INSTITUT NATIONAL DE RECHERCHE EN INFORMATIQUE ET EN AUTOMATIQUE. Pascal MORIN - Claude SAMSON N March apport de recherche ISSN

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1 INSTITUT NATIONAL DE RECHERCHE EN INFORMATIQUE ET EN AUTOMATIQUE Control of Nonlinear Chained Systems. From the Routh-Hurwitz Stability Criterion to Time-Varying Exponential Stabilizers Pascal MORIN - Claude SAMSON N 326 March 997 THÈME 4 apport de recherche ISSN

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3 Control of Nonlinear Chained Systems. From the Routh-Hurwitz Stability Criterion to Time-Varying Exponential Stabilizers Pascal MORIN - Claude SAMSON Th me 4 Simulation et optimisation de syst mes complexes Projet ICARE Rapport de recherche n326 March pages Abstract: We show how any linear feedback law which asymptotically stabilizes the origin of a linear integrator system of order (n?) induces a simple continuous timevarying feedback law which exponentially stabilizes the origin of a nonlinear (2; n) single-chain system. The proposed control design method is related to, and extends in the specic case of chained systems, a recent method developed by M'Closkey and Murray [9] for driftless systems in order to transform smooth feedback stabilizers yielding slow polynomial convergence into continuous homogeneous ones which ensure faster exponential convergence. Key-words: Chained system, time-varying control, homogeneous system, continuous feedback. (R sum : tsvp) From September 996 to August 997, P. Morin is with the Control and Dynamical Systems Department, California Institute of Technology Pasadena, CA 925, USA. He contributed to this work while he was with INRIA. pmorin@cds.caltech.edu Claude.Samson@inria.fr Unité de recherche INRIA Sophia Antipolis 2004 route des Lucioles, BP 93, SOPHIA ANTIPOLIS Cedex (France) Téléphone : Télécopie :

4 Commande de Syst mes Cha n s. Du Crit re de Stabilit de Routh-Hurwitz des Retours d'etat Exponentiellement Stabilisant R sum : Nous montrons que tout retour d' tat lin aire qui stabilise asymptotiquement l'origine d'un int grateur lin aire d'ordre (n? ) induit un retour d' tat instationnaire continu simple qui stabilise l'origine d'un syst me cha n non-lin aire d'ordre n deux entr es. La m thode propos e pour la construction de la loi de commande est apparent e, et tend dans le cas particulier des syst mes cha n s, une m thode d velopp e r cemment par M'Closkey et Murray [9] pour les syst mes sans d rive an de transformer des retours d' tat stabilisant di rentiables conduisant une convergence polynomiale lente en des retours d' tat continus homog nes assurant une convergence exponentielle. Mots-cl : Syst me cha n, commande instationnaire, syst me homog ne, retour d' tat continu.

5 Control of chained systems... 3 Introduction Control systems in the so-called chained form have been extensively studied in the past recent years. This research interest partly stems from the fact that the kinematic equations of many nonholonomic mechanical systems, such as these arising in mobile robotics (unicycle-type carts, car-like vehicles with trailers,...), can be converted into this form [3, 7, 9]. Systems in the chained form thus oer a general framework for studying the control of these mechanical systems. The present paper addresses the problem of asymptotic stabilization of a given equilibrium point (which corresponds to a xed desired conguration for a mechanical system). Since chained systems (with state and control vectors denoted as x and u respectively) do not satisfy Brockett's necessary condition [], they cannot be asymptotically stabilized, with respect to any equilibrium point, by means of a continuous pure state feedback u(x). In [6], one of the authors proposed and derived smooth time-varying feedback laws u(x; t) for the stabilization of a unicycle-type vehicle, the equations of which can be converted into a three-dimensionnal chained system. This showed how the topological obstruction raised by Brocket could be dodged, and was the starting point of other studies about time-varying feedbacks. In [3, 4], Coron established that most controllable systems can be asymptotically stabilized with this type of feedback. The literature on the subject has since then mostly focused on the problem of explicit design of such stabilizing control laws. Smooth feedback laws, yielding slow (polynomial) asymptotic convergence, have rst been developed using either a Lyapunov approach ([4, 6, 7],...) or center manifold techniques ([0, 8, 20, 2],...). In order to obtain a faster (exponential) rate of convergence, which cannot be achieved via smooth feedback for systems whose linearization is not controllable, M'Closkey and Murray have used in [7] the properties associated with homogeneous systems. This yields time-varying feedback laws which are only continuous everywhere. This approach has since then been further investigated by other authors, in the specic case of chained systems ([7, 2, 5],...) and for more general driftless controllable systems [8, ]. Recently, M'Closkey and Murray have also presented in [9] a method for transforming smooth time-varying stabilizers into homogeneous continuous ones. The method is best suited for driftless systems for which it applies systematically. The construction of the exponential stabilizer relies upon the initial knowledge of an adequate Lyapunov function coupled with a smooth stabilizing feedback law. More precisely, the exponential stabilizer is obtained by scaling the size of the smooth control inputs on a level set of the Lyapunov function. The continuous time-varying feedbacks derived in the present paper have been obtained by adaptating and com- RR n326

6 4 P. Morin and C. Samson bining the core of this method to the control design method earlier proposed by Samson in [7] for the smooth feedback stabilization of chained systems. Although our approach is specic to chained systems, and therefore, in some respect, less general than the work reported in [9], it also carries with it two important improvements with respect to this work. The rst one is that the knowledge of a (denite negative) Lyapunov function coupled with a smooth stabilizing feedback is not needed. In fact, instead of going thru the intermediary stage consisting of nding a stabilizing smooth time-varying feedback and a corresponding Lyapunov function for the controlled system, we go one step further and show that the knowledge of a linear feedback which stabilizes a linear integrator system whose struture is reminiscent of the one of the chained system, and of a Lyapunov function the derivative of which is only semi-negative along the solutions of the stabilized linear integrator, are sucient. This makes a signicant dierence because nding a good" Lyapunov function for a chained system of order larger than three is not such a simple task. Moreover, for general controllable driftless systems, the design of a smooth time-varying stabilizer may in fact be more dicult than the direct construction of a continuous homogeneous time-varying stabilizer. As a matter of fact, no general control design method has so far been developed in the smooth case, while one already exists in the continuous homogeneous case []. The second improvement is related to the scaling factor used to transform the smooth feedback into a continuous exponentially stabilizing one. In [9], this factor is implicitely dened as the positive real solution to an equation involving the considered Lyapunov function. The uniqueness of this solution along the controlled system's trajectories is required and depends on a transversality condition the satisfaction of which itself depends on the candidate Lyapunov function and has to be checked beforehand. Solving such an equation will usually have to be performed numerically. The rst continuous time-varying feedback law proposed in the present study is of this type. However, we also show in a second result that this scaling factor may in fact be replaced by an adequate explicit function. The implementation of the resulting control law is consequently simplied. This paper is organized as follows. In Section 2, a few technical results used further for the design of the control laws are recalled. In particular, useful relationships between the Routh-Hurwitz stability criterion for linear systems and the transformation of a companion matrix into the so-called Schwartz matrix are reviewed. The two main results and proposed control laws are presented in Section 3 in the form of two propositions. In the rst one, the aforementionned scaling factor is still INRIA

