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Math. Systems Theory 21, 63-83 (1988) MathematicalSystems Theory1988 Sonnger-Verlag New York Inc.

E x a c t L i n e a r i z a t i o n o f N o n l i n e a r S y s t e m s w i t h O u t p u t s *D. Cheng, ~ A. Isidori, 2 W. Respondek, 3 and T. J. Tarn 4= nstitute of Systems Science, Academyof Sciences,Beijing, People's Republic of China2 Department of Information and Systems Science, Universita di Roma "'La Sapienza',Rome, Italy3 Institute of Mathematics, Polish Academyof Sciences, Warsaw, Poland4 Department of Systems Sciences and Mathematics, Washington University,St. Louis, Missouri, USA

A b s t r a c t . This paper discusses the problem of using feedback and coordin-ates transformation in order to transform a given nonlinear system withoutputs into a controllable and observable linear one. We discuss separatelythe effect of change of coordinates and, successively, the effect of both changeof coordinates and feedback transformation. One of the main results of thepaper is to show what extra conditions are needed, in addition to thoserequired for input-output-wise linearization, in order to achieve full linearityof both state-space equations and output map.

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

In the last years there has been an increasing interest in the problem of compensat-ing the nonlinearities of a nonlinear control system. Among all the problems ofthis kind, the mos t natural question is that of when there exists a local (or global)change of coordinates, i.e., a diffeomorphism, in the state space that carries thegiven nonlinear system into a linear one. Krener [1] showed the importance ofthe Lie algebra of vector fields associated with the system in studying such aquestion and gave an answer to this problem. Brockett [2] enlarged the class of

*This research was supported in part by the National Science Foundation under GrantsECS-8515899, DMC-8309527, and INT-8519654.

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64 D. Cheng , A. Isidori, W. Respon dek, and T. J. Ta mt r a n s f o r m a t i o n s b y a ls o a l lo w i n g a c e r t a in f o r m o f f e e d b a c k . F o l lo w i n g B r o c k e t t 'sp a p e r , J a k u b c z y k a n d R e s p o n d e k [ 3] , S u [4 ] , a n d H u n t a n d S u [ 5] c o n s i d e r e dt h e f u ll f e e d b a c k g r o u p a n d g a v e n e c e s s a r y a n d s u ff ic i en t c o n d i t i o n s f o r l in e a r i-z a t io n o f t h e s ta t e - sp a c e e q u a t io n s . S t u d y i n g t h e i n p u t - o u t p u t b e h a v i o r , C l a u d ee t a l . [ 1 2 ] g a v e s u ff ic i en t c o n d i t i o n s f o r th e i m m e r s i o n o f a s y s t e m i n t o a l in e a ro n e a n d I s i d o r i a n d R u b e r t i [ 1 3 ] f o u n d 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 s ,t o g e t h e r w i th a c o n s t r u c ti v e p r o c e d u r e , f o r l in e a r i z at io n o f t h e i n p u t - o u t p u tr e s p o n s e .

T h e g r o w i n g l i t e r a t u r e c o n c e r n i n g l i n e a r i z a t i o n n o w i n c l u d e s , b e s i d e s t h ea b o v e p r o b l e m , t h e c o n s i d e r a t io n o f d if f er e n t c la s se s o f t r a n s f o r m a t i o n s [ 6] ,g l o b a l q u e s t i o n s [ 7 ] - [ 1 1 ] , [ 2 8 ] , [ 2 9 ] , p a r t i a l l i n e a r i z a t i o n [ 1 4 ] - [ 1 6 ] , [ 3 4 ] , a n dm a n y o t h e r s [ 2 2 ] , [ 2 3] . A s u r v e y o f l i n e a r i z a t i o n p r o b l e m s c a n b e f o u n d i n [ 2 5]a n d [ 3 6 ] .

I n th i s p a p e r w e c o n s i d e r a s y s te m o f t h e f o r mg = f ( x ) + ~ g , ( x ) u i ,

i = 1 (1 )Yi = h i ( x ) , i = 1 , . . . , p ,w h e r e f , g~ . . . . , g m a r e C ~ v e c t o r f i e l d s i n R n, h = ( h l h p ) r i s a C ~ v e c t o r -v a l u e d f u n c t i o n .

W e a s s u m e w e a r e g i v e n a n i n i ti a l s t a te X o ~ R n a r o u n d w h i c h w e c o n s i d e ro u r s y s t e m a n d w e a ls o a s s u m e , t h r o u g h o u t t h e p a p e r , t h a t f ( x o ) = 0 a n d h ( x o ) = O.W e s t u d y v a r i o u s k i n d s o f e q u i v a le n c e o f th e a b o v e s y s t e m t o a l i n e a r s y s t e m o ft h e f o r m

= A z + ~ b i ui ,i= l (2 )y i = C i z , i = l , . . . , p .

W i t h o u t t h e a s s u m p t i o n f ( x o ) = O , h ( x o ) = 0 a l l r e s u l t s s t i l l r e m a i n v a l i d b u t i t i sn e c e s s a r y t o a d d c o n s t a n t t e r m s t o A z a n d C ~ z , r e s p e c t i v e l y .

A l l t h e r e s u lt s s h o w n b e l o w a r e l o ca l . I t m e a n s t h a t t h e c o n d i t i o n s w e s t a ten e e d t o h o l d l o c a l ly a r o u n d t h e i n it ia l s t a t e Xo a n d t h a t t h e l i n e a r iz i n g t r a n s f o r m a -t i o n s e x i s t l o c a l l y a r o u n d t h i s p o i n t .

S e c t i o n 2 d e a l s w i t h l i n e a r i z a t i o n w i t h o u t f e e d b a c k . W e s t a t e a s e r i e s o fc o n d i t i o n s w h i c h d e s c r ib e t h o s e n o n l i n e a r s y s t e m s w h i c h c a n b e t r a n s f o r m e d ,v i a l o c a l d i f f e o m o r p h i s m s , t o c o n t r o l l a b l e a n d o b s e r v a b l e l i n e a r o n e s . W e c a l lt h i s e x a c t l i n e a r i z a t i o n . I n S e c t i o n 3 w e c o n s i d e r a s i n g l e - i n p u t s y s t e m a n d w ed e a l w i th t h e s e a r c h o f a f e e d b a c k a n d c o o r d i n a t e s c h a n g e i n th e s t a t e s p a c ew h i c h t r a n s f o r m s b o t h t h e d y n a m i c s a n d t h e o u t p u t m a p s i n t o a l i n e a r f o r m . W ec a ll t h is e x a c t l in e a r i z a t io n v i a f e e d b a c k . I n S e c t i o n 4 w e p r o v i d e a n i m p r o v e dv e r s io n o f a p r e v i o u s t e st [1 3 ] f o r l i n e a ri z a t i o n o f th e i n p u t - o u t p u t r e s p o n s e . I nS e c t io n 5 w e u s e th is i m p r o v e d v e r s i o n in o r d e r t o s o l v e t h e p r o b l e m o f e x a c tl i n e a r i z a t i o n v i a f e e d b a c k f o r a m u l t i - i n p u t s y s t e m . I n p a r t i c u l a r , w e i d e n t i f y t h ee x t r a c o n d i t i o n s n e e d e d , i n a d d i t i o n t o t h o s e r e q u i r e d f o r l i n e a r i z a t i o n o f t h ei n p u t - o u t p u t r e s p o n s e , i n o r d e r t o a c h i e v e f u l l l i n e a r i t y a t a s t a t e - s p a c e l e v e l .

