BULLETIN DE L ’ACADEMIE POLONAISE DES SCIENCES Serie des sciences mathematiques

Vol. XXVIII, No. 9—10, 1980

SYSTEMS, CONTROL

On Linearization of Control Systemsby

Bronisław JAKUBCZYK and Witold RESPONDEK

Presented by C. OLECH on December 7, 1979

Summary. Necessary and sufficient conditions are given for the linearization of control system o f the form

x - f { x ) + I u,g, ( a )under a class of transformations including changing of coordinates in the state and input spaces and feedback.

1. Introduction. Consider a class of nonlinear control systems o f the formk

(1.1) *= /(* )+ y] m, gi w,/=i

where x e R ";f, g {, ..., gk are C® vector fields on Rn and / (0)=0 . Let G be a group of transformations o f systems (1.1) generated by(i) change o f coordinates in the state space /, gi-xp* (/ ) , <P* (g,), where

tp: Rn-*Rn is a diffeomorphism preserving Oe Rn.(ii) linear change of coordinates in the input space Rk, nonlinearly dependingon A"

k£, - > V htJgj, k, where H (x )= (h ij (x ))

j= i

is a k x k matrix o f class C00, nonsingular at 0 e Rn.k

(iii) feedback o f the form: /-+/+ V at gh where a, are C® functions, a,: Ra-+Ri= I

and y.i (0)=0.. We consider a problem of linearizing system (1.1) by means of transformations

from the group G, i.e. transforming it via (i)-(iii) in a neighbourhood o f zero to the form

k(1.2) x = A x + ^ Ui b( ,

i= i

where matrix A and vector fields bh /= 1, ..., k are constant (in fact we are interested in controllable systems only). We give necessary and sufficient conditions for system( l . l ) to be locally linearizable.

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518 B. Jakubczyk, W. Respondek

Our paper has been inspired by the paper of Brockett [1], where a criterion o f linearization of system (1.1) with scalar control is given. Our criterion is similar to that of [1], however Brockett considers a slightly different equivalence relation (the functions hu ( a*) in (ii) are assumed to be constant).

The class of transformations G is a natural modification of the class used by Brunovsky [2] for the classification of linear controllable systems (1.2). Our lineari­zation procedure leads to the Brunovsky canonical form and, as in the linear case the transformations can by taken linear, it gives also a proof of his theorem.

For the problem of linearization o f systems (1.1) under the transformations (i) the reader is referred to [4]* and [5] (in the latter paper the case/ = 0 is considered). The problem of generic classification o f systems (1.1) with / = 0 under the transfor­mations (i) and (ii) was studied in [3].

2. The main result. Let V ( Rn) denote the family of all C® vector fields on Rn. For /, g e V (Rn) denote adf g = [/ , g] and inductively adlf g—[f , cid'f ~l g] /'= 1, 2,... . where [ •, • ] denotes the Lie bracket. To state the theorem we shall need the following conditions. Below j denotes a nonnegative integer satisfying ( K K « - 1 .

(A 1) For any O ^p^ j and l^ y , there are functions

ai(Ie C x (R") such that [adpf gs, adj gt]= V aiqadjgiOis; i*Uc

(A 2) dim span {adj g ; (a )|0^<7^j, l^ iś :k }= r j (.v) = const.

( A 3 ) dim span {aaąf gt (x)\§^q^n— 1, l^ i^ k } = r n_ 1 (x )—n.

Those conditions we express now' in an invariant form. For any subset A c:V (Rn) we denote by (A ) a submodule o f V (Rn) generated by A over the ring o f smooth functions. Put 3/=<gj, ..., g,,), ^ f—f + ^ = { f+ g \ g € and by induction

J/° = 27, J4j = < Y$f , ~ 1 ], ~1 > ,

where [A, B ]= {[a ,b ] a e A, b e B} lor any A, B a V (Rn). Let M J (* )= .{g (a ), g e eJ/J}czTx Rn and denote rrij (x )=d im J/j (a ). It is easy to see that subsets (ś, Vf <=v (Rn) are invariant under the transformations (ii), (iii). Thus, since the Lie bracket is invariant under (i), the following conditions are invariant under the group G (j denotes a nonnegative integer satysfying O ^ j^ n — 1)

(B 1) J/J are closed with respect to Lie bracket i.e.

g \ g "e J 4 J=>\g',g"]e„#J,

(B 2) nij (a ) = const,

(B 3) mn_ , (x )=n .

We will use the following notations p0 W and Pj (x )= rJ (x ) — rJr.1 (a )

for 7= 1, ..., n -1 . It can be easily seen that p0 (x )^ p , (x )^ ...^ p n- i (x)^0.To introduce the Brunovsky canonical form let us denote by N the smallest

number such that rv (x )—n. It exists if (A 3) holds. It will be proved that condition

*) In [4] the linearization condition is stated with an error.

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Linearization of Control Systems 519

(B 1) implies rs (.x )—mj (x ) and in this case { (x) } and the number N can also be defined using {nij (x )} instead of {/*,• (x)}. It is obvious that for j= N , N + l , n— 1 we have r} (x)=/2.

