On stability analysis of linear discrete-time switched systems using quadratic Lyapunov functions
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1 On stability analysis of linear discrete-time switched systems using quadratic Lyapunov functions Paolo Mason, Mario Sigalotti, and Jamal Daafouz Abstract he paper deals with the stability properties of linear discrete-time switched systems with polytopic sets of dynamics. he most classical and viable way of studying the uniform asymptotic stability of such a system is to check for the existence of a quadratic Lyapunov function. It is known from the literature that letting the Lyapunov function depend on the time-varying dynamic improves the chance that a quadratic Lyapunov function exists. We prove that the dependence on the dynamic can be actually assumed to be linear, with no prejudice on the effectiveness of the method. Moreover, we show that no gain in the sensibility is obtained if we allow the Lyapunov function to depend on the time as well. We conclude by showing that Lyapunov quadratic stability is a strictly stronger notion that uniform asymptotic stability. I. INRODUCION his paper is devoted to linear discrete-time parametervarying polytopic systems of the type xk + 1) = A ξk) xk). hese systems are characterized by the fact that the statespace matrices A ξ depend on some time-varying parameters and evolve in a convex polytope. Control theorists have devoted considerable attention to the study of this class of dynamical systems. he main interest for this quite general class is its capability of representing system dynamics for which a static state-space matrix would be too rough and unnatural. For instance, when the parameters denote physical quantities real-time measurable, these systems are called linear parameter-varying systems for which parameter dependent controllers can be designed [1]. hey also correspond to uncertain time-varying systems in the case of uncertain parameters, that is, parameters that can not be considered precisely known, [16]. Finally, switching systems can also be described by polytopic systems and in this case the parameters play the role of switching functions [14]. Stability analysis is a fundamental question widely investigated in the literature. he increase of interest in testing the stability of a polytope of matrices in general is demonstrated by the large number of papers on the subject see, e.g., [3], [13] and references therein). Historically, a particular attention has been given to a stability concept denominated quadratic stability, which we shall call static J. Daafouz is with CRAN, 2 avenue de la forêt de Haye, Vandœuvre-lès-Nancy cedex, France. Jamal.Daafouz@ensem.inpl-nancy.fr. P. Mason and M. Sigalotti are with INRIA, Institut Élie Cartan, UMR 7502 Nancy-Université/CNRS/INRIA, POB 239, Vandœuvre-lès- Nancy cedex, France. s: paolo.mason@iecn.u-nancy.fr, mario.sigalotti@inria.fr. quadratic stability to avoid confusion with what follows. his notion was inspired by [2] where Lyapunov functions quadratic in the state and independent of the parameters where used for the first time. he main advantage in using such a particular Lyapunov function is the fact that necessary and sufficient conditions for quadratic stability can be formulated in terms of algebraic Riccati equations or linear matrix inequalities LMI) [7]. he available solvers make the solutions proposed in this context numerically tractable. Looking for more general Lyapunov functionals has received special attention during the last decades in order to derive stability conditions that are weaker than static quadratic stability. A way to achieve this goal consists in using quadratic Lyapunov functions with an explicit dependence on the parameters leading to what are called parameter dependent Lyapunov functions PDLF) [12]. Regarding linear discrete-time parameter-varying polytopic systems, which constitute the main concern of this note, LMI stability conditions using Lyapunov functions quadratic in the state and polytopic in the parameters have been developed in [8]. hese conditions are proved to be necessary and sufficient for the existence of this kind of Lyapunov functions and can also be used for design problems control, state reconstruction, etc). Recently, stability analysis has also been carried out in the framework of the so-called joint spectral radius: a measure of the maximal asymptotic growth rate [4]. Despite its natural interpretation, the joint spectral radius is difficult to compute. A procedure for approximating the joint spectral radius with arbitrary high accuracy is provided for the case of finite sets of matrices; however, of course, higher accuracy comes at larger computational cost. A question of interest is the following: can one expect an improvement of the results in [8] by considering other quadratic PDLF not necessarily polytopic with respect to the parameters and not necessarily time-independent)? In the case of linear time-varying LV) systems k ξk) fixed) it is known that stability is equivalent to the existence of a time-varying quadratic Lyapunov function see [17]). For LV s, therefore, time-varying quadratic Lyapunov functions are not equivalent to time-invariant ones as a tools to check stability. Answering whether this is still the case for linear discrete-time parameter-varying polytopic systems does not seem to be immediate. o this end, we focus in this paper on three criterions of stability. he first one is called Parameter Dependent quadratic stability PD-quadratic stability). It refers to checking stability by mean of Lyapunov function quadratic in the state and parameter dependent without any