7 Control of chained systems... 5 implicitely dened. The second proposition is an adaptation of the rst one in order to get rid of the implicit denition of the scaling factor. The proofs of these results are reported Section 4, and the proofs of the technical lemmas introduced along the paper are given in the paper's Appendix. 2 Preliminary recalls 2. Stabilization of a multi-order integrator and the Routh-Hurwitz criterion Consider the following linear (n? )-order integrator: whose equivalent controllable state realization is: Any linear feedback control 8 >< >: (n?) d x 2 = u () dt _x 2 = x 3 _x 3 = x 4 _x n? = x n _x n = u. X (2) i=n u =? a i x i (3) i=2 asymptotically (and exponentially) stabilizes the origin of this system provided that all roots of the characteristic polynomial p(s) = s n? + a n s n?2 + : : : + a 3 s + a 2 associated with the closed-loop system have strictly negative real parts. The Routh- Hurwitz table associated with this polynomial is a n? a n?3 : : : : : : a n a n?2 a n?4 : : : 0 b n b n?2 : : : : : : 0 c n c n?2 : : : : : : 0 d n d n?2 : : : : : : : : : (4) RR n326

8 6 P. Morin and C. Samson with a k = 0 for k < 2 b k =? a n (a k?2? a n a k? ) =? a n a k? a n a k?2 ; c k =? (a n b k?2? b n a k?2 ) =? a n a k?2 ; a n b n a n b n b n b k?2 d k =? (b n c k?2? c n b k?2 ) =? b n b k?2 b n c n b n c n c n c k?2. (5) Let k = (k 2 ; : : : ; k n ) be dened from the rst column of the Routh-Hurwitz table as follows: k n = a n k n? = b n k n?2 = c (6) n Then, we have the following two lemmas whose proofs, which may be found in several control textbooks (see [2] for example), are given in the Appendix for the sake of completeness. Lemma Let X 2 = (x 2 ; x 3 ; : : : ; x n ) T and consider the linear change of coordinates X 2 7?! Z 2 = (z 2 ; z 3 ; : : : ; z n ) T = k X 2 dened by. z 2 = x 2 z 3 = x 3 z j+3 = k j+ z j+ + L f z j+2 for j = ; : : : ; n? 3 ; (7) where L f z i 2 f stands for the Lie-derivative of the function z i (X 2 ) along f (X 2 ) = (x 3 ; x 4 ; : : : ; x n ; 0) T. Then, in the coordinates Z 2, the controlled system (2)-(3) becomes 8 _z 2 = z 3 _z 3 =?k 2 z 2 + z 4 >< >:. _z j+ =?k j z j + z j+. _z n? =?k n?2 z n?2 + z n _z n =?k n? z n?? k n z n (8) INRIA

9 Control of chained systems... 7 Using the fact that the time-derivative of the quadratic function V z (Z 2 ) = Z T 2 diag(; k 2 ; along any solution of the system (8) is ; : : : ; Q k 2 k i=n? )Z 2 (9) 3 i=2 k i one easily establishes: _V z (Z 2 ) =?2 k n Qi=n? i=2 k i z 2 n ; (0) Lemma 2 The origin Z 2 = 0 of the linear system (8) is asymptotically stable if and only if k i > 0 for i = 2; : : : ; n. A corollary of the above two lemmas is the well-known Routh-Hurwitz stability criterion: Corollary (Routh-Hurwitz stability criterion) All roots of the polynomial p(s) = s n? + a n s n?2 + : : : + a 3 s + a 2 have strictly negative real parts if and only if k i > 0 for i = 2; : : : ; n. 2.2 Non-exponential time-varying feedback stabilization of chained systems Beyond the interest of recalling a rather simple method for proving the Routh- Hurwitz stability criterion, the prime objective of the previous section was to point out the algebraic operations which transform the chain of integrators involved in the system (2)-(3) into the skew-symmetric representation (8) to which the simple Lyapunov function (9) can be associated. The objective was also to recall the oneto-one correspondance between the two sets of control parameters a i (i = 2; : : : ; n) and Routh-Hurwitz parameters k i (i = 2; : : : ; n) respectively involved in these two equivalent system's representations. In [7], the structural similitude between the linear n-order integrator system 2 and the following nonlinear (2; n) single-chain system: RR n326 8 >< >: _x = u _x 2 = u x 3 _x 3 = u x 4. _x n? = u x n _x n = u 2 ()

10 8 P. Morin and C. Samson has been used, with the aforementionned transformations, to prove the following stabilization result. Proposition ([7, Prop. 2.2]) Let a i (i = 2; : : : ; n) be a set of parameters for which the origin of the linear system (2)-(3) is asymptotically stable. Then, the continuous time-varying feedback control 8 >< >: u (x; t) =?k x + g(x 2 ) sin t u 2 (x; t) =?u (x; t) i=n X i=2 a i sign(u ) n+?i x i ; with k > 0 and g(x 2 ) a continuous function which vanishes at X 2 = 0 (i.e. g(0) = 0) and is strictly positive elsewhere, applied to the chained system () i) makes the positive function V x (X 2 ) X T 2 T k diag(; k 2 ; non-increasing along any solution of this system, (2) ; : : : ; Q k 2 k i=n? ) k X 2 (= V z (Z 2 )) (3) 3 i=2 k i ii) globally asymptotically stabilizes the origin x = 0 of this system. This result clearly indicates how any linear feedback control which, by application of the Routh-Hurwitz stability criterion, asymptotically stabilizes the origin of the linear (n? )-order integrator system (2) induces a simple continuous time-varying feedback law which globally asymptotically stabilizes the origin the corresponding chained system (). However, as pointed out in [7], a shortcoming of the feedback law (2) is that it yields slow (polynomial) asymptotic convergence to zero for most of the system's solutions. The main contribution of this paper is to show how this time-varying control may itself be simply modied in order to render the controlled chained system homogeneous of degree zero with respect to some dilation and ensure uniform exponential convergence. Note that the method proposed by M'Closkey and Murray in [9] to transform a smooth stabilizer into a continuous homogeneous one does not apply directly in the present case because i) the control (2) is not smooth since it is not dierentiable on the set X 2 = 0, ii) Lyapunov functions for the controlled system are not known, and iii) the degrees of homogeneity of the two control inputs u and u 2 are not equal. INRIA