Let fj (x) and ps (x) be constant (it is the case when (A 2) is satisfied). We will omit in notations the dependence on x and we will write r} and pj. Let (x t, x2, x „ ) = x denote the coordinates of Rn. We shall denote x = (x ° , x 1. xN), where x° consists o f the first p0 coordinates o f x, x 1 consists o f the next py coordi­nates , xN consists o f the last ps coordinates of y. In any group x ' denote by xJ the first Pj + x coordinates. System (1.1) of the form

(2.1)

gi=0, o, ..., 0), g2= (0, 1, 0, ..., 0), gro= (0, ..., 0, 1, 0,(r0— 1)—times

gi=(0, 0, ..., 0) i= r0 + l, k,

/= (0 » --, 0 , x ° ,

r0 — times

0)

will be called system in the Brunovsky canonical form.

Theorem 1. The following conditions are equivalent, locally around 0 e Rn:(a) system (1.1) is G-linearizable to a controllable system (1.2),

(b) system (1.1) is G-equivalent to a system in the Brunovsky canonical form,(c) conditions (A 1)-(A 3) are satisfied,(d) conditions (B 1)-(B 3) are satisfied.The Theorem can also be formulated in a more abstract way. Namely

Corollary 1. Let Ti be a finitely generated submodule o f V (Rn) and where/ e V {Rn),f (0 ) = 0. Then there exists a system o f coordinates in a neighbourhood o f 0 such that Tt and f can be expresed by (g 't, g*> and f + <&’, respectively with fand g' o f the form (2.1) i f and only i f the conditions (B 1)-(B 3) are satisfied in a neighbourhood o f 0.

Remark: 1. Conditions (A 1), (A 3) are straightforward modifications of the conditions of Brockett [1]. In [1] the s-um on the right hand side o f (A 1) is taken only for g< j. This is due to the fact that Brockett assumed the functions hu (x ) in (ii) to be constant. In the scalar control case (like in [1]) the condition (A 2) follows from (A 3).

R emark 2. The Theorem also holds for the class of systems (1.1) not satisfying f (0 )=0 . In this case we have to use a larger class o f transformations admitting in(iii) a/ not equal zero at zero. The field / in (2.1) is then, in general, a sum of constant vector field and that of (2.1).

In the proof of Theorem 1 we shall use the following modification o f the Fro- benius theorem.

\ ,L e m m a 1. Let Dq c I), cr... c:Ds denote a sequence o f involutive O r distributions

on a manifold M (dim M =n ) having constant dimensions Po^P) respecti­

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520 B. Jakubczyk, W. Respondek

vely. Then, around any point x0eM, there exists a coordinate system ( x , , ..., x„) such that the integral manifolds o f D, around x0 are o f the form

xi= C i i = P j + 1, n,. Ct— constant.

Proo f. Let f lt ...,f*0 be vector fields which generate the distribution D0 around x0i A , -..,fMi generate D iy ...,fu ...,fUfl generate the distribution DN around x0. Choose vector fields f ly + f, ...,/„ in such a way that f x, ..., f n generate TXo M.

Consider the map T

( / i , t„)— -+etlfl 0...0 e,nfn (x0) e M,

where eufl denotes the (local) flow generated by f . Let V denote a cube neighbour­hood o f Os Rn such that T: V -*U =T (V ) is a diffeomorphism. Then (U ,T~ l ) is a desired coordinate system.

According to the Frobenius theorem Dj defines in U a foliation with ///-di­mensional leaves (maximal integral manifolds o f Df). These leaves are exactly the leaves of the foliation 26 constructed as follows. For any point y e U o f the form y=e,nj+if uj+i o ... o etnfH(x0) we define a submanifoldNy which passes through;'by

N={p e U\p=e,ifl o ... o e,t,j f 'tj (y) j

(it is a pj dimensional submanifold by the regularity o f T).To prove the coincidence o f sd and 26 note that integral manifolds of Dj are

equal to orbits o f the family o f vector fields generating Dj (Sussmann [6]), so Nr is contained in the maximal integral manifold o f Dj passing through y. Moreover, N y is closed (trivial) and open (by the equality of dimensions) subset o f the maximal integral manifold of Dj passing through y. Therefore, they are equal.

Since T is bijective it follows that Z ,, Z 2 e U belong to the same leaf o f the folia­tion 26 if and only if /, (Z ,)= r , (Z 2) /=//, + 1, ..., n. This and the fact that sd and 26 coincide concludes the proof that tx, ..., tH are the desired coordinates.

3. P r o o f o f Theorem 1. The implication (b)=>(a) is trivial. To prove (a) >(d) let us notice that condition (d) is G-invariant and for the linear system (1.2) we have # = < 6 ,, ..., bky ^ '= ( A ą bi\0^q^j, IśH^ky, i.e. for this system condition (d) is satisfied.