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Exact Linearization of Nonlinear Systems w ith Outputs 652 . L i n e a r i z a t i o n W i t h o u t F e e d b a c k

I n th i s s e c t i o n w e il l u st r at e s o l u t i o n s t o t h e p r o b l e m o f tu r n i n g ( 1 ) i n t o a l i n e a rs y s t e m b y m e a n s o f c h a n g e o f c o o r d i n a t e s o n l y . W e a s s u m e t h e r e a d e r i s f a m i l i a rw i t h t h e n o t i o n s o f a L ie b r a c k e t o f v e c t o r f ie l ds , a L i e d e r i v a t iv e o f a f u n c t i o na l o n g a v e c t o r f i el d, o n e - f o r m s , a n d d i s t r ib u t i o n s a n d c o d i s t r ib u t i o n s . I n t r o d u c t i o nt o t h e s e s u b j e c t s m a y b e f o u n d , e .g ., in [ 2 7 ] , w h e r e s p e c i al e m p h a s i s o n t h e u s eo f th e s e c o n c e p t s i n c o n t r o l t h e o r y i s g i v e n .

S i n c e w e a r e o n l y in t e re s t ed i n c o n t r o l l a b l e a n d o b s e r v a b l e l i n e a r s y s te m s ,w e i n t r o d u c e a " c o n t r o l l a b i l i ty " c o n d i t i o n ( C ) a n d a n " o b s e r v a b i l i t y " c o n d i t i o n( O ) i n t h e f o l l o w i n g w a y .

L e t D j d e n o t e t h e d i s t r ib u t i o n s p a n n e d b y t h e s e t o f v e c t o r f i el ds{ ad~ -1 g i : l < - i < - m , l < - q < - j }

L e t E j d e n o t e t h e c o d is t r i b u t i o n s p a n n e d b y t h e se t o f o n e - f o r m s{ d L ~ - - ~ h i : l < _ i < _ p , l < _ q < _ j } .( C ) T h e d i s t r i b u t io n s D j, 1 < - j < - n , h a v e c o n s t a n t d im e n s i o n s a r o u n d

x o a n d d i m D " ( X o ) = n .( O ) T h e c o d i s t ri b u t i o n s E j, 1 - < j - n , h a v e c o n s t a n t d i m e n s i o n s a r o u n d

X o a n d d i m E " ( X o ) = n .S i n c e D ~ c D ~+~ b y d e f i n i t io n t h e n u n d e r ( C ) w e c a n d e f i n e a s e q u e n c e o f

i n t e g e r s t q , . . . , v , b y s e t t i n gv 1 = d i m D l ,

D j~,j = d i m DS_-----i, j > 1.W i t h th e h e l p o f ( v l , - - . , ~ ',) w e d e f i n e a n o t h e r s e q u e n c e o f in t e g er s K l . . . . , K ,.b y s e t t i n g

K~ = m a x (k : ~'k -> j ) . ( 3 )mO b s e r v e t h a t K1 - > . > - Km > 0 a n d ~ i = 1 K i = n . N o t e t h a t , f o r a l i n e a r s y s t e m , ( C )

r e p r e s e n t s t h e c o n t r o l l a b i l i ty c o n d i t i o n e x a c t l y a n d (K ~ . . . . , K in) t h e s e t o f c o n -t r o l l a b i l i t y ( o r B r u n o v s k y ) i n d i c e s [ 2 1 ] , [ 3 2 ] .

O b s e r v e t h a t i f ( C ) h o l d s t h e n w e c a n r e o r d e r t h e g ~'s i n s u c h a w a y t h a tD j = S p { a d ~ - 1 gi: 1

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66 D. Chen g, A. lsidori, W. Respondek, and T. J. TamA s f o r t h e c o n d i t i o n ( O ) , s i n c e E J c E j+~ b y d e f i n i ti o n , w e c a n d e f i n e a

s e q u e n c e o f i n t e g e r s o h , . , w , b y s e t t in gt r t = d im E 1 ,

E ~= d i m E------S, j > 1,a n d t h en a n o t h e r s e q u e n c e / z , , . . . , / ~ p g iv e n b y

/zj = m a x ( k : O'k -->j).O b s e r v e t h a t / z t - > - . . - - p . p _ - > 0 a n d ~ = l ~ = n . N o t e t h a t , f o r a l i n e a r s y s te m ,( O ) r e p r e s e n t s t h e o b s e r v a b i l i t y c o n d i t i o n e x a c t l y a n d (/~ , . . . . ,/ ~p ) t h e s e t o fo b s e r v a b i l i t y i n d i c e s [ 3 2 ] .I f ( O ) h o l d s w e c a n r e o r d e r t h e h i 's i n s u c h a w a y t h a t

E j = Sp {dL ~-~ h~ : 1

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Exact Linearization of Nonlinear Systems with Outputs 67P r o o f T h e e q u i v a l e n c e ( a ) c : > ( b ) h a s b e e n p r o v e n b y N i j m e i j e r [ 1 8 ] . W e s h o wt h e im p l ic a t io n ( b ) ~ ( c ) ~ ( d ) ~ ( e ) ~ ( a ) .

( b ) ~ ( c ) . F o r a n y s m o o t h v e c t o r f ie ld s X , Y , a n y s m o o t h f u n c t io n h , a n da n y i n t e g e r s - > 0 w e h a v e [ 1 7 ] , [ 2 6 ]Lad~:yh= k = 0 ~ ( - - 1 ) k ( S k )sx-kLYLkh" (4 )

U s i n g t h i s f o r m u l a a n d ( G 1 ) w e o b t a i nL~d~g,(L g L ~h~) = L ) L g L )L gL yh , =Ok = o ~ \ k J

f o rl

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68 D. Cheng, A. lsidori, W. Respondek, and T. J. Ta mH e n c e

L x , ( C i ( X ) ) = 0 f o r j = 1 , . . . , n ,i .e . , c i ( x ) = c o n s t a n t .

N o w d e n o t e Xq~ = a d ~ g~ , 1 < - i - m , 0 - < q - K~ - 1 . W e u s e t h e a b o v e r e s u l tf o r Y = a d ~ J g a n d { X t , . . . , X , } = { X q i : l < - i < - m , O < - q < - K i - 1 } . F r o m ( G 3 ) w eh a v e

[ X q , , a d ~ g ~] = 0a n d h e n c e

a d ? g j = ~ c " X , , (5 )f o r s u i t a b l e c o n s t a n t s c a ( d e p e n d i n g o n j ) .A s s u m e n o w t h a t

a d ~ g j = ~ . c " X t , .C o m p u t e

a d ~ +1 g j = [ f , a d ) g ~] = E c " [ f , X , , ] = Y . c " [ f , a d ) g , ] . ( 6 )W e h a v e t + 1 -< x~. T h u s , f r o m ( 5) a n d f r o m t h e d e f i n i t i o n o f X , , e v e r y s u m m a n do n t h e r i g h t - h a n d s i d e o f (6 ) b e l o n g s t o l i n { X , }. T h e r e f o r e , b y a n i n d u c t i o na r g u m e n t ,

a d ~ g j ~ l i n{ X a : 1 - i - m , 0 - t - K j - 1} (7 )f o r e v e r y 1 < - j - < m a n d s > - 0 .

N e x t s i n c e ( G 3 ) g i v e sL s ~ L ~ h ~ = c o n s t a n t f o r l < - i < - p , l < - j < - m , a n d O < - q < - K j - 1 ,

w e c a n s e e , u s i n g f o r m u l a ( 4 ) a g a i n , t h a tL a d ~ g , ( h ~ ) = c o n s t a n t f o r l < - i < - p , l < - j < - m , a n d O < - q < - K j - 1 . (8 )

A c c o r d i n g t o ( 7 ), e q u a t i o n ( 8 ) r e m a i n s t r u e f o r a n y q -> 0 . H e n c e i t is a l s o t r u e t h a tL a a T g , ( L ~ h ~ ) = c o n s t a n t f o r l < - i < - p , l < - j < - m , a n d a n y q , s > - O .