11 Control of chained systems Homogeneity and exponential stabilization The set of nonlinear systems which are homogeneous of degree zero with respect to some dilation constitutes a fairly natural extension of the set of linear systems. Some properties of these systems, that will be used in the sequel, are briey recalled hereafter. For more details, the reader is referred to [5] or [6], for example. For any > 0 and any set of real parameters r i > 0 (i = ; : : : ; n), one denes a dilation operator (; :) : IR n 7?! IR n by (; x ; : : : ; x n ) = ( r x ; : : : ; rn x n ) A function f 2 C 0 (IR n IR; IR) is homogeneous of degree 0 with respect to the (family of) dilations (; :) if : 8 > 0; f ((; x); t) = f (x; t): An homogeneous norm associated with this dilation operator is a function from IR n to IR, homogeneous of degre one with respect to the dilation, positive ((x) 0, 8x), and proper ((x) tends to innity when jxj, the euclidean norm of x, tends to innity). A consequence of this denition is that (x) tends to zero only when jxj tends itself to zero. An example of homogeneous norm is: p (x) = ( nx j= p r jx j j j ) p with p > 0: (4) A dierential system _x = f (x; t) (or a vector eld f), with f 2 C 0 (IR n IR; IR n ), is homogeneous of degree 0 with respect to the dilation (; :) if for any i = ; : : : ; n, the ith component f i of the vector eld f is homogeneous of degree + r i. Finally, let f 2 C 0 (IR n IR; IR n ), with f (x; :) T periodic, dene an homogeneous vector eld of degree 0 with respect to the dilation (; :). Then, the two following properties are equivalent: i) the origin x = 0 of the system _x = f (x; t) is asymptotically stable, ii) x = 0 is globally exponentially stable in the sense that there exists > 0 and, for any homogeneous norm, a value K such that along any trajectory x(t) (t t 0 ) of the system _x = f (x; t), (x(t)) K(x(t 0 )) e?(t?t 0) RR n326

12 0 P. Morin and C. Samson 3 Main results Let us consider the chained system () and dene a family of dilations q (; X 2 ) = ( r 2 x 2 ; : : : ; rn x n ) indexed by the integer q 2 IN via the dilation weights r i chosen as follows: r i = n? i + q for i = 2; : : : ; n : (5) Let us also consider a set of parameters a i (i = 2; : : : ; n) chosen so that the linear control (3) asymptotically stabilizes the origin of the linear system 2. The corresponding positive Routh-Hurwitz parameters are denoted as before as k i (i = 2; : : : ; n), and the regular square matrix associated with the change of coordinates dened in Lemma is again denoted as k. The rst result involves a specic homogeneous norm q (X 2 ) which satises the following equality: V x ( q ( q (X 2 )? ; X 2 )) = C ; 8X 2 6= 0 (6) where C is a positive real number and V x is the quadratic positive function introduced in Proposition. The next lemma asserts that q (X 2 ) is uniquely dened by the polynomial equation (6), provided that q is chosen large enough. Lemma 3 There exists q 0 > such that, for any q q 0 (q 2 IN), i) 8X 2 6= 0, the equation V x ( q (; X 2 )) = C admits a unique positive solution (X 2 ), ii) the function q, from IR n? to IR +, dened by q (X 2 ) = (X 2 )? when X 2 6= 0 and q (0) = 0, is smooth on IR n?? f0g and homogeneous of degree one with respect to the family of dilations q (; :). In view of the previous notations and denitions we are now ready to state the rst main result in the following proposition. Proposition 2 The continuous time-varying feedback control: 8 >< >: u (x; t) =?k x + q (X 2 ) sin(t) k > 0; q q 0 u 2 (x; t) =?u (x; t) applied to the chained system () i=n X i=2 a i sign(u ) n+?i x i = q (X 2 ) n+?i (7) INRIA

13 Control of chained systems... i) makes the controlled system homogeneous of degree zero with respect to the dilation q (; x) = (x ; q (; X 2 )), ii) makes q (X 2 (t)) non-increasing along any solution of the controlled system, iii) globally exponentially stabilizes the origin x = 0 of this system. The proof of this Proposition is given in Section 4. Remarks By imposing q to be larger than one, although the inverse of q (X 2 ) is not x dened for X 2 = 0, each term i q(x 2 involved in the control u ) n+?i 2 (x; t) is homogeneous of positive degree and tends to zero when X 2 tends to zero. Therefore, u 2 (x; t) is, by continuity, well dened on IR n IR. In the control expression (7), q may be seen as a design parameter whose value equals the degree of homogeneity of the control input function u 2 (x; t), while the degree of homogeneity of u (x; t) is equal to one. The possibility of assigning non-equal degrees of homogeneity for the control inputs results from the specic structure of the chained system and represents an extra degree of freedom at the control design level which had not been considered in [9]. The minimal value q 0 of q, for which the homogeneous norm q (X 2 ) is uniquely dened, depends a priori on the constant C, the system's dimension n, and the set of parameters a i. The existence of a value of q 0 which, for given values of C and n, would not depend on the choice of the parameters a i is a pending question which we have not yet explored. The condition imposed on the size of q 0 is directly related to the satisfaction of the transversality condition described in [9, Th. 5.]. The connection appears explicitely in the proof of Proposition 2. The homogeneous norm q (X 2 ) plays the same role as the quadratic function V x (X 2 ) in the case of the non-homogeneous controls (2). In particular, the asymptotic stability of the origin of the controlled system stems from the nonincrease of this function along any system's solution. A practical diculty with the control (7) is that the calculation of q (X 2 ) requires solving the polynomial equation V x ( q (? q ; X 2 )) = C. In general, this will RR n326

14 2 P. Morin and C. Samson have to be done numerically. However, this diculty can be avoided by considering another homogeneous norm such as X i=n p p;q (X 2 ) = ( jx i j n?i+q ) p ; with p > 0 (8) i=2 and using this function in the control expression, instead of q (X 2 ). This statement is precised in the following proposition which is the second result of this paper. Proposition 3 There exists q 0 > such that if q q 0 and p > n? 2 + q then the continuous time-varying feedback control 8 >< >: u (x; t) =?k x (sin 2 t + sign(x ) sin t)? k n+ p;q (X 2 ) sin t; k > 0; k n+ > 0 u 2 (x; t) =?u (x; t) i=n X i=2 applied to the chained system () a i sign(u ) n+?i x i = p;q (X 2 ) n+?i (9) i) ensures that along any solution of the controlled system, V x (Y 2 ((k + ))) (q)v x (Y 2 (k)) 8k 2 IN ; (20) x 2 where (q) < and Y 2 = ( p;q (X 2 ) ; x 3 n?2 p;q (X 2 ) ; : : : ; x n? n?3 p;q (X 2 ) ; x n) T, ii) globally exponentially stabilizes the origin x = 0 of this system. The proof of this proposition is given in Section 4. Remark: Contrary to Propositions and 2, the stability proof no longer relies upon the knowledge of a positive function which is non-increasing along the system's solutions. It uses instead the fact that V x (Y 2 (t)), evaluated at periodic time-instants, is decreasing. As shown in the proof of the proposition, this property itself comes from the particular choice of the control u (x; t) which is such that ju (x; t)j k n+ p;q (X 2 )j sin tj, with the sign of u (x; t) changing periodically as the sign of sin t. Although the slightly more simple control u (x; t) =?k x? k n+ p;q (X 2 ) sin(t) does not satisfy this inequality, so that the stability proof does not hold without modication in this case, we conjecture that this control, combined with the control u 2 (x; t) of 9, also ensures that the origin of the control system is g.a.s. Exponential convergence of the solutions to zero would follow all the same since the controlled systems remains homogeneous of degree zero with respect to the family of dilations dened by q (; x) (x ; q (; X 2 )). INRIA

