Sequential estimation in distributed parameter systems with time delays

Abstract

A b s t r a c t This paper considers the problem of o p t i m a l e s t i m a t i o n o f t h e s t a t e s of a class o f d i s c r e t e d i s t r i b u t e d p a r a m e t e r s y s t e m s s u b j e c t t o m u l t i p l e time d e l a y s . The time d e l a y s are assumed t o o c c u r i n b o t h t h e s y s t e m a n d t h e o b s e r v a t i o n models. Algorithms are presented for obtaining estimates a n d t h e a s s o c i a t e d e r r o r v a r i a n c e s s e q u e n t i a l l y . I n t r o d u c t i o n The problem of s t a t e and parameter estimation i n lumped and d i s t r i b u t e d p a r a m e t e r s y s t e m s h a s b e e n s t u d i e d e x t e n s i v e l y i n t h e r e c e n t p a s t 11-41. Kwakernaak [SI and r e c e n t l y Koivo [6] have. conside r e d t h e p r o b l e m o f e s t i m a t i o n o f c o n t i n u o u s t i m e s i g n a l s w i t h time delay. For a p r o c e s s d e s c r i b e d by 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 e q u a t i o n s , t h e e q u at i o n s f o r t h e o p t i m a l e s t i m a t e s a n d a s s o c i a t e d e r r o r v a r i a n c e s a r e i n t h e f o r m of p a r t i a l i n t e g r od i f f e r e n t i a l e q u a t i o n s [ 4 ] w h i c h , d u e t o l a c k o f p e r t i n e n t t h e o r y , are n o t e a s i l y s o l v a b l e . The problem becomes even more complicated i f t h e r e a r e time d e l a y s a s s o c i a t e d w i t h e i t h e r t h e m e s s a g e model o r t h e o b s e r v a t i o n p r o c e s s o r b o t h . One a p p r o a c h t o s o l v e t h e s e e q u a t i o n s would b e t o d i s c r e t i z e them b u t t h i s , d e p e n d i n g upon t h e s y s t e m , may b e q u i t e cumbersome and may e v e n l e a d t o e r r on e o u s r e s u l t s . An a l t e r n a t i v e a p p r o a c h w o u l d b e t o d i s c r e t i z e t h e model i t s e l f and obtain the sequent i a l e s t i m a t i o n a l g o r i t h m . T h i s p a p e r c o n s i d e r s t h e p r o b l e m o f o b t a i n i n g e s t i m a t i o n a l g o r i t h m s f o r a d i s c r e t e d i s t r i b u t e d p a r a m e t e r s y s t e m s u b j e c t t o m u l t i p l e time d e l a y s i n t h e s y s t e m a s well a s o b s e r v a t i o n m o d e l , i n a sequential manner. Problem Formulation L e t 0 b e a n open subspace of a n-dimensional Eucledean space En. The N-dimensional s t a t e v e c t o r f u n c t i o n X ( x , k ) d e f i n e d on fi i s assumed t o evolve from the delay-difference equation w h e r e t h e i n t e g e r s h i t s r e p r e s e n t t h e d e l a y s which occur in the system and are o r d e r e d as 0ho < hl < . , . < h. A s s o c i a t e d w i t h t h e i t h t i m e d e l a y a n d k t h s t a g e is gxk, a l i n e a r N x N m a t r i x d e l a y -d i f f e r e n c o p e r a t o r w i t h r e s p e c t t o t h e s p a c e c o o r d i n a t e X' = (Xl,X2,. , .,%) and W(x,k) is t h e Nxl v e c t o r G a u s s i a n n o i s e i n p u t w i t h z e r o mean and covariance matrix, V x , y~n...