Wp-6- 1 DETERM IN ISTDC CONTROL OF U NCERTAI-N-SJSTEES A.S.I. Zinober

Department of A plied and Computational Mathematics fIniversitv of Sheffield

Sheffield S I O 2TN, U.K.

Abstract

I n Lhe d e t e r m l n i s t l c c o n t r o l of u n c e r t a i n m u l t i v a r i a b l e s y s t e m s r o b u s t c l o s e d - l o o p c o n t r o l i s a c h i e v e d for a s p e c i f i e d m a g n i t u d e r a n g e o f a c l a s s o f p a r a m e t e r v a r i a t i o n s a n d d j s t u r b a n c e s . T h e s e n o n l i n e a r t i m e - v a r y i n g s y s t e m s employ n o n l j n e a r c o n t r o l f u n c t i o n s . T h e two majn a p p r o a c h e s are Lyapunov c o n t r o l a n d v a r l a b l e s t r u c L u r e c o n t r o l . T h e d e v e l o p m e n t of t h e t h e o r y of t h i s l a t t e r class of c o n t r o l s y s t e m s w i l l b e b r i e f l y r e v i e w e d h e r e a n d p a r t i c u l a r r e f e r e n c e w i l l b e made t o t h e e i g e n s t r u c t u r e a s s i g n m e n t a p p r o a c h . T h e u s e of a CAD p a c k a g e VASSYD allows for s t r a i g h t f o r w a r d d e s i g n and s i m u l a t i o n s t u d i e s . D e s i r a b l e r o b u s t n e s s a n d i n v a r j a n c e p r o p e r t i e s are r e a d i l y o b t a j n e d .

I n t r o d u c t i o n

T h e d e t e r m i n i s t i c c o n t r o l of u n c e r t a i n t ime-vary j n g s y s t e m s i s a c h i e v e d u s i n g n o n l i n e a r f e e d b a c k c o n t r o l f u n c t i o n s , w h i c h o p e r a t e e f f e c t i v e l y o v e r a s p e c i f i e d m a g n i t u d e r a n g e of a class of s y s t e m p a r a m e t e r v a r i a t i o n s , w i t h o u t t h e n e e d for o n - l i n e i d e n t i f i c a t i o n of t h e v a l u e s of t h e p a r a m e t e r s . S t a t i s t i c a l i n f o r m a t i o n of t h e s y s t e m variations is not r e q u i r e d . If t h e p a r a m e t e r v a r l a t i o n s s a t i s f y c e r t a i n m a t c h i n g c o n d i t i o n s , t o t a l i n v a r i a n c e c a n b e a c h i e v e d . T h i s c o n t r o l p h i l o s o p h y c o n t r a s t s s h a r p l y w i t h s t o c h a s t i c a d a p t i v e c o n t r o l s y s t e m s i n w h i c h t h e c o n t r o l l a w is a l L e r e d w h i l s t s y s t e m p a r a m e t e r v a l u e s are c a l c u l a t e d u s i n g o n l i n e i d e n t i f i c a t i o n a l g o r i t h m s .

The two main a p p r o a c h e s are Lyapunov c o n t r o l , a n d v a r i a b l e s t r u c t u r e c o n t r o l (VSC) . Lyapunov c o n t r o l f o l l o w s t h a r o a c h o f r e s e a r c h e s s s u c h as Gutman a n d P a l m o r , 4 C o r l e s s a n d Lej tmann , Ryan3, Garofalo a n d G l i e l m o , a n d many o L h e r r e s e a r c h e r s . U s i n g a Lyapunov f u n c t i o n a n d s p e c i f i e d m a g n i t u d e b o u n d s on t h e u n c e r t a i n t i e s , a n o n l i n e a r c o n t r o l law is d e v e t o p e d t o e n s u r e u n i f o r m u l t i m a t e b o u n d e d n e s s o f t h e c l o s e d - l o o p f e e d b a c k t r a j e c t o r y t o a c h i e v e s u f f i c i e n t a c c u r a c y . The r e s u l t i n g c o n t r o l l e r is a d i s c o n t i n u o u s c o n t r o l f u n c t i o n , w i t h g e n e r a l l y a c o n t i n u o u s c o n t r o l i n a b o u n d a r y l a y e r i n t h e n e i g h b o u r h o o d of t h e s w i t c h i n g s u r f a c e . T h e b o u n d a r y l a y e r c o n t r o l p r e v e n t s t h e e x c i t a t i o n of h j g h - f r e q u e n c y unmodel l e d p a r a s j t i c d y n a m i c s . Lack o f s p a c e p r e v e n t s a d e t a i l e d a n a l y s i s h e r e o f Lhe e x t e n s i v e t h e o r y o f t h i s i m p o r t a n t f i e l d . The r e a d e r is i n v j t e d t o s t u d y f u r t h e r d e t a i l s i n o t h e r p a p e r s i n t h j s s e s s i o n , c h a p t e r s i n t h e book5 a n d numerous p u b l i c a t i o n s i n t h j s g r o w i n g area of r e s e a r c h . A c o m p a r a t i v e s t u d y is p r e s e n t e d by D e C a r l o e t a1 . I n t h i s p a p e r p a r t i c u l a r a t t e n f i o n w i l l be p a i d t o t h e VSC a p p r o a c h . I n f a r t Lhe r e s u l t i n g c o n t r o l l e r is o f t e n i d e n t i c a l t o Lyapunov c o n t r o l . I t i s s i m p l y t h e d e s j g n p h l l o s o p h y w h i c h d i f f e r s . I n VSC t h e d e s j r e d c l o s e d - l o o p dynamic p r o p e r t i e s are

Fi pp

6

WP-6- -1-

UP-6-1

d i r e r L l y s p e c i f ' i e d arid s u b s e c j w n t . l y a s u i t a b l e ( o n l , r o l f u c L i o n is d e t e r m i n e d 1.0 e n s u r e t h e at .Lainment a n d e x j s l . e n r e of' f he s l i d i n g mode ( d e s r r i b e d b e l o w ) .

