International audienceThe global stabilization problem of a nonlinear control system with an integrator is considered when the initial system (subsystem without integrator) is stabilizable. It is supposed that the stabilizing feedback and the Lyapunov function for the initial system satisfy the Barbashin-Krasovski\u ı asymptotic stability theorem. Explicit formulae for stabilizing feedback of a nonlinear system with an integrator are derived. The use of Lyapunov functions with derivatives of constant but not fixed signs significantly simplifies the computing of stabilizing feedback. This is confirmed by examples. The global stabilization of a nonlinear control system of the form \dot x= f(x, y), \qquad \qquad \dot y=u \tag 1 is studied, where x∈ \bbfRⁿ, y∈ \bbfR^p, u∈ \bbfR^p and f is a smooth vector field such that f (0, 0) =0. It is proved that to find a feedback stabilizer for this system we do not need to have a strict Lyapunov function for the subsystem (2) \dot x= f(x, v), where v is the input. Moreover, it is proved how to asymptotically stabilize system (1) without stabilizing system (2)

Iggidr, Abderrahman

Sallet, Gauthier

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Nonlinear stabilization by adding integrators

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. 

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HAL Id: hal-01860809https://hal.archives-ouvertes.fr/hal-01860809Submitted on 23 Aug 2018HAL is a multi-disciplinary open accessarchive for the deposit and dissemination of sci-entific research documents, whether they are pub-lished or not. The documents may come fromteaching and research institutions in France orabroad, or from public or private research centers.L’archive ouverte pluridisciplinaire HAL, estdestinée au dépôt et à la diffusion de documentsscientifiques de niveau recherche, publiés ou non,émanant des établissements d’enseignement et derecherche français ou étrangers, des laboratoirespublics ou privés.Nonlinear stabilization by adding integratorsAbderrahman Iggidr, Gauthier SalletTo cite this version:Abderrahman Iggidr, Gauthier Sallet. Nonlinear stabilization by adding integrators. Kybernetika,Institute of Information Theory and Automation, 1994, 30 (5), pp.499–506. ￿hal-01860809￿KY BERNET I K A — V OL UME 3 0 ( 1 9 9 4 ) , N UM B ER 5 , PAGE S 4 9 9 – 5 0 6NONLINEAR STABILIZATIONBY ADDING INTEGRATORS1Abderrahman Iggidr and Gauthier SalletIn this paper, we study the global stabilization, by means of smooth state feedback, ofsystems (S) obtained by adding an integrator to a general nonlinear system (P). We showhow to compute the stabilizing feedback for (S) when a stricte Lyapunov function for (P)is difficult to find.1. INTRODUCTIONIn this paper we deal with the global stabilization of nonlinear control systems ofthe form: {ẋ = f(x, y)ẏ = u (1)where x ∈ IRn, y ∈ IRp, u ∈ IRp and f is a smooth vector field such that f(0, 0) = 0.It is well known [3, 6, 7] that if the subsystem:ẋ = f(x, v) (2)where v is the input, is globally asymptotically stabilizable (G.A.S) by means offeedback law v = k(x), where k is of class Cr, r ≥ 1 then system(1) is G.A.S.Moreover, if it is possible to