c t r i c a l Engineering We accept t h i s thesis as conforming to the required standard Research Supervisor Members of Committee Head of Department Members of the Department of E l e c t r i c a l Engineering THE UNIVERSITY OF BRITISH COLUMBIA August, 1974 In p r e s e n t i n g t h i s t h e s i s in p a r t i a l f u l f i l m e n t o f the r equ i r ements 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 C o l u m b i a , 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 r e f e r e n c e and s tudy . I f u r t h e r agree t h a t p e r m i s s i o n f o r e x t e n s i v e c o p y i n g o f 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 g r a n t e d by the Head o f my Department o r by h i s r e p r e s e n t a t i v e s . It i s unde r s tood that c o p y i n g o r p u b l i c a t i o n o f t h i s t h e s i s f o r f i n a n c i a l g a i n s h a l l not be a l l owed w i thout my w r i t t e n p e r m i s s i o n . Depa rtment The U n i v e r s i t y o f B r i t i s h Co lumbia Vancouver 8, Canada ABSTRACT This t h e s i s i s concerned w i t h e s t i m a t i o n and c o n t r o l of l i n e a r d i s t r i b u t e d parameter systems. For the e s t i m a t i o n of l i n e a r d e t e r m i n i s t i c continuous-time d i s t r i b u t e d parameter systems, a l i n e a r d e t e r m i n i s t i c d i s t r i b u t e d p a r a -meter f i l t e r that y i e l d s the s t a t e estimate based on n o i s e l e s s l i n e a r measurements a v a i l a b l e over the complete occupied s p a t i a l domain i s de-r i v e d by c o n s i d e r a t i o n of a Lyapunov type of s t a b i l i t y . The general r e s u l t s are then s p e c i a l i z e d to the case when n o i s e l e s s l i n e a r measure-ments are a v a i l a b l e at only s e v e r a l p o i n t s i n the s p a t i a l domain. A numerical example i l l u s t r a t e s i t s use i n an o v e r a l l c o n t r o l scheme. For the e s t i m a t i o n of l i n e a r s t o c h a s t i c d i s c r e t e - t i m e d i s t r i b -uted parameter systems, a l i n e a r d i s c r e t e - t i m e d i s t r i b u t e d parameter f i l t e r having a p r e d i c t o r - c o r r e c t o r s t r u c t u r e , that y i e l d s the minimum-variance estimate of the s t a t e based on n o i s e - c o r r u p t e d l i n e a r measure-ments assumed a v a i l a b l e at only s e v e r a l s p a t i a l l o c a t i o n s , i s d e r i v e d . The f i l t e r e d estimate and the f i l t e r i n g e r r o r are shown to s a t i s f y an orthogonal p r o j e c t i o n lemma, whence a Wiener-Hopf equation i s d e r i v e d . The f i l t e r i s implementable o n - l i n e and a n umerical example i l l u s t r a t e s i t s use. The optimal pointwise r e g u l a t i o n c o n t r o l problem f o r l i n e a r s t o c h a s t i c d i s c r e t e - t i m e d i s t r i b u t e d parameter systems i s t r e a t e d through a p p l i c a t i o n of dynamic programming. The s e p a r a t i o n of the com-p l e t e c o n t r o l scheme i n t o the e s t i m a t i o n and c o n t r o l subsystems i s shown. I t s usefulness i s i l l u s t r a t e d i n a numerical example. By f i r s t expanding Green's f u n c t i o n and then c o n s i d e r i n g the i i i l i m i t i n g behaviour of the corresponding d i s c r e t e - t i m e r e s u l t s on estima-t i o n and c o n t r o l obtained p r e v i o u s l y , s o l u t i o n s of the continuous-time l i n e a r minimum-variance f i l t e r i n g e s t i m a t i o n and o p t i m a l p o i n t w i s e regu-l a t i o n c o n t r o l problems f o r l i n e a r s t o c h a s t i c continuous-time d i s t r i b u t e d parameter systems are obtained. F u r t h e r , a s e p a r a t i o n theorem i s o b t a i n -ed and Kalman's d u a l i t y theorem extended. For the pointwise r e g u l a t i o n c o n t r o l problem of l i n e a r s t o c h -a s t i c d i s c r e t e - t i m e d i s t r i b u t e d parameter systems, the case of unknown noise c h a r a c t e r i s t i c s i s t r e a t e d . Based on an examination of the open-lo o p - o p t i m a l feedback c o n t r o l approach, a suboptimal c o n t r o l scheme i s proposed. A f i l t e r t h a t i s a d a p t i v e l y s e l e c t e d o n - l i n e based on m i n i -m i z i n g an instantaneous cost f u n c t i o n a l so de r i v e d from the o r i g i n a l one as to r e a l i z e a t r a d e - o f f between c o n t r o l and e s t i m a t i o n costs i s put c o n t r o l scheme i n comparison w i t h the opt i m a l one. i v TABLE OF CONTENTS Page ABSTRACT . .• H i TABLE OF CONTENTS v LIST OF TABLES i x LIST OF FIGURES x ACKNOWLEDGEMENT x i i 1. INTRODUCTION 1 1.1 I n t r o d u c t i o n 1 1.2 Thesis O b j e c t i v e and Layout 3 2. FILTERING ESTIMATION OF LINEAR CONTINUOUS-TIME DETERMINISTIC DISTRIBUTED PARAMETER SYSTEMS 7 2.1 I n t r o d u c t i o n 7 2.2 Problem For m u l a t i o n . ••• 8 2.3 D e r i v a t i o n of the D e t e r m i n i s t i c F i l t e r 9 2.3.1 S t r u c t u r e of the D e t e r m i n i s t i c F i l t e r 10 2.3.2 D e r i v a t i o n of the D i s t r i b u t e d F i l t e r Gain 11 2.4 Discrete-space Measurements 19 2.5 Feedback Considerations 24 2.6 Numerical Example 25 2.6.1 Problem Formulation 25 2.6.2 S o l u t i o n of the R i c c a t i Equations 31 2.6.3 S i m u l a t i o n Study 35 . 2.6.3.1 System Response 35 2.6.3.2 Estimator Response 41 2.6.3.3 O b s e r v a b i l i t y 48 v 2.6.4 Conclusion 2.7 Conclusion 52 3. OPTIMAL PREDICTION AND FILTERING OF LINEAR DISCRETE-TIME STOCHASTIC DISTRIBUTED PARAMETER SYSTEMS BASED ON DISCRETE-SPACE MEASUREMENTS" 53 3.1 I n t r o d u c t i o n 53 3.2 Problem Formulation 55 3.3 On the L i n e a r D i s c r e t e - t i m e S t o c h a s t i c D i s t r i b u t e d Parameter Model 58 3.4 D e r i v a t i o n of the L i n e a r D i s c r e t e - t i m e D i s t r i b u t i v e Kalman F i l t e r 61 3.4.1 F i l t e r i n g E r r o r 64 3.4.2 P r e d i c t i o n E r r o r Covariance 67 3. A.3 F i l t e r i n g E r r o r Covariance 68 3.4.4 Optimal D i s t r i b u t e d F i l t e r Gain 73 3.5 Orthogonal P r o j e c t i o n 74 3.6 Numerical Example 76 3.7 Conclusion 85 4. OPTIMAL POINTWISE REGULATION CONTROL OF LINEAR DISCRETE-TIME STOCHASTIC DISTRIBUTED PARAMETER SYSTEMS 91 4.1 I n t r o d u c t i o n 91 4.2 Problem Formulation . 93 4.3 On the L i n e a r D i s c r e t e - t i m e D i s t r i b u t e d Parameter Model w i t h P o i n t w i s e C o n t r o l 97 4.4 D e r i v a t i o n of the S t o c h a s t i c Optimal P o i n t w i s e C o n t r o l 101 4.4.1 P e r f e c t l y Observed St a t e 102 .4.4.2 Noisy State Measurements 105 v i 4.5 Numerical Example i u y 4.6 Conclusion 126 5. OPTIMAL FILTERING ESTIMATION AND POINTWISE REGULATION CONTROL OF LINEAR CONTINUOUS-TIME STOCHASTIC DISTRIBUTED PARAMETER SYSTEMS . 128 5.1 I n t r o d u c t i o n 128 5.2 L i n e a r Continuous-time D i s t r i b u t i v e Kalman F i l t e r 128 5.2.1 Problem Formulation 128 5.2.2 S t r u c t u r e of the F i l t e r 133 5.2.3 Optimal D i s t r i b u t e d F i l t e r Gain 135 5.2.4 F i l t e r i n g E r r o r Covariance 137 5.3 S t o c h a s t i c Optimal Po i n t w i s e R e g u l a t i o n C o n t r o l f o r L i n e a r Continuous-time D i s t r i b u t e d Parameter Systems 138 5.3.1 Problem Formulation 138 5.3.2 Cost F u n c t i o n a l 145 5.3.3 Optimal Po i n t w i s e C o n t r o l 151 5.4 D u a l i t y 153 5.5 Conclusion 155 6. SUBOPTIMAL POINTWISE REGULATION CONTROL OF LINEAR DISCRETE-TIME STOCHASTIC DISTRIBUTED PARAMETER SYSTEMS 157 6.1 I n t r o d u c t i o n 157 6.2 Problem Formulation 159 6.3 Background to Open-loop-optimal Feedback C o n t r o l 163 6.4 Proposed Open-loop-suboptimal Feedback C o n t r o l 170 6.4.1 Co n t r o l Problem 171 6.4.2 F i l t e r i n g Problem 172 6.4.3 Equivalent Cost F u n c t i o n a l 178 6.5 Asymptotic S t a b i l i t y of the F i l t e r 179 v i i 6.5.1 S t a b i l i t y C o n s i d e r a t i o n 179 6.5.2 Bounds on the S c a l a r Gain 182 6.6 D e r i v a t i o n of the C o n t r o l A l g o r i t h m 185 6.7 Numerical Example 188 6.8 Conclusion 202 7. CONCLUSION 204 APPENDIX I 208 APPENDIX I I 211 REFERENCES 215 v i i i LIST OF TABLES Page Table (.2.1) F i r s t f i v e r oots o f . p ^ t g p. = 1 32 Table (2.2) S o l u t i o n of eqn. (2.6.31) w i t h M=3, 3=1 and B=0.1 33 Table C2.3) S o l u t i o n of eqn. (.2.6.39) 34 CD M=3, 3=1, r £ = l and Q £=I 3 34 C2) M=3, 3=1, r =10 and 0^=^ 34 (3) M=3, 3=1, r^=10 and Q £=10I 3 34 Table (2.4) Zero-crossings of e i g e n f u n c t i o n s eqn. (2.6.27) 51 Table (3.1) Value of the s t a t e t r a n s i t i o n m a t r i x ^^ +^ k °f eqn. (3.6.3) 78 Table (3.2) Value of the disturbance t r a n s i t i o n m a t r i x ^ of eqn. (3.6.3) 79 T p V i l o f i . l l V a l n o n f the* f n r i t r n i t - r a n p i ' t - i n n m a f - r i v V n f - • - v • •• y • - • . . . , k+i ,k eqn. (4.5.6) I l l Table (4.2) Breakdown of the cost C 126 u Table (6.1) Values of the average cost of eqn. (6.7.4) 201 i x LIST OF FIGURES Page F i g . C2.1) Block diagram f o r 2.6.3.1 37 F i g . C.2.2) Temperature p r o f i l e u ^ ( t , x ) a t x=0., 0.5, 1 38 F i g . (.2.3) Optimal c o n t r o l f°.('t) 39 F i g . (2 .4 ) Cost f u n c t i o n a l J 40 F i g . C2.5) Block diagram f o r 2.6.3.2 43 F i g . (2.6) Measurement r e s i d u a l y ( t , 0 . 5 ) - u ( t , 0 . 5 ) 44 F i g . (2.7) Response d i f f e r e n c e u A ( t , x ) - u A ( t , x ) at x=0., 0.5, 1 47 (a) Response d i f f e r e n c e u A ( t , 0 . ) - u ^ ( t , 0 . ) 45 (b) Response d i f f e r e n c e u ^ ( t , . 5 ) - u ^ ( t , . 5 ) 46 (c) Response d i f f e r e n c e u ^ ( t , 1 . ) - u ^ ( t , 1 . ) 47 F i g . (2.8) C o n t r o l d i f f e r e n c e f°(t)-f°(t) 49 F i g . (2.9) Cost d i f f e r e n c e J - J 50 F i g . (3.1) Comparison of true u^^) a n d estimate u^(x) at x=0 86 F i g . (3.2) Comparison of tr u e u ^ ( x ) and estimate u^(x) at x=0.5 .... 87 F i g . (3.3) Comparison of true uk_( x) and estimate u^(x) at x=l 88 F i g . (3.4) F i l t e r i n g e r r o r v a r i a n c e P (x,x) at x=.2 and x=.7 89 K. F i g . (4.1) Graph of V ^ ( x , l . ) w i t h k as parameter 117 F i g . (4.2) Graph of V ( l . , l . ) 118 F i g . (4.3) Comparison of d e t e r m i n i s t i c u A^.(x) and estimate u A ^ ( x ) a t x=0 121 F i g . (4.4) Comparison of d e t e r m i n i s t i c u ^ ( x ) and estimate u ^ O O at x=0.5 122 F i g . (4.5) Comparison of d e t e r m i n i s t i c u A ^ ( x ) and estimate u ^ ( x ) a t x=l. 123 ^ o F i g . (4.6) Comparison of d e t e r m i n i s t i c f ^ and s t o c h a s t i c f ^ •••••• 124 x Fig. (.6. 1) 195 Fig. (6. 2) Comparison of deterministic u., (x) and estimate *k at 197 Fig. (.6. • 3) Comparison of deterministic u A k ( x ) and estimate at .198 Fig. (6. .4) Comparison of deterministic u,, (x) and estimate at 199 Fig. (.6 • 5) Comparison of deterministic f° and stochastic *k f * k ••• 200 xi ACKNOWLEDGEMENT I wish to thank Dr. A. C. Soudack. f o r being my t h e s i s s u p e r v i s o r . F u r t h e r , I wish, to express my a p p r e c i a t i o n to Dr. E. V. Bohn f o r c r i t i c a l l y reading the t h e s i s . F i n a l l y , I g r a t e f u l l y acknowledge the f i n a n c i a l a s s i s t a n c e from the N a t i o n a l Research C o u n c i l of Canada f o r the years 1968-69 and 1973—74 (grant 67-3138) and i t s 1969-72 Postgraduate S c h o l a r s h i p (1560). x i i To my parents To King Keat and Ing Min 1 1. INTRODUCTION 1 • 1 I n t r o d u c t i o n Many a p h y s i c a l process i n nature can only be adequately described by p a r t i a l d i f f e r e n t i a l equations, i n t e g r a l equations or i n t e g r o - p a r t i a l d i f f e r e n t i a l equations. Such processes where the system s t a t e and/or c o n t r o l i s a f u n c t i o n of both time and space are termed d i s t r i b u t e d parameter systems Much i n t e r e s t has been shown i n recent years i n developing a u n i f i e d theory and techniques to c o n t r o l such d i s t r i b u t e d parameter systems. Examples of these, which can be 1 2 3 4 found f o r example i n ' ' ' , i n c l u d e the c o n t r o l of thermal processes, m e t a l l u r g i c a l r e f i n i n g and heat exchangers. Now, there are two p h i l o s o p h i c a l approaches to the study of <H s1*r"t h n f - p r l n s r a t n o f p i " R t r s l - e - m oat- l it ia t-1 n n / r n t i t r n T n r n M pmo . T h o f-i r e f -i c to e s t a b l i s h some or use some e s t a b l i s h e d v a l i d approximation methods so as to set up i n the f i r s t p l a c e a corresponding lumped parameter system from the d i s t r i b u t e d parameter model and then to apply the w e l l -developed lumped parameter e s t i m a t i o n / c o n t r o l theory. However, such an approach, though f o r some d i s t r i b u t e d parameter systems may enjoy the advantage of r e t a i n i n g c e r t a i n p h y s i c a l m i c r o s c o p i c s t r u c t u r e of the system, tends i n general to obscure the d i s t r i b u t e d nature of the system s t a t e and c o n t r o l s t r u c t u r e s and t h e i r f u n c t i o n a l r e l a t i o n s h i p through a d i s t r i b u t e d parameter cost f u n c t i o n a l . Moreover, i t u s u a l l y i s not c o n c e p t u a l l y c l e a r , a p r i o r i , which s p e c i f i c approximation method i s most s u i t e d to the p a r t i c u l a r system under c o n s i d e r a t i o n such t h a t the s o l u t i o n of the e s t i m a t i o n / c o n t r o l problem based on the approximation model so derived may conceivably d e v i a t e c o n s i d e r a b l y from the a c t u a l 2 s o l u t i o n . Furthermore, i t i s c l e a r that one approximation method developed or found s u i t a b l e f or one p a r t i c u l a r class of d i s t r i b u t e d parameter systems may turn out not to be so for another such that i t s general usefulness i s but l i m i t e d . Therefore, i n t h i s t h e s i s , we adopt the second approach, which i s to formulate the estimation/control problem and to develop the corresponding estimation/control theory within the framework of d i s t r i b u t e d parameter system and then to proceed to obtain solutions f o r s p e c i f i c problems of the various equations, u s u a l l y p a r t i a l d i f f e r e n t i a l or i n t e g r o - p a r t i a l d i f f e r e n t i a l equations, derived i n the theory. Moreover, we have here the advantage of being i n a p o s i t i o n to choose, a p o s t e r i o r i , the approximation scheme that i s most su i t a b l e from the computational viewpoint f o r the s p e c i f i c problem i n question to obtain meaningful numerical r e s u l t s from i t s d i s t r i b u t e d parameter s o l u t i o n . The o v e r a l l c o n t r o l scheme for a p h y s i c a l process generally involves estimation and c o n t r o l of i t s behaviour, subject to p o s s i b l e random disturbance and/or random measurement e r r o r , such that some c r i t e r i o n of performance i s optimized. Moreover, they are r e l a t e d to each other, as estimation being the f i r s t step y i e l d s the information on process behaviour needed by the c o n t r o l before i t can be e f f e c t i v e l y applied. This thesis i s therefore concerned with c e r t a i n aspects of the two r e l a t e d problems of estimation and c o n t r o l involved i n the c o n t r o l l i n g of l i n e a r d i s t r i b u t e d parameter systems. The t h e s i s layout together with background material i s presented i n the next s e c t i o n . 3 1.2 Thesis Objective and Layout For the closed-loop c o n t r o l of a l i n e a r d e t e r m i n i s t i c continuous-time d i s t r i b u t e d parameter system, the s t a t e of the system 3 4 u s u a l l y needs to be k now ' . The system s t a t e must t h e r e f o r e be rec o n s t r u c t e d from a v a i l a b l e n o i s e l e s s measurements made. Such a pro-5 7 8 blem has been i n v e s t i g a t e d ' ' . However, as y e t , there are no general r e s u l t s corresponding to the d e t e r m i n i s t i c f i l t e r i n lumped 9 10 parameter system ' . In Chapter 2, we t h e r e f o r e f i r s t d e r i v e a d e t e r m i n i s t i c d i s t r i b u t e d f i l t e r , shown to be l i n e a r i n s t r u c t u r e , by c o n s i d e r a t i o n of a Lyapunov type of s t a b i l i t y , assuming that n o i s e l e s s l i n e a r measurements are a v a i l a b l e over the whole occupied s p a t i a l domain. Next, we s p e c i a l i z e the general r e s u l t s to the p r a c t i c a l case when n o i s e l e s s measurements are assumed a v a i l a b l e at o n l y s e v e r a l p o i n t s i n In a p r a c t i c a l s i t u a t i o n , a d i s t r i b u t e d parameter c o n t r o l system, however, i s more o f t e n than not a f f e c t e d by time and/or space dependent random f l u c t u a t i o n s that enter i n t o the p i c t u r e e i t h e r as u n c o n t r o l l e d d i s t r i b u t e d and/or boundary input d isturbances or as s t o c h a s t i c v a r i a t i o n s of the system parameters or as measurement e r r o r . For such s t o c h a s t i c d i s t r i b u t e d parameter systems, some s t u d i e s have 11—18 been made on t h e i r e s t i m a t i o n and c o n t r o l . However, p r a c t i c a l l y a l l of them are concerned w i t h the continuous-time case. Now, i n the process c o n t r o l f i e l d , i t i s d e s i r a b l e , w i t h the advent of d i g i t a l computers implementable o n - l i n e , to c o n t r o l as f a r as p o s s i b l e the system o n - l i n e . We are t h e r e f o r e motivated to co n s i d e r d i s c r e t e - t i m e d i s t r i b u t e d parameter systems. 4 For the estimation of s t o c h a s t i c discrete-time d i s t r i b u t e d parameter systems from noisy measurements made on the state, i t i s u n r e a l i s t i c to assume that measurements are a v a i l a b l e over the complete occupied s p a t i a l domain. We are therefore motivated fur t h e r to consider, i n Chapter 3, the f i l t e r i n g estimation problem of l i n e a r s t o c h a s t i c discrete-time d i s t r i b u t e d parameter systems based on discrete-time noise-corrupted l i n e a r measurements a v a i l a b l e only at several points i n the s p a t i a l domain. We derive, a f t e r a consideration of the l i n e a r s t o c h a s t i c discrete-time d i s t r i b u t e d parameter model, i n t h i s chapter a l i n e a r minimum-variance d i s t r i b u t e d parameter f i l t e r s u i t a b l e for on-line implementation. Moreover, we show that the. f i l t e r e d estimate and the f i l t e r i n g e r r o r s a t i s f y an orthogonal p r o j e c t i o n lemma, corresponding to 19 that i n lumped parameter systems . A Wiener-Hopf equation r e l a t i o n i s then further shown to be derivable. In the c o n t r o l of d i s t r i b u t e d parameter systems, though i t i s p o s s i b l e i n theory to have a c o n t r o l d i s t r i b u t e d over the complete occupied s p a t i a l domain, i t has been pointed out by Athans i n a l u c i d l y 21 w r i t t e n paper that such a c o n t r o l i s i m p r a c t i c a l from the implemen-t a t i o n point of view. As a p o s s i b l e p r a c t i c a l a l t e r n a t i v e , we are therefore motivated to consider discrete-time c o n t r o l concentrated at c e r t a i n points i n the s p a t i a l domain, that i s , pointwise c o n t r o l . Further, i t i s of i n t e r e s t to note that included w i t h i n the pointwise c o n t r o l problem are a l l the one-dimensional boundary c o n t r o l ones. Hence, i n Chapter 4, we develop the s t o c h a s t i c c o n t r o l theory of optimal pointwise c o n t r o l of l i n e a r s t o c h a s t i c discrete-time d i s t r i b u t e d parameter systems, the average cost f u n c t i o n a l considered being quad-r a t i c both i n the state and the c o n t r o l e f f o r t , through the a p p l i c a t i o n 22 23 of the dynamic programming technique 5 . The main r e s u l t obtained i s the s e p a r a t i o n of e s t i m a t i o n and c o n t r o l subsystems, corresponding . . . , 24,25,26 to that i n lumped parameter systems In Chapter 5, we f i r s t develop and then u t i l i z e the approach of f i r s t expanding the Green's f u n c t i o n i n a T a y l o r ' s s e r i e s i n time and then a l l o w i n g the sampling time p e r i o d to go to zero of the d i s c r e t e -time r e s u l t s i n Chapters 3 and 4 on the l i n e a r minimum-variance d i s -t r i b u t e d parameter f i l t e r and s t o c h a s t i c o p t i m a l p o i n t w i s e r e g u l a t i o n c o n t r o l , such that by so doing we d e r i v e the corresponding continuous-time analogues. Besides, the other main r e s u l t s f o r the continuous-time cases of e s t i m a t i o n and c o n t r o l are a s e p a r a t i o n theorem and an 20 extension of Kalman's d u a l i t y i n lumped parameter systems For d i s t r i b u t e d parameter systems that are d i s t u r b e d both i n the p l a n t and the measurement process by n o i s e , the n o i s e c h a r a c t e r i s -27 t i c s are seldom e x a c t l y known. Though we may extend methods th a t are a v a i l a b l e f o r lumped parameter systems to i d e n t i f y the m i s s i n g s t a t i s t i c a l moments, the procedure of f i r s t i d e n t i f i c a t i o n and then use of the a v a i l a b l e o ptimal c o n t r o l theory poses c o n t r o l scheme complexity problems. Hence, we have adopted the approach of combined e s t i m a t i o n and c o n t r o l f o r the s o l u t i o n of the p o i n t w i s e r e g u l a t i o n c o n t r o l problem i n Chapter 6. To t h i s end, an open-loop-optimal feedback c o n t r o l method 28 favoured i n lumped parameter systems i s examined i n the present d i s -t r i b u t e d parameter context. Based on i t s a p p r a i s a l , a suboptimal c o n t r o l scheme that r e t a i n s the e s t i m a t i o n - c o n t r o l sybsystems s t r u c t u r a l d i v i s i o n i s proposed. For the e s t i m a t i o n subsystem, an adaptive f i l t e r charac-t e r i z e d by a s c a l a r gain to be determined o n - l i n e i s proposed to generate the s t a t e estimate based on past n o i s y measurements assumed a v a i l a b l e at only s e v e r a l p o i n t s i n the s p a t i a l domain. The other parameters of the f i l t e r can be determined o f f - l i n e , r e q u i r i n g no d e t a i l e d a p r i o r i knowledge of the n o i s e c h a r a c t e r i s t i c s . The c o n t r o l subsystem c o n s i s t s of a time-varying feedback gain which i s to m u l t i p l y the instantaneous s t a t e estimate. Furthermore, the feedback gain i s independent of the noise c h a r a c t e r i s t i c s i n i t s determination. The s c a l a r g a i n of the f i l t e r i s updated i n an adaptive f a s h i o n based on m i n i m i z i n g the instantaneous cost f u n c t i o n a l , which i s so de r i v e d from the o r i g i n a l average qu a d r a t i c cost f u n c t i o n a l as to achieve v i a the c o u p l i n g s c a l a r g a i n a t r a d e - o f f between c o n t r o l and e s t i m a t i o n c o s t s . The proposed suboptimal c o n t r o l scheme i s s t r a i g h t f o r w a r d i n i t s imple-mentation and s u i t a b l e f o r o n - l i n e a p p l i c a t i o n . I t s e f f e c t i v e n e s s compared w i t h the optimal c o n t r o l scheme i s shown, v i s - a - v i s a p a r t i c u -l a r numerical example. 7 2. FILTERING ESTIMATION OF LINEAR CONTINUOUS-TIME DETERMINISTIC DISTRIBUTED PARAMETER SYSTEMS 2.1 I n t r o d u c t i o n In the c o n t r o l of d i s t r i b u t e d parameter systems, occasions a r i s e where a s t a t e feedback c o n t r o l scheme i s i n v o l v e d , such as i n the implementation of the op t i m a l l i n e a r c o n t r o l law to systems whose 3 4 performance c r i t e r i o n i s qu a d r a t i c both i n the s t a t e and c o n t r o l ' However, i n many cases the e n t i r e s t a t e of a given system i s not a v a i l -able f o r feedback purposes as a d i r e c t output q u a n t i t y . Some technique to o b t a i n an estimate of the e n t i r e s t a t e i s t h e r e f o r e c a l l e d f o r . In the c o n t r o l of lumped parameter systems, such problems of r e c o n s t r u c t i n g the e n t i r e s t a t e from a v a i l a b l e measurements have been 29 30 9 handled ' ' by making use of l i n e a r systems c a l l e d observers or d e t e r m i n i s t i c f i l t e r s . S i m i l a r problems i n d i s t r i b u t e d parameter systems have been 7 8 5 7 8 considered i n ' ' . Cannon and K l e i n have i n v e s t i g a t e d i n ' some p a r t i c u l a r cases of d i s t r i b u t e d parameter system e s t i m a t i o n . They, by f i r s t l y assuming that the s o l u t i o n of the d i s t r i b u t e d parameter system can be expanded i n t o a s e r i e s i n v o l v i n g the system e i g e n f u n c t i o n s , develop some estimator scheme w i t h r espect to the p a r t i c u l a r d i s t r i b u -ted parameter system considered to determine the c o e f f i c i e n t s of expansion u t i l i z i n g the a d d i t i o n a l i n f o r m a t i o n y i e l d e d by measurements. 5 4 Howland i n f i r s t l y shows that o b s e r v a b i l i t y i n the sense of Wang i s not a p p l i c a b l e to the general c l a s s of d i s t r i b u t e d parameter systems he c o n s i d e r s . He then proposes the approach of f i r s t approximating the d i s t r i b u t e d parameter system by an e i g e n v a l u e - e i g e n f u n c t i o n expansion 8 method i n t o a lumped parameter system and proceeds to design observers using the standard Luenberger observer theory to observe the lumped parameter model. We note that the estimators/observers they develop are r e s t r i c t e d i n usefulness by the p a r t i c u l a r approximation methods used "at-the-beginning" of s o l v i n g the e s t i m a t i o n problems. In t h i s chapter, a c l a s s of l i n e a r d i s t r i b u t e d f i l t e r s 9 u t i l i z i n g the concept contained i n i s developed. The theory i s developed as f a r as p o s s i b l e i n the d i s t r i b u t e d parameter domain. We see that by employing i t i n concrete cases u s i n g whichever s u i t a b l e approximation method "at-the-end" to s o l v e the p a r t i a l d i f f e r e n t i a l equations i n v o l v e d , the c o n s t r a i n t s imposed on the e s t i m a t i o n schemes otherwise obtained by "at-the-beginning" approximation methods are thereby removed. A f t e r f o r m u l a t i n g the d e t e r m i n i s t i c e s t i m a t i o n problem i n 2.2. the d e t e r m i n i s t i c f i l t e r i s d e r i v e d i n 2.3 by c o n s i d e r -31 4 ing the uniform asymptotic s t a b i l i t y i n the sense of Lyapunov ' of the e s t i m a t i o n e r r o r . The r e s u l t s are then s p e c i a l i z e d i n 2.4 to t r e a t the case where measurements are a v a i l a b l e only at a f i n i t e number of d i s c r e t e s p a t i a l l o c a t i o n s . In 2.5, some c o n s i d e r a t i o n i s given to the s t a b i l i t y e f f e c t of i n c l u d i n g the d i s t r i b u t e d d e t e r m i n i s t i c f i l t e r i n the system to generate a s t a t e estimate to be used i n the feedback c o n t r o l law i n the place of the t r u e s t a t e . F i n a l l y , i n 2.6, a numerical example on a one-dimensional boundary c o n t r o l problem i s considered i n some d e t a i l to i l l u s t r a t e i n p a r t i c u l a r the theory developed i n 2.4. 2.2 Problem Formulation Let D be a bounded open set i n a E u c l i d e a n r-space E w i t h 1 r piece-wise smooth boundary 3 D , and l e t x = (x , x ) denote a generic p o i n t i n D. Let t denote time d e f i n e d on the f i x e d i n t e r v a l T = [ t Q , t f ] . Consider a l i n e a r continuous-time d i s t r i b u t e d parameter system described by a v e c t o r p a r t i a l d i f f e r e n t i a l equation of the form 3u(t,x)_ = A U ( t > x ) + B ( t , x ) F , ( t ( x ) , xeD (2.2.1) 3 1 x u defined f o r t^>t>tg on the s p a t i a l domain D, where u ( t s x ) = c o l ( u ^ ( t s x ) . ..u ( t , x ) ) i s the n-dimensional s t a t e v e c t o r f u n c t i o n of the system, A i s a n x known l i n e a r nxn .matrix p a r t i a l - d i f f e r e n t i a l operator w i t h respect to x, B^(t,x) i s a known nxp m a t r i x and F,(t,x) i s the p x l c o n t r o l v e c t o r d i s t r i b u t e d d over the s p a t i a l domain. The i n i t i a l and boundary c o n d i t i o n s of eqn.(2.2.1) are given as and u ( t 0 , x ) = u 0 ( x ) , xeD (2,2.2) 3 u(t,x)=0, xe3D, (2.2.3) The system i s assumed to be observed by the measurement equation y ( t , x ) = M x u ( t , x ) , (2.2.4) where y ( t , x ) i s the mxl measurement v e c t o r d i s t r i b u t e d over D and M i s a s p a t i a l mxn m a t r i x measurement operator over D. The d e t e r m i n i s t i c e s t i m a t i o n problem can now be s t a t e d as f o l l o w s : Given the system of eqns (2.2.1), (2.2.3) and (2.2.4), and the measure-ments { y ( T , X ) , t <x<t} and the c o n t r o l s { F , ( x , x ) , t„<x<t>, (J— - d U— — f i n d a l i n e a r estimate u ( t , x ) that approaches u ( t , x ) a s y m p t o t i c a l l y . 2.3 D e r i v a t i o n of the D e t e r m i n i s t i c F i l t e r In 2.3.1, the s t r u c t u r e of the l i n e a r d e t e r m i n i s t i c f i l t e r i s f i r s t l y d e r i v e d and i s shown to be a Kalman-type f i l t e r w i t h u n s p e c i f i e d d i s t r i b u t e d f i l t e r g a i n . The f o l l o w i n g 2.3.2 i s then.devoted to s e l e c t -i n g the f i l t e r gain such t h a t , c o n d i t i o n a l only on o b t a i n i n g the 10 u n i f o r m l y bounded p o s i t i v e - d e f i n i t e s o l u t i o n of a R i c c a t i - t y p e i n t e g r o -p a r t i a l d i f f e r e n t i a l equation, the estimate u ( t , x ) given by the d e t e r m i n i s t i c f i l t e r a s y m p t o t i c a l l y approaches the true s t a t e u ( t , x ) independent of the i n i t i a l e s t i m a t i o n e r r o r . 2.3.1 S t r u c t u r e of the D e t e r m i n i s t i c F i l t e r Consider the s t r u c t u r e of the f o l l o w i n g l i n e a r e stimator 8 Q ^ , X ) =K 1u(t,x)+/ D K 2 ( t , x , x ' ) y ( t , x ' ) d D x , + K 3 F d ( t B x ) , t>t , xeD, C2,3 5l) w i t h boundary c o n d i t i o n s & u(t,x)=0, xe3D, (2.3.2) where K^, K (t,x,x') and K 3 are f r e e design m a t r i x operators of dimensions nxn, nxm and nxp, r e s p e c t i v e l y , With the J _ r j j „ , u ( t , x ) = u ( t , x ) - Q ( t , x ) , (2,3,3) s u b t r a c t i n g the s t a t e estimate eqn, (2.3.1) from the s t a t e eqn. (2,2,1) y i e l d s the i n t e g r o - p a r t i a l d i f f e r e n t i a l equation f o r the e s t i m a t i o n e r r o r u ( t , x ) as 9 G3 ^ , X ) = A x u ( t , x ) - K 1 u ( t , x ) - / D K 2 ( t , x , x ' ) y ( t , x ' ) d D x , + ( B d ( t , x ) - K 3 ) F d ( t , x ) =K 1u(t,x) + ( - K 1 u ( t , x ) + A x u ( t , x ) - / D K 2 ( t , x s x l ) M x , u ( t , x ' ) d D x , ) + ( B d ( t , x ) - K )F ( t , x ) , xeD, (2.3,4) w i t h boundary c o n d i t i o n s B u(t,x)=0, xe8D, (2.3.5) where i n d e r i v i n g the l a s t step of eqn. (2.3.4) the measurement equation (2.2.4) i s used. The requirement that the n u l l s t a t e of the e s t i m a t i o n e r r o r equation (2.3.4) be the e q u i l i b r i u m s t a t e y i e l d s u(t,x)=0, 3Q(.t,x) and 3 t ' (2.3.6) Applying condition (2.3.6) to eqn. (2.3.4) y i e l d s the following r e l a t i o n s : K^=B J(t,x) (2.3.7) 3 d and, because i n general u ( t , x ) ^ 0 , -K.u(t,x)+A u ( t , x ) - / K (t,x,x')M ,u(t,x')dD ,=0. (2,3.8) JL X D Z. X X Hence, from eqn. (2.3 . 8 ) , the matrix s p a t i a l operator i s given by K l ( ' ) = A x ( ' ) _ / D K 2 ( t , x , x , ) M x , ( - ) d D x l . (2.3.9) U t i l i z i n g eqns. (2.3.7) and (2.3.9) i n eqn. ( 2 . 3 . 1 ) , we have 8 Q3( ^ X ) =A xu(t,x)+/ D K 2 ( t , x > x ' ) [ y ( t , x , ) - M x , a ( t , x , ) ] d D x , + B d ( t , x ) F d ( t , x ) , xeD, (2.3.10) with boundary condition 6 xu(t,x)=0, xe8D. (2.3.11) An examination of the estimator given by equation (2.3.11) reveals that i t s structure i s equivalent to that of the d i s t r i b u t i v e 20 12 15 Kalman f i l t e r ' ' but with as yet unspecified f i l t e r gain matrix K ( t , x , x ' ) . 2.3.2 Derivation of the D i s t r i b u t e d F i l t e r Gain S u b s t i t u t i n g eqns. (2.3.7) and (2.3.9) i n t o eqn. ( 2 . 3 . 