V a r i a b l e s t r u c l . u r e f.onl.r*oI (VSC) w i t h a S I i d i n g mode was f i r s t . d e s c r i b e d by R u s s i a n a L h o r s i n t .he 1960's. A s u r v e y p a p e r by ULkjri ( 1977)' r e f e r e n c e s many o f t h e e a r l y r o n t r i b u L i o n s a v a i l a b l e i n t r a n s l a t i o n . Two r e c e n t , s u r v e y a n d f,u o r i a l p a p e r s w i t h numerous r e f e r e n c e s are by ULkin' a n d De Carlo e t a16 . A recent book by B u h l e r e x p l a i n s t h e s u b j e c t i n F r e n c h ' . D r a z e n o v i c " e s t a b l i s h e d some e a r l y r e s u l t s o n Lhe i n v a r i a n c e o f VSC i n t h e s l i d i n g mode f,o r e r f , a i n d i s t u r b a n c e s a n d p a r a m e f , e r v a r i a t i o n s . More r e c e n t l y t h e s u b j e c t h a s a t t racted great i n t e r e s t b e c a u s e of t h e e x c e l lent. i n v a r i a n c e p r o p e r t i e s . C o n s e q u e n t l y , VSC is p a r l . i c . u l a r l y s u i f , e d t o l ,he d e t e r m i n i s t j c c o n t r o l of u n c . e r t a j n c o n t r o l sysf ,ems . Some o f t h e major i n t e r e s t s h a v e been {,he u s e o f VSC a n d a 1 l i e d t e c h n i q u e s .in m o d e l - f o l l o w i n g a n d model r e f e r e n c e a d a p t i v e c o n t r o l , t r a c k j n g c o n t r o l a n d o b s e r v e r s y s t e m s .

T h e e s s e n t i a l f e a t u r e o f VSC is tha t , t h e n o n l j n e a r f e e d b a c k c o n t r o l h a s a d i s c o n t , j n u i t y on o n e o r more m a n i f o l d s i n t h e sLate s p a c e . Thus Lhe s t r u c t u r e o f t h e f e e d b a c k s y s t e m is al tered or s w i t c h e d as i ts s ta te crosses e a c h d i s c o n t i n u i t y s u r f a c e : i n c o n s e q u e n c e o f w h i c h , Lhe' c l o s e d - l o o p s y s t e m is d e s c r i b e d as a v a r i a b l e s t r u c f , u r e s y s t e m . A VSC s y s t e m may b e r e g a r d e d as a c o m b i n a t i o n o f s u b s y s t e m s , e a c h w i t h a f i x e d s t r u c t u r e and e a c h o p e r a t j n g j n a s p e c j f i e d r e g i o n of t h e s t a t e s p a c e .

I n t h i s p a p e r aLtent,jon w j I 1 be f o c u s s e d upon t h e VSC r e g u l a t o r . The aim of t h e d e s i g n is t o r e g u l a 1 , e t h e s y s t e m sf.ate t o z e r o . Much o f f.he Lheory may b e d i r e c 1 . l y a p p l j e d Lo model - fo l l o w i n g a n d L r a c k i n g c o n f . r o 1 sysLems.

S l i d i n g m o t i o n i n VSC

T h e c e n t r a l fea1.ur.e of VSC j s s l i d i n g m o t ion. T h i s occurs when !,he sysLern sf.al e r e p e a L e d l y crosses a n d j m m e d i a t e l y re-crosses a s w i t c , h i n g s u r f a r , e , b ~ ' c aiise al I m o t i o n i n $.he r i e i g h b o u r h o o d o f l,he m a n i f o l d i s d i r e c t e d i n w a r d s ( i . e . f.owards t h e m a n i f o l d ) . D e p e n d i n g o n t h e f'orm o f 1,he c o n t r o l law s e l e c l . e d , Lhis S I i d i n g rnof,ion may o c r u r on i n d i v i d u a l s w i f , c h i t i g s u r f a r , e s i n 1 h e sl,ai,e s p a c e , on a s e l e c t io f i o f a s u r f a c e s , or o n a l l Lhe s w i r c h i n g m a n i f o l d s a t o g e t . h e r . When 1,he last, o f t h e s e cases of urs, Lhe s y s f ~ m is s a i d 1.0 b e i r i t h e s l i d i n g mode.

The term ' s l i d i n g mode' is o f t e n a l so u s e d f o r l,he s l i d i n g s u b s p a r e . 'The dynamic mof.ion of t h e sysl .em

1

is t h e n e f f e c t i v e l y c o n s t r a i n e d t o l i e w i t h i n a c e r t a i n s u b s p a c e of t h e f u l l s t a t e s p a c e . The s y s t e m is t h u s f o r m a l l y e q u i v a l e n t , t o a n u n f o r c e d s y s t e m of lower o r d e r , t e r m e d t h e e q u i v a i e n t s y s t e m . The m o t i o n of t h i s e q u i v a l e n t , sysl.em is d i f f e r e n t from t h a t of e a c h of t h e consl. i I.iierif

s u b s y s t e m s .

The d e s i g n of a VSC s y s t e m e n t a i Is f h e c h o i c e ot' t h e s w i t c h i n g s u r f a c e s , t h e s p e c i f i t a t ion o f t.he

d i s c o n t i n u o u s c o n t r o l f u n c t i o n s a n d Ltie d e t e r m i n a t i o n of t h e s w i t c h i n g log ic . assoc i a l . e d w i t h t h e d i s c o n t i n u i t , y s u r f a c e s . ' l e s w i t r h i n g surfaces are u s u a l l y f i x e d h y p e r p l a n e s i n t . h e st.at.e s p a c e p a s s i n g t h r o u g h t h e s t a t e s p a r . e o r i g i n , t h e i n t e r s e c t i o n of w h i c h forms t h e sl i d i r i g s u b s p a c e . T h e o b j e c t i v e of t h e d e s i g n is t o d r i v e l.he s1.al.e of t h e s y s t e m from a n a r b i 1 , r a r y i n i t . i a i r o r i d i f . i o n t o t h e i n t e r s e c t , i o n of t h e s w i t c h i n g s u r f a c e s . O n c e t h e s tate starts s l i d i n g , t h e a c t i o n of t h e c . o n t r o l is r e q u i r e d on:y t o m a i n t a i n t h e s t a t e o n (o r i n t h e n e i g h b o u r h o o d of t h e i n t e r s e c L i o n m a n i f o l d .