15 Control of chained systems Proofs of the main results We report in this section the main steps of the proofs of Propositions 2 and 3. For the sake of conciseness the proofs of a few intermediary technical lemmas are omited. They are of course available from the authors to the interested reader. 4. Proof of Proposition 2 Let us assume that q > q 0 so that, according to Lemma 3, the equation: V x ( q (; X 2 )) = C (X 2 6= 0) (2) has a unique positive solution denoted, from now on, as (X 2 ). Dierentiating with respect to time both members of the above equality, one obtains with V x; (y) _ + V x;x (y) q (; _X 2 ) = 0 (22) y = (y 2 ; y 3 ; : : : ; y n ) T = q ((X 2 ); X 2 ) ; V x;x (y) 2 (y); : : (y)) n ; 0 r 2 y 2 V x; (y) = V x;x(y) r n y n From the chained system equations () and the denition of y, one easily veries that 0 u (x; t)y 3 q (; _X 2 ) = C A : u (x; t)y n q? u 2 (x; t) so that one deduces from (22), taking the expression of u 2 (x; t) into account, that _ =?u (x; t)v x;x (y) 0 y 3. y n C A? P i=n i=2 sign(u ) n+?i a i y i C A =V x;(y) : (23) In the proof of Lemma 3, it is shown that for any q large enough (i.e., q q 0 ), there exists two strictly positive numbers C and C 2 such that C (X 2 ) V x;(y) C 2 (X 2 ) (8X 2 ) (24) RR n326

16 4 P. Morin and C. Samson Therefore V x; (y) is strictly positive away from the origin. This corresponds to the tranversality condition introduced in [9, Th. 5.]. Let us now distinguish two cases. Case : u 0 In this case, V x;x (y) 0 y 3. y n? P i=n i=2 sign(u ) n+?i a i y i C A = V x;x(y)f x (y) (25) with X! i=n T f x (y) = y 3 ; : : : ; y n ;? a i y i : (26) In view of Lemma, and using the fact that V x (y) = V z ( k y), V x;x (y)f x (y) = V z;z (w)f z (w) =?2 with w = (w 2 ; : : : ; w n ) T = k y. Therefore, in view of (23)-(27), _ = 2ju j k n i=2 k n Qi=n? i=2 k i w 2 n ( 0) (27) Qi=n? i=2 k i w 2 n =V x;(y) ( 0) (28) This establishes that (X 2 (t)) is non-decreasing when u (x(t); t) is positive. Case 2: u < 0 Let us dene the change of coordinates : IR n? 7! IR n? as follows: (y) (y 2 ;?y 3 ; : : : ; (?) i y i ; : : : ; (?) n y n ) T : (29) From (26) and (29), it is simple to verify that if u is negative, [f x ( (y))] =? 0 y 3. y n? P i=n i=2 sign(u ) n+?i a i y i C A : (30) INRIA

17 Control of chained systems... 5 Therefore, using (30) in (23), _ =?ju (x; t)jv x;x (y) [f x ( (y))]=v x; (y) : (3) Using the denition of the matrix k of change of coordinates introduced in Lemma, it can also be shown that k (y) = ( k y) : (32) Since V z is a quadratic polynomial function with zero cross terms, V z ( [ k y]) = V z ( k y) (= V x (y)) (33) By using (32) and (33), it is not dicult to show that so that, in view of (27), (3), and (34), V x;x (y) [f x ( (y))] = V x;x (y)f x (y) (34) _ = 2ju j k n Qi=n? i=2 k i w 2 n=v x; (y) ( 0) : (35) This establishes that (X 2 (t)) is non-decreasing when u (x(t); t) is negative. Therefore, whatever the sign of u, the relation (35) is satised and _ is nondecreasing along any solution of the controlled system. This implies that the positive function q (X 2 ) p (= (X 2 )?p ), with p > n? 2 + q so as to ensure that this function is of class C on IR n?, is non-increasing along any solution of the controlled system. This in turn implies, from the expression of u (x; t), that x is bounded along any trajectory of the system. Thus, solutions of the controlled system exist for t 2 [0; +). In order to prove that x = 0 is asymptotically stable in the sense of Lyapunov, there only remains to show that every solution x(t) asymptotically converges to zero. To this purpose, one can apply Lasalle's invariance principle for time-periodic systems. Application of Lasalle's invariance principle: One deduces from what precedes that all solutions converge to the largest invariance set M contained in the set E = f x : d dt q(x 2 ) p = 0 g (36) RR n326

18 6 P. Morin and C. Samson with, in view of (35), d dt q(x 2 ) p =?2pju (x; t)j q (X 2 ) p Qi=n? wn=v 2 x; (y) : (37) i=2 k i Using the fact that C q (X 2 ) V x; (y) C 2 q (X 2 ) and that all coecients k i are strictly positive: E = f x : ju (x; t)j q (X 2 ) p? w 2 n = 0 g : (38) Let us consider a solution x(t) within the set E. If X 2 (0) = 0, then q (X 2 (t)) = q (X 2 (0)) = 0, 8t 0, and therefore X 2 (t) = 0, 8t 0. If X 2 (0) 6= 0, then q (X 2 (t)) is constant and dierent from zero, so that X 2 (t) 6= 0, 8t 0. From the expression of u (x; t), this in turn implies that u (x(t); t) cannot be identically equal to zero. Let (t ; t 2 ) denote a non-empty time interval on which u 6= 0. Without loss of generality, one can assume that u (x(t); t) is positive on (t ; t 2 ). Then it comes that w n (t) = 0 for t 2 (t ; t 2 ), since x(t) belongs to the set E. Now, since _ X 2 = u f x (X 2 ) when u is positive, and since (X 2 (t)) (= q (X 2 (t)? ) is constant and dierent from zero, one also has on the interval (t ; t 2 ) and with In particular, f z (w) = 0 k n _y = u f x (y) (39) _w = u f z (w) (40) w 3?k 2 w 2 + w 4.?k n?2 w n?2 + w n?k n? w n?? k n w n C A (4) w_ n = u (?k n? w n?? k n w n ) : (42) Since w n (t) = 0 when t 2 (t ; t 2 ), with and u strictly positive, one deduces from 42 that w n? (t) = 0 when t 2 (t ; t 2 ). By repeating the same reasoning for the other components of the vector w, one iteratively establishes that w i (t) = 0 for i = 2; : : : ; n and t 2 (t ; t 2 ). Therefore, w(t) = y(t) = X 2 (t) = 0 on the interval (t ; t 2 ), so that q (X 2 (t)) = 0 on this interval, thus yielding a contradiction with the initial assumption according to which q (X 2 (t)) is constant and dierent from zero for t 2 [0; +). The largest invariant INRIA

