Fault diagnosis based on analytical models for linear and nonlinear systems - a Tutorial

8/7/2019 Fault diagnosis based on analytical models for linear and nonlinear systems - a Tutorial

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F A U L T D I A G N O S I S B A S E D O N A N A L Y T I C A LM O D E L S F O R L I N E A R A N D N O N L I N E A R

S Y S T E M S - A T U T O R I A LM i c h e l K i n n a e r t

Dpt. of Control Engineering and System Analysis, UniversitdLibre de BruxeUes. CP 165/55, 50 Av. F. D. Roosevelt , B-1050Brussels , BELGIUM

Abs t r a c t : T he d i a gnos i s sys t e m s c ons ide r e d in t h i s pa pe r r e ly on the i nc ons i s t e nc yb e t w e e n t h e a c t u a l p r o c e s s b e h a v i o u r a n d i t s e x p e c t e d b e h a v i o u r a s d e s c r i b e d b ya n a na ly t i c a l m ode l . T he inc ons i s t e nc y i s e xh ib i t e d i n s igna l s c a l l e d r e s idua l s . T wom e t h o d s f o r r e s i d u a l g e n e r a t i o n a r e p r e s e n t e d i n a t u t o r i a l w a y : t h e p a r i t y s p a c ea nd the obse r ve r ba se d a ppr oa c he s . L ine a r a nd non l ine a r m ode l s a r e suc c e ss ive lyc ons ide r e d a s a ba s i s f o r t he de s ign o f t he r e s idua l ge ne r a to r s .

Ke ywor ds : Fa u l t de t e c t i on a nd i so l a t i on , l i ne a r a nd non l ine a r sys t e m s , obse r ve r ,a n a l y t i c a l r e d u n d a n c y , r e s i d u a l g e n e r a t i o n , p a r i t y s p a c e m e t h o d .

1 . I N T R O D U C T I O N

F o r t h e c o m p l e x h i g h l y a u t o m a t e d s y s t e m s e n -c oun te r e d in p r oc e ss i ndus t r i e s , i n a e r ona u t i c s o rpowe r p l a n t s f o r i ns t a nc e , i t i s f unda m e nta l t ob e a b l e t o m o n i t o r t h e h e a l t h o f t h e i n s t a l l a ti o na t e a c h t im e ins t a n t i n o r de r t o de t e c t i nc ip i e n tf a u l t s a n d t o l o c a t e t h e d e t e r i o r a t e d c o m p o n e n t s .I nde e d , f o r s a f e ty c r i t i c a l sys t e m s suc h a s nuc l e a rpowe r p l a n t s a nd a i r p l a ne s , t he t o t a l f a i l u r e o fa c o m p o n e n t c a n c a u s e h a z a r d t o p e r s o n n e l o rt o t h e p o p u l a t i o n a n d h e n c e i t m u s t b e a v o i d e da t a ny c os t . M or e ove r t he e a r ly de t e c t i on o f am a l f unc t ion a l l ows one to p l a n t he r e qu i r e d m a in-t e n a n c e a c t i o n s a n d d e c r ea s e s t h e n u m b e r o f em e r -ge nc y shu tdow ns o f a p roc e ss , whic h a r e o f t e n ve r ycost ly .For sa f e ty c r i t i c a l sys t e m s , t he p r e se n t m oni to r ingt o o l s h e a v i l y r e l y o n h a r d w a r e r e d u n d a n c y . F o rins t a nc e t h r e e t o f our s e nsor s a r e o f t e n use d tom e a s u r e t h e s a m e q u a n t i t y , a n d a v o t i n g s c h e m ede te r m ine s i t s m os t l i ke ly va lue . T h i s r e qu i r e sa d d i t i o n a l s p a c e t o a c c o m m o d a t e t h e e q u i p m e n t ,

i t increases the weight which i s a c r i t ica l i ssue insom e a pp l i c a t i ons , a nd i t i s qu i t e c os t l y .M o s t i n d u s t r i a l m o n i t o r i n g s y s t e m s t h a t a r e n o tb a s e d o n h a r d w a r e r e d u n d a n c y r e l y o n t h e c o m -pa r i son o f m e a sur e d s igna l s t o spe c if i ed t h r e sh -o lds . T he y do no t e xp lo i t t he c or r e l a t i on e x i s t i ngbe twe e n the d i f f e r e n t m e a sur e d s igna l s . T he r e f o r et h e y o n l y a l lo w t h e d e t e c t i o n o f i m p o r t a n t d e v i-a t i o n s f r o m n o r m a l o p e r a t i n g c o n d i t i o n s a n d n o tinc ip i e n t c ha nge s .T h e s e d r a w b a c k s m o t i v a t e t h e d e v e l o p m e n t o f d i -a g n o s i s s y s t e m s b a s e d o n a n a l y t i c a l m o d e l s . T h em a i n i d e a b e h i n d s u c h s y s t e m s i s t o c h e c k t h ec o n s i s te n c y b e t w e e n t h e m e a s u r e m e n t s o f d if fe r en tva r i a b l e s t a ke n on the supe r v i se d p r oc e ss , a ndt h e e x p e c t e d b e h a v i o r o f t h i s p r o c e s s a s d e s c r ib e dby a n a na ly t i c a l m ode l . As shown in f i gur e 1 ,t y p i c a l d ia g n o s is s y s t e m s a r e m a d e o f t w o p a r t s : ar e s id u a l g e n e r a t o r a n d a d e c is i o n m a k i n g m o d u l e .T h e r e s i d u a l s a r e s i g n a l s t h a t a r e g e n e r a t e d b yp r o c e s si n g t h e m e a s u r e d p l a n t i n p u t s a n d o u t p u t s .I n t he a bse nc e o f f a u l t , t he y de v ia t e f r om z e r o on lyd u e t o m o d e l l i n g u n c e r t a i n t i e s a n d m e a s u r e m e n t

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no i se . Whe n a f a u l t oc c ur s , t he de v i a t i on f r omz e r o i s suc h t ha t t he ne w c ond i t i on c a n be d i s t i n -gu i she d f r om t he f a u l t l e s s w or k i ng mode . The r o l eo f t h e d e c i si o n m o d u l e i s t o d e t e r m i n e w h e t h e r t h er e s i dua l s d i f f e r s i gn i f i c a n t l y f r om z e r o , a nd , f r omt he pa t t e r n o f z e r o a nd non - z e r o r e s idua l s , t o de -c i de w h i c h a r e t he mos t l i ke l y f a u l t y c ompone n t s .

M e a s u r e dinputs

U n k n o w ni n p u t s

. . _ 1 S u p e r v i s e d- - I p r o c e s s

R e s i d u a lg e n e r a t o rE,, ~ R e s i d u a l s

D e c i s i o nm o d u l eF a u l t yc o n l p o n e n t ( s )

M e a s u r e dO U t p U t D _ _

F i g . 1 . S t r u c t u r e o f a d ia gnos i s sys t e m

I n t h i s pa pe r t he ma i n f oc us i s on t he de s i gn o fr e s i dua l ge ne r a t o r s f o r sys t e m s de sc r i be d by de t e r -mi n i s t i c a na l y t i c a l mode l s . The f i r s t pa r t c on t a i nsa t u t o r i a l p r e s e n t a t i o n o f b o t h p a r i t y s p a c e a n dobse r ve r ba se d a ppr oa c he s t o t he d e s i gn o f r e s id -ua l ge ne r a t o r s f o r l i ne a r sys t e ms . A ve r y l i mi t e dk n o w l e d g e o f s y s t e m t h e o r y i s e x p e c t e d f r o m t h er e a de r f o r t h i s pa r t ( s e c t i ons 2 t o 4 ) . A s mos tp r oc e s se s a r e i nhe r e n t l y non l i ne a r , t he r e s i dua lg e n e r a t o r s b a s e d o n l i n e a r m o d e l s w o r k p r o p e r l yp r o v i d e d t h e p r o c e s s i s r u n n i n g i n t h e o p e r a t i n gr e g i on f o r w h i c h t he l i ne a r a ppr ox i m a t i on i s va l id .To r e l a x t h i s c ond i t i on , i t i s ne c e s sa r y e i t he r t o a c -c oun t f o r mode l l i ng unc e r t a i n t i e s due t o t he l i ne a ra p p r o x i m a t i o n o r t o e x p l o i t t h e n o n l i n e a r m o d e lo f t he p r oc e s s i n s t e a d o f i t s l ine a r a pp r ox i m a t i ona r ound a n e qu i l i b r i um po i n t . A f t e r a d i s c us s i onof d if f e r e n t w a ys t o h a nd l e mo de l l i ng e r r o r s i nse c t i on 5 , va r i ous c la s se s o f non l i ne a r mod e l s a r ec ons i de r e d a nd t he w a ys t o ge ne r a l i z e e i t he r t hep a r i t y s p a c e m e t h o d o r t h e o b s e r v e r b a s ed m e t h o df o r r e s i dua l ge ne r a t i on a r e p r e se n t e d ( s e e s e c t i ons6 a n d 7 ) . S o m e o p e n p r o b l e m s a r e p o i n t e d o u t i nt he c onc l us i on .

2. M O D E L O F T H E S U P E R V I S E D P R O C E S SUsing ph ys ica l laws , a la rge c lass of engine er ingsys t e ms c a n be mode l l e d by d i f f e r e n t i a l e qua t i onso f t he f o r m

w h e r e g a n d h a r e s m o o t h 1 n o n l i n e a r f u n c t i o n s o ft h e i r a r g u m e n t s , x ( t ) i s t he s t a t e ve c t o r , u ( t ) th ev e c to r o f k n o w n i n p u t s ( t h e m a n i p u l a t e d i n p u t sf o r i n s t a nc e ) , y ( t ) t h e v e c t o r o f m e a s u r e d o u t p u t s ,d ( t ) t h e v e c t o r o f u n k n o w n i n p u t s a n d f ( t ) th eve c t o r o f f a u l t s. T he d i m e ns i on s o f t he d i f f er e n tve c t o r s a r e r e spe c t i ve l y n , m , p , r i d , a nd h i . I nf a u l t l e s s w or k i ng mode , f ( t ) i s zero for al l t .T h e u n k n o w n i n p u t s c o r r e s p o n d t o u n m e a s u r e dd i s t u r ba nc e s suc h a s w a ve s a c t i ng on a sh i p o rw i nd gus t s on a n a i r p l a ne f o r i n s t a nc e .W h e n s y s t e m ( 1 ) , (2 ) is w o r k i n g a r o u n d n o m i n a lo p e r a t i n g c o n d i t i o n s t h a t a r e c h a r a c t e r i z e d b y a ne q u i l i b r i u m s t a t e , i t s b e h a v i o u r c a n b e d e s c r i b e dby a li ne a r t i me i nva r i a n t ( LTI ) m ode l o f t he f o r m

g c(t) = A x ( t ) + B u ( t ) + E d d ( t ) + E / / ( t ) (3 )y ( t ) = C x ( t ) + D u ( t ) + G d d ( t ) + G / f ( t) (4 )