construct a Lyapunov function V such that〈∇V, f(x, k(x)〉 < 0 ∀x ∈ IRn, x 6= 0 (3)(V is said a strict Lyapunov function for system (2) and exists by the Lyapunovinverse theorem) then a stabilizing feedback u(x, y), which depends on k and V , isexplicitly given. However it is not easy to find a function V satisfying (3) even ifone knows that the origin is a globally asymptotically stable equilibrium point forthe closed loop systemẋ = f(x, k(x)). (4)The goal of this paper is to weaken these hypotheses. Theorem 1 shows that, tofind a feedback stabilizer for system (1), we do not need to have a strict Lyapunov1Presented at the IFAC Workshop on System Structure and Control held in Prague on September3–5, 1992.500 A. IGGIDR AND G. SALLETfunction for (2). Theorem 2 shows how to asymptotically stabilize system(1) withoutstabilizing system (2).We recall that the relationship between the stabilizability of (2) and (1) is anopen problem when system (2) is stabilizable by means of continuous feedback (notC1). This problem was addressed in [1] and [2] from the local stabilization pointof view. The authors proved that the local stabilizability of (2) is equivalent to thelocal stabilizability of (1) if n = p = 1 and f is a real analytic function.2. MAIN RESULTSBefore 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 thereexist:1. a function k : IRn → IRp of class Cr(r ≥ 1) with k(0) = 02. a function V : IRn → IR of class C1, definite positive and proper such that:i) X · V (x) ≤ 0 ∀x ∈ 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) = 〈∇V, X(x)〉 where 〈·, ·〉 is the inner product in IRn)ii) The largest invariant set contained in E = {x ∈ IRn|X · V (x) = 0} is theorigin of IRn.Definition 2. A continuously differentiable scalar function V : IRn → IR is a weakLyapunov function forẋ = X(x)if V is positive definite proper andX · V (x) ≤ 0 ∀x ∈ IRn.By a proper function we mean a functionV : IRn → IRsuch that {x ∈ IRn|V (x) ≤ ξ} is compact for each ξ > 0.Through this paper ‖ . ‖ will denote the usual Euclidian norm in IRp, Xt(·) is theflow of the vector field X defined on IRn.Remarks.1. System (2) is of (L-T) ⇔ ẋ = 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 501Theorem 1. If system (2) is of (L-T) then system (1) is also of (L-T) and thestabilizing feedback isu = −y + k(x) + dk(x).f(x, y)−G(x, y)T · ∇V (x). (5)(T = transpose)wheref(x, y) = f(x, k(x)) + G(x, y) · (y − k(x)). (6)Remark. One can choose for G the matrix:G(x, y) =∫ 10∂f∂y(x, ty + (1− 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 abovedefinition. Let us denote X(x) = f(x, k(x)) and Ω the largest invariant set by Xcontained in E = {x ∈ IRn|X · V (x) = 0}. By hypotheses Ω = {0}.LetZ(x, y) =(f(x, y)u(x, y))whereu(x, y) = −y + k(x) + dk(x) · f(x, y)−G(x, y)T · ∇V (x)and define (see [7])W (x, y) = V (x) +12‖y − k(x)‖2W is of class C1, definite positive and properẆ (x, y) = Z ·W (x, y) = 〈Z(x, y),∇W (x, y)〉 = X · V (x)− ‖y − k(x)‖2 ≤ 0Note that all trajectories of the closed-loop system are bounded because W is properand its derivative is nonpositive.LetẼ = {(x, y) ∈ IRn+p |Z ·W (x, y) = 0}= {(x, y) ∈ IRn+p |y = k(x) and X · V (x) = 0}.According to LaSalle’s theorem (see [4] pp. 66–67) all solutions tend to Ω̃ the largestinvariant set by Z contained in Ẽ. To prove Theorem 1 it remains to that Ω̃ is theorigin of IRn+p.On Ω̃ the vector field Z is given by:Z(x, k(x)) =(X(x)Y (x))where Y (x) = dk(x) · f(x, k(x))−G(x, k(x))T · ∇V (x) and X(x) = f(x, k(x)){ẋ = f(x, k(x)) = X(x)ẏ = dk(x) · f(x, k(x))−G(x, k(x))T · ∇V (x).502 A. IGGIDR AND G. SALLETLet (x(t), y(t)) be a solution of the above system starting at (x(0), y(0)), the initialcondition ∈ Ω̃. Since Ω̃ is Z-invariant we have (x(t), y(t)) ∈ Ω̃ for all t ≥ 0 but wehaveddt(x(t)) = X(x(t))hence x(t) = Xt(x) where Xt(·) is the flow of the vector field X defined on IRn.Consider, now, the following set:M = {x ∈ IRn|(x, k(x)) ∈ Ω̃}.If x ∈ M then (x, k(x)) ∈ Ω̃ and (x(t), y(t)) ∈ Ω̃ since Ω̃ is invariant, this implies(Xt(x), y(t)) ∈ Ω̃ but y(t) = k(Xt(x)).So we have shown: x ∈ M ⇒ (Xt(x), k(Xt(x))) ∈ Ω̃ ⇒ Xt(x) ∈ M .This proves that M is X-invariant and since M is contained in E we have M = {0}and then Ω̃ = {(0, 0)} which completes the proof of Theorem 1. 2Example 1. Consider the following system which evolves in IR4:ẋ1 = x2x3 + b1yẋ2 = −x1x3 + b2yẋ3 = b3yẏ = u(8)where b3 6= 0 and (b1, b2) 6= (0, 0).The subsystem ẋ1 = x2x3 + b1yẋ2 = −x1x3 + b2yẋ3 = b3y(9)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 functionsk = −b3 x3 −(x12 + x22 + x3) (2b3 + 4 (b1x1 + b2x2))P (x3)2−b3(x12 + x22 + x3)2P ′(x3)2V =x32 +(x12 + x22 + x3)2P (x3)2whereP (x) =b344(b12 + b22) + b32 x + 2(b12 + b22)x2Nonlinear Stabilization by Adding Integrators 503so all the assumptions of Theorem 1 are satisfied and then our result can be thestabilizing feedback for system (8) using the formula (5). Note that a strict Lyapunovfunction for system (9) has never been found and may be difficult to construct sothe results of [3, 6, 7] cannot be applied to stabilize system (8).Generally, to compute the stabilizing feedback for systems of the form:ẋ = f(x) + g(x)yẏ = ux ∈ IRn, y ∈ IRp, g(x) ∈ Mn,p(C∞(IRn, IR))where the subsystem ẋ = f(x) + g(x)y is stabilizable via the Jurdjevic–Quinn’smethod, one can apply Theorem 1 but not the results based on strict Lyapunovfunction.For the following result we suppose that f is a smooth (i. e. C∞) vector field.Theorem 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, suchthat:i) X · V (x) ≤ 0 ∀x ∈ IRn where X(x) = f(x, k(x))ii) The setS = {x ∈ IRn|Xs+1 · V (x) = Xs · Yi · V (x) = 0, s ∈ IN, i = 1, . . . , p}where Yi =∂f∂yi(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)T · ∇V (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)T · ∇V (x) + v where v is a newinput so system(1) can be written:(ẋy)= Z(x, y) + B(x, y) · v = Z(x, y) +p∑i=1viBiwhereZ(x, y) =(f(x, k(x)) + G(x, y) · (y − k(x))dk(x) · f(x, y)−G(x, y)T · ∇V (x))and Bi(x, y) = en+i.IntroduceW (x, y) = V (x) +12‖y − k(x)‖2.504 A. IGGIDR AND G. SALLETW is smooth, definite positive and properZ ·W (x, y) = X · V (x) ≤ 0.According to [5], the above system is globally asymptotically stabilizable by thefeedbackv = −B ·W (x, y)if the setA ={(x, y) ∈ IRn+p|Zs+1 ·W (x, y) = Zs ·Bi ·W (x, y) = 0, s ∈ IN, i = 1, . . . , p}is reduced to the origin of IRn+p.Since Z.W (x, y) = X · V (x) and B ·W = y − k(x) we