Iggidr, Abderrahman

Sallet, Gauthier

English

Institute of Mathematics AS CR, v. v. i.

KybernetikaAbderrahman Iggidr; Gauthier SalletNonlinear stabilization by adding integratorsKybernetika, Vol. 30 (1994), No. 5, 499--506Persistent URL: http://dml.cz/dmlcz/125138Terms of use:© Institute of Information Theory and Automation AS CR, 1994Institute of Mathematics of the Academy of Sciences of the Czech Republic provides access to digitizeddocuments strictly for personal use. Each copy of any part of this document must contain theseTerms of use.This paper has been digitized, optimized for electronic delivery and stamped withdigital signature within the project DML-CZ: The Czech Digital Mathematics Libraryhttp://project.dml.czK Y B E R N E T I K A — V O L U M E 30 ( 1 9 9 4 ) , N U M B E R 5, P A G E S 4 9 9 - 5 0 6 NONLINEAR STABILIZATION BY ADDING INTEGRATORS1 ABDERRAHMAN IGGIDR AND GAUTHIER SALLET In this paper, we study the global stabilization, by means of smooth state feedback, of systems (5) obtained by adding an integrator to a general nonlinear system (^2)- We show how to compute the stabilizing feedback for (S) when a stricte Lyapunov function for ( £ ) is difficult to find. 1. INTRODUCTION In this paper we deal with the global stabilization of nonlinear control systems of the form: x = f(x,y) yz=u where x G IRn, y G Htp, u G IRP and / is a smooth vector field such that /(0,0) = 0. It is well known [3, 6, 7] that if the subsystem: x = f(x,v) (2) where v is the input, is globally asymptotically stabilizable (G.A.S) by means of feedback law v = k(x), where k is of class C, r > 1 then system(l) is G.A.S. Moreover, if it is possible to construct a Lyapunov function V such that (VV,f(x,k(x)) <Q VxGlR", x # 0 (3) (V is said a strict Lyapunov function for system (2) and exists by the Lyapunov inverse theorem) then a stabilizing feedback u(x,y), which depends on k and V, is explicitly given. However it is not easy to find a function V satisfying (3) even if one knows that the origin is a globally asymptotically stable equilibrium point for the closed loop system x = f(x,k(x)). (A) The goal of this paper is to weaken these hypotheses. Theorem 1 shows that, to find a feedback stabilizer for system (1), we do not need to have a strict Lyapunov 1 Presented at the IFAC Workshop on System Structure and Control held in Prague on September 3-5, 1992. 500 A. IGGIDR AND G. SALLET function for (2). Theorem 2 shows how to asymptotically stabilize system(l) without stabilizing system (2). We recall that the relationship between the stabilizability of (2) and (1) is an open problem when system (2) is stabilizable by means of continuous feedback (not C1). This problem was addressed in [1] and [2] from the local stabilization point of view. The authors proved that the local stabilizability of (2) is equivalent to the local stabilizability of (1) if n = p = 1 and / is a real analytic function. 