A l t hough i t i s no t i nd i c a t e d e xp l i c i t l y , i t i s a s -s u m e d t h a t x ( t ) , u ( t ) , d ( t ) , f ( t ) a n d y ( t ) r e p r e s e n tde v i a t i ons f r om t he i r va l ue a t e qu i l i b r i um i n ( 3 ) ,( 4 ). T h e l i ne a r m o d e l i s o b t a i n e d b y p e r f o r m -i ng a Ta y l o r s er i es e xpa n s i on o f t he f unc t i onsg ( x ( t ) , u ( t ) , d ( t ) , f ( t ) ) a n d h ( x ( t ) , u ( t ) , d ( t ), f ( t ) )a r o u n d t h e e q u i l i b r i u m p o i n t , a n d k e e p i n g t h ef i r s t o r de r t e r ms on l y .T h e f a u l t f ( t ) a p p e a r s a s a n a d d i t i o n a l i n p u t i nt he l i ne a r mode l , a nd he nc e i t i s c a l l e d a n a dd i t i vef a u l t ( a s oppose d t o a mul t i p l i c a t i ve f a u l t w h i c hc ons i s t s o f a c ha nge i n t he e n t r i e s o f t he m a t r i c e sA , B , C , . . . ) . I t m i g h t c o r r e s p o n d t o a s e n s o r bi a s,t h e j a m m i n g o f a n a c t u a t o r o r a l e a k fo r in s t a n c e.I nde e d , f o r a s e nso r b i a s on t he f i r s t me a su r e me n t ,oc c ur r i ng a t t i me t o , i t su f f i c e s t o t a ke E! = 0 ,G I = ( 1 0 " " 0 ) T , a n d f ( t ) = a l ( t - t 0 ) w h er e" a " i s t h e b i a s m a g n i t u d e a n d l ( t ) i s t h e u n i ts t e p f unc t i on ( l ( t ) = 0 f o r t < 0 , a nd l ( t ) = 1f o r t > 0 ) . T h e j a m m i n g o f t h e f i rs t a c t u a t o rc a n b e m o d e l le d w i t h E / = B . ,1 , G / = 0 , a n df ( t ) = ~ 1 - U l ( t ) , w he r e B . , 1 de no t e s t he f i r s tc o l u m n o f m a t r i x B , u l ( t ) , t h e f i r s t c o m p o n e n to f ve c t o r u ( t ) , a nd U l i s t he va l ue a t w h i c h t hea c t ua t o r i s s t uc k . A l e a k i n a n hydr a u l i c sys t e mc a n b e m o d e l l e d b y a s u p p l e m e n t a r y i n p u t i n amod e l , na m e l y t he f low o f t he l e a k i ng f l ui d . I t c a nt hus b e e xpr e s se d i n t he f o r m o f mode l ( 3 ) , ( 4 )w he n a l i ne a r mode l i s u se d .Mul t i p l i c a t i ve f a u l t s a r e no t c ons i de r e d i n t h i spa pe r . Th e r e a de r i s r e f e r r e d t o ( I s e r m a nn , 1993)a n d ( Z h a n g et a l . , 1994) f o r a n i n t r oduc t i on t ot h i s t op i c .

~ ( t ) = g ( x ( t ) , u ( t ) , d( t ) , f ( t ) ) (1 )y ( t ) = h ( x ( t ) , u ( t ) , d( t ) , f ( t ) ) (2 ) 1 A s m o o t h f u n c t i o n h a s c o n t i n u o u s p a r t i a l d e r i v a t i v e s o fa n y o r d e r w i t h r e s p e c t t o i t s a r g u m e n t s .

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3. P A R I T Y S P A C E M E T H O D F O R L T IS Y S T E M S

3.1 I n t r o d u c t i o nA f u n d a m e n t a l n o t i o n o n w h i c h d i a g n o s i s s y s -t e m s b a s e d o n a n a l y t i c a l m o d e l s r e l y i s a n a l y t i c a lr e d u n d a n c y . A n a l y t i c a l r e d u n d a n c y r e l a t i o n s a r ee q u a t i o n s t h a t a r e d e d u c e d f r o m a n a n a l y t ic a lm o d e l a n d w h i c h s o l e l y i n v o l v e m e a s u r e d v a r i -a b l e s . T h e y m u s t b e f u l f i l l e d i n t h e a b s e n c e o ff au l t .A s a v e r y s i m p l e e x a m p l e , c o n s i d e r t h e r e l a t i o nb e t w e e n t h e v e l o c i t y a n d t h e a c c e l e r a t i o n o f am o b i l e d ev i ce :

, ) ( t ) = a ( t ) ( 5 )

I f b o t h v a r i a b l e s a r e m e a s u r e d p e r f e c t l y , a n d t h ed e r i v a t i v e o f t h e v e l o c i t y c a n b e d e t e r m i n e d e x -a c t l y , t h e q u a n t i t y

r ( t ) = i ~ m ( t) - a m ( t ) ( 6 )

m a y b e u s e d t o d e t e c t s e n s o r f a u l t s ( t h e i n d e xm i n d i c a t e s m e a s u r e m e n t ) , r ( t ) i s z e r o f or a l lt in t h e a b s e n c e o f f a u l t a n d i t b e c o m e s n o n -z e r o u p o n o c c u r r e n c e o f s e n s o r f a u lt s . I n d e e d , l e tv m ( t ) = v ( t ) + f v ( t ) d e n o t e t h e m e a s u r e d v e l o c i t y ,w h e r e f v ( t ) i s n o n - z e r o u p o n o c c u r r e n c e o f as en s o r f au l t , an d co n s i d e r a s i m i l a r ex p r e s s i o n f o rt h e a c c e l e r a t io n : a m ( t ) = a ( t ) + A ( t) . S u b s t i t u t i n gt h es e ex p r e s s i o n s i n ( 6 ) y i e l d s

= ] . ( t ) - A ( t ) ( 7 )

w h i c h t a k e s a t l e a s t t e m p o r a r i l y n o n - z e r o v a l u e sw h e n I v ( t ) a n d / o r f a ( t ) i s n o n - ze r o 2 . r ( t ) i s s a i dt o b e a r e s i d u a l t h a t a l lo w s d e t e c t i o n o f f a u l t s o nt h e v e l o c i t y a n d a c c e l e r a t i o n s e n s o r s a n d ( 5) is a na n a l y t i c a l r e d u n d a n c y r e l a t i o n .

3 .2 A n a l y t i c a l r e d u n d a n c y r e l a t io n s a n d p a r i t yf u n c t i o n sO u r a i m i s n o w t o d e t e r m i n e h o w t o g e n e r a t e a n -a l y t i c a l r e d u n d a n c y r e l a t i o n s t o m o n i t o r s y s t e m sd es c r i b ed b y a m o d e l o f t h e f o r m ( 3 ) , ( 4) . T o t h i sen d l e t u s co n s i d e r t h e s u cces s i v e d e r i v a t i v e s o ft h e o u t p u t y ( t ) . F o r t h e s ak e o f s im p l i c i t y , t h ec a s e w h e r e t h e r e a r e n o u n k n o w n i n p u t ( E d - - 0 )a n d n o f a u l t ( f ( t ) = 0 ) is f i r st co n s i d e r ed . F r o m( 3 ) an d ( 4 ) , o n e d ed u ces

2 The pathological case w here both faults com pensateeach other is not considered as it is improbable in practice.

y = C x + D u (8 )= C ~ + D it = C A x + C B u + D it (9 )

~) = C A 2 x + C A B s + C B i t + D i i (10)

y(S ) = C A S x + C A S - I B u + . . .+ C B u ( 8 -1 ) + D u (~) (11)

w h e r e t h e t i m e a r g u m e n t h a s b e e n o m i t t e d f o rc o m p a c t n e s s o f t h e n o t a t i o n s . L e t t i n g Y a ( t ) ( U s ( t ) )d e n o t e t h e v e c t o r o b t a i n e d b y s t a c k in g y ( t ) ( u ( t ) )a n d i t s d e r i v a t i v e s u p t o o r d e r s , T8~' b e t h e l o w e rt r i a n g u l a r b l o c k T o e p l i t z 3 m a t r i x w i t h f ir s t c o l-u m n [ D T ( C B ) T . . . ( C A ( 8 - 1 ) B ) T ] T a n d 0 8[ C T ( C A ) T . . . ( C A S ) T ] T , e q u a t i o n s ( S ) - ( l l ) c a nb e w r i t t e n i n a c o m p a c t f o r m a s

Y s ( t) = 0 8 x ( t ) + ~ " U s ( t ) (12)A s s u m i n g t h a t t h e d e r i v a t i v e s o f u ( t ) a n d y ( t ) c a nb e o b t a i n e d f r o m t h e a v a i l ab l e m e a s u r e m e n t s ( se es ec t i o n 3 .4 f o r t h i s i s s u e ) , t h e o n l y u n k n o w n i n( 12 ) i s t h e s t a t e v e c t o r x ( t ) . W h e n s is c h o s e ns u f f ic i en t l y la r g e , t h e l e f t n u l l s p ace o f m a t r i x O , ,n a m e l y t h e s p a c e g e n e r a t e d b y t h e r o w v e c to r s w 8which fu l f i l

= 0 , ( l a )i s n o n - ze r o . M u l t i p l y i n g ( 1 2 ) o n t h e l e f t b y o n es u c h v e c t o r y i e l d s

(14)( 14 ) i s a n a n a l y t i c a l r e d u n d a n c y r e l a t i o n , n a m e l ya r e l a t i o n t h a t o n l y i n v o l v e s k n o w n v a r i a b l e s a n dd e r i v a t i v e s o f k n o w n v a r i a b l e s . T h i s m o t i v a t e s t h ed e f i n it i o n o f t h e q u a n t i t y r s ( t ) a s

= - ( 1 5 )

O b v i o u s l y , r s ( t ) i s n u l l i n t h e a b s en ce o f fau l t . T od e t e r m i n e i t s v a l u e i n f a u l t y o p e r a t i n g c o n d i t i o n ,t h e e q u a t i o n c o r r e s p o n d i n g t o ( 1 2 ) i n t h e p r e s e n c eof f au l t i s used :

Y ~ (t) = O , x ( t ) + T 2 U , ( t ) + T I F f ( t ) (16)w h e r e T / i s t h e l o w e r t r i a n g u l a r b l o c k T o e p l i t zm a t r i x w i t h f i r s t c o l u m n[ G ~ ( C E I ) T . . . ( C A ( 8 - 1 ) E f ) T ] T a n d F s ( t ) is av e c t o r m a d e o f f ( t ) a n d i t s d e r i v a t i v e s u p t o o r d e rs . S u b s t i t u t i n g ( 1 6 ) f o r Y s ( t ) i n t o ( 1 5 ) y i e l d s

= ( 1 7 )