2 specified structure. he second one called Parameter and ime Dependent quadratic stability PD-quadratic stability). It refers to Lyapunov functions that are quadratic in the state and depend explicitly on both the time and the parameters. he last one, and a priori the less costlier to check, is the so called poly-quadratic stability used in [8] and which refers to Lyapunov functions quadratic in the state and polytopic in the parameters. Our main contribution is to prove that all these criterions are equivalent. he result can be rephrased as: quadratic stability needs only to be tested on quadratic polytopic Lyapunov functions. A partial result of the same type, guaranteeing the equivalence of PD- and poly-quadratic stability under a further technical restriction on the corresponding LMI s namely, G i = G for every i, using the notations of [8]) can be found in [11]. A related question which naturally arises is whether uniform asymptotic stability is equivalent to quadratic stability. he counterpart of this question for continuoustime switched systems was addressed by Dayawansa and Martin in [10], where it was shown that uniformly asymptotically stable switched systems exist which do not admit quadratic Lyapunov functions see also [6]). he result has been further strengthened in [15] where it is proved that, even if for every uniformly asymptotically stable time-continuous switched system a polynomial Lyapunov function can be found, this becomes impossible if we impose a uniform bound on its degree. We extend here the result in [10] to the case of discrete-time switched systems, showing that a system can be uniformly asymptotically stable without being poly-quadratically stable or PD-quadratically stable, nor PD-quadratically stable). Recall that the polytopic approach presented in [8] was applied to stability analysis and switched output feedback control design of discrete time switched linear systems in [9]. Owing to the results presented here, one can see that the stability analysis problems investigated in [8] and [9] are equivalent. he paper is organized as follows. In Section II we introduce the main definitions and we discuss the equivalence of asymptotic stability under convexification of the set of state-space matrices. Section III forms the core of the paper and is devoted to the proof of the equivalence between the three stability criterions introduced above. Section IV shows that uniformly asymptotically stable polytopic systems are not necessarily poly-quadratically stable. We end the paper by a conclusion. II. NOAIONS AND MAIN DEFINIIONS For every n N, let {e 1,..., e n } be the canonical basis of R n and denote by M n n the set of all real n n matrices. Given A, B M n n, we write A B to denote that A B is negative semidefinite. A function with values in M n n is convex if it is so with respect to such partial) order. he Euclidian norm in R n and that induced in M n n are both denoted by. A function w : ε wε) M n n defined for all ε > 0 is said to be of order k k N) if lim sup ε 0 wε) ε k is finite. In this case we write wε) = Oε). Let n, m be two integer numbers, Ξ be a subset of R m and A : ξ A ξ be a map from Ξ to M n n. Consider the system xk + 1) = A ξk) xk), 1) where ξk) Ξ for every k N. We will refer to k ξk) as to a switching function. Definition 1: We say that 1) is uniformly asymptotically stable UAS) if for every x0) R n the solution to 1) converges to zero uniformly with respect to {ξk)} k N Ξ i.e., for every ε > 0 there exists K N such that for every {ξk)} k N Ξ we have xk) < ε for k K) and if, moreover, for every R > 0 there exists r > 0 such that xk) < R for every {ξk)} k N Ξ and every k N, provided that x0) < r. Due to the linear nature of the discrete) dynamics of 1), it is well known that 1) is UAS stable if and only if it is uniformly exponentially stable. he most widespread tool for proving that a system of the form 1) is UAS is to exhibit a quadratic Lyapunov function, which in general can depend on ξ. Definition 2: We say that 1) is parameter-dependent quadratically stable PD-quadratically stable) if there exist three positive constants α 0, α 1, α 2 and a Lyapunov function such that V x, ξ) = x Pξ)x 2) α 1 x 2 V x, ξ) α 2 x 2 and whose difference along the solutions of 1) is negative definite decreasing, that is, for every x0) R n, every {ξk)} k N Ξ, and every k N, we have V xk + 1), ξk + 1)) V xk), ξk)) α 0 xk) 2. Motivated by the characterization of asymptotic stability of LV s in terms of existence of time-varying quadratic Lyapunov functions see [17]), we introduce the following a priori weaker) notion. Definition 3: We say that 1) is parameter- and timedependent quadratically stable PD-quadratically stable) if there exist three positive constants α 0, α 1, α 2 and a Lyapunov function such that V k, x, ξ) = x Pk, ξ)x 3) α 1 x 2 V k, x, ξ) α 2 x 2 4) and for every x0) R n, every {ξk)} k N Ξ, and every k N, we have V k+1, xk+1), ξk+1)) V k, xk), ξk)) α 0 xk) 2.
