On dynamic feedback linearization

Systems & Control Letters 13 (1989) 143-151 North-Holland

143

On dynamic feedback linearization B. C H A R L E T

*, J. L I ~ V I N E

C.A.L /Section A utomatique, Ecole Nationale Sup~rieure des Mines de Paris, 35 rue Saint HonorS, 77305 Fontainebleau, France R. MARINO

**

Dipartimento di Ingegneria Elettronica, Seconda Uniuersitgt di Roma 'Tot Vergata" Via O. Raimondo, 00173 Roma, Italia

Received 21 December 1987 Revised 26 May 1989 Abstract: We address the problem of transforming a nonlinear multi-input system into a linear controllable one via nonsingular dynamic feedback and (extended) state space diffeomorphism. We show that, in the single input case, if a system is dynamic feedback linearizable it is also static feedback linearizable. We then give sufficient conditions for a class of multi-input systems to be dynamic feedback linearizable. All three-dimensional systems with two controls and a controllable linear approximation about the origin are shown to belong to such a class. Keywords: Nonlinear systems; feedback linearization; dynamic compensator.

1. Introduction The study of transformations of nonlinear control systems m

~.=f(z)+ ~_,gi(z)ui(t)=f(z)+G(z)u, z~R",

u~R",

(1)

i=1

with f(0) = 0 and rank G(0) = m into linear controllable ones 2 = Ax + By,

x ~ R n',

U E Rm,

(2)

provides a classification of nonlinear systems and a simplification of analysis and control of those systems (1) w h i c h a r e t r a n s f o r m a b l e i n t o (2). T h e first t r a n s f o r m a t i o n to b e s t u d i e d w a s t h e n o n l i n e a r c h a n g e o f c o o r d i n a t e s (q~ is a d i f f e o m o r p h i s m a n d n ' = n ) in t h e state s p a c e (see [9])

x

n',

°.

(3)

Necessary and sufficient conditions which identify those nonlinear systems (1) transformable by nonlinear change of coordinates into systems (2) with v = u can be found in [17]. In [1] it was proposed to enlarge the class of transformations (3) by adding also state feedback transformations (/3 is a constant invertible m × m matrix) u=a(z)+flo,

(4)

o~n".

T h e y w e r e l a t e r g e n e r a l i z e d in [4] a n d [8] b y i n t r o d u c i n g t h e s t a t e f e e d b a c k t r a n s f o r m a t i o n s

u =

+/3(z)v,

v

n m,

* On leave from D.G.A. Minist6re de la D6fense Nationale. * * Research partially supported by Ministero Pubblica Istruzione (Fondi 40%). 0167-6911/89/$3.50 © 1989, Elsevier Science Publishers B.V. (North-Holland)

(5)

144

B. Charlet et al. / Dynamicfeedback linearization

where fl(z) is a m × m invertible matrix. Necessary and sufficient conditions were obtained in [4] and [8] which identify those systems (1) which are transformable by nonlinear change of coordinates (3) and state feedback (5) into systems (2). The more general class of singular state feedback transformations (fl(z) singular and of rank m' < m in (5)) has not yet been investigated. In order to measure the degree of nonlinearity of (1), those systems (1) which are transformable by (3) and (5) into systems of the following type with (A, B) a controllable pair Yc~1) = A x ~) + Bv, .~2) = y(x~l), x~2)) + 8 ( x O ) , x~2))v,

x ~ ~ R P, x~2)~ n , _ p ,

(6)

were studied first in [10] in the case m = 1 and then in [14] in the general case (see also [11] and [15]). In particular it is shown in [14] that a set of controllability indices (k~ . . . . . . kin) is associated with any nonlinear system (1) and that the integer p = ~ , m l k ~ gives the dimension of the largest feedback linearizable subsystem. A natural generalization of static state feedback transformations (5) is given by dynamic state feedback transformations (a(0, 0) = 0) ~=a(z, w)+B(z, w)v, u = c~(z, w ) + f l ( z , w ) v ,

w ~ n q, v ~ am.