19 Control of chained systems... 7 set within E is thus contained in the set fx : X 2 = 0g, so that any solution x(t) is such that X 2 (t) asymptotically converges to zero. From the expression of u (x; t) and the system's equation _x = u, it follows that x (t) also converges to zero. Finally, the exponential rate of convergence of x(t) to zero simply comes from the fact that the controlled system is homogeneous of degree zero with respect to the dilation q (; x) = (x ; q (; X 2 )). 4.2 Proof of Proposition 3 This proof makes use of the following three technical lemmas. x Lemma 4 The application Y 2 : X 2 7! ( 2 x p;q(x 2 ; 3 x ) n?2 p;q(x 2 ; : : : ; n? ) n?3 ; x p;q(x 2 ) n) T, with p > n? 2 + q and q > 0, is an homeomorphism on IR n?, and a C dieomorphism on IR n?? f0g provided that q is large enough. Moreover, Y 2 (X 2 ) = 0 if and only if X 2 = 0, and lim jx2 j!+jy 2 (X 2 )j = +. Lemma 5 Let h(u ; X 2 ) = (u x 3 ; u x 4 ; : : : ; u x n ; u 2 (u ; X 2 )) T. There exist two continuous functions q; (X 2 ) and q; (X 2 ) from IR n?? f0g to IR such that ( u i) 8X 2 6= 0, L h(u ;X 2 ) p;q (X q; (X 2 ) if u > 0 2 ) = u q;2 (X 2 ) if u < 0 ii) lim sup q! + X 2 6= 0 j q;i (X 2 )j = 0, (i = ; 2). Lemma 6 Consider the system _y = (t)(a + (y; t)b)y (43) with y 2 IR n?, (:) a continuous function from IR to IR +, A a Hurwitz-stable matrix, (:; :) a uniformly bounded continuous function from (IR n?? f0g) IR to IR. Let P denote a symmetric positive denite (s.p.d.) matrix such that P A+A T P 0 (such a matrix exists since A is stable), and y(t) denote a maximal solution of (43). Then, given: i) a function 0 (:) from IR to IR + such that 0 (t) > 0 on some non-empty interval (t ; t 2 ), RR n326

20 8 P. Morin and C. Samson ii) a real number 2 (0; t 2? t ], there exists 2 (0; ) and > 0 such that: (t) 0 (t) ; 8t 2 (t ; t 2 ) kk with kk = Supf(x; t) : (x; t) 2 (IR n?? f0g) IRg. ) =) y(t) T P y(t) (?)y(t ) T P y(t ) ; 8t 2 [t +; t 2 ] The proof of Proposition 3 involves two steps. In the rst one, we show that if q is large enough and if x(t) is a solution such that X 2 (t) 6= 0 on some time interval, then there exist two quadratic positive functions V + (Y 2 ) and V? (Y 2 ) such that, at any time-instant of this time interval, one of them is non-increasing. This implies that any other solution which crosses, at some time, the set X 2 = 0 (the same as the set Y 2 = 0, in view of Lemma 4) remains in this set everafter. For such a solution, the rst state variable satises, after a nite time, the equation _x =?k x (sin 2 t + sign(x ) sin t), and this implies that x (t) asymptotically converges to zero (see [2], for example). Therefore the only solutions which may not converge to zero are those which never cross the set X 2 = 0. The second step of the proof thus consists in showing that any of these solutions asymptotically converges to zero. Exponential stability then simply results from the (easily veriable) fact that the controlled system is homogeneous of degree zero with respect to the dilation q (; x) = (x ; q (; X 2 )). (44) Step : If X 2 (t) 6= 0, the derivative of Y 2 (X 2 ) at time t is well dened and such that _Y 2 = u p;q 0 y 3 y 4. y n? P i=n i=2 a isign(u ) n+?i y i C A + L h p;q p;q 0 2)y 2?(n? 3)y 3.?y n? 0 C A (45) with y i+ = x i+ denoting the ith component of the vector Y n?i? 2. p;q Let us assume that X 2 (t) 6= 0 on some interval [t 0 ; t ). The function u p;q is well dened on this interval. Moreover, in view of the expression of the control u (x; t), j u p;q j k n+ j sin tj with the sign of u being the opposite of the sign of sin(t). The sign of u thus changes periodically. The time interval [t 0 ; t ) is also the union of intervals j (j = 0; ; : : :) such that j sin tj 6= 0 when t belongs to the interior of j. Without lack of generality, we may assume that sin t is negative on the intervals 2k INRIA

21 Control of chained systems... 9 and positive on the intervals 2k+. Let us distinguish two cases. First case: t 2 2k () u (x(t); t) 0). In this case, using the result of Lemma 5, the equation (45) may be rewritten as with _Y 2 = ju (x; t)j p;q (X 2 ) (A + + q; (X 2 )B)Y 2 (46) : : : : : : : : : 0 A + = : : : 0 0?a 2?a 3 : : : : : :?a n? a n B =?diagfn? 2; n? 3; : : : ; ; 0g: Since the matrix A + is stable, there exists two s.p.d. matrices P + and Q + such that P + A + + A T +P + =?Q +. Let us then consider the positive quadratic function V + (Y 2 ) = Y T 2 P +Y 2. Its time-derivative is C A ; (47) _V + = u (x; t) p;q (X 2 ) Y T 2 (?Q + + q; (X 2 )(P + B + B T P + ))Y 2 : (48) Clearly, this time-derivative is non-positive when q; is small enough. In view of Lemma 5, this can be achieved by choosing q large enough. Therefore, V + (Y 2 (t)) is non-increasing on 2k when q is large enough. Second case: t 2 2k+ () u (x(t); t) 0). In this case, we obtain _Y 2 = ju (x; t)j p;q (X 2 ) (A?? q;2 (X 2 )B)Y 2 (49) with A? = 0 0? 0 : : : : : : 0 0 0? 0 : : : : : : 0? 0 C A : (50) (?) n? a 2 (?) n?2 a 3 : : : : : : (?) 2 a n??a n RR n326