T h e e q u i v a l e n t , s y s t e m m u s t b e a s y m p t o t i c a l l y s t a b l e t o e n s u r e t h a t t h e s tate a p p r o a c h e s t h e s t a t e s p a c e o r i g i n w j t h i n t h e s l i d i n g mode. S t a b l e s l i d i n g m o t i o n is a s s u r e d by t h e s u i t a b l e s e l e c t i o n of t h e s w i t c h i n g h y p e r p ; a n e s , t h e t a s k w h i c h forms t h e first stage of t h e VSC d e s i g n p r o c e s s . T h i s p r o b l e m of d e t e r m i n i n g a se t of h y p e r p l a n e s p r o v i d i n g s u i t a b l e b e h a v i o u r i n t h e s l i d i n g mode is d e s c r i b e d as t h e e x i s t e n c e p r o b l e m . It is c o n c e r n e d s o l e l y w i t h f i x i n g a s e t of h y p e r p l a n e s s u c h t h a t t h e s l i d i n g mode o n t h e i r i n t e r s e c t i o n ( i f i t is r e a c h e d ) g i v e s a s p e c i f i e d p e r f o r m a n c e t o t h e e q u i v a l e n t l o w e r o r d e r s y s t e m . T h e s o l u t i o n of t h e e x i s t e n c e p r o b l e m may b e c o m p l e t e d wi thout , a n y a s s u m p t i o n s o n t h e form of t h e c o n t r o l f u n c t i o n s t o b e e m p l o y e d by t h e s y s t e m .

Once t h e e x i s t e n c e p r o b l e m h a s b e e n s o l v e d , t h e s e c o n d stage of t h e d e s i g n p r o c e d u r e i n v o l v e s t h e s e l e c t i o n of t h e c o n t r o l w h i c h w i l l e n s u r e t h a t . t h e c h o s e n s l i d i n g mode is a t t a i n e d a n d m a i n t a i n e d . F o r t h i s r e a s o n , t h e p r o b l e m of d e t e r m i n i n g a c o n t r o l s t r u c t u r e a n d a s s o c i a t e d g a i n s w h i c h e n s u r e t h e r e a c h i n g o r h i t t i n g of t h e s l i d i n g mode, is c a l l e d t h e r e a c h a b i l i t y p r o b l e m . The s o l u t i o n of t h e r e a c h a b i l i t y p r o b l e m is d e p e n d e n t o n t h e s w i t c h i n g h y p e r p l a n e s ( a s m i g h t b e e x p e c t e d , s i n c e t h e c o n t r o l f u n c t i o n s are r e q u i r e d t o b e d i s c o n t i n u o u s o n t h e s w i t c h i n g s u r f a c e s ) , a n d so c a n n o t b e s o l v e d u n t i l t h e e x i s t . e n c e p r o b l e m h a s b e e n r e s o l v e d .

The t r a n s j e n t rno t ion t h e r e f o r e c o n s i s t s of two i n d e p e n d e n t stages:

a ( p r e f e r a b l y r a p i d ) m o t i o n b r i n g i n g t h e s t a t e of t h e s y s t e m t o t.he m a n i f o l d i n w h i c h s l i d i n g o c c u r s ;

a slower s l i d i n g mot,ion d u r i n g w h i c h t,he s ta te s l i d e s t o w a r d s t h e s ta te s p a c e o r i g i n w h i l e r e m a i n i n g i n t h e s l i d i n g s u b s p a c e .

T h i s two stage b e h a v i o u r c a n h e l p t,o r e s o l v e t h e c o n f l i c t bet,ween t.he o p p o s i n g requi remen1.s of s t a t i c a n d dynamic a c c u r a c y w h i c h are e n c o u n t e r e d when d e s i g n i n g a l j n e a r coril.ro1 s y s t e m , b e c a u s e a VSC s y s t e m may b e d e s i g n e d 1.0 g i v e : a r a p i d

r e s p o n s e w i t h n o loss of s L a b i I i L y ; asyrnpf,c,t.ic s ta te r e g u l a t i o n ; i n s e n s i t . i v i t . y t o a c lass of p a r a m e t e r v a r i a t i o n s ; a n d i n v a r i a n r e 10 c e r t a i n e x t e r n a 1 d i s t u r b a n c e s .

D u r i n g t h e s l i d i n g mode t h e d i s c o n t . i n u o u s c o n t . r o l r hat,t.er,s about, t h e swiI . r .h ing s i i r f a r p at t i igh f r e q u e r i r y . T h i s phenomenori is usilal l y u r l d e s i r : i b l e f o r m o s t . p r a c t i c a l a p p l i ( al. i o n s arid ii sinooc.hed c.ont i n u o u s non1 i n e a r con1 r ' o l ( ari b e subsf, i t u t , e d wit.h i i t t ] ? a l t e r a t i o n i r i t tie dynamic b e h a v i o u i , of Lhe syat.em . T h e smoof.hed ( o r i t rol ier a lso p r e v o n t s Lhe exr i t a t i o n of h i g h - f r e q u e n c y unmode l e d d y n a m i c s ( c f t h e b o u n d a r y l a y e r i n t h e Lyapiinov a p p r o a c h 1 .

T h e R e g u l a t o r S y s t e m

Let, u s c o n s i d e r t h e r e g u l a t o r syst ,em w h i c h may he e x p r e s s e d i n i t s most g e n e r a l f o r m as

:( 1.) :[ P.+AA( t, 1 ]X ( t +[ B+AB( t , ) 1 U ( t ) + D f ( t. 1 ( 1 )

w h e r e x is t h e s t a t e n - v e c t o r , U is t h e c o n ' . r o l m-vectcir a n d f is a p - v e c t o r of d i s t u r b a n c e s . I . is assumecl t h a t n>m, B is of f u l l r a n k m a n d t h a t 1,he p a i r ( A , B ) i s c o m p 1 e L e l y c o n t r o l l a b l e . T h e ma'.vix A A r e p r e s e n t s t h e v a r i a t i o n s a n d u n c e r t a l n t . i e s i n t h e p l a n t p a r a m e t e r s , AB is t h e p l a n t / r o n l . r o I i n t e r f a c e u n c e r t a i n t y a n d f r e p r e s e n t s ext.ei-na1 d i s t u r b a n c e s . T h e o v e r a l l aim of a v a r i a b l e s t r u c t u r e r e g u l a t o r c o n t r o l d e s i g n is t o r e g u : a t . e t h e z ,ys tem s t a t e from a n a r b i t r a r y i n i ' , i a l c o n d i t i o n x(0) :x t o t h e s t a t e o r i g i n a s y m p t o t i c a l ~ y as ?-+W.