T h i s , i n p a r t i c u l a r , i m p l i e s t h e s e c o n d p a r t o f ( G 4 ) , i .e .,L g j L ~ ( h ~ ) = c o n s t a n t f o r l < - i < - p , l < - j < - m , a n d 0 - < s < - / z ~ - l .T o p r o v e t h e f ir st p a r t o f ( G 4 ) d e n o t e( g l , , a d.7 ~-~ g ~ , g2 . . . . , g in , - , a d . 7 " - ~ g , , ) = a ( X ~ , . . . , X , ) = X

a n d( d h l . . . . . d L ~ '- ~ h ~ , d h 2 , . . . , d h p , . . . , d L ; , - ~ h p ) T = H .

T h e n w e h a v eH ( x ) . X ( x ) = E (9 )

a n dd L ~ , h , ( x ) . X ( x ) = e , , (10 )

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Exact Linearization of Non linear Systems with Outputs 69w h e r e E i s a n n x n n o n s i n g u l a r c o n s t a n t m a t r i x a n d e~ i s a 1 x n c o n s t a n t r o wv e c t o r . S i n c e X ( x ) i s n o n s i n g u l a r t h e n ( 9 ) a n d ( 1 0 ) i m p l y th e f ir st p a r t o f ( G 4 ) .

( e ) ~ ( a ) . I n t ro d u c e n e w l oc a l c o o r d i n a te s a s ( z l , . . . , z~) r =( z , 1 , . . . , z l , , , Z E t , . . . , Zp,,,) T s u c h t h a tz i q = L ~ - l h ~ , l < _ i < _ p , l < _ q< _ l.~ i.

T h i s is a v a l id c h a n g e o f c o o r d i n a t e s b e c a u s e o f t h e o b s e r v a b i l it y a s s u m p t i o n( O ) . N o w ( G 4 ) i m p l i e s th a t in t he s e c o o r d i n a t e s w e h a v e

y~ = z i l ,~iq "~ Ziq+ l ~- ~ iqj g j ( z ) , 1 < q < ~ - 1,j= l~ i ~ , , = Y ~ c ~ ' z s , + ~ i ~ ,j g j ' ( z ) ,

j = lw h e r e g jq d e n o t e s t h e c o m p o n e n t s o f g j. F r o m

L s j L ~ h i = c o n s t a n t , l < - - j < - m , l < _ i < _ p , 0 _ < q _ < / z i - 1 ,w e d e d u c e t h a t g j-q (z) = c o n s t a n t a n d t h u s i n z c o o r d i n a t e s ( 1) ta k e s t h e l i n e a rf o r m ( 2 ). [ ]

R e m a r k 2 . C o n d i t i o n ( G 2 ) s e e m s t o b e s t a te d in t h e m o s t e l eg a n t a n d c o m p a c tf o r m . H o w e v e r , w e a ls o f o r m u l a t e ( G 3 ) a n d ( G 4 ) b e c a u s e t h e y a re h e l p f u l ins t u d y i n g f e e d b a c k i i n e a r i z a t i o n a s w e w i l l s e e l a t e r . M e a n w h i l e , f r o m t h e p r o o fo f T h e o r e m 1 w e c a n s e e t h a t c o n d i t i o n ( G 3 ) ( r e sp . ( G 4 ) ) c a n b e u s e d f o r t h ee q u i v a l e n c e o f ( 1) t o a c o n t r o l l a b l e ( r e s p . o b s e r v a b l e ) l i n e a r sy s t e m . W e s t a t e ita s f o l l o w s .C o r o l l a r y 2 . T h e n o n l i n e a r s y s t e m (1 ) i s e q u i v a l e n t t o a c o n t r o l la b l e l i n e a r s y s t e mo f t he f o r m (2 ) w i t h n o n z e r o C i i f a n d o n l y i f c o n d i t i o n s ( C ) a n d ( G 3 ) h o l d , a n dL g jL ~ h~ ~ O o r s o m e 1 < - j < - m , k < K j. R e s p e c t i v e l y , (1 ) is e q u i v a l e n t t o a n o b s e r v a b l el in e a r s y s t e m o f t he f o r m (2 ) w i t h n o n z e r o b j i f a n d o n l y i f c o n d i t io n s ( 0 ) a n d ( G 4 )h o l d , a n d L g , L ~ h i ~ 0 f o r s o m e 1 < - i < - p , k < I Z~ .R e m a r k 3 . I t c a n b e e a s il y d e d u c e d f r o m t h e p r o o f t h a t w e m a y st a te c o n d i t io n s( c) a n d ( d ) o f T h e o r e m 1 in a n e q u i v a le n t f o r m w h e n r e p l a c i n g e v e r y / z i a n d kjb y n . T h i s c r e at e s s o m e r e d u n d a n c y b u t a v o i d s c o m p u t i n g t h e c o n t r o l l ab i l it ya n d / o r o b s e r v a b i l i t y i n d ic e s . A n e q u i v a l e n t o f C o r o l l a r y 2 c a n a l so b e s t a t e d int h i s f o r m .

3 . F e e d b a c k L i n e a r iz a t io n o f S i n g l e - In p u t S i n g l e - O u t p u t S y s te m sI n t h i s a n d i n t h e f o l l o w i n g s e c t i o n s w e d i s c u s s l i n e a r i z a t i o n b y m e a n s o fd i f f e o m o r p h i s m t o g e t h e r w i t h s ta t e f e e d b a c k c o n t r o l

u=~ ( x ) +~ ( x ) v , ( 1 1 )

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70 D. Cheng , A. lsidori, W. Respon dek, and T. J. Ta mw h e r e a is a s m o o t h ( m v e c t o r ) - v a l u e d f u n c t i o n , / 3 is a s m o o t h ( m x m m a t r ix ) -v a l u e d f u n c t i o n , b o t h d e f i n e d i n a n e i g h b o r h o o d o f X o. I t is a l s o a s s u m e d t h a t/ 3 ( x ) is n o n s i n g u l a r f o r a ll x i n t h is n e i g h b o r h o o d .N e c e s s a r y a n d s u ff ic ie n t c o n d i t io n s f o r l i n e a ri z a t i o n o f t h e i n p u t - o u t p u tb e h a v i o r h a v e b e e n e s t ab l is h e d in [ 1 3]. I n t h is p a p e r w e s h o w h o w t h e s e c o n d i t io n sh a v e t o b e s t r e n g t h e n e d i n o r d e r t o o b t a i n , v i a f e e d b a c k a n d l o c a l s t a t e - s p a c ed i f f e o m o r p h i s m , a f u l l y l i n e a r s t a t e - s p a c e d e s c r i p t i o n .

W e d e a l f ir st w i t h th e c a s e o f a s i n g l e - i n p u t s i n g l e - o u t p u t s y s t e m , w h i c h ism u c h e a s i e r . I n t h i s c a s e w e r e c a ll t h a t a s y s t e m

= f ( x ) + g ( x ) u , u e R , (12)y = h ( x ) , y ~ R ,

i s s a i d t o h a v e c h a r a c t e r i s t i c n u m b e r S a t Xo i fL g L ~ h ( x ) = 0 f o r a ll x n e a r X o, f o r a ll k < 8 ,LgL~h(xo) ~ O.U s i n g t h i s n o t a t i o n , w e c a n s t a t e t h e f o l l o w i n g r e s u l t .

T h e o r e m 3 . S ys t em (12) is equivalent to the control lable l inear sys tem ( 2 ) , withnonzero C, under a d i f feomorph ism ~b and a s ta te f ee db ac k o f the fo rm ( 1 1 ) , i f a n donly i f :

( i) g, a d y g , . . . , a d 7 - ~ g are l inear ly independen t a t Xo .( i i ) The sys tem has charac ter i st ic num ber 8 < n a t Xo .

