Published in IET Control Theory and Applications Received on 5th September 2008 Revised on 18th December 2008 doi: 10.1049/iet-cta.2008.0393 ISSN 1751-8644 Special time-varying Lyapunov function for robust stability analysis of linear parameter varying systems with bounded parameter variation R.C.L.F. Oliveira 1 M.C. de Oliveira 2 P.L.D. Peres 1 1 School of Electrical and Computer Engineering, University of Campinas – UNICAMP, P.O. Box 6101, 13083-970 Campinas, SP, Brazil 2 Department of Mechanical and Aerospace Engineering, University of California San Diego, La Jolla, CA 92093-0411, USA E-mail: peres@dt.fee.unicamp.br Abstract: The robust stability of linear continuous-time uncertain systems in polytopic domains is investigated. The uncertain parameters are assumed as time varying with bounded rates of variation. The robust stability conditions are obtained from the definition of a Lyapunov function with a particular structure, depending on integer powers k of the dynamic uncertain time-varying matrix of the system and on a parameter-dependent matrix to be determined. As a consequence, parametrised linear matrix inequality conditions can be derived in terms of k for a particular structure of the decision variables. As k grows, the robust stability conditions can take into account bounds on the successive time derivatives of the uncertain parameters whenever this information is available, reducing the conservativeness of the evaluations. Numerical examples illustrate the effectiveness of the proposed methodology. 1 Introduction The study of robust stability of linear time-varying uncertain system is an issue where much attention has been paid during the last years. When nothing is known about the rates of variation of the parameters, a common approach is the well-known quadratic stability, where a quadratic in the state Lyapunov function with parameter-independent matrix is employed. Recent generalisations of quadratic stability are the approaches of [1–3], where the Lyapunov function can be homogeneous of arbitrary degree in the state. It is worth mentioning the class of methods based on piecewise quadratic Lyapunov functions [4] and polyhedral Lyapunov functions [5]. In the case where bounds on the variation rate of the uncertain parameters are known a priori, there exist several results using quadratic in the state Lyapunov functions with affine [6–8], quadratic [9], polynomial [10] and homogeneous polynomial [3] parameter dependence. Note that Chesi et al. [3] can also cope with homogeneous polynomial state dependence. See also [11, 12] in the context of time-varying parameters that admit a linear fractional transformation representation. The problem investigated here is the robust stability analysis of uncertain time-varying linear systems in polytopic domains with bounded rates of variation [3, 7, 8]. The aim is to use a quadratic in the state Lyapunov function with a particular polynomial parameter dependence, which has already been studied in the context of time-invariant polytopic linear systems [13– 15]. In this framework, the robust stability conditions can take into account bounds on the successive time derivatives of the uncertain parameters and are expressed in terms of parametrised linear matrix inequality (LMI) relaxations. The results immediately applies in the special case when the time-varying parameter depends exponentially on time, as investigated in [8]. An 1448 IET Control Theory Appl., 2009, Vol. 3, Iss. 10, pp. 1448–1461 & The Institution of Engineering and Technology 2009 doi: 10.1049/iet-cta.2008.0393 www.ietdl.org