System identification for multivariable control

SYSTEM IDENTIFICATION FOR

MULTIVARIABLE CONTROL

•

G.A. van Zee

• •

-Delft University Press

-SYSTEM IDENTIFICATION FOR

MULTIVARIABLE CONTROL

SYSTEM IDENTIFICATION FOR

MULTIVARIABLE CONTROL

PROEFSCHRIFT ter verkrijging van

de graad van doctor in de

technische wetenschappen

aan de Technische Hogeschool Delft

op gezag van de rector magnificus,

prof. ir. B.RTh. Veltman

voor een commissie aangewezen

door het college van dekanen

te verdedigen op

woensdag 6 mei 1981

te 14.00 uur door

GERRIT ADRIANUS VAN ZEE

natuurkundig ingenieur

geboren te Amersfoort

U

Dit proefschrift is goedgekeurd door de promotor:

PROF. IR. R.G. BOITEN

6

A C K N O W L E D G E M E N T

I t i s a p l e a s u r e f o r me t o thank

O.H. B o s g r a f o r t h e i n s p i r i n g d i s c u s s i o n s w h i c h have been o f much h e l p i n o b t a i n i n g a d e e p e r u n d e r s t a n d i n g o f t h e s t u d i e d p r o b l e m s ,

t h e s t u d e n t s J.C. Van d e r P o e l , G.L.M. K o o t , H.E. Hermans, CP.A.M. B e y e r s b e r g e n , J . B a r e n d r e g t , W. G o u t , A. De J o n g , F . A z o d i , R.J.M. K o o l , F.J.H. B a a r s f o r t h e i r c o n t r i b u t i o n t o t h i s r e s e a r c h ,

F . J . Vergouwen f o r t h e d e s i g n and C . J . K r e m e r s , R. Van Overbeek, and A. Huisman f o r t h e b u i l d i n g and t h e m a i n t e n a n c e o f t h e a p p a r a t u s ,

F.R. Van d e r V l u g t , P. V a l k , and W. Postma f o r t h e i r h e l p i n d e s i g n i n g and s u p p o r t i n g t h e s o f t w a r e .

F u r t h e r I am o b l i g e d t o E.E. L i c h t who t y p e d t h e m a n u s c r i p t and A. G u t t e l i n g who drew t h e d i a g r a m s .

SUMMARY

T h i s s t u d y c o n c e r n s t h e a p p l i c a t i o n o f s y s t e m i d e n t i f i c a t i o n methods and modern c o n t r o l t h e o r y t o i n d u s t r i a l p r o c e s s e s . T h e s e p r o c e s s e s must o f t e n be c o n t r o l l e d i n o r d e r t o meet c e r t a i n r e q u i r e m e n t s w i t h r e s p e c t t o t h e p r o d u c t q u a l i t y , s a f e t y , e n e r g y c o n s u m p t i o n and e n v i r o n m e n t a l l o a d . Some o f t h e s e r e q u i r e m e n t s have r e c e n t l y become more r e s t r i c t i v e and c a n no l o n g e r be s a t i s f i e d i f

c o n v e n t i o n a l s i n g l e l o o p c o n t r o l i s a p p l i e d . An a p p r o a c h t o o b t a i n b e t t e r c o n t r o l i s t o use modern c o n t r o l s y s t e m d e s i g n methods w h i c h t a k e t h e o c c u r r i n g

i n t e r a c t i o n phenomena and s t o c h a s t i c d i s t u r b a n c e s i n t o a c c o u n t . The main c o n c e r n i s t h e n t o o b t a i n a s u f f i c i e n t l y a c c u r a t e dynamic m a t h e m a t i c a l model o f t h e p r o c e s s , by t h e o r e t i c a l m o d e l l i n g a n d / o r by s y s t e m i d e n t i f i c a t i o n . The p u r p o s e o f t h i s r e s e a r c h i s t o s t u d y t h e c o m p u t a t i o n a l a s p e c t s o f two i m p o r t a n t t y p e s o f i d e n t i f i c a t i o n s methods, i . e . s t o c h a s t i c r e a l i z a t i o n and p r e d i c t i o n e r r o r b a s e d p a r a m e t e r e s t i m a t i o n . The s t u d i e d c o m p u t a t i o n a l a s p e c t s a r e t h e r o b u s t n e s s , t h e a c c u r a c y , and t h e c o m p u t a t i o n a l c o s t s o f t h e methods. The s t u d y c o n s i s t s o f t h e o r e t i c a l a n a l y s e s and o f a p p l i c a t i o n s t o a m u l t i v a r i a b l e p i l o t s c a l e p r o c e s s , o p e r a t i n g u n d e r c l o s e d l o o p c o n d i t i o n s . S t o c h a s t i c r e a l i z a t i o n p r o c e d u r e s f o r s y s t e m s i n c l o s e d l o o p c o n s i s t o f t h r e e b a s i c s t e p s , i . e . r e a l i z a t i o n , s p e c t r a l f a c t o r i z a t i o n and d e c o m p o s i t i o n i n t o open l o o p m o d e l s . In p r i n c i p l e t h e s e s t e p s may be p e r f o r m e d i n d i f f e r e n t c o n s e c u t i v e o r d e r s . However, i t i s shown i n t h i s s t u d y t h a t t h e c o m p u t a t i o n a l p r o p e r t i e s o f t h e r e s u l t i n g p r o c e d u r e s a r e s i g n i f i c a n t l y d i f f e r e n t . A l l p r o c e d u r e s a p p e a r t o be r o b u s t f o r t h e u s e o f a s u i t a b l e sample c o v a r i a n c e f u n c t i o n , b u t o n l y t h e p r o c e d u r e s w h i c h s t a r t w i t h t h e s p e c t r a l f a c t o r i z a t i o n s t e p a p p e a r t o be r o b u s t f o r t h e e f f e c t s o f low o r d e r a p p r o x i m a t i o n s i n t h e r e a l i z a t i o n s t e p . F u r t h e r i t i s shown t h a t t h e most a c c u r a t e r e s u l t s a r e o b t a i n e d i f t h e d e c o m p o s i t i o n s t e p i s p e r f o r m e d n e x t a f t e r t h e s p e c t r a l f a c t o r i z a t i o n s t e p . F i n a l l y , t h e f e a s i b i l i t y i s d e m o n s t r a t e d o f a r e a l i z a t i o n method w h i c h i s b a s e d upon a s i n g u l a r v a l u e d e c o m p o s i t i o n , t o o b t a i n a p p r o x i m a t e and low o r d e r s t a t e s p a c e m o d e l s . The r e s u l t s by s t o c h a s t i c r e a l i z a t i o n have a l i m i t e d

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One may e x p e c t t o o b t a i n more a c c u r a t e r e s u l t s by p r e d i c t i o n e r r o r b a s e d p a r a m e t e r e s t i m a t i o n , " where a more d i r e c t r e l a t i o n e x i s t s between t h e d a t a and t h e e s t i m a t e d p a r a m e t e r s .

Some c o m p u t a t i o n a l a s p e c t s o f n o n l i n e a r p a r a m e t e r e s t i m a t i o n a r e

d i s c u s s e d i n d e t a i l . A method i s p r o p o s e d t o o b t a i n t h e most f a v o u r a b l e p a r a m e t r i z a t i o n f o r a g i v e n i n i t i a l model. F u r t h e r the s i m p l i c i t y and

c o m p u t a t i o n a l e f f i c i e n c y i s d e m o n s t r a t e d o f a dynamic programming a p p r o a c h t o compute g r a d i e n t s , combined w i t h a v a r i a b l e m e t r i c method t o compute

a p p r o x i m a t e H e s s i a n s . U s i n g t h e r e s u l t i n g method, t h e p a r a m e t e r s i n t h e models as o b t a i n e d by s t o c h a s t i c r e a l i z a t i o n a r e a d j u s t e d t o m i n i m i z e a o n e s t e p -ahead p r e d i c t i o n e r r o r c r i t e r i o n . The a c c u r a c y o f s t o c h a s t i c r e a l i z a t i o n and p r e d i c t i o n e r r o r b a s e d p a r a m e t e r e s t i m a t i o n i s j u d g e d by t h e v a l i d a t i o n o f t h e r e s u l t i n g m o d e l s . S i n c e t h e models a r e i n t e n d e d t o be u s e d f o r t h e d e s i g n o f c o n t r o l s y s t e m s , t h e y a r e e v a l u a t e d u s i n g t h e a c c u r a c y o f t h e c l o s e d l o o p p e r f o r m a n c e p r e d i c t i o n s . T h i s a c c u r a c y measure i s o b t a i n e d by c o m p a r i n g t h e r e s p o n s e s o f t h e c o n t r o l l e d model w i t h t h e r e s p o n s e s o f t h e a c t u a l c o n t r o l l e d p r o c e s s . I f t h e u s u a l o n e - s t e p - a h e a d p r e d i c t i o n e r r o r c r i t e r i o n i s m i n i m i z e d , t h e n t h e a c c u r a c y o f t h e models a p p e a r s t o become worse. T h i s l e a d s t o t h e c o n c l u s i o n t h a t i n t h e c a s e o f a p p r o x i m a t e m o d e l l i n g t h e c o n s i d e r e d c r i t e r i o n i s p o s s i b l y n o t a s u i t a b l e measure f o r t h e model a c c u r a c y . F u r t h e r a n a l y s i s o f t h i s p r o b l e m shows t h a t i t i s c a u s e d by t h e low f r e q u e n c y e r r o r s w h i c h a r e p r e s e n t i n t h e m o d e l s . To o b t a i n m u l t i v a r i a b l e c o n t r o l s y s t e m s w h i c h a r e r o b u s t f o r t h e p r e s e n c e o f c o n s t a n t d i s t u r b a n c e s and w h i c h a d m i t t h e d e f i n i t i o n o f n o n - z e r o s e t p o i n t s , t h e c o n t r o l systems must i n c l u d e i n t e g r a t i n g a c t i o n . F o r t h e d e s i g n o f c o n t r o l s y s t e m s w i t h i n t e g r a t i n g a c t i o n t h e low f r e q u e n c y model e r r o r s a p p e a r t o be more c r i t i c a l t h a n i n p r o p o r t i o n a l c o n t r o l s y s t e m d e s i g n . T h e r e f o r e we p r o p o s e a m o d i f i e d e s t i m a t i o n c r i t e r i o n w h i c h r e d u c e s e s p e c i a l l y t h e low