can write:A = {(x, y) ∈ IRn+p|y = k(x) and x ∈ C}where:C ={x ∈ IRn|X · V (x) = Zs+1 ·W (x, k(x)) = Zs ·Bi ·W (x, k(x)) = 0,s ≥ 1, i = 1, . . . , p}.We shall show that C = S. We haveB ·W = y − k(x)Z ·Bi ·W = Z · (yi − ki(x)) = 〈Z,∇(yi − ki(x))〉 .For (x, y) ∈ A the vector field Z is (since y = k(x)):Z =(X(x)dk(x) · f(x, k(x))−G(x, k(x))T · ∇V (x))so〈Z,∇(yi − ki(x))〉 = −〈X,∇ki(x)〉+〈dk(x) · f(x, k(x))−G(x, k(x))T · ∇V (x), en+i〉.But 〈dk(x) · f(x, k(x))−G(x, k(x))T · ∇V (x), en+i〉=n∑j=1∂ki∂xj(x)fj(x, k(x))−n∑j=1∂fj∂yi(x, k(x))∂V∂xj(x)= 〈∇ki(x), X(x)〉 −〈∇V (x), ∂f∂yi(x, k(x))〉.ThusZ ·Bi ·W (x, y) = −Yi · V (x)whereYi =∂f∂yi(x, k(x)).Nonlinear Stabilization by Adding Integrators 505Now Z2 ·Bi ·W (x, y) = Z · (Z ·Bi ·W )(x, y) = −Z · (Yi · V (x))and since Yi · V (x) is independent of y we have:Z · (Yi · V (x)) = X · Yi · V (x)soZ2 ·Bi ·W (x, y) = −X · Yi · V (x)and by induction we prove that for any integer s ≥ 1 and any (x, y) ∈ A:Zs ·Bi ·W (x, y) = −Xs−1 · Yi · V (x), i = 1, . . . , p. (11)A similar computation shows that we have also for any integer s and any (x, y) ∈ A:Zs+1 ·W (x, y) = Xs+1 · V (x) (12)The equalities (11) and (12) show that C = S so Theorem 2 is proved. 2Example 2. Consider the following system:ẋ = sin(xy) = f(x, y)ẏ = u(x, y) ∈ IR2, u ∈ IR.(13)To stabilize this system we can try to stabilize first the systemẋ = 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 stabilizesystem (13) but the feedback resulting is complicated.Alternatively we can stabilize system (13) by a simpler smooth feedback if we applyTheorem 2 as follow:IntroduceV (x) =12x2 and k(x) = 0we haveX · V (x) = 0, Y = ∂f∂y(x, k(x)) = x, Y · V (x) = 0 ⇔ x = 0.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:u(x, y) =−y − x sin(xy)yif y 6= 0−x2 otherwise506 A. IGGIDR AND G. SALLETExample 3. ẋ1 = y(x1 − x2)ẋ2 = y(x2 + y)ẏ = vx = (x1, x2) ∈ IR2, y ∈ IR, v ∈ IR.(14)First, consider the reduced systemẋ1 = u(x1 − x2)ẋ2 = u(x2 + u)(15)where u is regarded as the control. Heref(x, u) =(u(x1 − x2)u(x2 + u)), and∂f∂u(x, u) =(x1 − x2x2 + 2u).If we choose u = k(x1, x2) = 0 and V (x1, x2) =12(x21 + x22) then all the hypothesesof Theorem 2 are satisfied. In fact, with the notations used in the proof, we haveX.V (x1, x2) = 0, Y =∂f∂u(x, k(x)) =(x1 − x2x2)〈∂f∂u(x, k(x)),∇V (x)〉= x21 − x1x2 + x22.So Y · V (x) = 0 ⇔ x1 = x2 = 0.A stabilizer for (14), computed using (10), isu = −y − x1(x1 − x2)− x2(x2 + y).(Received March 23, 1993.)REFE REN CES[1] J.M. Coron and L. Praly: Adding an integrator for the stabilization problem. SystemsControl Lett. 17 (1991), 89–104.[2] W.P. Dayawansa, C. F. Martin and G. Knowles: Asymptotic stabilization of a class ofsmooth 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 stabilizationof 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 linearcomposed systems. SIAM J. Control Optim. 28 (1990), 1491–1503.[7] E. Sontag and H. J. Sussmann: Further comments on the stabilizability of the angularvelocity 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 57 070 Metz. France.

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Nonlinear stabilization by adding integrators

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