4 ) , we have the estimation error equation as iHgj*). = A Q ( t,x ) -r K 9Ct,x,x')M u(t,x')dD , x e D , (2.3.12) d t X U Z. X X with boundary condition 3 u ( t , x)=0, xe9D. (2,3.13) Before proceeding f u r t h e r i n determining ( t , x , x ' ) , c onsider an i n t e g r a l transform of u ( t , x ) of the f o l l o w i n g form u a ( t , x ) = / D P a ( t > x , x , ) u ( t , x ' ) d D x , (2.3.14) where p^(t,x,x') i s the i n v e r s e of a d i s t r i b u t e d parameter m a t r i x + 13 P^(t,x,x') r e l a t e d through the i n t e g r a l equations of the forms / D P £(t,x,x")p£(t,x",x')dD x l I=l6(x~x'), x.x'eD, (2.3.15) and / D p J(t,x,x")P J l(t,x",x')dD x, l=l6(x-x'),'x.x'eD, (2.3.16) where I denotes the u n i t y m a t r i x and 6(«) the D i r a c d e l t a f u n c t i o n . By p r e m u l t i p l y i n g both s i d e s of eqn. (2.3.14) by P ^ ( t i x " i x ) and i n t e g r a t i n g w i t h respect to x, we have, a f t e r the a p p l i c a t i o n of eqn. (2.3.15), u ( t , x " ) which, w i t h the n o t a t i o n a l changes x"->x and x->x' , can be r e w r i t t e n as u ( t , x ) = / D P A ( t > x , x ' ) u £ ( t , x ' ) d D x , ( (2.3.17) Now p a r t i a l - d i f f e r e n t i a t i n g eqn. (2.3.17) w i t h respect to t and s u b s t i t u t i n g eqn. (2.3.12) f o r the l e f t hand s i d e , we have A x i i ( t , x ) -/ D K 2 ( t , x,x' )M x, u ( t , x') dD x, 8P (t,x,x') 9u ( t , x ) =/D -St V ^ ' ^ X ' + ' D P £ ( t , x , x ' ) 3 t dD x, (2.3.18) S u b s t i t u t i n g eqn. (2.3.17) i n t o the l e f t hand s i d e of eqn. (2.3.18) and re a r r a n g i n g , we have 13 3u Ct,x') ' D ^ C t , x , x ' ) dD x, 3P (t,x,x') _ /D T tT V t , x ' ) d D x , + /D ^ V ^ ' ^ V ^ ' ^ x ' "VD K 2 ( t ' X ' X ' ) M x ' P £ C t : ' X , ' X " : ) G J l ( t : ' X " ) d D x , d D x " (2.3.19) •j-By p r e m u l t i p l y l n g both s i d e s of eqn. (2.3.19) by P^(t,x"',x) and i n t e g r a t i n g w i t h respect to x, we have, a f t e r u t i l i z i n g eqn. (2.3.16), u^(t,x"') which, w i t h the n o t a t i o n a l changes x"'->-x, x-*x' , x'-*-x" and x" -J-x1" , can be r e w r i t t e n as Su (t , x ) 9P (t . x ' . x " ) -VD V ^ * ' * ' ) "hi — " £(t,x")dD x,dD x„ +VD pJ( t, x» x')A x,P J l(t,x',x , ,)u j l(t,x , ,)dD x tdD x„ "VD'D pJ(t,x,x')K 2(t,x ,,x")M x, IP £(t,x",x"')u J l(t,x ,")dD x ldD x I 1dD x, 1, (2,3.20) This then i s the equation s a t i s f i e d by the transformed f i l t e r i n g e r r o r u^(t,x) d e f i n e d by eqn. (2.3.14). We now proceed to consider the choices of K^O^Xjx') and P ( t , x , x ' ) . The d i s t r i b u t e d f i l t e r gain K ( t , x , x ' ) can be chosen t o ensure t h a t the e q u i l i b r i u m s o l u t i o n of the transformed s t a t e e s t i m a t i o n e r r o r equation (2.3.20) i s u n i f o r m l y a s y m p t o t i c a l l y s t a b l e i n the sense 31 4 of Lyapunov ' such t h a t the estimate u(t,x) of u(t,x) w i l l be 14 a s y m p t o t i c a l l y c o r r e c t r e g a r d l e s s of i n i t i a l s t a t e e s t i m a t i o n e r r o r . Consider the choice of a Lyapunov f u n c t i o n a l V ( t ) such that i t s d e r i v a t i v e w i t h respect to t along the t r a j e c t o r y u^Qt,x) i s given by V(t)= -/ D/ D u ^ t s x ) Q £ ( t , x , x ' ) u A ( t , x ' ) d D x d D x f (2.3.21) where Q^(t,x,x') i s an a r b i t r a r y s e l f - a d j o i n t p o s i t i v e - d e f i n i t e m a t r i x . We can assume th a t the s o l u t i o n to eqn. (2.3.21) i s given by V(t)= / D / D u ^ ( t , x ) P £ ( t , x , x ' ) u £ ( t , x ' ) d D x d D x , (2.3.22) where P (t,x,x') i s p r e s e n t l y to be determined. D i f f e r e n t i a t i n g eqn. (2.3.22) w i t h respect to t , we have 9u]"(t,x) V(T)= VD 9t P , ( t , x , x ' ) u £ ( t , x ' ) d D x d D x , 3P ( t . x . x 1 ) +VD V C' X ) — Tt A£ ( T' X' ) DV Dx-T 9u ( t , x ' ) +VD \ ( t , x ) P £ ( t , x , x > ) _ dD xdD x, • (2.3.23) U t i l i z i n g eqn. (2.3.20) i n the f i r s t term of eqn. (2.3.23) and ap p l y i n g eqn. (2.3.16) y i e l d 9flp(t,x) VD 9t P t ( t , x , x ' ) a l ( t l * 1 ) d D x d D x I =VD FL*(T'X) 9P (t x x') - \ t ' + P A ( t , x , x ' ) A x , - / D P J l ( t , x , x , , ) M ^ 1 1 K T ( t , x ' ,x")dDx„} • ^ ( t , x ' ) d D x d D x , (2.3.24) T It i s noted that , the transpose of the matrix s p a t i a l d i f f e r e n t i a l T T T T operator A , i s defined by the r e l a t i o n CA u) =u A such that A i s ob-X X X . X v i o u s l y an operator to the l e f t . Moreover, u t i l i z i n g eqn. C2.3.19) i n t h i r d term of eqn. (.2.3.23) and applying eqn. (2.3.15) y i e l d T 3u (t,x>) VD Vt,*)P A(t,x,x')—Tt D W VD a I ( t ' x ) 9P (t,x,x') ' { + A x P £ ( t ' X ' X ' ) _ / D K 2 ( t » x ' x M ) M x " P J l ( , : » X , , ' X , ) d D x " } •u £(t,x')dD xdD x, (2.3.25) Substituting eqns. (2.3.24) and (2.3.25) i n t o eqn. (2.3.23) and equating i t to eqn. (2.3.21), we have, a f t e r rearranging terms, 9P (t,x,x') T -Si = A x P 4 ( t , x , x ' ) + P £ ( t . x i x ' ) A x -/ D K 2(t,x,x")M x„P £(t,x ,*,x')dD x„ -; D P £(t,x,x")M^„K^(t,x',x")dD x„ +Q £(t,x,x') (2.3.26) 16 Now, s e l e c t the f i l t e r gain matrix K^Ct^x^x') such that K2Ct,x,x')= j / D P^Ct.x.x'OM^R^Ct.x'^x^dD^, (2.3.27) where R^Ct,x,x') i s a s e l f - a d j o i n t p o s i t i v e - d e f i n i t e matrix f o r a l l t,x and x' of i n t e r e s t . S u b s t i t u t i n g eqn. (2.3.27) into eqn. (2.3.26) f we have 3P Ct,x,x«) a~E = A^Ct.x.x^+P^Ct^.x^A^, - / D / D P £ (t . x . x ' ^ M ^ (t ,x" ,x" ' )M x„ ,P A (t ,x" ' ,x' )dD x„dD x„ , + QACt,x,x') . (2.3.28) The boundary condition of eqn. (2.3.28) can be obtained from eqns. (2.3.13) and (2.3.17) as g xP^(t,x,x')=0, xe3D, x'eD. (2.3.29) Sub s t i t u t i n g eqn. (2.3.27) in t o eqn. (2.3.10), the determinis-t i c f i l t e r i s therefore given as = A x u ( t , x ) + / D / D | p £ ( t , x , x " ) M ^ ( t , x " , x > ) •[y(t,x')-M x,u(t,x ,)]dD x,dD x„ + B d ( t , x ) F d ( t , x ) , xeD (2.3.30) with boundary condition given by eqn. (2.3.2) as 3 ti(t,x)=0, xe3D X (2.3.31) and an a r b i t r a r y i n i t i a l estimate QCt0,x)=u0Cx) . (2.3.32) 17 Hence, c o n d i t i o n a l only on obtaining the uniformly bounded 32 p o s i t i v e - d e f i n i t e s o l u t i o n of eqn. (2.3.28), assuming that an a r b i -t r a r y p o s i t i v e - d e f i n i t e , s e l f - a d j o i n t i n i t i a l P £Ct 0,x,x')=P £ 0(x,x') , x,x'eD ( 2 3 3 3 ) i s given, we can always f i n d a d i s t r i b u t e d f i l t e r gain ^ ( t p X . x ' ) as given by eqn. (2.3.27) so that the estimate u(t,x) given by eqn. (2.3.30) approaches the true state u(t,x) asymptotically independent of the i n i t i a l estimation, e r r o r . The r e l a t i o n between a uniformly bounded p o s i t i v e - d e f i n i t e s e l f - a d j o i n t s o l u t i o n of the R i c c a t i eqn. (2.3.28) and i n t r i n s i c con-t r o l system properties such as c o n t r o l l a b i l i t y and o b s e r v a b i l i t y turns out to be very d i f f i c u l t to e s t a b l i s h and i s i n f a c t a problem i n i t s e l f . We observe that the R i c c a t i eqn. (2.3.28) appears to be s i m i -l a r to the one usually encountered i n lumped parameter theory. In f a c t , the lumped parameter R i c c a t i equation i s i n t i m a t e l y r e l a t e d to the uniform asymptotic s t a b i l i t y of the lumped parameter f i l t e r . We there-fore n a t u r a l l y r a i s e the question as to whether c h a r a c t e r i z a t i o n s of the conditions for lumped parameter f i l t e r uniform asymptotic s t a b i l i t y have counter-parts i n d i s t r i b u t e d parameter theory. In lumped parameter estimation, Kalman has considered i n ^ the c o n d i t i o n a l mean (optimal) f i l t e r i n g problem f o r the system of a l i n e a r system and l i n e a r measurement equation under gaussian assumptions and has obtained a R i c c a t i equation which the symmetrical c o n d i t i o n a l f i l t e r i n g e r r o r covariance matrix s a t i s f i e d . He has also shown that, i f the two condi-tions of uniform complete c o n t r o l l a b i l i t y and uniform complete observa-b i l i t y characterized by the corresponding p o s i t i v e - d e f i n i t e matrices being uniformly bounded over any t i m e - i n t e r v a l are met by the lumped parameter system, the s o l u t i o n of the R i c c a t i equation i s uniformly bounded away from the n u l l matrix given an a r b i t r a r y p o s i t i v e - d e f i n i t e i n i t i a l c o n d i t i o n a l f i l t e r i n g e r r o r covariance matrix. Moreover, by using t h i s s o l u t i o n as the weight i n a quadratic function, he has 31 established that i t i s a Lyapunov function , that i s , i t i s uniformly bounded away from zero and decreases along the t r a j e c t o r y of motion with increase of time, f o r the uniform asymptotic s t a b i l i t y of the adjoint of the o r i g i n a l system and hence that of the system. The main point here of course i s that the existence of the uniformly bounded p o s i t i v e - d e f i n i t e s o l u t i o n of the R i c c a t i equation i n lumped parameter theory i s c o n d i t i o n a l upon the f a c t that the system e x h i b i t s i n t r i n s i c uniform complete c o n t r o l l a b i l i t y and o b s e r v a b i l i t y system p r o p e r t i e s . However, i n d i s t r i b u t e d parameter theory, the d e f i n i t i o n s of c o n t r o l l -4 a b i l i t y and o b s e r v a b i l i t y given by Wang have been found to be too r e s t r i c t i v e 34,5,1-1 therefore s t i l l are open questions where research i s c u r r e n t l y being done with regard to t h e i r d e f i n i t i o n s and character-i z a t i o n s . Hence, as to exactly what the s o l u t i o n of the R i c c a t i eqn. (2.3.28) i s to be r e l a t e d to becomes d i f f i c u l t to say, whence without r e s o r t i n g to r e f e r r i n g to c o n t r o l l a b i l i t y and o b s e r v a b i l i t y , we make the unconditional statement that, i f eqn. (2.3.28) on s o l v i n g y i e l d s a uniformly bounded p o s i t i v e - d e f i n i t e s o l u t i o n , then the f i l t e r of eqn. (2.3.30) e x h i b i t s the desired property of uniform asymptotic s t a b i l i t y . Furthermore, i n the numerical s o l u t i o n of d i s t r i b u t e d parameter problems, the p a r t i a l d i f f e r e n t i a l equations involved need to be approximated by one scheme or the other, whichever i s to be applied depending h e a v i l y on the problem type and the degree of accuracy required. We observe that though a uniformly bounded p o s i t i v e -d e f i n i t e s o l u t i o n of the R i c c a t i equation (2.3.28) i s d i f f i c u l t to 19 r e l a t e to some c o n t r o l system properties, the f i l t e r equation (2.3.30) and eqn. (2.3.28) together nevertheless o f f e r a common framework wit h i n which the goodness of various approximation schemes can be compared and, when employed i n a concrete case, t h e i r approximations can always be checked by standard lumped parameter procedure f o r s t a b i l i t y , boundedness and p o s i t i v e - d e f i n i t e n e s s . 2.4 Discrete-space Measurements In many p r a c t i c a l estimation problems of d i s t r i b u t e d parameter systems, measurements d i s t r i b u t e d a l l over the s p a t i a l domain may not be av a i l a b l e . Moreover, i t i s more r e a l i s t i c to consider the s i t u a t i o n where measuring devices are located at d i s c r e t e points i n the s p a t i a l domain. The p a r t i c u l a r state estimation problem wherein measurements , .1 - L - „ 1 —1 „ A n «-V.„ Cl.-t— llOOU.il.l^ t^ \-\J u C - V J^- J—uUu J->— (-1 i_ V^i.ju_^Jr ^ *" -| , , . . . , , w.A — s p a t i a l domain can be considered i n the following manner. Define f i r s t l y the measurement operator M^ to be m M (•) = /"„ E M(t,x,x.)6(x'-x.) (-)dD , (2.4.1) i = i J where M(t,x,x..) i s a d^ .xn matrix. S u b s t i t u t i n g eqn. (2.4.1) in t o the measurement equation (2.2.4) y i e l d s the d i s t r i b u t e d measurement vector y(t,x) due to a l i n e a r summation of the state response u(t,x_.), each weighted by the measurement matrix M(t,x,x^), at d i s c r e t e points x_. , j=l,...,m, i n the s p a t i a l domain to be m y(t,x)= I M(t,x,x.)u(t,x.) j = l 1 1 (2.4.2) Hence, from eqn. (2.4.2), the output y(t,x_.) of the measuring device located at the point x^ i n the s p a t i a l domain i s given by m y(t,x-i_)= £ M(t,x i ?x.)u(t,x.) j = l 3 2 Define, for not a t i o n a l convenience, the d-vector (2.4.3) n ( t ) = m y ( t , x m ) where d= Z , the p a r t i t i o n e d matrix i = l A M(t) = M(t,x 1,x 1); ' ;M(t,x 1 5X m) M(t,x . x j - • «M(t,x ,x ) m l . . ' m m C2.4.4) o f dimensions dxmn, and the mn-vector y(t ) = u C t j x D u(t,x ) m Augmenting y('t,x ) given by eqn. (2.4.3) f or i=l,2,...,m and using the d e f i n i t i o n s (2.4.4) for n ( t ) , M(t) and y ( t ) , we have n(t)=M(t)p(t) . (2.4.5) The state estimation problem i n t h i s case can therefore be stated as follows: Given the system of eqns. (2.2.1), (2.2.3) and (2.4.5), and the measure-ments {n(T), t^<T<t} and the. controls {F^(j,xYt t^<;T<t}, determine a l i n e a r estimate u(t,x) that approaches u(t,x) asymptotically. The measurement r e s i d u a l i n the f i l t e r equation (2,3.30) must now be weighted only at the d i s c r e t e measuring lo c a t i o n s x^, j=l,2,.,,,m, such that we set m R £ ( t , x , x ' > Z R^(t,x.,x )6(x-x.)6(x'-x ) (2.4.6) where R„(t,x.,x.) i s a d.xd. symmetric p o s i t i v e - d e f i n i t e matrix. Define 21 fo r n o t a t i o n a l convenience, the p a r t i t i o n e d matrix A Vt>= R £ ( t , x 1 , x 1 ) ; . i / t , ^ , ^ ) (2.4.7) which i s of dimensions dxd. Let us focus our a t t e n t i o n on the f i l t e r equation (2.3.30) T f i r s t . S u b s t i t u t i n g eqn. (2.4.1), r e c a l l i n g that M i s an operator to the l e f t , and eqn. (2.4.6) into the second term of the d e t e r m i n i s t i c f i l t e r eqn. (2.3.30) y i e l d s ~ / D / D P j l(t 5x,x")M x, 1R £(t,x , ,,x')M x,u(t,x ,)dD v,dD v l, X X m 5VD \ ^ a ) x I x . ) M T ( t ) x " ) x i ) i = l m • E R ^ ( t , x a , x b ) 6 ( x " ~ x a ) 5 ( x ' - x b ) a,b=l (2.4.8) m E M(t,x',x )u (t,x )dD ,dD „ m m =| E P £ ( t , x , x . ) [ E M T ( t , x a , x . ) R J l ( t , x a , x b ) M ( t , x b , x j ) ] u ( t , x j ) i , j = l a,b=l Define now for n o t a t i o n a l convenience the p a r t i t i o n e d matrix to be P,(t,x,x m) ] (2.4.9) which i s of dimension nxmn. Using t h i s matrix /ir (t,x) » we can write eqn. (2.4.8) as IVD P £ ( t , x , x " ) M ^ ( t , x " , x ' ) M x I u ( t , x - ) d D x , d D x l t (2.4.10) =|TT % (t,x)M T (t)R^ (t )M(t) y (t) 22 since the term wit h i n the square brackets i n eqn. (2.4.8) i s the i j - t h T submatrix of the matrix product M ( t ) R £ ( t ) M ( t ) . Moreover, from the d e f i n i t i o n (2.3.3) of estimation e r r o r u(t,x) ^ t n e d e f i n i t i o n (2.4.4) of y(t) and the measurement eqn. (2.4.5), we have M(t)y(t)=n(t)-M(t)y(t) (2.4.11) where tf(t) i s the estimate of P ( t ) . I t i s seen that eqn. (2.4.11) i s i n f a c t the measurement r e s i d u a l . S u b s t i t u t i n g eqns. (2.4.10) and (2.4.11) i n t o the d e t e r m i n i s t i c f i l t e r eqn. (2.3.30), we have 3 \ ( t , x ) = A u(t ,x)+K (t ,x) [ n(t)-M(t)(i(t) ] d t X JL +B ( t , x ) F d ( t , x ) , X E D , (2.4.12) with boundary and i n i t i a l conditions given by eqn. (2.3.31 and (2.3.32) r e s p e c t i v e l y , B xu(t,x)=0 , xe9D, u(t 0,x)=u 0(x) , xeD, (2.4.13) where i n eqn. (2.4.12) ^ ( t j x ) i s now the d i s t r i b u t e d f i l t e r gain given by K 2 ( t , x ) = i 1 T J l ( t , x ) M T ( t ) R £ ( t ) . (2.4.14) Let us now turn our a t t e n t i o n to the non-linear term i n the R i c c a t i eqn. (2.3.28). Su b s t i t u t i n g eqns. (2.4.1) and (2.4.6) i n t o the non-linear term of eqn. (2.3.28), we have / D / D P £ ( t .x.x-^.R^Ct ,x",x" • )M x„ , P A ( t ,x" ' ,x' )dD x I 1dD x„ , m T =VD e p £ ( t » x > x 1 ) M Ct,x", X i) i = l m • Z R 0Ct,x ,x )SCx"-x )S(x'"-x, ) , . Jc a D a b a,b=l m Z M(t,x ,",x.)P„(t,x.,x')dD „dD „, (2.4.15) 1=1 m m „ = Z P Ct.x.x^I Z M ( t , x a , x 1 ) R £ ( t 5 x a , x b ) M ( t , x b , x ) ] P £ ( t , x ,x') i , j = l a,b=l Again, using the d e f i n i t i o n (2.4.9) f o r TT^(t,x) i n eqn. (2.4.15), and recognizing that the term i n the square brackets i s the i j - t h submatrix T of the matrix product M (t)R^(t)M(t) , we have VD P£(t >X,X")MXIIV<= , X " , X " ' ) M X " . p £ ( t ' ,x' )dD x I IdD x„, = T i j i(t,x)M T(t)R J i(t)M(t)^(t,x') (2.4.16) Substituting eqn. (2.4.16) in t o the R i c c a t i eqn. (2.3.18) y i e l d s the p a r t i a l d i f f e r e n t i a l equation f o r the kernel P„(t,x,x') as A , 9P (t,x,x*) . — B i = A xP 4(t,x,x')+P £(t,x,x')A x I -TT £(t,x)M T(t)R £(t)M(t)TT^(t,x') +Q £(t,x,x') . (2.4.17) The i n i t i a l and boundary conditions remain given by eqns. (2.3.29) and (2.3.33), r e s p e c t i v e l y , such that P (t x,x')=P (x.x 1) , x,x'eD, £ 0 10 (2.4.18) and ^ P ^ t . x . x ' ^ O , xe9D, x'eD. (2.4.19) 24 2.5 Feedback Considerations With the stable state estimator constructed f o r the l i n e a r d i s t r i b u t e d parameter system to give an. estimate of i t s state, the next question that needs to be answered i s what the e f f e c t on the system, e s p e c i a l l y with regard to i t s closed-loop s t a b i l i t y p r o p e r t i e s , using t h i s state estimate i n the place of the true state i n r e a l i z i n g a st a t e feedback c o n t r o l scheme i s . I t i s pointed out below that the state e s t i -mate used i n place of the state i n the feedback c o n t r o l law does not a l t e r the closed-loop s t a b i l i t y p r o p e r t i e s . The l i n e a r state feedback c o n t r o l law using the true state u(t,x) may be expressed as F d ( t , x ) = / D K c(t,x,x')u(t,x')dD x, (2.5.1) W l l C J - C i K ; X C J c t i i O.X U±. C i-Ot Xy C O U L I U J . ^ U J L U I U C L L. JL . U U U O L -i- C <J-t- J-l.i.£3 c eqn. (2.5.1) in t o the state equation (2.2.1), we have ^ 0 „ V ( t > x ) + / D K d(t,x,x')u(t,x')dD x, , ( 2 5 2 ) where we define f o r convenience K d(t,x,x') as K,(t,x,x')=B (t,x)K (t,x,x') . d d c ( 2 5 3 ) I f the estimate u(t,x) output from the estimator eqn. (2.3.30) or eqn. (2.4.12) i s used i n the place of the true state u ( t , x ) , then the co n t r o l law eqn. (2.5.1) becomes F d ( t ' x ) = / D K c(t,x,x')u(t,x')dD x, . ( 2 > 5 > 4 ) Hence, the state equation (2.2.1) using the co n t r o l law eqn. (2.5.4) becomes 3 U 3 t ? X ) = A x u ( t ' x ) + / D K d ( t > x > x ' ) u ( t , x ' ) d D x , =A xu(t,x)+/ D K d ( t , x , x ' ) u ( t , x ' ) d D x , - ; D K d(t,x,x')u(t,x')dD x, , (2.5.5) where u(t,x) i s the state estimation e r r o r defined by eqn. (2.3.3) and s a t i s f i e s , from eqn. (2.3.12), ^ B A j t u ( t , x ) - / D K 2 ( t , x > x ' ) M x , u ( t , x - ) d D x , . ( 2 > 5 > 6 ) Eqns. (2.5.5) and (2.5.6) together can be considered to make up the set of state equations f o r the o v e r a l l system of closed-loop estimation and c o n t r o l . Now, whether we can f i n d a c o n t r o l law of the form of eqn. (2.5.1) such that the c o n t r o l l e d system eqn. (2.5.2) i s uniformly asymptotically stable i n the sense of Lyapunov i s , i n general, a d i f f i c u l t question to answer. However, i t has been shown that i n c e r t a i n cases, 33 there can be found controls to insure the uniform asymptotical s t a b i l -i t y of the d i s t r i b u t e d parameter system. Moreover, we are to consider i n t h i s s e ction the e f f e c t on closed-loop s t a b i l i t y of using the s t a t e estimate i n the place of the true state by i n c l u d i n g the estimator i n the loop. Therefore, we can assume that the c o n t r o l l e d system eqn. (2.5.2) i s uniformly asymptotically stable i n the f i r s t place. I t i s seen that, since K2(t,x,x') i n eqn. (2.5.6) i s so chosen as to insure the uniform asymptotic s t a b i l i t y of the e q u i l i b r i u m of i i ( t , x ) , the state response of eqn. (2.5.5) i s but that of a uniform asymptotic s t a b l e free system excited by the t r a n sient estimation e r r o r u ( t , x ) . The s t a b i l i t y proper-t i e s of the system are therefore not disturbed by i n c l u s i o n of the state estimator as compared with state feedback c o n t r o l . 2.6 Numerical Example 2.6.1 Problem Formulation Consider the process of one-sided heating of metal which r e -s u l t s i n a 1-dimensional heat-conduction system described by the 26 d i f f u s i o n equation 9u A(t,x) 9 2 u A ( t , x ) 3 t = 9 x 2 _ (2.6.1) where u A ( t , x ) , as a function of the time v a r i a b l e t>0 and the s p a t i a l coordinate x, 0<x <1 , represents the temperature d i s t r i b u t i o n of the metal slab . The s p a t i a l coordinate x i s normalized with respect to the thickness of the slab and t i s normalized such that the c o e f f i c i e n t corresponding to thermal d i f f u s i v i t y i s unity. The i n i t i a l and boundary conditions are given as u A(0,x)=0, (2.6.2) 3u A(t,0) 3 x u A ( t , x ) ] = — ^ = 0 , (2.6.3) x=0 3 x u A ( t , x ) ] = 3 u A ( t , l ) 4 — = S f * ( t ) , (2.6.4) x=l where 3 i s the heat trans f e r c o e f f i c i e n t , assumed''to be a constant, and f A ( t ) i s a manipulable external temperature source. The boundary cond i t i o n eqn. (2.6.4) i s but a restatement of Newton's law of cooling, that i s , the temperature gradient at the surface x=l i s p r o p o r t i o n a l to the temperature di f f e r e n c e between the surface x=l and the ambient temperature. And the boundary condition eqn. (2.6.3) shows that the other surface x=0 has i t s temperature gradient equal zero, that i s , i t i s i n s u l a t e d . The system i s assumed to be observed at the s i n g l e point x=0.5 i n the s p a t i a l domain by the measurement equation, corresponding to eqn. (2.4.3) with m=l and M(t,0.5,0.5)=1, y A(t,0.5)=u A(t,0.5) . (2.6.5) 27 I t i s now required to obtain the optimal c o n t r o l f ^ ( t ) such that the cost f u n c t i o n a l Co 1 i n J= / / C u*(t,x ) - u o o) 2dxdt+B/ ( f A ( t ) - f s g ) 2 d t 0 0 " A A 0 " B s (2.6.6) i s minimized, where B i s a weighting c o e f f i c i e n t and f i s the required steady-state input to the system to maintain the prescribed e q u i l i b r i u m state u of the state u ( t , x ) . The cost f u n c t i o n a l weighs both the temp-erature d e v i a t i o n and excessive input. This problem i s then seen to be a regulation problem for a class of d i s t r i b u t e d parameter systems. For the equilibrium temperature condition u , i t follows from s s eqn. (2.6.4) that the steady-state value of the input f A ( t ) i s given by f =u SS SS Consider now the transformations u(t,x)=u. (t,x)-u ss f ( t ) = f + C t ) - f ss (2.6.7) (2.6.8) (2.6.9) Substituting eqn. (2.6.8) into eqns. (2.6.1), (2.6.2), (2.6.3) and, together with eqn. (2.6.9), into eqn. (2.6.4), we have 3u(t,x)^3 2u(t,x) 3t 3x 2 u(0,x)=-u g s , 3u(t,0) 3 x u ( t ' x ) ] n 3x x=0 =0 , B„u(t,x)] =gu(t,l)+ 8 U 8 ( ^ 1 ) =Bf(t) x=l (2.6.10) 28 34 Now, using Brogan's concept of extended operators i n order to convert the inhomogeneous boundary condition at x=l i n eqn. (2.6.10) into the homogeneous one, we have the co n t r o l f ( t ) now appearing i n the state equation such that eqn.(2.6.10) becomes " " ^ + 6 ( x - l ) g f ( t ) 3t 3x 2 3u(t,x) _ 3*-u(t,x) u(0,x)=-u (2.6.11) ss x=0 B u ( t , x ) ] =6u(t,l) + 3 " C t ' i : > = 0 . X x=l d x From eqn. (2.6.5), we can define a new measurement equation using eqn. (2.6.8) as y(t,0.5)=y A(t,0.5)-u s s=u(t,0.5) . (2.6.12) Further, s u b s t i t u t i n g eqns. (2.6.8) and (2.6.9) in t o eqn. (2.6.6), we have the cost f u n c t i o n a l expressed i n terms of the new v a r i a b l e s as CO ^ 0 OO o 3=1 f u 2(t,x)dxdt+B/ f 2 ( t ) d t 0 0 0 (2.6.13) The o r i g i n a l optimal c o n t r o l problem characterized by eqns. (2.6.1), (2.6.2), (2.6.3), (2.6.4) and (2.6.6) has now been cast i n t o the equivalent one s p e c i f i e d by eqns. (2.6.11) and (2.6.13), which, by the very nature of the c o n t r o l term entering into the state equation i n eqn. (2.6.11), i s seen to be but an optimal pointwise r e g u l a t i o n c o n t r o l pro-blem and can therefore be solved by s p e c i a l i z i n g the general r e s u l t s presented i n Chapter 5. I t i s noted i n passing that t h i s p a r t i c u l a r 1-dimensional c o n t r o l problem characterized by eqns. (2.6.10) and (2.6.13) can also be viewed as a boundary c o n t r o l problem which can be solved by 35 the method presented i n r e s u l t i n g n a t u r a l l y i n the same c o n t r o l pre-29 sented below. However, the equivalence between pointwise c o n t r o l and boundary con t r o l i s po s s i b l e only for l^dimensional cases because of the fact that the boundary consists of but two points, e.g., the boundary 3D of t h i s problem i s 3D={x: x=0, x=l}. The optimal control f°(t) for the problem defined by eqns. (2.6.11) and 2.6.13) i s , from Chapter 5, given by f°(t)==- | / s ( x , l ) u ( t , x ) d x , B 0 (2.6.14) where s ( x , l ) i s the s o l u t i o n s(x,x') at x'=l of the steady-state R i c c a t i equation As(x,x')- £ s(x,l)s(l,x ' ) + 6(x-x')=0 , and A i s the two-dimensional Laplacian. The boundary conditions of eqn. (2.6.15) are given as (2.6.15) . , 3s(0.x') p a »,x,x ) } - ^ -u x=0 x B xs(x,x')] =gs(l,x')+ 3 % ( ^ X , ) =0 x=l Lfor a l l x' (2.6.16) and B,s(x,x')] - ^ z O i - o > X =0 fo r a l l x (2.6.17) 6 ,s(x,x')] =3s(x,l)+ 3 S ^ ; 1 } =0 X x'=l X The associated estimation problem f o r the system defined by eqns. (2.6.11) and (2.6.12) i s solved by using the r e s u l t s of t h i s chapter, i n p a r t i c u l a r 2.4. The estimator f o r the system of eqns. (2.6.11) and (2.6.12) i s , from eqn. (2.4.12), therefore gug^O . 3 2u(t,x) + R ( t , x ) [ y ( t i 0 . 5 ) - f l ( t , 0 . 5 ) ] 3t (2.6.18) +6(x-l)Bf°(t) , 30 where f°(t) i s the optimal c o n t r o l using the estimate u(t,x) i n the place of u ( t , x ) . The i n i t i a l and boundary conditions f o r the estimator eqn. (2.6.18) are, r e s p e c t i v e l y , u(0,x)=-u ss (2.6.19) and D M 3u(t,0) _n B xu(t,x)] -0 , x ~ ° (2.6.20) 3 xu(t,x)] = (3u(t,l) + - ^ ( ~ 1 1 =0 • x=l 92 Now, because the s p a t i a l operator A.= — of the state equation X 3x2 and i n eqn. (2.6.11) i s time-invariant, we can choose P (t,x,x') Q^(t,x,x') i n eqns. (2.3.22) and (2.3.21) to be time-invariant such that the R i c c a t i equation (2.4.17) reduces to the steady-state one. Let us denote the time-invariant P (t,x,x') and Q^(t,x,x') by p^(x,x') and q^(x,x') re s p e c t i v e l y . Moreover, R f l(t) i n eqn. (2.4.14) can be chosen to be a constant r such that the f i l t e r gain K (t,x) i s now time-variant. The f i l t e r gain K (t,x) i n eqn. (2.6.18) i s , from eqn. (2.4.14), given by K 2(t,x)= - 2-p^(x,0.5)r J L , (2.6.21) where p (x,.5) i s the s o l u t i o n p (x,x') at x=0.5 of the steady-state R i c c a t i equation with the boundary conditions Ap^(x,x')-r 0p 0(x,.5)p 0(.5,x')+q 5(x,x')=0 , (2.6.22) and B X P £ ( X ' X ' ) ] x=0 e xp £(x,x')] x=l 3 ,p (x,x')] x * x'=0 B .P (x,x')] X * x ' = l 3p £(0 , x ' ) 3x =0 3p ? ( l , x ' ) Bp £(l,x')+ - A =0 3x f o r a l l (2,6.23) 3p £(x,0) -0 • ap.Cx.i) 3p.(x,l)+ — i =0 * 3x' for a l l (2.6.24) x . 31 Eqns. (2.6.15) and (2.6.22), with t h e i r associated respective boundary conditions, are steady-state R i c c a t i equations, the former for the pointwise c o n t r o l and the l a t t e r f o r the gain of the f i l t e r using the discrete-space measurement obtained at one s i n g l e measurement l o c a t i o n . The R i c c a t i equations must therefore be solved i n order to implement the optimal c o n t r o l eqn. (2.6.14) and the d e t e r m i n i s t i c f i l t e r eqn. (2.6.18) 2.6.2 Solution of the R i c c a t i Equations It i s known that i n a regular region D the operator A x ^ 2 oo has a complete set of orthonormal eigenfunctions {<f>_^ } ._^ and an associated set of eigenvalues { A } , . Moreover, the eigenvalue A i s negative-l i = l i d e f i n i t e and decreases without bound as i - x o . In t h i s case, we have ( v. Appendix I ) , by considering the eigenvalue-egenfunction problem associated with eqn. (2.6.11), the eigenvalues A.=-p? , (2.6.25) x r x where p^ are the roots of the transcendental equation p. tg p.= 6 , (2.6.26) x x the f i r s t f i v e roots of which, with g»l, are shown i n Table (2.1). The transcendental equation (2.6.26) moreover shows that f o r large values of p^, the sine of the angle approaches zero so that we have approximately, for large i , p^=(i-l)ir ". Besides, we have the eigenf unction where <(>i(x)=/g(pi) cos p ±x , (2.6.27) 2(1+P 2) g(P-)= • (2.6.28) 2+p| 32 The f u n c t i o n a l equation (2.6.15) with boundary conditions (2.6.16) and (2.6.17) can be solved by expanding s(x,x') i n a CO convergent double s e r i e s of ^i( x)'-{=]_ o f t h e f o r m where s(x,x')= E s . .()>. (x)(j>. (x) , i , j = l 1 1 i j x s..=// <f>. (x)s(x5x')c}>. (x')dxdx' 4 0 0 1 J (2.6.29) (2.6.30) A , 1 2 3 4 5 0.8603 3.4256 6.4373 9.5293 12.6453 -0.7402 -11.7350 -41.4388 -90.8082 -159.9032 Table (2.1) F i r s t f i v e roots of p ± tg p ± =1. I f the f i r s t M terms are retained i n the s e r i e s eqn. (2.6.29), eqn. (2.6.15) reduces to the following set of coupled, non-linear algebraic lumped parameter R i c c a t i equation i n v o l v i n g the c o e f f i c i e n t s ' i j ' where AS+SA- g S$ M(1)S+I M=0 , S11*" S1M G • • • O Ml MM A=diag[A 1 • • A^J , $^(1) i s the value at x=l of $ M 0 0 defined by <(>1(x)4'1(x) ••• (|)1(x)4)M(x) + M(x)4» 1(x)"» <f>M(x)(J)M(x) (2.6.31) (2.6.32) (2.6.33) (2.6.34) 37 and 1^ i s the i d e n t i t y matrix of order M. Eqn. (2.6.31) i s solved by using Bingulac's method ~" and the r e s u l t s for M=3, 3=1 and B=0.1 are shown i n Table J(2.2). The f u n c t i o n a l equation (2.6.22) w i l l be solved f o r the case CO when q^(x,x') can be expanded i n a convergent double s e r i e s i n {<J>^ (x)} of the form ' q £(x,X<)= _ E q M j * . ( x ) ^ ( x ' ) , 1 > 3 I where the c o e f f i c i e n t q .. i s given by the formula 3 1 1 q.,,=/ / T (x)q.(x,x')(|> (x')dxdx' X , 1 J 0 0 1 ^ J (2.6.35) (2.6.36) 0.326398 0.008714 -0.000852 0.008714 0.041633 0.000142 -0.000852 0.000142 0.012036 Table (2.2) Solution of eqn. (2.6.31) with M=3, g=l and B =0.1 Then, eqn. (2.6.22) with the boundary conditions (2.6.23) and (2.6.24) can be solved by, again, expanding p^(x,x') i n the double s e r i e s where P £(x,x')= _ i P ^ C X H J C X ' ) , P m E s / 1 / 1 *,(x)p p(x,x')<j) (x')dxdx' . (2.6.37) (2.6.38) I f the f i r s t M terms are retained i n eqns. (2.6.35) and (2.6.37), eqn. (2.6.22) reduces to the following a l g e b r a i c R i c c a t i equation: A P*. +V-V*V°- 5 ) P£ + V ° » (2.6.39) 34 where and P£ll*"° P£1M q A l l * *" qUM q£Ml" " q £ M M (2.6.40) (2.6.41) Eqn. (2.6.39) i s again solved by using Bingulac's method and the r e s u l t s with M=3, 3=1 for (1) r £ = l , ', (2) r £=10, Q £=I 3 and (3) vf:10' Q £ = 1 0 I 3 a r e s h o w n i n T a b l e (2.3) (1), (2) and (3), r e s p e c t i v e l y . 0.499156 0.000323 0.000201 0.000323 0.042605 -0.000002 Table (2.3) (1) M=3, 0=1, r =1 and Q = I, 0.247330 0.001368 0.000943 0.001368 0.042588 -0.000020 Table (2.3) (2) M=3, 3=1, r £=10 and Q £ = I 3 0.000201 -0.000002 0.012063 0.000943 -0.000020 0.012036 0.951514 0.033898 0.029578 0.033898 0.425191 -0.001155 0.029578 -0.001155 0.118474 Table (2.3) (3) M=3, 3=1, r £=10 and Q £ = 1013 Table (2.3) Solution of eqn. (2.6.39) 35 2.6.3 Simulation Study The tracking a b i l i t y of the d e t e r m i n i s t i c f i l t e r developed needs to be studied and we approach t h i s problem i n two steps. In 2.6.3.1, the given d i s t r i b u t e d parameter system alone i s simulated and i t s response obtained. In 2.6.3.2, the d e t e r m i n i s t i c f i l t e r i s cas-caded to the simulated system and the f i l t e r response i s obtained. The f i l t e r response i s then compared with the system response. In both cases, the simulation i s c a r r i e d out on the d i g i t a l computer IBM 360/67 using Continuous System Modelling Programme (CSMP). 2.6.3.1 System Response For the purpose of the d i g i t a l computer simulation study of the given d i s t r i b u t e d parameter system using CSMP, the following 38 discrete-space continuous-time (DSCT) scheme i s adopted. The c e n t r a l d i f f e r e n c e formula 3 2u(t,x) u i + 1 ( t ) - 2 u ± ( t ) + u i _ 1 ( t ) 3x 2 (Ax) 2 (2.6.42) i s applied i n the domain 0 < x < 1, the forward d i f f e r e n c e formula 3u(t,x) u ( t ) - u (t) B u C t . x ) ] = = X x=0 3x Ax (2.6.43) i s applied to the x=0 boundary, but the condition at the x=l boundary i s . „ . , 39, p.274 approximated by u.,, +u. u. n-u. D • . \ - i „ l + l l , i + l l 3 u(t,x)] =3 5 + — r - , x x=l 2 A x (2.6.44) where u i + 1(t)=u(t,x+Ax) , U i ( t ) = u ( t , x ) , (2.6.45) and u ±_ 1(t)=u(t,x-Ax) 36 By choosing Ax=0.1, there r e s u l t s a t o t a l of 11 ordinary d i f f e r e n t i a l equations representing the system eqn. (2.6.11) such that we have d u x ( t ) u 2 ( t ) - u 1 ( t ) dt (Ax)' du.(t) u (t)-2u (t)+u (t) X l - r l X X—1 , i=2,...,10, dt (Ax) 2 ' ' ' (2.6.46) du, (t) (2+33Ax) 1 2g and = u (t)+ u l n ( t ) + f ( t ) • dt (2+3Ax)(Ax) 2 x (Ax) 2 (2+3Ax)Ax Even though CSMP, when used, automatically d i s c r e t i z e s the time v a r i a b l e such that the DSCT scheme i s i n a c t u a l i t y a discrete-space discrete-time one, nevertheless, we can s t i l l conceptually look at the problem as i f only the s p a t i a l v a r i a b l e i s d i s c r e t i z e d and the time v a r i a b l e not. Since both s p a t i a l and time v a r i a b l e s are u l t i m a t e l y d i s -40 c r e t i z e d , the condition At (2.6.47) < 0.5 . (Ax) 2 that guarantees s t a b i l i t y of the numerical scheme must be observed and, therefore, with Ax set to be 0.1, we choose At=0.002. With the a v a i l a b i l i t y of the above simulation scheme, the set-up of which i n block diagram i s shown i n F i g . (2.1), the output (temperature) u(t,x) of the system eqn. (2.6.11), with u =1, 3=1 and B=0.1, i s obtained at x=0.,0.5,1.0 corresponding to u^(t) , u ^ ( t ) , u^Ct) and the o r i g i n a l u A(t,x)=u(t,x)+u s g at these values of x are displayed i n F i g . (2.2). ^ n e optimal control function f°(t) given by. eqn. (2.6.14) i s obtained by i n t e g r a t i o n by Simpson's r u l e using with. x'=l the s(x,x') obtained, using the f i r s t M=3 terms, according to eqn. (2.6.29) and the d i s c r e t e space components u ^ ( t ) , 1=1,2,...,11, of u ( t , x ) . The o r i g i n a l c o n t r o l function f°(t)=f°(t)+f i s displayed i n F i g . (2.3). The cost f u n c t i o n a l J given by eqn. (2.6.13) i s c a l c u l a t e d again using Simpson's rul e based on the u(t,x) and f°(t) obtained above and i s shown as a f u n c t i o n of time i n F i g . (2.4). C O i l U i u i gaxn eqn. (2.6.14) f°(t) d i s t r i b u t e d parameter system eqn. (2.6.10) u(t,x) > F i g . (2.1) Block-diagram for 2.6.3.1. u, v(t,l.) t F i g . (2.2) Temperature p r o f i l e u^(t,x) at x=0., 0.5, 1. t -»• F i g . C2.4) Cost functional J . o 41 2.6.3.2 . Estimator Response The d i s t r i b u t e d parameter system eqn. C2.6.11) i s simulated'by 11 ordinary d i f f e r e n t i a l equations obtained by d i s c r e t i z i n g the s p a t i a l domain, 0<x<l, into 10 equal i n t e r v a l s , the simulation scheme being d e t a i l e d i n 2.6.3.1. Cascaded to the system i s the d i s t r i b u t e d parameter estimator eqn. (2.6.18) f o r which the following approximation scheme i s adopted. CO We expand the estimate u(t,x) i n the s e r i e s of (<}>^ (x) °f t n e f orm u(t,x) = Z u(t)(f> (x) , 1=1 (2.6.48) where u 1 ( t ) = / u(t,x)c|>. (x)dx 0 1 (2.6.49) If the f i r s t M terms are retained i n the expansion eqn. (2.6.48), we have, from eqn. (2.6.18), the following vector-matrix ordinary d i f f e r -e n t i a l equation for the c o e f f i c i e n t s u^Ct) , i = l , 2 , . ... ,M, u(t) = Au(t) + K 2ty(t,0.5) - **(0.5)u(t)] + * i r(1.0)3f°(t) , (2.6.50) v where u(t) = u X ( t ) u (t) (2.6.51) A K2 = r K 1  K2 4 C2.6.52) i d = / K0(t,x)cj> (x)dx, i=l,2,...,M, 2 o 1 x (2.6.53) and $ v(0.5) and $ v(1.0) are the values at x=0.5 and x=1.0, r e s p e c t i v e l y , 42 of the vector function $ (x) defined by v .(x) GO C2 .6 .54) From eqn. (2.6.21) f o r the d i s t r i b u t e d f i l t e r gain K ^ ( t , x ) , i t i s not d i f f i c u l t to see that the f i l t e r gain K 2 f o r the lumped parameter f i l t e r eqn. (2.6.50) i s given as K2 -\ W ° ' 5 ) r * • (2'6-55) The f o r c i n g function f°(t) i n t h i s approximation scheme i s then, from eqn. (2.6.14), given by f°(t) = - i e$ (l.O)Su(t) . (2.6.56) D V The i n i t i a l conditions for the lumped parameter f i l t e r eqn. (2.6.50) are from eqns. (2.6.19) and (2.6.49) given as u(0) = / -u 0 (x)dx . 0 ss v (2.6.5.7) There are two inputs to the estimator eqn. (2.6.50) with M=3, 3=1 and B=0.1, v i z . , the measurement r e s i d u a l at x=0.5, i . e . , y(t,0.5)-u(t,0.5), generated on-line and the f o r c i n g function f°(t) given by eqn. (2.6.56). The set-up i n block diagram i s shown i n F i g . (2.5). F i g . (2.6) shows the measure-ment r e s i d u a l y(t,0.5)-u(t,0.5) at x=0.5 input to the estimator eqn. (2.6.50) for the three cases of (1) r =1, Q ^ y (2) rj,= 1 0> Q £ = I 3 a n d O) r£=10> Q =10I 3- F i g . (2.7) (a), (b), (c) show at x=0., 0.5, 1., r e s p e c t i v e l y , the d i f f e r e n c e between the estimated response (t ,x)=tl (t ,x)+u g s and the a c t u a l system response u A ( t , x ) obtained i n 2.6.3.1 f o r the same three sets of para-meters c i t e d above. We see from F i g . (2.7) that the d i f f e r e n c e u ^ ( t , x ) - u A ( t , x ) does approach zero asymptotically as t increases, that i s , tiA(t,x) approaches, f ° ( t ) u ( t , x ) d i s t r i b u t e d parameter system eqn. (2.6.10) -A measuring device 1 / <5(x-x.) (-)dx 0 1 + f i l t e r gain eqn. (2.6.21) u C t ^ . ) control gain eqn. (2.6.14) deterministic f i l t e r eqn. (2.6.18) Fi g . C2.5) Block diagram for 2.6.3.2. y(t,o.5)-u(t,0.5) o.oos-h -O.OOO---O.OOS-J --0.010-^ \ -0.035-j V \ -0.020-^ \ -0.02S-^ -0.030-^ -0.035-J -0.04] --0.045-t -> -I 1 1 1 1 ! T--T 0.5 1 I ' ' 1.0 — i — i — i — i — i — i — i — i — i — i — i — i — i — i — ' — 1 i I 1.5 2.0 ..-.«-- C3) ^ C2) CD \ ^ -\ \ F i g . (2.6) Measurement re s i d u a l yCt ,0.5)-ti(t ,0.5) . a. (t,o.)-u. (t,o.) 0.08 -0.02 F i g . (2.7) (a) Response difference u,,(t,0.)-uA(t,0.) u J t Ct,.5)-u i k Ct,.5) 0.030-0.025H 0.020H 0.0I5H 0.010 0.005H / / •v. "s. (1) w i ? - j i •\ ii \ 1) •#r-i—i—i—i—i—i—i—i—I—'—i—TXnr—i I r tot °-5 -0.005--0.010-1 (2) (3) - i — i — i — i — i — i " r . j i — i — I — i — i — i — i — " — : — i 1 1 \ V t F i g . (2.7) (b) Response difference u^(t, .5)-u j V(t, .5) u ^ ( t , l . ) - u A ( t , l . ) + O.lO-i 0.08-0.05-1 0.04-4 4 -D.OO-fp-r-r--0.02--0.04--0.06--0.08--0.10-t -> -i—i—i—r-0.5 -i—i—i—i—i— i—|— i—i—i—I— i—i—i—'—r 1.0 -i—i—i—i—i—r-,J_g_ -=.-:..=:.==-S!.*=,«J1' - C D "(2) (3) F i g . (2.7) (c) Response difference u A(t,1.)-u^(t,1.) F i g . (2.7) Response di f f e r e n c e u A ( t ,x)-u^(t ,x) at x=0., .5, 1. 48 as required by theory, u^(t,x) asymptotically as t increases. F i g . (2.8) o ^ o ^ o shows the d i f f e r e n c e between the c o n t r o l function f,(t)=f (t)+f with, f (t) x ss using the estimated state u('t,x) and f * ( t ) obtained i n 2.6.3.1 f o r the same three sets of parameters c i t e d above.' And F i g . (2,9) cttows the d i f f e r e n c e between the cost f u n c t i o n a l J according to eqn. (.2.6.13) using the estimated state u(t,x) and the c o n t r o l f°(t) and J obtained i n 2.6.3.1 again f o r the same three sets of parameters c i t e d above. I t i s seen from F i g . (2.9) that, i n the worst case of (1) with r £ = l , Q£=I^, J i s l e s s than J by approximately 10%. This moderate decrease of the cost i s a r e s u l t of the los s of informat-ion due to truncation of the i n f i n i t e s e r i e s by using the approximation scheme for the d i s t r i b u t e d parameter f i l t e r i n t h i s s e c t i o n and seems none-theless to i n d i c a t e that the approximation scheme i s an acceptable one. An examinations of F i g s . (2.6) to (2.9) i n d i c a t e s that f o r t h i s system the choice - % ^ - • 3 — -but goes to show that i n general r £ and Q£, or the corresponding R £ ( t ) and Q^(t,x,x') can be so chosen as to obtain an a r b i t r a r y f i l t e r s e t t l i n g time. 2.6.3.3 Observability In r e a l i t y a point sensor that occupies no room does not e x i s t and i t then becomes important to have an idea how much we can observe using a sensor that approximates an i d e a l point sensor. With regard to the estimation problem considered here, a concept of o b s e r v a b i l i t y expounded upon by Goodson 41 and K l e i n i s of use. This i s the s o - c a l l e d N-mode o b s e r v a b i l i t y . A d i s t r i b u t e d parameter system i s said to be N-mode observable i f the unique-ness of the f i r s t N modes (eigenfunctions) i s guaranteed by the measurement data. Now, by c a l c u l a t i n g the zero crossings of the eigenfunctions given by eqn. (2.6.27), a l i s t of these zero crossings for up to the f i f t h mode being tabulated' i n Table (2.4), i t i s found that t h i s example problem i s N-mode O.Ol-i -o.oo--o.oi-t -o.ozHi -0.03 -h -0.04H?; ~ i — i — i — i — i — i — i — i — i — j — i — i — i — i — i — i — r > v f | — i — i — i — i — i ! _ j , . - f " T | i i i r -0.05-0.5 1.0 1.5 -I 1 ! 1 1 -2-.0 y C3) / /C2> s s s -0.05H 1 * % / V -/ / / / / / / / / / OS; -0.07-J F i g . (2.8) Control difference f°(t)-f°(t), -0.00 -0.01 -0.02 -0.03 -0.04 51 Eigenf unction Values of x at which, number zero-crossings occur 1 None 2 0,4585 3 0.2440 0.7320 4 0.1648 0.4945 0.8242 . 5 0.1242 0.3727 0.6211 0.8695 Table (2.4) Zero-crossings of eigenfunctions eqn. (2.6.27). observable up to the t h i r d mode at the given measurement l o c a t i o n x=0.5 even i f the sensor averages over ±0.55% of the s p a t i a l domain. 2.6.4 Conclusion In t h i s s e c t i o n , the state estimation problem given discrete-space measurement data of a p a r t i c u l a r 1-dimensional, stationary," heat-conduction d i s t r i b u t e d parameter system i s completely solved by making use of the theory developed. The de t e r m i n i s t i c estimator i s approximately modelled by a lumped parameter system using the eigenvalues and the eigenfunctions associated with the system s p a t i a l operator defined on the given s p a t i a l domain. The numerical r e s u l t s show that the estimator does f i r s t l y give a good estimate of the complete system state from a v a i l a b l e measurement data such that the estimated state can be used f o r the purpose of state-feedback c o n t r o l and secondly does not introduce s t a b i l i t y problems to the o v e r a l l system made up of the given system and the estimator. In r e a l - l i f e conditions, we can v i s u a l i z e a s i t u a t i o n that, given a d i s t r i b u t e d parameter system, we can, with a reasonably accurate knowledge of the model of the d i s t r i b u t e d parameter system, construct a state estimator and store i t i n an on-line computer. Then by taking measurements of the 52 given.distributed parameter.system and making use of these data i n the state estimator by the computer, i.t i s possible to obtain an estimate of the state s u i t a b l e e i t h e r f or an end i n i t s e l f or f o r the purpose of state—feedback. 