f r e q u e n c y model e r r o r s . The r e s u l t i n g models a p p e a r t o y i e l d s a t i s f a c t o r y a p p l i c a t i o n s o f o p t i m a l c o n t r o l compared t o s i n g l e l o o p and c a s c a d e r e f e r e n c e c a s e s . The models w h i c h have been o b t a i n e d by s t o c h a s t i c r e a l i z a t i o n a p p e a r t o

be n o t s u f f i c i e n t l y a c c u r a t e f o r t h e d e s i g n o f m u l t i v a r i a b l e c o n t r o l s y s t e m s , b u t a r e i n d i s p e n s a b l e a s i n i t i a l models f o r p a r a m e t e r e s t i m a t i o n .

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S A M E N V A T T I N G

Het onderwerp v a n d e z e s t u d i e i s de t o e p a s s i n g van s y s t e e m i d e n t i f i c a t i e metho-den en moderne r e g e l t h e o r i e op industriële p r o c e s s e n . Deze p r o c e s s e n moeten vaak g e r e g e l d worden om aan b e p a a l d e e i s e n t e v o l d o e n wat b e t r e f t de p r o d u k t k w a l i t e i t de v e i l i g h e i d , h e t e n e r g i e g e b r u i k en de i n v l o e d op h e t m i l i e u . Een a a n t a l v a n deze e i s e n z i j n de l a a t s t e t i j d s t r e n g e r geworden en kunnen n i e t meer worden v e r v u l d i n d i e n g e b r u i k w o r d t gemaakt v a n c o n v e n t i o n e l e e n k e l v o u d i g e r e g e l k r i n g e n Een b e t e r e r e g e l i n g kan worden v e r k r e g e n d o o r g e b r u i k t e maken van moderne o n t werpmethoden v o o r r e g e l s y s t e m e n , w a a r i n r e k e n i n g w o r d t gehouden met de o p t r e d e n -de i n t e r a c t i e en s t o c h a s t i s c h e v e r s t o r i n g e n . H e t b e l a n g r i j k s t e p r o b l e e m i s dan om een v o l d o e n d e n a u w k e u r i g d y n a m i s c h m a t h e m a t i s c h model van h e t p r o c e s t e v e r k r i j g e n d o o r m i d d e l van t h e o r e t i s c h e m o d e l v o r m i n g e n / o f d o o r m i d d e l van s y s t e e m -i d e n t -i f -i c a t -i e . H e t d o e l van d -i t o n d e r z o e k -i s om de r e k e n t e c h n -i s c h e a s p e c t e n t e b e s t u d e r e n van twee b e l a n g r i j k e t y p e n methoden v o o r s y s t e e m i d e n t i f i c a t i e , t e weten s t o c h a s t i s c h e r e a l i s a t i e en p a r a m e t e r s c h a t t i n g g e b a s e e r d op p r e d i c t i o n e r r o r s . De r e k e n t e c h n i s c h e a s p e c t e n d i e worden b e s t u d e e r d z i j n de r o b u u s t h e i d , de n a u w k e u r i g h e i d en de r e k e n k o s t e n van de methoden. De s t u d i e b e s t a a t u i t t h e o r e t i s c h e a n a l y s e s en u i t t o e p a s s i n g e n op een d o o r t e r u g k o p p e l i n g g e r e g e l d m u l t i -v a r i a b e l p r o c e s op l a b o r a t o r i u m s c h a a l . S t o c h a s t i s c h e r e a l i s a t i e m e t h o d e n v o o r systemen i n een g e s l o t e n k e t e n b e s t a a n u i t d r i e e l e m e n t a i r e s t a p p e n , n a m e l i j k r e a l i s a t i e , s p e c t r a l e f a c t o r i s e r i n g en decom-p o s i t i e i n odecom-pen k e t e n m o d e l l e n . I n decom-p r i n c i decom-p e kunnen d e z e s t a decom-p decom-p e n i n v e r s c h i l l e n d e v o l g o r d e n worden u i t g e v o e r d . In d e z e s t u d i e w o r d t e c h t e r a a n g e t o o n d d a t de r e k e n t e c h n i s c h e e i g e n s c h a p p e n van de r e s u l t e r e n d e methoden a a n z i e n l i j k v e r s c h i l l e n d z i j n . A l l e methoden b l i j k e n r o b u u s t t e z i j n v o o r h e t g e b r u i k van een g e s c h i k t e c o v a r i a n t i e s c h a t t e r , maar a l l e e n de methoden d i e b e g i n n e n met de s p e c t r a l e f a c -t o r i s e r i n g s s -t a p b l i j k e n r o b u u s -t -t e z i j n v o o r de g e v o l g e n van l a g e o r d e benader i n g e n i n de benader e a l i s a t i e s t a p . V e benader d e benader w o benader d t a a n g e t o o n d d a t de n a u w k e u benader i g s t e benader e s u l t a t e n worden v e r k r e g e n a l s de d e c o m p o s i t i e s t a p d i r e c t na de s p e c t r a l e f a c t o r i s e -r i n g s s t a p w o -r d t u i t g e v o e -r d . T e n s l o t t e l a t e n we z i e n d a t een -r e a l i s a t i e m e t h o d e ge b a s e e r d op een s i n g u l a r v a l u e d e c o m p o s i t i e g e s c h i k t i s om b e n a d e r e n d e t o e s t a n d s m o d e l l e n van een l a g e o r d e t e v e r k r i j g e n . De r e s u l t a t e n van s t o c h a s t i s c h e r e a l i

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s a t i e hebben een b e p e r k t e n a u w k e u r i g h e i d door de c u m u l a t i e v e u i t w e r k i n g van de f o u t e n i n de t u s s e n r e s u l t a t e n . Men mag n a u w k e u r i g e r r e s u l t a t e n v e r w a c h t e n b i j p a r a m e t e r s c h a t t i n g g e b a s e e r d op p r e d i c t i o n e r r o r s , waar de r e l a t i e t u s s e n de d a t a en de g e s c h a t t e p a r a m e t e r s d i r e c t e r i s .

E n k e l e r e k e n t e c h n i s c h e a s p e c t e n van n i e t - l i n e a i r e p a r a m e t e r s c h a t t i n g worden i n d e t a i l b e s p r o k e n . E r w o r d t een methode v o o r g e s t e l d om de g u n s t i g s t e p a r a m e t r i -s e r i n g t e v e r k r i j g e n v o o r een gegeven -s t a r t m o d e l . V e r d e r worden de eenvoud en de r e k e n t e c h n i s c h e d o e l m a t i g h e i d a a n g e t o o n d van een aanpak met b e h u l p van dynamisch programmeren om gradiënten t e b e r e k e n e n , g e c o m b i n e e r d met een v a r i a b l e m e t r i c methode om b e n a d e r i n g e n van Hesse m a t r i c e s t e b e r e k e n e n . Met b e h u l p van de r e s u l t e r e n d e methode worden de p a r a m e t e r s i n de m o d e l l e n , d i e v i a s t o c h a s t i -sche r e a l i s a t i e z i j n v e r k r e g e n , b i j g e s t e l d om een 'one-step-ahead p r e d i c t i o n e r r o r ' c r i t e r i u m t e m i n i m a l i s e r e n .

De n a u w k e u r i g h e i d van s t o c h a s t i s c h e r e a l i s a t i e en p a r a m e t e r s c h a t t i n g g e b a s e e r d op p r e d i c t i o n e r r o r s wordt b e o o r d e e l d op g r o n d van de v a l i d a t i e van de r e s u l t e r e n d e m o d e l l e n . Omdat de m o d e l l e n b e d o e l d z i j n v o o r h e t ontwerpen van r e g e l systemen, worden z i j geëvalueerd met b e h u l p van de n a u w k e u r i g h e i d van de v o o r -s p e l l i n g van h e t g e d r a g van de g e -s l o t e n k e t e n . Deze n a u w k e u r i g h e i d -s m a a t wordt v e r k r e g e n d o o r de r e s p o n s i e s van h e t g e r e g e l d e model t e v e r g e l i j k e n met de r e s -p o n s i e s van h e t w e r k e l i j k e g e r e g e l d e -p r o c e s .