3 t h e m a t r i x

D 0 . . . . . . 0G B D 0 . . . 0

G A ( S - 1 )B C B D

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As long asw, ' l # 0 ,

r s ( t ) i s a n i n d i c a t o r o f t he p r e se nc e o f a f a u l t; i ti s ca l led a r es idua l . (15) , (16) and (17) def ine apa r i t y f unc t i on o r pa r i t y c he c k , r s ( Y s ( t ) , U s ( t ) ) ,na me l y a f unc t i on o f y ( t ) , u ( t ) a nd t he i r de r i va -t i ve s t ha t i s e qua l t o z e r o i n t he a bse nc e o f f a u l ta nd i s non- z e r o i n t he p r e se nc e o f a f a u l t . The s e to f a ll pa r i t y f unc t i ons i nvo l v ing s i gna l de r i va t i ve sup t o o r de r s c a n be ob t a i ne d by l i ne a r c ombi -na t i ons o f t he e n t r i e s o f t he p a r i t y ve c t o r 4 ~ 8(t )def ined as

= a , ( Y . ( t ) - (19)whe re ~8 deno tes a bas is 5 for the le f t null spaceof O h . Th e d i m e ns i on o f t h i s subspa c e i s np = ( s +1 ) p - r a n k ( 9 8 , s mus t t hus be c hose n su f f i c i e n t l yl a r ge t o a s su r e t ha t np i s pos i t i ve . A ny su i t a b l erow vec tor w8 has the form w8 = vsgts , wherevs i s a n np - d i me ns i ona l r ow ve c t o r c hose n so t ha tvsgt~T~I ~ 0. As gt8 is no t u niqu e, the re ex ists ani n f i n it e num be r o f pa r i t y ve c t o r s ~ 8 . Th e spa c ege ne r a t e d by t he se ve c t o r s i s c a l l e d t he pa r i t yspace .Whe n unknow n i npu t s a r e p r e se n t , i t i s ne c e s sa r yt o a s su r e t ha t t he r e s i dua l i s no t a f f e c t e d by suc hi npu t s . Wi t h Ed ~ 0 , ( 16 ) be c ome s

Y a ( t ) = O s x ( t ) + T s u U a ( t ) + T s d D s ( t )( 2 0 )

w he r e t he ne w no t a t i ons T8d and D8 ( t ) a re def inedin a s imi la r way as TsI and F s ( t ) . S u b s t i t u t i n g t h i se xpr e s s i on i n t o ( 15 ) now y i e l ds

r . ( t) = w s ( ~ f F . ( t ) + % d D . ( t )) (21)T o a s s u r e t h a t r s ( t ) i s not a f fec ted by d ( t ) , t h ec ond i t i on

= o ( 2 2 )mus t be i mpose d on w 8 be s i de s ( 13 ) a nd ( 18 ) .A ne c e s sa r y c on d i t i on f o r t he e x i s t e nc e o f a r owve c t o r w 8 t ha t me e t s t he t h r e e c ond i t i ons i s t ha tt h e n u m b e r o f u n k n o w n i n p u t s n d be l ow e r t ha nt h e n u m b e r o f m e a s u r e d o u t p u t s p .R e m a r kA n a l y t i c a l r e d u n d a n c y r e l a t i o n s w e r e p r e s e n t e din (Chow and Wil l sky, 1984) for l inear d isc re te

4 I t i s a s s u m e d t h a t t h e r e i s n o s t a t e v a r i a b l e t h a t i se x c l u s i v e l y c o n t r o l l a b l e f r o m t h e f a u l t ( s e e ( N y b e r g , 1 9 9 9) ,p a g e 2 0 9 ).5 S u c h a b a s i s c a n b e c o m p u t e d b y p e r f o r m i n g a s i n g u l a rv a l u e d e c o m p o s i t i o n o f m a t r i x ( 98 .

Ta b l e 1 . Ef f e c t s o f t he f a u l t s on t her e s i d u a l s - " d i a g o n a l " c o d i n g s e t

f~ /2 f3r l 1 0 0r2 0 1 0r3 0 0 1

t i m e s y s t em s . I n t h a t f r a m e w o r k , t h e y a r e de f in e da s r e l a t i o n s b e t w e e n t h e m e a s u r e d i n p u t s a n dou t pu t s ove r a f i xe d t i me hor i z on , t ha t mus t beful f i l led in the absence of f aul t . Here we havec hose n t o w or k w i t h Con t i nuous - t i me mode l s t oe a s e t h e g e n e r a l i z a t i o n t o n o n l i n e a r s y s t e m s . T h ep a r i t y s p a c e a p p r o a c h t o r e s i d u a l g e n e r a t i o n w a sa l so ge ne r a l i z e d t o t r a ns f e r f unc t i on mode l s , i nw h i c h c a s e t h e r e s i d u a l m a y d e p e n d o n t h e w h o l es e t o f p r i o r d a t a , a n d n o t o n l y o n t h e d a t a o v e ra f i x e d t i m e h o r i z o n ( M a s s o u m n i a a n d V a n d e rVelde , 1988) , (Ger t le r , 1993) , (Ger t le r , 1998)

3.3 F a u l t i s o l a t i o nA d i a gnos i s sys t e m shou l d no t on l y de t e c t f a u l t sb u t a l s o i s o l a t e t h e m , n a m e l y d e t e r m i n e w h i c hf a u l t ( s ) oc c ur r e d . To t h i s e nd , r e s i dua l s t ha t a r ese ns i t i ve t o c e r t a i n f a u l t s a nd i n se ns i t i ve t o o t he r sm u s t b e d e s i g n e d . A s s u m e f o r i n s t a n c e t h a t n if a u l t s ha ve to be i so l a t e d . I f n i r e s i dua l s c a n bede s i gne d i n suc h a w a y t ha t t he i th res idua l i s onlya f f e c t e d by t he i th f a u l t , t he n f a u l t i so l a t i on c a nbe a c h i e ve d e a s i l y . The c o r r e spond i ng s i t ua t i on i sde p i c t e d i n t a b l e 1 i n t he c a se o f t h r e e f a u l ts . A1 i nd i c a t e s t ha t t he f a u l t i n t he c o r r e sp ond i ngc o l um n a f f ec t s t he r e s i dua l o f t he c o r r e spond i ngr ow . Ea c h c o l umn o f t he t a b l e de f i ne s a c odea s soc i a t e d t o t he c o r r e spond i ng f a u l t . The s e t o ff a u l t c ode s f o r ms a c od i ng s e t .F o r t he c od i ng s e t o f t a b l e 1 t o be a c h i e va b le , t henum be r o f f a u l t s t o be iso l a t e d c a nn o t be l a r ge rt h a n t h e n u m b e r o f m e a s u r e d o u t p u t s , a s w i ll b ec l ea r f r o m t h e n e x t p a r a g r a p h . I f t h i s c o n d i t io n i sno t me t , i t mi gh t s t i l l be pos s i b l e t o pe r f o r m f a u l ti s o l a t i o n , b u t u n d e r t h e h y p o t h e s i s t h a t o n l y as i ng l e f a u l t c a n oc c ur a t a t i me . I n t h i s c a se , n Ir e s i dua l s m a y be de s i gne d so t ha t t he i *h r e s i dua li s a f fec ted by a l l f aul t s , except the i th one ( seet a b l e 2 ) . C l e a r l y , w he n s i mul t a ne ous f a u l t s oc c ur ,a ll r e s idua l s be c ome non- z e r o f o r th i s t y pe o f c od-i ng se t , w h i c h p r e v e n t s i so l a t i on 6 . M or e ge ne r a l ly ,i so l a ti on o f s ing l e f a u l t s c a n be a c h i e ve d by c hoos -i ng a c od i ng s e t w i t h no i de n t i c a l c ode s . Y e t , f o ra vo i d i ng e r r one ous i so l a t i on w he n t he r e a c t i on o fo n e r e s id u a l i s n o t a s e x p e c t e d u p o n o c c u r r e n c e o fa f a u l t ( due t o no i se , mode l unc e r t a i n t y o r t i meh i s t o r y o f t he f a u l t s i gna l ) , some c od i ng s e t s a r e

6 A n a l t e r n a t i v e a p p r o a c h t o h a n d l e m u l t i p l e fa u l t s m i g h tb e t o d e f i n e a d i f f e r e n t c o d e f o r e a c h f a u l t c o m b i n a t i o n b u tt h i s w o u l d l e a d t o l a r g e r e s i d u a l s e t s .

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Ta b l e 2 . Ef f e c ts o f t he f a u l t s on t h er e s i du a l s - i so l a t i on o f s i ng l e f a u l t s

A / 2 1 ' 3rl 0 1 1r2 1 0 1r3 1 1 0

pr e f e r a b l e t o o t he r s a s d i s c us se d i n ( G e r t l e r a ndS i nge r , 1990) , ( G e r t l e r a nd K unw e r , 1993) . Re s i d -ua l s w h i c h a r e de s i gne d a c c or d i ng t o a spe c i f i cc od i ng s e t a r e c a l l e d s t r uc t u r e d r e s i dua l s .T h e m e t h o d t o d e s i g n r e s i d u a l s a c c o r d i n g t o ag i ve n c od i ng s e t i s a s t r a i gh t f o r w a r d e x t e ns i on o ft he r e su l t s p r e se n t e d i n t he p r e v i ous s e c t i on . I n -de e d , f o r a g ive n r e s i dua l , t he f a u l t ve c t o r f ( t ) c a nb e s e p a r a t e d i n t o t w o p a r t s : f a ( t ) t h e f a u l t s t h a tshou l d a f f e c t t he r e s i dua l , a nd f n a ( t ) t he f a u l t st h a t s h o u l d n o t a f fe c t t h e r e s id u a l . T h e m e t h o d o ft h e p r e v i o u s s e c t io n c a n t h e n b e a p p l i e d t o d e s i g na n a p p r o p r i a t e r e s i d u a l b y s u b s t i t u t i n g r e s p e c -t ive ly f a ( t ) a n d [ d ( t) T f n a ( t ) T ] T f o r t he ve c t o r sf ( t ) a n d d ( t) i n s e c t i on 3 .2 . F r om t he e x i s t e nc ec ond i t i on s t a t e d a t t he e nd o f t ha t s e c t i on , i t i ss e en t h a t t h e d i m e n s i o n o f [ d ( t) T f n a ( t ) T ] T m u s tb e l o w er t h a n t h e n u m b e r o f o u t p u t s f o r a s o l u t i o nt o e x i s t .