2.7 Conclusion In t h i s chapter, we have considered the d e r i v a t i o n of the deter-m i n i s t i c f i l t e r that y i e l d s an estimate of the state f o r l i n e a r d e t e r m i n i s t i c cont-nuous-time d i s t r i b u t e d parameter systems based on measurement data, i n the f i r s t case, obtained over the whole occupied s p a t i a l domain and, i n the second case, a v a i l a b l e at only a f i n i t e number of s p a t i a l l o c a t i o n s . For the f i r s t case, the d e r i v a t i o n begins with e s t a b l i s h i n g that the f i l t e r i s l i n e a r i n s tructure, and has a d i s t r i b u t i v e f i l t e r gain yet to be s e l e c t e d . By con-s i d e r a t i o n of uniform asymptotic s t a b i l i t y i n the sense of Lyapunov of the transformed f i l t e r i n g e r r or, the d i s t r i b u t i v e f i l t e r gain i s shown to be chosen trom tne s o l u t i o n of a k i c c a t i - i i k e i n t e g r o - p a r c i a l d i f f e r e n c i a l equat-ion. For the second case, the general r e s u l t s obtained previously are s p e c i a l -i z e d , and the d i s t r i b u t i v e f i l t e r gain i s shown now to be obtained from the s o l u t i o n of R i c c a t i - l i k e p a r t i a l d i f f e r e n t i a l equation. A numerical example on a one-dimensional boundary c o n t r o l problem i s presented to i l l u s t r a t e the use i n an o v e r a l l c o n t r o l scheme of the determin-i s t i c f i l t e r f o r the second case mentioned above. It i s based on expanding the space-dependent functions i n terms of the eigenfunctions of the system s p a t i a l operator that the R i c c a t i equation i s solved to y i e l d the f i l t e r gain. 53 3. OPTIMAL PREDICTION AND FILTERING OF LINEAR DISCRETE-TIME STOCHASTIC DISTRIBUTED PARAMETER SYSTEMS BASED ON DISCRETE-SPACE MEASUREMENTS 3.1 Introduction In recent years, much i n t e r e s t has been shown i n extending, by various approaches with d i f f e r e n t degrees of refinement and r i g o u r , the op-19 20 timal l i n e a r estimation theory as developed by Kalman and Bucy ' to the 11 12 15 17 state estimation problem of l i n e a r d i s t r i b u t e d parameter systems ' ' ' ' 18,42-45 M . 12,15,17,42,43 , _ u • , Much of the e f f o r t i s devoted to the case i n which the noise-corrupted measurement data are assumed obtainable at any point over a l l the s p a t i a l domain, except + ^ . Thau has considered the f i l t e r -ing problem of a simple s c a l a r continuous-time d i s t r i b u t e d parameter system for the two cases of discrete-time and continuous-time measurements, both ob-18 tained at but one measuring point l o c a t i o n . Sakawa has considered the case of continuous-time d i s t r i b u t e d parameter system and continuous-time 44 45 measurements. Both Matsumoto et a l . and Hassan et a l . have considered the case of discrete-time d i s t r i b u t e d parameter system and discrete-time measurements, but they have not included system noise i n t h e i r r e s pective problem formulations. It i s the purpose of t h i s chapter to develop a l i n e a r minimum-variance algorithm f o r the sequential estimation of the state of a gen-e r a l class of discrete-time s t o c h a s t i c d i s t r i b u t e d parameter systems, under the assumptions that noise-corrupted measurements can be obtained at only several f i x e d points i n the s p a t i a l domain and that, say f o r reasons of economy i n instrumentation, measurements of the system are 44 45 taken only at discrete-time i n s t a n t s . The major d i f f e r e n c e from ' i s that system noise i s included i n the problem formulation. The algorithm developed i s s u i t a b l e f o r real-time on-line implementation. Moreover, because of the p a r t i c u l a r form of the discrete-time d i s t r i b u t e d parameter model used, Green's functions are involved. In engineering a p p l i c a t i o n s , the Green's functions can be determined experimentally ^ . Hence, e i t h e r by approximation of the i n t e g r a l s i n v o l v i n g the experiment-a l l y determined Green's functions by quadrature formulae or by determin-ing experimentally the f i r s t few s i g n i f i c a n t c o e f f i c i e n t s of expansions i n terms of eigenfunctions of the Green's functions, the r e s u l t a n t r e -cursive variance equation can be solved. We observe that the undesir-21 able on-line s o l u t i o n of complex p a r t i a l d i f f e r e n t i a l equations that would otherwise have been obtained had the d i s t r i b u t e d parameter system been formulated ab i n i t i o as a continuous-time one i s therefore avoided. r. . , i j . t . — - -• _ i r .i .. -i j . ... - »_i JT _ -i -i : „ ^ .LUC: t C i t l d l l i O C i . VJX U i l - L O C l l c i p i - ^ i - -J- O KX*. V c . i C i sections. In 3.2, the ( o p t i m a l ) l i n e a r minimum-variance d i s t r i b u t e d f i l t e r i n g problem f o r the state estimation of a l i n e a r discrete-time s t o c h a s t i c d i s t r i b u t e d parameter system i s f i r s t l y formulated. We then give an example i n 3.3 on how the discrete-time s t o c h a s t i c d i s t r i b u t e d parameter model i n the present problem formulation n a t u r a l l y a r i s e s when sampling i s introduced i n t o a continuous-time s t o c h a s t i c d i s t r i b u t e d parameter system as demanded by the f a c t that measurements are being taken only at discrete-time i n s t a n t s . In 3.4, we present the d e t a i l e d s o l u t i o n of the problem formulated. In 3.5, as a consequence of the optimality of the d i s t r i b u t i v e f i l t e r gain obtained i n 3.4, an orthogonal 19 p r o j e c t i o n lemma corresponding to that i n lumped parameter estimation theory i s shown to hold true for t h i s p a r t i c u l a r case of d i s t r i b u t e d parameter estimation. F i n a l l y , a numerical example on estimation i s 55 presented.in 3.6 to i l l u s t r a t e the use of the algorithm developed. 3.2 Problem' Formulation Let xeD denote a generic point i n the fix e d s p a t i a l domain D of the r~dimensional Euclidean space E . Let the closed t i m e - i n t e r v a l [ t ^ t ^ ] be d i s c r e t i z e d into N subintervals. Further, l e t L ={k: k=0,1,...,N-1} be a discrete-time index set and {tg , t ^ , . . . , t }^ be the corresponding d i s c r e t e -time set. Consider a l i n e a r discrete-time s t o c h a s t i c d i s t r i b u t e d parameter system which can be described by the following vector d i f f e r e n c e - i n t e g r a l equation U k + l C x ) = ' D G k + l , k C x ' X ' ) u k ( x ' ) d D x V + ' D F k + l , k ( x ' X ' ) w k ( x ' ) d D x ' ' ° - 2 - 1 ) k e l , where u (x) i s the n-vector state at t , G (x,x') i s the nxn kernel of the state t r a n s i t i o n matrix i n t e g r a l operator, F . , (x,x') i s the nxc IC*T1 , ic kernel of the disturbance t r a n s i t i o n matrix i n t e g r a l operator, dD^, i s the elemental volume about the point x'eD, and t'w ( X 1 ) , k=U,!,...,N-l} i s a c-vector input disturbance gaussian sequence. The state i n i t i a l condition u^(x) to eqn. (3.2.1) i s assumed to be given as a gaussian random n-vector with mean E[u Q(x)] = G Q(x) = 0 , (3.2.2) and nxn nonnegative-definite covariance matrix E [U 0 ( X ) UQ ( X ' ) ] = P 0(x,x') , (3.2.3) where E[*] denotes the expectation operator. We assume that measurements of the system state are taken at d i s c r e t e instants of time by zero-memory transducers. The transducers are assumed to be located at m d i s t i n c t points x., i e l , i n the s p a t i a l domain, where l m I ={i: 1=1,2,...,m} i s a d i s c r e t e measurement l o c a t i o n index set. Let the m output at t, ,,, k e l , of the transducer located at x. be a d.-vector and be k+1 t i x 56 denoted by y, .• The measurement y . i s now assumed to consist of the state responses at x_. , j e: 1^, l i n e a r l y combined together and the a d d i t i v e measurement noise v, ,, .. Thus, we have the measurement equation f o r the fc+1,1. transducer located at x . t l as i m m (3.2.4) where y k + l , i = .1 M k + l , i 3 U k + l C x j ) + V k + l , i ' i s a d.xn measurement matrix r e l a t i n g the state response at x ^ c + l ^ j i = ' * 3 to the output of the transducer located at x. and (v ., k=0,1,...,N-1} i s a d^-vector measurement error gaussian sequence. It i s desirable f o r n o t a t i o n a l convenience to augment eqn. (3.2.4) f o r i=l,2,...,m into one si n g l e equation. To t h i s end, we define f i r s t l y the following notations: 7k+l,1 'k+1 yk+l,m A. +1 "k+1, 11' \ + l , lm k+1 ml' +1 ,mm " k + l 6 ^ (3.2.5) 57 k+1,1 v k+1 v. k+l,m A M where we observe that n, ,, i s a d-vector, d= E d ., M_ ,, a dxmn matrix, u k+1 . , i k+1 k+1 i = l a mn-vector and v, ,, a d-vector. k+1 Using the above d e f i n i t i o n s , we can write by augmenting eqn. (3.2.4) for i=l,2,....m the si n g l e measurement equation \ + l = \ + l U k + l + Vk+1 (3.2.6) Equations. (3.2.1) and (3.2.6) together are now the system and ob-servation models f o r estimation consideration i n the sequel. The system noise w^(x) and the measurement noise v^ +T a r e assumed each to be zero mean and gaussian white i n time, i n ad d i t i o n to being un-correlated with each other and the given state i n i t i a l c ondition u^(x). Thus, E[w k(x)] = E[v f c + 1] = 0 , E[wk(x)wJ(x')] = Q k(x,x')6 k ;, , E [ v k + l V £ + l ] = ' (3.2.7) E t W k ( x ) v £ + l ] = ° > E[w k(x )u J(x')] = E [ v k + 1 u J ( x ) ] = 0, for a l l k,£el , where Q (x,x') i s a nonnegative-definite s e l f - a d j o i n t matrix t K. s a t i s f y i n g Q k(x,x') = Q k(x',x) , (3.2.8) Rk i s a symmetric p o s i t i v e - d e f i n i t e matrix, and i s the Kronecker d e l t a . 58 Now,.let the c o l l e c t i o n of the given state i n i t i a l c ondition mean UQ ( X) and measurements r\^, n^,..., n be denoted by n 3 . The estimate of the true state u, ,, (x) based on the measurement data n 3, i , k e l , w i l l be denoted k + 1 ' J ' t by u, i , (x). I f i<k+l, the estimate ii, ,. i . (x) i s termed the predicted k+1 1 j k + 1 1 j F estiamte. If i=k+l, the estimate u, ,,(x) i s termed the f i l t e r e d estimate J ' k + 1 | k + 1 and w i l l be denoted by u^^Cx) f ° r n o t a t i o n a l s i m p l i c i t y . Further, the estimate u, , . i . f x ) i s s a i d to be l i n e a r i f the estimate u, , , i . ( x ) i s obtained k + 1 | 3 ' k + 1 | j from a l i n e a r operation on the measurement data n 3. We w i l l be concerned here with the two cases of j<k+l only. The problem to be considered i n the sequel i s the (optimal) l i n e a r minimum-variance d i s t r i b u t e d f i l t e r i n g problem which can now be stated as follows: k+1 Given the measurement data n , k e l , an i n i t i a l estimate UQ ( X ) of the i n i t i a l s tate u.Cx) and i t s covariance P_(x.x'). f i n d a l i n e a r f i l t e r e d estimate U u u. n (x) of u (x) which minimizes the f i l t e r i n g error variance K~T-L K.+1 E [ | | u k + 1 ( x ) | | 2 ] , (3.2.9) where u^ +^(x) i s the f i l t e r i n g error defined as \ + l ( x ) = \ + l ( x > - \ + l ( x ) ( 3 - 2 ' 1 0 ) and ||*|| denotes the Euclidean norm. 3.3 On the Linear Discrete-time Stochastic D i s t r i b u t e d Parameter Model Before proceeding further to obtain the s o l u t i o n of the problem formulated i n 3.2, we gaive an example i n t h i s s e c t i o n to show how the discre t e -time stochastic d i s t r i b u t e d parameter model eqn. (3.2.1) n a t u r a l l y a r i s e s . The example to be considered i s that of a continuous-time s t o c h a s t i c d i s -t r i b u t e d parameter system being sampled as demanded by the f a c t that the measurements made on i t are assumed to be taken only at discrete-time instants by transducers located at d i s t i n c t s p a t i a l points. Consider a l i n e a r continuous-time s t o c h a s t i c d i s t r i b u t e d parameter system described by the vector p a r t i a l d i f f e r e n t i a l equation 3u(.t,x) = A ^ u ( t > x ) + B(t,x)wCt,x) , (3.3.1) defined for t^>t>t^ on a s p a t i a l domain D of the r-dimensional Euclidean space E , where u(t,x) i s an n-dimensional state vector function of the system, w(t,x) i s the c-dimensional stoc h a s t i c disturbance input, i s a known l i n e a r matrix p a r t i a l d i f f e r e n t i a l operator, with respect to the s p a t i a l v a r i a b l e xeD, whose parameters may depend on both t and x, and B(t,x) i s a known nxc matrix function. The i n i t i a l and boundary conditions are given by u ( t Q , x ) = u Q(x) , (3.3.2) 3 xu(t,x) = 0 , XE3D, (3.3.3) where u Q ( x ) i s a stochastic vector function, 3D i s the boundary surface of the s p a t i a l domain and 3^ i s a l i n e a r p a r t i a l d i f f e r e n t i a l operator i n xe3D of order le s s than A . x Since we are interes t e d only i n the values of u(t,x) at the d i s c r e t e -time instants t. .. , k e l , as demanded by the measurement equation (3.2.6), l e t K. T" JL t us therefore consider i n the following how eqn. (3.3.1) can be reduced to the form of eqn. (3.2.1). The s o l u t i o n to eqns. (3.3.1) with the given i n i t i a l and boundary 4 34 conditions (3.3.2) and (3.3.3) r e s p e c t i v e l y can be expressed as ' u(.t,x) = / G(t,x;t n,x')u n(x')dD , + f t . / _ G(.t ,x; t' ,x' )B(.t 1 ,x' ) w ( t , x ' )dD ,dt' u U U X U X 0 (3.3.4) where the Green's function G(t,x;t',x') of dimension nxn and defined f o r t>0 and xeD=DU3D can be shown to s a t i s f y the following conditions: 60 |- G(t,x;t' ,x') .= A G(t,x;t' ,x') , C3.3.5) dt X g G(t,x;t',x') = 0 , x e3D, C3.3.6) and G(t,x;t,x') = I6(x-x') , (3.3.7) where <$(•) i s the Dirac d e l t a function. Now, l e t the closed t i m e - i n t e r v a l [tQ,t^] be p a r t i t i o n e d i n t o N sub-i n t e r v a l s such that the f i r s t N time points correspond to the discrete-time index set I i n 3.2. Consider the system eqn. (3.3.1) i n the time-subinterval t^<t<t^ +^ for some k e l ^ . We assume that u(t^,x) i s given. We must, moreover, assume that the input disturbance w(t,x) i s a piecewise continuous f u n c t i o n of time and that changes i n the values of w(t,x) occur only at the time instants t=t, , that i s , k w(t,x) = w k(x) , t k - t < t k + l * It then follows immediatelv from eqn. (3.3.4) that u C t k + l ' x ) = ;D G C t k + 1 , x ; t k > x ' ) u ( t k > x ' ) d D x f fck+l / D [/ G ( t k + 1 , x ; t ' , x , ) B ( t , , x ' ) d t , ] w k ( x ' ) d D x , . t k (3.3.8) By de f i n i n g u k + 1 ( x ) = u ( t k + 1 , x ) , u k ( x ) = u ( t k , x ) , A (3.3.9) G k + l , k ( x ' X , ) = G ( t k + l ' x ; t k ' x , ) > F k + 1 ) k ( x , x ' ) = / k + 1 G ( t k + 1 , x ; t ' , x ' ) B ( t ' , x ' ) d t ' ., we can rewrite eqn. (3.3.8) as U k + l ^ = / D G k + l , k C x ' X ' ) u k ( x ' ) d D x ' + / D F k + l , k ( x ' X ' > k ( x , ) d D x ' ' C 3 - 3 ' 1 0 ) f o r k e l . We see that eqn. (3.3.10) i s the same as eqn. (3.2.1). Eqn. (3.3.1) has been shown to be reducible to eqn. (3.2.1), but the converse c l e a r l y i s g e n e r a l l y not true. 3.4 Derivation of the Linear Discrete-time D i s t r i b u t i v e Kalman F i l t e r We begin the d e r i v a t i o n of the l i n e a r minimum-variance discrete-time d i s t r i b u t i v e f i l t e r with the observation that, by repeated a p p l i c a t i o n of eqn. (3.2.1), u^^Cx) i s expressible as a l i n e a r sum of u^(x) , w_^(x), w^ _^  (x) , . . . , w^(x) f o r some i<k. To be exact, we have u k + 1 ( x ) = / D 6 k + 1 > 1(x,x')u 1(x')dD x, k+1 + VD G k + i , j ( x ' x , ) F j , j - i ( x , ' x " ) w j - i ( x " , ) d r ) x ' d D x « « ( 3 - 4 ' 1 ) where the kernel G ,(x,x') of the state t r a n s i t i o n i n t e g r a l operator k + 1 , i s a t i s f i e s the following t r a n s i t i o n a l property G k + l , i ( x ' x J ) = ' D G k + l , j ( x ' X ' : ) U j , i ( x : : » X : ) a V ' ( 5 - 4 - 2 ) and G. .(x,x*) = I6(x~x') , (3.4.3) for a l l i<j<k, i , j , k e l . Consider the system at time indexed by k+1. We assume that a l l past k measurements n and the f i l t e r e d estiamte u k(x) are a v a i l a b l e . I t i s c l e a r from eqn. (3.4.1) that, since noise cannot be predicted, the single-stage pre-k dieted estiamte ^ k + 1 j k ( x ) based on the past measurements n i s given by \ + l | k ( x ) = / D G k + l , k ( x ' X , ) \ ( x , ) d D x ' • ( 3 - 4 ' 4 ) Further, i t i s c l e a r from eqn. (3.2.4) that, since noise cannot be predicted, the single-stage predicted measurement based on the past measurements n i s m (x,). We define the d i f f e r e n c e between the actu a l measurement E Mi . L i - u i J.I i i ex.: ^ k+l,i.j k+l|k 2 taken at t, ' , v i z . , y, .," . and the predicted measurement to be the measurement k+1' k+1,I r e s i d u a l z,,-. ., that i s , K~r_L j 1 62 m Jk+1 ,t y k + l , i . l x \ + . l , i j U k + l | k C x j ) > (3.4.5) i e l . Consider the following recursive l i n e a r discrete-time d i s t r i b u t i v e m f i l t e r : \ + l C x ) = \ + l | k C x ) + . \ K k + l , i C x ) z k + l , i ' i = l C3.4.6) i s where k e l , i e l , i n i t i a l u_ (x)=E[u r i(x) ] i s assumed given, and K, - . (x)  the nxd. d i s t r i b u t e d f i l t e r gain matrix at x.. l i Let us f i r s t examine the structure of the linear, f i l t e r given by eqn. (3.4.6). Here, the p r e d i c t o r - c o r r e c t o r concept i s employed. At time t, , the previous f i l t e r e d estimate u. (x) i s predicted forward one stage to k+1 K give the pred i c t o r term \ + 1 | k C x ) - T h e predicted estimate \ + 1 | k ( x ) i s f u r ~ ther used to obtain the best estimate of the act u a l current measurement y k + 1 ± r e s i d u a l z, ,., . at x. becomes the innovative piece of information a v a i l a b l e at k + 1 , 1 1 x.. The measurement residuals z, , n . each weighted by the d i s t r i b u t e d f i l t e r l k + l , i gain K, n .(x) i s then summed over a l l i e l to give the corrector term which KTJ. 5 X TCI i s added to the predictor term uk+]_|k(x) t 0 obtain the current f i l t e r e d es-timate 6 ^ + ^ ( x ) . For notational convenience, we define a mn-vector £. . n i . as k+11 j | U k + l | j ( x i ) k+1 j u, . , I . (x ) k+11J m (3.4.7) a d-vector measurement Ck+-^ a s 63 'k+1,1 'k+1 (3.4.8) 'k+1,m and a nxd d i s t r i b u t e d f i l t e r gain matrix K ^ ^ O O A S ' W x ) = ' w 0 0 : • :WCx)] • °-4-9) Using the above d e f i n i t i o n s f o r (J , ? ( 1 and K (x) , we can K+1 K. K+1 K+1 rewrite the f i l t e r equation (3.4.6) and the measurement r e s i d u a l equation ob-tained by augmenting eqn. (3.4.5) f or i=l,2,...,m as V L 1Cx) = fn G,^ J x . x ^ u J x ^ d D ^ , + K ^ C X ) ^ , (3.4.10) and 'k+l v"' 'D ~k+l,k""'" '~k x" '""x' ' k + l v '"k+1 **k+l nk+l \ + l ^ k + l | k ' (3.4.11) re s p e c t i v e l y . We observe that, by repeated a p p l i c a t i o n of eqn. (3.4.4), we have \ + l | i ( x ) " ' D G k + l , i ( x ' X , ) V X ' ) d D x ' (3.4.12) for a l l i<k, i , k e l t . Further, by repeated a p p l i c a t i o n of eqn. (3.4.10), we have k+1 \ + l ( x ) " ' D G k + l , i C x ' X , ) V X ' ) d D x ' + 'D G k + l , j ( x ' X ' ) K j ( X ' ) S d D x ' ' (3.4.13) fo r a l l i<j<k, i . j . k e l . U t i l i z i n g eqns. (3.4.12) and (3.4.11) i n eqn. (3.4.13), we have k+1 \ + l C x ) = \ + l | i ( x ) + l+1 fV G k + l , j C x ' X ' ) K j C x ' ) ^ j " V ; j | i ) d Dx" 0.4.14) for a l l i<j<k, i , j , k e l . Setting 1=0 i n eqn. (3.4.14), we obtain k+1 \ + l ( x ) = \ + l | 0 C x ) + *ml f B G k + l , j C x ' X , ' ) K j ( x ' > ^ j - - V j | 0 ) d D x ' - ( 3 ' 4 - 1 5 ) 64 Now, i t i s clear from eqn. (3.4.12) that u\| Q(x) and £j|g ^ o r a n ^ J e I t > J^°> depend on u (X)=EIUQ(X)J only such that u^^OO i s a function of the past and k+1 current measurements n as required of a f i l t e r e d estimate. With the f i l t e r s tructure established, the problem of determining the optimal f i l t e r e d estimate u (x) so as to minimize the f i l t e r i n g e r r o r variance E[ | l u^ + 1_ 0 0 I I 21 reduces to one of optimizing the "parameters" ^+1 ^ ' For m a n i p u l a t i v e convenience, define the usual f i l t e r i n g error covariance matrix P k + 1 ( x , x ' ) = E [ u k + 1 ( x ) u £ + 1 ( x ' ) ] . (3.4.16) C l e a r l y , from i t s d e f i n i t i o n , P k + l ( x ' X , ) = P k + l ( x ' ' X > • ( 3 ' 4 - 1 7 ) Then, equivalently, the optimal d i s t r i b u t e d f i l t e r gain K k + T . 0 0 i s chosen to minimize t r \ + 1 ( x , x ) = t r P k + 1 ( x , x ' ) ] x , = x , ' (3.4.18) where t r stands f o r "the trace of". It i s seen that eqn. (3.4.10) with the optimal d i s t r i b u t e d f i l t e r gain K j c + 2 ^ x ^ i s d e f a c t 0 t n e ( l i n e a r ) Kalman f i l t e r f o r t h i s l i n e a r d i s c r e t e -time stoc h a s t i c d i s t r i b u t e d parameter system case given discrete-time d i s -crete-space noisy measurements. 3.4.1 F i l t e r i n g Error In order to obtain a recursive r e l a t i o n s h i p f o r ^ . j . ^ 0 * > x 1 ) , that f o r fl, ,, (x) needs to be obtained f i r s t . k+1 From the d e f i n i t i o n (3.2.10), subtracting eqn. (3.4.6) from eqn. (3.2.1) and using eqn. (3.4.4) r e s u l t i n the f i l t e r i n g error as Q k + ! ( X ) = ;D G k + l , k C x ' X ' ) a k C x ' ) d D x ' + ;D F k + l , k C x > X ' ) w k C x ' ) d D x ' m - . \ K k + l , i ( x ) z k + l , i • < 3 - ^ > i = l By s u b s t i t u t i n g the measurement equation (3,2,4) i n t o eqn. (3 .4.5) and making use of the', s t a t e equation (3.2.1), the measurement r e s i d u a l at x_^  i s obtained as m z, , . = Z M. .. . . /_ G, ... , (x.,x')u, (x')dD , + v. ,. . k + l , i ^ Tv--i-l,ij D k + l , k v j ' k x' k + l , i m + I M. L 1 .. /„ F, . (x. ,x')w, (x')dD , . C3.4.20) ^ k+1, i j D k+l,k j k x' Define, f o r n o t a t i o n a l convenience, a nxn m a t r i x T^-fi k^ X' X'^ a S A m M , k ( x ' X , ) = G k + l , k ( x ' X , ) - . * W ( x ) J W i W ( X 3 ' X , ) * ( 3 - 4 ' 2 1 ) 1 > 3 1 Then, combining eqns. (3.4.19) and (3.4.20), the r e c u r s i v e r e l a t i o n f o r the f i l t e r i n g e r r o r i s \ + l ( x ) = 'D T k + l , k ( x > X , ) \ C x ' ) d D x ' + ' D F k + l , k ( x ' X , ) w k ( x ' ) d D x ' m + T, K. . . . (x) I M F. . . f x . . x ' k (x')dD . . ._. K.-ri,x u K.-rx,ij K-rl,K. j K. x' y 3 — 1 m " .\ K k + l , i ( x ) v k + l , i ' ( 3 ' 4 - 2 2 ) 1=1 Besides, we show below the two p r o p e r t i e s on the c o r r e l a t i o n between f i l t e r i n g e r r o r and nois e . We f i r s t of a l l see t h a t , by s e t t i n g i=0 i n eqn. (3.4.1), u ^ + ^ ( x ) i s e x p r e s s i b l e as a l i n e a r sum of UQ ( X ) , WQ ( X), w^(x),..., W ^ ( X), that i s U k + l ( x ) = ' D G k + l , 0 ( x ' X ' ) u 0 ( x ' ) d D x ' k+1 + jf1 VD G k + i , j ( x ' x , > F j , j - i ^ x , ' x " ) w j - i ( x " ) d D x ' d I ) x - ( 3' 4' 2 3> Hence, given the zero mean gaussian'assumptions of u gCx) and each w j_ -^Cx) , j = l , 2 , . . . ,k+l, u k + j _ C x ) i s a l s o zero mean gaussian f o r each k=0,l,.... Now, by t a k i n g the covariance of u ^ ^ O O w i t h w £ ( x ! ) and u s i n g the second and f i f t h 66 relations in eqn. (3.2.7), we have E[u k + 1(x)wJ(x')] = 0 , (3.4.24) for a l l £>k, where k,£el . Further, by taking the covariance of u^ + 1(x) with v £ + ^ and using the fourth and f i f t h relations in eqn. (3.2.7), we have E [ u k + ! ( X ) V I + 1 ] " ° > . ( 3-'- 2 5 ) for a l l k and £, k,£el . We secondly observe that, from eqns. (3.2.6) and (3.4.23), n k + i . i s expressible as a linear sum of u^(x), w^(x), w^(x),..., w^(x) and v^ +^. Hence, given the zero mean gaussian assumptions of u Q ( X ) and each W j _ ^ ( x ) > j=l,2,...,k+l, and v k + 1 > \ + i i s a x s o zero mean gaussian at each k=0,l, Now, by taking the covariance of nk+-^ and w^(x) and using eqn. (3.4.24) and the fourth relation in eqn. (3.2.7), we have E['i k + 1w £(x) ] = 0 , (3.4.26) for a l l £>k, where k,£el . Further, by taking the covariance of n k + 1 and v £ + ^ and using eqn. (3.4.25) and the third relation in eqn. (3.2.7), we have ^ k + l ^ f l l ' ° > ( 3 ' 4 - 2 7 ) for a l l £>k, k,£el . We thirdly r e c a l l from eqn. (3.4.15) that " k + 1 ( x ) i s a linear k+1 function of measurements n Now, by taking the covariance of u^ + 1(x) with w^(x') and using the f i f t h relation in eqn. (3.2.7) and eqn. (3.4.26), we have E[u k + 1(x)w^(x')] = 0 , (3.4.28). for a l l £>k, k,£el t. Hence, we see from eqns. (3.4.24) and (3.4.28) that the f i l t e r i n g error u, .. (x) is uncorrelated with the current and future system noise w^(x), that i s , E [ u k + ] (x)w*(x')J = 0 , C3.4.29) for a l l l>k, k,£el . t Moreover, by taking the covariance of ^ ^ ( x ) a n c* v£+i a n c ' u s i r i § t n e f i f t h r e l a t i o n i n eqn. (3.2.7) and eqn. (3.4.27), we have E[u k + 1(.x)v£ + 1] = 0 , (3.4.30) for a l l £>k, k,£el . Hence, we see from eqns. (3.4.25) and (3.4.30) that the f i l t e r i n g error u k + ^ ( x ) i s uncorrelated with the future measurement noise v ^ + ^ that i s , E<wx)vL.> =0 . ' (3-4-31) f o r a l l £>k, k,£el , 3.4.2 P r e d i c t i o n Error Covariance As an intermediate step i n obtaining P ^(x,x'), the single-stage p r e d i c t i o n error covariance l J k + ^ | k ( x ,x !) defined by P k + l | k ( X ' X ' ) = E [ G k + l | k ( x ) G k + l | k ( x ' > 5 > ( 3 ' 4 - 3 2 ) where the p r e d i c t i o n error u ^ + ^ j ^ (x) i s defined as \ + i | k ( x ) = W x ) " \ + i | k ( x ) > ( 3 - 4 - 3 3 ) i s obtained below. Subtracting eqn. (3.4.4) from eqn. (3.2.1) y i e l d s the r e c u r s i v e r e -l a t i o n of p r e d i c t i o n e r r o r as u. -h (x) = / G . (x,x')u, (x')dD , + / F . (x,x')w, (x')dD , (3.4.34) k+l|k D k+l,k ' k x' D k+l,k k x' T By forming the product u k + ^ | k ^ G k + l |k^ X' ^  ' W S ^ a v e t * i e f o i l ° w i n S expression: 0 k + l | k C x ) a k + l [ k C x , ) 68 + VD \+I,^>*"H^ + VD W ^ ' ^ k ^ ^ (3,4.35) where x" and x"' are dummy v a r i a b l e s . Now, on taking the expectation of eqn. (.3.4.35) i n order to obtain the predicted error covariance P^+i |k^ X' X' ^  ' W S see that the second and t h i r d terms vanish since ^ ( x ) i s uncorrelated with w £(x') for £>k, kyUel^, as contained i n eqn. (3.4.29). Hence, the predicted error covariance P i (x,x') i s given as k+111c p k + i [ k C x ' x , ) " VD \+i,k ( x ' x , , ) \ ( x ^ x ' M ) G L i,k C x ,' x , M ) d\" d\"' + VD \+i,k(x'xM)\^^x,M>FLi,k(X,'X"')DVdDx"' (3.4.36) From the sel f - a d j o i n t n e s s of P k(x,x') and Q k(x,x'), we see that Fk+l|k^ X' X'^ "*"S a^" s o s e i f ~ a d j o i n t , that i s , P k + l | k ( x > X , ) = P k + l [ k ( x ' ' X > ' • ( 3 - 4 ' 3 7 ) 3.4.3 F i l t e r i n g Error Covariance In order to obtain the recursive r e l a t i o n f o r the f i l t e r i n g error covariance P (x,x') as defined by eqn. (3.4.16), we f i r s t of a l l form the iC i JL ~T product u (x)u (x') and then take i t s expectation. K"T"_L K .T 1 We r e c a l l that the f i l t e r i n g e r r o r u^ +^(x) i s given by eqn. (3.4.22). T The product ( x ) u k + ^ ( x ' ) i s the following expression i n v o l v i n g altogether sixteen terms: 69 u. , 1 (x)u (x' ) k+1. k+1. • T, ., , Cx,x,r] [uT, (x")u|r Cx"1) ] (x1 ,x" ') dD „dD „ , DD k+l,k k+1, le' x x + /„/„ T,^ , Cx,x")[u, C*")vl&")]** . fc',x"')dD ,,dD .„ DD k+l,kv m 1 c v " '"kv" ' J~k+l,k v T T X X 'VD . E •i- > 3 -f" m fD * T k + l , k ^ ' x M ) t \ ^ " ) v k + l , i ] \ + l , i ^ ' ) d V + VD \ + i 5 k ( x ' x " ) [ ^ ( x ' , ) Q k ( x " , ) ] T k + i , k ( x ^ x " ' ) d D x " d V -+ VD \+i,k(x'x,)^(xM)wk^M>JFLi,k^^x",)d\"dV' m T D'D . * W ^ ' ^ X ^ ^ W i > 3 1 ' D .| F k + l , k ( x ' X ' , ) [ w k ( x , , ) v k + l , i ] < + l , i ( x , ) d D x " m VD . * W ( x ) W j W ( V x H ) [ ^ 1 > J -L ra D'D . E W ( x ) W A + i , k ( x r x ' ' X ^ 1 > 3 -L m m r E K , n .(x) E M. - . F ,^ . (x ,x")[w,(x")w,(x , M)] D D . . n k+l,i , , Tc+l,ia k+l,k a' k k i,3=l a,b=l * Fk+l ,k ( xb >X"' K+l, ,J ( X' ) d D x " d D x " • m m + /. D . E W ^ W j W ^ ' ^ \ ^ X + l ^ W ^ ' ^ x " i,3=l J J a=l D j Kk+l,i^ )[\+l,l\&t"»ILl,k^ '-X">dDX" m i=l m 1=1 a,b=l m 70 x>3 1. where x" and x"' are dummy v a r i a b l e s . Now, on taking the expectation of eqn. (3.4.38), we see that f i r s t l y the second, t h i r d , f i f t h and ni n t h terms vanish since u, (x) i s uncorrelated with w (x') f o r £>k, k,£el , as contained i n eqn. (3.4.29), secondly the fourth and t h i r t e e n t h terms vanish since u^(x) i s un-correlated with v. for l>k, k,fc£l as contained i n eqn. (3.4.31) and l a s t l y the A/ L eighth, twelfth, fourteenth and f i f t e e n t h terms vanish because df the un-c o r r e l a t i o n assumption between w^(x) and v £ + ^ for a l l k and I, k,£el t, as given by the fourth r e l a t i o n i n eqn. (3.2.7). We are therefore l e f t with altogether s i x terms only. Hence, the f i l t e r i n g e r r o r covariance P k +^(x,x') i s given as P k + 1 ( x , x ' ) = VD \ + i,k ( x ' x , , ) \ ( x "' x ' , , ) T Li,k ( x'' x , M ) d Dx" d Dx-+ VD ***it^>*">\*">^ ~ f n f n S F , .^ , ( X , X " ) Q , ( X " , X " ' ) F L , ( X J > X " ' ) M L ,,K,T^ ,(x')dD x I IdD x I 1 I VD . E , V i l k ( ^ x M ) \ ( x ^ x l , , ) F w > k ^ ' 5 t , , , ^ i l i j i i + i , i ( x , : m rnf„ E K, . (x)M,,. . .F, , (x. ,x")Q. ( x " , x , , , ) F 1 , . .( D D . V K L , i Tt+l,ij k+l,k j ' x k ' k+l,k >3 1 m m + VD . Z , W ° ° I , M k + l , i a F k + l , k ^ a ' x , , ) \ ( x , , » X , , , ) i , j = l a,b=l ' F k + 1 ,k ( xb >X"' K + l , 3 b C l ,3 ( X ' ) d D x " d D x " • m . * K k + l , l C x ) R k + l , l j K k + l , J . ( X , ) i > 3 1 (3.4.39) Using the d e f i n i t i o n (3.4.21) of T (x,x') i n the f i r s t term on K."T"X y K. the r i g h t hand side of eqn. (3.4.39), we have on grouping terms P^ + 1(x,x') as P k + 1Cx,x') m E K. . .Cx)[ E ( /_/ n G C x , , x")P 1 (x' ,x"' ) G T (x. . - lc+ 1 , 1 . . D D k + 1 , k k k + 1 , k 3 i = l 3 = 1 ,x"')dD dD , + VD \ + i , k ( x ^ x M ) \ ( x ^ x " , ) F L i , k ^ r x " , ) d \ " d D x " - u ] T m m - * [ . V VD \ + i j c ( x ' x " ) \ ( x ^ x ' ' , ) \ + i , k ( x j ' x , M ) d D x " D V ' 1=1 3=1 + VD \ + i 5 k ^ ' x M ) \ ( x ^ x " , ) F L i , k ( x r x M , ) d D x " d D x " ' ) Mi!+i,i3 ] K k + i , i ( x , ) m + * K k + l , i ( x ) a,b=l "•-»•'— - - "•-»- - - ~ - -T T + VD V l . k ^ a ^ ' ^ k ^ ' ^ ' ^ ^ + l . k ^ b ^ ' ^ ^ x - ^ x - ^ l . j b + \ + l , i 3 ] K k + l , 3 ( x , ) * (3.4.40) For n o t a t i o n a l convenience, we define a nxn matrix ct k +^(x,x') as <WX>X,) -pk+i|k^»x,) > (3-4-41) a nxd p a r t i t i o n e d matrix 3 ^ + 1 a s B, ,-, Cx) = [6 . + 1 . (x) ' . • • • Cx)] , (3.4.42) k+1 k+1,1 . k+1,m where i t s i - t h nxd. submatrix B, . , .(x) i s defined as i k+1,i A m T K,, --Cx) = Z K^U.(*,xjK.,-> ,, > (3.4.43) W , i C x ) = ^ P k + l | k C x ' X 3 ) M k + l , i j ' 72 and a dxd p a r t i t i o n e d matrix Y k + 1 as Yk+1 Y k + 1 , 1 1 Y k + 1 , lm Y k + l , m l Y k + l , m m C3.4.44) where i t s i i - t h d.xd. submatrix i s defined as J x x m f k + l , i j ~ I , \ + l , i a P k + l | k ( x a ' X b ) M k + l , j b + \+l,±i ' J a,b=l ' (3.4.45) From the se l f - a d j o i n t n e s s of p k + 1 | k ( x > x ' ) a n d \ + i 5 i j ' w e s e e f r o m eqn. (3.4.45) that Y k + l , i j Y k + l , j i (3.4.46) Hence, from eqn. (3.4.44), Yk+1 Yk+1 (3.4.47) Further, we assume that Y k + 1 > ° ' (3.4.48) that i s , Y k + 1 i s p o s i t i v e - d e f i n i t e . By using eqn. (3.4.36) for P k + 1 | f c ( x , x ' ) , the d e f i n i t i o n s f o r a, . . ( x . x 1 ) , (3.4.43) for B, .(x) and (3.4.45) f o r y . . i n eqn. (3.4.40), we have p k + i C x ' x , ) = a k + i ( x » x , ) - j x t K k + i , i C x ) C , i + P k + i , i ^ + i , t < x , ) ] m + . S , ^ k + l . l ^ k + l . l d ^ + l . j ^ ' 0 * (3.4.49) F i n a l l y , by using the d e f i n i t i o n s C3.4.9) f o r K , (3.4.42) for B k + 1 and (3.4.44) for y , we can rewrite the above equation compactly as 73 p ; + 1 cx,x') = «k+1cx,x') - vi . w C i ( ! t , ) - W x ) K k + i ( x , ) We have therefore obtained the recursive r e l a t i o n f o r the f i l t e r i n g error covariance P k ^ ( x , x ' ) with a non-optimal d i s t r i b u t e d f i l t e r gain K k +^(x). 3.4.4 Optimal D i s t r i b u t e d F i l t e r Gain To obtain the optimal d i s t r i b u t e d f i l t e r gain, we f i r s t replace x' by x i n eqn. (3.4.50) to y i e l d P k + 1 ( x , x ) = <x k + 1 Cx,x) - ^ k + l ( x ) 3 k + l ( x ) - \ + l ( x ) K k + l ( x ) + K k + l C x ) Y k + l K k + l ( x ) * (3.4.51) 46 Then, applying the r e s u l t s of concerning the d i f f e r e n t i a t i o n of the trace of a matrix product by one of i t s constituent matrices, we have, f o r eqn. (3.4.51), t r K (x)B, (x) 3 K k + 1 ( x ) " ~ k + l ^ y M k + l T t r 3 (X)K, (x) 3 K k + 1 ( x ) " M k + l w k+1 = 3 k + 1 (x) , (3.4.52) a n d 3K ^  (x) t r K k + l ( x ) Y k + l K k + l ( x ) k+1 = 2 K k + l ( x ) \ + l > ( 3 ' 4 - 5 3 ) where i n deriving the l a s t step, the r e l a t i o n (3.4.47) has been used. Hence, upon p a r t i a l d i f f e r e n t i a t i n g t r P k +^(x,x) with respect to K (x) , we have lc-rl 9 t r P, . . C x . x ) = -2B. . . ( x ) .+ 2K. . N CX)Y,,.1 • (3.4.54) 3 K . . . (X) " ^k+l v"'~' ^ k + l w " " ~ k + l w , k + l KrrX Now, to s a t i s f y the necessary condition f o r the o p t i m a l i t y of 74 K (x), we equate eqn. (3.4.54) to zero to obtain k+1 The f i l t e r gain K CX) as given by eqn. C3.4.55) i s indeed the K.+1 su i t a b l e candidate to minimize t r P . ^ ( x , x ) , since T V V T t r p k + i C x > x ) = 2\+i > 0 (3.4.56) from eqn. (3.4.48). The recursive r e l a t i o n s h i p for the f i l t e r i n g error covariance using the optimal f i l t e r gain K k + l ^ x ^ t a e n becomes, from eqn. (3.4.50), p k + i ( x ' x , ) = \ + i ( x ' x , ) " W x ) \ + i e k + i ( x , ) • ( 3 - A - 5 7 ) with the i n i t i a l condition c l e a r l y being PQ ( X , X ' ) . 3.5 Orthogonal P r o j e c t i o n In t h i s s e c t i o n , we show that a lemma corresponding to what i s commonly known i n lumped parameter estimation theory as the orthogonal pro-j e c t i o n lemma i s s a t i s f i e d by the l i n e a r d i s t r i b u t i v e f i l t e r e d estimate, eqn. (3.4.10), with the optimal d i s t r i b u t e d f i l t e r gain s p e c i f i e d by eqn. (3.4.55). T We assume temporarily that E[u. (x)u (x')]=0. We r e c a l l that the f i l t e r e d estimate u k + 1 ^ ( x ) i s given by eqn. (3.4.10) and that the r e c u r s i v e r e l a t i o n f o r the f i l t e r i n g error u^ +^(x) i s given by eqn. (3.4.22). On taking the covariance of u (x) with ^ ( x 1 ) , we make use of the u n c o r r e l a t i o n be-tween the estimate u^(x) with the noise w^(x') and v ^ + ^ for £>k, k,£.£lt, as contained i n eqns. (3.4.28) and (3.4.30) so as to obtain B i V i < i & , ) ] = \ + i ^ ) E t . W k + i C x , ) ] • C 3- 5- 1 ) We r e c a l l that the measurement r e s i d u a l z, ,. . at x. i s expressible by eqn. k+l,x l 75 (3.4.20). Now, on taking the covariance of z, .1 . and u, ,,(x), we make use of eqns. (3.4.29) and (3.4.31) and further the uncorr e l a t i o n between the noises w Cx) and v , for a l l k,£el i n the fourth r e l a t i o n of eqn. C3-2.7) so as to obtain E [ z k + i , i ^ + i C x » - P L i,i C x ) -1 ± \ - M , i J l < k + i , J C x ) • ' ( 3 ' 5 - 2 ) From the d e f i n i t i o n s (3.4.9) f o r K k + 1 ( x ) , (3.4.8) for £ , (3.4.42) f o r B k + 1 and (3.4.44) for Y k +^> eqn. (3.5.2) can be augmented f or i=l,2,...,m i n t o E [ W V i ( x ) ] = W x > - W k + i w • ( 3- 5- 3 ) Hence, s u b s t i t u t i o n of eqn. (3.5.3) into eqn. (3.5.1) y i e l d s E [ \ + i ( x ) a k + i ( x ' ) ] = W x ) e k + i ( x , ) - K k + i ( x ) Y k + i K k + i ( x , ) • ( 3 - 5 - 4 ) The choice of optimal d i s t r i b u t e d f i l t e r gain K (x) as given by eqn. (3.4.55) y X t i i U D JLliliiiciLi .i-O. C O l_y E c\ +i ( x ) f l Li ( x , ) ] =0 • ( 3 - 5 - 5 ) T Since E[UQ(X)QQ(x')]=0, eqn. (3.5.5) i s true f o r a l l k e l t , that i s , the optimal f i l t e r e d estimate i s orthogonal to the f i l t e r i n g e r r o r . As the estimate u k + ^ ( x ) i s a l i n e a r function of U Q ( X ) , U^, n^,..., nk +^> then, i n view of eqn. (3.5.5), i t i s true that E [ \ + i Q L i ( x ) ] =0 > ( 3- 5- 6 ) &>k, k,£el t > that i s , the f i l t e r i n g e r r o r i s orthogonal to a l l the past and currently a v a i l a b l e discrete-space measurements. A l t e r n a t i v e l y , eqn. (3.5.6) can be rewritten, using d e f i n i t i o n (3.2.10) f o r ii . (x) , as K.-TX E [ \ + i u L C x ) ] = E [ \ + A + i ^ > C3-5-7) £>k, k,£el t. Relation (3.5.7) i n f a c t corresponds to the s o - c a l l e d d i s c r e t e 76 Wiener-Hopf equation i n lumped parameter f i l t e r i n g estimation theory. Eqns. (3.5.5) and (3.5.6) are the a l t e r n a t i v e expressions of the orthogonal p r o j e c t i o n lemma f o r the f i l t e r i n g estimation of l i n e a r d i s c r e t e -time d i s t r i b u t e d parameter systems from discrete-time measurements a v a i l a b l e at only several points i n the s p a t i a l domain. 3.6 Numerical Example Consider the problem of estimating the state of the process char-acteriz e d by the one-dimensional s c a l a r d i f f u s i o n equation 3u(t,x) 8 2u(t,x) , . f r t , — — — - = + w(t,x) , (3.6.1) 3x" at , r Z where u ( t , x ) , the state, i s at time t>0 the temperature d i s t r i b u t e d over the s p a t i a l domain 0<x<l and w(t,x) the gaussian noise added to the process. The boundary conditions are assumed given as A 3u(t.O) (3.6.2) P x u ^ , x ; j x = 0 - — = u 3 x u ( t , x ) ] x = 1 = f3u(t,l) + 9 u 9 ( x ? 