( i i i ) The vector f ie lds f an d ~, def ined byf ( x ) = f ( x ) + g ( x ) a ( x ) , (13)i f ( x ) = g ( x ) / 3 ( x ) , (14)w i th a ( x ) a n d / 3 ( x ) g iv en b y

- L ~ + ' h ( x ) 1a ( x ) = L g L } h ( x ) ' / 3 ( x ) = L~L~h(x)~ ' (15)are such tha t the n + 1 vec tor f i e lds ~ , a d ? if, . . . . a d ~ f f a r e co mm u ta t ive .

P r o o f ( N e c e s s i t y ) . W i t h o u t l o ss o f g e n e r a l i t y , w e m a y a s s u m e th a t ( 2 ) is aB r u n o v s k y c a n o n i c a l f o r m [ 2 1 ]. S e t z = 6 ( x ) . I t f o l lo w s t h a t

f ' ( z ) = ( 6 , 2 9 6 - ' ( z ) = ( a ( z ) , z l , . . . , z n - , ) T ,

g ' ( z ) = ( ~ b . f ) o 6 - 1 ( z ) = ( b ( z ) , 0 , . . . , 0 ) T,h ' ( z ) = h o 6 - 1 ( z ) = c ~ z~ + . . . + c . z n ,

w h e r e 4 ) . d e n o t e s t h e d if f e re n t ia l o f 6 , a a n d b a re s m o o t h f u n c t i o n s , b ( z ) # 0is n o n z e r o f o r a ll z n e a r 6 ( X o ), a n d C l , . . . , c , a r e r e a l n u m b e r s . F r o m t h is , as i m p l e c o m p u t a t i o n s h o w s t h a t ( i) is t r u e .

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Exact Linearization of No nline ar Systems with Ou tputs 71A s f o r ( i i ) , n o t e t h a t L g L ~ h ( x ) = k ,L g , L } , h ) o 4 9 ( x ). A s s u m e c l . . . . ck = 0

a n d Ck+1 ~ O. A n e a s y c o m p u t a t i o n s h o w s t h a t ~ 'g ,L ' f .h ( z ) = 0 f o r a l l i -< k - 1 a n dL , . L ~ , h ' ( z ) = C k + l b ( z ) . T h e l a t t e r is n o n z e r o a t z = ~ b(X o) a n d t h e r e f o r e B = k 6 + 1 .

T h u s , L g L ~ h = c o n s t a n t f o r a l l 0 - k - n - 1.T h e s y s t e m

= 7 ( x ) + ~ ( x ) v ,y = h ( x )

s a ti s f ie s c o n d i t i o n ( C ) ( w h i c h is f e e d b a c k i n v a r i a n t [ 3 ]) a n d c o n d i t i o n ( G 3 ) o fT h e o r e m 1. H e n c e ( c o m p a r e C o r o l l a r y 2 ) i t i s e q u i v a l e n t t o a c o n t r o l l a b l e l i n e a rs y s t e m v i a a d i f f e o m o r p h i s m . T h i s c o m p l e t e s t h e p r o o f . [ ]

I n . c a s e t h e s y s t e m h a s o n e i n p u t a n d p > 1 o u t p u t s , t h e p r e v i o u s r e s u l t c a nb e e a s i l y e x t e n d e d t o y i e l d t h e f o l l o w i n g s t a t e m e n t .

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7 2 D . C h e n g , A . I s i d o ri , W . R e s p o n d e k , a n d T . J . T a m

Corollary 4 . S u p p o s e m = 1 . T h e n s y s t e m (1) i s e q u i v a l e n t t o a c o n t r o l l a b l e l i n e a rs y s t e m (2 ) , w i t h n o n z e ro C , u n d e r a d i f f e o m o r p h i s m b a n d a s ta t e f e e d b a c k o f t h ef o r m ( I 1 ) , i f a n d o n l y i f :( i) g , a d f g , . . . , a d ~ - ' g a r e li n e a rl y i n d e p e n d e n t a t X o.

(ii) F o r s o m e j , 1 < - j < - p , t h e t r ip l e t { f , g , h i} h a s c h a r a c t e r i s ti c n u m b e r 8 j < na t x o .( i i i ) The v e c t o r f i e l d s (13) a n d (14) , w i t h a a n d f l s t il l a s i n (15) b u t w i t h hr e p l a c e d b y h i , a r e s u c h t h a t t h e n + 1 v e c t o r f i e l d s g , ad ~ , . . . , a d ~ ~ a r ec o m m u t a t i v e .

( iv) L ~ L ~ h i = c o n s t a n t f o r a l l 0 < - k < _ n - 1 a n d a l l 1 < - i < - p , i # j .

4. Input-Output Linearization, via Feedback, o f M ultivariable SystemsI n o r d e r t o d e v e l o p a n e x t e n s i o n o f T h e o r e m 3 to s y s te m s w i t h m > 1 i n p u t s a n dp > 1 o u t p u t s , i t i s c o n v e n i e n t to s u m m a r i z e f ir st s o m e k n o w n f a c t s a b o u t l i n e a ri z -a t i o n o f t h e i n p u t - o u t p u t b e h a v i o r . W e s y n t h e s i z e h e r e s o m e r e s u lt s f r o m [ 1 3]a n d , i n a d d i t i o n , w e s h o w t h a t t h e t e s t f o r i n p u t - o u t p u t l i n e a r iz a b i l i ty c o n t a i n e dt h e r e i n c a n b e s u b s t a n t i a l l y i m p r o v e d .W e r e c a ll t h a t a sy s t e m o f t h e f o r m ( 1) i s s a id t o b e i n p u t - o u t p u t i i n e a r iz a b l ea r o u n d Xo b y a f e e d b a c k o f t h e f o r m ( 11 ) i f t h e i m p l e m e n t a t i o n o f s u c h a f e e d b a c ky i e ld s a n i n p u t - o u t p u t b e h a v i o r o f t h e f o r m

y ( t ) = q ( t , X o )+ ~ k i ( t - r ) v i ( 1 " ) d r.i= 1F r o m t h i s d e f in i t io n , i n vi e w o f s o m e w e l l - k n o w n p r o p e r t i e s o f t h e i n p u t -

o u tp u t ex p an s io n s o f s y s t em s o f t h e fo rm (1 ) ( s ee , e. g ., [2 0 ] ) , it i s n ece s s a ry t os t a t e t h e fo l l o win g r e s u l t .L e m m a 5 [ 1 3 ]. S y s t e m (1) i s i n p u t - o u t p u t l in e a r iz a b le a r o u n d X o b y t h e f e e d b a c k( 1 1 ) i f a n d o n l y i f L ~ L ~ h ( x ) = c o n s t a n t a r o u n d X o f o r a l l k >- O , w h e r e f = f + g aa n d p, = g B .