3 Most of the paper will focus on the case where AΞ) is a convex polytope, i.e., AΞ) = conv{a 1,..., A M } for some M N and A 1,..., A M M n n. Without loss of generality, this is equivalent to say that m = M, Ξ is the simplex conv{e 1,..., e M }, and that A is the linear map satisfying A i = Ae i ). In this case we will say that the system 1) is convex. Definition 4: Let 1) be convex. We say that 1) is polyquadratically stable if a Lyapunov function V as in 2) can be found with P ) linear with respect to ξ. Remark 5: Let 1) be convex. hen 1) is stable if and only if the system where Ξ is replaced by {e 1,..., e M } is stable. Indeed, let us recall that for a bounded set of matrices AΞ) the uniform asymptotic stability of 1) is equivalent to the property that the joint spectral radius ρaξ)) = lim sup h max ξ 1,...,ξ h Ξ A ξ 1 A ξ h 1 h is strictly smaller than one. We want to see that ρconv{a 1,..., A M }) = ρ{a 1... A M }). For this purpose it is enough to observe that for any positive integer N and any choice of convex combinations B j = M λj) i A i, j = 1,..., N, the matrix product B N B 1 is the convex combination herefore, B N B 1 = B N B 1 i 1,...,i N =1 i 1,...,i N =1 λ 1) i 1 λ N) i N A in A i1. λ 1) i 1 λ N) i N A in A i1 max A i N A i1. 1i 1,...,i N M From the definition of the joint spectral radius it trivially follows that ρconv{a 1,..., A M }) ρ{a 1... A M }), which proves the claim. III. EQUIVALENCE BEWEEN DIFFEREN NOIONS OF QUADRAIC SABILIY heorem 6: Let 1) be convex. hen 1) is PDquadratically stable if and only if it is PD-quadratically stable if and only if it is poly-quadratically stable. Proof. It is clear by the definitions given in the previous section that if 1) is poly-quadratically stable then it is PDquadratically stable and that if it is PD-quadratically stable then it is PD-quadratically stable. We are therefore left to prove that a PD-quadratically stable system is polyquadratically stable. Assume that 1) is PD-quadratically stable and fix V k, ξ, x) and P k, ξ) as in Definition 3. he first step consists in showing that P can be assumed to be linear with respect to ξ. o this extent, define P k,i = P k, e i ) for k N and i = 1,..., M. Let, moreover, Π : N Ξ M n n be defined by Π k, ξ) = ξ i P k,i, where the ξ i s denote the components of ξ, i.e., ξ = M ξ ie i. Let us restrict our attention for a moment to switching functions with values in the set of vertices of Ξ. Requiring that for every {ξk)} k N {e 1,..., e M }, every k N, and every x0) R n, we have α 0 xk) 2 xk+1) P k+1, ξk+1))xk+1) xk) P k, ξk))xk) = xk) A ξk) P k + 1, ξk + 1))A ξk) P k, ξk)))xk) is equivalent to asking that, for every k N and i, j {1,..., M}, the following matrix relation holds A i P k+1,j A i P k,i α 0 Id. 5) Fix j {1,..., M} and let F j : N Ξ M n n be defined by F j k, ξ) = A ξ P k,j A ξ. Notice that F j is convex with respect to ξ, since it is the composition of a linear map with f : A A P k,j A, whose convexity is proved by the matrix relation fλa + 1 λ)b) λfa) 1 λ)fb) = = λ1 λ)fa B) 0, which holds for every A, B M n n and every λ [0, 1]). herefore, ) F j k + 1, ξ) = F k j + 1, ξ i e i = ξ i F j k + 1, e i ) ξ i A i P k+1,j A i α 0 Id + Π k, ξ) where the last inequality follows from 5) and the definition of Π. We are ready to prove that Π is a time-dependent Lyapunov function stabilizing 1). Indeed, for every ξ 1, ξ 2 Ξ and every k N, we have = j=1 A ξ 1Π k + 1, ξ 2) A ξ 1 Πk, ξ 1 ) = ξ 2 j A ξ 1P k+1,j A ξ 1 Πk, ξ 1 ) ) α 0 Id. Define now Ωk) = P k,1,..., P k,m ) for every k N. Notice that if there exist k 1, k 2 N such that k 1 < k 2