T h e m c . o n t r o 1 c o m p o n e n t s U . of t h e c o n t r o ; v e c t o r U are s ta te d e p e n d e n t ( f e e d b d c k ) f u n c t i o n s , U . :U. X I . The s w i t c h i n g ( o r s l i d i n g ) s u r f a c d s arc i n t e r s e c t i n g h y p e r p l a n e s M . p a s s i n g t h r o u g h t h e staLe s p a c e o r i g i n a n d d e f i n e d by M.=:{x:r. .::=a> (j=1,2 ,..., m) w h e r e c . is a row n - v d c t o r . J The ( i d e a l ) s l i d i n g modeJ o c c u r s when t h e s t a t e iie.7 s i m u l L a n e o u s l y i n e a c h of t h e h y p e r p l a n e s M . f o r j = 1 , ..., m . T h i s is a c h i e v e d when t h e sLat.e r4ar :hes a n d r e m a i n s i n t h e m a n i f o l d M, M={x:Cx=O] ~ w h i c h is t h e i n t e r s e c t i o n of t h e m h y p e r p l a n e s . I n g e o m e t r j c a l terms t h e s u b s p a c e M is t h e n u l l s p a c e ( o r k e r n e l ) of C , d e n o t e d N ( C ) .

F o r s i m p l i c j t y of p r e s e n t a t i o n l e t u s now r e s t : - i r . t o u r attent. ion t o t h e n o m i n a l s y s t e m w i t h o u t u n c e r t a i n t ies

= A x ( t ) + Bu(t , ) ( 2 )

i.e. A A = O , AR=O, f = O . T h e a n a l y s i s may b e c,ars-ied out. s i m i l a r l y for Lhe more g e n e r a l case b u t wi1.h greater n o t a t j o n a l c o m p l e x i t y . T h e s l i d i n g rnode may b e de1,ermined froin t h e d e f i n i n g c o n d i t i o n C x ( . . ) = O for t>,t w h e r e 1. is t h e time when l,he SI i t1 ing mani f o l dS is r e a c h e d .'D i f f e r e n t i a t i n g r e s p e c t , to t i m e a n d i n s e r t i n g (2) g i v e s

w i t h

C i ( j , ) = CAx(t . )+CBu( t , ) = 0 (1 . >/ 1, ) . ( 3 )

An e q i i i v a l e n L c o n t r o l U may b e e x p r e s s e d i n t h e 1 i n e a r f e e d b a c k form ueq ( 1 . ) I -Kx ( 1,) . mx n f e e d b a c k m a t r i x K is g r v e r i by K=(CB:"FA. T h e c l o s e d - l o o p s y s t e m d y n a m i c s i n t h e s l i d i n g mode are t h e n d e s ( r i bed by Lhe s y s t e m e q u a t i o n

UP-6-1 -2-

It s h o u l d b e n o t e d t h a t t h i s motian is i n d e p e n d e n t of the a c t u a l control U a n d d e p e n d s o n l y o n t h e c h o i c e af C. The f u n c t i o n of t h e control U IS t o d r i v e t h e state i n t o t h e s l i d i n g s u b s p a c e M , and t h e r e a f t e r t o m a i n t a i n i t w i t h i n M. The c o n v e r g e n c e of t h e s t a t e v e c t o r t o t h e o r i g i n is e n s u r e d by s u i t a b l e c h o i c e of the f e e d b a c k maLr ix K . The d e t e r m i n a t i o n of t h e m a t r i x K or a l t e r n a t i v e l y , t h e d e t e r m i n a t i o n of t h e matrix C d e f i n i n g Lhe s u b s p a c e M, may b e a c h i e v e d w i t h o u t p r i o r knowledge of Lhe form of t h e c o n t r o l v e c t o c U. ( T h e r e v e r s e I S not, t r u e ) . The n u l l s p a c e of C, N(C1, and t h e r a n g e s p a c e of B, R ( B ) , are complemen ta ry s u b s p a c e s ; i . e . N(C)AR(B) = EO]. S i n c e m o t i o n l ies e n t j r e l y w i t h i n N ( C ) d u r i n g t h e i d e a l s l i d i n g mode, t h e dynamic b e h a v i o u r of t h e s y s t e m d u r i n g s l i d i n g is u n a f f e c t e d by t h e c o n t r o l s b e c a u s e t h e y act o n l y w i t h i n R ( B ) .

7

Sys tem T r a n s f o r m a t i o n

The deve lopmen t of t h e t h e o r y and d e s i g n p r i n c i p l e s is s i m p l i f i e d by u s i n g a p a r t i c u l a r c a n o n i c a l form for t h e s y s t e m . T h i s form is c l o s e l y r e l a t e d t o t h e c o n t r o l l a b i l i t y c a n o n i c a l form for a m u l t i v a r i a b l e l i n e a r s y s t e m . By a s s u m p t i o n t h e m a t r i x B h a s f u l l r a n k m , so t h a t t h e r e e x i s t s a n o r t h o g o n a l nxn t r a n s f o r m a t i o n m a t r i x T s u c h t h a t

TB = [I2] ( 5 )

where B is mxm and n o n - s i n g u l a r . T h e o r t h o g o n a l i t y r e s t r i c E i o n is imposed o n T fo r r e a s o n s of n u m e r i c a l s t a b l l i t y and t o remove t h e p r o b l e m of i n v e r t i n g T when t r a n s f o r m i n g b a c k t o t h e o r i g i n a l s y s t e m i n t h e CAD package!'

The t r a n s f o r m e d state is y = Tx and t h e s t a t e e q u a t i o n (2) becomes

i ( t ) = TATTy(t) + T B u ( t ) . ( 6 ) T The s l i d i n g c o n d i t i o n is C T y ( t ) = O . I f t h e

t r a n s f o r m e d s ta te y is now p a r t i t i o n e d as T T

Y ( Y , Y:) ; y 1 y2"' ( 7 ) T T and t h e matrices TAT , T B and CT are p a r t i t i o n e d

a c c o r d i n g l y , t h e n

' l ( t ) = A , , y l ( t ) + A I 2 y 2 ( t )

>,(t) = A 2 1 y l ( t ) + A 2 2 ~ 2 ( t ) + B2U(t) ( 8 )

and

C , y l ( t ) + C2Y2( t ) = 0 (9)

and C2 is n o n - s i n g u l a r (from CB n o n - s i n g u l a r ) . The more g e n e r a + s y s t e m ( I ) c a n b e s i m i l a r l y e x p r e s s e d , i f T(AA)T and T(AB) are a l s o p a r t i t i o n e d c o m p a t i b l y w i t h y1 a n d y2 .

