ROBUST STABILITY OF TIME-DELAY CONTINUOUS-TIME SYSTEMS IN POLYTOPIC DOMAINS P. L. D. Peres ⋆ , S. Tarbouriech ⋄ , G. Garcia ⋄,† , V. J. S. Leite ♭ ⋆ DT/FEEC/UNICAMP, University of Campinas, CP 6101, 13081-970 Campinas - SP - Brazil, peres@dt.fee.unicamp.br ⋄ LAAS du CNRS — 7, Avenue du Colonel Roche, 31077 Toulouse CEDEX 4 — France, {tarbour,garcia}@laas.fr † Also with INSA, 135 Avenue de Rangueil, 31077 Toulouse CEDEX 4, France. ♭ UnED Divin´ opolis – CEFET-MG, R. Monte Santo, 319 - 35502-036, Divin´ opolis - MG - Brazil, valter@div.cefetmg.br Keywords: Robust stability; Parameter-dependent quadratic functions; Time-delay systems; Continuous-time systems; Lin- ear matrix inequality. Abstract Most of the existent linear matrix inequality based conditions for robust stability of time-delay systems in polytopic domains are expressed in terms of constant Lyapunov-Krasovskii func- tions. This note presents a simple way to extend these condi- tions in order to construct parameter-dependent functions that provide less conservative results, in both delay-independent and delay-dependent situations. 1 Introduction During the last decades several works have dealt with the prob- lem of stability of time-delay systems [9], [14], [18], [23]. One of the most popular techniques for the stability analysis of this kind of linear systems is undoubtedly the one based on Lyapunov-Krasovskii functionals [17], [32]. Since the numerical efficiency of these conditions for stability is a major concern, most of them have been rewritten as lin- ear matrix inequalities (LMIs) which can nowadays be solved by polynomial time interior point algorithms [2], [11]. Sev- eral LMI conditions assuring robust stability appeared, in both delay-independent (i.e. the stability does not depend on the size of the time delay) [4], [21], [31], [34] and delay-dependent sit- uations [3], [12], [19], [20]. For stabilizability purposes, including H ∞ or H 2 norm opti- mization criteria, several results were developed as extensions of the stability analysis based on Lyapunov-Krasovskii func- tions. As a natural consequence, for the uncertain linear sys- tems with time-delay, quadratic stability and quadratic stabiliz- ability concepts [1] were used to accomplish with robust sta- bility analysis, robust control and robust filter design [6], [7], [15], [16], [19], [22], [24], [26]. For uncertain systems in poly- topic domains, a simple evaluation of the feasibility of a set of LMIs defined at the vertices of the polytope provides sufficient conditions for the existence of a robust feedback gain or a full order linear filter. However, the analysis of stability based on constant Lyapunov functions can sometimes provide very con- servative results. Some recent works introduced the analysis of robust stability and other closed-loop properties by means of parameter dependent Lyapunov functions [8], [10], [13]. Very recently, systematic ways to test for the existence of pa- rameter dependent Lyapunov functions have appeared, as in [5], [28], where an augmented LMI formulation with extra ma- trix variables yields sufficient conditions for the robust stability of a polytope of matrices. Another simple and efficient way to construct such Lyapunov matrices can be found in [29], [30]. This note exploits the methodology first introduced in [29], [30] to provide less conservative sufficient conditions for the robust stability of time-delay systems with uncertain parameters in polytopic type domains. The key idea is to use homogeneity properties of the LMIs and simple alge- braic manipulations to derive sufficient conditions for the negative definiteness of the Lyapunov-Krasovskii functional time-derivative associated to the time-delayed system. As a result, a feasibility LMI test formulated at the vertices of the uncertainty polytope provides parameter dependent matrices for that functional. To illustrate the technique proposed, some existent LMI conditions for the stability of time-delay systems are here extended to cope with robust stability in both delay-dependent and delay-independent cases. The conditions proposed encompass previous results based on quadratic stability and are illustrated by means of some examples. Notations. ℜ + is the set of nonnegative real numbers. I denotes the identity matrix of appropriate dimensions. C τ = C ([−τ, 0], ℜ n ) denotes the Banach space of contin- uous vector functions mapping the interval [−τ, 0] into ℜ n with the topology of uniform convergence. ‖ · ‖ refers to either the Euclidean vector norm or the induced ma- trix 2-norm. ‖ φ ‖ c = sup −τ ≤t≤0 ‖ φ(t) ‖ stands for the norm of a function φ ∈C τ . When the delay is finite then “sup” can be replaced by “max”. C v τ is the set defined by C v τ = {φ ∈C τ ; || φ || c < v, v > 0}. The symbol ⋆ stands for symmetric blocks in the LMIs. 2 Preliminaries Consider a continuous-time linear system given by ˙ x(t)= Ax(t)+ A τ x(t − τ ) (1) with the initial conditions x(t 0 + θ)= φ(θ), ∀θ ∈ [−τ, 0],t 0 ,φ ∈ℜ + ×C v τ (2)