1 ) = 0 , where 3 i s the heat transfer c o e f f i c i e n t assumed to be unity. The process eqn. (3.6.1) with boundary conditions (3.6.2) w i l l be simulated d i g i t a l l y . To t h i s end, the following approach i s adopted. F i r s t -l y , the discrete-space continuous-time scheme d e t a i l e d i n 2.6 i s used to ob-t a i n the corresponding continuous-time lumped parameter model. This i s done by d i v i d i n g the s p a t i a l domain into N =10 equal s l i c e s each of f i x e d length sp Ax=0.1 such that the system of eqns. (3.6.1) and (3.6.2) i s represented by a t o t a l of N =N +1=11 ordinary d i f f e r e n t i a l equations, one at each node. Then eq sp the continuous-time lumped parameter system i s converted using a time sub-i n t e r v a l of At=0.1 into the following equivalent discrete-time lumped para-meter system: 77 k + 1 . k+l,k k k+ljk k C3.6.3) where u, and w, are each. N column vectors whose i ~ t h elements are k k eq u k=u(kAt, (i.-l) Ax) and wk=w(kA.t, (i-1) Ax) , r e s p e c t i v e l y , and $ k + 1 k and r k +-^ k are the state and disturbance t r a n s i t i o n matrices 3 r e s p e c t i v e l y , each of dimensions N xN . For the simulation lumped parameter model, the element eq eq values of the'matrices of $, , , and F, ,, , are ca l c u l a t e d and shown i n Tables k+l,k k+l,k (.3.1) and (3.2), r e s p e c t i v e l y . It i s noted that * k + l k a n c * ^k+1 k a r e c o n s ~ tant matrices f o r a l l k. The process i s assumed to be observed at two given s p a t i a l l o c a t i o n s x=x^ and x = X £ , 0<x^,X2<l, corresponding to eqn. (3.2.6) with m=2 and ^ k +-^ i ± = x and Mk_^ i j = ^ > > ^ y t n e measurement equation (3.6.4) ) i s the gaussian noise added to the measurement equation. The measurement noise i s assumed to be of zero mean and i t s co-variance characterized by y k + l , l Vk+1,1 = + yk+l,2 U k + l ( x 2 } . Vk+1,2_ where v, , , = 0 0 1 (v, . , v, , „ k + 1 k + 1 , 1 k + 1 , 2 +1 v (3.6.5) where C i s a constant 2x2 matrix, v Following the argument presented i n 3.3, eqn. (3.6.1) together with the boundary conditions (3.6.2) can be, by d i v i d i n g the time i n t e r v a l [ t ^ . t ^ ] into N equal i n t e r v a l s each of fixed duration At, cast i n t o the following discrete-time representation 1 0 U k + l C x ) = 0 G k + l , k C x ' X , ) V X ' ) d x ' + { F k + 1 > k ( x , x ' ) w k ( x ' ) d x ' , (3.6.6) where k, k=0,1,...,N-1, i s the discrete-time index. In the above equation, 3 2 the Green's function that corresponds to the s p a t i a l operator A = — and x 9x2 0.1773 0.1685 0.1523 0.1310 0.1072 0.0836 0.0623 0.0446 0.0313 0.0223 0.0175 0.1685 0.1611 0.1472 0.1285 0.1074 0.0859 0.0659 0.0489 0.0356 0,0265 0.0214-0.1523 0.1472 0.1373 0.1236 0.1072 0.0897 0.0725 0.0569 0.0441 0.0348 0.0294 0.1310 0.1285 0.1236 0.1160 0.1059 0.0939 0.0807 0.0677 0.0561 0.0471 0.04x4 0.1072 0.1074 0.1072 0.1059 0.1026 0.0969 0.0891 0.0799 0.0707 0.0627 0.0571 0.0836 0.0859 0.0897 0.0939 0.0969 0.0978 0.0961 0.0920 0.0865 0.0807 0.0756 0.0623 0.0659 0.0725 0.0807 0.0891 0.0961 0.1008 0.1027 0.1020 0.0994 0.0956 0.0446 0.0489 0.0569 0.0677 0.0799 0.0920 0.1027 0.1108 0.1156 0.1170 0.1149 0.0313 0.0356 0.0441 0.0561 0.0707 0.0865 0.1020 0.1156 0.1258 0.1311 0.1308 0.0223 0.0265 0.0348 0.0471 0.0627 0.0807 0.0994 0.1170 0.1311 0.1395 0.1408 0.0175 0.0214 0.0294 0.0414 0.0571 0.0756 0.0956 0.1149 0.1308 0.1408 0.1428 Table (3.1) Value of the state t r a n s i t i o n matrix $^+1 k °f e c I n - (3.6.3). 0.0309 0.0227 0.0161 0.0111 0.0074 0.0048 0.0030 0.0018 0.0011 0.0007 0.0005 0.0227 0.0244 0.0177 0.0124 0.0085 0.0056 0.0036 0.0022 0.0014 0.0009 0.0006 0.0161 0.0177 0.0207 0.0150 0.0106 0.0G73 0.0049 0.0032 0.0021 0.0014 0.0010 0.0111 0-0124 0.0150 0.0189 0.0139 0.0C99 0.0069 0.0047 0.0032 0.0022 0.0017 0.0074 0.0085 0.0106 0.0139 0.0181 0.0134 0.0097 0.0069 0.0048 0.0035 0.0028 0.0048 0.0056 0.0073 0.0099 0.0134 0.0179 0.0134 0.0099 0.0072 0.0054 0.0045 0.0030 0.0036 0.0049 0.0069 0.0097 0.0134 0.0181 0.0138 0.0105 0.0082 0.0069 0.0018 0.0022 0.0032 0.0047 0.0069 0.0099 0.0138 0.0187 0.0147 0.0119 0.0103 0.0011 0.0014 0.0021 0.0032 0.0048 0.0072 0.0105 0.0147 0.0201 0.0168 0.0148 0.0007 0.0009 0.0014 0.0022 0.0035 0.0C54 0.0082 0.0119 0.0168 0.0230 0.0206 0.0005 0.0006 0.0010 0.0017 0.0028 0.0C45 0.0069 0.0103 0.0148 0.0206 0.0279 Table (3.2) Value of the disturbance t i a n s i t i o n matrix ^ of eqn. (3.6.3) 80 i t s associated boundary conditions (3.6.2). ± s given by (v. Appendix I f o r i t s derivation) °° A. At G k + l , k 0 ' » X , ) = . \ & X • i W * i C x ) » i = l where A ( ^ ^"P^ ) a n d ^ G O , 1=1,2,. functions of the s p a t i a l operator, and , (k+1)At (3.6.7) are the eigenvalues and eigen-Fk+1 k C x ' x , ) = S ' k+i,ic k A t G((k+l)At,x;t',x')dt' (3.6.8) The process noise i s assumed to be of zero mean and white i n space with covariance characterized by Q. (x,x*) = C 6(x-x') , (3.6.9) K W where C i s a constant. Further, the plant and measurement noises are assumed w to be uncorrelated with each other and with the state i n i t i a l c o n d i t i o n u^(x). eqn. (3.6.3) from the av a i l a b l e measurements provided by eqn. (3.6.4) i s , from eqn. (3.4.10), given by u k + 1 ( x ) = / G k + l 5 k ( x , x ' ) \ ( x ' ) d x ' + K k + 1 ( x ) c k + 1 , (3.6.10) where ^ ( x ) = [ ^ ( x ) K ^ C x ) ] , (3.6.11) a 1x2 vector as given by the formula (3.4.55), i s the d i s t r i b u t i v e f i l t e r gain and, corresponding to eqn. (3.4.11), 'k+1 i s the measurement r e s i d u a l . z k + l , l y k + l , l zk+l,2 yk+l,2_ { G k + i , k C x i ' x , ) V x ' ) d x ' / G. (x 0,x')u. (x')dx' 0 k+1,k 2 k (3.6.12) In order to obtain the d i s t r i b u t i v e f i l t e r gain K k + 1 ( x ) > t n e f i l t e r -ing error covariance given r e c u r s i v e l y by eqn. (3.4.57) must be solved. The 81 discrete-time R i c c a t i equation (3.4.57) can be solved i n the.following manner. F i r s t l y , we.observe that we'have the following expansions i n terms of the eigenfunctions (cf>.(x)}. oa G (x,x>) = E g ^ ^ i W ^ C x ' ) , C3.6.13) CO WCx'x,) = .\ C ^ i ^ ^ ^ ) > ( 3 - 6 - 1 4 ) ' 1 = 1 CO P k(x,x') = E p£j<|> (x)cf> (x') , (3.6.15) i , j = l CO Q (x,x') = E q j V (x)<f>. (x>) . (3.6.16) 1,3=1 By r e t a i n i n g the f i r s t M terms i n the above expansions, we have M A T G k + l . k k ' x , > S A 4 + l , k * i ( x ) * i ( x , ) • V x ) G k + l , k V X F k + i , k C x ' x , ) ~- f i + i , k * i ( x ) * i ( x , ) - $ > ) ] w v x , ) ' ( 3 - 6 - 1 8 ) M A T . P k(x,x') - E ^ p^.^M.^') = * ^ ( x ) P k * v ( x ' ) , (3.6.19) i,3=l M Q k(x,x') - E qkJ<f».(x)(f, (x') = *^(x)Q k* v(x') , (3.6.20) i,3=l where $ v(x) i s as defined by eqn. (2.6.54), and the matrices G k +^ k > F^+i k » P, and Q, , each of dimensions MxM, have t h e i r i i - t h elements as k k' J 4il,k-8k+l,k6ij» fk +l,k° fkil,k 6ij» P k J a n d V 1 ' respectively, i . j - 1 , 2 , . . . , M . • Substituting eqns. (3.6.17) to (3.6.20) into eqn. (3.4.36), we have the expansion i n terms of the f i r s t M eigenfunctions f o r the p r e d i c t i o n co-variance as 82 w C x ' x , ) = $v(x)[Gk+i5k.\GLi,k+ Fk+i,k\Fk+i,kJ\Cx,)' °-6-2l) whence, P k h l | k , whose i j - t h element i s defined to be P k + 1 | k » i s g i v e n as X „ -- T Pk+.l|k = Gk+l,kPkGk+l,k + \+l,kQkFk+l,k (3.6.22) Further, s u b s t i t u t i n g eqn. (3.6.22) into eqn. (3.4.43), we have (3.6.23) m 8, . (x) = * (x)P. I, [ E M. . .* (x.) ] k + l , i v k+l|k L k + l 5 i j v j such that by augmenting eqn. (3.6.23) f o r i=l,2,...,m according to d e f i n i t i o n (3.4.42), we have (3.6.24) W x ) = *v° c ) Pk+l|k Dk+l ' where the mxM matrix k+1 k+1,1 D k+1 ,m (3.6.25) has i t s i j - t h row I>k+1_ i defined by m A *" T k+l,i . Tc+l.ij v j (3.6.26) And, into the d e f i n i t i o n (3.4.45), s u b s t i t u t i n g eqn. (3.6.21) and then using d e f i n i t i o n (3.6.26) y i e l d Y i . n •• = D v . i i D ? . , . + R , . n (3.6.27) k + l , i j k + l , i k+1 k k+l,j k + l , i j such that i n accordance to d e f i n i t i o n s (3.4.44) and (3.6.25), we have \ + l ' D k + l P k + l k°k+l + \ + l ' C 3 - 6 ' 2 8 ) The d i s t r i b u t i v e f i l t e r gain <,,,(x) can now be approximately obtained K.« J-from eqn. (3.4.55) u t i l i z i n g eqns. (3.6.24) and (3.6.28) as 83 T T -1 K. ,. (x) = <S> (x)P I D v k+1. V k+11 k. k+1 k+1 = ^ ( x R , . , (3.6.29) v k+1 where we define K k + l = P k + l | k D k + l Y k + l • C 3 - 6 - 3 0 ) Further, the degenerate kernel ? k +-^ i n the expansion (3.6.19) of P (x,x') i s now obtained by s u b s t i t u t i n g eqns. (3.6.21), (3.6.29) and (3.6.2 (3.6.24) into eqn. (3.4.57) as Pk+1 " Pk+11 k " Pk+11 k D k + l \ + l D k + l P k + l | k • ( 3 * 6 ' 3 1 ) The estimator eqn. (3.6.10) can now be solved i n the following manner. F i r s t l y , compatible with the expansions (3.6.13) to (3.6.16), expand <L (x) i n terms of the eigenfunctions such that we have co u k(x) = E uyf^ Cx) , (3.6.32) i = l i 1 where u, = / u, (x)<J>.(x)dx . (3.6.33) K Q LC X By r e t a i n i n g the f i r s t M terms i n the expansion (3.6.33) for the approximate so l u t i o n of eqn. (3.6.10), we have M u. (x) = E <xY(x) = $ T(x)u, , (3.6.34) tC * i K. 1 V K. x=l where tl, i s a Mxl vector whose i - t h element i s i i , 1 . k k Using the truncated expansions (3.6.17), (3.6.34) and (3.6.29) i n eqn. (3.6.10), we have the c o e f f i c i e n t s i n eqn. (3.6.34) r e c u r s i v e l y given by \ + i = G k + i , k \ + W k + i ' ( 3 - 6 - 3 5 ) where, by s u b s t i t u t i n g eqns. (3.6.7) and (3.6.34) in t o eqn. (3.6.12), the measurement r e s i d u a l ? k + ^ 1 S now approximately given by 84 \ + l \ + i " \ + l G k + l , A C3.6-.36) With, the approximation, scheme for the s o l u t i o n of the. state f i l t e r (3.6.10) -now completely given, we proceed to implement i t on-line. The system eqns. (3.6.1) and C3.6.2) are simulated by eqn. (3.6.3) with 0, , and I\ ,, , given i n Tables (3.1) and (3.2), r e s p e c t i v e l y . The k+j_,k k+l,k simulated system i s assumed to be observed by eqn. (3.6.4) with the two measurement locations gaiven as x^=0.2 and X2=0.7. The plant noise covariance i s assumed to be Q (x,x') = 0.26(x-x') K. and the measurement noise covariance \ + l = 0.03 (3.6.37) (3.6.38) 1.0 0.1 0.1 1.0 For the f i l t e r eqn. (3.6.10), the number of eigenfunctions found s u f f i c i e n t to approximate i t i s M=3. The values of G k + 1 k a n d Fk+l,k are com-puted and shown to be Jk+l,k and k+l,k 0.9287 0.0 0.0 0.0964 0.0 0.0 0.0 0.3093 0.0 0.0 0.0589 0.0 0.0 0.0 0.0159 0.0 0.0 0.0237 (3.6.39) (3.6.40) Moreover, D, ,, i s found to be k+1 D. k+1 1 . 1 1 0 4 1.0545 0.3908 0.9287 -1.0023 -0.2863 C3.6.41) Assume the process to be at time k when u k and are obtained. The 85 c o e f f i c i e n t estimate.u, i s then p r e m u l t i p l i e d by G. ,, . to y i e l d the predicted c o e f f i c i e n t estimate u, i . . Knowing P. , Q. , G, ,, , and F, ,, , » ,, i, i s k+i|k. k. tc k+l,k k.+l,k. k+l|k calculated according to eqn. C3.6.22). and the f i l t e r gain c o e f f i c i e n t Ky+^ i s then conrouted using t h i s P, i. , D, and R, according to eqn. (3.6.30). k+1|k k+1 K+1 . Assume the process now to be at time k+1 when the measurement r\ ,, i s taken. K.+.I. To form the measurement r e s i d u a l £^'1' Quantity "kn|] c P r e m u l t i p l i e . d by ^k+l 1 S s u b t r a c t e d from the current measurement n ^ + ^ . The current c o e f f i c i e n t estimate u, ,.. i s then given by the addition to u, , , i . of the c o r r e c t i o n C, ,N k+1 6 J k+11 k k+1 premultiplied by K k + i ' The estimator response i s u ^(x) computed from eqn. (3.6.32). The quantities u, , a n d P, ,., , calculated by u t i l i z i n g P. ,.. i , , K ,., 1 k+1 k+1 k+1|k k+1 and Vy+X i n eqn. (3.6.31), are stored. The above chain of events i s then r e -peated. The state i n i t i a l condition i s assumed to be characterized by . °* • (3.6.42) P (x.x 1) = O.Olfi(x-x') . The optimal estimate u (x) i s shown as u, (0.), u, (0.5) and iL (1.) i n k. tc Ic tc Figs. (3.1), (3.2) and (3.3), r e s p e c t i v e l y , i n which the s o l i d l i n e s are the corresponding d e t e r m i n i s t i c system response u k ( x ) at x=0., 0.5 and 1. Further, the f i l t e r i n g error variance P^(x,x) at the measurement l o c a t i o n s x^=0.2 and x2=0.7 are shown i n F i g . (3.4). 3.7 Conclusion In t h i s chapter, we have considered the d e r i v a t i o n of the f i l t e r that y i e l d s an optimal l i n e a r estimate of the state of l i n e a r s t o c h a s t i c d i s c r e t e -time d i s t r i b u t e d parameter systems based on noise-corrupted measurements ob-tained at a f i n i t e number of s p a t i a l l o c a t i o n s i n the domain over which the system i s defined. The plant and measurement noises are assumed to be gaussian d i s t r i b u t e d and, f o r s i m p l i c i t y . i n presentation, uncorrelated with each other and with the state i n i t i a l c ondition, also assumed gaussian. That the f i l t e r -ing estimate i s optimal i s i n the sense of minimizing the variance of the f i l t e r i n g e r r o r . The d e r i v a t i o n of the optimal f i l t e r ' b e g a i n s with the im-p o s i t i o n that i t s structure be l i n e a r and that the p r e d i c t o r - c o r r e c t o r concept exhibited i n l i n e a r lumped parameter Kalman f i l t e r s be adhered to. Repeated a p p l i c a t i o n of the f i l t e r which i s recursive shows that the current estimate given by the f i l t e r i s i n f a c t obtained v i a a l i n e a r operation on current and a l l past measurement data. Next, with the structure of the f i l t e r e s t a b l i s h e d , the current-stage d i s t r i b u t i v e f i l t e r gain that weights the measurement r e s i d u a l , the product of the two being the corrector term, i s obtained so as to minimize the current-stage f i l t e r i n g e r r o r variance, shown to be ex-p r e s s i b l e r e c u r s i v e l y i n terms of i t s previous-stage value. An algorithm has been derived for the d i s t r i b u t i v e f i l t e r gain i n v o l v i n g a p a i r of simultaneous i n t e g r a l - d i f f e r e n c e equations which could be solved o n - l i n e . It i s seen that the d e r i v a t i o n of the optimal l i n e a r minimum-variance f i l t e r i s i n e f f e c t one of "parameter" optimization, the "parameters" being the d i s t r i b u t i v e f i l t e r gain. F i n a l l y , we show that the optimal f i l t e r -ed estimate i s orthogonal to the f i l t e r i n g e r r o r , that i s , they s a t i s f y a r e l a t i o n s h i p s i m i l a r to what i s known i n lumped parameter estimation theory as the orthogonal p r o j e c t i o n lemma, whence a Wiener-Hopf equation r e l a t i o n can be derived. A numerical example has been presented to demonstrate the use of the algorithm. Based on expansions of space-dependent functions i n terms of the eigenfunctions of the system s p a t i a l operator, the computational a l -gorithm i s reducible to algebraic r e c u r s i v e equations i n v o l v i n g the c o e f f i c i e n t s of expansions. This fa c t reduces computational e f f o r t and makes the implementability on-line of the estimator f e a s i b l e . 91 4. OPTIMAL PQLNTWLSE REGULATION CONTROL OF LINEAR DISCRETE-TIME STOCHASTIC DISTRIBUTED PARAMETER SYSTEMS 4.1 Introduction Much of the e f f o r t devoted i n recent years to the i n v e s t i g a t i o n of the various optimal c o n t r o l aspects of d i s t r i b u t e d parameter systems con-1 3 4 centrales on continuous-time d e t e r m i n i s t i c problems, see, for example ' ' ' 34 35 47-52 ' ' . Tzafestas and Nightingale, nevertheless, have considered a gen-e r a l continuous-time stochastic optimal (regulation) c o n t r o l problem i n X ^ where they have assumed noise-corrupted measurement data are a v a i l a b l e over the complete s p a t i a l domain. The controls i n the problems considered i n a l l the references c i t e d above are d i s t r i b u t e d and/or boundary c o n t r o l s . Such controls i n general vary continuously with respect to time and space and con-sequently pose severe computational and instrumentational d i f f i c u l t i e s . A possible and simpler a l t e r n a t i v e to c o n t r o l l i n g a d i s t r i b u t e d parameter system i s to use pointwise c o n t r o l which v a r i e s with respect to time only. We note i n passing that pointwise c o n t r o l includes the case of one-dimensional bound-ary c o n t r o l . Such a problem of optimally c o n t r o l l i n g a d i s t r i b u t e d parameter system by pointwise c o n t r o l has received some at t e n t i o n r e c e n t l y i n the l i t -erature ^ ' - ^ Hassan and Solberg ^ 3 have considered the case of a d i s c r e t e -time d e t e r m i n i s t i c d i s t r i b u t e d parameter system. They have t a c i t l y assumed that complete and e r r o r - f r e e measurement data on the system state are a v a i l -44 able. However, Matsumoto and Ito have considered i n an optimal .pointwise regulation c o n t r o l problem of a s c a l a r discrete-time d e t e r m i n i s t i c d i s t r i b u t e d parameter system and a separate estimation problem of the same system with discrete-time measurements assumed obtainable at several f i x e d s p a t i a l points, 92 but they have not included system noise i n e i t h e r of t h e i r problem formulat-ions and, f u r t h e r , have not shown the o p t i m a l i t y of the o v e r a l l c o n t r o l l e r -estimator control scheme. It i s the purpose of t h i s chapter to coneider the optimal pointwise regulation c o n t r o l problem of a s t o c h a s t i c discrete-time d i s t r i b u t e d parameter-system together with a measurement system y i e l d i n g noise-corrupted d i s c r e t e -time measurement data at only several f i x e d s p a t i a l points. The major d i f ~ 44 53 ference i n the present problem formulation from ' i s the i n c l u s i o n of system noise and, because of which, the use of an average cost f u n c t i o n a l as a measure of performance. The algorithm developed to compute the optimal con-t r o l p o l i c y i s a recursive one but, because of the f a c t that the algorithm i n i t i a l i z a t i o n has to be at f i n a l time, the computation must be done o f f - l i n e . S t i l l , j u s t as has been indi c a t e d i n 3.1, the s o l u t i o n of complex p a r t i a l d i f -f e r e n t i a l equations has been avoided because of the form of the discrete-time d i s t r i b u t e d parameter model used. The remainder of t h i s chapter i s divided into the following sections. In 4.2, the discrete-time l i n e a r s t o c h a s t i c pointwise r e g u l a t i o n c o n t r o l prob-lem i s f i r s t l y formulated. We then give an example i n 4.3 on how-the l i n e a r discrete—time sto c h a s t i c d i s t r i b u t e d parameter model i n the present problem formulation n a t u r a l l y a r i s e s when sampling i s introduced into a continuous-time stochastic d i s t r i b u t e d parameter system as demanded by the f a c t that the co n t r o l input to the system i s piecewise continuous with respect to time and that measurements are being taken at discrete-time i n s t a n t s . The c o n t r o l prob-lem posed i s then solved i n 4.4 by following an inductive approach. S t a r t i n g i n 4.4.1 with the assumption that state measurements are p e r f e c t , the optimal 22 23 co n t r o l and average cost are obtained v i a dynamic programming ' , and then i n 4.4.2, the case of imperfect state measurements i s treated using the (optimal) l i n e a r minimum-variance d i s t r i b u t i v e f i l t e r derived i n Chapter 3 but s u i t a b l y modified to accommodate presence of c o n t r o l . Moreover, we point 53 out i n 4.4.1 that r e s u l t s of Hassan and Solberg are the d e t e r m i n i s t i c counterparts obtainable from the sol u t i o n s of the ptesent s t o c h a s t i c problem formulation and shown i n 4.4.2, i n p a r t i c u l a r , the separation (uncoupling) of the optimal f i l t e r i n g estimator and the optimal c o n t r o l l e r , corresponding 24 25 26 to that ' ' i n lumped parameter theory. The chapter concludes i n 4.5 with anumerical example i l l u s t r a t i n g the use of the o v e r a l l c o n t r o l l e r -estimator control scheme. 4.2 Problem Formulation For convenience of reference i n t h i s chapter, we res t a t e below where appropriate the d e f i n i t i o n s of the various pertinent notations used i n Chapter 3. N i s the number of the subintervals the closed t i m e - i n t e r v a l [t„,t,_] i s d i s c r e t i z e d i n t o . I ={k: k=0,1,...,N-1} i s a discrete-time index set and {tn,t^,...,t^_^} i s the corresponding discrete-time set. The notation xeD denotes a generic point i n the f i x e d s p a t i a l domain D of the r-dimensional r A space E . Now, l e t I^={£: £=l,2,...,q} be a c o n t r o l l o c a t i o n index set and {x^,x^,...>Xq} be the corresponding c o n t r o l l o c a t i o n set. The c o n t r o l l e r s are assumed to be located at q d i s t i n c t points x^, ^ d ^ s i n the closed s p a t i a l domain D. Further, l e t the c o n t r o l at t , k e l , output by the c o n t r o l l e r k t £ located at x , £el , be a p -vector and be denoted by f . We term the c o n t r o l £ q K f, the pointwise c o n t r o l at x=x . We assume that the t o t a l pointwise c o n t r o l AC 36 . £ input to the system i s a l i n e a r combination of the controls f ^ , £=l,2,...,q. Consider a l i n e a r discrete-time s t o c h a s t i c d i s t r i b u t e d parameter system subject to pointwise c o n t r o l which can be described by the fo l l o w i n g vector d i f f e r e n c e - i n t e g r a l equation 9 4 u k + i ( x ) = 'D W ( x ' x ' ) u k ( x ' ) d D x « + * < + i 5 k ( - x ) f k + 'D F k + i , k ( x ' x ' ) w k ( x ' ) d D x ' ' ( A ' 2 - 1 ) k e l , where u (x.) i s the n-vector s t a t e at t, , G, n , (x,x') i s the nxn ker n e l of the state t r a n s i t i o n matrix i n t e g r a l operator, dD^, i s the elemental volume £ £ about the point x'eD, x s t n e n xPj?, c o n t r ° x t r a n s i t i o n matrix f o r f ^ , {f^, k = 0 , l , 2 , . . . , N - 1 } i s a p^-vector input pointwise c o n t r o l sequence, F . ,(x,x') i s the nxc kernel of the disturbance t r a n s i t i o n matrix i n t e g r a l operator, and {w (x), k = 0 , 1 , . . . , N - 1 } i s a c-vector input disturbance gaussian sequence. I ={i: i = l , 2 , . . . ,m} i s a d i s c r e t e measurement l o c a t i o n index s e t . m We assume that the set {x., i e l } i s d i s t i n c t from the set {x„, £ E I }. The x m £ q measurements of the system state are taken at d i s c r e t e instants of time by zero-memory transducers located at d i s t i n c t points x., i e l , i n the s p a t i a l J r x m domain. The measurement equation f o r the transducer located at x., """^ m' ^ s assumed to be given by m y k + l , i * . f 1 M k + l , i j U k + l ( x j ) + V k + l , i ' ( 4 - 2 > 2 ) where y, ,, . i s the d.-vector measurements taken at t, . - , k e l , by the zero-Jk+l,x x k + 1 ' t ' memory transducer located at x., i e l , M, ., .. i s a d.xn measurement matrix J x m lc+l,X3 x and {v, ., k = 0 , l , . . . , N - 1 } i s a d.-vector gaussian measurement err o r sequence. K. i X j 1 X £ We observe that by s e t t i n g £ ^ . = 0 f o r a l l £ = 1 , 2 , . . . , q and a l l k = 0 , l , . . . , N - 1 , eqns. ( 4 . 2 . 1 ) and ( 4 . 2 . 2 ) reduce to eqns. ( 3 . 2 . 1 ) and ( 3 . 2 . 4 ) f o r which the optimal f i l t e r i n g problem has been solved i n Chapter 3 . For n o t a t i o n a l convenience, we define an augmented nxp c o n t r o l t r a n -s i t i o n matrix k^ X^ as 95 W ( x ) = [ W ° ° : (4.2.3) and an augmented p-vector pointwise-control f^. as .1 -i where the number p i s given by (4.2.4) 1=1 *• (4.2.5) By using the d e f i n i t i o n s (4.2.3) and (4.2.4), we can conveniently rewrite eqn. (4.2.1) as U k + ! ( X ) 'D G k + l , k ( x ' X , ) u k C x ' ) d D x ' + \ + l , k ( x ) f k + ' D F k + l , k C x ' X ' ) w k C x ' ) d D x ' (4.2.6) Besides, we r e c a l l that, by using the d e f i n i t i o n s of the notations n k + ^ , Mjc+]_» u, and v. i n eqn. (3.2.5), we can augment eqn. (4.2.2) in t o the s i n g l e k+1 k+1 measurement equation \ + l = W k + l + Vk+1 (4.2.7) The system of eqns. (4.2.6) and (4.2.7) i s the one we w i l l use i n the sequel. The state i n i t i a l c ondition i s given as u^(x) whose mean and co-variance are E[ u Q ( x ) ] = u 0(x) = 0 , E [ u 0 C x ) u J C x ' ) ] = P Q C X , X ' ) . (4.2.8) The assumptions on the system and measurement noises w ^Cx) and v ^ + ^ are the same as given by eqn. (3.2.7). 96 Let the c o l l e c t i o n of pointwise controls f ^ , f ^ , . . . , f j _ T be denoted by X and r e c a l l that the c o l l e c t i o n of the state i n i t i a l c o n d i t i o n mean — i u 0(x) and the measurement data n, , n ... n. i s denoted by n . The c l a s s of 0 1 2 3 c o n t r o l p o l i c i e s we are to r e s t r i c t our a t t e n t i o n tc i s that of closed—loop c o n t r o l p o l i c i e s , that i s , c o n t r o l p o l i c i e s such that the c o n t r o l f^ at time tj> J £ I t > 1 S t o depend only on information that i s p h y s i c a l l y a v a i l a b l e f o r processing at that time, v i z . , the past and current measurement data and past controls f-' x . Since the past controls f ^ , f ^ , . . . , f _^ depend on U QC X ) , n^,..., n ^ , f.. depends on U Q ( X ) , n^,..., T K . I t i s c l e a r that the c o n t r o l f . at t. can be written i n the form J 3 f j =  f 3 < ' T ) i ' f : i _ 1 ) (4.2.9) for every j e l ^ . . Relation (4.2.9) expresses the f a c t that the c o n t r o l i s p h y s i c a l l y r e a l i z a b l e and also that i t i s closed-loop. The problem to be considered i n the sequel i s the s t o c h a s t i c optimal pointwise regulation c o n t r o l problem which can now be stated as follows: Given the system of eqns. (4.2.6) and (4.2.7), f i n d a p h y s i c a l l y r e a l i z a b l e c o n t r o l law f , j=0,l,...,N-1, of the form i n eqn. (4.2.9) which minimizes the average cost f u n c t i o n a l I ^ E t y , C4.2.10) where and L j + 1 ( U j + 1 ( x ) , f . ) - V D V i w V i ( x - x , ) V i ( x ' ) d V D x ' + f j V j (4.2.12) The matrices A^ + 1(x,x') and are s e l f - a d j o i n t p o s i t i v e - d e f i n i t e weighting 97 mar ices of dimensions nxn and qxq r e s p e c t i v e l y . 4.3 On the Linear Discrete-time Stochastic D i s t r i b u t e d Parameter Model with  Pointwise Control We give i n t h i s section an example of a discrete-time s t o c h a s t i c d i s t r i b u t e d parameter modal eqn. (4.2.6) which r e s u l t s from sampling a continu-ous-time system as required by the discrete-time measurement equation (4.2.7). We begin with the assumption that there are q c o n t r o l l e r s located at q d i s t i n c t points i n the closed s p a t i a l domain D. Let the c o n t r o l output £ by the c o n t r o l l e r located at x^, ^ ^ q ' ^ e a P^-vector and be denoted by f ( t ) . We assume that the t o t a l c o n t r o l input to the system i s a l i n e a r combination £ of the controls f ( t ) , £=1,2,...,q. Consider a l i n e a r continuous-time i s t o c h a s t i c d i s t r i b u t e d parameter system subject to pointwise c o n t r o l which can be described by the following vector p a r t i a l d i f f e r e n t i a l equation = A u(t,x) + i. tf,.(t)r (t;o(x-x„) t. JHt ,x)w(t ,x) (4.5.1) at £ = 1 f £ defined f o r t >t>t-. on D E , where u(t,x) i s the n-vector s t a t e , A i s a known f - - 0 x l i n e a r matrix p a r t i a l d i f f e r e n t i a l operator with respect to the s p a t i a l v a r i -£ able xeD, whose parameters may depend on both t and x, B^(t) i s a known rixp^ £ weighting matrix function for the p^-vector c o n t r o l f (t) impulsed at the s p a t i a l point x=x £, B(t,x) i s a known nxc matrix function and w(t,x) i s a c-vector s t o c h a s t i c disturbance input. The i n i t i a l and boundary conditions are assumed to be given as u ( t Q , x ) = u Q(x) , (4.3.2) g xu(t,x) = 0 , xe9D, (4.3.3) where u^(x) i s a stoc h a s t i c vector funct i o n , 3D i s the boundary surface of the s p a t i a l domain and 3 i s a l i n e a r p a r t i a l d i f f e r e n t i a l operator i n xe3D of order less than A . x 98 We observe that by .de f i n i ng an augmented nxp matr ix B^Ct) as r . A r I r s B f Ct ) = [B fCt.) (4.3.4) and a p-vector F(.t,x) as F(t,x) = f J "C t ) «Cx-x 1 ) f q ( t ) 6 C x - x q ) (4.3.5) where p= T, p , we can conveniently rewrite eqn. (4.3.1) as i = l i H i ^ 2 0 = A u C t : ) X ) + B f C t)F ( t , x ) + B ( t ,x )w(t ,x ) (4.3.6) We note that the cont r o l F(t,x) i s separable into two parts: the part depend-ent on time i s manipulable and the other dependent on space i s not. Further, by s e t t i n g F(t,x)=0, we obtain the unforced system f o r which i t s d i s c r e t e -time model has been obtained i n 3.3. We consider i n the following how eqn. (4.3.1) can be reduced to the form of eqn. (4.2.6) since we are again i n t e r e s t e d only i n the values of u(t,x) at the discrete-time instants t, ,. , k e l , as demanded by the discrete-time k+1 t measurement equation (4.2.7). The s o l u t i o n to eqn. (4.3.1) with the given i n i t i a l and boundary 4 34 conditions (4.3.2) and (4.3.3) r e s p e c t i v e l y can be expressed ' as u(t,x) = fL G ( . t , x ; t 0 , x , ) u 0 C x , ) d D x , + / t /_ G C t , x ; t ' , x ' ) Z B ^ C t ' ^ C t ' ^ C x ' - x )dD ,dt« t Q ° 4=1 f + / /_ G ( t , x ; t \ x ' ) B ( t \ x ' ) w ( t \ x ' ) d D ,dt' , t 0 (4.3.7) 99 where G(t,x;t',x') i s the nxn matrix Green's function s a t i s f y i n g the cond-i t i o n s (3.3.5) to (3.3.7). Now, l e t the closed t i m e - i n t e r v a l [t ,t ] be p a r t i t i o n e d into N subintervals such that the f i r s t N time points correspond to the d i s c r e t e -time index set I i n 4.2. Consider the system eqn. (4.3.1) i n the time sub-i n t e r v a l t, <t<t, ,, for some k e l . We assume that u(t, ,x) i s given. We must, k- - k+1 t k' & a moreover, assume that both the c o n t r o l input f (t) and the disturbance input w(t,x) are each a piecewise continuous function of time and that t h e i r changes of values occur simultaneously and only at the time instants t = t k > that i s , f*ct) = 4 > V t < t : k + i > and w(t,x) = w k(x) , ^ ' k + l -It then follows immediately from eqn. (4.3.7) that U ( t k + ! ' X ) = JD G ( t k + 1 , x ; t k , x ' ) u ( t k , x ' ) d D x , + [ / k + 1 / D G ( t k + 1 , x ; t ' , x ' ) E B j ( t ' ) 6 ( x ' - x £ ) d D x , d t ' ] f k  t k ^ L + / D [/ k + 1 G ( t k + 1 , x ; t ' , x ' ) B ( t ' , x ' ) d t ' ] w k ( x ' ) d D x , t k = ;D G ( t k + l ' X ; t k ' X ' ) u ( t k ' X ' ) d D x ' q fck+l I + E [/ K + L G(t ,x;t',x ) B * C t ' ) d t ' ] f k 1=1 t. k + / D [ / k + 1 G ( t k + 1 , x ; t ' , x ' ) B ( t ' , x ' ) d t ' ] w k ( x , ) d D x , . (4.3.8) C k Hence, by using the d e f i n i t i o n s of u k + ^ ( x ) , U K 0 0 > ^k+1 k^ X' X'^ F (x,x') i n eqn. (3.3.9) and d e f i n i n g an nxp c o n t r o l t r a n s i t i o n matrix 100 H k + l , k ( x ) a s H k + i , k C x ) " ; k + 1 G C \ + r x ; t ' 5 x )B^Ct')dt'- , t k we can conveniently rewrite eqn. (.4.3.8) as • q £ £ U k + ! ( X ) = ' D G k + l > k ( x ' X , ) u k ( x , ) d D x > + ^ V l , k ( x ) f k + 'D F k + l , k ( x ' X ' ) w k ( x , ) d D x -(4.3.9) ) (4.3.10) for k el. We observe that eqn. (4.3.10) i s the same as eqn. (4.2.1). To further reduce eqn. (4.3.10) to the following form (eqn. (4.2.6)) U k + ! ( X ) - ' D G k + l , k ( x ' X ' ) u k ( x ' ) d D x ' + + / D F k - , - l , k ( x ' X ' ) W k ( x ' ) d D x ' ' ( 4 - 3 - 1 L ) where f, i s as defined by eqn. (4.2.4), we see that H, ,, ,,(x) as defined by ix eqn. (4.2.3) has to be given by t. i x i -L. • rv "k+1 \ + l , k ( x ) = f Y ( t k + 1 , f ,x )3(t')dt' , (4.3.12) where the nxn 2 matrix function y ( t j + ^ , t ' ,x) i s defined as Y ( t k + 1 , t * ,x) = [ G ( t k + 1 , x ; t ' , X ; L) and the n 2xp matrix function 3(t) i s defined as B^(t)-ICO = •G(t k + 1,x;t',x )] (4.3.13) (4.3.14) ; B q ( t ) We have shown as an example how the discrete-time d i s t r i b u t e d para-meter model eqn. (5.3.11) (eqn. (4.2.6)) can a r i s e by sampling the input of a 101 continuous-time system eqn. (4.3.1). Though eqn. (4.3.1) i s reducible to eqn. (4.3.11), the converse c l e a r l y i s generally not true. . - \ 4.4 Derivation of the Stochastic. Optimal Pointwise Control The procedure developed to obtain the s o l u t i o n to the s t o c h a s t i c optimal closed-loop pointx^ise c o n t r o l problem formulated i n 4.2 i s as follows. F i r s t l y , the simpler case where the state i s exactly measured i s treated using the dynamic programming approach. Next, the case with noisy discrete-space state measurements req u i r i n g the reconstruction of the state using the d i s -t r i b u t i v e f i l t e r developed i n Chapter 3 i s considered and the o v e r a l l c o n t r o l scheme optimality shown. F i r s t of a l l , we denote the cost f o r the l a s t N-k stages by where I N _ k = V L (u ( x ) ^ . ^ ^ ^ - 1 ) ) , (4.4.1) 3=k and by d e f i n i t i o n I Q = 0 . (4.4.2) From the r e l a t i o n between c o n d i t i o n a l and unconditional expectations, we can write TN = E [ E [ I N - k l k ] ] ' ( 4 - 4 < 3 ) where .|k denotes the t o t a l i t y of a v a i l a b l e information at time k. I t i s then noted immediately that l ^ _ k i s minimized by minimizing the c o n d i t i o n a l ex-pectation of I„ , , that i s , r N-k JCL,N-k " E [ V k l k ] N-i ; . . , = E[ E. L. C u . + 1 ( x ) , f . ( t r 1 , ^ i ) )|k] . (4.4.4) 22 23 Now, the dynamic programming approach ' i s as follows. Pro-ceeding f i r s t ' t o the last-but-one stage, the cost-to-go J i s minimized by choosing the' optimal c o n t r o l f° ; the r e s u l t a n t minimum of J . i s denoted JN—X Oij, X by J° . At time M-2, the cost-to-go J by usiiig any f 0 but with the optimal control f° , i s N-l JCL,2 - E t W V - l ( x ) » f N - 2 > + J C L , l l N - 2 ] ' The optimal c o n t r o l f ^ _ 9 that minimizes J „ i s then chosen. The procedure of backward stepping and minimization i s repeated, thus leading to the recur-s i v e r e l a t i o n J° CL,N-k " m f i n E [ J C L , N - k - i + V D u k + i ( x ) \ + i ( x ' x ' ) u k + i ( x , ) d D x d D x - + f k B k F J K ] > k (4.4.5) k=N-l,N-2,...,0. The s o l u t i o n to eqn. (4.4.5) can be assumed to be of the following form: JCL,N-k = V D U k C x ) V N - k ( x ' X ' ) u k ( x ' ) d D x d D x ' + 3N-k ' ( 4 ' 4 - 6 ) For notational convenience, define S N_ k(x,x') = A k(x,x') + V N_ k(x,x') . (4.4.7) The v a l i d i t y of t h i s assumed s o l u t i o n f o r cases where measurements are a v a i l a b l e w i l l be confirmed by s u b s t i t u t i o n of eqn. (4.4.6) in t o eqn. (4.4.5) and, moreover, recursive r e l a t i o n s f o r V^_ k(x,x') and k obtained. 4.4.1 P e r f e c t l y Observed State In t h i s case, i t i s assumed that the measurements, at time k, are the past and current states u_. (x) , j=0,l,.. . ,k, somehow measured exactly and a v a i l a b l e . Let the c o l l e c t i o n of these states be denoted by 103 u * Cx) = {u Cx)»• . . jU^Cx)}. • The optimal problem, then, i s that of choosing the fun c t i o n a l r e l a t i o n s f .=f .Cu J(x) ?f 3 L ) , coresponding to eqn. (4.2.9), so as to minimize the con d i t i o n a l expected cost-to-go, corresponding to eqn. (4.4.4), JCL N-k = E [ 1 L x ( u tCx),f CuJCx),fJ 1 ) ) | k ] , (4.4.8), ' j=k J J " J k subject to state constraints eqn. (4.2.6). Here, since u (x) are the a v a i l -able measurements, E[*|k] i s understood to be E [ * | u k ( x ) , f k ^ ] . Substituting the assumed s o l u t i o n (4.4.6) with, k replace by k+1 in t o the recursive r e l a t i o n (4.4.5) and i n s e r t i n g the state eqn. (4.2.6) y i e l d JCL,N-k - E V D V D u^n>GLi,^  + 2 /D V D ^ x ' ^ l / x ^ W l ( x ' X , V I k ( j , ) f k d ] , A ' d D x " + f k ( V D H ^ + l , k C x ) S N - k - l ( X ' X , ) \ + l , k ( x ' ) + \ } f k + Vk - i l k ] • • (4.4.9) To obtain the optimal c o n t r o l f° that minimizes J T , , eqn. (4.4.9) i s d i f f e r e n t i a t e d with respect to f and the d e r i v a t i v e equated to zero such that f k = _ / D \ 1 S k ( x ) u k ( x ) d D x ' • ' (4.4.10) where \ = V D <+i,kCx)SN-k-i(x'x,)\+i,kCx')dDxdDx' + B k - ( 4 - 4 - n ) and VX) = V D < + l , k ( x , ) S N - k - l ( x , ' X " ) G k + l , k ( x " ' X > d D x ' D V / ( 4 - 4 ' 1 2 ) 104 Further, s u b s t i t u t i o n of f° from eqn. (_4.4.10) into. eqn. (4.4.9) f or R. J _ T „ , y i e l d s an expression f o r "J° „ , such, that, by. comparison with i t s CL,N-k 1 v CL,N-k ' 1 * assumed'solution (4.4.6), the following recursive r e l a t i o n s f o r V^_ k(x,x') and aN-k a r e ° ^ t a i n e d ' r e s p e c t i v e l y , V k C x > x , ) V D G k + i , k C x " ' * > V k - i ( ^ ^ V D < + i , k ( x ^ x ) V k - i ( x ^ x n , ) \ + i , k ( x , , ' ) d \ " d V ' [ V D < + i , k ( x M ) V k - i C x ^ x , , , ) H k + i , k ( x " ' ) d D x " d D x - + V"1 V D i i A C x , ' ) V w ( x l ' x ' M ) G k + i ) k ( x ' ' ' ' x , ) d D x " d D x - ' ( 4 ' 4 - 1 3 ) and aN-k a N - k - l + T R L V D V D ^ k + l . k ^ ' ^ V k - l ^ (4.4.14) k=N-l,N-2,...,0. From condition (4.4.2), the " i n i t i a l " conditions f o r V N_ k(x,x') and a„, , are r e s p e c t i v e l y N-k and V 0(x,x') = 0 , (4.4.15) a Q = 0 . (4.4.16) Thus., at time k, the minimal expected value of the cost-to-go using the optimal control f° i s , from eqns. (4.4.3) and (4.4.6), ^ - k = E [ V D U k ( x ) V N - k C x ' X ' ) u k ( x ' ) d D x d D x ' ] + aN-k > ( 4 ' 4 - 1 7 ) /• . . . . where V^_ k(x,x') and a^_ k are given by eqns. (4.4.13) and (4.4.14), r e s p e c t i v e -l y . To c a l c u l a t e V N__.(x,x'), s u b s t i t u t e the i n i t i a l c ondition (4.4.15) in t o 105 eqn..