I n o r d e r t o d e r iv e e x is t e n c e c o n d i t i o n s , a n d a c o n s t r u c t iv e p r o c e d u r e y i e l d i n gs u c h a f e e d b a c k , i t i s u s e f u l t o p r o c e e d i n t h e f o l l o w i n g w a y . F i r s t , w e i m p r o v eL e m m a 5 b y s h o w i n g t h a t a n y l i n e a r i z i n g f e e d b a c k i s c h a r a c t e r i z e d b y a f i n i t es e t o f c o n s t r a i n t s.L e m m a 6. C o n s i d e r a s y s t e m

~ = f ( x ) + g ( x ) u = f ( x ) + ~ g i ( x ) u i ,i= l

y = h ( x ) = ( h i ( x ) , . . . , h , ( x ) ) Tw i t h x ~ M , a n n - d i m e n s i o n a l m a n i f o ld . I f

L s L ~ h ( x ) = c o n s t a n tf o r a l l 0

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Exact Linearization of Non linear Systems with Outputs 73P r o o f . R e c a l l t h a t ( se e p . 4 9 o f [ 2 7 ]) , o n a n o p e n a n d d e n s e s u b m a n i f o l dM ' ~ M , th e la r g es t c o d i s tr ib u t io n Q in v a r i a n t u n d e r f , g ~ , . . . , g m w h i c hc o n t a i n s s p a n { d h ~ , . . . , d h p } i s l o c a l l y s p a n n e d b y e x a c t o n e - f o r m s s u c ha s t o = d ( L g , o . . . L g , h j ) w h e r e r < - n - 1 a n d O < - i k < - - m , w i t h g o = f S i n c e ( 1 7 )h o l d s w e s e e , in p a r t ic u l a r , th a t Q is l o c a l ly s p a n n e d b y e x a c t o n e - f o r m s su c ha s t o = d L ~ h j , l < - j < - p , O < - k< - rj . L et d i m ( Q ) = q a n d c h o o se n e wc o o r d i n a t e s ( z ~ , . . . , Zq, ~ , . . . , 2 , _ q ) = 4 , ( x ) w i t h t h e z / s f u n c t i o n s i n t h e s e t{ L ~ h j ; l < - j < - p , 0 < - k _< ~ }. U s i n g t h e s e c o o r d i n a t e s s y s t e m ( 1 6) b e c o m e s ( s e e ,e .g ., p p . 2 9 - 3 0 o f [ 2 7 ] )

= f ~ ( z ) + g ~ ( z ) u , ( 1 8 a )~" = fz (z , ~ ) + g2(z , 2)u , (18 b)y = h , ( z ) . ( 1 8 c )

C l e a r l y , ( L g L ~ h ) o c b - ~ ( z ) = L g L ~ , h ~ ( z ) f o r a ll k - 0 . F r o m t h is a n d ( 1 7 ) i t isp o s s i b l e t o d e d u c e t h a t

a d ~ , g l r ( Z ) = c o n s t a n t , l < - r < - m , O < - s < - n . ( 1 9 )A s a m a t t e r o f fa c t , a c o n s e q u e n c e o f ( 1 7) i s t h a t

c o n s t a n t = Lg, L~ ,L~,h l j 5 k ( -1) ~(d L~ -h~ j , a s( - 1 ) L a d ) ,g , L ~ h ~ j = d f , g l r ) .S i n c e e a c h o f t h e z ~ 's h a s a f o r m z~=Lkrh~: ( w i t h k < _ n - 1 ) t h e l a s t e q u a t i o np r o v e s t h a t

(dz~, a d } , g ~ ,) = i t h e l e m e n t o f a d~ , g~, = c o n s t a n t f o r a l l s -< n ,a s r e q u i r e d .

R e t u r n n o w t o e q u a t i o n s ( 1 8 a ) , ( 1 8 c ) a n d n o t e t h a t t h e i t h r o w o f g ~ ( x ) ,h a v i n g a f o r m (d z~ , g ) = L g L : h ~ , i s c o n s t a n t , i .e . , g ~ ( x ) = B , w h e r e B is a c o n s t a n tq x rn m a t r i x . T h e e l e m e n t s o f h ~ ( z ) a r e p r e c i s e l y s o m e o f th e z i ' s a n d t h e r e f o r ea l s o h ~ ( z ) = C z z , w h e r e C ~ i s a c o n s t a n t p x q m a t r i x . U s i n g ( 1 9 ) w e d e d u c e as i m i l a r p r o p e r t y f o r ( a t l e a st a d e c o m p o s i t i o n o f ) f ~ ( z ).

L e t R d e n o t e th e s m a l le s t d i s tr i b u ti o n i n v a r i a n t u n d e r f ~ , g ~ , . . . , g tm w h i c hc o n t a i n s s p a n { g H , . . . , g ~r~}- A n e a s y c o m p u t a t i o n ( s ee , e.g ., p p . 3 3 - 3 9 o f [2 7 ] )s h o w s t h a t i n t h is c a s e , b e c a u s e o f ( 1 9) , R h a s t h e e x p r e s s i o n

R = S p a n { a d ~ , g t, ; 1

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74 D. Cheng, A. Isidori, W. Respondek, and T. J. TamIn addition, it is easy to see thatf ' ( z ' , z " ) = A ' z ' + y ( z " ),

where A is a constant matrix, because all the vectors in (19) (including the onecorr espo ndin g to s = n) are constant. F rom (20) and (18b), a direct calculationshows that

L g L ~ h ( x ) = C ' ( A ' ) k B ' for all k->0.Thus, L , L ~ h ( x ) is constant around any point of M', an open and dense subsetof M. Being a smooth function, it is cons tant for each x ~ M. []

This result is rather important because it shows that in order to find feedbackwhich linearizes the input-output behavior of (1) we only have to satisfy theconditions

L. L~ h ( x ) =constantfor all k -< 2n - 1.

To check whether or not such a feedback exists and to calculate it, we recallthe results of [13]. Set

T k ( X ) = L g L ~ h ( x )and consider a sequence of Toeplitz matrices

" To (x ) T , ( x ) . . . T k ( X ) ]M k ( X ) = 0 T o ( x) " ' " T k - , ( X ) I

. . . . . . . . . . . . . . . . . . . . . . , , . I0 0 " ' " T o ( x ) ]with k -< 2n - t.

We say that Xo is a regular point for M , if the rank of M k ( X ) is constant forall x near Xo. If this is the case we denote the r ank of M k ( X o ) by r K ( M k ) . WithM k we associate another integer, the dimension of the vector space generatedover the field R by the rows of M k , and we denote this number by r n ( M k ) .Clearly, rR (M k) >-- rK (M k) .

Using this notation it is possible to state the following result, a synthesis ofthe main theorem of [13] and Lemma 6.Theorem 7. S y s t e m (1) is i n p u t - o u t p u t l in e a r iz a b le a r o u n d X o b y t h e f e e d b a c k(11) i f a n d o n l y if, f o r a l l k

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Exact Linearization of Nonlinear Systems with Outputs 75Algorithm. Step 1. Suppose Xo is a regular po int of Mo. If and only if rR(Mo) =rn (Mo) there exists a nonsingular matrix of real number s, denoted by

where P~ performs row permutations, such thatV , T o ( x ) = [ S ' ( o X ) ] ,

where S , ( x ) is an ro x m matrix and rank S , ( x o ) = t o . Set6, = ro,7,(x) = P , h ( x ) ,~ / , ( x ) = K l h ( x )

and note thatL ~ y , ( x ) = S , ( x ) ,L g ~ , ( x ) = 0 .

I f T o ( x ) = O, t h e n P~ m u s t b e c o n s i d e r e d a s a m a t r i x w i t h n o r o w s a n d K 1 is t h eidentity matrix.