11. w i l l b e a s sumed t h r o u g h o u t (,he r e m a i n d e r of t h i s p a p e r t h a L t h e u n c e r t a i n t i e s i n t h e p l a n t c o n t r o l . i n t e r f a c e o c c u r o n l y o n t h e a c t u a l jnpui, c h a n n e l s , i .e. r a n k ( B ) = r a n k ( B AB). Under t h i s a s s u m p t i o n , AB1 ( t ) = O , and y , becomes . i ndependen t of t h e c o n t r o l U.

'The c a n o n i c a l form is c e n t r a l 1.0 t h e h y p e r p l a n e d e s j g n me thods d e s c r i b e d be low. I f . a l s o p l a y s a s i g n i f i c , a n t r o l e in t h e s o I u L i o n of t h e r e a c h a b i l i t y p r o b l e m , i .e. t h e d e t e r m i n a L i o n of t h e c o n t r o l form e n s u r i n g Lhe a t t a i n m e n t of t h e s l i d i n g mode, wh ich w i l l b e d i s c u s s e d la1.er. EquaLion ( 9 ) d e f i n i n g t h e S I i d i n g mode is e q u i v a l e n t , t o

where t h e mx(n-m) riraLrix F is d e f i n e d by

- 1 (12) 2 c l F = C

so t o y l . The s l i d i n g mode sa t i s f ies t h e e q u a t i o n s

t h a t i n t h e s l i d i n g mode y2 is r e l a t , e d l i n e a r l y

( 1 3 )

T h i s r e p r e s e n t s a n (n -m) - th o r d e r s y s t e m i n wh ich y2 p l a y s t h e role of a s t a t e f e e d b a c k c o n t r o l . Thus

( 1 4 )

so t h a t t h e d e s i g n of a s t a b l e s l i d i n g mode s u c h t h a t y-+O as t 3 w r e q u i r e s t h e d e t e r m i n a t i o n of t h e g a i n m a t r i x F s u c h t h a t A -A F h a s n-m l e f t - h a n d h a l f - p l a n e e i g e n v a l u e s . " T h l g may b e a c h i e v e d by u s i n g a m o d i f i e d form of a n y s t a n d a r d d e s i g n method g i v i n g a l i n e a r f e e d b a c k conf , ro l l e r for a l i n e a r d y n a m i c a l s y s t e m . The main me thods are t h o s e b a s e d o n t h e m i n i m i s a t i o n of a n i n t e g r a l cost f u n c t i o n a l w i t h q u a d r a t i c i n t e g r a n d , and d i r e c t p o l e p l a c e m e n t .

H y p e r p l a n e D e s i g n Sy Direct E i g e n v a l u e Ass jgnmen t

The m a j o r i t y of VSC d e s i g n t e c h n i q u e s u s e e i g e n v a l u e a s s j g n m e n t me thods i n s e l c t j n g i.he s l j d i n g mode. F o r s c a l a r - c o n t r o l l e d e x a m p l e s w i t h a s i n g l e swi t c h j n g h y p e r p l a n e , s p e c i f i c a t i o n of t h e n-1 e i g e n v a l u e s to b e a s s o c j a t e d w i t h t h e s l i d i n g mode complei ,e ly d e t e r m i n e s t h e f e e d b a c k maLr ix F wh ich is a n (n -1 ) - row v e c t o r . F o r Lhe m u l t i p l e input . case U t k i n a n d Yang'' show t h a t t h e p a i r

is c o n t r o l l a b l e and t h a t e i g e n v a l u e

a s s j g n m e n t is f e a s i b l e . I t is w e l l known, however , thal , t h e a s s i g n m e n t of e i g e n v a l u e s of a n n t h o r d e r m-jnpul, s y s t e m r e q u i r e s o n l y n of t h e nm d e g r e e s of f r eedom ( d . 0 . f . ) a v a j l a b l e i n c h o o s i n g t h e f e e d b a r k g a i n m a t r i x . The rema i n i rig ( n - m ) d . 0 . f . may b e u t i l i s e d i n p a r t i a l l y a s s i g n i n g t h e a s s o c i a t e d e i g e n v e c t o r s . T h i s c a p a b i I i t y o f a s s i g n i n g Lhe e i g e n v e c t o r s may b e u s e d i n two complemen ta ry ways: to s h a p e Lhe r e s p o n s e of t h e c l o s e d loop s y s t e m ; or t o max imise t h e r o b u s t n e s s o f t h e e i g e n v a l u e p l a c e m e n t . We assume t h r o u g h o u t L h a t t h e non-ze ro s l i d j n g mode e i g e n v a l u e s are d i s l i n c t from e a c h o t h e r and from t h e e i g e n v a l u e s of A l l .

Our f irst a p p r o a c h r e v o l v e s a r o u n d t h e i d e a of

( A 1 1 ? A t , )

W-6-1 -3-

a s s i g n i n g t h e e i g e n v e c t o r s of' ( 14) d i r e c . l . 1 ~ i n o r d e r 1.0 s h a p e t h e syst.em r e s p o n s e d u r i n g sl i d i r l g . I f . is c l e a r t h a t t h e e j g e n v e c l o r s s h a p e t h e r e s p o n s e of a n u n f o r c e d a u t o n o m o u s 1 i n e a r sysf etit k = Px. I f t,he t , r a n s i i . i o n mat,rix is w r i l . 1 eri i r i I he modal form

e x p ( p t ) = x e x p ( O t ) X-' ( 1 ' ) )

w h e r e Q is t h e J o r d a n normal form of P a n d X i s f h e e i g e n v e c i , o r mat , r ix a s s o c i a % e d wj1.h Q .