2. MAIN RESULTS Before stating the first theorem we introduce the following notations and definitions: Definition 1. We shall say that system (2) is of LA SALLE-Type (L-T) if there exist: 1. a function k : IR" -> IRP of class Cr(r > 1) with k(0) = 0 2. a function V : IRn —* IR of class C1, definite positive and proper such that: i) X • V(x) < 0 V i G IRn where X(x) = f(x, k(x)) and X • V is the Lie-derivative of V along the trajectories of the vector field X (here: X • V(x) = (W,X(x)) where (•, •) is the inner product in IRn) ii) The largest invariant set contained in E = {x G IRn |X • V(x) = 0} is the origin of IR". Definition 2. A continuously differentiable scalar function V : IRn —• IR is a weak Lyapunov function for x = X(x) if V is positive definite proper and XV(x)<0 VxGlR". By a proper function we mean a function V : IR" — IR such that {x G lRn| V(x) < £} is compact for each £ > 0. Through this paper | | . || will denote the usual Euclidian norm in IRP, Xt(-) is the flow of the vector field X defined on IRn. Remarks . 1. System (2) is of (L-T) o- x = f(x, k(x)) is globally asymptotically stable (see [4])-2. It is often easier to find V satisfying (i) and (ii) then a function V satisfying (3) (Mechanical systems are well known examples of this situation). Nonlinear Stabilization by Adding Integrators 501 T h e o r e m 1. If system (2) is of (L-T) then system (1) is also of (L-T) and the stabilizing feedback is u = -y + k(x) + dk(x).f(x, y) - G(x, y)T • VV(x). (5) (T = transpose) where f(x, y) = f(x, k(x)) + G(x, y)-(y- k(x)). (6) Remark . One can choose for G the matrix: G(x,y) = J ^(x,ty+(l-t)k(x))dt. (7) P r o o f . (2) is of L-T so there exist k and V satisfying i) and ii) of the above definition. Let us denote X(x) = f(x,k(x)) and U the largest invariant set by X contained in E = {x G M.n\X • V(x) = 0}. By hypotheses Q = {0}. Let "/(*,V) Z(x.y) = . , . K y> \ u(x,y) where u(x, y) = -y + k(x) + dk(x) • f(x, y) - G(x, y)T • VV(x) and define (see [7]) W(x,y) = V(x)+h\y-k(x)\\2 W is of class C1, definite positive and proper W(x, y) = Z- W(x, y) = (Z(x, y),VW(x, y)) = X • V(x) - \\y - k(x)\\2 < 0 Note that all trajectories of the closed-loop system are bounded because W is proper and its derivative is nonpositive. Let E ={(x,y)eJRn+p\Z-W(x,y) = 0} = {(x, y) e Etn + P \y = k(x) and X • V(x) = 0}. According to LaSalle's theorem (see [4] pp. 66-67) all solutions tend to Cl the largest invariant set by Z contained in E. To prove Theorem 1 it remains to that Cl is the origin of I T + P . On Q the vector field Z is given by: / X(x) where Y(x) = dk(x) • f(x,k(x)) - G(x, k(x))T • VV(x) and X(x) = f(x, k(x)) x = f(x,k(x)) = X(x) y = dk(x) • f(x, k(x)) - G(x, k(x))T • VV(x). 502 A. IGGIDR AND G. SALLET Let (x(t),y(t)) be a solution of the above system starting at (x(0),y(0)), the initial condition G Q. Since Q. is ^-invariant we have (x(t),y(t)) G Q, for all i > 0 but we have ±(x(t)) = X(x(t)) hence x(t) = Xt(x) where Xt(-) is the flow of the vector field X defined on IR". Consider, now, the following set: M = {x £lRn\(x,k(x)) € &}. If x G M then (x,k(x)) G fi and (x(t),y(t)) G Q. since Q. is invariant, this implies (Xt(x),y(t)) G fi but y(t) = fc(X.