22 20 P. Morin and C. Samson Since the matrices A + and A? share the same characteristic polynomial, the matrix A? is also Hurwitz-stable and there exists two s.p.d. matrices P? and Q? such that P? A? +A T?P? =?Q?. Proceeding as in the rst case, we obtain that the quadratic positive function V + (Y 2 ) = Y2 T P?Y 2 is non-increasing on the time-interval 2k+ when q is large enough. Step 2: We now consider a solution such that X 2 (t) 6= 0, 8t 0. In this case, the interval [t 0 ; t ) considered in Step coincides with [0; +), itself the union of intervals i+ = [i; (i + )), i 2 IN. Let us thus again consider two cases, according to whether u is positive or negative. First case: t 2 2k () u (x(t); t) 0). The equation of evolution of Y 2 (t) is given by (46). Note that (t) = ju (x(t);t)j p;q(x 2 (t)) is strictly positive and continuous inside 2k, and that it is larger than 0 (t) = k n+ j sin(t)j. In view of Lemma, it is also simple to verify that P A + + A T + P = diagf0; : : : ; 0;? Q 2kn i=n? g, with P = T k k diagf; i=2 i k 2 ; : : : ; Q i=n? g k. Therefore k i=2 i P A + + A T +P is a semi-negative matrix. Note also that the quadratic function V x dened in (3) is such that V x (Y 2 ) = Y T 2 P Y 2. Therefore, by application of Lemma 6, there exists q > 0 and q 2 (0; ) such that, if q q, V x (Y 2 (2k)) (? q )V x (Y 2 ((2k? ))) : (5) Second case: t 2 2k+ () u (x(t); t) 0). The equation of evolution of Y 2 (t) is given by (49). It has previously been established (see proof of Proposition 2) that P A? + A T? P = P A + + A T + P. By applying again Lemma 6, one deduces that there exists q 2 > 0 and q2 2 (0; ) such that, if q q 2, From (5) and (52), we obtain that V x (Y 2 ((2k + ))) (? q2 )V x (Y 2 ((2k))) : (52) V x (Y 2 ((i + ))) V x (Y 2 (i)) 8i 2 IN ; (53) with = sup(? q ;? q2 ) (2 (0; )), provided that q sup(q ; q 2 ). This relation, plus the fact, proved in Step, that V + (Y 2 (t)) is non-increasing on the intervals 2k and that V? (Y 2 (t)) is non-increasing on the intervals 2k+, clearly imply that Y 2 (t) asymptotically converges to zero. The convergence of x (t) to zero then easily follows from the rst system's equation _x = u and the expression of the control u (x; t) (see [2] for example). INRIA

23 Control of chained systems... 2 Appendix Proof of Lemma The (n? 2) rst equations of the system (8) directly stem from the denition of the change of coodinates between X 2 and Z 2, after remarking that _z j+ 2 _X 2 = L f z j+. Let p(s) = s n? + a n s n?2 + : : : + a 3 s + a 2 denote the characteristic polynomial associated with the linear system (2)(3). Interpreting s as the time-derivative operator, one has p(s)x 2 = 0. In order to prove that the last equation of (8) is correct, one only has to show that the characteristic polynomial q(s) associated with the system (8), i.e. the polynomial with leading coecient equal to one and such that q(s)z 2 = 0, coincides with the polynomial p(s). Let B i (i = 2; : : : ; n) denote the operator between z 2 and z i, i.e. the operator such that z i = B i z 2. In view of (8), one has: B 2 = B 3 = s B i = sb i? + k i?2 B i?2 i = 4; : : : ; n (54) From (54) it is simple to verify that each B i is a polynomial in the following form, according to whether i is par or odd: B 2j = s 2j?2 + j?2 X B 2j+ = s 2j? + k=0 j?2 Using the last equation of (8), one also has: b 2j;k s 2k X k=0 b 2j+;k s 2k+ (55) q(s) = (s + k n )B n (s) + k n? B n? (s) (= B n+ (s) + k n B n (s)) (56) In order to show that q(s) = p(s), one only has to show that the Routh-Hurwitz tables associated with these two polynomials are the same. To this purpose, it is sucient to prove that the rst column of both tables are identical. Indeed, as easily seen from the triangular structure of the Routh-Hurwitz table, each column of the table can be calculated form the previous column, starting with the bottom element of the column and continuing with the upper elements of this column. Let R q (resp. R p ) denote the Routh-Hurwitz table associated with the polynomial RR n326

24 22 P. Morin and C. Samson q(s) (resp. p(s)). In view of (6), and since the leading coecient of the polynomials B i (s) (i = 2; : : : ; n) is equal to one, it is sucient to prove that, for j = 2; : : : ; n, the elements of the jth row of R q, denoted as R q;j, are equal to the coecients of the polynomial k n?j+2 B n?j+2 ordered by decreasing power of s. Let us proceed by induction. By construction of R q, and since q = B n+ + k n B n, the elements of the second row of R q are equal to the coecients of k n B n. The property is thus satised for j = 2. Assume now that it is satised for j = 2; : : : ; m with m < n, and dene k n+ =. This assumption implies, in view of (54), that the elements of R q;j (j = 2; : : : ; m) are equal to the coecients of the polynomial k n?j+2 (sb n?j+ +k n?j B n?j ). In particular, the elements of R q;m? are equal to the coecients of the polynomial k n?m+3 (sb n?m+2 + k n?m+ B n?m+ ) so that the ith coecient of the polynomial k n?m+ B n?m+ is equal to: k n?m+3 R i+ q;m?? k n?m+2 R i+ q;m =? k n?m+3 k n?m+2 (k n?m+3 R i+ q;m? k n?m+2r i+ q;m? ) where R i q;j denotes the ith term of the row R q;j. In view of (5), this coecient is also, by construction of the Routh-Hurwitz table, equal to R i q;m+. Therefore the elements of R q;m+ are equal to the coecients of k n?m+ B n?m+, and the property is satised for j = m +. Proof of Lemma 2 Consider the set of quadratic functions: V 2 (Z 2 ) = 2 z2 2 V k (Z 2 ) = 2 (P k? i=2 (Q k? j=i k j)z 2 i + z2 k ) ; k = 3; : : : ; n One easily veries that along any solution Z 2 (t) of the system (8): _V k (Z 2 (t)) = z k (t)z k+ (t) ; k = 2; : : : ; n? _V n (Z 2 (t)) =?k n z n (t) 2 a) (k i > 0 for i = 2; : : : n))(the origin of (8) is asymptotically stable) This part of the lemma easily follows from the application of Lasalle's invariance principle, with V n (Z 2 ) taken as a Lyapunov function for the system (8). b) (the origin of (8) is asymptotically stable))(k i > 0 for i = 2; : : : n) Assume that k n 0, then _ V n (Z 2 (t)) 0 so that V n (Z 2 (t)) > 0 8t if z n (0) 6= 0 and z i (0) = 0 for i = 2; : : : ; n?. This implies that the coresponding solution Z 2 (t) does INRIA