Step i. Consider the matrixL~v,(x)

= r s ' - ' ~ x : ' lL g y , - , ( x ) L L g L : 9 , - , ( x ) JL , L : y , _ , ( x )and let Xo be a regular point of this matrix. If and only if

r s , - , 1 r ' , - , 1R L L ~ L [ 9 , - , J = rK L L = L y ~ , _ , Jthere exists a nonsingular matrix of real numbers, denoted byL o 0 1

-. 0 PiK ' , . - . K I - , ~ C l

where P~ performs row permutations, such that[ L g 3 q . ( x )~ ' l ~ , , , : , ~ x , _ _ s f , ] ,, [ L g L 1 ~ , - , ( x )

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76 D. Cheng , A. Isidori, W. Respon dek, and T. J. Ta mw h e r e S i ( x ) i s a n r i -i x m m a t r i x a n d r a n k S i ( x o ) = r ~ _ ~ . S e t

~ i = r i - I - - r i - 2 ,yi (X) = P i L f ' Y i - i ( x ) ,~ , ( x ) = K 'I y , ( x ) + " " " + K I - , y , - , ( x ) + K ~ L f f / , _ , (x )

a n d n o t e th a t

= S i ( x ) ,L g y , ( x )

L ~ i ( x ) = O.I f t h e r a n k c o n d i t i o n is s a t is f i e d b u t t h e l a s t p - r~ -2 r o w s o f t h e m a t r i x d e p e n do n t h e f ir s t r~ -2 , t h e n t h e s t e p d e g e n e r a t e s , P~ m u s t b e c o n s i d e r e d a s a m a t r i xw i t h n o r o w s , K I t h e i d e n t i t y m a t r i x , 6~ = 0 , a n d S ~ ( x ) = S ~ _ J x ) .

A f t e r t h is a l g o r i th m h a s b e e n c y c l e d 2 n t i m e s , a s et o f fu n c t i o n s Y ~ , . . . , "/2 ,is d e f i n e d ( n o t e t h a t s o m e o f t h e m a r e o n l y " f o r m a l l y " c o u n t e d i n t h e s e q u e n c e ,t h e o n e s c o r r e s p o n d i n g t o d e g e n e r a t e s t e p s o f th e a l g o r i th m , w h i c h d o n o t a c t u a l l ye x i s t ) f r o m w h i c h a l i n e a r i z i n g f e e d b a c k c a n b e c o n s t r u c t e d . S e t

F ( x ) = [ y ' (: x ) ] ( 21 )L j

a n d r e c a l l t h a t S 2 . = L g F i s a n r z . _ l X m m a t r i x , o f r a n k r 2 . _ 1 a t X o. T h e n t h ee q u a t i o n s

[ L g F ( x ) ] a ( x ) = - L j F ( x ) , (2 2)[L~r(x)] /3(x) = [ I ~ . _ , 0 ] ( 2 3 )

c a n b e s o l v e d fo r a a n d / 3 i n a n e i g h b o r h o o d o f X o. T h e s e a a n d / 3 a r e s u c h a st o i m p o s e ( s e e [ 1 3 ] )L e L ~ h ( x ) = c o n s t a n t

f o r a ll k - < 2 n - 1 a n d t h e r e f o r e s o lv e t h e i n p u t - o u t p u t l in e a r iz a t io n p r o b l e m .

5. Feedb ack Linearization of M ulti -Inpu t-M ulti -Ou tputNonlinear Systems

I n t h is s e c t i o n w e c o n s i d e r n o n l i n e a r c o n t r o l s y s t e m s o f t h e f o r mY c = f ( x ) + g ( x ) u = f ( x ) + ~ I d i g i ( x ) , x ~ R " , x ( O ) = x o ,i = 1y = h ( x ) = ( h i ( x ) , , h p ( x ) ) " r. ( 2 4 )

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Exact Linearization of Non linear Systems with Outputs 77W e s t u d y t h e f e e d b a c k e x a c t l i n e a r i z a t i o n p r o b l e m , i . e . , w e s e e k a f e e d b a c kp a i r ( a , / 3 ) s u c h t h a t t h e f e e d b a c k m o d i f i e d s y s te m i s e q u i v a l e n t , b y a ( l o c a l )c o o r d i n a t e s c h a n g e , t o a l in e a r sy s t em o f t h e f o r m

~ = A z + Bu = A z + ~ u ib~,i = ! (25)y = C z .

T h e m a i n r e s u l t o f t h is s e c t io n s h o w s t h a t i f s u c h a f e e d b a c k e x i s ts t h e n i t c a nb e c o m p u t e d v i a t h e s t r u c tu r e a l g o r i t h m g i v e n i n S e c t i o n 4 , th u s g e n e r a l i z i n g th er e s u l t o f S e c t i o n 3 . I n th e c a s e o f u n i q u e n e s s o f f e e d b a c k ( 2 2 ) - ( 2 3 ) w h i c hl i n e a r iz e s th e i n p u t - o u t p u t m a p ( i .e ., w h e n r an k L g F ( x o ) = m ) t h i s g i v e s av e r i f ia b l e c o n d i t i o n f o r f e e d b a c k e x a c t l in e a r i z a t i o n . T h e n w e f o r m u l a t e a n dp r o v e a r e s u l t s t at in g t h a t a n y c o n t r o l l a b l e l i n e a r s y s t e m c a n b e m a d e o b s e r v a b l ev i a s t a t i c f e e d b a c k . T h i s l e a d s u s t o a f e e d b a c k e x a c t l i n e a r i z a t i o n c o n d i t i o n ,d u a l i n a s e n s e t o th e m a i n t h e o r e m , w h i c h d o e s n o t i n v o l v e c o m p u t i n g t h e L i eb r a c k e t s .

W e h a v e t h e f o l l o w i n g .T h e o r e m 8 . S ys t em (2 4 ) i s f eedb ack exac t l ineari zab le to a con tro l lab le sys tem o ft h e f o r m (2 5 ) i f and on ly i f :

( i ) ( 2 4 ) satisfies the controllability condition ( C ) ,( i i ) (24) i s inpu t -o u tpu t l inear izab le in a ne ighborhood o f xo ,

( i i i ) there exis ts a solut ion (a , [3) o f (2 2 ) , ( 2 3 ) such tha t[ a d ~ g , , a d ~ f f j ] = 0 , f o r l< - i, j < - m , O < - q + r < - 2 n - 1 ,

w he re f = f + g a , g = g /3 . (2 6 )R e m a r k 4 . I f t h e s o l u t i o n o f ( 2 2 ) , ( 2 3 ) i s u n i q u e ( i. e ., w h e n r a n k L g F ( x o ) = m ,a n d t h i s i n c l u d e s , i n p a r t i c u l a r , s i n g l e - i n p u t m u l t i - o u t p u t s y s t e m s , s e e S e c t i o n 3 )t h e a b o v e r e s u l t g iv e s a v e r if ia b l e c o n d i t i o n f o r f e e d b a c k e x a c t l in e a r i z a t io n .R e m a r k 5 . T h e r e is a l o t o f r e d u n d a n c y i n ( 2 6 ). T o a v o i d i t w e m a y f o r m u l a t et h e a b o v e t h e o r e m , a s w ill b e c le a r f r o m t h e p r o o f , i n th e f o l l o w i n g f o r m : ( 2 4)is f e e d b a c k e x a c t l i n e ar i z ab l e t o a c o n t r o l l a b l e s y s t e m o f t h e f o r m ( 2 5 ) i f a n do n l y i f :

( ii ) ( 2 4 ) i s i n p u t - o u t p u t l i n e a r i z a b le in a n e i g h b o r h o o d o f X o,( i i ') t h e r e e x i s t s a s o l u t i o n ( a , [3 ) o f ( 2 2 ) , ( 2 3 ) s u c h t h a t t h e s y s t e m

.~ = f + ~u (2 7 )s a ti sf ie s th e c o n t r o l l a b i l it y a s s u m p t i o n ( C ) a n d[ a d ~ g ~ , a d } f f j ] = O , I

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7 8 D . C h e n g , A . I s i d o r i , W . R e s p o n d e k , a n d T . J . T a m