4 and Ωk 1 ) = Ωk 2 ), then the theorem is proved. Indeed, by defining Π ξ) = k 2 1 we have, for every ξ 1, ξ 2 Ξ, A ξ 1Π ξ 2 )A ξ 1 Π ξ 1 ) = k=k 1 Πk, ξ), = A ξ Πk 1, ξ 2 ) A 1 }{{} ξ 1 Πk 2 1, ξ 1 ) + =Πk 2,ξ 2 ) + k 2 2 k=k 1 k 2 k 1 )α 0 Id. A ξ 1Πk + 1, ξ 2 )A ξ 1 Πk, ξ 1 ) ) Notice now that {Ωk)} k N is a bounded sequence in M n n ) M, due to 4). We can thus extract a converging subsequence {Ωk l )} l N. he idea is to take as timeindependent Lyapunov function that corresponding to Π l, ξ) = k l+1 1 k=k l Πk, ξ), with l large enough. Indeed, let l be such that α 0 2 Id A i P kl+1,j P kl,j)a i α 0 2 Id for every i, j {1,..., M}. hen the same computations as above show that A ξ 1Πl, ξ 2 )A ξ 1 Π l, ξ 1 ) A ξ 1Πk l, ξ 2 ) Πk l+1, ξ 2 ))A ξ 1 k l+1 k l )α 0 Id α 0 2 Id, and the proof of heorem 6 is finished. IV. ASYMPOIC VS QUADRAIC SABILIY Lemma 7: here exist stable systems of type 1) which are not PD-quadratically stable. Proof. he proof works by contradiction. he idea is to consider a continuous-time switched system with suitable properties and to time-discretize it. For every time-step the system that is obtained is stable, but the assumption that all of them are PD-quadratically stable leads to a contradiction when the time-step goes to zero and the discrete systems converge, morally, to the continuous one). ake n = 2 and Ξ = {1, 0), 0, 1)} and assume by contradiction that if 1) is stable, then there exist two positive definite matrices P 1, P 2 such that, setting V x, ξ) = x P ξ x, we have that V is strictly decreasing along a non-trivial path xk), ξk)) for every x0) R 2 and every {ξk)} k N {1, 2}. Consider a bidimensional continuous switched system of the type ẋ = ut)c 1 x + 1 ut))c 2 x, ut) {0, 1}, 6) and assume that C 1 and C 2 are Hurwitz matrices with nonreal eigenvalues. It follows from the results by Dayawansa and Martin see [10]) that we can assume in addition that the system is uniformly exponentially stable and that there exist no quadratic function which is positive definite and whose derivative along the trajectories of 6) is always nonpositive. Notice that we are not restating the exact result appearing in [10], which proves the possible nonexistence, for a uniformly exponentially stable system, of a quadratic Lyapunov function. Here we ask a slightly stronger property, since we want to rule out the possibility of a positive definite quadratic function which is non-increasing along trajectories of 6). he result, however, directly follows from the reasoning in [10].) Let us define, for every ε > 0, A ε 1 = e εc1, A ε 2 = e εc2, and consider the corresponding family of discrete systems xk + 1) = A ε ξk) xk), ξk) {1, 2}. 7) Every such system is obviously stable, because of the uniform exponential stability of 6). According to the contradiction hypothesis, let us assume the existence, for every ε > 0, of two positive definite matrices P ε 1, P ε 2 such that A ε i ) P ε i A ε i P ε i < 0 i = 1, 2, 8) A ε i ) P ε j A ε i P ε i < 0 i j. 9) Up to a rescaling we can assume max,2 P ε i = 1 for every ε > 0. By a aylor expansion with respect to ε we have e εci ) Pi ε e εci = = Id + εci + Oε 2 ) ) Pi ε Id + εci + Oε 2 ) ) = Pi ε + εci Pi ε + Pi ε C i ) + Oε 2 ) so that, according to inequality 8), C i Pi ε + Pi ε ) A ε C i + Oε) = i ) Pi εaε i P i ε < 0. 10) ε By compactness of the set of matrices Pi ε we can find a suitable sequence ε h 0 such that P ε h i P i for i = 1, 2 for some positive semidefinite matrices P 1 and P 2. Moreover, at least one among P 1 and P 2 has norm equal to one. hen, passing to the limit as ε h goes to 0 in 10), we obtain C i P i + P i C i 0, i = 1, 2. Similarly, from 9) we get that P ε j P ε i + Oε) < 0 if i j i.e. P ε 1 P ε 2 = Oε) and therefore P 1 = P 2 =: P. Observe that P is not only semidefinite, but it must be positive definite. Indeed, since P = 1, if it is semidefinite then there exists one dimensional subspace Λ such that x P x > 0 if and only if x Λ. hen the only way not to increase x P x along a trajectory starting from Λ would