(7)

where q is the order of the compensator. Dynamic compensation (7) was introduced in [16] in the study of input-output decoupling of nonlinear systems (1) with outputs y = h ( x ) , y ~ R s, (see also more recent works [2], [3] on the same subject). In [6] and [7] sufficient conditions are given under which a system (1) with outputs is simultaneously input-output decoupled and linearized by a dynamic compensator. In this paper we investigate the transformations of system (1) into system (2) via nonsingular dynamic compensators of type (7) and extended-state diffeomorphisms (n' = n + q) x = ~ ( z , w).

(8)

We first establish a negative result: if system (1) with m = 1 cannot be transformed into system (2) via (3) and (5) then it cannot be transformed into (2) via (8) and (7). In other words dynamic compensation is of no use in order to linearize a single-input system of type (1). Then we give sufficient conditions for the existence of transformations (7)-(8) which take (1) into (2) for systems (1) such that the integer p equals n - 1. As a corollary we show that any system (1) such that rank G(0) = n - 1 and its linear approximation about the origin is controllable is dynamic feedback linearizable.

2. Preliminaries and definitions

Consider the nonlinear system (1) introduced in Section 1 with f(0) = 0, where f , g~ . . . . . gm are real analytic vector fields on U0, a neighborhood of the origin, gl(0) . . . . . gin(0) are linearly independent. There is no loss of generality in considering systems in which the inputs appear linearly since the more general system 2 =f(z,

(9)

u)

with f(0, 0) = 0 can be transformed into a system of type (1) via the dynamic compensator ul = va . . . . .

~,, = vm.

(10)

Since we are precisely addressing the problem of making the system (1) linear and controllable via dynamic compensation and (extended) state diffeomorphism, if sufficient conditions are obtained for systems of type (1) they can also be applied to systems of type (9) by using the preliminary compensation

B. Charlet et al. / Dynamicfeedback linearization

145

(10). On the other hand it was shown in [18] that system (9) can be transformed into (2) by a state feedback u = a(z, v), a(0, 0) = 0, and a state diffeomorphism (3) if and only if system (9)-(10) is static feedback linearizable (see definition 2.3). Hence we restrict ourselves to system (1). If we define the distribution = sp{ gl . . . . . g,. }, system (1) can be described in a coordinate free way ([13]) by the pair ( f , ~). On the other hand, given a basis for ~, (ga . . . . . gin), and a local system of coordinates (zl . . . . . zm), (1) is a local coordinatization of the pair ( f , ~). Definition 2.1. Two systems ( f , re) and (f~ ~ ) such that f(0) = f ( 0 ) = 0 are said to be feedback equivalent if (i) ~ = ~; (ii) f + g = f for some vector field g ~ ~ such that g(O) = O, in other words f - f - ~ ft. The transformation ( f , ~ ) ~ ( f + g, ~), g ~ ~, is called a feedback transformation.

If (f~ ~ ) is described in local coordinates as } = f ' ( ~ ) + ~ g(,~)vi(t ) = f ( e ) + G ( ~ ) o ( t )

(11)

i=1

and is feedback equivalent to (1), then there exists a static state feedback ( f l ( z ) nonsingular, a ( 0 ) = 0)

u=a(z) + fl(z)v such that the closed loop system 2 =f(z) + G(z)a(z) + G(z)fl(z)v(t) is transformed into (11) by the diffeomorphism Y = ~(z). Definition 2.2. The system ( f , if) is said to be partially static feedback linearizable with controllability indices k~' _> • • • >_ kp, p < m, if ( f , if) is feedback equivalent to (f~ if) for which there exists a basis for if, (gl . . . . . gin), and state coordinates (x 1. . . . . xn) in U0 such that (f~ if) is described as P

•(1)