A l s h e t g e b r u i k e l i j k e ' o n e s t e p a h e a d p r e d i c t i o n e r r o r ' c r i t e r i u m wordt g e m i n i -m a l i s e e r d , dan b l i j k t de n a u w k e u r i g h e i d van de -m o d e l l e n s l e c h t e r t e worden. D i t l e i d t t o t de c o n c l u s i e d a t i n h e t g e v a l van benaderende modelvorming d i t c r i t e -r i u m m o g e l i j k geen g e s c h i k t e maat i s v o o -r de m o d e l n a u w k e u -r i g h e i d . Een v e -r d e -r e a n a l y s e van d i t p r o b l e e m t o o n t aan d a t h e t wordt v e r o o r z a a k t d o o r de l a a g f r e -q u e n t e m o d e l f o u t e n . Om m u l t i v a r i a b e l e r e g e l s y s t e m e n t e v e r k r i j g e n d i e r o b u u s t z i j n v o o r de a a n w e z i g h e i d van c o n s t a n t e v e r s t o r i n g e n en d i e de d e f i n i t i e van s e t p o i n t s o n g e l i j k aan n u l t o e l a t e n , moet i n t e g r e r e n d e a c t i e worden t o e g e v o e g d aan de r e g e l s y s t e m e n . V o o r h e t ontwerpen van r e g e l s y s t e m e n met i n t e g r e r e n d e a c t i e b l i j k e n de l a a g f r e q u e n t e m o d e l f o u t e n k r i t i s c h e r t e z i j n dan v o o r h e t o n t werpen van p r o p o r t i o n e l e r e g e l s y s t e m e n . Daarom s t e l l e n we een g e w i j z i g d s c h a t t i n g s c r i t e r i u m v o o r d a t v o o r a l de l a a g f r e q u e n t e m o d e l f o u t e n r e d u c e e r t . De r e s u l -t e r e n d e m o d e l l e n b l i j k e n b e v r e d i g e n d e r e s u l -t a -t e n op -t e l e v e r e n van o p -t i m a l e r e g e l i n g , v e r g e l e k e n met de r e f e r e n t i e g e v a l l e n waar e n k e l v o u d i g e r e g e l k r i n g e n en c a s c a d e r e g e l i n g z i j n t o e g e p a s t . De m o d e l l e n d i e d o o r m i d d e l van s t o c h a s t i -sche r e a l i s a t i e z i j n v e r k r e g e n b l i j k e n n i e t v o l d o e n d e n a u w k e u r i g t e z i j n v o o r h e t ontwerpen van m u l t i v a r i a b e l e r e g e l s y s t e m e n , maar z i j n o n m i s b a a r a l s s t a r t -m o d e l l e n v o o r p a r a -m e t e r s c h a t t i n g .

12 C O N T E N T S page ACKNOWLEDGEMENT 6 SUMMARY 8 SAMENVATTING 10 1 INTRODUCTION 15 1.1 The f i e l d o f i n v e s t i g a t i o n 15 1.2 P r o b l e m s t a t e m e n t 16 1.3 O r g a n i z a t i o n o f t h e t h e s i s 17 2 STOCHASTIC REALIZATION 19 2.1 I n t r o d u c t i o n 19 2.2 S p e c t r a l f a c t o r i z a t i o n 24 2.2.1 S p e c t r a l f a c t o r i z a t i o n i n s t a t e space form 24 2.2.2 S p e c t r a l f a c t o r i z a t i o n i n i m p u l s e r e s p o n s e f o r m 26 2.2.3 C o m p u t a t i o n a l a s p e c t s 28 2.3 D e c o m p o s i t i o n i n t o open l o o p models 37 2.3.1 D e c o m p o s i t i o n i n s t a t e s p a c e f o r m 37 2.3.2 D e c o m p o s i t i o n i n i m p u l s e r e s p o n s e form 38 2.3.3 C o m p u t a t i o n a l a s p e c t s 38 2.4 R e a l i z a t i o n 44 2.4.1 R e a l i z a t i o n methods 44 2.4.2 A p p r o x i m a t e r e a l i z a t i o n b y t h e s i n g u l a r v a l u e d e c o m p o s i t i o n 46 2.5 C o n c l u s i o n s 50

13 page 3 PARAMETER ESTIMATION 51 3.1 I n t r o d u c t i o n 51 3.2 C a n o n i c a l forms 54 3.2.1 I n v a r i a n t s 54 3.2.2 C o m p u t a t i o n a l a s p e c t s 56 3.3 P a r a m e t e r a d j u s t m e n t 57 3.3.1 C o m p u t a t i o n a l c o s t s o f a d j u s t m e n t methods 57 3.3.2 G r a d i e n t c o m p u t a t i o n b y dynamic programming 59 3.4 C o n c l u s i o n s 62 4 MULTIVARIABLE CONTROL 63 4.1 I n t r o d u c t i o n 63 4.2 O p t i m a l c o n t r o l 63 4.3 P e r f o r m a n c e p r e d i c t i o n 67 4.3.1 F o r m u l a t i o n o f c o n t r o l p r o b l e m s 67 4.3.2 R e s u l t s 71 4.4 A m o d i f i e d e s t i m a t i o n c r i t e r i o n 76 4.4.1 I n t e g r a t e d r e s i d u a l s 76 4.4.2 R e s u l t s 79 4.5 C o n c l u s i o n s 79 5 CONCLUSIONS 84 APPENDICES 87 A A p i l o t s c a l e p r o c e s s 87 B I d e n t i f i c a t i o n e x p e r i m e n t s 90 C P r o o f s 95 D D e f i n i t i o n s 103 E N o t a t i o n s a n d symbols 107 REFERENCES 110 CURRICULUM VITAE 118

15 1 I N T R O D U C T I O N 1.1 T h e f i e l d o f i n v e s t i g a t i o n In t h i s t h e s i s we a r e c o n c e r n e d w i t h the a p p l i c a t i o n o f modern c o n t r o l t h e o r y t o i n d u s t r i a l p r o c e s s e s , s u c h as c h e m i c a l p l a n t s o r power s t a t i o n s . The t a s k o f a c o n t r o l s y s t e m i s t o keep a number o f p r o c e s s v a r i a b l e s w i t h i n c e r t a i n l i m i t s o f o p e r a t i o n . T h e s e o p e r a t i o n a l r e q u i r e m e n t s may o r i g i n a t e from demands upon p r o d u c t q u a l i t y , s a f e t y , e n e r g y c o n s u m p t i o n o r e n v i r o n m e n t a l l o a d . E s p e c i a l l y the l a t t e r two a r e o f g r o w i n g i m p o r t a n c e and f o r c e t h e d e s i g n e r o f a c o n t r o l s y s t e m t o a r e c o n s i d e r a t i o n o f c o n v e n t i o n a l s o l u t i o n s . I n the c o n v e n t i o n a l a p p r o a c h a p l a n t i s c o n c e i v e d as b e i n g d e c o u p l e d i n t o s i n g l e -i n p u t s -i n g l e - o u t p u t u n -i t s , w h -i c h a r e s e p a r a t e l y c o n t r o l l e d . The p a r a m e t e r s o f t h e c o n t r o l l e r s a r e c h o s e n on the b a s i s o f s c a l a r dynamic m o d e l s , o r a r e e x p e r i m e n t a l l y t u n e d , e.g. a c c o r d i n g t o Z i e g l e r , N i c h o l s ( 1 9 4 2 ) . T h i s w i l l g e n e r a l l y y i e l d s a t i s f a c t o r y r e s u l t s i f the c o n t r o l l o o p s a r e l o o s e l y t u n e d . However, i f t h e c o n t r o l must be t i g h t , t h e n the i n t e r a c t i o n between t h e l o o p s may c a u s e s e r i o u s p r o b l e m s . The s o l u t i o n o f t h e s e p r o b l e m s r e q u i r e s t h e a v a i l a b i l i t y o f m u l t i v a r i a b l e models w h i c h i n c l u d e t h e i n t e r a c t i o n phenomena, and o f c o n t r o l s y s t e m d e s i g n methods f o r m u l t i v a r i a b l e s y s t e m s . Not o n l y t h e i n t e r a c t i o n , b u t a l s o the o c c u r r e n c e o f random d i s t u r b a n c e s i n a p r o c e s s may i n h i b i t the improvement o f a c o n t r o l s y s t e m by c o n v e n t i o n a l d e s i g n methods. To r e d u c e the a d v e r s e e f f e c t s o f d i s t u r b a n c e s on the p e r f o r m a n c e o f a c o n t r o l system, i t i s n e c e s s a r y t o t a k e a s t o c h a s t i c a p p r o a c h t o t h e m o d e l l i n g and t o t h e c o n t r o l s y s t e m d e s i g n .

D u r i n g the l a s t d e c a d e s p o w e r f u l m u l t i v a r i a b l e e s t i m a t i o n and c o n t r o l t h e o r i e s have been d e v e l o p e d ( S p e c i a l i s s u e IEEE T r a n s . Autom. C o n t r o l , 1971; R o s e n b r o c k , 1969; Wonham, 1974; M a c F a r l a n e , 1979) . However, t h e r e have been o n l y a few a p p l i c a t i o n s i n p r o c e s c o n t r o l ( R i j n s d o r p , S e b o r g , 1976). T h i s h o l d o f f o f i n d u s t r i a l a p p l i c a t i o n s o f t h e t h e o r y i s m a i n l y due t o t h e l a c k o f

s u f f i c i e n t l y a c c u r a t e models o f t h e p r o c e s s e s ( F o s s , 1973) . F o r many p r o c e s s e s the p h y s i c a l knowledge i s l i m i t e d , and hence the models must be c o m p l e t e l y o r p a r t l y d e t e r m i n e d f r o m e x p e r i m e n t s , i . e . by s y s t e m i d e n t i f i c a t i o n . Thus system i d e n t i f i c a t i o n p l a y s a c r u c i a l r o l e i n the a p p l i c a t i o n o f modern c o n t r o l t h e o r y

16

t o i n d u s t r i a l p r o c e s s e s .