(x)). So we have shown: Z G M =>> (Xt(:c), &(X((z))) G fi => Xt(x) G M. This proves that M is X-invariant and since M is contained in E we have M -~ {0} and then Q = {(0,0)} which completes the proof of Theorem 1. D Example 1. Consider the following system which evolves in IR4: f i i =x2ai3 + 6iy x2 = -X1X3 +b2y xз = bзУ ý = u (8) where 63 ^ 0 and (61, 62) ^ (0,0). The subsystem { i i = £20:3 + 612/ x2 = -X1X3 +b2y (9) x3 = b3y is the "famous" system of the angular velocity of a symmetric rigid body. In [5, 7] it is shown that (9) is smoothly globally asymptotically stabilizable. Furthermore, it is shown in [5] that the system (9) is of L-T with the two polynomial functions (Xl2 + x22 + x3) (263 + 4 ( M i + b2x2))P(x3) k = -63 x3  630Kl2 + Z22 + Z3)2I3'(Z3) X32 + (Xl2 + X22 + X3Y P(X3) 2 where , 4 P<*> = 77TTTTT, + * 3 2 - + 2 (612 + 622) x2 4 (61 + 62 J Nonlinear Stabilization by Adding Integrators 503 so all the assumptions of Theorem 1 are satisfied and then our result can be the stabilizing feedback for system (8) using the formula (5). Note that a strict Lyapunov function for system (9) has never been found and may be difficult to construct so the results of [3, 6, 7] cannot be applied to stabilize system (8). Generally, to compute the stabilizing feedback for systems of the form: x = f(x) + g(x)y i) = u xem", ye mr, g(x) e M„,p(coo(iRn,iR)) where the subsystem x = f(x) + g(x)y is stabilizable via the Jurdjevic-Quinn's method, one can apply Theorem 1 but not the results based on strict Lyapunov function. For the following result we suppose that / is a smooth (i.e. C°°) vector field. T h e o r e m 2. If there exist a smooth feedback k (not necessarily a stabilizing one) for system (2) and a smooth function V, which is definite positive and proper, such that: i) X • V(x) < 0 V i € E " where X(x) = f(x, k(x)) ii) The set S = {x eMn\Xs+1 -V(x) = X! Yi-V(x) = Q, s e IN, i=l,...,P} where Yi = -r—(x,k(x)), is reduced to the set {0}. then system (1) is G.A.S and the stabilizing feedback: u = -y+ k(x) + dk(x) • f(x, y) - G(x, y)r • W(x) (10) where G is defined by the formula (7). P r o o f . We take u = dk(x) • f(x,y) — G(x,y)r • W(x) + v where v is a new input so system(l) can be written: where (X) = Z(x, y) + B(x, y)-v = Z(x, y) + J^ v ^ WJ , = i f(x,k(x)) + G(x,y)-(y-k(x)) 1 dk(x).f(x,y)-G(x,yf-VV(x) and Bi(x,y) = en+i. Introduce W(x,y) = V(x)+\\\y-h(x) 504 A. IGGIDR AND G. SALLET W is smooth, definite positive and proper ZW(x,y) = XV(x)<0. According to [5], the above system is globally asymptotically stabilizable by the feedback v = -BW(x,y) if the set A = {(x, y) G IR n + p |Z s + 1 • W(x, y) = Zs • B. • W(x, y) = 0, s € IN,i = 1,.... ,p} is reduced to the origin of ]Rn+p. Since Z.W(x, y) = X • V(x) and B • W = y — k(x) we can write: A = {(x,y) G IRn+p |y = k(x) and x G C} where: C = {x e JR." \X • V(x) = Zs+1 • W(x, k(x)) = Zs • Bi • W(x, k(x)) = 0, s> 1, i= l,...,p}. We shall show that C = S. We have BW = y- k(x) ZBiW = Z-(vi- ki(x)) = (Z, V(y i - *,•(-.))). For (x,y) G A the vector field Z is (since y = k(x)): X(x) Z= . dk(x) • f(x, k(x)) - G(x, k(x))T • W ( x ) so (Z,V(yi-ki(x)))= -(X,Vki(x)} + (dk(x) • f(x, k(x)) - G(x, k(x))T • VV(x), en+i) But (dk{x) • f(x, k(x)) - G(x, k(x))T • VV(x), en+i) = t^o/i(«.*(«))-£g(-.