4. O B S E R V E R B A S E D A P P R O A C H F O R L T IS Y S T E M S

4.1 R e f r e s h e r o n L u e n b e r g e r s t a t e o b s e r v e rL e t u s c o n s i d e r a L T I s y s t e m w i t h n o u n k n o w ni npu t a nd i n f a u l t l e s s ope r a t i on . I t i s de sc r i be d byt h e s t a t e s p a c e m o d e l

2 ( t ) = A x ( t ) + B u ( t ) (24)y ( t ) = C x ( t ) + D u ( t ) (25)

A s t a t e obse r ve r i s a f i l t e r t ha t r e c e i ve s a s i npu t st he s i gna ls u ( t ) a n d y ( t ) a n d g e n e r a t e s a n e s t i m a t eo f t h e s t a t e x ( t ) d e n o t e d 2 ( t ) . A L u e n b e r g e r s t a t eobse r ve r i s de sc r i be d by t he f o l l ow i ng d i f f e r e n t i a le qua t i on :

~ ( t ) = A ] c (t ) + B u ( t ) + L ( y ( t ) - C 2 ( t ) - D u ( t ) )= ( 2 6 )

w h e r e L i s a n x p c o n s t a n t m a t r i x o f w h i ch t h ee n t r i e s a r e c hose n i n suc h a w a y t ha t t he t r a ns i e n ti n t h e o b s e r v a t i o n e r r o r e z ( t ) = x ( t ) - ~ ( t ) de c a yssu f f ic i e n tl y f a s t t o z e r o . Th e dyn a m i c s o f t he e r r o ri s o b t a i n e d b y s u b t r a c t i n g ( 2 6) f r o m ( 24 ) a n ds u b s t i t u t i n g ( 2 5 ) f o r y ( t ) . I t y i e l ds

3 .4 I m p l e m e n t a t i o n o f p a r i t y c he c k sD ue t o t he p r e se nc e o f de r i va t i ve s i n ( 15 ) , r s ( t )c a n n o t b e g e n e r a t e d a s s u c h f r o m y ( t ) a n d u ( t ) .O n l y a f i l t er e d ve r s i on , s a y r , / ( t ) , o f th i s s i gna lc a n be ob t a i ne d . A s a n e xa mpl e , u s i ng La p l a c et r a n s f o r m e x p r es s io n s , r ~ ( s ) c a n b e g e n e r a t e d b y

r s ( s ) ( 23 )= + z ) .

w h e r e / 3 E I R+ i s c hose n i n suc h a w a y t ha t t heef fect of noise i s suf f ic ient ly f i l t e red 7 . I t has beens h o w n t h a t t h e s a m e s i g n a l r [ (t ) c a n b e g e n e r a t e du s i n g a n o b s e r v e r b a s e d a p p r o a c h ( M a g n i a n dMouyon , 1994) . A l t hough t h i s e qu i va l e nc e i s w e l le s t a b l i she d f o r l i ne a r sys t e ms , i t i s no t t he c a sef o r n o n l i n e a r s y s t em s . A s t h e p a r i t y s p a c e a n d t h eo b s e r v e r b a se d m e t h o d s w i ll b o t h b e p r e s e n t e d f o rt he de s i gn o f r e s i dua l ge ne r a t o r s f r om non l i ne a rmode l s , a n i n t r oduc t i on i s g i ve n t o obse r ve r ba se dr e s i dua l ge ne r a t i on f o r LTI sys t e ms f i r s t .

~ z ( t) = ( A - L C ) e x ( t ) (27)e ( O ) = z o -

W h e n t h e s y s t e m i s o b s e r v a b l e 8 , m a t r i x L c a n b ed e t e r m i n e d s o t h a t t h e e i g e n v a l u e s o f ( A - L C )t a ke f i xe d va l ue s . I mpos i ng t ha t a l l e i ge nva l ue sh a v e n e g a t i v e r e a l p a r t e n s u r e s t h a t e x ( t ) a s y m p -t o t i c a l l y de c a ys t o z e r o . F r om ( 25) , a n e s t i ma t e~ ( t ) o f t h e o u t p u t i s d e d u c e d a s

9 ( t) = C 2 ( t ) + D u ( t ) ( 2 s )

T h e o u t p u t e s t i m a t i o n e r r o re ~ ( t ) = y ( t ) - 9 ( t ) (29)

i s d i rec t ly l inked to e z ( t ) , s ince e u ( t ) - C e x ( t ) .

4.2 S t a t e o b s e r v e r f o r f a u l t d e t e c t i o n a n d i s o l a t i o nA L T I s y s t e m w i t h o u t u n k n o w n i n p u t i s c o n -s i d e r e d . I n f a u l t y w o r k i n g m o d e , t h e s y s t e m i sde sc r i be d by ( 3 ) , ( 4 ) w i t h E d = 0 a n d G d = O ,a n d t h e e r r o r e q u a t i o n s ( 2 7 ) , ( 2 9 ) b e c o m e

7 A bold character has been used here for the Laplaceoperator in order to avoid any confusion with the order ofthe derivatives and the order of the fi l ter.

8 A system is observable if the ou tput trajectories corre-sponding to distinct init ial states differ for at least sometime instant.

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~ x (t ) = ( A - L C ) e x ( t ) + ( E I - L G f ) f ( t ) ( 3 0 )( 0 ) = x o -

e u (t ) = C e x ( t ) 4 - G f f ( t ) (31)T he t r a n s f e r f unc t ion be tw e e n f a nd e u is ob-t a i n e d b y t a k i n g t h e L a p l a c e t r a n s f o r m o f ( 3 0 ) ,(31); i t yields:H i e , ( S ) = C ( s I - A + L C ) - I ( E I - L G I ) + G I .

A s l o n g a s n o n e o f t h e c o l u m n s o f H s e ~ ( s ) i s equa lto z e ro , a ny non z e r o c om pon e n t o f ] ( t ) w i l l a f fe c ta t l e a s t t e m p o r a r i l y e v ( t ) . T h u s a f t er t h e t r a n s i e n tdue t o i n i t i a l c ond i t i ons ha s va n i she d , e u ( t ) c a nbe use d to de t e c t t h e oc c ur r e nc e o f a f a u l t i nthe sys t e m . I nde e d , i n t he a bse nc e o f f a u l t % ( t )i s c lose t o z e r o a nd i t be c om e s d i s t i ngu i sha b lydi f fe rent f rom zero when f ( t ) devia tes s igni f icant lyf rom zero. e u ( t ) thus qua l i f ies as a res idua l s igna lan d (26) , (28) , (29) is the as soc ia ted res idu a lg e n e r a t o r . N o t i c e t h a t e u ( t ) n o w d e p e n d s o n t h ewho le t ime h is to ry of y( r ) , u(~ ') , for r E [0, t ] .T he r e a r e tw o a ppr oa c he s t o a c h i e ve f a u l t i so -l a t i on us ing L ue nbe r ge r obse r ve r s . T he f i r s t onea m o u n t s t o c h o o s i n g t h e o b s e r v e r g a i n m a t r i x Li n s u c h a w a y t h a t t h e t r a j e c t o r i e s i n d u c e d b ye a c h type o f f a u l t s a r e c onf ine d to i nde pe nde n tsubspa c e s o f t he e y - spa ce . By p r o j e c t i ng e ~ ( t )i n to e a c h o f t he se subsp a c e s a n d a n a lys ing them a g n i t u d e o f e a c h p r o j e c t i o n , o n e c a n a c h ie v ef a u l t i so l a t i on ( M a ssoum nia , 1986) , ( Whi t e a ndSpe ye r , 1987) . We sha l l no t go a long th i s l i nehe r e , a s t he se c ond a ppr oa c h , ba se d on a ba nkof observe rs , of fe rs mo re f lexibi li ty . We prese nt i ti n t he pa r t i c u l a r c a se o f s e nsor f a u l t s . T he g e ne r a lc a se i s m or e i nvo lve d a nd i t w i l l be de sc r ibe d inthe ne x t s e c t i on .T hu s c ons ide r a sys t e m of t he f o r m ( 3 ), ( 4 ) ,w i t h o u t u n k n o w n i n p u t , i n w h i c h o n l y s e n s o rf a u l t s a r e t a ke n in to a c c oun t

5 c(t) = A x ( t ) + B u ( t )y ( t ) = C x ( t ) + D u ( t ) + f ( t )

(32)(33)

T h e o u t p u t e q u a t i o n c a n b e w r i t t e n c o m p o n e n t -wise as

y i ( t ) = C i , . x ( t ) + D i , . u ( t ) + f i ( t )i = 1 , . . . , p ( 3 4 )

w he r e C i , . ( D i , . ) de no te s t he i th r o w o f m a t r i x C(D ) .I f ( C i , . , A ) , i = 1 , . . . , p a r e o b s e r v ab l e p a i rs , pL u e n b e r g e r s t a t e o b s e r v e rs c a n b e d e s ig n e d o n t h eba s i s o f e qu a t ion ( 32), ( 3 4 ) , one f o r e a c h ou tpu t ,a s de p i c t e d i n F ig . 2 . T he y ha ve t he f o r m ( 26)

w i t h y i ( t ) s u b s t i t u t e d f o r y ( t ) . C o m p u t i n g t h ee s t im a t ion e r r o r a s i n ( 30) , ( 31) , one de duc e s t ha tt h e i th O utpu t e s t im a t ion e r r o r i s on ly a f f e c t e d byth e i th f a u l t, he nc e t he poss ib i l i t y t o i so l a t e s e nsorf a u lt s b y a n a l y z i n g t h e p a t t e r n o f z e ro a n d n o n -z e r o r e s idua ls . T h i s t y pe o f d i a gnos i s sys t e m i sc a l l e d t he de d ic a t e d obse r ve r s c he m e ( F r a nk a ndW u n n e n b e r g , 1 9 8 9 ) .

U

J

J ,q

Y l

system ,

I Observer 1 e ~

Observer p e , ;

Fig . 2 . D e d ic a t e d obse r ve r s c he m eI f som e o f t he pa i r s ( C~ ,. , A ) a r e no t obse r va b le ,a n o t h e r o p t i o n i s t o d e s i g n p L u e n b e r g e r o b -servers , the i th o n e b e i n g b a s e d o n a l l m e a s u r e -m e n t s b u t t h e i eh o u t p u t y ~ ( t ) . I n t h i s w a y , uponoc c ur r e nc e o f a s i ng l e se nsor f a u l t , s a y on se nsorj , a l l r e s idua l s w i ll be c o m e non z e r o e xc e p t f o r thej t h one . T h i s i s t he ge ne r a l i z e d obse r ve r s c he m ew hic h a l l ow s one t o i so l a t e s i ng l e f a u l t s , bu t no ts im ul t a ne ous f a u l t s .T h e a b o v e t w o a p p r o a c h e s c o r r e s p o n d r e s p e c -t i ve ly t o c od ing se t s o f t he t ype de sc r ibe d in t a b l e1 a nd 2 . Bo th sc he m e s m a ke use o f f u l l o r de r( o r f u l l - d im e ns iona l ) s t a t e obse r ve r s . H ow e ve r , i ti s no t ge ne r a l l y ne c e ssa r y t o e s t im a te t he w holes t a t e ve c to r i n o r de r t o ob t a i n r e s idua l s i gna l s . Re -c o n s t r u c t i n g t h e w h o l e s t a t e i m p o s e s m o r e s t r i n -g e n t c o n d i t i o n s t h a n n e e d e d , a n d t h i s m o t i v a t e sthe use o f f unc t iona l obse r ve r s , n a m e ly obse r ve r stha t e s t im a te on ly g ( w i th g < n ) spe c i f i c l i ne a rc o m b i n a t i o n s o f t h e s t a t e v a r ia b l e s .