From t h e v a s t l i t e r a t u r e on t h i s s u b j e c t (Astrom, E y k h o f f , 1971) one m i g h t e x p e c t t h a t t h e e x p e r i m e n t a l m o d e l l i n g o f a p r o c e s s p r o c e e d s a l o n g w e l l known p a t h s . T h i s i s however n o t t h e c a s e , due t o s e v e r a l p r o b l e m s w h i c h o c c u r i n t h e a p p l i c a t i o n o f s y s t e m i d e n t i f i c a t i o n methods. F i r s t l y t h e r e i s t h e r o b u s t n e s s p r o b l e m . F o l l o w i n g t h e d e f i n i t i o n o f Huber (1972), we c a l l an i d e n t i f i c a t i o n method r o b u s t i f s m a l l d e v i a t i o n s from t h e h y p o t h e s e s u n d e r w h i c h i t has been d e r i v e d , c a u s e o n l y s m a l l e r r o r s i n t h e r e s u l t i n g model. Some i d e n t i f i c a t i o n methods a p p e a r t o be n o t s u f f i c i e n t l y r o b u s t t o a d m i t t h e i r p r a c t i c a l a p p l i c a t i o n .

S e c o n d l y t h e r e i s t h e a c c u r a c y p r o b l e m . The a c c u r a c y o f i d e n t i f i c a t i o n methods has been s t u d i e d by a n a l y s i s o f t h e i r s t a t i s t i c a l p r o p e r t i e s ( G u s t a v s s o n , 1972; Mehra, 1974; S o d e r s t r o m , L j u n g , G u s t a v s s o n , 1975) and has been e v a l u a t e d i n c a s e s t u d i e s ( G u s t a v s s o n , 1969; E k l u n d , G u s t a v s s o n , 1 9 7 3 ; O l s s o n , 1973). However, t h e r e seems t o be l i t t l e knowledge a b o u t t h e r e l a t i o n between t h e a c c u r a c y w h i c h i s p u r s u e d i n i d e n t i f i c a t i o n methods and t h e s u i t a b i l i t y o f t h e r e s u l t i n g models f o r t h e d e s i g n o f c o n t r o l s y s t e m s . I n a d a p t i v e c o n t r o l s y s t e m s , i d e n t i f i c a t i o n and c o n t r o l s y s t e m d e s i g n a r e c l o s e l y r e l a t e d t o e a c h o t h e r ( W i e s l a n d e r , W i t t e n m a r k , 1971; A s t r o m , B o r i s s o n , L j u n g , W i t t e n m a r k , 1977). However, t h e i d e n t i f i c a t i o n i s t h e n r e s t r i c t e d t o o n - l i n e methods f o r w h i c h , e s p e c i a l l y i f t h e s y s t e m i s l a r g e , an i n i t i a l model i s i n d i s p e n s a b l e . Thus i t r e m a i n s i m p o r t a n t t o o b t a i n i n s i g h t i n s e p a r a t e and p o s s i b l y o f f - l i n e i d e n t i f i c a t i o n methods. T h i r d l y t h e r e a r e t h e c o m p u t a t i o n a l c o s t s . In p r i n c i p l e one may e x p e c t t o o b t a i n a c c u r a t e models i f a l l r e l e v a n t v a r i a b l e s and d i s t u r b a n c e s a r e measured and i f t h e d e s c r i p t i o n o f t h e i r dynamic b e h a v i o u r i s i n c l u d e d i n t h e m o d e l . However, t h e c o m p u t a t i o n a l c o s t s w i l l t h e n e a s i l y become p r o h i b i t i v e . A c c u r a c y and c o m p u t a t i o n a l e f f i c i e n c y a r e o f t e n c o n f l i c t i n g o b j e c t i v e s i n s y s t e m

i d e n t i f i c a t i o n .

1.2 P r o b l e m s t a t e m e n t

The above m e n t i o n e d p r o b l e m s may be c h a r a c t e r i z e d a s c o m p u t a t i o n a l p r o b l e m s o c c u r i n g i n the a p p l i c a t i o n o f s y s t e m i d e n t i f i c a t i o n . The p u r p o s e o f t h i s s t u d y i s t o i n v e s t i g a t e t h e s e p r o b l e m s and t h e i r i m p a c t on t h e f e a s i b i l i t y o f

17

s y s t e m i d e n t i f i c a t i o n methods t o a r r i v e a t a p p l i c a t i o n s o f modern c o n t r o l t h e o r y t o i n d u s t r i a l p r o c e s s e s .

One p o s s i b l e c l a s s i f i c a t i o n o f i d e n t i f i c a t i o n methods i s a d i v i s i o n by t h e t y p e o f i m p l e m e n t a t i o n , i n t o e x p l i c i t methods and p a r a m e t e r a d j u s t m e n t methods

( E y k h o f f , 1974). T h i s c l a s s i f i c a t i o n i s u s e f u l from t h e c o m p u t a t i o n a l p o i n t o f v i e w , e s p e c i a l l y w i t h r e s p e c t t o t h e c o s t s . F o r t h e t y p e o f systems t h a t we c o n s i d e r i n t h i s s t u d y , t h e c o m p u t a t i o n a l c o s t s o f p a r a m e t e r a d j u s t m e n t methods a r e h i g h . E x p l i c i t i d e n t i f i c a t i o n methods a r e g e n e r a l l y l e s s a c c u r a t e

t h a n p a r a m e t e r a d j u s t m e n t methods, b u t a r e c o m p u t a t i o n a l l y more e f f i c i e n t and a r e i n d i s p e n s a b l e t o o b t a i n i n i t i a l p a r a m e t e r e s t i m a t e s ( C a i n e s , R i s s a n e n , 1974; T s e , W e i n e r t , 1975; A k a i k e , 1976). T h i s s t u d y w i l l be r e s t r i c t e d t o s t o c h a s t i c r e a l i z a t i o n , w h i c h i s an i m p o r t a n t e x p l i c i t method, and t o p r e d i c t i o n e r r o r b a s e d p a r a m e t e r e s t i m a t i o n , w h i c h i s an i m p o r t a n t p a r a m e t e r a d j u s t m e n t method. The d e s i g n o f m u l t i v a r i a b l e c o n t r o l s y s t e m s w i l l be r e s t r i c t e d t o l i n e a r q u a d r a t i c o p t i m a l e s t i m a t i o n and c o n t r o l . Thus b o t h t h e d e t e r m i n i s t i c and t h e s t o c h a s t i c p a r t o f t h e models w i l l be u s e d i n a f o r m a l i z e d way t o end up w i t h a c o n t r o l s y s t e m . We w i l l use l i n e a r

t i m e - i n v a r i a n t d i s c r e t e - t i m e s t a t e s p a c e m o d e l s . The use o f l i n e a r s t a t e space models i s r e q u i r e d by t h e c h o s e n c o n t r o l system d e s i g n method, the t i m e

-i n v a r -i a n c e r e s t r -i c t -i o n o r -i g -i n a t e s from t h e c o n s -i d e r e d -i d e n t -i f -i c a t -i o n methods, and t h e d i s c r e t e - t i m e r e s t r i c t i o n i s m o t i v a t e d by t h e computer c o n t r o l i m p l e m e n t a t i o n . 1.3 O r g a n i s a t i o n o f t h e t h e s i s The k e r n e l o f t h e t h e s i s c o n s i s t s o f t h e c h a p t e r s 2 - 4 . I n c h a p t e r 2 t h r e e p o s s i b l e p r o c e d u r e s a r e c o n s i d e r e d f o r t h e i d e n t i f i c a t i o n o f systems i n c l o s e d l o o p by s t o c h a s t i c r e a l i z a t i o n . The p r o c e d u r e s c o n s i s t e a c h o f t h e f u n d a m e n t a l s t e p s o f r e a l i z a t i o n , s p e c t r a l f a c t o r i z a t i o n , and d e c o m p o s i t i o n i n t o open l o o p m o d e l s . The c o m p u t a t i o n a l a s p e c t s o f each o f t h e s e s t e p s a r e s t u d i e d i n d e t a i l . I n c h a p t e r 3 t h e c o m p u t a t i o n a l c o s t s o f d i f f e r e n t p a r a m e t e r a d j u s t m e n t methods a r e compared and t h e s u i t a b i l i t y o f a dynamic programming f o r m u l a t i o n t o r e d u c e t h e s e c o s t s i s i n v e s t i g a t e d . F u r t h e r we c o n s i d e r t h e p r o b l e m t o o b t a i n a s u i t a b l e p a r a m e t r i z a t i o n . I n c h a p t e r 4 t h e r e s u l t i n g models a r e e v a l u a t e d by comparing c l o s e d l o o p r e s p o n s e s o f t h e model w i t h t h o s e o f t h e a c t u a l p r o c e s s , u s i n g t h e same f e e d b a c k c o n t r o l s y s t e m . Thus we compare t h e

a c c u r a c y measures w h i c h a r e u s e d i n t h e i d e n t i f i c a t i o n methods, w i t h a measure w h i c h i s b a s e d on t h e i n t e n d e d use o f t h e m o d e l s .