=

Acx(a) + y, Bivi = Acx 0) + Bcv,

(12)

i=l

= a(x%

x

+

b,(x

= a(x%

x

+

6(x (1),

o,

i=1

where x (1) = (x I . . . . . Xq), x (2) = ( / q + l . . . . . xn) and (Ac, B~) is a pair of matrices in Brunovsky canonical form with controllability indices kf" _> • • • > kp, p < m, F.f=lk* = q, namely A c = block diag (A~ . . . . . A~ ),

{ii

Bc = block diag ( B~ . . . . . Be ),

OI 0

k,*xk~*

i}

k ~*xl

B. Charlet et al. / Dynamic feedback linearization

146

Definition 2.3. The system ( f , ~ ) is said to be static feedback linearizable if it is partially feedback linearizable with controllability indices kl* -> • • • _> k * such that ~"~=ak~* = n . Consider the distributions (.~ denotes the involutive closure of .~) ~o = if,

*~,+l = ~ , + ad~+l-~o.

(13)

Assume that the distributions *~i and .~ have constant rank in U0. A sequence of integers ( r 0, r 1, r 2 . . . . ) is then defined as r 0 = dim ~0,

ri = dim *~i - dim -~i-1,

(14)

and is shown in [14] to be a nonincreasing one. Then a set of integers (k~ . . . . . kin) is uniquely associated with system (1) as follows: k~=card{jirj>i,

j>0}.

(15)

Recall that for a linear system £ = A x + Bu we have ~, = ~ i = I m ( B , A B , . . . , A ' B ) .

(16)

It is shown in [14] that the distributions .~i and the controllability indices ( k I . . . . . kin) are invariant under feedback transformations and characterize the largest linearizable subsystem in the following sense.

Theorem 2.1 [14]. System ( f , ~ ) is partially feedback linearizable with indices ( k 1. . . . . kin). I f system ( f , f~) is partially feedback linearizable with indices (k~' . . . . . k ~ ) , p < m, then it must be k * < k i for l
3. Dynamic feedback linearization for single input systems Consider the extended system made of system (1) controlled by a dynamic compensator (7),

=f(z) + a(z)~(z, w) + 6(z):(z, w)v,

•=a(z,

w)+ B(z, w)v,

(17)

which can be written as

~=f(~) + d(~)v =f(~) + ~: g,(~)v

(18)

i=l

with ~ = (z, w) being the extended state and

a(~, w)

'

¢= (G(~)~(z, w) 1 8(z, w) ]

(19)

B. Charlet et al. / Dynamic feedback linearization

147

If u j = %(~)-}-Eim=lflj,i(Z)Ui, 1 < j < m, are viewed as m outputs for system (18), we can define m characteristic indices vI . . . . . vm in the usual way [6]:

vj =

0

if flj,~ =~0 for some i, 1 < i < m,

min{r [L~L~--aaj(E) ~ 0 for some i, 1 < i < m }

if flj,i(E) = 0 for every i.

(20)

We set vj = + o¢ if flj,~-= 0 for every i and L~L~aj =- 0 for every i, 1 < i _< m, and every k > 0. Let =

r i,,(5) LgL)- % ( e )

if vj=O,

(21)

if v j > 0 .

When all vj are finite, the m × m matrix

D(E) = (6j,i(e))

(22)

is called the decoupling matrix of the compensator (7) for system (1). Definition 3.1. A dynamic compensator (7) is said to be regular for system (1) if the corresponding

decoupling matrix D(~) is nonsingular in a neighborhood V0 of the origin in R n+q. Note that the static state feedback (5) with fl(z) nonsingular is a nonsingular compensator according to the above definition. Definition 3.2. System (1) is said to be dynamic feedback linearizable if there exists a nonsingular dynamic

compensator (7) such that the closed loop system (17) is locally transformable into a linear and controllable system (2) by an extended state space diffeomorphism (3). Theorem 3.1. Consider a single input system (1) (m = 1). The following statements are equivalent: (i) (1) is static feedback linearizable. (ii) (1) is dynamic feedback linearizable. Proof. (i) ==, (ii) is obvious. (ii) = (i): By assumption (ii) there exists a linearizing compensator (7) for system (1) with m = 1 and characteristic index v. We first establish a relationship between the distributions ~i defined by (13) for system (1) and the distributions .qi defined for system (18) with m = 1 obtained by using the linearizing compensator:

*~0 -- sP{ g},

C~im:

*~,'~-~,--1+ sp{ad}-g }.

f(o)

ad)g = / (yad)-2g + X/)

(23)

if i < v , (24) if i > v,

where Xi E.~i_,_ 1 and y - - ( - 1 ) ~ 6 , 6 being defined by (21). Proof of the Claim: By induction on i. For i = 0 we have

w i t h f l = - t and Xo = O i f v = O a n d f l = O i f

v>l.

B. Charlet et al. / Dynamicfeedback linearization

148

We assume that the claim holds true for an integer i: we shall prove it for i + 1. We consider two cases: i>_g; i < u . If i > v ,

where

0x,

Xi+ 1 = [ f , ~ ] + a [ g , X,] + (L/3')ad)-~g - (Lad,~a)g + ayadgad)-~g + --~-wa. Since X , ~ _ ~ _ 1 and y = ( - 1 ) ~ 3 , Xi can be written Xi(z, w ) - F v o j ( z , w ) ~ . , ( z ) , with ~ , ~ , _ ~ thus ((}Xi/()w)a = Y~j( Laoj)Fj, i ~ i - ~ , - 1 and Xi+ 1 belongs to ~ i - , . If i < v, by assumption ad}-g=( 0 )

1, and

ad;+lg=(-(Lad'/ga)g).,

so

Since 0 Lad)'ga =

ifi
i

( - - 1 ) LgL'f-a

if i = v - 1,

it follows that

ad}+ lg =

* Y,g)

if i = v - l .

The claim is proved. We now show, by contradiction, that the distributions .~i must be involutive. Suppose that the distributions ~ are not involutive for each i, 0 < i < n - 1. Let k < n be the smallest integer such that "qk is not involutive; then for some j, 0 < j < k - 1, lad}g, ad~g] ~*~k.

(25)

But according to the claim,

where ~ + , ~ j _ I = N j _ I and X k + ~ k _ l = ~ k _ l : it follows from (25) and (26) that ~k+~ is not involutive and this contradicts the assumption (ii). We established that ~i is involutive for every i, 0 < i < n - 1. Suppose that ..~ is not of constant rank for some i < n. Since ~s is involutive for every i, 0 < i < n, then ~i+~ cannot be of constant rank and this contradicts asumption (ii). Finally we need to show that rank N,_ 1 = n. If assumption (ii) holds then, according to the claim, we have by (24) T~r (*~k+~)=*~k where T~r is the tangent map to the projection (z, w) ,-%z. Thus if .~, = U~=o~, and d , = U~=od~ we have T~r(.~,) =.~,. Assumption (ii) implies that d , = TR "+q, so we have ~ , = T ~ " , but since .~, = ~ , _ ~ (z ~ ~ " ), we have rank .~,_ 1 = n . [ ]

B. Charlet et aL / Dynamic feedback linearization

149

4. Sufficient conditions

In Section 3 we have established that dynamic feedback linearization is of no interest for single input systems. In this section we establish sufficient conditions for dynamic feedback linearization of systems such that the sum of their controllability indices defined by (14) and (15) is equal to n - 1. The conditions are to be checked once system (1) has been transformed into a 'pseudocanonical' form (Definition 2.4). Theorem 4.1. A s s u m e F . ~ l k ~= n - 1 for system (1), with k i defined by (13), (14), and (15). Let

=/(x) = E g,(x)o,(t)

(27)

i=l

be a 'pseudocanonical' form for system (1). If, for some j , 1 < j < m,

ad~j ~ ~

(28)

A =-~kj_2 + sp(ad~-agj }

(29)

with

where -A denotes the involutive closure of A. Then the system (1) is dynamic feedback linearizable by a nonsingular compensator o f order m - 1.