4 .3 F u n c t io n a l o b s e r v e r f o r F D II n t h i s se c t io n a n d i n th e r e m a i n i n g p a r t o f th epa pe r , i t w i l l be a s sum e d tha t t he f a u l t ve c to r f ( t )d o e s n o t e n t e r t h e o u t p u t e q u a t i o n . T h i s a m o u n t sto r e p r e se n t ing se nsor f a u l t s f s ( t ) = G f f ( t ) ( s e e( 4 ) ) by pse udo a c tua to r f a u l t s . I t c a n be a c h i e ve d

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by c ons i de r i ng f s ( t ) a s t h e o u t p u t o f a L T I s y s t e md r i v e n b y a n a p p r o p r i a t e i n p u t , a n d i m p o s i n gt h a t t h e L T I s y s t e m h a s n o d i r e c t f e e d t h r o u g ht e r m ( M a s s o u m n i a et a l . , 1989) . Th i s me t hod f o rha nd l i ng s e nso r f a u l t s i s c ons i de r e d i n a l l t hel i t e r a t u r e o n t h e g e o m e t r i c a p p r o a c h t o f a u l t d e -t e c t i on a nd i so l a t i on . I t i s a dop t e d he r e i n o r de r t op r e se n t a un i f i e d f r a me w or k e nc ompa ss i ng r e su l t so n F D I f o r n o n l i n e a r s y s t e m s , s o m e o f w h ic h a r ee xc l us i ve l y ba se d on ge ome t r i c sys t e m t he or y .A s w a s s e e n a t t he e nd o f s e c t i on 3 .3 , onc e c od i ngse t s a r e c hose n , t he de s i gn o f r e s i dua l ge ne r a t o r sf o r F D I a m o u n t s t o d e t e r m i n i n g a s e t o f f il te r st ha t so l ve spe c i f i c f a u l t de t e c t i on p r ob l e ms . Ea c hf i lt e r ( r e s i dua l ge ne r a t o r ) i s de s i gne d f o r a ppr opr i -a t e v e c t o r s o f u n k n o w n i n p u t s a n d f a u l t s. H e n c et h e f u n d a m e n t a l p r o b l e m t o b e s o lv e d is b a s e d o na m o d e l o f t h e f o r m

~ c(t) = A x ( t ) + B u ( t ) + E d d ( t ) + E s f ( t ) (35)y ( t ) = C x ( t ) (36)

Th e t y pe o f fi l te r t ha t w i l l be c ons i de r e d i n o r de rt o de s i gn a r e s i dua l ge ne r a t o r i s :

i c e( t) = A ~ x ~ ( t ) + B ~ u ( t ) + M ~ y ( t )r ( t ) = C ~ x r ( t ) + D ~ u ( t ) + N ~ y ( t )

(37)( 3 8 )

T h e p r o b l e m w h i c h h a s a l r e a d y b e e n c o n s i d e r e di n t h e s e c t i o n o n t h e p a r i t y s p a c e m e t h o d i s n o ws t a t e d i n a mor e f o r ma l w a y ; i t i s c a l l e d t hef u n d a m e n t a l p r o b l e m o f r e si d u a l g e n e r a ti o n .F u n d a m e n t a l p r o b l e m o f r e s id u a l g e n er a -t i o n ( F P R G )D e t e r mi n e a f i lt e r o f t he f o r m ( 37) , ( 38 ) suc h t ha t( 1 ) W h e n f ( t ) = 0 for al l t , r ( t ) a s y m p t o t i c a l l y

de c a ys t o z e r o f o r a ny i npu t s u ( t ) a n d d ( t ) .( 2 ) I n t he p r e se nc e o f a f a u lt ( na m e l y w he nf ( t ) ~ 0 for a l l t > to) , to be ing the faul t

oc c ur r e nc e t i me ) , r ( t ) i s a f f ec t e d by t he f a u l t ;i t t a ke s a non- z e r o va l ue f o r a t l e a s t somet > t o , w h a t e ve r t he i n i t i a l c ond i t i ons o f t hef i l t e r a nd t he p r oc e s s .

Whe n a so l u t i on e x i s t s , t he p r oc e dur e t o so l ve t heF P R G i s m a d e o f t w o s te p s . F i r s t o n e e x t r a c tsf r om mode l ( 35 ) , ( 36 ) a n obse r va b l e subsys t e mw h i c h h a s n o t d ( t ) a s i n p u t , b u t f o r w h i c h f ( t ) isa n i npu t . The s e c ond s t e p c ons i s t s i n de s i gn i nga s t a t e obse r ve r f o r t he obse r va b l e subsys t e m.T h e o u t p u t e s t i m a t i o n e r r o r o f t h e o b s e r v e r i sa su i t a b l e r e s i dua l . The r e a r e s e ve r a l w a ys t oa c h i e ve t he f i r s t s t e p . O ne o f t he m r e li e s on t hef a c t t ha t a ne c e s sa r y a nd su f f i c i e n t c ond i t i on f o rt h e e x i s t en c e o f a s o l u t io n t o t h e F P R G i s t h e

e x i st e n ce o f c o n s t a n t m a t r i c e s P , A , B , L 1 a n d L 2 ,w i t h P , LI a nd L2 d i f f e r e n t f r om z e r o , suc h t ha t

P A - A P = B C (39)P E a = 0 ( 4 0 )

L s C - L 2 P = 0 (41)P E I ~ 0 (42)a nd t he pa i r ( L2 , A ) i s obse r va b l e .

I nde e d , w he n suc h ma t r i c e s e x i s t , one c a n de t e r -m i n e t h e r e q u i r e d o b s e r v a b l e s u b s y s t e m a s f o l -lows. Le t ~( t ) = P x ( t ) b e t h e s t a t e o f t h is s u b -sys t e m , t he n ( 35 ) y i e l ds

= P A x + P B u + P E d d + P E f f (43)S u bs t i t u t i ng ( 39 ) f o r P A a nd ( 40 ) f o r P Ed , ( 43 )b e c o m e s

= 7i~ + B C x + P B u + P E l f (4 4)B y ( 3 6 ) t h e s e c o n d t e r m i n t h e r i g h t h a n d s i d e i sn o t h i n g b u t B y ( t ) s o t ha t ( 44 ) f i na ll y be c om e s

= fi.~ + [ Ty + P B u + P E l f (45)D e f i n i ng y ( t ) = L l y ( t ) , e q u a t i o n ( 4 1 ) t o g e t h e rw i t h t h e de f i n i t i on o f ~ ( t ) i m p l y

= L 1 C x = L2~ (46)N ow sys t e m ( 45) , ( 46 ) i s a subsys t e m i n w h i c hi n p u t d ( t ) d o e s n o t a p p e a r a n y m o r e . B e s i d e s ,( L2 , A ) i s obse r va b l e . A Lu e nb e r ge r obse r ve r f o rt h i s s y s t e m t a k e s t h e f o r m

= fi ~ + :B y + P B u + L ( ~ - L2~) (47)a n d t h e o u t p u t e s t i m a t i o n e r r o r c a n b e u s e d a s ar e s i dua l

r ( t ) = r / ( t) - L2~ ( t ) (48)I n d e e d , c o m p u t i n g t h e s t a t e e s t i m a t i o n e r r o r e~ -

- ~ yield s

~ (t ) = ( f i~ - L L 2 ) e ~ ( t ) + P E l f ( t )r ( t ) = L2e~( t )

(49)( 5 0 )

I t i s c l e a r f r o m t h e s e e q u a t i o n s t h a t n e i t h e r u ( t )n o r d ( t ) can a f fec t r ( t ) , w hi l e f ( t ) wil l do so gen-era l ly s ince P E I i s non- z e r o b y ( 42 ) a nd ( L2 , A -L L 2 ) i s observable . I f P E r h a s f u l l c o l u m n r a n k ,sys t e m ( 49) , ( 50 ) i s a c t ua l l y i npu t obse r va b l e

9 The time arg um ent is om itted in x, ~, u, y, d and f forthe sake of compactness.

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w .r . t , f ( t ) ( M a s s o u m n i a e t a l . , 1989) ; t h i s me a nst h a t t h e m a g n i t u d e f o f a s te p - l ik e f a u lt f ( t ) =f l ( t ) c a n b e d e t e r m i n e d u n i q u e l y f r o m r ( t ) , t ~ _ 0w he n e ~ ( 0 ) - - 01 . S ys t e m ( 47) i s a f unc t i ona lobse r ve r f o r P x ( t ) , he nc e t h e t i t l e o f t h i s s e c t i on .Th e s e t o f e qua t i ons ( 39 ) - (41 ) c a n be so l ve d e i-t h e r b y a m e t h o d i n v o l v i n g t h e t r a n s f o r m a t i o n o ft h e s t a t e e q u a t i o n s t o K r o n e c k e r c a n o n i c a l f o r m( F r a n k a n d W u n n e n b e r g , 1 9 8 9 ) o r b y a n i t e r a -t i v e a l g o r i t h m b a s e d o n 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 ( K i n n a e r t , 1 9 9 9 ) . A n o t h e r a p p r o a c h t os o l v e t h e F P R G , i n w h i c h t h e t w o s t e p p r o c e d u r ei n t r oduc e d he r e doe s no t a ppe a r e xp l i c i t l y i s t hem e t h o d b a s e d o n o b s e r v e r e i g e n s t r u c t u r e a s s i g n -m e n t ( C h e n a n d P a t t o n , 1 9 9 9 ) .Y e t a n o t h e r m e t h o d t o s o l v e t h i s p r o b l e m i st he ge ome t r i c a ppr oa c h . I t s i n t e r e s t i s t ha t i tc a n be ge ne r a l i z e d t o a w i de c l as s o f non l i ne a rsys t e ms . I t y i e l ds bo t h a ne c e s sa r y a nd su f f i -c i e n t c ond i t i on f o r t he e x i s t e nc e o f a so l u t i ont o t h e F P R G a n d a s y s t e m a t i c d e s i g n m e t h o d .Th e r e su l t s a r e e xpr e s se d i n t e r ms o f subspa c e sc a ll ed ( C , A ) - u n o b s e r v a b i l i t y s u b s p a c es . T h e d e -s i gn me t hod a l so e s se n t i a l l y c ons i s t s i n e x t r a c t -i n g a n o b s e r v a b l e s u b s y s t e m w h i c h h a s n o t d ( t )a s i npu t ; t he ope r a t i ons r e qu i r e d t o a c h i e ve t h i sg o a l c a n b e p e r f o r m e d u s i n g b a c k w a r d s t a b l e a l -g o r i t h m s d e d u c e d f r o m ( V a n D o o r e n , 1 9 8 1 ) a ss h o w n i n ( A l e x a n d r e a n d K i n n a e r t , 1 9 9 3 ) .C o n s i d e r i n g a r b i t r a r y m a t r i c e s A , C , E d , E I of a p -p r o p r i a t e d i m e n s i o n s , t h e F P R G h a s a s o l u t i o n i fa n d o n l y i f n ! ~ - n d < n a n d n d < p ( M a s s o u m n i ae t a l . , 1989) .I n t h e a b o v e r e s u l t s , i t i s a s s u m e d t h a t t h e p l a n tm o d e l i s p e r f e c t l y k n o w n a n d t h a t t h e m e a s u r e -me n t s a xe e xa c t i n f a u l t f r e e w or k i ng mode . I np r a c t i c e t h i s doe s no t ho l d a s d i s c us se d i n t he n e x tsec t ion.