The main p o i n t s o f t h e t h e o r e t i c a l a n a l y s e s a r e f o r m u l a t e d i n theorems and r e s u l t s , o f w h i c h t h e f i r s t a r e o f a more g e n e r a l n a t u r e t h a n t h e l a t t e r , w h i c h a r e c o n c e r n e d w i t h some s p e c i f i c d e t a i l s o f t h e t h e o r y . I n e a c h o f t h e c h a p t e r s t h e t h e o r e t i c a l a n a l y s e s a r e f o l l o w e d by a p p l i c a t i o n s t o d a t a w h i c h have been o b t a i n e d f r o m a m u l t i v a r i a b l e p i l o t s c a l e p r o c e s s . The p r o c e s s , w h i c h i s d e s c r i b e d i n a p p e n d i x A, has some p r o p e r t i e s w h i c h a r e c h a r a c t e r i s t i c f o r i n d u s t r i a l p r o c e s s e s , s u c h as t h e o c c u r r e n c e o f s p a t i a l l y d i s t r i b u t e d phenomena, t h e i n t e r a c t i o n between t h e p r o c e s s v a r i a b l e s , and t h e p r e s e n c e o f s t o c h a s t i c d i s t u r b a n c e s . I t may t h e r e f o r e be e x p e c t e d t h a t t h e c o n c l u s i o n s from o u r e x p e r i m e n t s a r e o f i n t e r e s t f o r i n d u s t r i a l p r o c e s s e s . The i d e n t i f i c a t i o n e x p e r i m e n t s a r e d e s c r i b e d i n a p p e n d i x B.

P r o o f s o f new r e s u l t s a r e g i v e n i n a p p e n d i x C. Some i m p o r t a n t r e s u l t s and d e f i n i t i o n s f r o m t h e e x i s t i n g l i t e r a t u r e a r e g i v e n i n a p p e n d i x D. F o r t h e r e m a i n i n g r e s u l t s and d e f i n i t i o n s we r e f e r t o t h e l i t e r a t u r e . F i n a l l y , t h e n o t a t i o n s and symbols a r e d e f i n e d i n a p p e n d i x E.

2 S T O C H A S T I C R E A L I Z A T I O N 19 2.1 I n t r o d u c t i o n I n t h e f o l l o w i n g we w i l l t r e a t t h e p r i n c i p l e s o f r e a l i z a t i o n , s t o c h a s t i c r e a l i z a t i o n , and t h e i d e n t i f i c a t i o n o f c l o s e d l o o p systems by s t o c h a s t i c r e a l i z a t i o n .

C o n s i d e r t h e n - t h o r d e r s t a t e s p a c e model (A,B,C) d e s c r i b i n g a system by

x ( k + l ) = Ax(k) + Bu(k) (2 1 1 ) y ( k ) = Cx(k)

where x ( k ) e i ?n, u ( k ) e i ?m, and y(k)ei?~^ d e n o t e t h e s t a t e , i n p u t , and o u t p u t o f t h e

t h e s y s t e m a t t i m e i n s t a n t k. A p a r t from t h e i n i t i a l s t a t e , t h e s y s t e m may be e q u i v a l e n t l y d e s c r i b e d by y ( k ) = E h ( i ) u ( k - i ) (2.1.2) i = l where h ( i ) i s t h e i m p u l s e r e s p o n s e , s a t i s f y i n g h ( i ) = C A1 _ 1B , i = l , 2 , . . . (2.1.3)

The p r o b l e m o f f i n d i n g a s t a t e s p a c e model (A,B,C) s a t i s f y i n g e q u a t i o n ( 2 . 1 . 3 ) , w i t h h ( i ) a g i v e n i m p u l s e r e s p o n s e , i s known as t h e r e a l i z a t i o n p r o b l e m . I f n i s t h e l o w e s t p o s s i b l e o r d e r o f a model (A,B,C) s a t i s f y i n g ( 2 . 1 . 3 ) , t h e n (A,B,C) i s s a i d t o be a m i n i m a l r e a l i z a t i o n o f h ( i ) . The r e a l i z a t i o n p r o b l e m may be e q u i v a l e n t l y f o r m u l a t e d i n terms o f t h e t r a n s f e r f u n c t i o n H ( z ) , by z t r a n s f o r m a -t i o n ( J u r y , 1964) o f ( 2 . 1 . 3 ) , y i e l d i n g (Rosenbrock, 1970)

H(z) = C ( z I - A )_ 1B (2.1.4)

We w i l l now c o n s i d e r t h e m o d e l l i n g o f a r e a l zero-mean s t o c h a s t i c p r o c e s s v ( k ) ( a p p e n d i x D . l ) . We assume t h a t v ( k ) i s wide s e n s e s t a t i o n a r y and r a t i o n a l ( a p p e n d i x D.2, D.7). By t h e s e a s s u m p t i o n s v ( k ) a d m i t s a d e s c r i p t i o n a s t h e

o u t p u t o f a t i m e - i n v a r i a n t f i n i t e d i m e n s i o n a l s y s t e m h a v i n g a s q u a r e and p r o p e r ( a p p e n d i x D.18) t r a n s f e r f u n c t i o n G ( z ) and a w h i t e n o i s e i n p u t ( a p p e n d i x D.6). F u r t h e r t h e s p e c t r u m o f v ( k ) e x i s t s , i s r a t i o n a l , and c a n be w r i t t e n as Sv v( z ) = G ( z ) W GT( z_ 1) (2.1.5) w i t h W t h e c o v a r i a n c e o f t h e w h i t e n o i s e i n p u t ( a p p e n d i x D.3-D.5). I n a d d i t i o n we assume t h a t v ( k ) i s f u l l r a n k ( a p p e n d i x D.8), i . e . t h a t S (z) has f u l l w rank f o r a l m o s t a l l z. G(z) and W can g e n e r a l l y n o t be r e c o v e r e d f r o m a g i v e n s p e c t r u m S ( z ) , b e c a u s e w the f a c t o r i z a t i o n (2.1.5) i s n o t u n i q u e . A l l models G ( z ) , W w h i c h s a t i s f y (2.1.5 are e q u i v a l e n t w i t h r e s p e c t t o S ( z ) , and l e a d t o t h e same s o l u t i o n i n t h e

v v d e s i g n o f l i n e a r minimum v a r i a n c e e s t i m a t o r s (Sage, M e l s a , 1971). T h i s s o l u t i o n i s c l o s e l y r e l a t e d t o a p a r t i c u l a r s p e c t r a l f a c t o r , w h i c h i s d e s c r i b e d i n t h e f o l l o w i n g theorem. We d r o p t h e i n d i c e s o f S^{z) f o r n o t a t i o n a l c o n v e n i e n c e . Theorem 2.1.1 L e t S ( z ) be a r a t i o n a l s p e c t r u m h a v i n g f u l l rank f o r a l m o s t a l l z. Then t h e r e e x i s t s a u n i q u e s p e c t r a l f a c t o r i z a t i o n S ( z ) = G ( z ) W GT( z_ 1) , W=WT>0 s a t i s f y i n g 1 G(z) has a l l p o l e s i n |z|<l . ( a p p e n d i x D.16) 2 G (z) has a l l p o l e s i n |z|<l 3 l i m G(z) = I z-*» P r o o f : Y o u l a (1961); Gohberg, K r e i n (1966); A n d e r s o n , Moore (1979). V

G(z) i n theorem 2.1.1 i s known as t h e minimum phase s p e c t r a l f a c t o r o f S ( z ) o r as t h e i n n o v a t i o n s r e p r e s e n t a t i o n o f v ( k ) ( a p p e n d i x D.17). I f we assume i n a d d i t i o n t h a t S ( z ) has f u l l r a n k on | z | = l , t h e n t h e theorem c a n be s t r e n g t h e n e d i n t h e s e n s e t h a t G * ( z ) i s a s y m p t o t i c a l l y s t a b l e ( a p p e n d i x D.17). The i n n o v a t i o n s r e p r e s e n t a t i o n G(z) i s t h e n s a i d t o be c a u s a l l y i n v e r t i b l e , w i t h G ^ ( z ) t h e s t a t i o n a r y l i n e a r minimum v a r i a n c e p r e d i c t o r o f t h e s t o c h a s t i c p r o c e s s v ( k ) ( G e v e r s , K a i l a t h , 1973).

21 C o m b i n i n g r e a l i z a t i o n and s p e c t r a l f a c t o r i z a t i o n , t h e p r o b l e m t o f i n d a s t a t e s p a c e model (A,K,C) and W s a t i s f y i n g

G(z) = C ( z l - A ) K + I S ( z ) = G ( z ) W GT( z_ 1) , W=WT>0 (2.1.6) w i t h S ( z ) a g i v e n s p e c t r u m , i s known as t h e s t o c h a s t i c r e a l i z a t i o n p r o b l e m ( F a u r r e , 1976; R i s s a n e n , K a i l a t h , 1972). In t h e f o l l o w i n g we d r o p t h e argument T -1 * z f o r n o t a t i o n a l c o n v e n i e n c e , and d e n o t e G (z ) s h o r t l y as G . C o n s i d e r t h e c l o s e d l o o p s y s t e m o f f i g u r e 2.1.1 where Hp, HR r e p r e s e n t t h e d e t e r m i n i s t i c p a r t o f t h e p r o c e s s ( f o r w a r d system) and t h e f e e d b a c k ( r e v e r s e s y s t e m ) . H r H r

F i g . 2.1.1 Closed loop system

T T T

The unmeasured d i s t u r b a n c e s a r e r e p r e s e n t e d by v=(v^ [ v ) . We make t h e f o l l o w i n g a s s u m p t i o n s on t h e c l o s e d l o o p s y s t e m . A s s u m p t i o n 2.1.1 v can be d e s c r i b e d as

V

V

L

-

1° \ \

22

1 +

. J

F i g . 2.1.2 Closed loop system satisfying (2.1.7)

A s s u m p t i o n 2.1.2 The j o i n t i n p u t - o u t p u t s = ( yT | uT)T c a n be d e s c r i b e d as G2 1 G2 2 (2.1.9) o r s=Gw, w i t h G(z) r a t i o n a l , h a v i n g f u l l r a n k a l m o s t e v e r y w h e r e , and a s y m p t o t i c a l l y s t a b l e . By t h e s e a s s u m p t i o n s t h e s p e c t r u m S = GWG (2.1.10) e x i s t s , i s r a t i o n a l , and h a s f u l l r a n k a l m o s t e v e r y w h e r e . A s s u m p t i o n 2.1.3 F o r z-x» we have H (z) =0, H (z)=0, G ( z ) = I , G „ ( z ) = I . F R F R A s s u m p t i o n 2.1.4 L e t n , n and n be t h e o r d e r s o f m i n i m a l r e a l i z a t i o n s o f (H JG ) , (H | G ) F R F F R R and G. Then we assume n +n =n.