(4.4.13)' with k=N-l and use t h i s equation r e c u r s i v e l y backwards down to k=j. A s i m i l a r procedure i s followed to obtain a . with the appropriate i n i t i a l condition (4.4.16), We observe that a ^ as given by eqn. (4.4.14) includes the effect' of system disturbance from stages N-k to N. Moreover, a N _ k appears i n eqn. (4.4.17) as an a d d i t i o n a l term and therefore represents the extra cost we must pay because the system i s subject to s t o c h a s t i c disturbance. It i s of i n t e r e s t to note that by s e t t i n g w k(x)=0, k=0,1, . . . ,N-.l, i n eqn. (4.2.6) with the consequence that a ,=0, the corresponding deter-JN—R m i n i s t i c problem with exact state measurements i s obtained. The minimum value of the cost-to-go f o r t h i s d e t e r m i n i s t i c case i s therefore given by eqn. (4.4.17) with the term due to system s t o c h a s t i c disturbance omitted and the expectation operator E dropped, where V . (x,x') Is s t i l l given by eqn. IN K. (4.4.13) and the optimal nointwise c o n t r o l f.° by eqn. (4.4.10). Hence, the 53 r e s u l t s i n are but for a p a r t i c u l a r case of the present general one. 4.4.2 Noisy State measurements To emphasize the nature of the c o n t r o l used, the c o n t r o l w i l l appear e x p l i c i t l y i n the argument of the cost f u n c t i o n a l . Let, at stage k, I ^ _ k ( f ^ ) be the cost to complete the process using some p h y s i c a l l y r e a l i z a b l e c o n t r o l s a t i s f y i n g the f u n c t i o n a l r e l a t i o n (4.2.9), and I°_^(f°) be that using some feedback c o n t r o l f° not using the f i l t e r e d estimate u (x) but with the R R optimal feedback c o n t r o l gain - A ^ B (x) , where A^ and B^(x) are s p e c i f i e d by eqns. (4.4.11) and (4.4.12), r e s p e c t i v e l y . The d i f f e r e n c e between the two costs i s obtained by subtracting eqn. (4.4.9) but with f ^ replaced by f° from eqn. (4.4.9) such that = E[E [ J N _ k Cf k )|k] ] - E[E[J°_ k(f°)|k]] 106 - fkT{vD ^ l f k « wi^x,)Hk+i;"k0t,)di)xdDx'+ \}fk + 2 / D V D "k^^l,^^^ - fk)] C4.4.18) T Using eqn. (4.4.10) to remove u"(x") i n the l a s t term of eqn. (4.4.18), we have Vk(fk> - -^k(fk> " E t fkVk- f f Vk- 2 f f V f k- f k>J = E ^ f k - f k ) T \ ( f k - f k ^ > ( 4 - 4 - 1 9 ) where A^_ i s as given by eqn. (4.4.11). Since the control f° does not use the optimal f i l t e r e d estimate, i t can i n general be put into the following form f° = f° + f° . (4.4.20) k k k Here, where the optimal feedback con t r o l gain -A^B^Cx) i s s p e c i f i e d by eqns. (4.4.11) and (4.4.12) together, the optimal f i l t e r e d estimate u^(x) by eqn. (3.4.6): m \ ( x ) = \ | k - i ( x ) + A K k , i ( x ) z k , i * ( 4 - 4 - 2 2 ) where, to r e f l e c t the fac t that the co n t r o l f k _ j _ i s input to the system, the predicted estimate u k| k_^(x) i s now given by ; \ | k - l ( x ) = ' D G k , k - l & ' x , ) V l ( x , ) d D x ' + \ , k - l W f k - l ' (4-4.23) 107 and, moreover, where the f i l t e r i n g error u (x) i s defined by eqn. (3.2.10). It i s noted, since u ^ G O i s a l i n e a r function of UQ (X) , • . • , n^ j f ^ as given by eqn. (4.4.21) does s a t i s f y condition (4.2.9). Then, eqn. (4.4.19) can further be rewritten = E f ( f k - / k ) T \ ( f k - K> - 2 < f k - + • ( 4 - 4 - 2 5 ) Now, for t h i s case of the f i l t e r e d estimate xi (x) given by eqn. (4.4.22) having the co n t r o l f ^ - i i - n i t , we make the following observations. ' " - " k - i " ° ' " ' : • i<,.1. " i estimate, the measurement r e s i d u a l z, . remains given by eqn. (3.4.20). More-k , i over, since the control f ^ _ ^ appears i n both the state eqn. (4.2.6) and the f i l t e r eqn. (4.4.22), the f i l t e r i n g error u (x) remains given by eqn. (3.4.22).' K. Hence, i t i s clear that eqn. (3.5.3) remains true. Then, as a consequence of the fact that the estimate ^ ( x ) i s a l i n e a r f unction of UQ(X) ,r-|^ 5. . . ,1")^, the v a l i d i t y of eqn. (3.5.3) and the p h y s i c a l r e a l i z a b i l i t y c o n d i t i o n (4.2.9) f o r the c o n t r o l fj c_^> w e see that, upon the choice of the optimal d i s t r i b u t e d f i l t e r gain K k ( x ) given by eqn. (3.4.55), eqn. (3.5.6) s t i l l holds true i n t h i s case. Hence, f i s orthogonal to f°. Furthermore, f° as defined by eqn. (4.4.21) i s a l i n e a r function of ( x ) » • • • , s u c h that, again i n view of eqn. (3.5.6), f° i s also orthogonal to f°. Hence, eqn. (4.4.25) reduces to - E t ( f k - ^ T \ ( f k - $ + K X Q • ( " - 4 - 2 6 ) 108 We note immediately that the choice . f k - f° ' (4.4.27) r e s u l t s i n the d i f f e r e n c e , eqn. (4.4.26) being a minimum such that Wft - - W?\& . (4.4.28) Moreover, for any con t r o l f ^ not s a t i s f y i n g eqn. (4.4.27), we observe, by subtracting eqn. (4.4.28) from eqn. (4.4.26), that W V - -^k«k> = E [ ( f k - ^>\Cfk - K » > ( 4 - 4 - 2 9 ) whence, as the quantity on the r i g h t hand side of eqn. (4.4.29) i s p o s i t i v e , W V > • ( 4 - 4 / 3 0 ) Hence, the cost-to-go for the c o n t r o l f° i s the minimum cost-to-go so that V k ^ = *!U<fr > ( 4 - 4 ' 3 1 ) and f ^ i s therefore the optimal pointwise feedback c o n t r o l . From the d e f i n i t i o n (4.4.24) of f°, i t can be shown that E k^TVk] = T R V D X ( x ) ^ \ ( x , > p k ( x ' » x ) d D x d D x ' • ( 4 ' 4 ' 3 2 ) Hence, from eqns. (4.4.28), (4.4.31), (4.4.17) and (4.4.32), the cost-to-go f o r t h i s case with noisy measurements i s ^-k(^ = E [ V D Uk(x)VN-k(x'X')uk(x')dDxdDx'] + aN-k. • (4-4'33) where V N_ k(x,x') i s given by eqn. (4.4.13) and aN-k = V - k - l + ^ VD- 5k ( x )\\(X '^k C x'' x ) d Dx d Dx' (4.4.34) 109 The i n i t i a l conditions f o r V ^ (x,x'), eqn. (4.4.13), and a^ ^ , eqn. (4.4.34), are, again from condition (4.4.2)., r e s p e c t i v e l y , V Cx ,x ' ) =0 , (4.4.35) and a Q = 0 . (4.4.36) To c a l c u l a t e V .(x,x') 5 s u b s t i t u t e the i n i t i a l c ondition (4.4.35) ^ J into eqn. (4.4.13) with k=N-l and use t h i s equation r e c u r s i v e l y backwards down . to k=j. A s i m i l a r procedure with the appropriate i n i t i a l c ondition (4.4.36) i s followed to obtain a„ .. N-j It i s noted that a^_ k as given by eqn. (4.4.34) now includes, i n addition to the e f f e c t of system disturbance, the e f f e c t of estimation e r r o r , from stages N-k to N. The term a^_^ appearing i n eqn. (4.4.33) therefore now represents the extra cost we must pay f o r the system being subject to stocha-s t i c disturbance and for the system state not being measured exactly. Summarizing the r e s u l t s obtained, we can conclude from eqns. (4.4.27) and (4.4.21) that: the optimal c o n t r o l system for the l i n e a r s t o c h a s t i c point-wise re g u l a t i o n c o n t r o l problem consists of the optimal l i n e a r d i s t r i b u t i v e f i l t e r whose output i s operated upon by the optimal d i s t r i b u t e d feedback c o n t r o l gain obtained from solving the corresponding l i n e a r d e t e r m i n i s t i c pointwise regulation c o n t r o l problem. The two parts of the co n t r o l system are determined separately. The cost f u n c t i o n a l f o r the o v e r a l l c o n t r o l system i s governed by eqn. (4.4.33). 4.5 Numerical Example Consider the problem of c o n t r o l l i n g the one-sided metal slab heating process described by the following one-dimensional s c a l a r d i f f u s i o n equation 9u A(t,x) 9 2 u A ( t , x ) : = + w(t,x) , ' (4.5.1) 9t 3x 2 110 where u.,.(t,x) at t f>t>0 i s the.temperature d i s t r i b u t e d over the s p a t i a l domain 0<x<l and w(t,x) the additive, plant noise. The boundary conditions to eqn. (4.5.1) are assumed given as . 3 xu,(t,x)] = — — .= 0 , x=0 (4.5.2) 3u,.(t,l) B ^ C t . x ) ] = u ^ ( t , l ) + 3 - = 3f * C t ) , x=l where B i s the heat trans f e r c o e f f i c i e n t assumed to be unity and f, v(t) i s an external manipulable heat source. Assuming that the steady value of the co n t r o l to maintain the st a t e at the steady value u i s f , we see from the second r e l a t i o n of eqn. (4.5.2) ss ss that f = u . (4.5.3) ss ss i T r t j „ .c; — r u(t,x) = u. (t,x) - u S£ and (4.5.4) f (t) = f , (t) - f * ' s s we can write eqns. (4.5.1) and (4.5.2), by f i r s t s u b s t i t u t i n g into them r e l a t i o n (4.5.4) and then extending the d e f i n i t i o n of the s p a t i a l operator A = — 3 4 to convert the non-homogeneous boundary conditions into the homo-x 2 3x geneous ones, as 3 u ( t i x ) _ = 3 2u(t,x.) + w C t > x ) + 6 ( x . 1 ) f ( t ) - f 8 t 3x 2 M l i O l = 0 (4.5.5) oX u ( t , l ) f ^ I l = 0 . The o r i g i n a l boundary con t r o l problem cast i n t h i s form i s seen to I l l be.reduced to but a s p e c i f i c example of the pointwise c o n t r o l problem w i t h i n the framework of the theory- developed'in the previous parts.of t h i s chapter. For the simulation of the l i n e a r s t o c h a s t i c d i s t r i b u t e d system eqn. ( . 4 . 5 . 5 ) , the approach outlined i n 2 . 6 and 3 . 6 i s repeated here r e s u l t i n g i n the following equivalent discrete-time representation U T j . i = *i a.1 i u T + r i . i * w i + ¥ i j . i i f i » ( 4 . 5 . 6 ) k+1 k+1, k k k+1, tc k k+1, k k where $, , . , , r, ,, , , each of dimension N xN , and 1' ., _ , , of dimension N x l , k+l,k k+l,k eq eq k+l,k eq are the state, disturbance and c o n t r o l t r a n s i t i o n matrices, r e s p e c t i v e l y . For N =11, Ax=0.1 and At = 0 . 1 , the matrices $, ., , and r, ,., , have eq k+l,k k+l,k t h e i r values as shown i n Tables ( 3 . 1 ) and ( 3 . 2 ) , r e s p e c t i v e l y , and, f u r t h e r , ^k+l k ^ S c o m P u t e d t o ^ e a s shown i n Table ( 4 . 1 ) . 0 . 0 0 4 3 0 . 0 0 9 7 0 . 0 1 6 2 0 . 0 2 6 7 0 . 0 4 2 5 0 . 0 6 5 6 0 . 0 9 7 8 0 . 1 4 1 0 0 . 1 9 6 6 0 . 2 6 5 6 Table ( 4 . 1 ) Value of the con t r o l t r a n s i t i o n matrix ^ + 1 k of eqn. ( 4 . 5 . 6 ) . The discrete-time system eqn. ( 4 . 5 . 6 ) i s assumed to be observed at 112 two given s p a t i a l locations x=-x and x=x 2, .0<x 1,x 2<l, corresponding to eqn. (4.2.7) with m=2 and M, ,, ....=1 and M. . . = 0 , i ^ j ? by', t h e measurement equation tc*T~X j xx, KLI x j x j yk+l,1 U k + l ( x l } " " v k + l , l ~ = + yk+l,2 _Vk+l,2_ C4.5.7) where v, , =col (v, ... - v 0) i s the gaussian noise added to the measure-K.+1 K+J- , 1 K+1 , Z ment equation. The measurement noise i s assumed to be of zero mean and i t s covariance characterized by (4.5.8) \ + l = C v where C i s a 2x2 constant matrix, v Following the argument presented i n 3.2 and 4.3, eqn. (4.5.5) can be, by d i v i d i n g the time i n t e r v a l [ t ^ j t ^ ] i n t o N equal i n t e r v a l s each of f i x e d duration At, cast into the following discrete-time representation, *> 1 v. .-60 = f G. , Cx.xMn. fxMrfx 1 R+1 * ' 0 K+1,k K + i l F k + l , k ( x ' X , ) w k ( x ' ) d x ' + H k + l , k ( x ) f k ' ( 4 ' 5 - 9 ) where k, k=0,1,...,N-1, i s the discrete-time index. In the above equation, G. , (x,x') and F, ,.. , (x,x') are as given by eqns. (3.6.7) and (3.6.8), r e s -k+1, k k+1, k p e c t i v e l y , and ^(x), according to eqn. (4.3.9), i s given by Hi u.i , < x) = / 1 ( ^ + 1 ) A t / 1 G ( ( k + l ) A t , x ; t ' , x ' ) 6 ( x , - l ) d x ' d t . (4.5.10) k+l,k kAt 0 The plant noise i s assumed to be of zero mean and white i n space with covariance characterized by Q (x,x>) = C t 6 (X - X ' ) , (4.5.11) K W where C i s a constant. Moreover, the plant and measurement noises are w assumed to be uncorrelated with each other and with the state i n i t i a l c ondit-ion u Q(x) It i s now required to construct a c o n t r o l sequence f ^ , k-0,1,..,N-1, 113 which minimizes the cost f u n c t i o n a l N-1 i L = E[At L / u 2 (x)dx + B f 2 ] , (4.5.12) N • , = 0 o 3+1 3 where B i s a p o s i t i v e constant, and assumed to be equal to 0.1 and N=40. The optimal control sequence s a t i s f y i n g the p h y s i c a l r e a l i z a b i l i t y condition (4.2.9) i s shown to be. given according to eqn. (4.4.21) by K = ^ 1 5 k C x ) f l k ( x ) d x , (4.5.13) where and B^(x) are given by eqns. (4.4.11) and (4.4.12), r e s p e c t i v e l y . The estimate u. (x) i s output from the estimator K. \ + i ( x ) = { G k + i , k ( x ' x , ) \ ( x , ) d x ' + K k + i ( x ) ? k + i + W ( x ) * k • ( 4 ' 5 - 1 4 ) where K^ +^(X) i s the di s t r i b u t i v e , f i l t e r gain given by eqn. (3.4.55) and C k + ^ i s the measurement r e s i d u a l given by eqn. (3.6.12). It i s c l e a r that i n order to obtain the optimal c o n t r o l f°, A^ and B^(x) must f i r s t be computed which e n t a i l s the recursive backward s o l u t i o n of the discrete-time f u n c t i o n a l equation (4.4.13) to y i e l d S^_ k(x,x') for k=N-l,...,1,0. Since the computation of eqn. (4.4.13) i s to proceed backward i n time, the time h i s t o r y of S^_ k(x,x') must be determined p r i o r to system operation, that i s , i t must be precomputed and stored. Now, eqn. (4.4.13) can be solved i n the following manner. We ob-serve that we can expand, i n terms of the eigenfunctions {<f>^ (x) > besides G (x,x') as given by eqn. (3.6.13), the following K.*T"1 y K (4.5.15) V N_ k(x,x') = E v ^ . ( x ) 4 (x') , (4.5.16) i,3=l 114 S N Gc,x'-).= . E. s j 1 <(> ( x H (x') , C4.5.17) OO and A (x, x') = E. af j<j>. (x)<(.. (x') . (4.5.18) R i , j = l ^ J By r e t a i n i n g the f i r s t M terms i n the above expansions f o r the approximate s o l u t i o n of eqn. (4.4.13), we have W 0 0 * "\ *i+i,v*±™ = * > > W ' ( 4- 5- 1 9 ) 1=1 M V N_ k(x,x') = E v ^ . ( x ) c j ) (x') = ^(x)V N_ k$ v(x') , (4.5.20) i , j = l SN_k(x,x') = E s ^ . ( x ) * (x-) = ^ ( x ) S N _ k * v ( x « ) , (4.5.21) M A^Cx .x ' ) - E akJ4>.(xH (x*) = ^(x)A k$ v(x I) , (4.5.22) i>j=l where $ v(x) i s as defined by eqn. (2.6.54), and the matrix H k + 1 k of dimensions Mxl, has i t s i - t h element as h. . , and the matrices V,, . , S , and A. , each ' k+l,k N-k N-k Tc i i i i i i of dimensions MxM, have t h e i r i j - t h elements as v ^ k » s N _ k a n d aN_k> r e s ~ pe c t i v e l y , f o r i,j=l,2,...,M. Moreover, from eqn. (4.4.7), we see that Now, s u b s t i t u t i n g eqns. (3.6.17), (4.5.19) and (4.5.21) into eqn. (4.4.13), we have, keeping eqn. (4.5.23) i n mind, V k = SN-k " \ T Gk+l,k SN-k-l Gk+l,k T T —1 Gk+l,k SN-lc-l l Ik+l,k [ Hk+l,k SN-k-l\+l,k + \ ] T , Hk+l,k SN-k-l Gk+l,k ' (4.5.24) 115 k=N-l,...,1,0.. The condition t o . i n i t i a t e the backward s o l u t i o n of eqn. (4.5.24) i s seen from eqns. (4.4.35) and (4.4.8) a f t e r s u b s t i t u t i n g eqns. (4.5.21) and (4.5.22) to be S Q = \ . . (4.5.25) Further, s u b s t i t u t i n g eqns. (4.5.19) and (4.5.21) into eqn. (4.4.11). we have \ = < + l , k S N - k - l \ + l , k + B k > ( 4 - 5 ' 2 6 ) and s u b s t i t u t i n g eqns. (4.5.19), (4.5.21) and (3.6.17) into eqn. (4.4.12), we have \ C X ) = ^ + l , k S N - k - l G k + l , k * v ( x ) = B, * (x) , (4.5.27) k v where we define B, = H,T., , S„ ... ,G,.., . (4.5.28) i V IV'I JL * IV i-N"" iV"' J JL. IV t JL * IV Now, we have the expansion (3.6.32) of ^ ( x ) i n terms of the eigenfunctions {c(>^(x)K_^ which i s , i n addition to the expansions (3.6.13) to (3.6.16), com-p a t i b l e with the expansions (4.5.15) to (4.5.18). Hence, the truncated ex-pansion (3.6,34) of u, (x) i n terms of the f i r s t M eigenfunctions can s t i l l be used i n t h i s approximation scheme. Su b s t i t u t i n g eqns. (4.5.27), (4.5.28) and (3.6.34) into eqn. (4.5.13), we have the optimal c o n t r o l given as Substituting eqns. (3.6.17), (3.6.29) and (4.5.19) in t o eqn. (4.5.14), i t i s seen that the u. (x) c o e f f i c i e n t estimate i i i s output from the following lump-k k • ed parameter f i l t e r Vt-1 ~ "k+l,k"k ' '^k+l^k+l where the computation of the f i l t e r gain K ^  i s as that has been d e t a i l e d i n u, ,, = G. .. , fi. +<,.,?,., + H. ,. . f, , (4.5.30)  k+1 k+1 lc+l,k k 116 3.6 and the measurement r e s i d u a l ? k + ^ r e m a i j : i s given approximately by eqn. C3.6.36). With the approximation scheme f o r the s o l u t i o n of V k(x,x') a n ^ the f i l t e r eqn. (4.5.14) completed, we proceed to obtain the numerical s o l u t i o n of the stoc h a s t i c optimal pointwise c o n t r o l problem defined by eqns. (4.5.9), (4.5.7) and (4.5.12). It i s found that the number of eigenfunctions needed for t h i s numerical example i s M=3. The value of G has been computed i n 3.6 and i s given by eqn. (3.6.39). The con t r o l t r a n s i t i o n matrix i s computed and shown to be \ + l , k Using these numerical values of G^ + 1 and H v + 1 v i n eqn. (4.5.24) with A ^ I ^ A t and B k=BAt=0.lAt, we compute V k> k=N-l,...,1,0, the i n i t i a l c o n d i t i o n being VQ=0., and store them f o r l a t e r use. The s o l u t i o n VN_^(x,x') constructed according to eqn. (4.5.20) i s shown i n F i g . (4.1) as a function of x at x'=l. with k as the parameter and alos shown i n F i g . (4.2) as a function.of k with x=x'=l. The system (4.5.5) i s simulated by eqn. (4.5.6) where the matrices \ + l k' Fk+1 k a n d \ + l k a r e § i v e n i n T a D l e s (3-1), (3.2) and (4.1), respect-i v e l y . The simulated system i s assumed to be observed by the measurement equation (4.5.7) with the measurement lo c a t i o n s given as x^=0.2 and X2=0.7. The plant noise covariance i s assumed to be Q k(x,x') = 0.26(x-x') (4.5.32) and the measurement noise covariance 0.0708 -0.0769 0.0328 (4.5.31) 118 \ 119 \ - M = ° - 0 3 1.0 0.1 0,1 1.0 (4.5.33) For the estimation problem, the values of G^ +^ ^, F ^ ^ ^ and D^+^ have been computed i n 3.6 and are given by eqns. (3.6.39), (3.6.40) and (3.6.41), r e s -p e c t i v e l y . The procedure for c o n t r o l l i n g the system (4.5.6) with the feedback c o n t r o l l e r using the state f i l t e r e d estimate constructed from the a v a i l a b l e measurements i s as follows. Assume the process to be at time k, when u , P^ and f° are obtained. The c o e f f i c i e n t estimate u i s then p r e m u l t i p l i e d by K. K, G, .. and the optimal c o n t r o l f,° by 11 to y i e l d together the predicted k+1, k k k+.L, k c o e f f i c i e n t u j r + ^ | k - B y s u b s t i t u t i n g the given G^+^ ^, F^+i k ' P^ a n d i n t o eqn. (3.6.22), P^-f-lIk ± s computed and i s then used together with D^+i and B^-^ i n eqn. (3.6.30) to compute the f i l t e r gain c o e f f i c i e n t Kk+1" Assume the process then to be at time k+1 when the measurement n, ,, i s taken. The k+1 quantity u i premultiplied by D i s substracted from the current measure-K."rX IC K.~rX ment n. , to form the measurement r e s i d u a l , • The current c o e f f i c i e n t estimate u, , ., i s then given by the sum of u\ ,.. and the c o r r e c t i o n term ,.. k+1 b J k+1 k+1 premultiplied by t c k + 1 • The optimal c o n t r o l f^ +-^ i s given by ti^-fi p r e m u l t i p l i e d by the precomputed and stored -A^B^ i n accordance to eqn. (4.5.29). The estimator response i s u k + ^ ( x ) computed from eqn. (3.6.32). The q u a n t i t i e s tijc+1^ Pk+1' w n ^ - c n a r e calculated u t i l i z i n g P k + 1| k> Kk+1 a n d °k+l "*"n e q n " (3-6.31), and f,0,-, are stored. The above chain of events i s then repeated. We note K T 1 "o that f„ i s noted needed at k=N since the process terminates at k=N. N Assuming an i n i t i a l c ondition of the s t a t e characterized by the mean u A(x) = -u =-1.0 (4.5.34) u ss and the covariance P 0(x,x') = 0.016(x-x') , (4.5.35) 120 we perform the 'simulation study. A t y p i c a l run y i e l d s the estimate u^Cx)=u k(x)+u , where u^(x) i s the response of the estimator eqn. (4.5.30), of u^Cx) as shown'in F i g s . (4.3), (4.4) and (4.5) as u^(0.)} u^CO.5) and i-L, (1.), r e s p e c t i v e l y , and the piecewise continuous optimal c o n t r o l f? =f.°+f *k . '»"k k ss as shown i n F i g . (4.6). We also draw i n F i g s . (4.3) to (4.6) f o r comparison purposes i n s o l i d l i n e s the corresponding forced d e t e r m i n i s t i c system response u,, (x) at x=0.,0.5 and 1. and the optimal c o n t r o l f° for the system of eqn. «k 1 .. *k 1 (4.5.6) with w k(x)=0. for a l l k and eqn. (4.5.12) with the expectation operator removed for t h i s d e t e r m i n i s t i c case by using standard lumped parameter system 38 optimal c o n t r o l theory The minimum of the cost f u n c t i o n a l eqn. (4.5.12) used for t h i s numerical example i s obtained by s e t t i n g k=0 i n eqn. (4.4.31) to be *N = E[iV u 0 C x)V N ( x , x ' ) u 0 ( x ' ) d x d x ' ] + a N / / u n ( x ) V M ( x , x ' ) u n ( x , ) d x d x ' 0 0 0 N 0 1 1 + / / V N ( x , x , ) P 0 ( x ' , x ) d x d x l + a N . (4.5.36) Now, s u b s t i t u t i n g the truncated expansions (3.6.34) and (4.5.20) in t o the f i r s t term on the r i g h t side of eqn. (4.5.36), we have approximately s = "5vo • (4-5-37) We see that i s i n fact the cost to con t r o l the corresponding d e t e r m i n i s t i c system. Substituting the truncated expansions (4.5.20) and (3.6.19) in t o the second term on the r i g h t hand side of eqn. (4.5.37), we have approximately C x = t r V N P 0 . (4.5.38) We see that C i s the cont r i b u t i o n to the t o t a l cost due to uncertainty i n the s t a t e , i n i t i a l condition u^(x). to l - 1 1 2 5 Further, s u b s i t u t i n g the truncated expansions. ( 3 . 6 . 1 7 ) F ( 3 - 6 . 1 8 ) , ( 4 . 5 . 1 9 ) , ( 4 . 5 . 1 7 ) , ( 3 . 6 . 1 9 ) and ( 3 . 6 . 2 0 ) into the t h i r d term on the r i g h t hand' hand side of eqn. ( 4 . 5 . 3 6 ) , we have approximately T aN-k = aN-k-l + t r Vk-lFk+l,kVk+l?k T — I T + [IIk+l,kSN-k-lHk+l,k + Bk] \+l,kSN-k.-lGk+l,k T k"k+l ,k"N-k-l"k+1 ,k •P,G_ , S „ , H^. = a.T . . + t r S.T , , Q. F.r,. , + A. 1 B . P . B.T , ( 4 . 5 . 3 9 ) N-k-1 N-k-1 k+l,k Hk k+l,k 1 k k k k = N - l , . . . , 1 , 0 , where i n obtaining the l a s t expression eqns. ( 4 . 5 . 2 6 ) and. ( 4 . 5 . 2 8 ) have been used. By repeated a p p l i c a t i o n of eqn. ( 4 . 5 . 3 9 ) , i t i s easy to shown that N _ 1 T N - 1 =-1- =T a„ =• a. + Z t r S._ . .F. . Q. F . + Z A / B . P . B ; N 0 J = Q N - 3 - 1 3 + 1 , j x j 3 + 1 , 3 ^ Q 3 3 3 3 = a Q + C g + C E . ( 3 . 5 . 4 0 ) We see from eqn. ( 4 . 5 . 4 0 ) that C i s the c o n t r i b u t i o n to the t o t a l cost due to system disturbance and that due to f i l t e r i n g estimation e r r o r . To evaluate a^, e i t h e r eqn. ( 4 . 5 . 3 9 ) or eqn. ( 4 . 5 . 4 0 ) can be used. The cost due to the i n i t i a l condition uncertainty, system disturbance and f i l t e r i n g estimation error i s C u = C I + C S + C E and the t o t a l cost i s therefore ; N = C D + C U • For t h i s p a r t i c u l a r numerical example, we have found that 0 ^ = 0 . 2 7 4 6 6 0 and 0 ^ = 0 . 0 4 0 1 7 9 such that 1 ^ = 0 . 3 1 4 8 3 9 . Moreover, to a s c e r t a i n which of the three components i n C that contributes most to C , C . i s broken down i n t o i t s r u u u 126 three, components as shown, i n Table (4.2).. C l e a r l y , the system disturbance C 0.002900 • 7.22% C 0.030126 74.98% L5 C 0.007153 17.80% E To t a l 0.040179 100.00% Table (4.2) Breakdown of the cost C . u contributes most (74.98%) to the cost C^. 4.6 Conclusion In t h i s chapter, we have considered a case of s t o c h a s t i c optimal c o n t r o l of l i n e a r discrete-time d i s t r i b u t e d parameter systems. The plant and measurement, noises are ass timed to be gaussian d i s t r i b u t e d and uncorrelated with each other and with the state i n i t i a l c o n d i t i o n assumed gaussian too. The cost f u n c t i o n a l , with respect to which the performance of the c o n t r o l scheme over an ensemble of systems i s measured, used here i s quadratic both i n the state and the c o n t r o l e f f o r t . The c o n t r o l i s assumed to have i t s com-ponents each concentrated on a small area of the s p a t i a l domain such that i t , can be approximated by impulsive functions. The approach, to t h i s problem has been put forward i n two steps. F i r s t l y , assuming that the state i s exactly measured, the dynamic programming technique i s employed to obtain the feed-back c o n t r o l , which, because of the q u a d r a t i c i t y of the cost f u n c t i o n a l and the l i n e a r i t y of the system, i s shown to be l i n e a r i n the s t a t e . The d i s t r i -butive feedback gain i s shown to s a t i s f y an i n t e g r a l - d i f f e r e n c e equation which, because of the f a c t that the " i n i t i a l " c ondition i s given at terminal time, must be solved r e c u r s i v e l y backwards and therefore o f f - l i n e . The minimum of the cost f u n c t i o n a l i s shown to consist of two terms, one i s the cost expended 127 to b r i n g the system to the desired state as i f plant noise were not there, accounting for the contribution due to i n i t i a l condition uncertainty at the same time, the other accounts f o r the' c o n t r i b u t i o n due to plant noise. Second-l y , with the state now assumed to be measured but not exactly at only several s p a t i a l l o c a t i o n s , i t i s shown that, by' employing the optimal l i n e a r minimum-variance f i l t e r as developed i n Chapter 3 to obtain an estimate of the s t a t e , the feedback control obtained i n the f i r s t step i s optimal to the extent of only needing the exactly measured state to be replaced by the state estimate. We thereby observe that the d i s t r i b u t i v e f i l t e r gain and the d i s t r i b u t i v e feed-back gain are independent of each other and can therefore be determined separately. The consequence to the cost of employing the state estimator i s shown, to be the addition to the minimum of the cost obtained i n the previous step of a term that accounts for estimation e r r o r . A nuTneric?.! e^.!?.™.^^ h ? . v c be*?™, ^ r ^ s e n t e d to i l l . i i s t r a t R the t h e o r y . Based on expansions of space-dependent functions i n terms of the eigenfunctions of the system s p a t i a l operator, the i n t e g r a l - d i f f e r e n c e equation s a t i s f i e d by the d i s t r i b u t i v e feedback gain has been reduced to algebraic r e c u r s i v e equations i n v o l v i n g the c o e f f i c i e n t s of expansion, which must then be solved backwards, therefore o f f - l i n e , and stored. For the estimator, the computation-a l algorithm for the f i l t e r gain has been u t i l i z e d . An o f f - l i n e study of the cost has been done to break i t down i n t o i t s c o n t r i b u t i v e components such that an assessment of what fa c t o r that contributes most to i t can be made. More-over, we observe that i t i t s e l f i s a standard of reference versus which the performance of other control schemes can be compared. This f a c t i s c l e a r l y of importance i n c o n t r o l system a n a l y s i s . 128 5. OPTIMAL FILTERING ESTIMATION AND POINTWISE REGULATION CONTROL OF LINEAR CONTINUOUS-TIME STOCHASTIC DISTRIBUTED PARAMETER SYSTEMS 5.1 Introduction With the a v a i l a b i l i t y of the r e s u l t s presented i n the pre-ceeding two chapters for l i n e a r discrete-time d i s t r i b u t e d parameter system f i l t e r i n g estimation given discrete-space measurements and s t o c h a s t i c optimal pointwise regulation c o n t r o l , we are now i n a p o s i t i o n to obtain t h e i r continuous-time analogues. The approach developed here to obtain the continuous-time algorithms i s for both cases by considering formally the l i m i t i n g behaviour of the corresponding discrete-time ones as the t i m e - i n t e r v a l between sampling instants i s made a r b i t r a r i l y small. This approach i s s i m i l a r to 54 that developed by Kalman who used i t i n lumped parameter estimation theory to obtain the optimal f i l t e r i n g equations. This chapter i s divided into three sections. In 5.2, the l i n e a r continuous-time f i l t e r i n g estimation problem given discrete-space measure-ments i s treated. In 5.3, the continuous-time s t o c h a s t i c optimal p o i n t -wise regulation problem i s treated. F i n a l l y , i n 5.4, by comparing the 20 solutions obtained i n 5.2 and 5.3, Kalman's d u a l i t y i s extended to the present l i n e a r d i s t r i b u t e d parameter system estimation and c o n t r o l pro-blems . 5.2 Linear Continuous-time D i s t r i b u t i v e Kalman F i l t e r 5.2.1 Problem Formulation Consider a l i n e a r continuous-time s t o c h a s t i c d i s t r i b u t e d para-meter system which can be described by the vector p a r t i a l d i f f e r e n t i a l 129 equation 3u(t tx) ^ 3t A u(t,x) + B(t,x)w(t,x) (5.2.1) defined for t^>t>t n on a s p a t i a l domain DCE" , where u(t,x) i s the n-vector state, A i s a known l i n e a r matrix p a r t i a l d i f f e r e n t i a l operator with respect x . to the s p a t i a l v a r i a b l e xeD, whose parameters may depend on both t and x, B(t,x) i s a known nxc matrix and w(t,x) i s the c-vector s t o c h a s t i c d i s t u r b -ance input. We assume that continuous-time measurements only are being taken on the system. Moreover, the continuous-time measurements are assumed to be taken by zero-memory transducers located at m d i s t i n c t s p a t i a l points x^, i e l , where I ={i: i=l,2,...,m} i s the d i s c r e t e measurement l o c a t i o n index m m set. The measurement equation f o r the transducer located at x., i e l i s ^ 1 m assumed to be given as m y (t) = E M (t)u(t,x ) + v (t) , j = l 1 3 J (5.2.2) where v.( t ) i s a d.-vector measurements, M..(t) i s a d.xn measurement matrix ; i l i j i and v ^ ( t ) i s a d^-vector gaussian measurement error. For n o t a t i o n a l con-venience, we augment eqn. (5.2.2) for i=l,2,,..,m i n t o the s i n g l e measurement equation n(t) = M(t)u(t) .+ v ( t ) , where we define the d-vector measurements n(t) as (5.2.3) nCt) = y ^ t ) y m t t ) m the dxmn measurement matrix M(t) as 130 M(t) M u ( t ) . • .M l m(t) M ( t ) - • -M (t) ml mm the mn-vector u(t) as r \  A u(t) = u ( t , x 1 ) u ( t , x m ) the d-vector measurement error as v ( t ) v 1 ( t ) m and the number d as A m d = E d. 1=1 1 The state i n i t i a l c ondition u^(x) of eqn.. (5.2.1) i s assumed given as a gaussian function with i t s mean and covariance given as E[u Q(x)] = u Q(x) = 0 and (5.2.4) E[ U ( )(X ) U Q ( X')] = P Q(x,x') , where the nxn matrix PQ(X,X') i s nonnegative-definite and the boundary condi-t i o n as B xu(t,x) = 0 , xe9D, (5.2.5) where SD i s the boundary surface of the s p a t i a l domain and 3 v i s a l i n e a r p a r t i a l d i f f e r e n t i a l operator i n xe8D of order l e s s than A x We assume that UQ(X) i s uncorrelated with the system noise {w(t),  t>t:Q^ a m d with the measurement noise ( v ( t ) , t>0}. Moreover, the 131 l a t t e r two are assumed to s a t i s f y the following conditions: E[w(t,x)] = E [ v ( t ) ] = 0 , E[w(t,x)w T(s,x 1)] = Q(t,x,x')6(t-s) , (5.2.6) E[v(l : ) v T ( s ) ] = RCt)d(t-s) , E[w(t,x)v T(s)] = 0 . for a l l t,s>t A, where Q(t,x,x') i s a continuous nonnegative-definite s e l f -adjoint cxc matrix and R(t) i s a continuous symmetric p o s i t i v e - d e f i n i t e dxd matrix. We denote i n general an estimate of u(t^,x) at some time t - ^ t g based on measurements n(f) obtained over the ti m e - i n t e r v a l t A<x<t by u ( t ^ | t , x ) . Further, we define the estimation error by the expression i A i ti^tjjtjx) = u ( t 1 , x ) - u ( t 1 | t , x ) , and the error covariance by . . . .. A . , . . . T . i . . , f(_t^ | t , X , X " ) = Ji(.U IZ^ j Z,X)U 1 .j | t , X J i . ^J.A./J The estimate u(t^|t,x) i s termed a f i l t e r e d estimate i f t^=t. Since we are concerned only with the f i l t e r i n g problem here, we w i l l denote f o r n o t a t i o n -a l s i m p l i c i t y u(t|t,x) by u ( t , x ) , u(t|t,x) by u(t,x) and P(t|t,x,x') by P(t,x,x') . The problem to be considered i n t h i s s e c t i o n i s the continuous-time (optimal) l i n e a r minimum-variance f i l t e r i n g problem which can now be stated as follows: Given the system of eqns. (5.2.1), (5.2.5) and (5.2.3), and the measurements {TI(T), t N<T<t}, determine a l i n e a r estimate u(t,x) of u(t,x) which minimizes the f i l t e r i n g error variance E [ | | u ( t , x ) | | 2 ] = t r P(t,x,x') , where ||*|.| denotes the Euclidean norm and t r "the trace of". To tackle t h i s problem, we s h a l l make use of the f i l t e r r e s u l t s 132 obtained i n Chapter 3. Before doing so, we w i l l f i r s t examine the fol l o w i n g . Let the discrete-time indic e s k+1 and k, k e l , be denoted by t+At and t, r e s p e c t i v e l y , where At>0. Expanding the Green's function G(t+At,x;t,x'), given by eqn. (3.3.5) and s a t i s f y i n g eqns. (3.3.6) and (3.3.7), i n a Taylor, s e r i e s about t, G(t+At,x;t,x') = GCt,x;t,x*)+ ^^^t^)_ At + 0 ( A t 2 ) , (5.2.8) o t where 0(At 2) i s a nxn matrix of a l l those terms i n v o l v i n g ( A t ) 2 or higher. Applying conditions (3.3.5) and (3.3.7) to eqn. (5.3.8), we have G(t+At,x;t,x') = (I+AtA )l6(x-x') + 0 ( A t 2 ) . (5.2.9) X Moreover, from the fourth r e l a t i o n i n eqn. (3.3.9), i t can be shown using eqn. (5.2.9) that F(t+At,x;t,x') = AtI6(x-x')B(t,x') + 0 ( A t 2 ) . (5.2.10) Replacing k+1 and k by t+At and t r e s p e c t i v e l y i n eqn. (3.3.10) or eqn. (3.2.1), we have u(t+At,x) = / G(t+At,x;t,x')u(t,x')dD , + r F(t+At,x;t,x')w(t,x')dD (5.2.11) ' JJ X such that, s u b s t i t u t i n g eqns. (5.2.9) and (5.2.10) in t o eqn. (5.2.11) y i e l d s u(t+At,x) = u(t,x) + At[A xu(t,x) + B(t,x)w(t,x)] + 0 ( A t 2 ) , (5.2.12) from which 8u(t,x) .. u(t+At,x)-u(t,x) At->0 = A u(t,x) + , B(t,x)w(t,x) , (5.2.13) for t^>t>t^. Moreover, from con d i t i o n (3.3.6), the boundary c o n d i t i o n f or eqn. (5.2.13) i s seen to be 3 xu(t,x) = 0 , xe3D. (5.2.14) F i n a l l y , r e p l a c i n g k+1 by t+At i n eqn. (3.2.4), we have 133 m y.(t+At) = E M..(t+At)u(t+At,x.) + v.(t+At) (5.2.15) 1 j = l 1 1 3 1 which reduces to, as At-K), m y (t) = E M ( t ) u ( t , x ) + v (t) , (5.2.16) j=l J • 3 for i=l,2,...,m. We can augment eqn. (5.2.16) for i=l,2,...,m by using the d e f i n i t i o n s of n ( t ) , M(t), u(t) and v ( t ) immediately following eqn. (5.2.3) to obtain r,(t) = M(t)u(t) + v ( t ) . (5.2.17) So far we have shown that the discrete-time system and measurement models eqns. (3.2.1) and (3.2.4) reduce to the corresponding continuous-time ones eqns. (5.2.1) and (5.2.3) when the samples become dense or, a l t e r n a t e l y stated, when the time i n t e r v a l between sampling instants goes to zero. We w i l l therefore r e s o r t to t h i s procedure of f i r s t r e p l a c i n g k+1 by t+At, At>0, and k by t i n the discrete-time equations we are i n t e r e s t e d i n and then l e t At tend to zero i n order to obtain the corresponding continuous-time ones.. 5.2.2 Structure of the F i l t e r From Chapter 3, the optimal continuous-time l i n e a r d i s t r i b u t i v e f i l t e r i s derived from eqn. (3.4.6) by f i r s t w r i t i n g t+At and t f o r k+1 and k r e s p e c t i v e l y to y i e l d u(t+At,x) = / G(t+At,x;t,x')u(t,x')dD , m + E K.(t+At,x)z.(t+At) . (5.2.18) i = l 1 X Substituting eqn. (5.2.9) into eqn. (5.2.18) y i e l d s u(t+ t,x) = u(t,x) + AtA u(t,x) + 0 ( A t 2 ) . m + E K.(t+At,x)z.(t+At) , (5.2.19) i = l 1 1 whence 3u(t,x) ., u(t+At,x)-u(t,x) 8 t At-,0 A t m K.(t+At,x) = A rtS(t,x) + E (lim — — — ) (lira z.(t+At)). X i = l At-0 At-K) 1 (5.2.20) Now, define the continuous-time d i s t r i b u t e d f i l t e r gain . K (t+At,x) K (t,x) = l i m . (5.2.21) At->0 Moreover, from. eqns. (3.4.4) and (3.4.5), the measurement r e s i d u a l at x. i s x m z..(t+At) =y.(t+At)- E M. . (t+At)/ G(t+At,x;t,x')u(t,x')dD , 1 X , - IL"] JL) X 3=1 (5.2.22) Substituting eqn. (5.2.9) into eqn. (5.2.22) y i e l d s m z.(t+At) =y.(t+At)- E M..(t+At)(I+AtA )u(t,x.)+0(At 2), X X , - X "1 X, "] 3=1 J 3 (5.2.23) such that, l e t t i n g At-K), m lim z.(t+At) = y . ( t ) - E M..(t)u (t,x.) . (5.2.24) At+0 1 1 j = l 1 J J Combining eqns. (5.2.20), (5.2.21) and (5.2.24), we have . . in = A f l ( t . x ) + 1 K . ( t , x ) z . ( t ) . (5.2.25) t X ' . - X ' X x=l Hence, from eqn. (5.2.25), the optimal l i n e a r continuous-time d i s t r i b u t i v e f i l t e r i s compactly w r i t t e n as = A xa(t,x) + K(t,x)c(t) (5.2.26) where, by augmenting eqn. (5.2.24) for i=l,2,...,m, the measurement r e s i d u a l 1 3 5 ?(t) i s 'c(t) = n(t) - M ( t ) y ( t ) . ( 5 . 2 . 2 7 ) The i n i t i a l condition for eqn. ( 5 . 2 . 2 6 ) i s c l e a r l y u^ (X)=E[UQ(x)] and the boundary condition, from condition ( 3 . 3 . 6 ) , i s ti(t,x) = 0 , XG3D . ( 5 . 2 . 2 8 ) The continuous-time d i s t r i b u t e d f i l t e r gain K(t,x) i n eqn. ( 5 . 2 . 2 6 ) which i s as yet unspecified w i l l now be considered. 5 . 2 . 3 Optimal D i s t r i b u t e d F i l t e r Gain In t h i s s e c t i o n , the optimal d i s t r i b u t e d f i l t e r gain ic(t,x) i s derived. To that end, we , f i r s t of a l l , have from eqns. ( 3 . 4 . 4 1 ) and ( 3 . 4 . 3 6 ) that a(t+At,x,x') = P(t+At|t,x,x') = G(t+At,x;t,x")P(t,x ,',x*'*)G T(t+At,x' ;t,x'")dD „dD „, + / p / D F(t+At,x;t,x") Q^.*^.*'") F T(t+At,x';t,x ,")dD x„dD x„, . ( 5 . 2 . 2 9 ) I t should be noted that i n obtaining eqn. ( 5 . 2 . 2 9 ) , Q k(x,x') i n eqn. ( 3 . 4 . 3 6 ) 0 ( t x x') 54 has been replaced by — — by an argument s i m i l a r to that i n . Sub-s t i t u t i n g eqns. ( 5 . 2 . 9 ) and ( 5 . 2 . 1 0 ) into eqn. ( 5 . 2 . 2 9 ) and rearranging y i e l d s P(t+At|t,x,x T) = P(t,x,x') + AtA P(t,x,x') + At[A , P ( t , x ' , x ) ] T X X + AtB(t,x)Q(t,x,x')B T(t,x') + 0 ( A t 2 ) . ( 5 . 2 . 3 0 ) It i s then c l e a r that for t>t^ and t > 0 , l i m P(t+At|t,x,x') = P(t,x,x') . ( 5 . 2 . 3 1 ) At-K) Moreover, we have, from eqn. ( 3 . 4 . 4 3 ) , m • , T B (t+At ,x) = E P(t+At.|t,x,x.)MT.(t+At) ( 5 . 2 . 3 2 ) j = l 2 1 J and from eqn. ( 3 . 4 . 4 5 ) , . m Y..(t+At) = E M. (t+At)P(t+At|t,x ,x )M (t+At) i J a,b=l i a 3 .. b J b , ,R..(t+At) + ~12J-t » (5.2.33) 54 R i ^ t + A t ) where, again , ^ i n eqn. (3.4.45) has been replaced by — i n d e r i v i n g eqn. (5.2.33). The optimal d i s t r i b u t e d f i l t e r gain, from eqn. (3.4.55) i s given by K(t+At,x) = 3(t+At,X)Y _ 1(t+At) . (5.2.34) Postmultiplying eqn. (5.2.34) by y(t+At), we have K(t+At,X)Y(t+At) = 3(t+At,x) , (5.2.35) which can further be written, i n i t s component form, as m E K.