5. D E A L I N G W I T H M O D E L L I N GU N C E R T A IN T I ES A N D M E A S U R E M E N TN O I S E

The l i ne a r t i me i nva r i a n t ( LTI ) mode l u se d i n t hep r e v i ous s e c t i ons i s su i t a b l e t o de sc r i be t he p l a n tbe ha v i our a r ound a spe c i f i c s e t po i n t . A s mos tp l a n t s a r e n o n l i n e a r , t h i s L T I m o d e l c o r r e s p o n d st o a f i r st o r d e r a p p r o x i m a t i o n o f t h e n o n l i n e a r it yi n a Ta y l o r s e r i e s e xpa ns i on . The f u r t he r a w a yt h e p r o c e s s m o v e s f r o m i t s n o m i n a l o p e r a t i n gp o i n t , t h e m o r e i m p o r t a n t t h e e f f e c t d u e t o t h eh i g h e r o r d e r t e r m s b e c o m e s . I f s u p e r v i s io n m u s t

b e p e r f o r m e d a r o u n d d i f fe r e n t se t p o i n t s , t h e r e a r ese ve r a l w a ys t o a c c oun t f o r t he non l i ne a r e f f e c t .F o r i n s t a nc e , one c a n use non l i ne a r mode l s a se xp l a i ne d i n s e c t i ons 6 a nd 7 , o r u se a ba nkof l i ne a r mode l s , e a c h one c o r r e sp ond i ng t o ad i f f e r e n t s e t po i n t ( A r t e e t a l . , 1995) .A r o u n d a g iv e n s e t p o i n t , m o d e l l i n g u n c e r t a i n t i e sd u e t o t h e l i n e a r a p p r o x i m a t i o n a n d / o r t o t h ee r r o r in t h e e s t i m a t e o f t h e m o d e l p a r a m e t e r s c a nbe a c c oun t e d f o r i n d i f f e r e n t w a ys . O ne c a n

use a non l i ne a r mode l s u se a l i ne a r mo de l s a nd r e p r e se n t t he h i ghe r

o r d e r t e r m s a s u n s t r u c t u r e d u n c e r t a i n t i e s( M a n g o u b i , 1 9 9 8 ) , ( M a n g o u b i a n d E d e l -ma ye r , 2000) , ( F r a nk a nd D i ng , 1994) ,

u s e u n k n o w n i n p u t s t o r e p r e s e n t t h e e f f e c to f m o d e l u n c e r t a i n t i e s ( P a t t o n a n d C h e n ,1 9 9 3 ) , ( C h e n a n d P a t t o n , 1 9 9 9 ) , ( S a u t e r a n dH a me l i n , 1999) , u se a s e t o f mod e l s , e a c h one c o r r e spo nd i ng t oa d i ff e r e n t pos s i b l e s e t t i ng o f t he pa r a m e t e r s( L o u e t a l . , 1986) , ( K i nna e r t , 1996) ,

u se a mo de l i n t he f o r m o f t r a ns f e r f unc t i onsw i t h p a r a m e t e r s i n i n t e r v a l s ( H a m e l i n a n dS a u t e r , 2000)

T h e e x i st e n c e o f m e a s u r e m e n t n o is e , a n d p o s s i b l ys t a t e n o i s e t o a c c o u n t f o r d i s t u r b a n c e s w i t h ak n o w n s p e c t r u m a ls o r e q u i r es a d a p t a t i o n o f t h et h e o r y p r e s e n t e d i n t h e p r e v i o u s s e c t i o n . I n d e e d ,t he r e s i dua l c a n no l onge r de c a y t o z e r o i n t hep r e se nc e o f noi se; t hus a d i f f e r e n t ob j e c t i ve ha st o be i mpose d i n i t s de s i gn ( N i koukha h , 1994) ,( M a ngoub i , 1998) . Be s i de s , s i mpl e c omp a r i son o ft h e r e s i d u a l t o a t h r e s h o l d b e c o m e s i m p r a c t i c a -b l e w h e n no i se i s i m po r t a n t , a s t he e f f e ct o f af a u l t m a y b e b u r i e d i n n o i s e . O n e h a s t o r e s o r tt o s t a t i s t i c a l c h a n g e d e t e c t i o n a l g o r i t h m s i n t h i sc a se ( Wi l l sky a nd Jo ne s , 1976) , ( Ba s se v il l e a ndN i k if o ro v , 1 9 9 3 ). T h e p r o b l e m o f r o b u s t n e s s w i t hr e spe c t t o mo de l l i ng unc e r t a i n t i e s a n d no i se e f-f e c t s i s t hus a ve r y b r oa d a r e a .H e r e w e ha ve c hose n t o e xp l o i t non l i ne a r mode l si n o r de r t o de c r e a se t he e f fe c t o f mo de l l i ng un -c e r t a i n t ie s . The r e f o r e , t h e e x t e ns i on o f t he t he o r ypr e se n t e d f o r l i ne a r sys t e ms w i l l suc c e s s i ve l y bep e r f o r m e d f o r p o l y n o m i a l n o n l i n e a r s y s t e m s ( i nt h e p a r i t y s p a c e f r a m e w o r k ) , f o r b il i n ea r s y s t e m sa nd f o r c on t r o l a f f i ne non l i ne a r sys t e ms ( us i ng a no b s e r v e r b a s e d a p p r o a c h ) .

6. P A R I T Y S P A C E A P P R O A C H F O RP O L Y N O M I A L N O N L I N E A R S Y ST E M S

1The on ly type of fault that may not affect r(t) corre-sponds to inputs associated to the transmission zeros ofthe transfer matrix between f and r in this case.The p r e se n t a t i on i n t h i s s e c t i on i s i n l i ne w i t h( I s idor i e t al . , 2001) . A mod e l o f t he f o r m ( 1 ) ,( 2 ) i s c ons i de r e d , w he r e g ( x ( t ) , u ( t ) , d ( t ) , f ( t ) ) a n d

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h ( x ( t ) , u ( t ) , d ( t ) , f ( t ) ) a r e p o ly n o mia l f u n c t io n s o fth e i r a r g u men t s . Le t y j d en o te th e j t h o u t p u t :y j = h j ( x ( t ) , u ( t ) , d ( t ) , f ( t ) ) . B y d e r iv in g th i sex p r ess io n s j t imes an d su b s t i tu t in g ( 1 ) f o r t h ed e r iv a t iv e o f t h e s t a t e , o n e g e t s eq u a t io n s eq u iv -a len t to (8) - (11) in the l inear case (a s l igh t gen-era l iza t ion is considered here s ince the order ofth e d e r iv a t iv e mig h t d i f f e r f r o m o n e o u tp u t t oth e o th e r , wh ich was n o t t h e case in th e l i n ea rf r amewo r k ) . Th ese eq u a t io n s can b e wr i t t en in aco mp ac t f o r m as :

Y ~ = H i ( x , Us ~ , Ds j , F s~ ) (51)

Y] j i s the vector made of y j ( t ) and i t s der ivat ivesu p to o r d e r s j , Us~ , Ds~ an d Fs j - a r e d e f in ed a sin th e l i n ear ca se. Th e co n ca ten a t io n o f eq u a t io n s(51) for j 1 , p gives a set of P= . . . , E ~ = l ( S ~ + 1)eq u a t io n s

Y = H ( x , U s , D s , F s ) (52)

whe re s = m axj s j and Y = [y(sl )T y(s 2)T . . , y (sp)T] T .An aly t i ca l r ed u n d an cy r e l a t io n s a r e o b ta in ed b ye l imin a t in g x an d Ds f r o m ( 5 2 ) . Th i s y i e ld s p o ly -nomial re la t ions:

P ( Y , U s , F s ) = 0 (53)Du e to th e p o ly n o mia l s t r u c tu r e , t h i s ex p r ess io ncan b e d eco mp o sed a s

P ( Y , Us, F, ) = Pc(Y, Us ) - P c ( Y , Us , F s ) = 0(54)

wh er e an y t e r m in th e p o ly n o mia l s wh ich mak eth e r o ws o f P c( Y , Us , F s ) i s o f degree a t least onewi th res pect to a n en try of Fs . Pc (Y, Us, Fs) i s thusequal to zero in fau l t less work ing mode. HenceP c ( Y , U s ) can b e co n s id e r ed a s a p a r i ty v ec to r ,and one can def ine a res idual vector as

r ( t ) = P c ( Y ( t ) , U s ( t ) ) ( 5 5 )To achieve fau l t i so la t ion , s t ructured res idualsh av e to b e co mp u ted . Th i s amo u n t s to ch o o s in gco d in g se t s an d p e r f o r min g th e ab o v e d es ig n wi tha v e c to r o f u n k n o w n i n p u t s a u g m e n t e d b y t h efau l ts tha t should not af fec t the res idual .Th e r e a r e sev e r a l way s to p e r f o r m th e e l imin a t io no f x an d D , th a t l ead s to ( 5 3 ). On e ap p r o ac h r e -l ies on Groe bne r basis (Com tet-V arga , 1997) ; an-o th e r u ses ch a r ac te r i s t i c se t s an d R i t t ' s a lg o r i th m( Lju n g an d Glad , 1 9 9 4) . A s tu d y o f ex i s t en cecondi t ions for par i ty funct ion s an d of fau l t sensi-t i v i ty i s p e r f o r med in ( S ta r o swieck i an d C o mte t -Varga , 2001) ;

F in a l ly , l e t u s n o ti ce th a t t h e p r o b lem o f g en e r -a t i n g p o l y n o m i a l a n a l y t i c a l r e d u n d a n c y r e l a t i o n scan actual ly be so lved for a more genera l c lasso f n o n l in ea r sy s t ems , n am e ly sy s t em s d esc r ib edb y p o ly n o mia l d i f f e ren t i a l a lg eb r a i c eq u a t io n s . Anin f o rma l p r e sen ta t io n o f th e m e th o d b ased o nch a r ac te r i s t i c se t s f o r t h i s t y p e o f mo d e l i s p r e -sen ted in ( Zh an g et al . , 1998)

7. O B S E R V E R B A S E D A P P R O A C H F O RN O N L I N E A R S Y S TE M S

B ef o r e d ea l in g wi th f au l t d e t ec t io n , so me p r e r eq -u i s i t e s o n o b se r v ab i l i t y f o r n o n l in ea r sy s t ems a r ein t r o d u ced .