*(«))^« = (Vk((x),X(x)) - (vV(x),^-(x,k(x))y Thus Z • Bi • W(x, y) = -Yt • V(x) where ft t Yt-^ixM*)). Nonlinear Stabilization by Adding Integrators 505 Now Z2 • Bi • W(x, y) = Z(Z Bi- W)(x, y) =-Z • (Y> • V(x)) and since Yj • V(x) is independent of y we have: Z • (Yi • V(x)) = X -Y{- V(x) S° Z2-Bi-W(x,y) = -X-Yi-V(x) and by induct ion we prove t h a t for any integer s > 1 and any (x, y) 6 A : Zs •Bi-W(x,y) = -Xs~l -Yi-V(x), i=l,...,p. (11) A similar computa t ion shows tha t we have also for any integer s and any (x, y) G A : Zs+X W(x,y) = Xs+1 V(x) (12) The equalities (11) and (12) show that C = S so Theorem 2 is proved. • Example 2. Consider the following system: { X = sin(xy) = f(x,y) y = u (13) (x,y)eM2, u € l . To stabilize this system we can t ry to stabilize first the system x = sin(xy) where y is considered as the control. This system is obviously asymptotically stabi-lizable by a smooth function y = k(x) and after we can use Theorem 1 to stabilize system (13) but the feedback resulting is complicated. Alternatively we can stabilize system (13) by a simpler smooth feedback if we apply Theorem 2 as follow: Introduce V(x) = \-x2 and k(x) = 0 we have XV(x) = Q, Y = ^-(x,k(x)) = x, Y-V(x) = 0& x = Q. This shows that V and k satisfy the hypotheses of Theorem 2 so system (13) is (G.A.S) by means of the feedback law: ( sin(xy) , , I -y-x—i—'- if y ,£0 u(x,y)={ » y •x2 otherwise 506 A. IGGIDR AND G. SALLET Example 3. x\ = y(x\ - x2) (14) (15) x2 = y(x2 + y) y = v x = (x\, x2) £ B2, y 6 1R, v e IR. First, consider the reduced system i\ = u(xi - x2) x2 = u(x2 + u) where u is regarded as the control. Here .. . f u(x\ — x2) \ df . ( xi - x2 f(x,u)=[ ) :' 1, and —(x,u)=[ Jy ' > \ u(x2 + u) J ' 8uK ' \ x2 + 2u If we choose u = k(x\, x2) =- 0 and V(x\,x2) = - ( x j + x2) then all the hypotheses of Theorem 2 are satisfied. In fact, with the notations used in the proof, we have X.V(xx,x2) = Q, Y= y~{z,k(x)) = (** ~ a*2) 'df, du (x, k(x)),VV(x)) = x{- xlX2 + xi So Y • V(x) = 0 & xx =x2 = 0. A stabilizer for (14), computed using (10), is u = -y - xx(x\ - x2) - x2(x2 + y). (Received March 23, 1993.) REFERENCES [1] J. M. Coron and L. Praly: Adding an integrator for the stabilization problem. Systems Control Lett. jT7(1991), 89-104. [2] W. P. Dayawansa, C. F. Martin and G. Knowles: Asymptotic stabilization of a class of smooth two-dimensional systems. SIAM J. Control Optim. 28 (1990), 1321-1349. [3] P. V. Kokotovic and H.J. Sussmann: A positive real condition for global stabilization of nonlinear systems. Systems Control Lett. 13 (1989), 125-133. [4] J. LaSalle and S. Lefschetz: Stability by Liapunov's direct method with applications. Academic Press, New York 1961. [5] R. Outbib and G. Sallet: Stabilizability of the angular velocity of a rigid body revisited. Systems Control Lett. 18 (1992), 93-98. [6] A. Saberi, P. V. Kokotovic and H.J. Sussmann: Global stabilization of partially linear composed systems. SIAM J. Control Optim. 28 (1990), 1491-1503. [7] E. Sontag and H.J. Sussmann: Further comments on the stabilizability of the angular velocity of the rigid body. Systems Control Lett. 12 (1988), 213-217. Dr. Abderrahman Iggidr and Prof. Gauthier Sallet, CONGE Project, INRIA Lorraine & University of Metz 4, rue Marconi 57070 Metz. France. 

Nonlinear stabilization by adding integrators

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Nonlinear stabilization by adding integrators

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