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A b s t r a c t 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 is a p p l i e d t o l i n e a r time v a r y i n g s y s t e m s a n d t o n o n l i n e a r s y s t e m s a m e n a b l e t o v i r t u a l l i n e a r i z a t i o n . A s y m p t o t i c s t a b i li t y i s g u a r a n t e e d e v e n i f t h e p e r f e c t model f o l l o w i n g c o n d i t i o n s do n o t h o l d , p r o v i d e d t h a t some s u f f i c i e n t c o n d i t i o n s a r e s a t i s f i e d . S i m u l a t i o n s show t h e scheme t o b e c a p a b l e o f e f f e c t i v e l y c o n t r o l l i n g c e r t a i n nonl i n e a r s y s tems . 1. I n t r o d u c t i o n The command g e n e r a t o r t r a c k e r (CGT) concept [l] has been proven to be a v e r y u s e f u l t o o l i n t h e d e v e lopment of 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 (MRAC) a l g o r i t h m s [ 2 ] . In t h i s p a p e r , t h e t i m e v a r y i n g CGT concept [ 3 ] i s d e v e l o p e d , t h u s a l l o w i n g t h e a p p l i c a t i o n o f MRAC t o l i n e a r t i m e v a r y i n g s y s t e m s a n d t o n o n l i n e a r s y s t e m w h i c h c a n b e o p e r a t e d u p o n u s i n g v i r t u a l l i n e a r i z a t i o n . 2 . MIWC Algorithm The ?IRAC problem w i l l b e s o l v e d f o r t h e f o l l o w i n g p r o c e s s e q u a t i o n s :(1,t h e o b j e c t i v e is t o f i n d a c o n t r o l u ( t ) s u c h t h a t t h e p l a n t o u t p u t y f o l l o w s t h e o u t p u t y o f t h e r e f e r e n c e model :To f a c i l i t a t e t h e d e v e l o p m e n t o f t h e c o n t r o l l a w , t h e time varying CGT c o n c e p t i s p r e s e n t e d h e r e : when perf e c t t r a c k i n g o c c u r s , i . e y ( t ) = y ( t ) f o r t > t t h e c o r r e s p o n d i n g s t a t e a n d c o n t r o l v a r i a b l e s w i l l b e denoted x* and u* and w i l l s a t i s f y by d e f i n i t i o n : m 0 'iU*(t)j i lf21 (t) s22(') U,(t), J h A where the S ( t ) m a t r i c e s a r e a p p r o p r i a t e l y d i m e n s i o n e d t i m e v a r y i n g m a t r i c e s . i j P r o v i d e d t h a t u =0, t h e S i j ( t ) m a t r i c e s w i l l have m t o s a t i s f y t h e f o l l o w i n g s e t o f e q u a t i o n s ( 3 ) : i l l ( t ) = A ( t ) S l l ( t ) -S l l ( t ) Am + B(t) S21(t) (8) i12(t) = A(t) S12(t) -S l l ( t ) Bm + B ( t ) S 2 2 ( t ) ( 9 ) (10) c s ( t ) = cm 11 1 2 c s ( t ) = 0 The g e n e r a t i o n o f a s o l u t i o n c a n b e shown under r a t h e r m i l d c o n d i t i o n s . The c o n t r o l law i s chosen of the form:o r : u=Kr where K=(K e , Kx, Ku) and r =((y -y) , x u ) . T T T m m m K ( t ) i s g e n e r a t e d a c c o r d i n g t o t h e f o l l o w i n g a d a pt i v e r u l e : K ( t ) = K I ( t ) + K ( t ) P (12) i I ( t ) = c ( y m ( t ) -y ( t ) ) r ( t ) T, ~~( 0 ) = K Io(13) Kp(t) = C ( y m ( t ) -y ( t ) ) r T ( t ) T ( 1 4 ) T h ' i t h t h e e r r o r d e f i n e d a s : e ( t ) = x * ( t ) -x ( t ) , t h e e r r o r d y n a m i c s will b e : e ( t ) = x...

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