S u p p o s ~ tha l , Lhe s l i d i n g mode h a s commenced on N(C). 'Then

;(t) z (A-BK) x ( t ) (16)

w h e r e K = ( C B ) - CA. S i n c e s l i d i n g m o t i o n m u s t r e m a i n i n N ( C ) , we h a v e t h a t

1

C[A-BK]=O <=> R(A-BK) C N ( C ) . ( 1 7 )

L e t h,(i=l, ..., n) b e t h e e i g e n v a l u e s of A-BK w i t h c o r r e i p o n d i n g e i g e n v e c t o r s w i . Then ( 17) imp1 i e s t h a t

C [ A - B K I w . = hi C w . = O ( 1 8 )

so t h a t e i t , h e r w . is zero o r wic N ( C ) . Now = A h a s p r e c i s h l y m z e r o - v a l u e d e i g e n v a l u e s , so 1 8 A . ( j = l , . . . , E - m ) b e t h e n o n - z e r o e i g e n v a l u e s ( d i s t j n c t by a s s u m p t i o n ) . Then s p e c i f y i n g t h e c o r r e s p o n d i n g e i g e n v e c t o r s wi ( i = l , ..., n-m) f i x e s t h e n u l l s p a c e of C , s i n c e d im[N(C)]=n-m.

T h e d r a w b a c k t o t h i s a p p r o a c h is t h a t !,he e i g e n v e c t o r s of A-BK are not,, i n g e n e r a l , f r e e l y a s s j g n a b l e . A t most m e l e m e n t s of a n e i g e n v e c t o r may b e a s s i g n e d a r b i t r a r i l y ; t h e r e m a i n i n g n-m e l e m e n t s are Lhen f u l l y d e t e r m i n e d by the a s s i g n e d e l e m e n t s . T h u s o n e a p p r o a c h t o e i g e n v e c t o r a s s i g n e n t , would b e t o p i c k m e l e m e n t s a c c o r d i n g Lo some s c h e m e a n d a c c e p t t h e r e m a i n i n g e l e m e n t s as d e t e r m j n e d . T h i s m i g h t a l l o w a d e g r e e of a d j u s t m e n t t o b e c a r r i e d o u t by i n s p e c t , j o n .

An a l L e r n a t i v e method of e j g e n v e c t o r a s s j g n m e n t j s by c o n s i d e r a t . i o n of Lhe a s s i g n a b l e s u b s p a c e c o r r e s p o n d i n g t o a g i v e n e i g e n v a l u e . I f t h e c h o s e n v e c t o r js not. a s s j g n a b l e , i t is t h e n m o d i f i e d i n t h e c o m p u t e r a l g o r i t . h m by d e t e r m i n i n g t h e " c l o s e s t " a s s i g n a b l e e j g e n v e c t . o r ( j n t,he least s q u a r e s s e n s e ) for t h e p r e s e n t e j g e n v a l u e . A more d e t a i l e d t r e a t m e ? $ h a s b e e n presen1 ,ed by Dorljng a n d Z i n o b e r .

A-$7

Robust. E j g e n v a 1 u e Assi gnment

The f r e e d o m t o a s s i g n (ai. l e a s t p a r t i a l l y ) t h e e i g e n v e c t - o r s of t h e e q u i v a l e n t , s y s t e m may a l t e r n a t i v e l y b e u s e d 1.0 e n s u r e Lhat, Lhe e i g e n v a l u e s a s s o c i a t . e d w i t h t h e si i d i n g mode are m a x j m a l l y j n s e n s i l . i v e Lo p e r L u r b a t . i o n s i n f.he s y s t e m p a r a m e t e r m a t r i x A . T h i s r e d u c e s d e v i a t ion from t h e d e s i r e d d y n a m i r r e s p o n s e w h i r h may ar ise from14;q% p a r a m e t e r v a r i a t i o n s w h i c h do not, l i e i n R ( B )

I n c l u d i n g Lhe p a r a m e t e r v a r i a t i o n t.erms i n t h e c l o s e d l o o p f o r m of ( 1 4 ) g i v e s

wt1f-r.r i.tie r ~ x p I i ( i f 1, imp d e p e n d e n c e of' y I , A A i a n d A A has t,f:erl drvppecf. The s r r . ~ n d i,erfri or1 h e r,.t!,<s. o f ' ( < ' ( I ) r*epres?r i f , s t.he p e r t u r b a l . i o n of' t h e riot" i na l d u e Lo par,amet e r v a r i at. i 011s.

T h i s k r m IS , i n g c m e r a l , I irrW-vary ng , m d a r b i i . r a i - y ; 1 t r r r*otiusl. assigrirtlc~f approa< t1 aims f o m i r l j m i s r i ( :; 6 , f ' t ' w I S .

I f ' t,he [,:lr'arrifl e r v;it'ial i o t i s : i r ' f , l f l : j f ( tll-'d, i .e . r a r r k ( B ) j$;mk ( 1 13 A A I ) , 1 h r , r i 1 t i c , r~ i l111: ;1 , f on1 r o l d e s i g n s may tie emp I r i y ~ d 1.0 c ' f i : ; i i i ' < x 1 ti:31 f r i e SI a1 ('

r e a c h e s a n d rerna i n s w i I h i r i I t i r , :: I i r i i ti): mm i 1 ' 0 I d NIC). T h i s is e q u i v a l e n l I o t t t ; i i r i ~ ; ~ i f ~ i r i i { t h c 111

z e r o - v a l u e d e i g e n v a l u e s o f A , , , , , , r i r , r , r , . ~ ~ ~ f ~ r i r ~ i r i ~ l 1 ( J

t h e d y n a m i c s i n R ( B ) , ai. zf 'r ' r) . l ; o w ~ ~ v c ~ r ~ , i f t t i e s

p a r a m e t e r v a r i a t i o n s are t i r t ~ r ~ a l ~ + i c ~ t l , I k i w i I,& iteal sliding mode w i l l b e c i i sh l r , t i f - f i . 0 1 1 1 ' aim is t h e r e f o r e t.0 d e s i g n !.he f e e d t m k m a t t ' i x F suf h f hat. t h e n-m a s s i g n e d n o n - z e r o e i g f x i v a l u ~ ' s are maximal l y r o b u s t , t h e r e b y m i n j m i s i r i g t tie r f f w t,s of i.hf- p a r a m e t , e r v a r i a t i o n s i n t h e N ( C ) or l.h? sl iOine; mode d y n a m i c s .

The s l i d i n g mode s y s t e m as d e s r r i b e d by ( 8 ) r e p r e s e n t s a n (n-m)i,h o r d e r sysi .em wij,h m con .rol i n p u t s . . T h e i n t e r f a c e m a f , r j x A , ? h a s d i m e n s i o n (n-m)xrn a n d r a n k p w h i c h sa+, isf ies

sy:.t err1 ( I / I )

I 6 p 6 min @-m, m). ( 2 1 )

Note Lhat, p=O would imply ( A l l , A unconti"o1 l a b l e , c o n t r a d i c t i n g Lhe con1,rol l a b i 1 lgy of t h e o r i g i n a l syst ,em. I f p<m, t.he s l i d i n g mode s y s t e m h a s redundant , input , s whic,h may b e removed by r e o r d e r i n g the r.ont.r'o1 inpu1.s and s e l . t i n g par t . of F t o zer3.