5 be to stay on Λ, which is impossible because the C 1 and C 2 have non-real eigenvalues. hus P is positive definite and satisfies C i P + P C i 0 for i = 1, 2. his is impossible due the initial assumption made on C 1, C 2. We reached a contradiction and the lemma is proved. CONCLUSION We investigated the equivalence between different notions of stability for discrete-time switched systems of type 1). We first proved Section III) that looking for a quadratic Lyapunov function in its more general form V k, ξ, x) = x P k, ξ)x that is, in a class which depends on an infinite number of parameters) is equivalent to looking for it in the much smaller class V ξ, x) = x M ξ ip i )x which depends on a finite family of parameters). Recall that this last problem can be formulated in terms of LMI [8]). herefore, quadratic stability however defined) turns out to be equivalent to a set of LMI s. On the other hand, as suggested by [5], uniform asymptotic stability is a strictly stronger notion that quadratic stability Section IV). REFERENCES [1] P. Apkarian and P. Gahinet. A convex characterization of gainscheduled h controllers. IEEE rans. Automat. Contr., 405): , [2] R. Barmish. Stabilization of uncertain systems via linear control. IEEE rans. Automat. Contr., 288): , [3] F. Blanchini and S. Miani. Stabilization of lpv systems: State feedback, state estimation, and duality. SIAM journal on control and optimization, 401):76 97, [4] V. D. Blondel and Y. Nesterov. Computationally efficient approximations of the joint spectral radius. SIAM Journal of Matrix Analysis, 271): , [5] V. D. Blondel and J. N. sitsiklis. he boundedness of all products of a pair of matrices is undecidable. Systems Control Lett., 412): , [6] U. Boscain. Stability of planar switched systems: the linear single input case. SIAM journal on control and optimization, 41:89 112, [7] S. Boyd, L. El Ghaoui, E. Feron, and V. Balakrishnan. Linear Matrix Inequalities in System and Control heory. SIAM, Studies in Applied Mathematics, [8] J. Daafouz and J. Bernussou. Parameter dependent lyapunov functions for discrete time systems with time varying. Systems and Control Letters, 438): , [9] J. Daafouz, P. Riedinger, and C. Iung. Stability analysis and control synthesis for switched systems: a switched Lyapunov function approach. IEEE rans. Automat. Control, 4711): , [10] W. P. Dayawansa and C. F. Martin. A converse lyapunov theorem for a class of dynamical systems which undergo switching. IEEE rans. Automat. Control, 44: , [11] M. C. de Oliveira and J. C. Geromel. A class of robust stability conditions where linear parameter dependence of the Lyapunov function is a necessary condition for arbitrary parameter dependence. Systems Control Lett., 5411): , [12] E. Feron, P. Apkarian, and P. Gahinet. Analysis and synthesis of robust control systems via parameter-dependent lyapunov functions. IEEE rans. Automat. Contr., 417): , [13] J. C. Geromel and P. Colaneri. Robust stability of time varying polytopic systems. Systems and Control Letters, 551):81 85, [14] D. Liberzon. Switching in Systems and Control. Birkhauser, Boston, MA, Volume in series Systems and Control: Foundations and Applications., [15] P. Mason, U. Boscain, and Y. Chitour. Common polynomial lyapunov functions for linear switched systems. SIAM journal on control and optimization, 45: , [16] R. Ravi, R.M. Nagpal, and P. P. Khargonekar. H control of linear time-varying systems: A state-space approach. SIAM journal on control and optimization, 29: , [17] J. L. Willems. Stability heory of Dynamical Systems. NELSON, 1970.
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