7 . 1 0 b s e r v a b i l i t y o f n o n l in e a r s y s t e m sC o n s id e r th e n o n l in ea r sy s t em

= u ) ( 5 6 )y - h ( x ) (57)

wh er e g an d h a r e smo o th n o n l in ea r f u n c t io n so f th e i r a r g u m en t s . Le t U d en o te th e se t o f ad -missib le inputs . By def in i t ion system (56) , (57)is observable i f , fo r every pai r o f in i t ia l s ta tes(x , x l ) , x ~ x 1 , there ex is ts an a dmissib le inpu ts ignal u : [0 , T] ~ U an d a t im e inst an t t e [0 , T]s u c h t h a t y( x , u , t ) =/: y ( x 1 , u , t ) w h e r e y ( x , u , t )d en o tes th e u n iq u e o u tp u t t r a j ec to r y o f ( 5 6 ) , ( 5 7 )f o r th e in i t i a l co n d i t io n x an d th e in p u t u . Aninpu t u : [0 , T] ~ U tha t d is t ing uishes any pai r o fin i t ia l s ta tes i s a un iversa l in pu t on [0, T] . Sy stem(56) , (57) i s sa id to be un iformly observable i f , fo revery T > 0 , every adm issib le i npu t u : [0 , T] - - . Uis a un iversa l in put on [0 , T] .B i l in ea r sy s t ems a r e n o t g en e r a l ly u n i f o r mly o b -sevable . Indeed , consider the fo l lowing s imple ex-amp le :

= u x 2 ( 5 8 )= 0 ( 5 9 )

y = X l (60)Clearly, if u ( t ) - 0 for a l l t , no in format ion onx 2 ( t ) c a n b e d e d u c e d f r o m t h e m e a s u r e m e n t o fy ( t ) an d h en ce th i s s t a t e i s n o t o b se r v ab le . An yp iecewise - co n t in u o u s n o n - ze r o in p u t wi l l mak e x 2o b se r v ab le . An in p u t f o r wh ich th e sy s t em i s n o tobservable i s ca l led a s ingular input . A speci f icc l ass o f n o n s in g u la r i n p u t s wi ll b e u sed to s t a t ea convergence resu l t fo r b i l inear systems in then ex t sec t io n , n ame ly so - ca l l ed p e r s i s t en t ly ex c i t -in g in p u t s ( Kin n a e r t , 1 9 9 9) . R o u g h ly sp eak in g

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s u c h i n p u t s g u a r a n t e e t h e o b s e r v a b i l i t y o f t h es y s t e m o v e r s o m e h o r i z o n T a t a n y t i m e .

7 .2 F a u l t d e t e c t i o n f o r b i l i n e a r s y s t e m sC o n t i n u o u s - t i m e b i li n e a r s y s t e m s d e s c r i b e d b y t h ef o ll o w i n g e q u a t i o n a r e c o n s i d e r e d

nd~ ( t) = A ( u ) x ( t ) + B u ( t ) + E ( F ~ x ( t ) + E d ) d i (t )

i--1nI

+ E ( F / x ( t ) + E { ) f , ( t ) (61)i=1

y ( t ) = C x ( t ) (62)w h e r e A(u) = Ao + ~-]im__lu i A i a n d i t i s a s s u m e dt h a t t h e i n p u t v e c t o r s u ( t ) , d ( t ) a n d f ( t ) a r e s u c ht h a t e v e r y t ra j e c t o r y i s d e f in e d o n t h e w h o l e ti m ein te rval [0, oo[ .I n t h i s c a s e a g a i n , t h e d e s i g n o f a r e s i d u a l g e n -e r a t o r c a n b e s e p a r a t e d i n t w o s te p s . F i r s t a no b s e r va b l e s u b s y s t e m w h i c h h a s n o t d ( t ) a s i n p u t ,b u t f o r w h i c h f ( t ) i s a n i n p u t i s e x t r a c t e d . N e x ta s u i t a b l e o b s e r v e r i s d e s i g n e d f o r t h e r e s u l t i n gs u b s y s t e m .T w o t y p e s o f s u b s y s t e m s h a v e b e e n u s e d int h e l i t e r a t u r e : l i n e a r t i m e i n v a r i a n t s u b s y s t e m su p t o o u t p u t i n j e c t i o n ( Y u a n d S h i e l d s , 1 9 9 6 ) ,( M e c h m e c h e et al . , 1 9 9 4 ) , an d b i l i n ea r s u b s y s -t e m s u p t o o u t p u t i n j e c ti o n ( K i n n a e r t , 1 9 9 9 ),( H a m m o u r i et al . , 2 0 0 1 ) . T h e a p p r o a c h b a s e d o na L T I s u b s y s t e m l e a d s t o a L T I r e s i d u a l g e n e r a -t o r , w h i c h c a n b e i m p l e m e n t e d e a s i l y . H o w e v e r ,t h e c l a s s o f b i l i n e a r s y s t e m s f o r w h i c h a r e s i d -u a l g e n e r a t o r b a s e d o n a L T I s u b s y s t e m e x i s t si s s m a l l e r t h a n t h e c l a s s o f s y s t e m s f o r w h i c h ar e s i d u a l g e n e r a t o r b a s e d o n a b i l i n e a r s u b s y s t e me x i s t s ( K i n n a e r t , 1 9 9 9 ). T h e l a t t e r o p t i o n i s b r i ef l yp r e s e n t e d h e r e .W i t h o u t l os s o f g e n e r al i ty , t h e n ~ - d i m e n s i o n a ls t a t e o f t h i s s u b s y s t e m c a n b e t a k e n a s a s et o fl i n e a r c o m b i n a t i o n s o f t h e e n t r i e s o f x ( t )

~ B ( t) = p B x ( t ) (63)

T h e e x t r a c t i o n o f t h e b i l in e a r s u b s y s t e m c o n s is t si n d e t e r m i n i n g n ~ , t h e m a t r i c e s p B , A i , B i , i -0 , . . . , m , L B a n d L B w i t h p B , L B an d L B d i f f er-e n t f r o m z e r o s u c h t h a t

p B A i - - i i P B = B i C i = 0 , . . . , m (64)p s ( E d F i ~ ) = 0 i = 1 , . . . , n d (65)

L B c - L B p s = 0 (66)p B ( E [ F [ ) # 0 i = 1 , . . . , n I ( 67 )

a n d t h e s y s t e m

= ( 6 8 )r l ( t ) = L 2 ( t) ( 6 9 )

i s o b s e r v ab l e .F r o m ( 6 4 ) - ( 6 7 ) , o n e d e d u ce s

~ B ( t) = i ( u ) ~ B ( t) + p S B u ( t ) + [ ~ ( u ) y (t )n l

+ E p B ( F [ x ( t ) + E I ) f ( t ) (70)i=1

w h e r e 7 t ( u ) = A o + E i m = l A i u i a n d s i m i l a r l y f o r

D e f i n i n g r iB( t ) b y ~ B ( t ) = L B y ( t ) = L B 1 C x ( t ) ,o n e g e t s

~ B ( t) - - L B ~ B ( t ) (71)A s t h e d y n a m i c s o f ( 7 0 ) i s b i l i n e a r , o b s e r v e r sf o r b i l i n e a r s y s t e m s n o w h a v e t o b e c o n s i d e r e d .C o n t r a r y t o t h e l i n e a r c a se , o b s e r v a b i l i t y o f s y s -t e m ( 7 0 ) , ( 7 1 ) d e p e n d s o n t h e i n p u t u ( t ) a s w a sd i s c u s s e d i n s u b s e c t i o n 7 .1 . L e t u ( t ) b e a r e g u l a r l yp e r s i s t e n t i n p u t f o r s y s t e m ( 6 8 ), ( 6 9 ) . F o r s u c h a ni n p u t , i t c a n b e p r o v e d t h a t t h e f o l l o w i n g s y s t e m i sa n e x p o n e n t i a l o b s e r v e r 11 ( B o r n a r d et al . , 1988)f o r s y s t e m ( 7 0 ), ( 7 1 ) w h e n f ( t ) = 0 for al l t ::,B ( t) = 7 t( u )~ B ( t) + p B B u ( t ) + B ( u ) y ( t )

+ R ( t ) - I C T ( ~B ( t) -- L B ~ B ( t) ) (72)_ _ r B T r BR ( t ) = O R ( t ) 7 t (u ) T R ( t) R ( t ) f i . ( u ) + - 2 ~ 2

(73)where ~B (0) E ]Rn~ , R(0) is a nf x nf symmetricpositive definite matrix, and/9 E ]R+. The outputreconstruction error

r(t) = ~B (t) -- LB~ B (t)e x p o n e n t i a l l y d e c a y s t o z e r o w h a t e v e r d ( t ) , ~ ( O ) ,~ s ( 0 ) a n d u ( t ) p r o v i d e d t h e l a t t e r i s r e g u l a r l yp e r s i s t e n t . T h e r e a d e r i s r e f e r re d t o ( K i n n a e r t ,1 9 9 9 ) f o r a d i s c u s s i o n r e g a r d i n g t h e s e n s i t i v i t y o fr ( t ) t o t h e f a u l t f ( t ) .U s i n g a n a p p r o p r i a t e d e f i n i t i o n o f t h e s e n s i t i v i t yo f a re s i d u a l t o a f a u l t , a p r e c i se s t a t e m e n t o ft h e f u n d a m e n t a l p r o b l e m o f r e s i d u a l g e n e r a t i o nf o r b i l i n e a r s y s t e m s c a n b e o b t a i n e d . A n e c e s-s a r y a n d s u f f i c i e n t c o n d i t i o n f o r t h e e x i s t e n c e

l lTh is means tha t l im B (t)- ~B(t) l I O, wh ere A is a positive cons tant wh ich depends o n theinitial conditions ~ B(o), ~B (0), R(O), on th e upper boun dof Ilu(t)ll and on 0.

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of a so lu t i on t o t h i s p r ob le m c a n the n be e x -p r e s se d i n a ve r y s im i l a r f o r m a s f o r t he l i ne a rc a s e ( H a m m o u r i et a l . , 2 0 0 1 ) b y i n t r o d u c i n g t h eno t ion o f ( C , 4) - unobse r va b i l i t y subspa c e s w he r e, 4 s t a n d s f o r ( A 0 , . . - , A m ) . A n a l g o r i t h m t o c h e cks u c h a g e o m e t r i c c o n d i t i o n c a n b e d e t e r m i n e dus ing s t a nda r d a lge br a i c t oo l s .A s i m i la r m e t h o d o l o g y c a n b e u s e d t o h a n d l e s t a t ea ff ine sys t e m s , na m e ly sy s t e m s o f t he f o r m :

J

~ ( t ) = A ( u ) x ( t ) + E d ( x ) d ( t ) + E l ( x ) : f ( t ) ( 7 4 )y ( t ) = C ~ ( t ) ( 75 )

w h e r e A ( u ) , E d ( x ) a n d E i ( x ) a re m a t r i c e s o fw hic h the e n t r i e s a r e sm o oth f unc t ions o f t hec o m p o n e n t s o f u a n d x . T h e r e q u i r e d c o n c e p t sh a v e b e e n d e v e l o p e d i n ( H a m m o u r i et a l . , 1998),( H a m m o u r i et a l . , 2000) , (De Pers is and I s idor i ,2000) . T he m a in d i f f e r e nc e w i th t he b i l i ne a r c a sei s t h a t s y m b o l i c c o m p u t a t i o n s m u s t b e p e r f o r m e dt o i m p l e m e n t t h e d e s i g n m e t h o d .