(d)=>(c). We shall prove using an induction argument with respect to j that (B 1), /=0, 1 imply the equalities

(3.1) J (J (adqf g(\0^q^j, j = 0...... n - l .

Fory=0 this is trivial. Assume that this is true fo ry— 1. The inclusion “ =>” follows from the definition of J t j . To prove the converse one, note that from (B 1) we have [<S, Thus +c [ f J/j ~ 1 ] + J ij ~1. Therefore

J/J= ( [V f , J t j ~% J t j ~ *> c< [/ , J/j - 1 ], j/ j ~xy .

This and the induction assumption on the form o f . ^ ,_1 give the desired inclusion.

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Linearization of Control Systems 521

The conditions (B 1)-(B 3) and equality (3.1) imply easily the conditions

(A IM A 3).

(c)=>(b). We denote / = (/ ° ,/ \ ...,/*), where f ° consists of the first p0 coordinates of / = (/ x, ...,/„), f l consists of the next p t coordinates, consists of the lastpN coordinates. We shall also use the notations from (2.1).

The distributions D} generated by R j={adqf gi\0^q^j, 1 </^/c} satisfy the assumptions of Lemma 1. Let us choose the coordinates around 0 e Rn in such

a way that the integral manifolds o f Dj , j — 0, ..., N are given by x t= c h i = r j + 1,..., n. We assume that / and g\ s in (1.1) are expressed in this coordinate system. All

our future changes of coordinates will preserve the above integral manifolds. The form of the integral manifolds of Dj implies that the vector fields in Rj have zero components along the last n — rj coordinates. Using this fact and dim span Rj (x)=rj we shall prove that the coordinates o f f J do not depend on the variables

8fJx ° , ..., xJ~2 0 > 2 ) and rank (x )—pj . Note first that for g e R j - V we have

that g and so D g - f are equal to zero along the last n — coordinates. Thus, along these coordinates [ f , g ] is equal to —D f g. In particular if g' e Rj_2 then

we have Df Jg = 0 by the fact that [ f , g ' ] £ R j - 1- This gives that f J does not dependc f J

on x °,...,x J 2. The equality j - '{ ~ (x)—Pj is a consequence of the fact that

there are pj independent vectors among [ f ,g] , g e R j _ x.

Now we shall transform the vector field / = (/ ° , ..., f N) to the canonical form

(2.1). We shall use the same notations to express it in an old coordinate system and

in a new one. In the first step we introduce the new coordinates yN~* = ( f N, xN~l ),yJ= x J if j j ć N —1, where xN_1 denotes Pn- i ~Pn coordinates o f xN~1 chosen in

^yv~1 dfNsuch a way that rank ^n- T (P )—Pn- i (this is possible since rank ' ^xn-\ (Q)=

=pN). We obtain / in the form (/ °, x N~x), where we write x N~1 insteadof to use consequently notation jc for coordinates. In the second step we intro­

duce the new coordinates yN~2= ( / N_1 , and get / = ( / ° , . . . , f N~2, x N~2, x N~l). After N — 1 steps we obtain / = (/ ° , x ° , ..., x N~1).

By the form of the distribution D0 we can transform the vector fields gh i = 1,..., k to the canonical form g ,= (0, ..., 0, 1, 0,..., 0) for i = l , ..., r0 and ^ = (0 ,.. . , 0) for

( l - 1 ) - tim es

i= r 0 + 1 , k via transformations (ii). Finally, using feedback (iii) we obtain

/= (0 , Jc°,..., xN~l ). The proof is complete.

Added in proof. After this paper had been submitted, R. W. Brockett infomed

us that he had obtained an analogous result.

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522 B. Jakubczyk, W. Respondek

INSTITUTE OF MATHEMATICS, POLISH ACADEMY OF SCIENCES, ŚNIADECKICH 8, 00-950 WARSAW (INSTYTUT MATEMATYCZNY, PAN)INSTITUTE OF MATHEMATICS, TECHNICAL UNIVERSITY, PL. JEDNOŚCI ROBOTNICZEJ 1, 00-661

WARSAW(INSTYTUT MATEMATYKI, POLITECHNIKA WARSZAWSKA)

REFERENCES

[1] R. W . B rockett, Feedback invariants fo r nonlinear systems, in: Proc. IFAC Congress, Helsinki, 1978.

[2] P. B runovsky, A classification o f linear controllable systems, Kybernetica, 1970, 173-180.[3] B. Jakubczyk , F. P rzytyck i, Singularities o f k-tuples o f vector fields, Preprint No.

158, Institute of Mathematics, Polish Academy of Sciences, Warsaw, 1978.[4] A. J. K ren er, On the ec/uivalence o f control systems and linearization o f nonlinear systems,

SIAM J, Contr, and Optim., 11 (1973).[5] J. L. Sedw ick, D. L. E llio tt , Linearization o f analytic vector fields in the transitive case,

J. Diff. Equat., 25 (1977).[6] H. J. Sussinann, Orbits o f families o f vector fields and integrability o f distributions, Trans.

Am. Math. Soc., 180 (1973), 171-188.

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