23

By a s s u m p t i o n 2.1.3 we e x c l u d e some t r i v i a l a m b u i g i t i e s i n t l i e r e l a t i o n between G and (H ,G ,H ,G ).-By a s s u m p t i o n 2.1.4 we e x c l u d e t h e o c c u r r e n c e o f p o l e - z e r o

r r R K

c a n c e l l a t i o n s i n t h e c l o s e d l o o p s y s t e m . Now we c o n s i d e r t h e q u e s t i o n i f t h e open l o o p models a r e r e c o v e r a b l e from S.

D e f i n i t i o n 2.1.1

The open l o o p models (H ,G , H ,G ) a r e s a i d t o be r e c o v e r a b l e i f i t i s p o s s i b l e F F R R t o o b t a i n models (H',G' H',G') s a t i s f y i n g F F R R H' = H G'W'G1 = G W G F F F F F F F F * * (2.1.11) H' = H G'W'G' = G W G R R R R R R R R V F u r t h e r , w i t h G p a r t i t i o n e d c o n f o r m a b l y as ( 2 . 1 . 9 ) , we d e f i n e t h e d e c o m p o s i t i o n method HF = G1 2G2 2 % = S rG1 2G2 2G2 1 HR - G2 1G1 1 GR " G2 2 "G2 1G1 1G1 2 Theorem 2.1.2 S u b j e c t t o a s s u m p t i o n s 2.1.1 - 2.1.4, t h e open l o o p models (H ,G ,H ,G ) a r e F F R R r e c o v e r a b l e from S, b y a p p l y i n g t h e d e c o m p o s i t i o n method (2.1.12) t o any c o r r e c t l y p a r t i t i o n e d s p e c t r a l f a c t o r G o f S.

P r o o f : Goodwin, Payne (1977); S i n , Goodwin (1979). V

I n a d d i t i o n we have t h e f o l l o w i n g r e s u l t . R e s u l t 2.1.1 I f G i n (2.1.9) i s minimum p h a s e , t h e n G', G' a s o b t a i n e d b y (2.2.12) a r e F R minimum p h a s e . P r o o f : a p p e n d i x C.1. Thus i t i s p o s s i b l e t o o b t a i n models o f t h e d e t e r m i n i s t i c a n d t h e s t o c h a s t i c p a r t o f systems i n c l o s e d l o o p , from o n l y t h e s p e c t r u m o f t h e j o i n t i n p u t -o u t p u t . The meth-od i s n -o t -o n l y u s e f u l f -o r a p p l i c a t i -o n s t -o t e c h n i c a l s y s t e m s , b u t a l s o f o r a p p l i c a t i o n s t o e c o n o m i c a l and e n v i r o n m e n t a l s y s t e m s w i t h i n h e r e n t f e e d b a c k . I n o u r a p p l i c a t i o n s we w i l l o n l y be i n t e r e s t e d i n (H | G ) , b u t we F F w i l l s t u d y t h e method i n i t s g e n e r a l form.

24 I n s e c t i o n s 2.2 - 2.4 we w i l l s t u d y t h e c o m p u t a t i o n a l a s p e c t s o f r e a l i z a t i o n , s p e c t r a l f a c t o r i z a t i o n , a n d d e c o m p o s i t i o n i n t o o p e n l o o p m o d e l s . I f we l a b e l t h e s e s t e p s b y t h e s y m b o l s R, S a n d D, t h e n t h e c o n s e c u t i v e o r d e r s i n w h i c h t h e y may b e e x e c u t e d c a n b e w r i t t e n a s i . e . t h e p o s i t i o n o f t h e r e a l i z a t i o n s t e p may b e f r e e l y c h o o s e n . H o w e v e r , i f t h e c o m p u t a t i o n a l a s p e c t s o f t h e t h e o r y a r e c o n s i d e r e d , t h e n t h e r e a p p e a r t o b e i m p o r t a n t d i f f e r e n c e s b e t w e e n t h e t h r e e p o s s i b l e p r o c e d u r e s . I n s e c t i o n 2.2 we w i l l c o n s i d e r s p e c t r a l f a c t o r i z a t i o n i n s t a t e s p a c e f o r m a n d i n i m p u l s e r e s p o n s e f o r m i n o r d e r t o h a v e s u i t a b l e f o r m u l a t i o n s f o r t h e d i f f e r e n t p r o c e d u r e s ( 2 . 1 . 1 3 ) , w h i c h we w i l l c o m p a r e . F o r t h e s a m e r e a s o n we w i l l c o n s i d e r t h e d e c o m p o s i t i o n i n t o o p e n l o o p m o d e l s i n s t a t e s p a c e f o r m a n d i n i m p u l s e r e s p o n s e f o r m . I n o u r a n a l y s e s o f t h e c o m p u t a t i o n a l a s p e c t s , t h e s t a r t i n g p o i n t i s a s a m p l e c o v a r i a n c e f u n c t i o n a n d t h e u l t i m a t e o b j e c t i v e i s t o o b t a i n a l o w o r d e r a p p r o x i m a t e s t a t e s p a c e m o d e l o f t h e c o n s i d e r e d s y s t e m . I n s e c t i o n 2.4 we w i l l c o n s i d e r t h e f e a s i b i l i t y o f d i f f e r e n t r e a l i z a t i o n m e t h o d s t o o b t a i n a p p r o x i m a t e s t a t e s p a c e m o d e l s o f a n i m p u l s e r e s p o n s e w h i c h i s c o n t a m i n a t e d b y n o i s e . 2.2 S p e c t r a l f a c t o r i z a t i o n R-S-D S-R-D ( 2 . 1 . 1 3 ) S-D-R

2.2.1 Spectral factorization in state space form

L e t t h e s p e c t r u m S ( z ) b e d e f i n e d a s ( 2 . 2 . 1 ) ( 2 . 2 . 2 ) o r e q u i v a l e n t l y a s 00 S ( z ) = Z z KR ( k ) , R ( - k ) = RT( k ) ( 2 . 2 . 3 ) k=-oo

25

R(k) = C Ak 1B k>0

= D k=0

T

w i t h (A,B,C) a g i v e n m i n i m a l r e a l i z a t i o n , and w i t h D=D >0. Then, a s p e c t r a l f a c t o r o f S ( z ) c a n be w r i t t e n a s G(z) = C ( z I - A )- 1K + I (2.2.5) By w r i t i n g down t h e s t a t e s p a c e e q u a t i o n s o f G ( z ) , x ( k + l ) = Ax(k) + Kw(k) 2 6) y ( k ) = Cx(k) + w(k) t h e r e l a t i o n between t h e p a r a m e t e r s i n (2.2.4) and (2.2.6) a p p e a r s t o be (Anderson, 1967a; F a u r r e 1976) B T = APC

+

+

Theorem 2.2.1

L e t S ( z ) be a p a r a h e r m i t i a n and r a t i o n a l f u n c t i o n . Then t h e r e e x i s t s a s p e c t r a l f a c t o r i z a t i o n

S ( z ) = G ( z ) W GT( z_ 1) , W=WT>0 (2.2.9)

26 1 S ( z ) i s a n a l y t i c on | z [ = l ^2 2 1 Q j 2 S ( z ) i O on |z|=l P r o o f : A n d e r s o n (1967a); F a u r r e (1976); F a u r r e , C l e r g e t , Germain (1979). V The c o n d i t i o n (2.2.10) i s e q u i v a l e n t t o t h e c o n d i t i o n t h a t F ( z ) i n (2.2.2) i s p o s i t i v e - r e a l ( A n d e r s o n , 1967b). T h e r e f o r e we w i l l r e f e r t o (2.2.10) a s t h e p o s i t i v e - r e a l c o n d i t i o n . I f t h e p o s i t i v e - r e a l c o n d i t i o n i s s a t i s f i e d , t h e n t h e r e e x i s t s a s e t o f s o l u t i o n s P20 t o (2.2.7) s a t i s f y i n g ( W i l l e m s , 1971; K u c e r a , 1972) P~ < P < P+ (2.2.11)

w i t h P t h e minimum and P t h e maximum s o l u t i o n . I f we a d d t o (2.2.10) t h e c o n d i t i o n t h a t S ( z ) h a s f u l l r a n k f o r a l m o s t a l l z, t h e n W i n (2.2.9) s a t i s f i e s W>0. The e q u a t i o n s (2.2.7) may t h e n be r e w r i t t e n a s t h e a l g e b r a i c R i c c a t i e q u a t i o n P - A P AT - ( B - A P CT) ( D - C P CT)_ 1( B - A P CT)T = 0 (2.2.12) Moreover, t h e i n n o v a t i o n s r e p r e s e n t a t i o n t h e n e x i s t s and i s o b t a i n e d by W =D - C P " CT T _t (2.2.13) K = (B-AP C )W

P may be computed by s e v e r a l methods. I f P i s computed from t h e e i g e n v e c t o r s o f t h e H a m i l t o n m a t r i x b e i n g a s s o c i a t e d t o ( 2 . 2 . 7 ) , t h e n i n a d d i t i o n t o (2.2.10)

must have d e t ( AD) ^ 0 w i t h AD= A - C D_ 1B ( K u c e r a , 1972; M o l i n a r i , 1975). I f P~ i s

computed b y a l i n e a r l y ( F a u r r e , 1976) o r q u a d r a t i c a l l y (Hewer, 1971) c o n v e r g i n g i t e r a t i v e a l g o r i t h m w h i c h s o l v e s ( 2 . 2 . 1 2 ) , t h e n i n a d d i t i o n t o (2.2.10) S ( z ) must have f u l l r a n k f o r a l m o s t a l l z .