7.3 Fa u l t d e t ec t io n fo r co n tro l a ~ n e n o n l in ea rs y s t e mSys t e m s m ode l l e d by e qua t ion s o f t he f o r m ( 76) ,( 77) be low a r e now c ons ide r e d

= g o (x ) + g u ( x ) u + g d ( x )d + g f ( x ) f (76)y = h ( x ) (77)

w i th t he n - d im e ns iona l s t a t e x de f ine d in a ne igh-b o r h o o d X o f t h e o r i g in , i n p u t s u , d a n d o u t -p u t y w i t h t h e s a m e d i m e n s i o n a s b e f o r e , a n dsc a l a r f a u l t f . A l l t he e n t r i e s i n t he ve c to r sg o ( x ) , g l ( x ) , h ( x ) a n d m a t r i c e s gu(x ) , gd(x) ares m o o t h f u n c t i o n s o f t h e i r a r g u m e n t s a n d g 0 (0 ) =0 , h ( 0 ) = 0 . T he l a s t tw o c ond i t i ons c a n a lw a ysb e a c h i ev e d b y a n a p p r o p r i a t e t r a n s l a t i o n o f t h es t a t e a n d o u t p u t .T h e p r o b l e m t o b e s o l v e d i s t h e d e t e r m i n a t i o n o fa f i l te r of the form

i s a f f e c t e d by f , i s no t a f f e c t e d by d , a nd a sym p-to t i c a l l y de c a ys t o z e ro w he n f i s i de n t i c a l l y e qua lto z e r o , w ha te ve r t he i npu t u .By r e so r t i ng t o d i f f e r e n t i a l ge om e t r y , a nd in pa r -t i c u la r t o t h e n o t i o n o f u n o b s e r v a b i l i t y d i s t r ib u -t i on ( t he ge ne r a l i z a t i on t o t he non l ine a r f r a m e -w or k o f (C , A ) - un obse r va b i l i t y su bspa c e s ) , a ne c-e s sa r y c ond i t i on f o r t he e x i s t e nc e o f a so lu t i ont o t h i s p r o b l e m c a n b e o b t a i n e d ( D e P e r s i s a n dI s idor i , 2001) . A lgor i t hm s a r e a va i l a b l e t o c he c kt h i s c o n d it i o n , a n d a m e t h o d o l o g y c a n b e d e d u c e dto e x t r a c t f r om the o r ig ina l sys t e m ( 76) , ( 77) aloc a l ly w e a k ly ob se r va b le 12 sub sys t e m of t he f o l-l ow ing f o r m

= g 0 ~( ~ , y ) + g ~ ( ~ , y ) ~ + g ~ ( ~ , y , t ) Iq = h~(~)

( s 2 )(83)

w h e r e t h e i n p u t d d o e s n o t a p p e a r a n y m o r e .Shou ld a n a sym pto t i c obse r ve r e x i s t f o r t h i s sys -t e m , one c ou ld p r oc e e d a s f o r t he l i ne a r o r t heb i l i ne a r ca se t o ob t a in a r e s idua l ge n e r a to r . U n f or -tuna t e ly , l oc a l w e a k obse r va b i l i t y i s no t su f f i c i e n tto gua r a n t e e t he e x i s t e nc e o f suc h a n obse r ve r ,a n d e x t r a h y p o t h e s e s a r e n e e d e d a t t h i s s t a g e .For i ns t a nc e , l e t us a s sum e tha t sys t e m ( 82) , ( 83)w i th f = 0 , c a n be b r ough t by a g loba l c ha nge o fcoo rdin a tes v = ( I) (~) in to th e form

~)1 : 1 (V l , V2, y, u)V2 --- (~2(Vl, V2, V3, Y, U)o o

o

~ = ~ ( v l , v 2 , - . . , v ~ , y , u )q = h V ( v ) .

T h e n t h i s s y s t e m i s u n i f o r m l y o b s e r v a b l e , a n d ah i g h g a i n o b s e rv e r c a n b e d e s i g n e d t o e s t i m a t e t h es t a t e ~ a n d t h e o u t p u t q , u n d e r s o m e a d d i t i o n a lt e c hn ic a l c ond i t i ons ( D e Pe r s i s a nd I s idor i , 2001) .T he r e s idua l is a ga in ob t a ine d a s r = q - h~ (~ )w h e r e ~ is t h e e s t i m a t e p r o v i d e d b y t h e o b s e rv e r .

= g g ( z , y ) + g ~ ( z , y ) ~ ( T s )r - h r ( z , y ) ( 7 9 )

w i t h a s i m i l a r s m o o t h n e s s c o n d i t i o n o n t h e d i f -f e r e n t f unc t ions a s a bove , a nd w i th g~(0 , 0 ) = 0 ,h r ( 0 , 0 ) - 0 , suc h tha t t he o u tp u t r o f t he c a s-c a d e d s y s t e m

( ~ g 0 ( ~ ) g ~ , (~ , h ( ~ ) ) ) u~ = h~ ( z , h ( ~ ) ) ( S ~ )

8 . C O N C L U S I O N

T h e p a r i t y s p a c e a p p r o a c h a n d t h e o b s e r v e r b a se dm e t h o d h a v e b e e n p r e s e n t e d f o r t h e d e s i g n o fr e s idua l ge ne r a to r s , f i rs t on t he ba s i s o f a L T Im ode l , ne x t f r om a non l ine a r p l a n t m ode l . A l -t h o u g h b o t h a p p r o a c h e s a r e e q u i v a l e n t f o r L T I

1 2 R o u g h l y s p e a k i n g a s y s t e m i s w e a k l y o b s e r v a b l e i f a n ys t a t e can be d i s t i ngu i shed f rom i t s ne ighbor s , and i t i sl oca l ly obse rvab le i f a s t a t e can be d i s t i ngu i shed f rom i t sn e i g h b o r s w i t h a n i n p u t t h a t k e e p s t h e s t a t e t r a j e c t o r yc l o se to t h e i n i ti a l s t a t e ( H e r m a n n a n d K r e n e r , 1 9 7 7 ).

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sys tems, th is is not proved for nonl inear sys tems,in genera l .T he pa r i ty spa c e a pproa c h c a n be a pp l ie d to po ly -nomia l non l ine a r sys te ms . I t i s sys te ma t ic , a nde x is t ing symbol ic c omputa t ion so f twa re s c a n beuse d to pe r fo rm the e l im ina t ion o f the unk now nvar iables . However , even for models of modera tes ize (5 to 10 s ta tes ) , the par i ty func t ion resul t ingf rom the e l im ina t ion migh t c on ta in s e ve ra l t e nsof t e rms . T he se t e rms migh t ha ve ve ry d i f f e r -e n t o rde r s o f ma gn i tude , whic h c ou ld ma ke thee va lua t ion o f the r e s idua l i l l - c ond it ione d. An o the rd ra wb a c k o f the pa r i ty spa c e a pproa c h i s the ne e dfor the der iva t ives of the s igna ls . Such der iva t ivesare sens i t ive to noise . In prac t ice they have to beob ta ine d mos t o f te n f rom sa mple d da ta . Ob ta in -ing su i ta b le e s t ima te s fo r h igh o rde r de r iva t ive s inth i s c on te x t m igh t no t be a n e a sy ta sk .On the o the r ha nd , fo r the obse rve r ba se d a p-p roa c h , sys te ma t ic me th ods r e ly ing e xc lus ively ona lgebra ic opera t ions exis t for bi l inear sys tems asseen in sec t ion 7 .2 , and for sys tems with poly-nomia l nonl inear i t ies of higher order ( Is idor i e tal., 2001) . In the la t te r case however , qui te s t r in-ge n t c ond i t ions a re impose d on the sys te m tobe a b le to de s ign a n obse rve r w i th l ine a r e r ro rdyna mic s . For s ta te a nd c on t ro l a f f ine sys te ms ,the obse rve r ba se d a pproa c h re qu i re s symbol icc omputa t ions to e x t ra c t f rom the o r ig ina l sys te ma subsy s te m w hic h i s no t a f f ec te d by c e r ta in f a u l t s .Onc e the subsys te m i s found , the de s ign o f theobserver is sys tematic in the s ta te a f f ine case , butnot for control a f f ine sys tems.Only pe r fe c t de c oup l ing o f the r e s idua l f rom thefa u l t s no t to be de te c te d ha s be e n c ons ide re dhe re. T h is r e qu i re me nt c a n be r e la xe d by impos ingtha t the e f fe ct on the r e s idua l , o f the f a u l t s no tto be de tec ted be suff ic ient ly small . This yie ldsop t imiz a t ion p rob le ms tha t ha ve be e n c ons ide re dbo t h fo r line a r (F ra nk a nd Ding , 1997), (Che n a ndPa t ton , 1999) a nd non l ine a r mode ls (Be sa nqon ,1999) . Nonlinear adapt ive observers have a lsobe e n use d to e s t ima te f a u l t s igna l s ins te a d o fa s su r ing f a u l t de c oup l ing (Xu a nd Z ha ng , 2002) .T he non l ine a r mode ls c ons ide re d in th i s pa pe r a reof te n de duc e d f rom phys ic a l l aws . For c e r ta in p ro -cesses , phys ica l descr ipt ions might be too involvedor poor ly known, a n a b la c k box mode l l ike ane ura l ne twork migh t be a su i ta b le a l t e rna t ive .For e xa mple s on how to use ne ura l ne tworks in thede s ign o f f a u l t de te c t ion a nd i so lat ion sys te ms there a de r i s r e fe r re d to (Z ha ng et a l . , 2002) , (Chena nd Pa t ton , 1999) .Fa u l t de te c t ion a nd i so la t ion sys te ms ba se d onana lyt ica l models have been used for a var i-e ty o f a pp l ic a t ions inc lud ing a e rona u t ic s (De c ke r t

e t a l . , 1977) , a u tomobi le (Ge r t l e r et al., 1995),( I s e rma nn et al., 2000) , (Nyberg, 1999) , d iese le ng ine a c tua to r (Bla nke et al., 1995), process in-dus t r i e s (K inna e r t et al., 2000), . . .T he de s ign o f d ia gnos i s sys te ms ba se d on a na ly t i -c a l mode ls i s now re a c h ing a c e r ta in s ta g e o f ma tu -r i ty . However , severa l research direc t ions a re s t i l lw ide ly ope n . One o f the m i s the e v a lua t ion o f there l i ab i l i ty o f suc h d ia gnos i s sys te ms a nd the w a yto t a ke in to a c c oun t r e l i a b i l i ty r e qu i re me nts a t a ne a r ly s ta ge in the de s ign . Ano the r on e is the s tudyof the in te rp la y be tw e e n the d ia gnos i s sys te m a ndthe c on t ro l le r in a f a u l t to le ra n t c on t ro l s c he me .

9 . R E F E R E N C E SAle xa ndre , P . a nd M. Kinna e r t (1993) . Nume r -

ic a l ly r e l i a b le a lgor i thm fo r the syn the s i s o fl inear faul t de tec t ion and isola t ion f i l te rsba se d on the ge ome t r ic a pproa c h . In : P ro -ceed in g s o f t h e 1 9 9 3 IEE E C o n feren ce o n S ys -t ems , M a n a n d C yb ern e t i c s . Vol. 5. pp. 359-364.

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