(t+At,x)Y..(t+At) = 3.(t+At,x) . (5.2.36) i = l 1 1 J 3 Arranging eqn. (5.2.36) for the purpose of taking l i m i t s i n the following manner: m K.(t+At,x) E ( 1 . ) ( Y- .(t+At)At) = 3 (t+At,x) , (5.2.37) i = l 3 3 we see, as a consequence of eqn. (5.2.31) and the d e f i n i t i o n s (5.2.21), (5.2.32) and (5.2.33), that, i n l e t t i n g At->0, m m E K. (t,x)R. . (t) = E P(t,x,x,)M. (t) . (5.2.38) i = l 1 1 J b=l 3 Eqn. (5.2.38) can be compactly rewritten as K(t,x)R(t) = i T(t,x)M T(t) , (5.2.39) such that the optimal continuous-time d i s t r i b u t e d f i l t e r gain i s <(t,x) = T r(t,x)M T(t)R~ 1(t) , (5.2.40) where we define i r ( t , x ) = [ P ( t , x , x . ) P(t,x,x ) ] . (5.2.41) l . m In eqn. (5.2.40), ir(t,x) as defined by eqn. (5.2.41) needs to be 137 s p e c i f i e d . Obtaining the recursive equation pertinent to the f i l t e r i n g error covariance P(t,x,x') i s then the subject of the next subsection. 5.2.4 F i l t e r i n g Error Covariance The recursive equation f o r the f i l t e r i n g error covariance P(t,x,x') w i l l now be derived. We have, from eqn. (3.4.50), P(t+At,x,x') = a(t+At,x,x') - K(t+At,x)y(t+At)KT(t+At,x') , (5.2.42) Substituting eqn. (5.2.29) into eqn. (5.2.42) and rearranging y i e l d s ,T P(t+At,x,x') = P(t,x,x') + At{A P(t,x,x') + [A ,P(t,x',x)] + B(t.,x)Q(t,x,x')B T(t,x')} + 0(At 2) - At[ ^  K(t+At,x) Y(t+At)K T(t+At,x')] . (5.2.43) Consider the l a s t term i n eqn. (5.2.43) f i r s t ; • | K(t+At,x)Y(t+At)KT(t+At,x') 1 m T V Z K.(t+At,X )Y..(t+ t)K.(t+At,x.), (5.2.44) At . . . x v " i j j j as expressed i n i t s components using d e f i n i t i o n s (3.4.9) and (3.4.44). As a consequence of eqn. (5.2.31) and d e f i n i t i o n s (5.2.33) and (5.2.21), 1 T li m — K(t+At,X)Y(t+At)K (t+At,x') At-K) m X K.(t,x)R..(t)K.(t,x') (5.2.45) i , j = l 1 1 3 J = tc ( t , x ) R ( t ) i c T ( t , x 7 ) . (5.2.46) Now, s u b s t i t u t i n g the optimal gain K(t,x) given by eqn. (5.2.40) i n t o eqn. (5.2.46), we have 1 T li m — K(t+At,X)Y(t+At)K (t+At,x') At->0 = T T ( t , x ) M T ( t ) R _ 1 ( t ) M ( t ) 1 T T ( t , x ' ) . (5.2.47) Hence, from eqns. (5.2.43) and (5.2.47), we see that the f i l t e r i n g .138 error covariance P(t,x,x') i s governed by the p a r t i a l d i f f e r e n t i a l equation 3P(t,x,x') = ^ P(t+At,x,x') - P(t,x,x')  9 t = At->0 A t " " = A P(t,x,x') + [A , P ( t , x ' , x ) ] T - ^ ^(t,x)M T(t)R~ 1(t)M(t.)•^• T(t,x ,) + B(t,x)Q(t,x,x*)B T(t,x') . (5.2.48) The appropriate i n i t i a l c ondition for P(t,x,x') i s , from c o n d i t i o n (5.2.4) , P(t 0,x,x') = P 0(x,x') . (5.2.49) The boundary condition to eqn. (5.2.48) i s obtained thus: from eqn. (5.2.5) and (5.2.28), i t follows 3 u(t,x) = 0 , xe3D, (5.2.50) such that, from the d e f i n i t i o n (5.2.7) for P(t,x,x'), r> f >. — — f \ ' _ r» —. T \ I ^ - T \ / cr n r i N X 18 The r e s u l t s f or a s i m i l a r estimation problem obtained by Sakawa v i a the Wiener-Hopf equation approach are somewhat more general i n that he includes stochastic disturbance i n the system boundary condition as w e l l . However, the omission of system boundary condition noise o f f e r s no los s of g e n e r a l i t y as i t can be included i n the system noise by s u i t a b l y extending the d e f i n i t i o n of the system operator ' . Suppressing the system boundary condition noise, i t i s seen that Sakawa's r e s u l t s ' c o i n c i d e with those presented i n t h i s s e c t i o n . 5.3 Stochastic Optimal Pointwise Regulation Control f o r Linear Continuous- time D i s t r i b u t e d Parameter Systems : 5.3.1 Problem Formulation We assume that there are q c o n t r o l l e r s located at q d i s t i n c t points i n the closed s p a t i a l domain D. The c o n t r o l output by the c o n t r o l l e r located 139 at x„ , lei i s the p -vector f ^ ( t ) , where I ={l: £=1,2,...,q}'is the c o n t r o l l o c a t i o n index set. Further, we assume that the t o t a l c o n t r o l input to the £ d i s t r i b u t e d parameter system i s a l i n e a r combination of the controls f (t) , £=l,2,...,q. Consider a l i n e a r continuous-time s t o c h a s t i c d i s t r i b u t e d para-meter system subject to pointwise c o n t r o l which can be described by the follow-ing vector p a r t i a l d i f f e r e n t i a l equatioin 3u(t,x) = A _ _ u ( t > x ) + I B ^ ( t ) f £ ( t ) S ( x - x „ ) + B(t,x)w(t,x) 3 t 1=1 defined for t £>t>t_ on DCE r, where u(t,x) i s the n-vector s t a t e , A i s a r - — U x known l i n e a r matrix p a r t i a l d i f f e r e n t i a l operator with respect to the s p a t i a l £ v a r i a b l e xeD, whose parameters may depend on both t and x, B^(t) i s a known nxp. weighting matrix for the p -vector c o n t r o l f (t) impulsed at the s p a t i a l point x=x £, B(t,x) i s a known nxc matrix and w(t,x) i s the c-vector s t o c h a s t i c disturbance input. The above equation f i r s t appears in 4.3 where we consider i t s discrete-time model. For.notational s i m p l i c i t y , we rewrite as i n 4.3 the above equation as 3u (t ,x) 3t A xu(t,x) + B f ( t ) F ( t , x ) + B(t,x)w(t,x) (5.3.1) where we r e c a l l the d e f i n i t i o n of the nxp matrix B^(t) i s B, (t) = [ B h t ) - • • ' -B*(t)- • • • -BjCt)] ' f ^ ' l " f v " / . . f and that of the p-vector F(t,x) i s FCt,x) = f x ( t ) 6 ( x - x x ) f q ( t ) 6 ( x - x q ) (4.3.4) (4.3.5) and p= • E p . 1=1 1 We assume that continuous-time measurements only are being taken on 140 the system and that they are taken by zero-memory transducers located at m d i s t i n c t s p a t i a l points x., i£l . We further assume that the set {x., i e l } r 1 i ' m x m . i s d i s t i n c t from the set {x., £el }. The measurement equation f o r the tra n s -£ q ducer located at x., i e l , i s assumed to be given as x m m y (t) = E M (t) u ( t , x ) + v (t) j = l J 3 where y.( t ) i s a d.-vector measurements, ' 1 ..(t) i s a d.xn measurement matrix J X X X j X and v ^ ( t ) i s a d^-vector gaussian measurement error. Again, f o r n o t a t i o n a l convenience, we augment as i n 5.2 the above equation f o r i=l,2,...,m i n t o the si n g l e measurement equation n(t) = M(t)u(t) + v ( t ) (5.3.2) by using the d e f i n i t i o n s of the d-vector measurements n ( t ) , the dxmn measure-ment matrix M(t), the mn-vector u ( t ) , the d-vector measurement error v ( t ) and the number d immediately following eqn. (5.2.3). We mention i n passing that by s e t t i n g F(t,x)=0, the system of eqns. (5.3.1) and (5.3.2) reduces to that of eqns. (5.2.1) and (5.2.3) f o r which the optimal f i l t e r i n g problem has been considered i n 5.2. The i n i t i a l condition of eqn. (5.3.1) i s given as u ( t Q , x ) = u Q ( x ) , (5.3.3) and the boundary condition i s of the general form 3 u(t,x) .= 0 , xe9D, (5.3.4) where 3D i s the boundary surface of the s p a t i a l domain and 3 x i s a l i n e a r p a r t i a l d i f f e r e n t i a l operator i n xeD of order l e s s than A^. We assume that the state i n i t i a l c o ndition u A(x) i s gaussian s a t i s f y i n g condition (5.2.4): E [u 0 (x ) ] = u Q(x) = 0 , (5.3.5) E [u 0 ( x )u J (x ' ) ] = P Q(x,x') , 141 where the nxn matrix P Q ( X , X ' ) i s n o n n e g a t i v e - d e f i n i t e . Moreover, the stocha-s t i c processes w(t,x) and v ( t ) are assumed to s a t i s f y the f o l l o w i n g c o n d i t i o n s : E[w(t,x)] = E [ v ( t ) ] = 0 , E [ w ( t , x ) w r ( s , x T ) ] = Q(t , x , x ' ) S ( t - - s ) , E [ v ( t ) v T ( s ) ] = R ( t ) 5 ( t - s ) , (5.3.6) E [ w ( t , x ) v T ( s ) ] = 0 , E[w(t,x) UQ(x')] = E [ v ( t ) u J ( x ) ] - 0, f o r a l l t,s>tQ, where Q(t,x,x') i s a continuous n o n n e g a t i v e - d e f i n i t e s e l f -a d j o i n t cxc m a t r i x and R(t) i s a continuous symmetric p o s i t i v e - d e f i n i t e dxd mat r i x . For n o t a t i o n a l convenience, we d e f i n e a p-vector p o i n t w i s e c o n t r o l f ( t ) as f q ( t ) _ The pointwise c o n t r o l f ( t ) i s to be synthesized from a v a i l a b l e data such that the p h y s i c a l r e a l i z a b i l i t y c o n d i t i o n f ( t ) = f [ n ( T ) , t Q<x <t; u ( t Q ) ; t ] (5.3.7) i s s a t i s f i e d . The problem to be considered i n t h i s s e c t i o n i s the optimal s t o c h a s t i c d i s t r i b u t e d parameter system po i n t w i s e r e g u l a t i o n c o n t r o l problem which can now be s t a t e d as f o l l o w s : Given the system of eqns. (5.3.1), (5.3.4) and (5.3.2), determine a p h y s i c a l l y r e a l i z a b l e c o n t r o l law of the form i n eqn. (5.3.7) which minimizes the average cost f u n c t i o n a l I = E[a + 8] , (5.3.8) f ( t ) = 142 a = / / u T ( t ,x)A^(x,x')u(t ,x')dD-dD , L) L) IT JL JL X X + f j f n f n u T(t,x)A(t,x,x')u(t,x')dD dD ,dt , (5.3.9) g = /. f f T ( t ) E ( t ) f ( t ) d t , . (5.3.10) C0 where A^(x,x') and A(t,x,x') are s e l f - a d j o i n t nonnegative-definite nxn matrices, and B(t) i s a symmetric p o s i t i v e - d e f i n i t e pxp matrix, for a l l t,x and x' of i n t e r e s t . Both A(t,x,x') and B(t) are assumed to be continuous f o r a l l t e [ t Q , t f ] . To solve the above problem, we s h a l l make use of the r e s u l t s f o r the optimal pointwise c o n t r o l of a l i n e a r discrete-time s t o c h a s t i c d i s t r i b u t e d parameter system i n Chapter 4 and obtain the corresponding continuous-time ones by l e t t i n g the i n t e r v a l between sampling instants go to zero. To t h i s end, we need the expressions, i n addi t i o n to G(t+At,x;t,x') and F(t+At,x;t,x') a v a i l a b l e i n eqns. (5.2.9) and (5.2.10), for H(t+At,t,x) and the cost function-a l V Replacing k+1 and k, k e l t , i n eqn. (4.3.12) by t+At and t, r e s -p e c t i v e l y , we have H(t+At,t,x) = / ^ + A t y(t+At,t',x ) I(t')dt' = Aty(t+At,t,x)B(t) + 0(At 2 ) , (5.3.11) and s u b s t i t u t i n g eqn. (5.2.9) into eqn. (4.3.13) y i e l d s y(t+At,t,x) = [ ( I + A t A ) I 6 ( X - X ) (I+AtA )l6(x-x )] X -L » • X CJ + 0(At 2 ) . (5.3.12) Hence, u t i l i z i n g eqn. (5.3.12) i n eqn. (5.3.11), we further have II(t+At,t,x) = A t [ l 6 ( x - x 1 ) I6(x-x q) ] I ( t ) + 0(At 2 ) . (5.3.13) This indeed i s the required r e l a t i o n as can be seen from the recovery of the 143 continuous-time model eqn. .(.5.3,1) shown i n the foll o w i n g . Replacing k+1 and k, k e l by t+At and t, r e s p e c t i v e l y , i n eqn. (4.2.6), we have u(t+At,x) = G(t+At,x;t :.x')u(t,x ,)dD , + H(t+At,t,x)f (t) JU X + '/ F(t+At,x;t,x')w(t,x')dD x, (5.3.14) such that s u b s t i t u t i n g eqns. (5.2.9), (5.2.10) and (5.3.13) i n t o eqn. (5.3.14) y i e l d s u(t+At,x) = u(t,x) + AtA. u(t,x) X + At[I<5(x-x )• I6(x-x )]I(t)f(t) + AtB(t,x)w(t,x) + 0(A t 2 ) . (5.3.15) We note from the d e f i n i t i o n of f ( t ) and the d e f i n i t i o n s (4.3.4), (4.3.5) and (4.3.14) for B ( t ) , F(t,x) and g ( t ) , r e s p e c t i v e l y , that [I6(x-x ) I6(x-x ) ] p ( t ) f ( t ) = [ B f ) 1 ( t ) 6 ( x - x 1 ) - - - - - B f > q ( t ) 6 ( x - x q ) ] f ( t ) [ B f , i ( t ) : " - : B f , q ( t ) ] fi ( t ) 6 ( x - x 1 ) f q ( t ) 6 ( x - x q ) = B f ( t ) F ( t , x ) (5.3.16) Hence, u t i l i z i n g eqn. (5.3.16) i n eqn. (5.3.15), rearranging and' taking l i m i t s , we have 3u(t,x) _ . u(t+At,x) - u(t,x) 8 t = l l l o A t = A u(t,x) + B f ( t ) F ( t , x ) + B(t,x)w(t,x) (5.3.17) fo r t > t > t „ . The boundary condition to eqn. (5.3.17) i s obtained from f - - 0 144 condition C3.3.6) as B uCt,x> = 0. , xe9D. (5.3.18) We now turn to obtaining the cost f u n c t i o n a l for the equivalent discrete-time problem. F i r s t l y , d i v i d e the time i n t e r v a l [ t ^ t ^ ] into N sub-i n t e r v a l s each of length At= ( t ^ - t ^ ) /N and use j to index these s u b i n t e r v a l s . It i s required that as A.t-^ 0 and Nx», the l i m i t of NA t=t^-tQ=constant. Now using such a p a r t i t i o n of the time i n t e r v a l and d e f i n i n g the value of the time-integrand of a over the time s u b i n t e r v a l (t.,t.,..] to be a constant 3 1+1 equal to that at w e can express eqn. (5.3.9) as a = l i m a (5.3.19) At*0 where a N = / D / D u T(t Q+NAt,x)A f(x,x')u(t 0+NAt,x')dD xdD x, N-1 .. + E / D / D u i ( t 0 + ( j + l ) A t , x ) A ( t 0 + ( j + l ) A t , x , x ' ) A t u ( t 0 + ( j + l ) A t , x ' ) d D x d D x t 3=0 (5.3.20) Again using the same p a r t i t i o n of the. time i n t e r v a l [ t ^ t ^ ] and d e f i n i n g the value of f ( t ) over the subinterval [ t . , t . , , ) to be a constant equal to that 3 3+1 at t , x^ e can express eqn. (5.3.10) as 3 = l i m & , (5.3.21) At+0 N-*°° xrfiere N-1 B N = E f i ( t 0 + j A t ) B ( t 0 + j A t ) A t f ( t 0 + j A t ) . (5.3.22) 3=0 From eqns. (5.3.8), (5.3.19) and (5.3.20), i t follows that 145 I = E[ct + 3] = E l i m [a + 3 ] At+O = l i m E[a + 3 ] . (5.3.23) At-K) Hence, for a given N, define the cost f u n c t i o n a l for the equivalent d i s c r e t e -time problem to be @ ? N = E [ a N + 3 N ] > (5.3.24) where and 3^ are given by eqns. (5.3.20) and (5.3.22), r e s p e c t i v e l y . Now, comparing eqn. (5.3.24) with eqns. (4.2.10), (4.2.11) and (4.2.13), the cost f u n c t i o n a l for the l i n e a r discrete-time s t o c h a s t i c d i s t r i b u t e d parameter system pointwise r e g u l a t i o n c o n t r o l problem, we obtain the following r e l a t i o n s : A j + 1 ( x , x ' ) A(t 0+(j+l)At,x,x')At, j=0,l,...,N-2, A „ t N .1. ( C . . . „ l \ l i . 1 —T.T ^  1 ( C O O C \ and 5.3.2 Cost Functional B = 3( t Q + j A t ) A t , j=0,l,...,N-1. (5.3.26) The minimum value of the cost f u n c t i o n a l f o r optimal c o n t r o l over the time i n t e r v a l [ t , t ^ ] i s expressible, from the corresponding eqn. (4.4.33) by replacing k by t and N-k by t ^ - t , as I°(t f-t) = E [ / D / D u T ( t , x ) V ( t f - t , x , x ' ) u ( t , x ) d D x d D x J + a'Ct - t ) . (5.3.27) Let us consider the f i r s t term on the r i g h t hand side of eqn. (5.3.27) f i r s t , dropping the expectation operator f o r the moment. Replacing k, k+1, N-k and N-k-1 by t, t+At, t ^ - t and t ^ - t - A t , r e s p e c t i v e l y , i n eqn. (4.4.13), we have, a f t e r u t i l i z i n g r e l a t i o n (5.3.26), 146 V D u T C t , x ) V ( t f - t , x , x ' ) u ( t , x ) d D x d D x , = / D / D C/D G(t+At,x H;t,x)u(t,x)dD x] T • S ( t f ~ t - A t , x , , , x " ' ) [ / D G(t+At 5x , , r;t,x ,)u(t,x')dD x,]dD x„dD x, I I - / D / D t / D G(t+At >x";t,x)u(t,x)dD x] T • S(t f-t-At,x" 5x" ,)H(t+At,t sx'")dD x, IdD x„, • [ / / H T(t+At,t,x")S(t -t-At,x" )x"')HCt+At,t,x ,")dD „dD „,+B(t)At] JL) JL/ X X X • / D / D H T(t+At,t,x")S(t f-t-At,x",x n') * [ / D G(t+At,x , M ;t,x')u(t,x')dD ,]dD „dD ,,, . (5.3.28) From the d e f i n i t i o n (4.4.7), r e p l a c i n g k and N-k by t and t ^ - t , r e s p e c t i v e -l y , and u t i l i z i n g the r e l a t i o n (5.3.25), we have V(t f-t,x,x*) = S(t f-t,x,x') - A(t,x,x')At . (5.3.29) Substituting eqns. (5.3.29), (5.2.9) and (5.3.13) into eqn. (5.3.26) and rearranging to solve for S ( t ^ - t , x , x ' ) , we get r / u T(t,x)S(t,-t,x,x')u(t,x')dD dD , D L) X. X X = / / u T ( t , x ) S ( t -t-At,x,x')u(t,x')dD dD , D D JL X X + / D / D u T ( t , x ) S ( t f - t - A t , x , x ' ) A t A x , u ( t , x ' ) d D x d D x , + / D / D [ A t A x u ( t , x ) ] T S ( t f - t - A t , x , x ' ) u ( t , x ' ) d D x d D x , - [ A t / n u T ( t , x ) a ( t -t-At,x ) I(t)dD ] U I X • | t { A t I t ( t ) a ( t f - t - A t ) I ( t ) + B ( t ) } " 1 • [ A t / D I T ( t ) a T ( t f - t - A t , x ' ) u ( t , x ' ) d D x , ] + A(t,x,x')At + 0( A t 2 ) , (5.3.30; where i n the fourth term on the r i g h t hand side of eqn. (5.3.30) o(«,x) i s defined as a.(« ,x) =' [S(.« ,x,x 1) (5.3.31) an nxnq matrix, and a(*) i s defined as cf(') = S(«,x 1,x 1) S(«, X q,x 1) 'S(-, X p X ) •S(.,x q,x q) (5 .3 .32 ) an nqxnq matrix. Assuming that, as At~K), the l i m i t of S(t f~t-At,x,x') e x i s t s and i s f i n i t e , we see from eqn. (5.3.30) that . lim S(t -t-At,x,x') = S(t -t,x,x') At-H) (5.3.33) Consider now the concrete case of the s p a t i a l operator A x being a matrix second order p a r t i a l d i f f e r e n t i a l operator such that A = ( T. A C+-"x" ' ~ . . , i j >"~y „ i . i . , i a x ^ x 3 i = i 3x J xeD, where A ( t , x ) , B 1(t,x) and C(t,x) are known nxn matrices, and the boundary operator g being X -r . 3 (•) = (B0Ct,x) + x 3 ( t j X ) _ £ _ ) ( . ) f x e 9 D > X 3 = 1 J ax J where g.(t,x) = E A . . ( t , x ) ? , i = l , . . . , r , and £ i s the d i r e c t i o n cosine T -> XJ X i = l of the angle between the e x t e r i o r u n i t normal £ and the x - a x i s . I t i s shown i n Appendix II that the operator A^ adjoint to A i s defined by X X A T . ( t , x ) - I B T ( t , x ) + C T ( t , x ) ) ( - ) , A * 0 ) = c • E i T V 3 1 3 x , 3 = l 8x 8x i = i ax and the boundary operator 3 adjoint to 3 v.is defined by X X 3*(.) = (BnCt.x) + E B^(t,x) — )(•) x . . • j = l J 9x J xe3D. 148 Applying the compact matrix Green's i d e n t i t y ( v. Appendix II ) to the second term and the transpose of Green's i d e n t i t y to the t h i r d term on the r i g h t hand side of eqn. (5.3.30) and making use for both cases the boundary condition (5.3.4), we get S(t -t,x,x') - S ( t F - t - A t , x , x ' ) V D U ( t ' x ) [ A T — " ]uCt,x)dDdD x, = / D / D u T ( t , x ) [ A x , S ( t f - t - A t , x ' , x ) ] T u ( t s x * ) d D x d D x , + V s D ^ C t ' X ) [ R x ' S ( t f ~ t _ A t » x' x> J T u Ct ,x')d3D^, dD x  + V D u T ( t > x ) f \ S ( t f - t - A t , x » x ' ) ] u C t : ) x ' ) d D x d D x I +/ D/ 3 D u T ( t , x ) [ B x S ( t f - t - A t , x , x ' ) ] u ( t , x ' ) d 9 D x d D x , - / D / D u T ( t , x ) a ( t f - t - A t , x ) I ( t ) • l A t g T ( t ) a ( . t f - t - A t ) 3 ( t ) + B(t)}~"' • J T ( t ) a T ( t -t-At,x')u(t,x')dD^dD , + A(t,x,x') + 0(At 2) . - (5.3.34) Af t e r l e t t i n g At->0 i n eqn. (5.3.34), f i r s t choose S(t^-t,x,x') at xe9D to s a t i s f y the homogeneous adjoint boundary condition B*S(t f-t,x,x') = 0, xe9D, X ' E D , (5.3.35) such that the second and the fourth i n t e g r a l s on the r i g h t hand side of the equation vanish, and then equate what remains on the r i g h t hand side to i t s l e f t hand side to obtain the p a r t i a l d i f f e r e n t i a l equation f o r S(t^-t,x,x') as 9S(t -t,x,x') * • •• * T = A x S ( t f - t , x , x ' ) + [ A x , S ( t f - t , x ' , x ) ] - a ( t f - t , x ) B ( t ) B _ 1 ( t ) g T ( t ) a T ( t f - t , x ' ) + A(t,x,x') . (5.3.36) 149 The appropriate " i n i t i a l " condition f or eqn. (5.3.36) i s obtained i n the following manner. Substituting N for k i n the d e f i n i t i o n (4.4.7) and u t i l i z i n g r e l a t i o n (5.3.25) y i e l d s S(0,x,x') = V(0 5x,x') + A (x,x') + A(t ,x,x')At . (5.3.37) Now, from eqn. (4.4.35), we have V(0,x,x') = 0 , (5.3.38) such that, using t h i s r e s u l t i n eqn. (5.3.37) and considering i t s l i m i t as At-K), the " i n i t i a l " condition for eqn. (5.3.36) i s given by S(0,x,x') = A (x,x') . (5.3.39) Consider now the second term on the r i g h t hand side of eqn. (5.3.27). Replacing k, k+1, N-k and N-k-1 by t, t+At, t ^ - t and t ^ - t - A t , r e s p e c t i v e l y , i n eqn. (4.4.34) and u t i l i z i n g r e l a t i o n (5.3.26) y i e l d a ( t f - t ) a(t„-t- t) J: + t r / D / D / D / D F T ( t + A t , x ' ; t , x ) S ( t f - t - A t , x ' , x " ) F ( t + A t , x " ; t , x " ' ) Q ( l : , X ^ ' j X ) d D x d D x ' d D x " d D x ' " + E.{[/ / / D H T(t+At,t,x')S(t f-t-At,x',x")G(t+At,x";t,x)u(t,x)dD xdD x,dD x„] T • [ / n / n H T(t+At,t,x)S(t -t-At,x,x')H(t+At,t,x')dD dD , + B ( t ) A t ] - 1 • [ / D / D / D H T(t+At,t,x')S(t f-t-At,x',x")G(t+At,x";t,x)u(t,x)dD xdD x,dD x„]}. (5.3.40) where, i n obtaining the above equation, Q k(x,x') i n eqn. (4.4.34) has been 0(t x x') 54 substituted by x ^— by an argument s i m i l a r to that i n and the second term i s written i n the manner shown f o r c l a r i t y . S u b s tituting eqns. (5.2.9), (5.2.10) and (5.3.13) into eqn. (5.3.40) y i e l d s 150 a ( t f - t ) a ( t f - t - A t ) + At t r / / B ^ t ^ S C t . - t - A t . x . x ^ B C t . x ^ Q C t ^ x ' . x ^ D dD , D D Jl X X . + At t r / D / D o(f. f-t-At,x ) I(t) =T = = -1 •[At3 ( t ) a ( t f - t - A t ) 3 ( t ) + B(t)] • 3 T ( t ) c T ( t -t-At,x')P(t,x',x)dD dD , f x x + 0(At 2) . (5.3.41) Transposing a ( t ^ - t - A t ) to the l e f t hand side i n eqn. (5.3.41) and d i v i d i n g throughout by At, we have, i n l e t t i n g At-K) and u t i l i z i n g eqn. (5.3.33), that d a ( t f - t ) dt = t r / D/ D B^Ct.x^Ctj-t.x.x^BCt.x^QCt.x/ ,x)dD xdD x, + t r / D / D a ( t f - t , x ) 3 ( t ) B _ 1 ( t ) 3 T ( t ) a T ( t f - t , x ' ) P ( t , x ' , x ) d D x d D x , (5.3.42) for t^>t>tnt where S(t^-t,x,x') i s given by eqn. (5.3.36) and the f i l t e r i n g e rror covariance P(t,x,x !) given by eqn. (5.2.48). The appropriate " i n i t i a l " condition for eqn. (5.3.42) i s , from eqn. (4.4.36), seen to be a(0) = 0 . (5.3.43) Further, s u b s t i t u t i n g eqn. (5.3.29) in t o eqn. (5.3.27) y i e l d s I°(t f-t) = E [ / D / D u T ( t , x ) ( S ( t f - t , x , x ' ) - A ( t , x , x , ) A t ) u ( t , x ' ) d D x d D x I ] + a ( t f - t ) (5.3.44) which i n the l i m i t as At-K) becomes 151 I°(t f-t) = E [ / D / D u T(.t,x)S(t f-t.,x,x')u(t-,x*)dD xdD x,J + a ( t f - t ) , (5.3.45) t e [ t n , t f ] , assuming that lim a ( t f - t ) e x i s t s . At-K) The minimum value of the cost f u n c t i o n a l f o r the e n t i r e i n t e r v a l of c o n t r o l i s therefore obtained, by l e t t i n g t = t A i n eqn. (5.3.45), as I°(t f-t 0) = E [ / D / D u T ( t 0 , x ) S ( t f - t 0 , x , x ' ) u ( t 0 , x ) d D x d D x , ] + a ( t f - t Q ) , (5.3.46) that i s , I ° C t f - t 0 ) = t r / D / D S ( t f - t 0 , x , x ' ) P ( t 0 , x ' , x ) d D x d D x , + a ( t f - t 0 ) , (5.3.47) where S(t^~t A,x,x') i s obtained from the s o l u t i o n of eqn. (5.3.36), P(t n,x,x') i s given i n eqn. (5.3.5) as the covariance of the state i n i t i a l c ondition and aCt^- tg) i s obtained from the s o l u t i o n of eqn. (5.3.42). 5.3.3 Optimal Pointwise Control It has been shown i n 4.4.2 for the corresponding d i s c r e t e -time case with noise-corrupted discrete-space measurements that the p h y s i c a l l y r e a l i z a b l e optimal pointwise feedback c o n t r o l i s given by, as a consequence of eqn. (4.4.27), the three equations (4.4.21), (4.4.11) and (4.4.2) t o -gether. To obtain the continuous-time optimal pointwise c o n t r o l , therefore, f i r s t l y replace k, k+1 and N-k-1 by t, t+At and t f - t - A t , r e s p e c t i v e l y , i n eqns. (4.4.21), (4.4.11) and (4.4.12) and u t i l i z e r e l a t i o n (5.3.26) to y i e l d f ( t ) = - / n { / n / n H T(t+At,t,x)S(t,-t-At,x,x')H(t+At,t,x')dD dD ,+B(t)At}' • / H T ( t + A t , t , x ' ) S ( t -t-At,x',x")G(t+At,x";t,x)dD ,dD 152 •uCt,x)dD . (5.3.48) x In t h i s case, the f i l t e r e d estimate u ( t , x ) , which must account f o r the contr o l input as w e l l , i s obtained, by a l i m i t i n g procedure as i n 5.2 and 5.3.1 from eqns. (4.4.22) and (4.4.23), as lMjL*20 = A xu(t,x) + ,c(t,x)?(t) +"B F ( t ) F ( t , x ) , (5.3.49) for tp<t<t^, s a t i s f y i n g the boundary condition B u(t,x) = 0 , ' xe3D. (5.3.50) The i n i t i a l condition for eqn. (5.3.49) i s u ( t Q , x ) = UQCX ) . (5.3.51) Substituting eqns. (5.2.9) and (5.2.10) into eqn. (5.3.48) y i e l d s , a f t e r some manipulation, f (t) = - / D jti A t I T ( t ) a ( t f - t - A t ) B ( t ) -I- B ( t ) } " 1 • A t B T ( t ) a T ( t f - t - A . t , x ) u ( t , x ) d D x + 0(At 2) . ' (5.3.52) Hence, i n the l i m i t as At->0, the continuous-time optimal pointwise feedback con t r o l i s f ( t ) = - / D B _ 1 ( t ) B T ( t ) a T ( t f - t , x ) u ( t , x ) d D x (5.3.53) for t Q<t<t f. As expected, we have a separation p r i n c i p l e here as w e l l . We see from eqn. (5.3.36) that S ( t ^ - t ) , involved i n the optimal c o n t r o l f ( t ) given by eqn. (5.3.53), i s independent of the s t a t i s t i c a l parameters i n the problem, v i z . , P Q ( X , X ' ) , Q(t,x,x') and R ( t ) , and K(t,x) i n the optimal f i l t e r eqn. (5.3.49) as given by eqn. (5.2.40) i s independent of cost f u n c t i o n a l weight-ing matrices A^(x,x'), A(t,x,x') and B ( t ) such that S ( t ^ - t ) and K(t,x) can be determined independently of each other. 153 5.4 D u a l i t y 20 In t h i s s e c t i o n , i t i s shown that Kalman's d u a l i t y theorem can be extended to the cases of d i s t r i b u t e d parameter system optimal f i l t e r i n g estimation given discrete-space measurements and d e t e r m i n i s t i c optimal p o i n t -wise feedback c o n t r o l by comparing t h e i r r e s pective solutions obtained i n 5.2 and 5.3., Let us therefore consider the following two problems: (i) Optimal d i s t r i b u t e d .parameter system f i l t e r i n g estimation problem given discrete-space measurements: Given the d i s t r i b u t e d parameter system and the discrete-space measurement equation 8 U 3 t , X ) = A x u ( t j X ) + B(t,x)w(t,x) , xeD, (5.4.1) y ± ( t ) = M. i(t)u(t,x.) + v . ( t ) , (5.4.2) i=l,2,..,,m, where the system and measurement noises w(t,x) and v ( t ) s a t i s f y the following conditions: E[w(t,x)w T(s,x')] = Q(t,x,x')6(t-s) , E [ v ( t ) v T ( s ) ] « R ( t ) 6 ( t - s ) , (5.4.3) E[w(t,x)v T(s)] = 0 , f i n d an optimal f i l t e r e d estimate u(t,x) of u(t,x) s a t i s f y i n g the c o n d i t i o n u(t,x) = u [ n ( x ) , t Q < T <t] (5.4.4) so as to minimize t r P ( t , x , x ' ) ] x l = ! x (5.4.5) with given i n i t i a l f i l t e r e d estimate and f i l t e r i n g e r r o r covariance and u ( t Q , x ) = u Q ( x ) , (5.4.6) P(t 0,x,x') = P Q(x,x') . (5.4.7) 154 ( i i ) Deterministic d i s t r i b u t e d parameter system optimal pointwise feedback control problem; Given a de t e r m i n i s t i c d i s t r i b u t e d parameter system subject to pointwise control and measurement equation au+Ct+,xt) _ A t , t,,t t , , j - ^ t ^ ^ t .t N J - : J 3t A ' t u ' C t ' , x ' ) + B ' C t ' ) F C t ' , x ' ) , x e D 1 , ( 5 . 4 . 8 ) t . t v t , , t y' C t ' , x ' ) =' M (t' ,x')u ( t \ x ' ) , ( 5 . 4.9) where F ( t + , x + ) £ (t )6(x -x 1) f q ( t + ) 6 ( x + - x f ) ( 5 . 4 . 1 0 ) and f ( t ' ) f V ) f V ) ( 5 . 4 . 1 1 ) f i n d an optimal pointwise feedback co n t r o l f ° ( t 1 ) , s a t i s f y i n g the condi t i o n ( 5 . 4 . 1 2 ) which minimizes the quadratic cost f u n c t i o n a l f (t ) = f [u (t ,x )] , J = a + B > ( 5 . 4 . 1 3 ) where a + = y D y + T ( t J , x + ) A j ( x + , x + ' ) y + ( t J , x + ' ) d D x t d D x t , t + + / + ° / D / D y + T ( t + , x + ) A + ( t + , x t , x + ' ) y + ( t + , x + ' ) d D x t d D x t , d t + , and B + = /'° f T ( t + ) B + ( t + ) f ( t + ) d t + ( 5 . 4 . 1 4 ) ( 5 . 4 . 1 5 ) To obtain the equivalence between problems ( i ) and ( i i ) stated above, compare eqn. ( 5 . 2 . 4 8 ) for the f i l t e r i n g error covariance P(t,x,x') 155 1" *!* *1~ and eqn. (5.3.36) f o r S(t',x',x ') pertinent to optimal pointwise feedback con t r o l to e s t a b l i s h the following d u a l i t y r e l a t i o n s : t t t t. =-t, x =x, D =D t * A t = A x x M +(t 1' 5x +) = B T(t,x) (5.4.16) =t t T B'O: ) = M X(t) A + ( t + J x + , x + ' ) = Q(t,x,x') B + ( t + ) = R(t) * 34 where A , the adjoint of A , i s defined i n the extended operator sense X X such that /_ [u T(t,x ) A v(t,x) - v T ( t , x ) A * u ( t , x ) ] dD = 0, (5.4.17) JL) X X X and, i n the fourth r e l a t i o n , p (c ) i s as deiiined i n eqn. (4.5.14) and M(t), defined immediately following eqn. (5.2.3), has i t s o f f - d i a g o n a l submatrices M ± j ( t ) = 0 , ir-j , ' (5.4.18) due to the assumed structure of themeasurement equation (5.4.2). Then i t can be concluded that: ( D u a l i t y Theorem ) The optimal d i s t r i b u t e d parameter system f i l t e r i n g estimation problem given discrete-space measurements, problem ( i ) , and the d e t e r m i n i s t i c d i s t r i b u t e d parameter system optimal pointwise feedback c o n t r o l problem, problem ( i i ) , are duals of each other under the d u a l i t y r e l a t i o n s (eqn. (5.4.16)). 5.5 Conclusion In t h i s chapter, we have considered the optimal f i l t e r i n g estimation problem given discrete-space noise-corrupted l i n e a r measurements and the 156 optimal pointwise regulation c o n t r o l problem of l i n e a r continuous-time stocha-s t i c d i s t r i b u t e d parameter systems. The approach i n s o l v i n g the estimation and control problem stated above i s a u n i f i e d one and i s by considering the l i m i t i n g behaviour as the samples become dense of the corresponding d i s c r e t e -time r e s u l t s obtained i n Chapters 3 and 4. The r e s u l t s for the continuous-time l i n e a r minimum-variance f i l t e r 18 obtained turn out to coincide i n the main with those presented i n . The l a t t e r r e s u l t s , however, are a r r i v e d at v i a the e n t i r e l y d i f f e r e n t approach of employing the Wiener-Hopf equation. The s o l u t i o n of the continuous-time stoch a s t i c optimal pointx-zise c o n t r o l problem y i e l d s a feedback c o n t r o l law l i n e a r i n the state with the kernel of the d i s t r i b u t i v e feedback co n t r o l gain operator s a t i s f y i n g a d i s -t r i b u t e d Riccati-type p a r t i a l d i f f e r e n t i a l equation. Further, we obtain the important continuous-time analogue of the f a c t that d i s t r i b u t i v e f i l t e r gain and the d i s t r i b u t i v e feedback c o n t r o l gain can be determined separately as observed i n Chapter 4. L a s t l y , based on d i r e c t comparison of the solutions of the estimation and c o n t r o l problems, d u a l i t y between f i l t e r i n g estimation given d i s c r e t e -space measurements and d e t e r m i n i s t i c pointwise re g u l a t i o n c o n t r o l i s e s t a b l i s h -ed so that what can be said about estimation can also be said about c o n t r o l under the d u a l i t y r e l a t i o n s h i p , and v i c e versa. 157 6. SUBOPTIMAL POINTWISE REGULATION CONTROL OF LINEAR DISCRETE-TIME STOCHASTIC DISTRIBUTED PARAMETER SYSTEMS 6.1 Introduction I t . i s shown i n Chapter 4 that the optimal s t o c h a s t i c pointwise con t r o l for the discrete-time l i n e a r d i s t r i b u t e d parameter system-quadratic cost s t o c h a s t i c pointwise co n t r o l problem can be considered to co n s i s t of two parts: a l i n e a r d i s t r i b u t i v e Kalman-type f i l t e r to obtain the minimum-variance (optimal) estimate of the state and an optimal d e t e r m i n i s t i c c o n t r o l l e r , obtained.from the corresponding d e t e r m i n i s t i c c o n t r o l problem, using t h i s estimate. We remark here that the o v e r a l l c o n t r o l scheme can be viewed as to be divided s t r u c t u r a l l y into an estimator and a d e t e r m i n i s t i c c o n t r o l l e r i n cascade. We further observe that the optimal s t o c h a s t i c c o n t r o l l e r de-veloped i s implementable per se but under the e x p l i c i t given assumption that the system and measurement noise c h a r a c t e r i s t i c s are known exactly. However, fo r most cases of p r a c t i c a l i n t e r e s t , the noise c h a r a c t e r i s t i c s are not known exactly such that some suboptimal c o n t r o l scheme must be sought a f t e r . That • the suboptimal control scheme should be able to achieve some degree of com-promise between f i r s t l y a lack of complete and exact s t a t i s t i c a l s p e c i f i c a t -ions and secondly e f f i c i e n c y and reasonable s i m p l i c i t y i n i t s use i s c l e a r l y h i g h l y desirable. Some considerations on analogous problems i n the s t o c h a s t i c con-t r o l of lumped parameter systems where noise c h a r a c t e r i s t i c s are unknown have been given i n ^ 4 . Gracovetsky ^ has investigated the p o s s i b l e use of a suboptimal o v e r a l l c o n t r o l scheme where he proposes f o r the es-timator subsystem an adaptive f i l t e r with a purely s c a l a r f i l t e r gain and some cost f u n c t i o n a l with respect to the minimization of which at every stage the f i l t e r gain i s adjusted. Furthermore, the cost f u n c t i o n a l as a measure of performance i s used to replace the standard average cost f u n c t i o n a l quad-r a t i c both i n the c o n t r o l and the system e r r o r . The former cost f u n c t i o n a l i s a r r i v e d at based on an i n t e r p r e t a t i o n of that l a t t e r , but i s done never-. 6 A 63 theless h e u i s t i c a l l y . Bohn et a l . l a t e r r e f i n e d and furthered h i s idea to the use of a more general adaptive f i l t e r and a d i f f e r e n t cost 64 f u n c t i o n a l . In the present d i s t r i b u t e d parameter c o n t r o l context, t h e i r idea w i l l be exploited but within the framework of open-loop-optimal feed-back control approach. By employing the open-loop-optimal feedback approach, a s u i t a b l e cost f u n c t i o n a l f o r determination of the f i l t e r s c a l a r gain and therefore the f i l t e r adaptively i s derived under c e r t a i n s i m p l i f y i n g assumpt-ions l o g i c a l l y from and moreover seen to be equivalent to the quadratic one assumed o r i g i n a l l y . We mention i n passing that the open-loop-optimal feedr 28 back control approach as f i r s t approach advocated by Dreyfus i n 1964 has since been s u c c e s s f u l l y applied with various degrees of refinement and ge n e r a l i t y but so f a r only to the c o n t r o l of lumped parameter systems, see, n 56-61 for example, The remainder of t h i s chapter begins i n 6.2 with the problem formu-l a t i o n together with the incomplete s t a t i s t i c a l assumptions made. A f t e r an examination of the standard open-loop-optimal feedback co n t r o l approach i n the present d i s t r i b u t e d parameter context i n 6.3, an open-loop-suboptimal feedback control scheme i s proposed i n 6.4. The o v e r a l l c o n t r o l scheme r e -tains the esti m a t o r - c o n t r o l l e r s t r u c t u r a l d i v i s i o n . The c o n t r o l l e r i s sub-optimal through the e x p l i c i t dependence on the estimate of the state obtained by a s u i t a b l e suboptimal f i l t e r with an adaptively adjusted s c a l a r gain. The gain i s found to enter a l g e b r a i c a l l y into the equivalent cost f u n c t i o n a l shown i n d e t a i l how derived from the o r i g i n a l one. Moreover, the equivalent cost f u n c t i o n a l has but one unknown parameter which a r i s e s from the lack of com-159 plete s t a t i s t i c a l information and which must therefore e i t h e r be predeter-mined o f f - l i n e to adapt to a v a r i e t y of s t o c h a s t i c environment or tuned on-l i n e . Before the d e t a i l e d d e r i v a t i o n of the on-line algorithm to adjust the f i l t e r s c a l a r gain i n 6.6, the asymptotic s t a b i l i t y of the adaptive f i l t e r i s • studied'in 6.5 wherein bounds on the f i l t e r s c a l a r gain that ensures f i l t e r tracking of the state are obtained. Because of the p a r t i c u l a r choice of the equivalent cost f u n c t i o n a l , the o v e r a l l c o n t r o l scheme i s f e a s i b l e f o r ap-p l i c a t i o n to c o n t r o l l i n g the system i n a v a r i e t y of s t o c h a s t i c environments. F i n a l l y , a numerical example i l l u s t r a t i n g of the e f f i c i e n c y of the o v e r a l l control scheme i s presented i n 6.7. 6.2 Problem Formulation Consider the system of a l i n e a r discrete-time s t o c h a s t i c d i s t r i b u t e d parameter system subject to pointwise c o n t r o l which can be described by the following vector d i f f e r e n c e - i n t e g r a l equation W^'D G k + l , k ^ ' X ' ) u k ( x ' ) d D x ' + j x H k + l , k ( x ) f k + /D F k + l , k ( x ' V K ( x ' ) d D x ' ' ( 6 - 2 - 1 } and i t s observation at the s p a t i a l point x_^ , i = l , 2, . . . ,m by a zero-memory measuring transducer whose output i s given by the measurement equation as m (6.2.2) y k + i , i = 4 M k + i , i j u k + i ( x j ) + v k + i , i ' 1= k=0,1,...,N-1. The system of eqns. (6.2.1) and (6.2.2) has been employed before, for which the optimal estimation problem has been solved i n Chapter 3 under the assumption that for a l l k=0,1,. . . ,N-1 and a l l £=1,2, . . . ,q and the optimal stochastic pointwise r e g u l a t i o n c o n t r o l problem solved i n Chapter 4. For convenience of reference i n t h i s chapter, we review the d e f i n i t -1 6 0 ions of the various notations used i n eqns. (.6.2.1) and (6.2.2). N i s the number of subintervals the closed t i m e - i n t e r v a l [t A,t_^] i s d i s c r e t i z e d i n t o . I ={k: k=0,1,...,N-1} i s defined to be a discrete-time index set to which corresponds the discrete-time s e t . { t A , t ^ 4 , . . , t ^}. The notation xeD denotes a generic point i n the fi x e d s p a t i a l domain D of the r-dimensional Euclidean r A space E . I ={£.: £=l,2 s...,q} i s defined to ba a c o n t r o l l o c a t i o n index set to which corresponds the c o n t r o l l o c a t i o n set {x,,x„,...,x }. q i s the 1 2 q number of d i s t i n c t points x^, ^<eIq> i n the closed s p a t i a l domain D at which £ the c o n t r o l l e r s are located and f ^ denotes the p^-vector pointwise c o n t r o l at t^ , kel^., output by the c o n t r o l l e r located at x^ , ^ e l ^ - i n e a, n- (6.2.1), u^(x) i s the n-vector state at t^, Gjc+i k^ x' x'^ ± s t^ i e n x n k e r n e l °^ the state t r a n s i t i o n matrix i n t e g r a l operator, dD , i s the elemental volume about x £ the point xeD, H, ,., , (x) i s the.nxp„ c o n t r o l t r a n s i t i o n matrix f o r f. , r ' k+l,k ^£ k (x,x') i s the nxc kernel of the disturbance t r a n s i t i o n matrix i n t e g r a l operator and. {w, (x) , k=0,1,. . . ,N-1} i s a c-vector input disturbance gaussian K sequence. I = { i : i=l,2,...,m} i s a d i s c r e t e measurement l o c a t i o n index set m to which the measurement l o c a t i o n set {x,,x.,...,x } corresponds. We have 1 2 m assumed that the cont r o l l o c a t i o n set {x„, £el } i s d i s t i n c t from the measure-£ q ment l o c a t i o n set {x., i e l }. In eqn. (6.2.2), y. . i s the d.