Proof. Consider a 'pseudocanonical' form (27) for system (1) obtained through a feedback transformation and the nonsingular compensator of order m - 1 dv____L= , dt

vi,

iq:j,l


(30)

vj=vj.

If we denote xk,+ x = v i, i ~ j, 1 <_ i < m, from (12) the closed loop system is X•li-_- X l +i l ,

l <_l < k i

'

X.ik , + l = V i ,,

2 [ = XI+ j 1,

l -
•

+ E bi( )o;

--

1,

i--#j,l
(31)

"J=vj, Xk:

i=1

which can be written as =f(~) + ~ g,(~)v;.

(32)

i=1

One can easily compute gi

i , axk' +1

i~j,l
adt:gi = adt:-lgi,

gj=gj,

(33) i --/:j, 1 < i < m ,

adtfgj = a d ~ j .

By assumption (28) and the above relationship there exists a function ~k(Y) on the extended state space such that (dtp, a d ~ g j ) 4: 0,

(dtk, ad~-g~)= 0,

O
(34)

150

B. Charlet et al. / Dynamic feedback linearization

(q~ is chosen such that ~p(ff) = q,(x) and such that d+ ~ A ± , cf. (29)). Let us define y,=Lt/-l~b,

l
(35)

l.

The mapping ~ ~ (xj, yS; 1 < l < ki + 1, i ~ j , 1 _< i _< m, 1 < s < kj + 1) is a local diffeomorphism since the (n + m - 1) × (n + m - 1) matrix

dxl

dx~71,+l dy~ (

, (ad~'gl . . . . . g~-l, ad~gj . . . . . g~, ad~÷'gj+l . . . . . g . . ) )

(36)

dyG+I d x ( +~

dXkL+l is nonsingular. In new coordinates (31) becomes •i _ i XI--XI+I~

l <--l < k i ,

pl=Yl+l,

l <_l<_kj,

i~j '

.~ , Xki+l "-~'Ui~ i~j,

pks+l=L~,+l~b+ Y ~ ( L g L ~ , t ~ ) v " + ( L L L ~ . q ~ ) o j .

i 4=j

(37)

From (34), LLL~,q~ = ( - 1)k'(d~k, a d ~ g j ) ~ 0 and therefore

Ujt

ngsn~jl~l

(_n~j+ll~_

i~jE (ng'n~Jl~)u;q-u;')

(38)

along with (30) gives the dynamic compensator which solves the dynamic feedback linearization problem. [] Corollary 4.2. A system (1) with m = n - 1, Eim=lki = n - 1 and such that its linear approximation about the origin Of ~=o x + G(O)u = F x + Gu Yc = -~x

(rank G = n - 1)

(39)

is controllable (rank(G, FG) = n ), is dynamic feedback linearizable by a nonsingular dynamic compensator of order n - 2.

ProoL Consider any 'pseudocanonical' form for system (1), 2 =/(x) + ~ gi(x)vi(t).

i=1

(40)

Since the linear approximation about the origin is controllable then there exists j, 1 _
B. Charlet et al. / Dynamic feedback linearization

151

5. Conclusions This p a p e r shows that d y n a m i c feedback l i n e a r i z a t i o n is a m u l t i - i n p u t p h e n o m e n o n . T h e o r e m 3.1 proves that, for single i n p u t systems, if feedback l i n e a r i z a t i o n c a n b e achieved at all it can always be achieved b y a static state feedback. O n the other h a n d i n m u l t i - i n p u t systems the geometric structure of nested involutive d i s t r i b u t i o n s of c o n s t a n t r a n k required b y T h e o r e m 2.2 m a y be i n d u c e d b y i n t r o d u c i n g integrators which delay the effect of some i n p u t c h a n n e l s with respect to the others. I n Section 4 it is shown, for instance, that a n y system (1) in R 3 with two i n d e p e n d a n t controls which has a c o n t r o l l a b l e linear a p p r o x i m a t i o n a b o u t the origin is d y n a m i c a l l y feedback linearizable b y a first order (at most) n o n s i n g u l a r compensator.

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