2.2.2 Spectral factorization in impulse response form

T Now we s t a r t w i t h a s p e c t r u m S ( z ) a s d e f i n e d by ( 2 . 2 . 3 ) , w i t h R(k)=R (-k) a g i v e n r a t i o n a l and p a r a s y m m e t r i c f u n c t i o n , so t h a t S ( z ) i s r a t i o n a l and

27

p a r a h e r m i t i a n . T h i s i s t h e e a r l i e s t f o r m i n w h i c h s p e c t r a l f a c t o r i z a t i o n h a s been s t u d i e d (Wold, 1938; Kolmogorov, 1941). By theorem 2.2.1 S ( z ) a d m i t s a s p e c t r a l f a c t o r i z a t i o n i f t h e p o s i t i v e - r e a l c o n d i t i o n (2.2.10) i s s a t i s f i e d . I f i n a d d i t i o n S ( z ) has f u l l r a n k on | z | = l , t h e n t h e i n n o v a t i o n s r e p r e s e n t a t i o n can be o b t a i n e d i n t h e f o r m CO G(z) = I + Z z ~kg ( k ) (2.2.14) k=l by t h e C h o l e s k y d e c o m p o s i t i o n ( S t e w a r t , 1973) T = L Q LT (2.2.15) w i t h R(0) R * ( l ) R ( l ) R(0) (2.2.16) L = g , ( i ) i g ^ z ) g2( D i

o

i o

o

The a d d i t i o n a l c o n d i t i o n t h a t S ( z ) has f u l l r a n k on |z|=l i s imposed t o have T>0, w h i c h i s n e c e s s a r y f o r t h e e x i s t e n c e o f t h e C h o l e s k y d e c o m p o s i t i o n

( 2 . 2 . 1 5 ) . The r e l a t i o n between t h e s i g n a t u r e o f S ( z ) on |z|=l and T i s f o r m u l a t e d i n t h e f o l l o w i n g r e s u l t .

R e s u l t 2.2.1

L e t S ( z ) be d e f i n e d by (2.2.3) and l e t T be d e f i n e d by (2.2.16) . Then S ( z ) > 0 on |z|=l i f and o n l y i f T>0.

28 Note t h a t t h e r e s u l t i s n o t r e s t r i c t e d t o t h e r a t i o n a l c a s e as e . g . i n F a u r r e , C l e r g e t , Germain (1979). Theorem 2.2.2 L e t S ( z ) as d e f i n e d by (2.2.3) be a p a r a h e r m i t i a n and r a t i o n a l f u n c t i o n s a t i s f y i n g (2.2.10) and h a v i n g f u l l r a n k on | z | = l . Then t h e C h o l e s k y d e c o m p o s i t i o n (2.2.15) y i e l d s gi( k ) -> g(k) W. •* W as i-Ko, w i t h g(k) (2.2.14) and W d e f i n i n g t h e i n n o v a t i o n s r e p r e s e n t a t i o n . R i s s a n e n , K a i l a t h (1972) have p r o p o s e d a r e c u r s i o n t o o b t a i n W^, g ^ t k ) w i t h k = l , i , f o r 1=1,2,.., . A l a t e r a l g o r i t h m by R i s s a n e n (1973) p e r f o r m s t h e same r e c u r s i o n f o r i = l , 2 , . . . , p , w i t h p a p r i o r i f i x e d , r e q u i r i n g l e s s o p e r a t i o n s In f a c t t h e s e C h o l e s k y d e c o m p o s i t i o n a l g o r i t h m s a r e l i n e a r l y c o n v e r g i n g a l g o r i t h m s t o s o l v e t h e a s s o c i a t e d a l g e b r a i c R i c c a t i e q u a t i o n (Pagano, 1976). A r e l a t e d a l g o r i t h m by T u e l (1968) may be u s e d i f R(k)=0 f o r k>p. However, t h e a l g o r i t h m r e q u i r e s t h e i n v e r s i o n o f m a t r i c e s o f d i m e n s i o n pq, w i t h q t h e d i m e n s i o n o f t h e b l o c k s R ( k ) , and hence becomes e a s i l y c o m p u t a t i o n a l l y i n t r a c t a b l e . 2.2.3 Computational aspects I f R(k) i n (2.2.3) i s a c o v a r i a n c e f u n c t i o n , t h e n t h e p o s i t i v e - r e a l c o n d i t i o n (2.2.10) i s s a t i s f i e d and hence t h e s p e c t r a l f a c t o r i z a t i o n i n t h e p r o c e d u r e s (2.1.13) e x i s t s . I n t h e f o l l o w i n g we w i l l c o n s i d e r t h e r o b u s t n e s s q u e s t i o n w h e t h e r t h e p o s i t i v e - r e a l c o n d i t i o n i s s t i l l s a t i s f i e d i f 1 a sample c o v a r i a n c e f u n c t i o n i s t a k e n as a s t a r t i n g p o i n t 2 t h e r e a l i z a t i o n s t e p i n t h e p r o c e d u r e s (2.1.13) i s a p p r o x i m a t e A commonly u s e d c o v a r i a n c e f u n c t i o n e s t i m a t o r i s ( a p p e n d i x D.9) i P r o o f : R i s s a n e n , K a i l a t h (1972, Theorem 5.6a) V N-k-1 v ( i + k ) vT( i ) , k=0,N-l (2.2.19) i=0

29 w i t h VN= [ v ( 0 ) v ( 1 ) . . . v ( N - l ) ] a sample o f t h e s t o c h a s t i c p r o c e s s v ( k ) . R(k) i s a c o n s i s t e n t e s t i m a t o r b u t i s b i a s e d . A c o n s i s t e n t and u n b i a s e d e s t i m a t o r R (k) 1 1 i s o b t a i n e d i f — i n (2.2.19) i s r e p l a c e d by — — . U s u a l l y R(k) i s p r e f e r r e d N N-k above R u( k ) b e c a u s e i t y i e l d s s m a l l e r mean s q u a r e e s t i m a t i o n e r r o r s ( J e n k i n s , W a t t s , 1968) . T h e r e i s s t i l l a n o t h e r r e a s o n t o p r e f e r R ( k ) . R e s u l t 2.2.2 L e t R(k) be d e f i n e d by R(k) = R(k) k=0,N-l 2 = 0 k>N-l Then S ( z ) as d e f i n e d i n (2.2.3) s a t i s f i e s S(z)>0 on | z | = l . P r o o f : a p p e n d i x C.3 Note t h a t S ( z ) i n r e s u l t 2.2.2 i s an i n t e r p o l a t i o n on |z|=l o f t h e sample s p e c t r u m S ( f ) , as d e f i n e d i n a p p e n d i x D . l l . I t i s shown i n a p p e n d i x C.3 t h a t r e s u l t 2.2.2 i s n o t v a l i d i f R(k) i n (2.2.20) i s r e p l a c e d by Ru( k ) . By r e s u l t 2.2.2 t h e p o s i t i v e - r e a l c o n d i t i o n i s s a t i s f i e d i n t h e p r o c e d u r e s S-R-D, S-D-R ( 2 . 1 . 1 3 ) , i f R(k) i s d e f i n e d as i n ( 2 . 2 . 2 0 ) . I n t h e p r o c e d u r e R-S-D t h e p o s i t i v e - r e a l c o n d i t i o n i s s a t i s f i e d i f t h e r e a l i z a t i o n (2.2.4) d e f i n e s p r e c i s e l y R(k) i n ( 2 . 2 . 2 0 ) . However, t h i s i s n o t t h e i n t e n d e d use o f t h e p r o c e d u r e , b e c a u s e t h e n t h e o r d e r o f t h e f i n a l s t a t e s p a c e model w i l l be i n a c c e p t a b l y h i g h .

A more a p p r o p r i a t e use o f t h e p r o c e d u r e i s t o compute low o r d e r r e a l i z a t i o n s w i t h t h e p u r p o s e t o a p p r o x i m a t e l y r e c o v e r t h e u n d e r l y i n g c o v a r i a n c e f u n c t i o n f r o m a sample c o v a r i a n c e f u n c t i o n w h i c h i s c o n t a m i n a t e d by e s t i m a t i o n s e r r o r s . But i f t h e r e a l i z a t i o n i s a p p r o x i m a t e , t h e n t h e r e i s no g u a r a n t e e t h a t t h e p o s i t i v e - r e a l c o n d i t i o n i s s a t i s f i e d . To answer t h e q u e s t i o n whether t h e p o s i t i v e - r e a l c o n d i t i o n i s a c t u a l l y v i o l a t e d i n p r a c t i c e , we w i l l c o n s i d e r an a p p l i c a t i o n .