-vector x m - i v 17 k+1, x x measurement taken at t. ,., , k e l , by the transducer located at. x., i e l , k+1 t J l m' M. - .. i s the d.xn measurement matrix and {v . ., k=0,1,...,N-1} i s the K T I J 1J X K.T1 y X d^-vector measurement error gaussian sequence. For n o t a t i o n a l convenience, we rewrite eqn. (6.2.1) as U k + ! ( X ) = ' D G k + l , k ( x ' X ' ) d D x ' + \ + l , k ( x ) f k + ' D F k + l , k ( x ' X , ) w k ( x ' ) d D x ' (6.2.3) by using the d e f i n i t i o n s of an^ ^ R ^ n e c l n s " (4.2.3) and (4.2.4) r e s p e c t i v e l y and further augment eqn. (6.2.2) f o r i=l,2,...,m into the s i n g l e 1 6 1 measurement equation n k+l = W k + l + Vk+1 ' ( 6 . 2 . 4 ) . by using the d e f i n i t i o n s n k +-^, ^c+]_ > a n c * v k ' l ^ N E <^ N" ( 3 - 2 . 5 ) . The system of eqns. ( 6 . 2 . 3 ) and ( 6 . 2 . 4 ) i s the one we w i l l use i n the sequel. The state i n i t i a l c ondition u n(x) i s assumed given with i t s mean and covariance as E [ u Q ( x ) ] = u Q(x) = 0 , ( 6 . 2 . 5 ) and E [ U 0 ( X ) U Q ( X ' ) ] = PgCx.x') , ( 6 . 2 . 6 ) r e s p e c t i v e l y . The system nosie w, (x) and measurement nosie v. ., are assumed to k k+1 be each of zero mean and uncorrelated with each other and with the state i n i t i a l condition u^(x). Moreover, t h e i r respective covariances are assumed to be f i n i t e but otherwise unknown. Thus, E[w k(x)] = E [ v k + 1 ] =0 , • E[w k(x)w^(x*)] = Q k ( x , x ' ) 6 K £ E [vL ] = \ + i 6 k £ • <'6-2-7> E [ wk ( x ) vLi ] =0 > E[w k(x) UQ(x')] = E [ v K + 1 U Q ( x ) ] = 0 , where the unknown system and measurement nosie covariances Q (x,x') and R k k+X are s e l f - a d j o i n t nonnegative-definite and symmetric p o s i t i v e - d e f i n i t e matrices r e s p e c t i v e l y . Had the noise covariances Q k(x,x') and R^-i-]. ^ e e n s p e c i f i e d , the problem to be considered would be the optimal s t o c h a s t i c pointwise r e g u l a t i o n c o n t r o l problem of Chapter 4. The formulation of that problem we w i l l r e -state below for reference i n t h i s chapter. Before doing so, we f i r s t r e c a l l 162 that the notation n 3 i s used to denote the c o l l e c t i o n of the state i n i t i a l condition mean UQ(X) and the measurement data n^,"2 > • • • >nj a n c^ t n e n o t a t i ° n f 3 ~ ^ to' denote the c o l l e c t i o n of pointwise controls f n , f ^ , . . . , f . _ ^ . More-over, we r e c a l l that the closed-loop feedback co n t r o l law f , j=0,1, . . . ,]SM,, i s to s a t i s f y the condition f. = fjOr'.f-5""1) , C6.2.8) for every j e l . The problem of Chapter 4 can now be restated as follows: Given the system of eqns. (6.2.3) and (6.2.4), f i n d a p h y s i c a l l y r e a l i z a b l e c o n t r o l law f., j=0.1,...,N-1, of the form i n eqn. (6.2.8) which minimizes the average cost f u n c t i o n a l 1^ , where i< = E [ V L . + 1 ( u . + 1 ( x ) , f . ) J (6.2.9) and T.. .. A i . . . ( x } . f = /./_ u T._ (x)A. .. fx.xMu. .. (x')dD dD . J T M . ' J IM . J u u J - r x " j - r j . J T l A X + f T B . f . . (6.2.10) 3 3 3 The matrices A_._^(x,x') and B^ . are nxn and pxp weighting matrices r e s p s c t i v e -l y and assumed to be s e l f - a d j o i n t . We term t h i s problem to be the o r i g i n a l problem (OP) and r e c a l l that i t has been solved i n Chapter 4 under the assumption that the noise covariances Q k(x,x') and R^-^ are known. Since the noise covariances Qj (x,x') and R^-fi are unknown, we cannot apply the co n t r o l scheme developed i n Chapter 4 per se to solve the above-mentioned o r i g i n a l problem (OP). We therefore need to adopt a d i f f e r e n t approa-ch to determine the admissible c o n t r o l sequence {f . , f, , . . . ,f_, -}. One n 0 1 N-l possible approach i s to use the s o - c a l l e d open-loop-optimal feedback c o n t r o l 28 scheme . The s a l i e n t feature of t h i s p a r t i c u l a r approach i s that the condition (6.2.8) required on the c o n t r o l i s relaxed such that the current 163 and future controls f . , j>k, are functions of the presently a v a i l a b l e i n -k formation n only obtained at the current time t . Thus, f = f.Cn^P"1) , j>k. (6.2.11) Condition (6.2.11) not only expresses the f a c t that the c o n t r o l i s p h y s i c a l -l y r e a l i z a b l e but also that the c o n t r o l i s open-loop. Further d e t a i l s of the open-loop-optimal feedback c o n t r o l scheme i n the present d i s t r i b u t e d parameter context w i l l be examined i n the next sec t i o n . Before doing so, we observe that the o r i g i n a l problem (OP) must now be reformulated so as to accommodate the change of condition (6.2.8) to condition (6.2.11). Hence, the problem to be considered i n the sequel can now be stated as follows: Given the system of eqns. (6.2.3) and (6.2.4), f i n d a p h y s i c a l l y r e a l i z a b l e c o n t r o l law f , j=0,l,...,N-1, of the form i n eqn. (6.2.11) which minimizes an equivalent cost f u n c t i o n a l J associated with I'. eq,0 • 0 In the subsequent sections, we suggest, a f t e r an examination of the open-loop-optimal feedback approach i n the present d i s t r i b u t e d parameter context, the equivalent cost f u n c t i o n a l J „ to be used as a measure of eq,0 performance and put forward an adaptive c o n t r o l scheme to solve the derived problem (DP). 6.3 Background to Open-loop-optimal Feedback Control 28 In t h i s s e c t i o n , the open-loop-optimal feedback c o n t r o l approach i s examined i n the present d i s t r i b u t e d parameter context. We f i r s t of a l l denote the average cost f or the l a s t N-k stages by I', where k I k = E [ ^ L j + 1 ( u . + 1 ( x ) , f . ) ] , (6.3.1) and by d e f i n i t i o n 164 IjJ = 0 . (6.3.2) Also, we denote the estimate of the true state u^Cx) at t based on a v a i l a b l e controls f 3 ^ and measurements n^ at t, by u . i . (x). We define the estimation k j I k e r r o r u . i , (x) f o r i>k, i , k e l as j | k t u. , Cx) = u.Cx) - u. | (x) (6.3.3) 3 I k J 3 |k and the corresponding error covariance P^| k(x,x') conditioned on f 3 ^ and as P j j k(x,x') = E [ u j j k ( x ) u ^ j k ( x , ) | n k , f j _ 1 ] . (6.3.4) For the case of j=k, we w i l l use simply the notations ^ ( x ) f ° r ^ l ^ ^ ^ ' u j c ( x ) f or uJ C |J (.( x) and P k(x,x') for P k | k ( x , x ' ) . Consider the system of eqns. (6.2.3) and (6.2.4) at the present time k-1 indexed by k and assume that the past controls f " and the past and current measurements are a v a i l a b l e . We assume that a f i l t e r e d estimate u ^ ( x ) 1 S a v a i l a b l e and. emphasize the f a c t that the f i l t e r e d estimate ^ k ( x ) i s presently not assumed to be the optimal one. ; - As f a r as the c o n t r o l l e r i s concerned i n the open-loop-optimal feed-back approach, the closed-loop feedback p h y s i c a l r e a l i z a b i l i t y c o n d i t i o n (6.2.8) i s relaxed as has been mentioned i n 6.2 such that the current and future controls f . , j>k, are functions of the presently a v a i l a b l e information k-1 k f and n only. Thus, f , j>k, i s of the form i n eqn. (6.2.11). Now, the controls f , j>k, are so chosen as to minimize ( i n d i s t i n c t i o n to the expected value of I k given by eqn. (4.4.1) ) the open-loop average cost f u n c t i o n a l N-1 I k = E[ E L (u . + 1 ( x ) , f Cn,f J X ) ) ] , (6.3.5) j=k or, equivalently, by an argumeirt s i m i l a r to that leading to the minimization 165 of eqn. (4.4.4), the co n d i t i o n a l expectation J k = E [ * ^ j + l ^ j + l Cx),f Cnk,fj L ) ) | k ] (6.3.6) subject to the state and measurements constraints of eqns. (6.2.3) and (6.2,4). Condition (6.2.11) implies temporarily that there are no future measurements to be taken and s p e c i f i e r the current: and future controls to be given as functions of the " i n i t i a l " point and time. Hence, the control sequence so obtained i s open-loop and optimal from time k on given past and current measure-ments. However, rather than c o n t r o l l i n g the system then on using the co n t r o l sequence so calcul a t e d , feedback i s incorporated and a d d i t i o n a l measurements made use of i n the following manner: only the f i r s t member f ^ i s a c t u a l l y implemented thereby advancing the. time index to k+1,. when a new measurement n, , i s taken and i s included i n the recomputation of the open-loop c o n t r o l K.T J_ k+1 f , ^ as a function of n at time k+1. This procedure i s repeated u n t i l k=N-l. Further, from eqn. (6.3.6) and the d e f i n i t i o n (6.2.10) for L. .. (u . .. (x) ,f.) , the open-loop c o n d i t i o n a l cost f u n c t i o n a l w r i t t e n i n f u l l j + l 3+1 3 i s Since the future controls are assumed to be independent of the future a v a i l -able measurements, the l a s t term on the r i g h t hand side of eqn. (6.3.7) i s excluded from the expectation operation. S u b s t i t u t i o n of eqns. (6.3.3) and (6.3.4) inot eqn. (6.3.7) y i e l d s • (6.3.7) A J l , k + J 2,k A = J l,k + J 21,k + J 22,k (6.3.8) 166 where J i , k = V, V D V i ! i , ( x ) ^ + i ( X ' X , ) " 3 + i | k ( x ' ) d D x d D x - + fWr (6-3'9) J2i,k ^ 2 ^ E f V D V i | k ^ ) ^ + i . ^ ' x ' > a j + i [ k ( x , ) d V D x ' ] 1' k . = 2 V t r / D / D A j + 1(x,x')E[a j + l | k(x ')aJ + l | k(x)]dD xdD x l , (6.3.10) and J22,k = ** K E T VD r i k | k ( x ) A J + i ( x ' x ' ) a j + i | k ( X ' ) D V D x ' ] N-l = ^ t r ; D / D A j + 1 ( x , x ' ) P j + l | k ( x ' , x ) d D x d D x l . (6.3.H). The problem of minimizing J of eqn. (6.3.6) subject to the state and measurement constraints of eqns. (6.2.3) and (6.2.4) by s u i t a b l y choosing the open-loop p h y c i a l l y r e a l i z a b l e control sequence {f , f k + ^ , . . . , f ^ _ ^ } can now be interpreted as that of minimizing j " k i n eqn. (6.3.8) subject to the constraints of the state equations for the estimate + ^ j ^ ( x ) , the estimation error u j _ | _ i |it-(x) a n c* t n e c o n d i t i o n a l estimation error covariance P^ +^| k(x,x') for which are assumed given the f i l t e r e d estimate ^ ( x ) , the f i l t e r i n g e r r o r Q k(x) and the con d i t i o n a l error covariance P k(x,x') r e s p e c t i v e l y . We proceed to derive below the state equations of k ( x ) , u . + l ] k ( x ) and P j + 1 | k ( x , x ' ) . We f i r s t l y observe that, by repeated a p p l i c a t i o n of the state equation (6.2.3), u_.+^(x) can be wr i t t e n as u J + 1 0 0 = / D G j + l f k(x,x')u k(x')dD x, + ^ / D G ^ ^ C x . x ' ^ ^ C x ' ^ d D + .'l, V d G J + i , i ( x ' x ' ) F i a - i ( x , ' x " ) w i - i ( x " ) d D x ' d D x " > ( 6 - 3 ' 1 2 ) x=k+l 167 for j>k, k . i e r , where we r e c a l l that the kernel G.,, , (x,x') of the state - ' ' J t ' j+l,k ' t r a n s i t i o n i n t e g r a l operator s a t i s f i e s the following t r a n s i t i o n a l property G j + l f k f r , * ' 3 = / D G j + l 5.Cx,x")G i > k(x",x')dD x t l (6.3.13) and G k k(x,x') = I5(x-x () (6.3.14) for a l l k<i<j, k . i j j e l ^ . Since noise cannot be predicted, we obtain from eqn. (6.3.12) the estimate u.,,,, Cx) as 3+1 u . , l h (x) = / G (x,x')u. (x')dD , + l f G... .(x,x')H. . Ax')f. .dD 3+11 k D 3+1,k k x i-k+i D 3+-1'1 i . 1 - ! i - l (6.3.15) for j>k, k , j e l t . Bu using eqns. (6.3.13) and (6.3.14), we can write from eqn. (6.3.15) the state equation f o r the estimate u j +1.1 k *~X ^  a s V l | k ( x ) = 'D G 3 + l , 3 ( X ' X ' ) G 3 | k ( x ' ) d D x ' + H 3 + l , 3 ( x ) f 3 ' ( 6 ' 3 - 1 6 ) for j>k, k , j e l f c . Substracting eqn. (6.3.15) from eqn. (6.3.12) y i e l d s the fact that u_. +^j k(x) i s expressible as a l i n e a r sum of u k ( x ) , w k(x), w k +^(x),..., w_. (x) . To be exact, we have V i | k ( x ) = X D V u ( x ' x , ) a k ( x , ) d I ) x ' + "f V D G j + i , i ( x ' x , ) F i , i - i ( x , ' x " > w i - i ^ " > d D x - D V i=k+l J (6.3,17) for j>k, k j j e l ^ . E i t h e r by applying eqns. (6.3.13) and (6.3.14) or by sub-t r a c t i n g eqn. (6.3.16) from the state equation (6.2.3) f o r u.,,(x), we obtain 3+1 the state equation f o r the estimation error u.. +^| k(x) a s a 3 + l | k ( x ) = 7D V l . J ^ ^ ' ^ J l k 0 1 ' ^ ' + ^ F 3 + l , 3 ( X ' X , ) W 3 ( X ' ) d D x ' ( 6 - 3 - 1 8 ) 168 fo r j>k, k , j e l t . Using the above equation for u i (x) to take the c o n d i t i o n a l J +11 K covariance of u. ,., i , (x) and u, ,, i , C x ' ) . we obtain 1+11 k 3+11 k pj + i ! k ( ; x ' x , ) = G - a l . ( X , X " ) P , . ( X " , X M ' ) G 1 ' 1 .(x',x"')dDx"dDx"' P P J+1,3 3 k j+1 ,j + VD F j + 1 > j ( x , x " ) E [ W ^ + / / F (x sx")Q (x",x"')F^ (x',x"')dD dD r (6.3.19) for j>k, k , j E l t > So f a r , we have not mentioned anything about the f i l t e r e d estimate k-1 k u^(x) obtained from the a v a i l a b l e f and n . Once u k C x ) I s s p e c i f i e d , u k ( x ) and P, (x,x') can of course be computed according to t h e i r respective d e f i n i t -ions (6.3.3) and (6.3.4) . Now, standard a p p l i c a t i o n s of open-loop feedback approach i n the lumped parameter control theory employ the use of a f i l t e r that i s optimal i n the condi t i o n a l mean sense. The possible implications of the corresponding use of (optimal) c o n d i t i o n a l mean f i l t e r i n the present d i s t r i b u t e d parameter context deserves to be investigated. i k k-1 From the d e f i n i t i o n of the f i l t e r e d estimate as u k(x)=E[u k(x)|n ,f ] and eqn. (6.3.16) together and eqn. (6.3.12), we can write j + l | k v ' L j+1 from eqn. (6.3.20) for ^ that ti...,, ( x ) = E [ u . + 1 ( x ) | n k , f j ] f o r j>k. Hence, E [ u j + 1 | f c ( x ) | n ,f J]=0 and we see J n > k . O . (6.3.20) Now, the co n d i t i o n a l mean f i l t e r f o r a system of l i n e a r state and 169 and measurement euqattons under gaussian assumptions i s a l i n e a r f i l t e r and moreover i t s unconditional p r e d i c t i o n and f i l t e r i n g error covariances are the same as the con d i t i o n a l ones. Since the c o n d i t i o n a l mean f i l t e r i s a l i n e a r f i l t e r , u j c 0 0 1 3 a l i n e a r function of f^* ^  and n^. From,this f a c t and eqn. (6.3.15), u^.j k(x) i s a l i n e a r function of f^ ^  and n^. Since u_. (x) from eqn. (6.3.12) i s a l i n e a r function of UQ(X). , f^ and WQ (X) ,. . . ,w_. _^ (x) , u.i , (x)=u.., ( x ) - u . i . (x) i s therefore a l i n e a r function of f^ ^ , and 3 Ik 3 I k 3 I k w (x),...,w. (x). Further, since E[n w T(x)|n k,f^ 1]=0 and E[f._,w T(x)|n C,f J ] U J 1 k 3 3 -1- 3 =0, j>k, due to the zero mean assumption on the plant noise, we have T i k i-1 E[u. i n (x)w.(x') n ,f ]=0 upon using i n ad d i t i o n the second r e l a t i o n of eqn. 3 I k 3 T i k i (6.2.7) . Hence, E[u_. | ^  (x)w^. (x') | n ,f J]=0. Therefore, for t h i s case of a l i n e a r f i l t e r , P.., , (x,x') i s now given from eqn. (6.3.9) as ' 3+1 k V i | k ( x ' x , ) -00 ' j+J . ,3 • ' ' j|k*" ' j + i , j " *" ""V" + V D F. + l s.(x,x")Q. (x",x"')Fj + 1 J(x',x"')dD x„dD x, M ( 6 3 > 2 1 ) f o r a l l j>k, k j j e l ^ . With J„ n ,=0, we are now l e f t with , and J„„ . i n J, of eqn. 21, k l , k 22,k. k (6.3.8) . The controls f . , j>k, appear e x p l i c i t l y i n J . only. Furthermore, 3 ~ l>k the c o n d i t i o n a l covariance P . i (x,x') i n J i s given by eqn. (6.3.21) J "T"jL iC /. /. y K. and i s completely determined once P k(x,x') i s given, that i s , ^ i s n o t subjectable to co n t r o l a c t i o n . Hence, we make the observation that, as f a r as the c o n t r o l l i n g aspect i s concerned, the co n t r o l sequence { f ^ > » • • ' • ' ^ N - 1 ^ that minimizes J, under the constraint of the state equation (6.2.3) f o r u, ,, (x) k k+1" i s that which minimizes J.. . under the constraint of the state equation (6.3.16) l,k for u \ + 1 | k ( x ) . Now, the problem of choosing { f k > f k + 1 , . . . , f N _ 1 > to minimize J subject to state constraint eqn. (6.3.16) i s a d e t e r m i n i s t i c problem. J_ y K. 170 Given the l i n e a r i t y of the state equation (6.3.16) and the q u a d r a t i c i t y of k ^, f , j>k, i s c l e a r l y l i n e a r i n n . We see here that ^=0 decouples the o v e r a l l control problem into a d e t e r m i n i s t i c c o n t r o l problem and a separate estimation problem.. We w i l l e x p l o i t the idea of enforcing ^=0 by employing a s u i t a b l e (suboptimal and adaptive) l i n e a r f i l t e r and thereby f i n d i n 6.4 an equivalent cost f u n c t i o n a l J _ to associate with the open-loop c o n d i t i o n a l cost eq 50 f u n c t i o n a l J A and therefore the o r i g i n a l cost f u n c t i o n a l 1^. 6.4 Proposed Open-loop-suboptimal Feedback Control The open-loop-optimal feedback c o n t r o l could of course be obtained 60 i n a fashion s i m i l a r to that presented i n for lumped parameter systems. However, that would e n t a i l , f o r a system with non-random parameters, the generation of the co n d i t i o n a l mean (optimal) f i l t e r e d estimate, i n v o l v i n g o f f - l i n e c a l c u l a t i o n and storage of the f i l t e r i n g error covariance with respect to a p a i r of known system and measurement noise covariances, in. the'manner depicted i n Chapter 3. In t h i s s e c t i o n , then, the structure of the open-loop feedback c o n t r o l i s f i r s t imposed; the c o n t r o l i s optimal i f an optimal f i l t e r e d estimate i s used, otherwise, suboptimal. In order to be able to .generate an estimate on-line, a suboptimal f i l t e r that i s capable of adapting to d i f f e r -ent noisy environments i s used. That the f i l t e r should d e s i r a b l y have a l i n e a r p r e d i c t o r - c o r r e c t o r structure i s ensured. Consequential to the im-posed structures of the c o n t r o l l e r and the f i l t e r , the open-loop cost f u n c t i o n a l i s furth e r reduced to an equivalent one, and i t i s with r e f e r -ence to the (lo c a l ) minimization at every stage of the equivalent cost f u n c t i o n a l subject to f i l t e r and measurement constraints that the suboptimal f i l t e r i s adaptively chosen. Since the c o n t r o l and f i l t e r i n g problems are c l o s e l y r e l a t e d to 171 each other, no matter what form the f i l t e r takes, what i t gives as an estimate of the state a f f e c t s the q u a l i t y of the c o n t r o l . Consequently, the use of a suboptimal f i l t e r r e s u l t s i n the c o n t r o l being suboptimal and the scheme therefore becomes an open-loop-suboptimal feedback one. 6.4.1 Control Problem Let us reexamine the open-loop cost f u n c t i o n a l given by eqn. (6.3.8). The open-loop co n t r o l sequence f . , j>k, appears e x p l i c i t l y only i n J ~ the component ^ (eqn. (6.3.9)). In the component ^, J 2 ; L k (eqn. (6.3.10)) can be equated to zero by a s u i t a b l e choice of the suboptimal f i l t e r to be d e t a i l e d l a t e r i n the following subsection, and J 2 2 i c (eqn. (6.3.11)) i s not subjectable to co n t r o l action as, ignoring future measurements, the error covariance P_.+^ | ^ (x,x') i s completely s p e c i f i e d by P ^ C ^ x ' ) , assumed known, through eqn. (6.3.21). Thus, an equivalent c o n t r o l problem, as far as the c o n t r o l l i n g aspect i s concerned, can be stated i n the following manner: the c o n t r o l sequence f , j>k, i s so chosen as to minimize J l , k = ^ V D 0 ] + l j l c W A j + l ( x ' X ' ) G j + l | k ( x , ) d D x d D x ' + f > j f j ( 6 - 4 - 1 } . 0 0 = /. G 4 J - 1 , ( x , x ' ) u , h (x')dD, + -H,.- , ( x ) f , , (6.4.2) subject to state constraint l j + l | k ( x ) = ;D G j + l , j ( A ' A ' u j | k ^ '""x' ' j>k, where u (x) i s given. This i s a d e t e r m i n i s t i c c o n t r o l problem. Following a procedure s i m i l a r to that i n Chapter 4, the open-loop c o n t r o l i s shown to be f° = - fn A T 1 ! . ( x ) u . j , (x)dD , ' (6.4.3) J D j i j | k x where A^ and B^(x) are given by , r e s p e c t i v e l y , eqns. (4.4.11) and (4.4.12) with N-k replaced by k. As mentioned i n 6.3, only the f i r s t member f 1 of the. sequence f , 172 j>k, Is to be implemented. Me. therefore note immediately that, i f . the estimate u^Cx) i s defined to be the c o n d i t i o n a l mean estimate, f° as given by eqn. (.6.4.3) i s p r e c i s e l y the optimal feedback c o n t r o l given by eqn. (4.4.10) and r e s u l t s i n the "separation" of the c o n t r o l and f i l t e r i n g problems. Substituting f° from eqn. (6.4.3) into eqn. (6.4.1) y i e l d s the minimum of , as 1 ,k where V k(x,x') i s as given by eqn. (4.4.13) but with N-k replaced by k. Then, J° k 1 S considered to be the control cost portion of the equivalent cost f u n c t i o n a l . As such, the estimate u^ -OO i s yet to be defined s u i t a b l y . This i s the topic of the next subsection. 6.4.2 F i l t e r i n g . Problem' u. (x) of the true state u, (x) and the cost f u n c t i o n a l equivalent to J„ . . k k n 2,k Since the system and measurement noise covariances are unknown, the l i n e a r d i s t r i b u t i v e Kalman f i l t e r as developed i n Chapter 3 cannot be used per se. Rather, a suboptimal f i l t e r that i s capable of adapting to d i f f e r e n t system and measurement noises i s sought. Moreover, i t i s d e s i r a b l e to pre-serve the l i n e a r p r e d i c t o r - c o r r e c t o r structure of the Kalman f i l t e r . Thus, the estimate u (x) i s defined to be the output of the following f i l t e r : k • = a k | k - l ( x ) + § k L ( x ) ? k \ | k - l ( x ) = ' D G k , k - l ( x ' X , ) \ - l ( x ' ) d D x ' + \ , k - l ( x ) f k - l ( 6 - 4 ' 5 ) k>l, given the i n i t i a l estimate u A(x) and i n i t i a l s c a l a r gain g A , where the measurement r e s i d u a l ; ck - \ - Vkjk-i (6-4-6) i s the d i f f e r e n c e between the actual and predicted measurements, g^ i s a 173 s c a l a r function s a t i s f y i n g gk * 8 k ° V i ( x ) ' f k - i ' V > C 6 - 4 - 7 ) and L(x) i s a space-dependent weighting matrix. Examination of eqn. (6.4.5) reveals that compared with the optimal l i n e a r d i s t r i b u t i v e f i l t e r gain K Cx) i s replaced by the combination g^L^x), that i s , i t i s set, from eqn. (3.4.10) with k+1 replaced by k, K k(x) = g RL(x) . (6.4.8) The weighting matrix L(x) i s chosen, to weight the measurement r e s i d u a l i n the corrector term i n such a way that more c o r r e c t i o n i s given to the domin-ant component(s) of the state vector. This can be done by l e t t i n g L(x) be the steady state value of the ^ ( x ) c a l c u l a t e d f o r the system eqn. (6.2.3) i n accordance with eqn. (3.4.55) based on assumed small values of the system and measurement noise covariances Q k_j(x,x') and Rj^. The s c a l a r gain g k then i n 6.6 so that i t r e s u l t s i n the decrease of the equivalent cost f u n c t i o n a l stage-wise and, at the same time, i n the f i l t e r of eqn. (6.4.5) " l e a r n i n g " the true state u k ( x ) . In order to do so e f f i c i e n t l y , i t i s advantageous to consider i n the place of the cost f u n c t i o n a l k a simpler k -It i s reasonable to make the following two assumptions: the f i l t e r e d estimate and predicted estimate u_. +^| k(x) at xeD i s uncorrelated with the estimation error u . , - , 1 . (x') at x'eD, x^x', that i s , j +11 k EtVHic ( x )Vi|k ( x' ) ] = 0 • x ^ ' ' ( 6 - 4 - 9 ) and, the estimation error u , , , i , (x) at xeD i s uncorrelated with u . L l i , (x 1) J+11k 3+11k at x'eD, x^x', that i s , E t V l | k ( x ) a j + l | k ( x ' ) ] = ° ' X ^ X ' ' (6.4.10) A p p l i c a t i o n of the assumptions (6.4.9) and (6.4.10) to eqns. (6.3.10) 174 and C6.3.11) y i e l d s the cost f u n c t i o n a l T ' -— J f + J ' (6.4.11) J2,k J21,k. 22,k ' wnere N-1 T J21,k - 2 ^ " ' D A j + l C X ' X ) E [ V l | k C x ) V M l c C x ) l d D x ' ( 6 ' 4 , 1 2 ) and J 2 2 , k = t r 7D V l ( X ' X ) P J + l | k ( X ' x ) d D x ' . ( 6 - 4 - 1 3 ) We are then to consider J ' , and J ' , each i n turn. .21,k 22,k Consider the system at time k and the optimal f i l t e r e d estimate "k l ^ x ^ § l v e n s u c n that, from eqn. (3.5.5), E[Q k_ 1(x)u k_ 1(x')] = 0 . (6.4.14) Now, s u b s t i t u t i n g into eqn. (6.4.12) eqn. (6.3.16), with the optimal c o n t r o l f j given by eqn. (6.4.3), and eqn. (6.3.18) y i e l d s , a f t e r making use of the r e s u l t (3.4.28), J21,k = 2 t r fT) V X ' x ) E [ V x ) \ ( x ) ] d D x ' (6.4.15) and, u t i l i z i n g eqn. (3.5.4), J21 k = t r 7D V k ( x , x ) [ K k ( x ) g k ( x ) + 3 k(x )K k(x) - 2K k(x)Y k"< k(x)]dD x. (6.4.16) Substituting eqn. (6.4.8) into eqn. (6.4.16) y i e l d s J n,k = 2 b A - 2 \ 4 * ( 6 - 4 - 1 7 ) where a k = t r v k ( x » x ) L ( x ) \ L T ( x ) d D x » (6.4.18) and b k = \ t r / D V k(x,x)[L(x )3 k(x) + 3 R(x)L T(x)]dD x . (6.4.19) Define 175 gP = bja,. . C6.4..20) 'k k' k T It i s observed that, consistent with enforcing E [ u ( x ) u ( x ) ]==(), the choice g =g° i n eqn. (6.4.17) r e s u l t s i n min J ' = 0 . (6.4.21) o 21,k g k = 8 k Substituting eqn. (6.3.21) with x' replaced by x in t o eqn. (6.4.13) y i e l d s where J22 k = t r ;D C k ( x ' x ) P k ( x , x ) d D x + C ° n S t ' (6.4.22) const = V V R/ D K.+1(x,x)i.+1^ (6.4.23) and C (x,x) i s given with x' replaced by x by N-1 T C k(x,x') ~ / D / D G . + 1 ) k ( x " , x ) A j + 1 ( x " , x " - ) G j + 1 > k ( x ; " , x ; ) d D x „ d D x „ , (6.4.24) Define, for convenience, C k(x,x') = C k(x,x T) + A k(x,x') . (6.4.25) A recursive r e l a t i o n f o r C k(x,x') i s d e s i r a b l e and i t can be e a s i l y shown that, from eqns. (6.4.24) and (6.4.25), c k ( x > x , ) " V D ^ i . k ^ ' ^ k ^ + \ ( x ' x , ) (6.4.26) k=N-l,N-2,...,0, with the " i n i t i a l " c ondition . ; C^(x,x') = A N(x,x') . (6.4.27) The second term appearing i n eqn. (6.4.22) i n v o l v i n g the unknown but f i n i t e system noise covariance Q (x,x') i s but an a d d i t i v e constant. To i n t e r p r e t C k(x,x') f u r t h e r , we now make use of the d u a l i t y 176 r e l a t i o n s h i p between optimal c o n t r o l of l i n e a r d e t e r m i n i s t i c systems subject 19 to quadratic costs and ( f i l t e r e d ) state estimation and extend i t to t h i s d i s t r i b u t e d parameter case (. or use a d i s c r e t e v e r s i o n cf the continuous-time d i s t r i b u t e d parameter system d u a l i t y r e l a t i o n developed i n Chapter 5 ). Let us therefore make the following " i d e n t i f i c a t i o n " of the various matrices: T / G (x,x')(*)dD , = system t r a n s i t i o n matrix from k to k-1; U K. • X., X T H",, , (x) = measurement matrix at k; k-rjL, k A ^ X j x ' ) = (white) system noise covariance matrix at k; ^k = ( w n i t e ) measurement noise covariance matrix at k; S, .,(x,x') = p r e d i c t i o n e r r o r covariance matrix at k; K+1 S^(x,x') = A^(x,x') = covariance of state vector at N p r i o r to processing the measurement at N. From eqns. (4.4.7) and (4.4.13) with the time index N-k now replaced by k and N-k-1 by k+1 for t h i s context, we have S k(x,x') '*VD < + l ( x n ) \ + l ( x ' > M , ) \ + l , k ( x , M > d D x " d D x " ' + B k ^ S " + l , k ^ " ) S k + l ( x " ' X " , > ' G k + l , k ( x " ' ' X , ) d D x " d D x " ' + VX>X') ' (6.4.28) with the " i n i t i a l " condition S N(x,x') = A^x.x') , (6.4.29) k=N-l,N-2,...,0. It i s seen that eqn. (6.4.28) describes the evolution of the pre-d i c t i o n error covariance matrix as measurements come i n , with the time index 177 decreasing as more measurements are processed, whereas, eqn. ( 6 . 4 . 2 6 ) describes that of the error covariance without the ben e f i t of measurements. Hence, the. di f f e r e n c e C'(x,x')-S (x,x') represents the improvement i n covariance due to measurements and i s therefore always at l e a s t nonnegative-definite such that C k(x,x') - V k(x,x') = C k(x,x') -Sk(.x,x') >0 . ( 6 . 4 . 3 0 ) In eqn. ( 6 . 4 . 2 2 ) , C (x,x') appears as a weight to the estimation e r r o r . Hence, consistent with the desire to obtain a simpler cost f u n c t i o n a l than k , i t i s reasonable to replace C k(x,x) i n eqn. ( 6 . 4 . 2 2 ) by V k ( x , x ) . The resultant e f f e c t i s that, according to eqn. ( 6 . 4 . 2 2 ) , the estimation i s now weighted somewhat l e s s , which i n a p r a c t i c a l s i t u a t i o n i s not i n t o l e r a b l e . An a d d i t i o n a l advantage i s evident: using V k ( x , x ' ) , which needs f i r s t to be precomputed and stored f o r the on-line generation of the co n t r o l f.. given by eqn. ( 6 . 4 . 3 ) , obviates the further need to precompute eqn. ( 6 . 4 . 2 6 ) and store i t s s o l u t i o n . Hence, J'„ , of eqn. ( 6 . 4 . 2 2 ) should now read J 2 2 k = T R •'"D \ ( x > x ) p k ( x > x ) d D x + const , ( 6 . 4 . 3 1 ) where "const" i s s t i l l given by eqn. ( 6 . 4 . 2 3 ) . Replacing k by k - 1 i n eqn. ( 3 . 4 . 5 1 ) and putting i n eqn. ( 6 . 4 . 8 ) , the . f i l t e r i n g error covariance i s P k ( x , x ) = a f c ( x , x ) - g k[L(x ) 3 k(x) + 3 k ( x ) L T ( x ) ] + g 2 [ L ( x ) Y k L T ( x ) ] , k>l. ( 6 . 4 . 3 2 ) Then, s u b s t i t u t i n g eqn. ( 6 . 4 . 3 2 ) into eqn. ( 6 . 4 . 3 1 ) y i e l d s J 2 2,k " C k " 2 b k g k + \ g k + C ° n S t ' ( 6 ' 4 - 3 3 ) where a, and b, are as defined by eqns. ( 6 . 4 . 1 8 ) and ( 6 . 4 . 1 9 ) r e s p e c t i v e l y and k k . c, = t r / V. (x,x)a (x,x)dD . . . ( 6 . 4 . 3 4 ) I t i s i n t e r e s t i n g to note that with g k=g k> ^ 2 2 k a l s ° a t t a i n s i t s m i n i m u m -178 For a non-optimal g , given'g° pre.computed according to eqn. (6.4.20), eqn. (6.4.34) can be written'as J 2 2 , k = (^- 24 gk + d k ) a k + C O n S t ' ^ 6 - 4 - 3 5 ) where , A. , •k k k The estimation cost to be minimized can therefore be considered to be, from eqns. (6.4.21) and (6.4.35), J 2 , k = 2 g k g k + V a k + C ° n S t • ( 6 ' 4 - 3 6 ) 6.4.3 Equivalent Cost Functional Combining eqns. (6.4.4) and (6.4.36), we have J k = V D \ « V k ( x ' X , ) \ ( x ' ) d D x d D x ' + (g£ - 2g°g k + d k ) a k + const . (6.4.37) It i s seen that the o r i g i n a l cost f u n c t i o n a l 'J"k (eqn. (6.3.7)) f o r the open-loop feedback con t r o l problem, yet to be minimized, has now been reduced to an equivalent cost f u n c t i o n a l J' given.by eqn. (6.4.37) dependent on the sc a l a r gain g, . In conformity with the desire to minimize the o r i g i n a l cost J , g i s , k k. R then to be so chosen as to minimize J' subiect to the f i l t e r and measure-k J ment constraints eqns. (6.4.5) and (6.2.4). To do so on - l i n e , needs to be turned into an instantaneous cost f u n c t i o n a l such that the parameters i n eqn. (6.4.37) can be estimated as far as possible from the a v a i l a b l e measurements. , We f i r s t note that by taking the covariance of the measurement re s i d u a l z1 . at x., given by eqn. (3.4.5) but with k+1 replaced by k, and Kj 1 1 that z, . at x., we obtain E t Z k , i Z k 5 J ] = Y k , i J > ( 6 - 4 ' 3 8 ) where y. . . i s as defined by eqn. (3.4.45). Augmenting eqn. (6.4.38) over k , i j 179 i , j el. , we have m v E[^] • C6*4'39) Now, from the d e f i n i t i o n of a^ given by eqn. (6.4.18) and eqn. (6.4.39), we have a R = t r / D V k(x,x)LCx)E[C k? k]L TCx)dD x where Wk = / D L X ( x ) V k ( x , x ) L ( x ) d D x . (6.4.41) Hence, dropping the expectation operation i n eqn. (6.4.40) to obtain the instantaneous value, a k i n eqn. (6.4.37) can be approximated by \ = • ( 6- 4- 4 2> Moreover. l n nrde-r to maV.e the cost functional. J,' (ena . (6.4.37)) K. s u i t a b l e for a v a r i e t y of noisy environments, g° i s replaced by g. Here, g i s , for d i f f e r e n t combinations of system and measurement noise covariances Q k_^(x,x') and R^, the average of the steady values of g° as defined by eqn. (6.4.20), where consideration of the steady values alone obviates the need of sto r i n g g° for a l l k. And, g i s to be determined o f f - l i n e or tuned on-line. Hence, incl u d i n g the above two points into eqn. (6.4.37), the f i n a l equivalent cost f u n c t i o n a l to be used subsequently i s J e q , k = VD ^Cx)V k(x,x')u k(x')dD xdD x, + [ g j - 2 i g k + d k K k W k ? k + const . ' (6.4.43) 6.5 Asymptotic S t a b i l i t y of the F i l t e r 6.5.1 S t a b i l i t y Consideration This section i s concerned with the conditions on the sca l a r gain g under which the l o c a l minimization problem defined by eqns. (6.2.3), (6.2,4), 180 (6.4.5) and, (6.4.43). at time k. does.Hot produce a d e c i s i o n that r e s u l t s at a . l a t e r stage i n loss of f i l t e r . t r a c k i n g of the state and the res u l t a n t i n -crease of the estimation cost component i n J . . eq,k From eqns. (6.2.3) and (6.4.5), i t i s seen that the f i l t e r i n g e r r o r i s u k(x) = u k(x) - u k(x) , = ' D G k , k - l ( x ' X ' ) a k - l ( x ' ) d D x ' + ' D F k , k - l ( x ' X , ) W k - l ( x ' ) d D x ' - g k L ( . x ) C K . (6.5.1) From eqn. (3.4.20), f o r t h i s case too, the measurement r e s i d u a l at x^ i s given by m z, . = E M . . G. . . ( x . ^ ' K n(x')dD , + v. . k,x ic,i] D k , k ~ l j ' k-1 x k , i m + E M. . ./ n F. . .(x^x^w. ..(x')dD , . . . l c , i i D k , k - l ~j k-1 x 1=1 . (6.5.2) Define, f o r notational convenience, G k , k - ! ( X ) = G k , k - ! ( X 1 ' X ) a mnxn matrix, and * A \ , k - l ( x ) = G. , , (x ,x) k.,k-l m F k , k - ! ( X 1 ' X ) (6.5.3) (6.5.4) F. . » (x ,x) k, k - l m a mnxc matrix. Making use of the d e f i n i t i o n s (6.5.3) and (6.5.4), eqn. (6.5.2) can be augmented for i=l,2,...,m into Substituting eqn. (6.5.5) into eqn eqn. (6.5,1) therefore y i e l d s = ' D [ G k 3 k - l ( x ' X , ) - \ L ( x ) \ G k , k - l ( x ' ) ] V l ( x ' ) d D x > + fH [ F k , k - l ( x ' X , ) - 6 k L ^ V k , k - l ( x , ) ] w k - l ( x , ) d D x ' - g k L ( x ) v k . (6.5.6) Taking the expectation, c o n d i t i o n a l on g , of both sides of eqn. (6.5.6) and using the zero mean assumptions on the system and measurement noises i n eqn. (6.2.7), we have E [ u k ( x ) | g k ] = / D ( G ^ C x . x ' ) - g k L ( x ) M k 6 * j k _ 1 ( x ' ) ) E t u k _ 1 ( x ' ) | g k ] d D x I , (6.5.7) u i i a l . j-s , E[u k(x ) I g k ] = / D ( G k ) k _ 1 ( x , x ' ) - g k L ( x ) M k G * j k _ 1 ( x ' ) ) E [ u k _ 1 ( x ' ) | g k _ 1 ] d D x , , (6.5.8) since the c o n t r o l i s non-anticipative. It i s desirable that, given i n i t i a l l y EfuQlgglo^, the c o n d i t i o n a l expected value of the f i l t e r i n g error decreases with increase of time, that i s , eqn. (6.5.8) should be asymptotically stable i n the large. It i s the asymptotic s t a b i l i t y of eqn. (6.5.8), instead of eqn. (6.5.6), that i s studied, as i t , being a d e t e r m i n i s t i c equation, i s more amenable to a n a l y s i s . Now, from eqns. (6.4.5) and (6.4.6), the f i l t e r can be w r i t t e n as V X > = ' D 6 k , k - l C x ' X , ) V l ( x , ) d D x ' + V k . l ( x ) f k - I ' + * k L C x ) C \ - V k l k - i ) . <6-5-9> where the components * \ | k - l ^ X j ^ °^ t n e column vector A ^ j ^ - ^ defined as i n 182 eqn. (3.4.7) i a given by. CL, i, , (x.) = / G. . 7 ( x . , x , ) u . T C X ^ C I D " , + H. . 7 ( x . ) f . . (6.5.10) k|k.-l 2 D k , k - l j k-1 x k , k - l j j Hence, we see that, from eqns. C6.5.9) and (6.5.10), the unforced equation to eqn. (6.5.9) i s , a f t e r using d e f i n i t i o n (6.5.3), " ' D G k , k - l ( ^ X , ) - \ L W \ ( H ( x , ) ) V l ( x ' ) d D x - • ( 6 - 5 - 1 1 } It i s then noted, to within a notationa.1 change, that eqn. (6.5.11) i s the same as eqn. (6.5.8). Hence, the asymptotic s t a b i l i t y i n the large of the equation for the c o n d i t i o n a l expected value of the error, eqn. (6.5.8), can be analyzed to a l l intents and purposes by considering that of the un-forced f i l t e r equation (6.5.11). For n o t a t i o n a l convenience, define G k , k - l t - ) ( * > = ^ D G k , k - l ( x ' X , ) ( ' ) d D x ' ( 6 - 3 ' 1 2 ) anrl r k , k - l < - > ^ = ' D L W M k G k , k - l ^ ' ^ - > d D x > ' ( 6 - 5 ' 1 3 ) Using the d e f i n i t i o n s (6.5.12) and (6.5.13), eqn. (6.5.11) can be compactly written as v*> - ( G k , k - i - 6 k r k , k - i > V i < x ) • ( 6 - ^ 1 4 ) 6.5.2 Bounds on the Scalar Gain As has been mentioned previously, i t i s d e s i r a b l e that eqn. (6.5.8) or, equivalently, eqn.(6.5.11), be asymptotically stable i n the large so that the c o n d i t i o n a l expected value of the error decreases with increase of time. This w i l l be the case i f the largest member of the set, c o n s i s t i n g of a l l the lar g e s t vector u (x) components i n absolute value i n the s p a t i a l domain, de-creases with increase of time. Hence, define f o r a vector u(x) = (u"*" (x) . • • u^x))''" the norm j (uj | = max max ju^(x).| , xeD. (6.5.15) 183 Associated with the vector norm, eqn. (6.5.15), i s the norm f o r a square matrix M("x,x') H U ( x , x ' ) . ,nl r , x • M (x,x') of dimension nxn, •M l n Cx,x' ') * nn, ,x .M (x,x ) (6.5.16) | | M | I = max max Y. f |M1J (X,X' ) | dD^, , x.x'eD. 1 x j = l Taking norms of both sides of eqn. (6.5.14), (Gk,k-l ~ ^ k - l ^ k - l l (6.5.17) < iiGk,k-i - vik-ii ^ k - l 1 using the properties of norms. Further, repeated a p p l i c a t i o n of eqn. - U l t v ~ x • ix — a. , M u J I < { n M G 1 . . - g . r 1 ' . ,||}||u n|| . II k - 1 1 j , j - 1 63 J , J ~ l 1 1 " 0' Now, associate an index i£l ={1,2,...,k} such that Then, i t i s true from eqn. (6.5.19) that H \ l l M G i , i - i - « i r J . i - i " ) k | | a o l l • A s u f f i c i e n t condition for the asymptotic s t a b i l i t y i n the of eqn. (6.5.11) i s neUi-«irUiiiK1 • The condition (6.5.22) i s c e r t a i n to be s a t i s f i e d i f , f o r a l l k e l ^ , > ' H G k , k - i " gk r J,k- iN < 1 • (6.5.18) (6.5.18) (6.5.19) (6.5.20) (6.5.21) large (6.5.22) (6.5.23) 184 that i s . Making use of the properties of norms, we have from eqn. (6.5.23) that I ! l G k , k - i M " !