25 Control of chained systems not converge to zero and contradicts the asymptotic stability of Z 2 = 0. Therefore k n > 0. Assume now that at least one coecient k i, with i 2 (2; : : : ; n? ), is equal to zero, and let m be the smallest integer such that k m = 0. Consider a solution with initial conditions such that z m+ (0) = z m+2 (0) = : : : = z n (0) = 0 and V m (Z 2 (0)) 6= 0. Since (z m+ = 0; z m+2 = 0; : : : ; z n = 0) is an equilibrium point of the subsystem: _z m+ = z m+2 _z m+2 =?k m+ z m+ + z m+3. _z n? =?k n?2 z n?2 + z n _z n =?k n? z n?? k n z n one deduces that, along this solution, _ V m (Z 2 (t)) = 0. Since V m (Z 2 (t)) is constant and dierent from zero, this implies that Z 2 (t) does not converge to zero and contradicts the asymptotic stability of Z 2 = 0. Therefore none of the coecients k i can be equal to zero. Assume nally that some coecient k m, with m 2 (2; : : : ; n? ), is negative. Then there exists an integer l 2 (m; : : : ; n? ) such that Q n? j=l k j is negative. Consider a solution Z 2 (t) with initial conditions such that z n (0) 6= 0, z n (0) 2 + ( Q n? j=l k j )z l (0) 2 = 0, and z i (0) = 0 if i 6= n and i 6= l. Then V n (0) = 0, _ V n (Z 2 (t)) 0, and there exists > 0 such that _ V n (Z 2 (t)) <? knzn(0)2 2 for t 2 [0; ]. Therefore V n (Z 2 (t)) <? knzn(0) 2 < 0 for t, implying that Z 2 (t) does not converge to zero. Since this contradicts the asymptotic stability of Z 2 = 0, one deduces that none of the k i can be negative. Proof of Lemma 3 We rst proceed with the proof of part i). In view of (3), the function V x is a quadratic positive function which vanishes only at the origin. As a consequence, for any X 2 6= 0, V x ( q (; X 2 )) tends to zero as tends to zero, and to + as tends to +. This implies the existence of a solution (X 2 ) to the equation V x ( q (; X 2 )) = C with C > 0. Let us show that, for q large enough, this solution is unique. To this purpose, let us assume that there exists, for some X 2 6= 0, two dierent values and 2 such that V x ( q ( ; X 2 )) = V x ( q ( 2 ; X 2 )) = C. Without loss of generality, we can assume that 0 < < 2. By application of the mean value theorem, there exists 0 2 [ ; 2 ] such x( q (; X 2 )) ( 0 ) = RR n326

26 24 P. Morin and C. Samson x ( q (; X 2 )) ( 0 @ 2 ( q ( 0 ; X ( 0; X 2 2 ( q ( 0 ; X 2 )) 0 (r 2 r 2 0 x 2; : : : ; r n rn 0 x n) T one also has, in view of x( q(;x 2 ( 0 ) = q 2 ( q ( 0 ; X 2 ))( r 2 0 x 2; : : : ; rn 0 x n) T + x 2 ( q ( 0 ; X 2 ))( n?2 q r 2 0 x 2; : : : ; q r n? 0 x n? ; 0) T (57) with, using the fact that V x is a quadratic function: 2 ( q ( 0 ; X 2 ))( r 2 0 x 2; : : : ; rn 0 x n) T = 2V x ( q ( 0 ; X 2 )) (58) x ( q ( 0 ; X 2 ))( n? 2 r 2 2 q x 2; : : : ; q r n? 0 x n? ; 0) T j K q V x( q ( 0 ; X 2 )) (59) for some positive constant K whose value depends on the matrix involved in the quadratic function V x, and thus on the coecients k i, (i = 2; : : : ; n? ). Therefore, in view of (57)-(59): q (2? K 0 q )V x( q ( 0 ; X 2 x( q (; X 2 )) ( 0 ) q (2 + 0 q )V x( q ( 0 ; X 2 )) (60) so 2)) 0 ) (with X 2 6= 0) is strictly positive when q is larger than K. Since this contradicts the existence of two distincts values 2 and 2 such that V x ( q ( ; X 2 )) = V x ( q ( 2 ; X 2 )) = C, we have proved that the equation V x ( q (; X 2 )) = C, when X 2 6= 0, has a unique positive real solution. Note that (60) implies that, for 0 = (X 2 ): (2q? K)C (X 2 x( q ((X 2 ); X 2 (2q + K)C (X 2 ) The positivity 2);X 2 when q is large enough, corresponds to the tranversality condition considered in [9]. The smoothness of q (X 2 ) = (X 2 )? on IR n?? f0g simply results from the implicit function theorem, using that V x and q are smooth and, as shown above, 2)) 2 )) 6= 0 when X 2 6= 0. Finally, it is simple to verify that q (X 2 ) is homogeneous of degree one with respect to the dilation q (; X 2 ) by using the part i) of the lemma and the fact that q ( q (Y 2 )? ; Y 2 ) = q ( ; X q(y 2 ) 2) for Y 2 = q (; X 2 ). (6) INRIA

27 Control of chained systems Proof of Lemma 4 We rst note that the mapping Y 2 is such that: Y 2 ( q (; X 2 )) = q Y 2 (X 2 ) (62) This already implies that Y 2 is continuous at the origin. This also implies that Y 2 is onto since this mapping transforms any element of the set S = fx 2 : p;q (X 2 ) = g into itself. In order to show that Y 2 is injective, let us proceed by contradiction and assume the existence of two distinct non-zero vectors Z 2 = (z 2 ; : : : ; z n ) T and W 2 = (w 2 ; : : : ; w n ) T such that Y 2 (Z 2 ) = Y 2 (W 2 ). Let us denote as z ; w and Z 2 ; W 2 the coecients and vectors dened by: Z 2 = q ( z ; Z 2 ) p;q ( Z 2 ) = W 2 = q ( w ; W 2 ) p;q ( W 2 ) = Using (62) and the fact that Y 2 (Z 2 ) = Y 2 (W 2 ) one deduces that W 2 that: = ( z w ) q Z 2 so p;q ( 0 Z 2 ) = p;q ( Z 2 ) = (63) with 0 = ( z w ) q 6=. By application of the mean value theorem, this implies the existence of > 0 such that: On the other hand, and in view of the denition of p;q 2 ( Z 2 ) Z 2 2 ( Z 2 ) Z 2 = 0 (64) p;q ( Z 2 ) p? nx i=2 r i p r i? jz i j p r i (65) can be equal to zero only if Z 2 = 0. Since, W 2 = ( z ) q w Z 2, this would imply that W 2 = Z 2 = 0, thus yielding a contradiction. Therefore Y 2 is a continuous one-to-one function on IR n?. From (62), it is simple to prove that jy 2 (X 2 )j tends to innity when jx 2 j tends itself to innity, and from there that Y? 2 is also continuous. The fact that Y 2 and the inverse mapping are of class C on IR n?? f0g simply results from the implicit function theorem and the fact that the 2 2 regular on this set for q large enough. RR n326