The s i g n a t u r e o f S ( z ) on |z|=l may be i n v e s t i g a t e d by computing t h e

s i n g u l a r i t i e s o f S(z) on |z|=l and by i n v e s t i g a t i n g t h e s i g n a t u r e o f S ( z ) f o r one v a l u e o f z on |z|=l between e a c h two c o n s e c u t i v e s i n g u l a r i t i e s . T h i s may be done by c o m p u t i n g t h e e i g e n v a l u e s o f S ( z ) , w h i c h a r e r e a l b e c a u s e S ( z ) i s

h e r m i t i a n on |z|=l ( S t e w a r t , 1973). R e s u l t 2.2.3 S ( z ^ ) 2 0 on |z |=1, i f and o n l y i f t h e e i g e n v a l u e s o f S ( Z j ) a r e n o n n e g a t i v e . P r o o f : S t e w a r t (1973). V The s i n g u l a r i t i e s o f S ( z ) c a n be i n v e s t i g a t e d u s i n g t h e r e l a t i o n ( M o l i n a r i , 1975) d e t 8 ( i ) = ^(Z> , . . -1. (2.2.21) p ( z ) p ( z ) w i t h p ( z ) = d e t ( z I - A ) f ( z ) = (-z) d e t ( D ) d e t ( AD) d e t ( z I - M ) A = A - B D- 1C T -1 T -T T -1 - I T - T l M = A -BD B A C D BD B A D D D (2.2.22) —T T — 1 L AD C D C .-T Now t h e f o l l o w i n g r e s u l t i s o b t a i n e d by i n s p e c t i o n o f (2.2.21), (2.2.22) R e s u l t 2.2.4 L e t (A,B,C,D) be a m i n i m a l r e a l i z a t i o n ( 2 . 2 . 1 ) , (2.2.2) o f S ( z ) . F u r h e r l e t d e t ( D ) ^ 0 , d e t ( AD) ^ 0 . Then t h e s i n g u l a r i t i e s o f S ( z ) a r e t h e e i g e n v a l u e s o f M . A p p l i c a t i o n 2.2.1 In a p p e n d i x B two i d e n t i f i c a t i o n e x p e r i m e n t s a r e d e s c r i b e d f o r t h e p i l o t s c a l e p r o c e s s w h i c h i s d e s c r i b e d i n a p p e n d i x A. From t h e s e e x p e r i m e n t s we have computed t h e sample c o v a r i a n c e f u n c t i o n R(k) o f t h e j o i n t i n p u t - o u t p u t . U s i n g t h e a p p r o x i m a t e r e a l i z a t i o n method w h i c h w i l l be d e s c r i b e d i n s e c t i o n 2.4, r e a l i z a t i o n s o f d i f f e r e n t o r d e r s have been computed. F i g u r e 2.2.1 shows t h e r e s u l t s f o r n=13 and n=24, and g i v e s an i d e a o f t h e smoothness o f t h e

a p p r o x i m a t i o n s . F i g u r e a shows t h e d i a g o n a l o f t h e sample c o v a r i a n c e f u n c t i o n and f i g u r e s b and c show t h e d i a g o n a l s o f t h e f u n c t i o n s R ( k ) , as d e f i n e d by t h e r e a l i z a t i o n s ( 2 . 2 . 4 ) . F o r a l l r e a l i z a t i o n s from b o t h e x p e r i m e n t s , t h e i t e r a t i v e p r o c e d u r e s t o s o l v e t h e a l g e b r a i c R i c c a t i e q u a t i o n d i d n o t c o n v e r g e . T h i s i s

R(k)

- 12B 0 ^ 128

a b c

Fig. 2.2.1 Diagonal of

a sample covariance function, R(k) b approximate realization R ^ ( k ) , n=13 c approximate realization R (k) , n=24

i n agreement w i t h t h e f a c t t h a t t h e a s s o c i a t e d H a m i l t o n m a t r i c e s a p p e a r e d t o have e i g e n v a l u e s on t h e u n i t c i r c l e , where S ( z ) became

i n d e f i n i t e . Hence S ( z ) d i d n o t s a t i s f y t h e p o s i t i v e - r e a l c o n d i t i o n ( 2 . 2 . 1 0 ) . T h i s l e a d s t o t h e c o n c l u s i o n t h a t t h e p r o c e d u r e R-S-D i s n o t s u i t a b l e f o r t h e c o n s i d e r e d t y p e o f a p p l i c a t i o n s . V Now we r e t u r n t o t h e p r o c e d u r e s S-R-D, S-D-R, f o r w h i c h t h e p o s i t i v e - r e a l c o n d i t i o n h a s been shown t o be s a t i s f i e d . I f i n a d d i t i o n S ( z ) h a s f u l l r a n k on |z|=1, t h e n we c a n compute t h e s p e c t r a l f a c t o r i z a t i o n i n i m p u l s e r e s p o n s e form by t h e C h o l e s k y d e c o m p o s i t i o n ( 2 . 2 . 1 5 ) . A v i o l a t i o n o f t h i s c o n d i t i o n w i l l become m a n i f e s t by t h e f a i l u r e o f t h e C h o l e s k y d e c o m p o s i t i o n a l g o r i t h m s . A c o m p u t a t i o n a l p r o b l e m o f s p e c t r a l f a c t o r i z a t i o n i n i m p u l s e r e s p o n s e form i s t h a t many i t e r a t i o n s may be r e q u i r e d t o o b t a i n c o n v e r g e n c e . F i r s t l y t h e c o n v e r g e n c e o f t h e r e c u r r e n t a l g o r i t h m s t o compute g ^ ( k ) , i n (2.2.17), (2.2.18) i s l i n e a r (Pagano, 1976). S e c o n d l y , i f t h e c o n s i d e r e d s y s t e m c a n n o t be d e s c r i b e d by a m o v i n g - a v e r a g e model o f f i n i t e o r d e r , t h e n c o n v e r g e n c e c a n n o t o c c u r f o r a f i n i t e number o f i t e r a t i o n s p . I f t h e number o f samples N i s f i n i t e and i f R(k) i s d e f i n e d by ( 2 . 2 . 2 0 ) , t h e n we c a n have c o n v e r g e n c e f o r a f i n i t e p>N-l. However, p may t h e n s t i l l be i n t r a c t a b l y l a r g e . To a c c e l l e r a t e t h e c o n v e r g e n c e one c a n a p p l y a l a g window w(k) y i e l d i n g a windowed f u n c t i o n R (k) w h i c h s a t i s f i e s R (k)=0 f o r |k|>r, w i t h rSN-1. Some w w 1 1 l a g windows w h i c h a r e commonly u s e d i n s p e c t r a l a n a l y s i s ( J e n k i n s , W a t t s , 1 9 6 8 ) , a r e l i s t e d i n t a b l e 2.2.1 t o g e t h e r w i t h t h e i r z - t r a n f o r m s on | z | = l . Window w ( k ) , k <r w(k)=0, k >r W(to) , ÜJ <TT R e c t a n g u l a r B a r t l e t t Tukey P a r z e n ^k .5+.5cos 2r= s i n a r / 2 u r / 2 s i n u r wr l - ( u r / T r ) ' 3 / s i n u r / 4 \ 4 ur/4

Table 2.2.1 Some common windows

I t i s an i n t e r e s t i n g q u e s t i o n w h e t h e r t h e f u n c t i o n S (z) w h i c h i s t h u s w

33 R e s u l t 2.2.5 L e t R (k) be d e f i n e d by w R (k) = R(k)w(k) k <r w =0 I k I > r r<N-l (2.2.23)

where w(k) i s a l a g window h a v i n g t h e z - t r a n s f o r m W ( z ) . Then S (z) as d e f i n e d w by ( 2 . 2 . 3 ) , w i t h R(k) r e p l a c e d by R ( k ) , s a t i s f i e s S (z)>0 on |z|=l i f W(z)>0 w w on |z|=1. P r o o f : a p p e n d i x C.4. V U s i n g t h i s r e s u l t , i t f o l l o w s by i n s p e c t i o n o f t a b l e 2.2.1 t h a t t h e B a r t l e t t window and t h e P a r z e n window y i e l d S^(z)>0 on | z | = l , whereas t h i s i s p o s s i b l y n o t t r u e f o r t h e r e c t a n g u l a r window and t h e Tukey window.

The drawback o f c o n v e r g e n c e a c c e l l e r a t i o n by t h e a p p l i c a t i o n o f windows i s the i n c r e a s e d b i a s o f t h e r e s u l t i n g e s t i m a t e s . A n o t h e r p o s s i b l e a p p r o a c h t o l i m i t t h e number o f i t e r a t i o n s f o r t h e c o m p u t a t i o n o f t h e C h o l e s k y d e c o m p o s i t i o n , i s t o t r u n c a t e t h e r e c u r s i o n f o r some p where c o n v e r g e n c e has n o t y e t been o b t a i n e d , a c c e p t i n g an a p p r o x i m a t e s o l u t i o n . The e r r o r s w h i c h o c c u r i n t h i s a p p r o a c h a r e n o t n e c e s s a r i l y l a r g e r t h a n t h e e r r o r s w h i c h a r e c a u s e d by t h e a p p l i c a t i o n o f a window.

We w i l l now c o n s i d e r some e x p e r i m e n t a l r e s u l t s .

A p p l i c a t i o n 2.2.2

The sample c o v a r i a n c e f u n c t i o n s R(k) as computed f r o m two e x p e r i m e n t s w i t h N=4096 ( a p p e n d i x B) have been u s e d t o compose f i n i t e b l o c k T o e p l i t z m a t r i c e s

R (0) R (1) w w R (1) R (0) w w R (p) T Rww (PJ R (0) w (2.2.24) w i t h Rw( k ) d e f i n e d as i n r e s u l t 2.2.5 and w i t h r<p<N-l. We have a p p l i e d t h e

r e c t a n g u l a r window w (k) , t h e B a r t l e t t window w ( k ) , and no window, i n w h i c h R B