%! M rk, k-iM I < 1 • (-6-5-24) -1 < I 1 ^ 1 I - U k! I l r j ^ l l < 1 • (6-5.25) Furthermore, iTcom the state equation (6.2.3), the unforced equation i s \ W " ' D G k , k - l C x » x , ) V l f e ' ) d D x ' + 'D F k , k - l ( x ' X ' ) w k ( x ' ) d D x ' (6.5.26) Taking the expectation of both sides of eqn. (6.5.26), E[u k(x)] = / D G k j k _ 1 ( x , x ' ) E [ u k _ 1 ( x ' ) ] d D x , . (6.5.27) This i s of the form of eqn. (6.5.14). Hence, by a s i m i l a r argument that has been applied to eqn. (6.5.14) leading to the asymptotic s t a b i l i t y c o n d i t i o n (o.;>.2.>j , xc i s seen LIICIL eqn. u^._>.z./y x» <iaytm>LuLxLoxiy ataolc i n uhs ljLi"g,e i f IIG^II < 1 • (6-5.28) It i s seen from eqn. (6.5.25) bearing the condition (6.5.28) i n mind . that the righ t hand i n e q u a l i t y i s to give the lower bound on g k since, with increase of g k, the r i g h t hand i n e q u a l i t y i s s a t i s f i e d . Further, the l e f t hand i n e q u a l i t y i s to give the upper bound of g k since, with increase of g k > the l e f t hand in e q u a l i t y i s s a t i s f i e d . From the r i g h t hand i n e q u a l i t y i n eqn. (6.5.25), we extract the lower bound of g k to be min i i i H r k f k - l " and from the l e f t hand inequality the upper bound k > k 1 , (6.5.29) 185 i : + l l G J ^ i M g - • _Ji--£-i-. . C6.5.30) max i i r l - i i ! 1 ric,k.-:L 1 ! Hence, the f i l t e r eqn. (6.5.9) w i l l be asymptotically s t a b l e i n the large with respect to the norm defined by eqn, (6.5.17) only i f the c o n d i t i o n g • < g, < g (6.5.31) mm k max i s s a t i s f i e d f o r a l l k. 6.6 Derivation of the Control Algorithm This section i s concerned with the d e r i v a t i o n of the on-line a l -gorithm by which g, i s chosen to minimize J , of eqn. (6.4.43). °k eq,k Consider the system at the present time to be indexed by k. Assume that the current measurement n, and the s c a l a r gain g, , from the previous k k-1 stage are a v a i l a b l e . We expand' the cost f u n c t i o n a l given by eqn. (6.4.43) around g^ ._^ » keeping only the l i n e a r term, to obtain Jeq,k ( gk' _ Jeq,k ( gk-1 + S g k - l ' • WW? + H f ^ V g ^ k - i • <6:6-" where 6 g k - l ' g k ~ g k - l ' C 6 ' 6 - 2 ) From eqn. (6.6.1), i t i s c l e a r that a reduction i n cost r e s u l t s i f the con-d i t i o n i s s a t i s f i e d , that i s , the gradient [3J , /3g] and the incremental eq,k 5 g=g k - 1 gain step g^^  ^ m u s t ^ e °^ opposite signs. This suggests a steepest descent approach and, therefore, l e t the incremental step be-186 where Si i s some p o s i t i v e constant to be s u i t a b l y chosen. Moreover, i n order to prevent e r r a t i c change of the sc a l a r gain and to ensure that the global minimum of J be reached, constrain the step s i z e 6g ... by X - i — 6 * g 4 - i > C 6 - 6 - 5 ) with the p o s i t i v e constant Si s a t i s f y i n g the condition 0 < 5£ < 1 . (6.6.6) g -Cl e a r l y , eqns. (6.6.4) and (6.6.5) must be consistent with each other. Therefore, we substitute eqn. (6.6.4) i n t o eqn. (6.6.5) to y i e l d « - « g / ( ^ - i / [ % ' X ^ j • (6-6-7) Further, s u b s t i t u t i n g eqn. (6.6.6) into eqn. (6.6.4) y i e l d s , a f t e r taking into consideration condition (6.6.3), the gain step as 3J ^ - i - ' W i - ^ i ^ • ( 6 ' 6 - 8 ) Evaluating the gradient [9j i / 3 g ] _ according to eqns. (6.4.43) and then sq»k g gk_-^ (6.4.5), we have 3J . r eq>k, 3g g=g k_ x 3u^(x) _ „ = [ V D 2 ~ T g - V k(x,x')u k(x')dD xdD x, + 2(g - g ) ^ ^ ] ^ ^ = 2 ^ V D L T ( x ) V k ( x , x ' ) u k ( x ' ) d D x d D x , + ( g k _ x - I ) V k ] . (6.6.9) where " k ( x ) = ' D G k , k - l ( X ' X , ) \ - l C x , ) d D x > + H k , k - l C x ) f k - l + g k - l L ( X ) V ^ - 6 - 1 0 > Here, " k ( x ) can be considered to be the estimate of the true state u k 0 0 using 187 the prevloua-stage gain g^^« ' -From eqns-. ("6.6.2). and (6.6.8), the current gain i s therefore 3 J g = g - 6£ g s g n r - ^ - 1 2 - ! , (6,6,11) °k 6 k - l g e k - l b 3g S^g^T where 19J , /3gJ i s given by eqn. (6.6.9). e£I>k S_§k_-j The cost f u n c t i o n a l J . of eqn. (6.4.43) has two cost terms both eq,k dependent on g^; the f i r s t i s the cost incurred by the d e t e r m i n i s t i c c o n t r o l and the second that by the estimation error weighted by a^ defined by eqn. (6.4.42). Here, i t can be interpreted that a^ weights r e l a t i v e l y the c o n t r o l cost and the estimation cost. When the f i l t e r i n g error u ^ ^ O O i s large, the measurement r e s i d u a l ? as given by eqn. (6.5.5) also becomes large with the consequence that a^ from eqn. (6.4.42) i s large such that more weight i s automatically given to the estimation cost term i n choosing the s c a l a r gain gu s a t i s f y i n g condition (6.5.31) to minimize J _ . Conversely, when the f i l t e r i n g error i s acceptable, r e l a t i v e l y l e s s weight i s given to the estimat-ion cost term such that more weight i s given to the c o n t r o l term. Hence, i t can be concluded that the c o n t r o l scheme i s adaptive i n nature and that the s c a l a r gain g chosen i n such an adaptive fashion achieve a trade-off between the c o n t r o l and estimation costs. The proposed on-line c o n t r o l algorithm to be executed at each stage k i s summerized below: 1) Obtain, from the current a v a i l a b l e measurement n^, the measurement r e s i d u a l z, , eqn. (6.4.6); 2) Obtain, using the previous gain g| the estimate u^(x), eqn. (6.6.10); 3) Calculate W , eqn. (6.4.41); 4) Evaluate the gradient [3J v/3g] _ , eqn. (6.6.9); eq,k g - g k - l 5) Calculate the gain increment 6g , eqn. (6.6.8); k—1 188 6) Obtain the' scalar gain g from eqn. (6.6.11). Check whether condition (6.5.30) i s s a t i s f i e d ; i f not, replace g^ by i t s upper or lower bound appropriately; 7) The current estimate u.j(x) i s then given by eqn. (6.4.5) and the c o n t r o l f f o r the next stage calculated from eqn. (6.4.3). 6.7 Numerical Example As has been mentioned previously i n the text of the present chapter, the proposed open-loop-suboptimal feedback c o n t r o l scheme i s developed i n order to c o n t r o l the given d i s t r i b u t e d parameter system when the i n i t i a l c ondition, system and measurement noise covariances are not known exactly. Now, in. order to in v e s t i g a t e the e f f e c t i v e n e s s of the proposed c o n t r o l scheme, i t must be compared with the optimal c o n t r o l scheme developed i n Chapter 4 that has complete and accurate access to the information with regard to the covariances. For t h i s reason, the. numerical examnle considered i n 4.5 w i l l be used here. The three covariances w i l l be assumed to be the same but un-known to the proposed suboptimal c o n t r o l scheme. The system to be studied i s given by eqns. (4.5.1) and (4.5.2) which, as has been mentioned i n 4.5, can be cast i n t o the form of eqn. (4.5.5) s u i t a b l e , too, f o r consideration using the theory developed i n the present chapter. The system, eqn. (4.5.5), i s again simulated as a discrete-time lumped parameter system by eqn. (4.5.6), restated here f o r convenience, k+1 k+1,k k k+1,k k k+1,k k where $ , T and Y , with N =11, Ax=0.1 and At=0.1, have t h e i r • lC~rX y tC r C l I y K. K.~T~J_ y iC values as shown i n Tables (3.1), (3.2) and (4.1), r e s p e c t i v e l y . The discrete-time system of eqn. (6.7.1) i s again assumed to be observed at two s p a t i a l l o c a t i o n s x=x1 and x=x ?, 0<x ,x^<l, corresponding i n 189 t h i s case to eqn. (6 .2.4). with. m=2 and M.^  measurement equation , ,. . .-1 and ML , ..=0, i£j, by the k+1, ii. k+l,ij J 7 ^ y k + l , l + vk+l,1 y k + l , 2_ u k + l C x 2 } _ vk+l,2 (6.7.2) As has been pointed out p r e v i o u s l y i n 4.5, the system, eqn. (4.5.5) can be cast into the following discrete-time representation • W x ) = i W C x ' 3 C , ) u k ( * , ) d x ' + I F k + i , k ( x ' x ' ) w k ( x , ) d x ' ' (6.7.3) J0 xk+i v~ ~ '"  + \ + l , k ( x ) f k k=0,l, . . . ,N-1, where G^+1 k ( x , x ' ) , F f c + 1 k(x,x') and ^ ( x ) are as given by eqns. (3.6.7), (3.6.8) and (4.5.9), r e s p e c t i v e l y . The cost f u n c t i o n a l with respect to which the performance of the suboptimal c o n t r o l strategy i s measured i s again N-1 ! I = E[ At E /" u ? L l (x)dx + Bf? ] , N j = 0 o 3+1 3 (6.7.4) where B i s a p o s i t i v e constant and assumed to be equal to 0.1 and N-40. The open-loop-suboptimal feedback c o n t r o l i s from 6.4.1 given by f. = - / A."1!. (x)u, (x)dx . (6.7.5) k 0 'c k k The estimate u (x) i s obtained from the estimator, corresponding to eqn. (6.4.5), having the following s t r u c t u r e : u k(x) = / G k ) k _ 1 ( x , x ' ) u k _ 1 ( x ' ) d x ' + g k L ( x ) ^ + H ^ O O ^ , (6.7.6) where the measurement r e s i d u a l C k i s given as y k, l _yk,2_ i G k , k - i ( x r x , ) \ - i ( x , ) d x * (6.7.7) 190 rZesJc.] = 2 J [ / /' L T(x)V. (x,x')u, (x')dxdx' + (g - l ) \ ? k ] > (6.7.9) 3g g=g i , i k 0 0 k k R L Th.e algorithm aa developed in 6.6 that chooses g in eqn. (6.7.6) for the stage k, g satisfying the condition (6.5.31), is summarized below k for reference, g k ^ V i - 6 V k - i s g n [ % j £ ] g ^ g k _ 1 ' C 6 - 7 - 8 ) where 3 J _ . T 1 1 =k-l and \M = I G k ) k - i ( x ' x ' ) V i ( x ' ) d x ' + " k - i L ( x k i c + : " H , k - i ( x ) f k - i ' ~ :•  : (6.7.10) given g Q. Now, in order to predetermine L(x), g, g . and g and then to ' 1 & ' 5mm max i •- ...1.,- . . . _ - ! _ • 1 ,.1 I r, -I O S 4 - ~ . /<; "7 l O S «-V>„ ^nnrwi'nwfUn scheme compatible with those presented in 3.6 and 4.5 involving expansions of distributed functions in terms of the f i r s t M eigenfunctions of the spatial A 3 2 ' , operator A = — is used here as well. 3x z To determine L(x) that weights the measurement residual in the adaptive f i l t e r , eqn. (6.7.10), we f i r s t l y observe, by an argument similar to that which leads to eqn. (3.6.29), that L(x) = <^(x)L . (6.7.11) And, L i s to be obtained as the steady-value constant Mxn matrix in the same manner as K, ,_, k->M, is obtained i n 3.6 involving the recursive solution of k+1 the discrete-time Riccati equation (3.6.31). T With the av a i l a b i l i t y of L(x)-$ (x)L, we proceed to determine in this approximation scheme the gain threshold g that appears in eqn. (6.7.9). 191 Substituting eqns.. (4.5.20} and. (6.7.11) into. eqn. (6.4.18)., wahaye a r = / 4>T(x)V * (x)4»T(x)Ly1 L T$ (x)dx . (6.7.12) k Q v " k v v k v Further, s u b s t i t u t i n g eqn. (4.5.20), (6.7.11) and (3.6.24) in t o eqn. (6.4.17), we have b k - \ i < ( x ) v v ( x ) $ v ( : x ) [ L D k \ i k - i + p k i k - i D k L T ] \ ( x ) d x - ( 6 - 7 - i 3 ) The gain threshold g i s then given as the average of the steady values of the r a t i o b, /a, , as k-*°°, over d i f f e r e n t values of C 's and C 's. k k w v Given the norm d e f i n i t i o n (6.5.17), we have i n t h i s approximation scheme, using eqn. (3.6.17) f o r the operator defined by eqn. (6.5.12), MG* || =max / | * J ( x ) G * v(x')|dx« , (6.7.14) x u and, using eqns. (6.7.11) and (3.6.17) for that defined by eqn. (6.5.13), !l rv.v-JI = raax I l ^ x ) L \ V v - i V * ' ) l d x ' » (6.7.15) x " ' where i s defined by eqn. (3.6.25). The approximate bounds wit h i n which g^ i s allowed to vary for the k-th stage are then calculated from eqns. (6.5.29) and (6.5.30) using eqns. (6.7.14) and (6.7.15). The on-line algorithm eqns. (6.7.8) to (6.7.10) i n t h i s approximat-ion scheme i s as follows. We f i r s t l y observe that u ^ ( x ) c a n D e expanded i n terms of the f i r s t M eigenfunctions as w e l l as M _. u k(x) = Z uk<j>i(x) = $^(x)u k , (6.7.16) i = l where uf = / u. (x)(j). (x)dx (6.7.17) i s the i - t h element of the Mxl column vector u . Assuming the process has started and i s now at time indexed by k, the c o e f f i c i e n t vector u k i n the expansion of the current estimate u, (x) using the previous-stage s c a l a r gain 192 g i s obtained by s u b s t i t u t i n g eqns. (3.6.17), (3.6.34), (4.5.19) and ~ lc—1 (6.7.11) into eqn. (6.7.10) as V = G k , k - l \ ~ l + g k - l L C k + \ , k - l f k - l • (6.7.18) where the measurement r e s i d u a l c;^ , by s u b s t i t u t i n g into eqn. (6.7.7) the expansions (3.6.17) and (3.6.34), i s now approximately given as £ ' = n - D G u . (6.7.19) k k k k , k - l k-1 Then, s u b s t i t u t i n g eqns. (4.5.20), (6.7.11) and (6.7.16) into eqn. (6.7.9), the gradient i s now approximately given by 9 J ' C ~ ^ W 1 = 2 ? i A \ + 2 ? k ( g k - i - S > V k • ( 6 - 7 - 2 0 ) where the weight W , by s u b s t i t u t i n g eqns. (4.5.20) and (6.7.11) in t o eqn. (6.4.41), i s W, = f1 L T$ (x)<5^'(x)V1 i> (x)* T(x)Ldx . (6.7.21) With g^ chosen, the expansion c o e f f i c i e n t vector u^ of the f i l t e r e d s tate estimate u. (x) for the current stage k, by s u b s t i t u t i n g eqns. (3.6.17), K. (3.6.34), (4.5.19) and (6.7.11) in t o eqn. (6.7.6) i s \ - G k , k - l \ - l + g k ^ k + H k , k - l f k - l • ( 6 ' 7 - 2 2 ) And, s u b s t i t u t i n g eqns. (3.6.34) and (4.5.27) into eqn. (6.7.5), the c o n t r o l f to be applied over the time i n t e r v a l [ ck , t :k+l^ x s approximately given by f k = - \ 1 § k \ • ( 6 - 7 ' 2 3 ) where A^ and remain given by eqns. (4.5.26) and (4.5.28), r e s p e c t i v e l y , with N-k-1 replaced by k+1 for t h i s context. With the approximation scheme for the s o l u t i o n of the proposed open-loop-suboptimal feedback c o n t r o l of the system of eqns. (6.7.3), (6.7.7) and (6.7.4) now completed, we now proceed to obtain numerical r e s u l t s using the same parametric values as i n 4.5 f o r comparison purposes. 19: -3 -4 With M=3, x =0.2,. x,,-0.7, the choice of C =10 and C =10 I y i e l d s 1 i. w v ^ 0.163301. 0.010383 0.000009 0.137443 -0.010768 -0.000008 (6.7.24) To determine the gain threshold g, the i n t e g r a l s f o r a^ and given by eqns. (6.7.12) and (6.7.13) are evaluated according to Simpson's r u l e with 11 points using the matrix L c a l c u l a t e d previously and given by eqn. (6.7.24) and V R c a l c u l a t e d i n 4.5. For the choice of Cw=0.1, 0.2, 0.3 and C =0.021 , 0.15I ?, 0.25I 2 > 0.375I 2 > 0.5I 2 w i t h i n which the plant and measurement noise covariances assumed unknown i n t h i s suboptimal scheme are expected to f a l l , i t i s found that g=0.373324. To determine 'the s c a l a r gain bounds g . and g , the i n t e g r a l s mm '-max i n eqns. (6.7.14) and (6.7.15) are evaluated again according to Simpson's rul e with 11 points using , and L given r e s p e c t i v e l y by eqns. (3.6.39), (3.6.41) and (6.7.24). I t i s found that | | G* ^11=0.993103, at . x=0, and Mr , 1 , . I I =0.320690, at x=0. Using eqn. (6.5.29), i t i s c a l c u l a t e d tc, k-1 that g . =-0.021508 and, using eqn. (6.5.30), g =6.215042 such that we ' mxn max have the condition -0.021508 < g k < 6.215042 (6.7.25) g^ must s a t i s f y f o r a l l k. Further, the condition (6.5.23) i s checked with any g s a t i s f y i n g condition (6.7.25) to ensure that i t i s not v i o l a t e d . I t i s noted i n passing that the unforced v e r s i o n of the lumped parameter approximation eqn. (6.7.22), corresponding to eqn. (6.5.11), has a l l i t s eigenvalues i n s i d e the u n i t c i r c l e f o r any g s a t i s f y i n g the condition (6.5.25), that i s , the unforced version of eqn. (6.7.22) i s asymptotically stable i n the large too. As has been mentioned, the system eqn. (4.5.5) i s simulated by eqn. 194 (6.7.1) where the matrices. 3> . , 1- , . and V are as given i n Tables v k+I,k k+l,k k+l,k (3.1), (3.2). and (4.1), r e s p e c t i v e l y . The simulated system i s again assumed to be observed by the measure-ment equation (6.7.2) with the two measurement locatio n s given as x^=0.2 and x2=0.7, The plant noise covariance Q k(x,x') = 0.25(x~x') (6.7.26) and the measurement noise covariance •c+1 \ - =0.03 1.0 0.1 0.1 1.0 (6.7.27) though again used here as i n 4.5, w i l l be assumed unknown i n t h i s suboptimal scheme. For the estimation problem, the values of G^+^ ^, k a n d Dk+1 have been calculated i n 3.6 and are given by eqns. (3.6.39), (3.6.40) and (4.6.41), r e s p e c t i v e l y . Using L of eqn. (6.7.24) and c a l c u l a t e d i n 4.5, the symmetric 2x2 matrix as approximately given by eqn. (6.7.21) i s computed, the i n t e g r a l being evaluated using Simpson's r u l e with 11 points, and the values 3 1 1 2 21 22 of i t s elements " , ~ = ^ k a m * ^ k a r e s n o w n i n F i g . (6.1). The c h a r a c t e r i z a t i o n of the state i n i t i a l c o n d i t i o n UQ(X) by i t s mean being G.(x) = -u = -1 (6.7.28) 0 ss and i t s covariance being P Q(x,x') = O.OlS(x-x') (6.7.29) w i l l again be used here, though the i n i t i a l c o n d i t i o n covariance w i l l be assumed unknown i n t h i s suboptimal scheme. Consider now the choice of g^ and • The- i n i t i a l s c a l a r gain g^ can be conveniently set to be the mean of g . and g such that i t i s i n mm max 196 the middle of the region I t i s allowed to vary. Now, based on a knowledge of the behaviour of the d i s t r i b u t i v e f i l t e r gain K^CX) o r i - t s expansion c o e f f i c i e n t matrix using some a r b i t r a r y noise covariances and, moreover, L and g„, the gain increment Sl can be reasonably set. In t h i s case, 6£ 5 0> to g J G i s therefore set to be equal to 0.1, that i s , at one given stage, the gain change i s 10% of the previous-stage gain. The simulation study y i e l d s , for a t y p i c a l run, the estimate fi., (x)=u\ (x)+u , where u, (x) i s the response of the adaptive f i l t e r eqn. *k k ss' k F ^ (6.7.6), of u A J ( x ) as shown i n F i g s . (6.2), (6.3) and (6.4) as ti;Vk(.0.), 0.^^(0.5) and u y { (1.), r e s p e c t i v e l y , and the piecewise-continuous open-loop suboptimal feedback c o n t r o l f. =f,+f as shown i n F i g . (6.5). For com-* *k k ss parison purposes, we also draw i n F i g s . (6.2) to (6.5) i n s o l i d l i n e s the corresponding forced de t e r m i n i s t i c system response u.,.k(x) at x=0., 0.5 and 1.0 and the optimal c o n t r o l f°^ for the system of eqn. (6.7.1) with w1=0. for a l l k and eqn. (6.7.4) with the expectation operator removed f o r t h i s d e t e r m i n i s t i c case by using standard lumped parameter system c o n t r o l theory To evaluate the average cost (6.7.4), we f i r s t l y observe that the only unspecified quantity i n the measurement equation (6.7.2) i s the measure-ment noise sequence and that i n the simulated system eqn. (6.7.1) the 38 noise sequence w^ , and, f u r t h e r , the i n i t i a l c o n d i t i o n U q for the system i s i t s e l f a random quantity. The procedure adopted to obtain the average cost i s therefore as follows. A random sample of u ^ . i s f i r s t generated using the mean, from eqn. (6.7.28), equal to -1. and covariance, from eqn. (6.7.29), equal (0.01/Ax) 1^=0.11.^ . Then, for t h i s p a r t i c u l a r i n i t i a l c ondition value, the process i s run 25 times, each time f o r a d i f f e r e n t set of noise sequences w. and v . The noise sequence w i s generated with mean 0. and covariance, K. K.T"X K. from eqn. (6.7.26), equal to ••'(0.2/Ax)I =2.T , and v k + 1 with mean 0. and is3 o O 201 covariance given'by eqn. (6.7.27). The average cost over these runs.is then p r a c t i c a l l y independent of the noises. The process i s then repeated 40 times, each, time with a d i f f e r e n t random sample of the i n i t i a l condition u^ of the simulated system., generated s t a t i s t i c a l l y independently from the others but again with the same mean and covariance. Altogether, then, the average cost of eqn. (6.7.4) i s evaluated over 1,000 runs. The process i s f i r s t l y run according to the optimal c o n t r o l scheme d e t a i l e d i n Chapter 4 and then secondly run according to the suboptimal con-t r o l scheme presented i n the present chapter using, for both cases, i d e n t i c a l random i n i t i a l condition samples and noise sequences. The values of the aver-age cost for both cases are shown i n the following Table (6.1). I n c i d e n t a l l y , scheme Optimal c o n t r o l 0.385797 scheme Table (6.1) Values of the average cost of eqn. (6.7.4) the average cost for the optimal c o n t r o l scheme obtained by the method above i s larger than that by the o f f - l i n e method i n the l a s t part of 4.5, the l a t t e r , of value 0.314839, being 81.61% of the former. This i s so because the l a t t e r i s obtained using the approximation scheme. It i s clear then that f o r t h i s p a r t i c u l a r case the suboptimal c o n t r o l scheme having no access to noise i n -formation i s e f f i c i e n t i n that i t y i e l d s an average cost but 0.43% greater than that by the optimal scheme which has complete and accurate access to noise information. 202 om 6.8 Conclusion In t h i s chapter, we have considered a suboptimal s t o c h a s t i c point wise control scheme for l i n e a r discrete-time d i s t r i b u t e d parameter systems when the covariances of the plant noise, the measurement noise and the rand state i n i t i a l condition are not exactly known. Based on an examination i n the present d i s t r i b u t e d parameter context of the standard open-loop-optimal feedback c o n t r o l scheme which has been up to now used .exclusively f o r lumped parameter systems, an open-loop-suboptimal feedback c o n t r o l scheme i s put forward. The open-loop feedback c o n t r o l i s shown to be, because of non-randomness of the system parameters, the same i n form as the closed-loop feed-back c o n t r o l , whence the computational algorithm to y i e l d the d i s t r i b u t i v e feedback gain developed i n Chapter 4 i s of use here as w e l l . The state estimate needed i n the co n t r o l i s given by a l i n e a r estimator that preserves the tsredicto?—corrector strnr - t - . i j r e. Tn the corrector term, the weight of the measurement r e s i d u a l i s broken down into two product terms, one a matrix so chosen as to s u i t a b l y d i s t r i b u t e the measurement r e s i d u a l and the other a sc a l a r gain. In choosing the measurement r e s i d u a l d i s t r i b u t i o n matrix, which i s i n fact a d i s t r i b u t i v e f i l t e r gain, the computational algorithm as deve-loped i n Chapter 3 i s used. For the s c a l a r gain, an algorithm implementable on-line i s developed so as to minimize the equivalent cost f u n c t i o n a l shown to be derivable under c e r t a i n s i m p l i f y i n g assumptions from the open-loop cost f u n c t i o n a l . The algorithm e f f e c t i v e l y s e l e c t s a f i l t e r from'the cl a s s of stable f i l t e r s parametrized by the s c a l a r gain. To insure s t a b i l i t y of the adaptive f i l t e r , formulae for the bounds wit h i n which the f i l t e r s c a l a r gain must l i e have been obtained. To inv e s t i g a t e the effectiveness of the proposed open-loop-sub-optimal feedback co n t r o l scheme, the numerical example used i n Chapter 4 i s 203 used here for the purpose of comparison. Again, space-dependent functions are expanded i n terms of the eigenfunctions of the system s p a t i a l operator so that numerical sol u t i o n of the example problem i s p o s s i b l e . It i s shown that the proposed suboptimal c o n t r o l scheme i s e f f e c t i v e i n y i e l d i n g an av-erage cost but s l i g h t l y greater than that by the optimal one. 204 7. CONCLUSION This thesis i s mainly concerned with c e r t a i n aspects of estimation and control of l i n e a r d i s t r i b u t e d parameter systems. In Chapter 2, we have considered the f i l t e r i n g s t ate estimation problem of l i n e a r d e t e r m i n i s t i c continuous-time d i s t r i b u t e d parameter systems. The d i s t r i b u t e d parameter f i l t e r that y i e l d s the state estimate based on noiseless l i n e a r measurements assumed a v a i l a b l e over the whole occupied s p a t i a l domain i s f i r s t l y shown to be l i n e a r i n s t r u c t u r e and then derived by consideration of a Lyapunov type of s t a b i l i t y . It i s shown that the time- and space-varying f i l t e r gain i s to be chosen from the s o l u t i o n of a R i c c a t i - t y p e i n t e g r o - p a r t i a l d i f f e r e n t i a l equation. Next, the general r e s u l t s are s p e c i a l i z e d to the case when noisel e s s l i n e a r measurements are a v a i l a b l e at only several s p a t i a l l o c a t i o n s . The consequence for the f i l t e r gain i n t h i s case i s that i t i s now to be chosen from the s o l u t i o n of a R i c c a t i - t y p e p a r t i a l d i f f e r e n t i a l equation. A numerical example i s solved using CSMP to i l l u s t r a t e i t s use i n an o v e r a l l c o n t r o l scheme for a one-dimensional boundary c o n t r o l problem. In Chapter 3, we have derived the l i n e a r f i l t e r that y i e l d s the minimum-variance (optimal) estimate of the complete state f o r l i n e a r stochastic discrete-time d i s t r i b u t e d parameter systems based on noise-corrupted l i n e a r measurements obtained at only several points of the s p a t i a l domain over which the system i s defined. The algorithm to y i e l d the d i s t r i b u t i v e f i l t e r gain at every time-instant has been obtained. Furthermore, we have shown that the optimal f i l t e r e d estimate and the f i l t e r i n g error s a t i s f y an orthogonal p r o j e c t i o n lemma, whence a Wiener-Hopf r e l a t i o n i s derivable. A numerical example i s presented-to i l l u s t r a t e the estimation 205 scheme. With the a v a i l a b i l i t y of the estimator developed i n Chapter 3, we then proceed to consider i n Chapter 4 the optimal pointwise regulation c o n t r o l problem of l i n e a r s t o c h a s t i c discrete-time d i s t r i b u t e d parameter systems with the average cost f u n c t i o n a l quadratic both i n the state and the c o n t r o l . e f f o r t . By employing the dynamic programming technique, we have derived the recursive f u n c t i o n a l r e l a t i o n s h i p by which the optimal c o n t r o l , shown to be l i n e a r i n the state i f the measurement of which i s exact or i t s estimate i f not, can be ca l c u l a t e d . Further, the sto c h a s t i c optimal pointwise regulation c o n t r o l problem i s shown to be separable into two optimization problems i n that the computation of the optimal d i s t r i b u t i v e f i l t e r gain of the given estimation problem i s independent of that of the kernel of the optimal feedback gain operator of the deterministic pointwise regulation c o n t r o l problem. The cost f o r the complete control scheme i s shown to consist of three components, the f i r s t . i s made up of the cost expended to bring the system to the desired state as i f no noise were present and that due to uncertainty associated with the i n i t i a l state, another due to the system disturbance and the t h i r d due to .estimation e r r o r . A numerical example has been employed to i l l u s t r a t e the use of the complete control scheme wherein, i n add i t i o n , an evaluation of the cost contributions has been made. By considering the l i m i t i n g behaviour as the time i n t e r v a l between sampling instants i s made a r b i t r a r i l y small of the discrete-time estimation and stochastic c o n t r o l r e s u l t s presented i n Chapters 3 and 4, we have obtained the corresponding continuous-time analogues i n Chapter 5. The d i s t r i b u t i v e f i l t e r gain of the continuous-time l i n e a r minimum-variance 2 0 6 f i l t e r of the estimation problem given discrete-space noise-corrupted continuous-time l i n e a r measurements i s shown to s a t i s f y a non-linear p a r t i a l d i f f e r e n t i a l equation of the R i c c a t i - t y p e and, moreover, the kernel of the i n t e g r a l representation of the feedback operator i n the continuous-time stochastic optimal pointwise c o n t r o l problem i s shown to s a t i s f y a s i m i l a r equation. Again, for the s t o c h a s t i c c o n t r o l problem, i t s separation i n t o estimation and d e t e r m i n i s t i c c o n t r o l which are independent of each other i s observed. Furthermore, d i r e c t comparison of the R i c c a t i equations f o r the f i l t e r i n g estimation problem given discrete-space noise-corrupted continuous-time l i n e a r measurements and the d e t e r m i n i s t i c pointwise regulation c o n t r o l problem i s then shown to y i e l d the d u a l i t y r e l a t i o n s under which one i s the dual of the other. In the optimal estimation and pointwise r e g u l a t i o n c o n t r o l schemes for l i n e a r stochastic discrete-time d i s t r i b u t e d parameter systems developed i n Chapters 3 and 4, the noise c h a r a c t e r i s t i c s need to be known exactly. For cases when they are not, we have therefore put forward i n Chapter 6 an open-loop-suboptimal feedback c o n t r o l scheme based on an examination i n the present d i s t r i b u t e d parameter context of the open-loop-optimal feedback co n t r o l scheme. The open-loop feedback c o n t r o l i s shown to be of the same form as the closed-loop feedback co n t r o l obtained i n Chapter 4. From an examination of the r e s u l t s on f i l t e r i n g estimation presented i n Chapter 3, an adaptive f i l t e r which has an adjustable s c a l a r f i l t e r gain i s proposed i n order to y i e l d the estimate of the complete state needed i n the r e a l i z a t i o n of the open-loop feedback c o n t r o l . Because of the imposed structures of the c o n t r o l l e r and the f i l t e r , the open-loop cost f u n c t i o n a l , shown derived from the o r i g i n a l quadratic cost f u n c t i o n a l , i s shown reducible to an equivalent one i n t o which the s c a l a r f i l t e r gain 207 enters simply to couple the cost due to c o n t r o l expenditure and that due to estimation. The various parameters, except the s c a l a r f i l t e r gain, i n the equivalent cost f u n c t i o n a l are computable o f f - l i n e . Further, a steep-est descent algorithm s u i t a b l e f o r on-line implementation i s developed by which the scalar f i l t e r g a i n 'is chosen so as to minimize at every stage the equivalent cost f u n c t i o n a l . Moreover;, formulae f o r the bounds wit h i n which the scalar f i l t e r gain must l i e to ensure s t a b i l i t y of the adaptive f i l t e r have been obtained. By employing the numerical example used i n Chapter 4, i t i s shown that the suboptimal c o n t r o l scheme, investigated v i s - a - v i s the optimal co n t r o l scheme, i s e f f e c t i v e as evidenced by y i e l d i n g a cost but s l i g h t l y greater than that yielded by the optimal one. APPENDIX I In t h i s appendix, the eigenvalues and the eigenfunctions of a p a r t i c u l a r p a r t i a l d i f f e r e n t i a l equation formulated below are derived and i t s Green's function i s simply obtained upon using the formula developed by Brogan Consider the 1-dimensional, l i n e a r , s t a t i o n a r y parabolic p a r t i a l d i f f e r e n t i a l equation 3u(t,x) = 3 2u(t,x) ( A l . l ) with boundary conditions 3t 2 3x A„ , N 1 A 3u(t,0) . e x u ( t ' x ) ] n = "tr—= 0 x=0 B xu(t,x)] = 6u(t,l)+ =0 x=l (A1.2) where u(t,x) i s a scalar function of the time v a r i a b l e t>0 and the space coordinate x, 0<x<l, and g i s a given constant. The eigenvalue-eigenfunction problem associated with t h e . s p a t i a l 3 2 operator A = — of eqn. ( A l . l ) i s defined as X 3x2 d2(|>.(x) = A.<f),(x) (A1.3) dx2 with boundary conditions d<K0) = n dx (A1.4) where i s the eigenvalue and <j>^(x).the corresponding eigenfunction. The s o l u t i o n of eqns. (Al.3) and (A1.4) must be of the form d>. (x) =' a.cos /^X\x + V. s i n /-A~.x , (A1.5) X I X X X where a^ and b^ are a r b i t r a r y constants, since, i f A_^/0, there are no a_^  and b. s a t i s f y i n g the boundary conditions (A1.4). x Applying the boundary conditions (A1.4) to eqn. (A1.5), we have a. a r b i t r a r y x and b.-0 x such that where d>. (x) = a. cos p.x , (A1.6) x x x / T . (A1.7) x s a t i s f i e s the transcendental equation P i 1 p i t § P i = P " It i s e a s i l y seen ( v. Appendix II ) that the s p a t i a l (A1.8) d i f f e r e n t i a l operator A i s s e l f - a d i o i n t such that A =A and that the r x x x boundary operator 3 at the boundary x=0,l ( v. eqn. (A1.2) ) i s also X s e l f - a d i o i n t such that 3 =3 at x=0,l. J x x Hence, the adjoint eigenfunctions (x) s a t i s f y the same eqns. (A1.3) and (A1.4) as the eigenfunctions. We therefore have tjj. (x) = c.cos p.x , (A1.9) x x x where c_^  i s an a r b i t r a r y constant. To s a t i s f y the orthonormality con d i t i o n 1 < <i>."(x), ih(x) > = / <t>. (x)^.(x)dx =5.. , (ALIO) 1 D 0 . 1 J . X J where 8 i s the Kronecker d e l t a , we, on u t i l i z i n g eqns. (A1.6), (A1.9) and ( A L I O ) , have to choose the constants a^ and c_^  such that 2(p2+3 2) 2 = c2 = h > ( A L U ) a (pf+3 z) + 2.10 The Green's function for eqn. (A.l.l) with boundary conditions (A1.2) i s therefore simply X . O-t') G(t,x;t r,x') = E e 1 <f>. (x)ifi. (x') . i = l -L i - p 2 ( t - t ' ) = Z g(p.)e cos p.x cos P^x' , (A1.12) i = l where 2(p?+32) g(p,) = • (A1.13) 1 (P 2+32) + 3 2 1 1 APPENDIX I I In t h i s appendix, v/e f i r s t l y i n d i c a t e how a given matrix d i f f e r e n t i a l operator i s r e l a t e d to i t s adjoint and secondly obtain a matrix Green's i d e n t i t y for a general matrix second order d i f f e r e n t i a l operator. The d e r i v a t i o n of the l a t t e r i s akin to that of Sakawa though the r e s u l t i s somewhat more general and i n a form suited to t h i s t h e s i s . Let D be a connected open domain of a r-dimensional Euclidean space E , and l e t 3D denote the boundary of D. Let the s p a t i a l coordinates 1 2 r T T vector x=[x x • • • x ] denote a generic point i n D, where [•] i s "the transpose of [•]", and t denote time on the f i x e d i n t e r v a l T=[tp,t^]. Let u(t,x) and v(t,x) be n-dimensional vector functions defined on TxD. In the subsequent development, for n o t a t i o n a l s i m p l i c i t y , the arguments of a function w i l l not be stated when no confusion can a r i s e . Consider given a nxn matrix d i f f e r e n t i a l operator A . The adjoint operator it A of A i s defined such that x x T * T r BP v A u-(A v) u = d i v P(v,u) = E =• , ( A 2 . 1 ) X X i = l Sx 1 . or, a f t e r i n t e g r a t i o n and applying Gauss theorem to the r i g h t hand side to transform a volume i n t e g r a l into a surface one, /_ (v TA u-(A*v) Tu) dD =/,_ P£ d3D = Z P.K. d3D , ( A 2 . 2 ) D x x dD sD , . i i 1=1 where P i s a vector with components P , dD i s the elemental volume of the domain D, d3D i s the elemental area of the boundary surface 3D, and E, denotes the d i r e c t i o n cosine of the angle between-the outward normal u n i t vector £ on 3D and the x_^-axis. The component P_^  depends b i l i n e a r l y on the functions u, v and t h e i r d e r i v a t i v e s . Consider now we are given a matrix p a r t i a l d i f f e r e n t i a l operator A (•) -= ( E A..(t,x) — E B. (t,x) —.+C(t,x))'(r) x . . . in * „ i , i . - l „ 1 i,;i=l 9x 9x J i = l 9x (A2.3) where A „ ( t , x ) , B^(t,x) and C(t,x) are known nxn matrices. Moreover, to accommodate the fac t that the. order of j j a r t i a l d i f f e r e n t i a t i o n i n the terra : T i s interchangeable, we assume that A. ,=A... 9x 13x J , 1 J J 1 The adjoint operator A of A i s defined by X X A*(-) = ( E — T . A T . ( t , x ) - E — . B T ( t , x ) + C T ( t , x ) ) ( - ) X i , j = l 3x 13x J 1 3 i = l 3x'- 1 (A2.4) Carrying out the operation demanded by eqn. (A2.1), we have, 11ti.li55in.cr on r i B . (h?. and .W\ v A u-(A v) u x x .T 9 2 , r 9 2 < A i j ^ . E A. . — ~ T - ( E — . i , j = l 1 J S^Zx2 i , j = l 3x 9x J + v T E B.-^r - ( - ( E ±— ) Tu) 1 1 1 i = l 3x i = l 9x T T T + v Cu - (C v) u • r ~ r r p ~ r 3 (v A. . ) = E — . ( E v A. . -^.) - E i = l 3 X 1 j = l 1 J 3x J i , j = l Sx 1 3x J r a r 3(v TA. .) r 3(v TA. .) _ - E — . ( E ^ L - u) + E |u i = l Sx 1 j = l 3x J i , j = l 3x 3 Sx 1 r + E — . (v TB.u) . (A2.5) 1 1 1=1 3x 213 Now, applying the condition A_. =A^ .. to eqn. (A2.5), we see that the second and fourth terms cancel each other out such that T * T r 3 P i v A u - ( A v ) u = E — ~ , (A2.6) X X , _ - J. x=l 3x where x T DA P. =' E (v TA.. — . - — . A..u - v T - 4 ^ u)+ v TB.u . (A2.7) 1 j = l 1 J 3x 3 3x J 1 J Sx"1 Hence, u t i l i z i n g eqn. (A2.2), we have f (v TA u - (A*v) Tu) dD = / -P? d3D , (A2.8) D X X dD where r r T PC = v T E 3. - ( E ^-r 3.)u + v TQu , xe3D, ' (A2.9) j=l 2 3xJ j = l dx2 2 r r 3 A.. . ( t , x ) 0 = Oft.x> = E CR. (t -x) - E — y U . , xe»J>, (A2.J0) • "  ' ' . , ' :i. ' . .. cix i i = l " 1=1 and 3. = 3.(t,x) = E A . . ( t , x K . , xe3D, (A2..11) 2 2 i = l 1 J for j = l , 2 , . . . , r . Consider now given the boundary operator 3 x such that r 3 3 u(t,x) = (B n(t,x) + E 3.(t,x) — .)u(t,x), xe3D,(A2.12) X U j = l J 3x J where 3..(t,x) i s as defined by eqn. ( A 2 . l l ) . Substittiting eqn. (A2.12) in t o eqn. (A2.9), we have r T P£ = v T 3 u - v T 3 nu - ( E -^L. 3.)u + v TQu X 3=1 3x J J = v T 3 u - (3^)^ , (A2.13) X X where we define 214 •B*vCt,x) = (^'Ct,x) + E 3 T Ct,x) — .)v(t,x) , xs3D. (A2.14) X U j = l 3 3x J The boundary operator 3 can be understood as the adjoint of the boundary operator 3 . From eqns. (A2.8) and (A2.13), we have the following compact X matrix Green's i d e n t i t y fn (v TA u - (A Kv) Tu) dD = Cv TB u - (3* v ) T u ) d3D . (A2.15) D x x 3D x x For higher order s p a t i a l d i f f e r e n t i a l operators, an equivalent r e l a t i o n s h i p of the form of eqn. (A2.15) between A^ and A^ can s t i l l be found using the method depicted above though the manipulations become more 55 complicated 215 REFERENCES 1. 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