28 26 P. Morin and C. Samson Proof of Lemma 5 By denition of the Lie derivative: with, in view of the denition of p;q : L h(u ;X 2 ) p;q (X 2 ) 2 (X 2 )h(u ; X 2 ) (66) p;q (X 2 ) = ( r jx 2 j? 2 sign(x 2 ) ; : : : ; p jx n j rn? sign(x n 2 r 2 p;q (X 2 ) p? r n p;q (X 2 ) p? ) (67) and, in view of the expression of u 2 (u ; X 2 ; t): h(u ; X 2 ) = u (x 3 ; : : : ; x n ;? nx i=2 It is then simple to verify that the functions: a i sign(u ) n+?i p;q (X 2 ) n+?i )T (68) P q; (X 2 ) 2 (X 2 )(x 3 ; : : : ; x n ;? P i=2 a i ) T p;q(x 2 ) n+?i (69) q;2 (X 2 ) 2 (X 2 )(x 3 ; : : : ; x n ; i=2 a i (?) n?i x i ) T p;q(x 2 ) n+?i are homogeneous of degree zero with respect to the dilation q (; X 2 ) and, using the fact that r i tends to innity when q tends to innity, that they satisfy the properties of the lemma. Proof of Lemma 6 Dene: Z t g(t) = (s)ds t for t 2 [t ; t 2 ] (70) Since (t), the derivative of g(t), is strictly positive on (t ; t 2 ), g(t) monotically increases on the time interval (t ; t 2 ) so that the inverse of g, denoted as g?, is well dened on [0; g(t 2 )]. It is also dierentiable on (0; g(t 2 )). Now, dene: z( ) y(g? ( )) for 2 [0; g(t 2 )] (7) Dierentiating z( ) with respect to, one obtains in view of (43): d d z( ) = Az( ) + (z( ); g? ( ))Bz( ) for 2 [0; g(t 2 )] (72) x i x i INRIA

29 Control of chained systems Let z( ) denote the solution of the linear system d d z = Az with z(0) = z(0). One has z( ) = exp A z(0), with jexp A j K < + (since A is a stable matrix). Introducing the vector ~z = z( )? z( ), one deduces that: d d ~z( ) = A~z( ) + (z( ); g? ( ))B(~z( ) + exp A z(0)) (73) with ~z(0) = 0. Since A is stable, there exists a s.p.d. matrix Q such that QA + A T Q =?I, where I stands for the unit matrix. Therefore, using (73): d d (~z( )T Q~z( )) =?j~z( )j 2 + 2(z( ); g? ( ))~z( ) T QB(~z( ) + exp A z(0)) (74) and, denoting the upperbound of j(x; t)j as : d d (~z( )T Q~z( ))?(? 2jQjjBj)j~z( )j 2 + 2K jqjjbjjz(0)jj~z( )j (75) Using the fact that ~z(0) = 0, one deduces from the above inequality that, whenever is smaller than 4jQjjBj : ~z( ) T Q~z( ) 2 K 2 jz(0)j [0; g(t 2 )] (76) with K 2 = 6jQj 3 jbj 2 K 2. The above inequality in turn implies the existence of a nite positive number K 3 such that: ~z( ) T P ~z( ) z(0) T P z(0) K 3 2 (77) Since z( ) K jz(0)j, it also results from (76) that there exists a positive number K 4 such that: ~z( )P z( ) z(0) T P z(0) K 4 (78) Now, let us consider the function a( ) = z( )T P z( ) which is analytic, since z( ) = z(0) T P z(0) exp A z(0), and such that a(0) =. Since P A + A T P is semi-negative, a( ) is non increasing. In fact, a( ) <, 8 > 0. Otherwise there would exist > 0 such that a( ) =, and this would imply that a( ) = on the interval [0; ], and consequently that a( ) =, 8 > 0 (by the analyticity of a( )). This would in turn RR n326

30 28 P. Morin and C. Samson contradict the fact that the origin of the system d d z = Az is asymptotically stable. Setting: = Z t + one thus has a() <? 2 for some 2 (0; ), and: t 0 (s)ds (> 0) (79) z( ) T P z( ) z(0) T P z(0)? 2 for (80) Using that z T P z = ~z T P ~z + 2~z T P z + z T P z it comes from (77)-(80) that: z( ) T P z( ) z(0) T P z(0) K K 4 +? [; g(t 2 )] (8) Therefore, if is small enough, then: z( ) T P z( ) z(0) T P z(0)? 8 2 [; g(t 2)] (82) Recalling that z(0) = y(t ) and z( ) = y(g? ( )), the above inequality is the same as: y(t) T P y(t) y(t ) T P y(t )? 8t 2 [g? (); t 2 ] (83) By assumption (t) 0 (t), so that g(t + ) and, since g? is a monotonic increasing function, g? () t +. It is then clear that (83) implies the inequality in the right-hand side of (44). References [] R.W. Brockett, Asymptotic stability and feedback stabilization, in: R.W. Brockett, R.S. Millman and H.H. Sussmann Eds., Dierential Geometric Control Theory (983). [2] C.-T. Chen, Linear system theory and design, Holt, Rinehart and Winston, Inc., 984. [3] J.-M. Coron, Global Asymptotic Stabilization for Controllable Systems without Drift, Mathematics of Cont. Sign. and Systems Vol. 5, (992), INRIA

31 Control of chained systems [4] J.-M. Coron, On the stabilization in nite time of locally controllable systems by means of continuous time-varying feedback laws, SIAM J. Control and Optimization Vol.33, No.3 (995), pp [5] H. Hermes, Nilpotent and high-order approximations of vector elds systems, SIAM Review, Vol. 33, No. 2, (99), [6] M. Kawzki, Homogeneous stabilizing control laws, Control-Theory and Advanced Technology, Vol.6, No. 4, (990), [7] R. T. M'Closkey and R. M. Murray: Nonholonomic systems and exponential convergence: some analysis tools, IEEE Conf. on Decision and Control, San Antonio, , (993). [8] R. T. M'Closkey and R. M. Murray: Exponential stabilization of nonlinear driftless control systems via time-varying homogeneous feedback, IEEE Conf. on Decision and Control, Lake Buena Vista, (994), [9] R. T. M'Closkey and R. M. Murray, Exponential stabilization of driftless control systems using homogeneous feedback, to appear in IEEE Trans. Automat. Control. [0] P. Morin, C. Samson, J.-B. Pomet, Z.-P. Jiang, Time-varying feedback stabilization of the attitude of a rigid spacecraft with two controls, Systems & Control Letters, Vol. 25, pp , 995. [] P. Morin, J.-B. Pomet, C. Samson, Design of homogeneous time-varying stabilizing control laws for driftless controllable systems via oscillatory approximation of Lie brackets in closed-loop, Submitted to SIAM J. on Control and Optimization. [2] P. Morin and C. Samson, Application of Backstepping techniques to the timevarying exponential stabilization of chained form systems, to appear in European Journal of Control, (997). [3] R. M. Murray and S. S. Sastry, Nonholonomic motion planning: Steering using sinusoids, IEEE Trans. Automat. Control, Vol. 38, No. 5, (993) [4] J.-B. Pomet, Explicit design of time-varying stabilizing control laws for a class of controllable systems without drift, Syst.and Contr. Letters, 8, (992), RR n326