P r o o f (Su f f i c i en cy ) . W e h av e f = f + g a , ~ = .gfl, wh ere a an d / 3 a r e s o l u t i o n so f ( 2 2 ) a n d ( 23 ), r e s p e ct iv e l y . I t f o l l o w s f r o m t h e p r o o f o f T h e o r e m 7 ( se e [ 13 ]o r [ 2 7] ) t h a t ( a , f l ) s a ti s f y i n g t h o s e e q u a t i o n s y i e l d s ( c o m p a r e L e m m a 5 )L i L ~ h i = c o n s t a n t , l < _ i< - p, l < _ j ~ - m , O < - q < - n - 1 . (29)N o w c o n s i d e r t he s y s te m

= f ( x ) +~ , ( x ) u , y = h ( x ) . (30)O bser ve tha t (26) im pl ies , in par t i c u la r , t h a t / 5 j = sp{ ad ~- ' g'~, 1 -< i -< m, 1 < q < j}a r e i n v o l u t iv e fo r j = 1 , . . . , n a n d t h e r e f o r e ( s e e [3 ] f o r t h e p r o o f ) t h e y a ref e e d b a c k i n v a r i a n t . T h u s t h e c o n t r o l l a b i l it y a s s u m p t i o n ( C ) h o l d s f o r ( 30 ). T h e r e -fo re , (3 0 ) s a ti s fi e s co n d i t i o n s (C ) an d (G 3 ) ( s ee (2 8 ) an d (2 9 )) o f Th eo re m 1 an dt h u s ( c o m p a r e C o r o l l a r y 2 a n d R e m a r k 3 f o l l o w i n g it ) is l i n e a r i z a b l e v iac o o r d i n a t e s c h a n g e . H e n c e ( 24 ) is f e e d b a c k e x a c t l in e a r i z a b l e u n d e r t h e f e e d b a c ku = a + ~ v .

( N e c e s s i t y ) . I t i s w e l l k n o w n t h a t ( C ) i s a n e c e s s a r y c o n d i t i o n a n d i t i so b v i o u s t h a t i f ( 2 4 ) is f ee d b a ck ex ac t li n ea r i zab l e , t h en i t i s, i n p a r t i cu l a r , f e e d b a cki n p u t - o u t p u t l in e a r iz a b l e . T h i s g iv e s ( ii ).

Le t u s co n s i d e r co n d i t i o n ( i i i ) . As s u m e t h a t (2 4 ) i s ex ac t l y l i n ea r i zab l e v i af e e d b a c k u = ti + f i r . T h e r e f o r e t h e r e e x i st c o o r d i n a t e s z = ( z , , z2) s u c h t h a t

z l = A z , y = C z,~2 = v = - ( f ) - ' 5 + ( f ) - ' u . (3 1)

W e s h o w t h a t i n z - c o o r d i n a t e s t h e m a t r i x f u n c t i o n F ( z ) g i v e n b y ( 2 1 ) i s l i n e a r ,w i t h r e s p ec t t o z . To s ee t h is , o b s e rv e t h a t y~ an d ~ a re l i n ea r in z s i n ce t h eya r e r e o r d e r e d o u t p u t f u n c t i o n s . N o w r e c a ll t h a t s i n c e L g ~ _ , = 0 t h e n Lf@ ~_~ isl i n ea r i n z ( s ee th e fo rm o f f i n (3 1 ) ) an d t h u s , i n p a r t i cu l a r , y ~ i s a l s o l i n ea r inz . Th u s we h av e

F ( z ) = DlZ l + D'Z2a n d

L ~ r ( z ) = D 2 ( / 3 ) - ' .Ob s e rv e t h a t t h e S t ru c t u re "Al g o ri th m i m p l i e s r an k D~ = r_ ,,_ ,. Th e re fo re t h e ree x i s t m a t r i c e s D * a n d i n v e r t i b l e ( m x m ) / 5 * s u c h t h a t

D 2 D * = L 2 . _ , a n d Dz/5*=( lr2 ._ , 0 ) .D e f i n e t h e f o l l o w i n g f e e d b a c k

a = ( ~ - f D * D , A z ) , (32a)f l = f D * . . ( 3 2 b )

W e s h o w t h a t ( 3 2a , b ) s a t i s f y (2 2) a n d ( 2 3) , r e s p e c ti v e l y . C o m p u t e( L g F ) a = D 2 f-~ (t~ - f D ~ D ~ A z ) = D 2 f l -~& - D ~ m z

a n d- L f r = - ( D , A z - D 2 f - ' f f ) ,

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Exact Linearization of Nonlinear Systems with Outputs 79

w h i c h p r o v e s ( 2 2 ) . T o p r o v e ( 2 3 ) c o m p u t e( L ~ F ) f l = ( D 2 ~ - ' ) ( ~ O ' ~ ) = [ I , ~ . _ , o ].

N o w w e s h o w t h a t ( a , f l ) g i v e n b y ( 3 2 a , b ) s a t is f ie s ( ii i) . I n z - c o o r d i n a t e s w e h a v eA zg = g f l = ( ~ * 2 * ) a n d ] = f + g a = [ _ D . D , A z ] ,

b e c a u s e - f l - ~ + f l - ~ ( 6 - f l D * D ~ A z ) = - D * D ~ A z . T h e r e f o r e t h e s y st e m 2 =" f + ~ u i s l i n e a r i n z - c o o r d i n a t e s a n d t h u s s a ti s fi e s ( ii i) . [ ]

A s w e a l r e a d y m e n t i o n e d i n R e m a r k 4 , c o n d i t i o n ( ii i) o f T h e o r e m 8 is n o te a s y t o v e r i f y e x c e p t f o r t h e c a s e r2 ~_ ~ = m , i .e ., w h e n r a n k L g r ( x o ) e q u a l s m .T h e n t h e s o l u t i o n ( a , f l ) o f (2 2 ) , ( 2 3 ) is u n i q u e . B e s i d e s, t o v e r i f y t h e c o m m u t i v i t yo f a s e t o f v e c t o r fi el d s i s a t e d i o u s c o m p u t a t i o n . T o a v o i d i t, w e i n t r o d u c e a na l t e r n a ti v e t h e o r e m . F i rs t w e n e e d t h e f o l l o w i n g l e m m a w h i c h is it s e lf i n t er e st in g .L e m m a 9 . C o n s i d e r a l i n e a r s y s t e m

Yc = A x + B u ,y = C x . (33)

I f (A , B ) i s c o n t r o l l a b l e a n d C i s n o t i d e n t i c a l l y z e r o , t h e n t h e r e e x i s t s a f e e d b a c kF s u c h t h a t ( A + B F , C ) is o b s e r va b l e .P r o o f W i t h o u t l os s o f g e n e r a l it y , w e a s u m e t h e o u t p u t is si ng l e. S i n c e f o r t h ec o n t r o l l a b l e s y s t em t h e r e a lw a y s e x i st s a f e e d b a c k F s u c h t h a t ( A + B F , B ) h a sa B r u n o v s k y c a n o n i c a l f o r m [ 2 1 ], w e c a n a l s o a s s u m e t h a t t h e s y s t e m i s i n t h eB r u n o v s k y f o r m .

F i rs t w e p r o v e t h e l e m m a f o r si n g le i n p u t c a s e:

A=o l 10 10 0 - - - 0 1 ]

/o o . . - o o . I i ]=T a k e F = (A , 0 , . . . , 0 ), w h e r e A is a re a l p a r a m e t e r , t h e nI0 10 0 1A + B F . . . . . . . . . . . . . . . . . . . . . . .0 0 . . . 0 1h 0 . . . 0

U s i n g t h i s , t h e o b s e r v a b i l i t y m a t r i x I sI C l C 2 . . e n ]D : = A C n C ; " ' " C , - ] [ .