We now d e s c r i b e b r i e f l y Lhe h y p e r p l a n e d e s i g n s c h e m e for r o b u s t , e i g e n v a l u e a s s i g a m e n f , u n d e r t.he assumpl . icn 1,hat. p=m. Lei, ~ . G ( T ( A , ~ ;-fti12F), wi1.h c o r r e s p o n d i n g r i g h t . e i g e n v e c t o r v . cR a n d left e i g e n v e c t o r Then i t , i s ' w e l l -knownIq (see Wil l t inson, 1965) LhaL t.he s e n s i t j v i t , y of w . 1.0 p e r t , u r b a t i o n s i n A l 1 , A 1 2 a n d F d e p e n d s a p p r o x i m a t e l y I i n e p r l y o n t h e quarit.it.y 1 / r . , t.he s e c a n t of t,he a n g i e b e t w e e n t.t?e v e c f o r s ' v a n d ri so t h a t . l / c . 2 I . I n t h e c a s e o f a n e q u a l number' of sl.at,esl a n d i n p u L s for t h e S I i d i n g mode s y s t e m (n-m-m), we h a v e c o m p l e L e f r e e d o m i n a s s i g n i n g t h e e i g e n v e c t o r s of A - A i F a n d r x d o n o beLt ,e r Lhan selec 1. i n g a n or t .hogona? ' c ' i g e r l v e c L o r m a t r i x V= [ v , v ... v 1 w h i c h c l e a r l y i ? i v e s I / < , . = i , a n d i n p a r 1 ?! I I fa r w"e;ay f . a k e V= I

More g e n e r a l l y , i < m < r i - m a n d we s h o u l d aim t o makp I/c. as small as p o s s i b l e for e a c h i z l ,..., n-m. I f . is ' a lso wel l -known I,hal an u p p e r bound on e a i h oi' t h e s e n s i 1 , i v i t i e s j s t .he spet. t .ra1 c.ondii . ion number K ( V ) ( d e f i n e d by

n-m'

- I K ( V ) = I I V I I . I I V I I ( 2 2 )

w h e r e Lhe spec 1.r.a I mai,ri x norm i s ernp I o y e d . T h u s m i n i m i s i n g K(V) s h o u l d e n s u r e t.hal. !,he s e n s i t , i v i i . i e s o f a l l t.he a s s i g n e d e i g e n v a l u e s are a c c e p t a b l e ; a n d , adr i i l i o n a l l y e n s u r e !.hat. V is we1 I - tor id i t , ionec i w . i ' . t . i n v e r s i o n , sc: I.hal f h e s o I u l , i o n p rwess f ' o r . t.he f e e d b a r k mtf . r ix F is s t . a b l e arid t h e t)otmtls o n t h e magnj1,iides of' f ' a n d Lhe t r a n s i e n t , re :spor iw a r t m i n i m i s e d . I t s h o u d b e nol.ecl, h o w e v e r , f tiat. we a re n o t f ' ree (.hoo:;e o u r

UP-6- 1 -4-

e i g e n v e c t o r s V . a r b i t r a r i l y i f m<n-m, a n d t h a t , t h e minimum a t t a i n h b l e c o n d i t i o n number K ( V ) is t h e r e f o r e n o t n e c e s s a r i l y u n l t y for a g i v e n se t of e j g e n v a l u e s .

T e c h n i q u e s for d e t e r m i n i n g a s u i t a b l e se t of e i g e n v e c t o r s for a s p e c i f i e d s p e c t r u m i n t h e u s u a l l i n e a r f e e d b a c k s y s t e m h a v e been d e s c r i b e d i n r e f e r e n c e 17. T h e s e t e c h n i q u e s h a v e been a d a p t e d t o t h e r e s t r i c t e d p r o b l e m o f a s s i g n i n g t h e e i g e n v a l u e s of t h e s l i d i n g mode s y s t e m . The me thod d o e s n o t i n f a c t p r o d u c e a minimum f o r K ( V ) , b u t i t d o e s m i n i m i s e a c o n d i t i o n i n g m e a s u r e g i v i n g a good a p p r o x i m a t i o n t o t h e o p t i m a l c o n d i t i o n i n g .

C o n t r o l Scheme D e s i g n

H a v i n g s o l v e d t h e E x i s t e n c e P r o b l e m ( t h e d e t e r m i n a t i o n of C) a t t e n t i o n mus t b e t u r n e d to t h e R e a c h a b i l i t y Problem: t h e p r o b l e m of s e l e c t i n g a s ta te f e e d b a c k c o n t r o l f u n c t i o n w h i c h w i l l d r i v e t h e state x from a r b i t r a r y x i n t o t h e n u l l s p a c e of C and m a i n t a i n i t w i t h i n &is s p a c e t h e r e a f t e r . T h e r e are a w i d e v a r i e t y of c o n t r o l forms w h i c h h a v e been s t u d i e d . E a r l y VSC d e s i g n s u s e d r e l a y control w j t h f j x e d o r s t a t e - d e p e n d e n t g a i n s w i t h a d i s c o n t i n u i t y o n o n e or more of t h e h y p e r p l a n e s forming t h e m a n i f o l d M:N(C) . More r e c e n t l y t h e d e s i g n t e c h n i q u e h a s been s i m p l i f i e d by a r r a n g i n g fo r d i s c o n t i n u i t i e s t o occur o n l y o n t h e in tersect ion of a l l t h e h y p e r p l a n e s .