A C n - 1 A C n C l . . C n _ 2 [1Ac2 A c a " A c . c ~ l.I

( 3 4 )

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80 D. Cheng, A. Isidori, W. Respondek, and T. J. TamA s s u m e c~ ~ 0 . T h e n

de t D = + ( c~ ) ~ - ~+ l o w e r - o r d e r t e r m s ,s o w e c a n c h o o s e A s u c h t h a t d e t D ~ 0 . I f cn = c~ _~ . . . . . c n _ , = 0 , c ~ -r -~ # 0,r < n - 2 . T h e n

d e t D = + ( C r )~ A ' -~ + l o w e r - o r d e r t e r m s .H e n c e t h e r e a l s o e x is t s s u c h A w h i c h m a k e s d e t D # 0 . I f cn = c , - t . . . . . c2 = 0 ,t h e n c~ # 0 s i n c e C # 0 . S e t A = 0 , t h e n d e t D # 0 .

T h e m u l t i - in p u t c a s e f o l lo w s i m m e d i a t e l y f r o m t h e s i n g l e -i n p u t c a s ea n d H e y m a n ' s l e m m a [ 3 0 ] ( se e a ls o p . 4 9 o f [3 1 ] ) w h i c h s t a te s t h at , f o r a n y c o n -t r o l l a b l e ( A , B ) a n d a n y b e I m B , t h e r e e x i s ts F s u c h t h a t ( A + B F , b ) i s c o n t r o l -l a b l e . [ ]

L e m m a 9 i m m e d i a t e l y g i v e s t h e f o l l o w i n g .C o r o l l a r y 1 0 . S y s t e m ( 2 4 ) i s f e e d b a c k e x a c t l i n e a r i z a b l e t o a c o n t r o l la b l e a n do b s e r v a b le s y s t e m o f t h e f o r m ( 2 5 ) i f a n d o n l y i f ( 2 4 ) s a t i s f i e s ( C ) a n d c o n d i t i o n s( i i ) a n d ( i i i ) o f T h e o re m 8 .P r o o f . I f (2 4 ) s a ti s fi e s ( C ) , ( i i ), a n d ( ii i) , t h e n , a c c o r d i n g t o T h e o r e m 8 , i t c a nb e t r a n s f o r m e d v i a f e e d b a c k a n d c o o r d i n a t e s c h a n g e t o a c o n t r o l l a b l e l i n e a r f o r m( 25 ). T h e n a p p l y i n g m o r e f e e d b a c k t o t h e l i n e a r s y s t e m o b t a i n e d , a s i n d i c a t e db y L e m m a 9 , w e g et a c o n t r o l l a b l e a n d o b s e r v a b l e f o r m . T h e n e c e s s i ty p a r t iso b v i o u s . [ ]

N o w w e s t a te t h e d u a l l i n e a r i z a t i o n r e s u lt .Th e o r em 11 . S y s t e m ( 2 4 ) i s f e e d b a c k e x a c t l i n e a r i z a b l e t o a c o n t r o l la b l e s y s t e mo f th e f o r m ( 2 5 ) i f a n d o n l y i f :

( i ) ( 24 ) s a t i s f i e s ( C ) .( i i ) T h e r e e x i s t a f e e d b a c k , p a i r ( a , f l ) ( d e f i n e d i n a n e i g h b o r h o o d o f x o ) s u c ht h a t

L ~ L ~ h ~ = c o n s ta n t , l < j < - m , l < - i < -p , 0 - < q < - 2 n - 1 , ( 3 5 )a n d

d hr a n k . = n , ( 3 6 )I L ? / ' d h J x = x o

w h e r e f = f + g a a n d ~ = g fl .R e m a r k 6 . O b s e r v e t h a t th e a b o v e t h e o r e m ( s ee th e p r o o f ) , in f a c t, g i ve s t h en e c e s s a r y a n d s u ff ic ie n t c o n d i t i o n f o r ( 24 ) t o b e f e e d b a c k e x a c t l in e a r i z a b l e t oa c o n t r o l la b l e a n d o b s e r v a b l e s y s t e m o f t h e f o r m ( 25 ). T h e r e f o r e it c a n b e s e e na s a d u a l t o C o r o l l a r y 1 0 .

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Exact Linear izat ion of Nonl inear Sys tems with Ou tputs 81P r o o f ( S u f f ic i e n c y ) . W e s h o w t h e c o m m u t a t i v i t y o f { a d ~ g i, 1 -< i - m , 0 < q - < n }.U s i n g f o r m u l a ( 4 ) w e g e t

L[ad~ ,ad~ffi]L~hs = La d~ ,( Zad~ffl( ~ h s ) ) -- L ad ~,( Zad~.f.,(L~h ~))t q - t t k

- g a a ~ f , ( , = ~ o ( - a ) ' ( ; ) L } - t L ~ , L } L } h s ) = Of o r a n y l < - i , j < - m , l < - s< - p , O < - q , r < - n, O < - k < - - n - 1 , b e c a u s e o f (3 5 ) . S in c ea m o n g t h e f u n c t i o n s L~h~, 1

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82 D. Cheng, A. Isido ri , W. Respondek, an d T. J. Ta mCoro l l ary 1 2 . A s s u m e t h a t ( 2 4 ) s a t i s f i e s :

( i ) T h e c o n t r o l l a b i l i t y a s s u m p t i o n ( C ) .( i i ) T h e r e e x i s t s o l u t io n s a a n d f l o f e q u a t i o n s ( 2 2 ) a n d ( 2 3 ) , r e s p e c t i v e l y , s u c h

t h a t t h e r a n k c o n d i t i o n ( 3 6 ) i s s a t i s f i e d .T h e n ( 2 4 ) is f e e d b a c k e x a c t l i n e a r i z a b le t o a c o n tr o l la b l e a n d o b s e r v a b l e s y s t e m o ft h e f o r m ( 2 5 ) .P r o o f . S i n c e s o l u t i o n s o f ( 2 2 ), ( 2 3) s a t i s f y ( 3 5 ) t h e n t h is c o r o l l a r y f o l l o w si m m e d i a t e l y f r o m T h e o r e m 11 . [ ]

6 . Conc l u s i on sT h i s p a p e r d e a l s w i th t h e p r o b l e m o f n o n l in e a r i t i e s c o m p e n s a t i o n f o r a n o n l i n e a rc o n t r o l s y s t e m w i th o u t p u t s . W e g i v e a l is t o f e q u i v a l e n t c o n d i t i o n s w h i c hg u a r a n t e e l i n e a r i z a t io n v i a s t a t e - s p a c e c o o r d i n a t e s c h a n g e . T h e n w e s t u d y th ef e e d b a c k l i n e a r i z a t i o n p r o b l e m . W e p r o v e t h a t i f t h e r e e x i s t s a f e e d b a c k w h i c hl in e a r i z e s t h e d y n a m i c s a n d t h e o u t p u t t h e n t h i s c a n b e c o m p u t e d v i a a n o n l i n e a rv e r s i o n o f S i l v e rm a n ' s S t r u c t u r e A l g o r i t h m . F o r a s u b c l a s s o f n o n l i n e a r s y s te m st h is l e a d s t o a v e r i f i a b l e c o n d i t i o n f o r f e e d b a c k l i n e a r i z a t i o n . T h e o u t p u t f e e d b a c kl i n e a r i z a t i o n , w h i c h i s n o t t r e a t e d i n th i s p a p e r , i s s t u d i e d i n [ 2 5 ] a n d [ 3 5 ].

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S ys t e ms , Sys tems Control Le t t . , 4 ( t 984 ) , 17 -22 .[ 1 4 ] A . Is i d o r i a n d A . J . K r e n e r , O n F e e d b a c k E q u i v a l e n c e o f N o n l i n e a r S y s te m s , S y s t e m s C o n t ro l

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Rece ived September 22 , 1987.