I n g e n e r a l t h e v a r i a b l e s t r u c t u r e control law c o n s i s t s of two p a r t s : a l i n e a r c o n t r o l l aw U a n d a n o n - l i n e a r p a r t U wh ich are a d d e d t o form k. T h e Linear control 8s a l i n e a r s ta te f e e d b a c k c o n t r o l l e r , u L ( x ) = L x , w h i l e t h e n o n l i n e a r f e e d b a c k c o n t r o l l e r U i n c o r p o r a t e s t h e d i s c o n t i n u o u s o r c o n t i n u o u s n o n l i n e a r elemerits of t h e c o n t r o l law. T h e s e n o n - l i n e a r i t i e s may l n c l u d e r e l a y s w i t h c o n s t a n t g a i n s , r e l a y s w i t h s t a t e - d e p e n d e n t g a i n s , l i n e a r f e e d b a c k w i t h s w i t c h e d g a i n s a n d s c a l e d u n i t - v e c t o r n o n - l i n e a r i t y , U ( x ) I p Cx / I I C x l l , ( p > O ) . The f irst t h r e e forms are d i s c o n t i n u o u s on t h e i n d i v i d u a l s w i t c h i n g h y p e r p l a n e s a p d r e q u i r e a h i e r a r c h i c a l d e s i g n s t r u c t u r e f o r m>l? I n t h e u n i t - v e c t o r case t h e i n d i v i d u a l c o n t r o l s are c o n t i n u o u s e x c e p t on t h e f i n a l j n t e r s e c t i o n , N ( C ) , o f t h e h y p e r p l a n e s , where a l l t h e c o n t r o l s are d l s c o n t i n u o u s t o g e t h e r .

To e l i m i n a t e t h e u n d e s i r a b l e c h a t t e r mot ion, c a u s e d by t h e a p p l i c a t i o n of d i s c o n t , i n u o u s c o n t r o l , f h e d i s c o n t i n u i t y c a n b e " s o f t e n e d " a n d r e p l a c e d 1 2 b y a I lboundary l a y e r " c o n t i n u o u s approximat , i o n . F o r e x a m p l e , a r e l a y m i g h t b e r e p l a c e d by a s a t u r a t i n g a m p l i f i e r , g i v j n g a small r e g l o n of u n s a t u r a t e d c o n t r o l effor t i n t h e n e i g h b o u r h o o d of t h e s w i t c h i n g s u r f a c e . T h i s e l i m i n a t e s c h a L t e r m o t i o n and y i e l d s 18smoothed" conk rol. Such a c o n t r o l s y s t e m is not a VSC s y s t e m i n t h e o r i g i n a l s e n s e of t h e term, a l t h o u g h i t works on t h e same p r i n c i p l e : a r a p i d a t t a i n m e n t of t h e s l i d i n g m a n i f o l d is f o l l o w e d by a t r a n s i e n t i n tthe n e i g h b o u r h o o d o f t h e m a n j f o l d , wh ich c a n b e made

N .

N

a r b i t r a r i l y c l o s e t o t h e i d e a l i s e d

The c o n t r o l s t r u c t u r e d e v e l o p e d i n form

s l i d i n g mode.

ref. h a s t h e

u ( x l = LX + p ( x ) N x / ( / /Mx(( + d ) (231

where t h e n u l l s p a c e s of N , M a n d C are c o i n c i d e n t : N ( N ) = N ( M ) = N ( C ) . T h e l i n e a r c o n t r o l law U s e r v e s o n l y t o d r i v e t h e st,ate to N(C) a s y m p k o t i c a l l y ; t o a t t a i n N ( C ) i n f i n i t e time, t h e n o n - l i n e a r componen t uN js r e q u i r e d . T h i s c o n t , r o l s1 , ruc ty t je h a s b e e n i n c l u d e d i n t h e CAD p a c k a g e VASSYD . When d i s t u r b a n c e s a n d u n c e r t , a i n t , ies are n o t p r e s e n t , a s c a l a r c o n s t a n t may b e e m p l o y e d . I n t h e p r e s e n c e of u n c e r t a i n t i e s t h e s c a l a r d is r e p l a c e d by a t j m e - v a r y i n g , s t a t e - d e p e n d e n t , f u n c t j o n i n c o r p o r a t l n g d e s i g n p a r a m e t e r s . The d e s i g n e r n e e d s t o s p e c i f y t h e r a n g e of m a g n i t u d e s of e x p e c t e d p a r a m e t e r v a r i a f , i ons a n d d i s t u r b a n c e s . F o r a f u l l e r e x p o s i t j o n of t h e t h e o r y a n d a d e t a i l e d e x p l a n a t i o n o f t h e d e s i g n p r o ~ ~ s ~ 3 , ! , ~ ; , 8 r e a d e r js r e f e r r e d f.0

r e f e r e n c e s

C o n c l u s i o n

Two a p p r o a c h e s t o t h e n o n l j n e a r d e t e r m i n i s t i c c o n t r o l o f u n c e r t a i n t i m e - v a r y j n g s y s t e m s are Lyapunov a n d v a r i a b l e s t r u c t u r e c o n t , r o l . The main p r o p e r t y is j n v a r i a n c e t o m a t c h e d p a r a m e t e r v a r i a t i o n s . T h i s p a p e r h a s c o n c e n t r a t e d on t h e d e s i g n of v a r i a b l e s t r u c t u r e c o n t r o l s y s t e m u s i n g a n e i g e n s t r u c t u r e a p p r o a c h . It n a s b e e n shown t h a t t h e h y p e r p l a n e d e s i g n p r o b l e m r e d u c e s t o a l i n e a r f e e d b a c k d e s j g n p r o b l e m f o r a lower o r d e r s y s t e m . t h e r e are two b a s i c algorithms; e i g e n v a l u e a n d p a r t i a l e i g e n v e c t o r a s s i g n m e n t t o s h a p e t h e c l o s e d - l o o p r e s p o n s e , a n d a r o b u s t e i g e n v a l u e ass ignment , a p p r o a c h t o m i n i m j s e t h e sensitivity of t h e e i g e n v a l u e s t o unmatched p a r a m e t e r v a r i a t i o n s . The CAD F o r t r a n 77 p a c k a g e , named VASSYD ( V a r j a b l e S t r u c t u r e S y s t e m D e s i g n 1 , p r o v j d e s faci I i ti es fo r o n - l j n e VSCS d e s j g n , s i m u t a t i o n of t r a n s i e n t r e s p o n s e a n d t h e storage or s y s t e m d a t a . It j n c o r p o r a t e s a n e x t e n s i v e ' h e l p ' modu le p r o v i d i n g t h e u s e r a t a l l stages w j t h d e t a j l e d i n s t u c t i o n s r e g a r d i n g t h e a p p l i c a b l e t h e o r y a n d t h e mode of o p e r a t i o n of t h e p a c k a g e .

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WF-6- 1 -&*