Robust Controller Design: Polynomially Parameter Dependent Lyapunov Function Approach

The robust stability analysis and robust controller design for linear systems subject to time-invariant uncertainties have attracted considerable attention in robust control theory. Undoubtedly, the Lyapunov theory is one of the main approaches to deal with this problem. Description of uncertain linear time-invariant systems using convex polytope-type uncertainty has found its natural framework in the linear matrix inequality (LMI) Boyd et al [3]. The first LMI stability analysis and robust controller design has been based upon the notion of quadratic stability. To reduce quadratic stability conservatism in the robust controller design procedure the parameter dependent Lyapunov function (PDLF) has been introduced (Apkarian et al [1], de Oliveira et al [11, 12], Henrion et al [9], Peaucelle et al [14] and others). In the existing studies, however, the PDLFs mostly employed are restricted to those affine in the uncertain parameters. To get around the conservatism arising from affine PDLFs, more recently, promising LMI based conditions using Polynomially Parameter-Dependent Lyapunov Function (PPDLF) have been proposed in [4, 10, 13]. In this paper we pursue the idea of Ebihara et al [4], where robust sufficient stability condition for the existence of such PPDLFs in terms of finitely many LMIs evaluated on the vertex of the polytope of continuous LTI system has been developed. The results in the present paper have been obtained by modifying the results of [4], to linear time-invariant discrete-time systems and adding to them the conditions which in the robust controller design ensure guaranteed cost in the D–LMI region . We use the following notations in this paper. Matrix P = P > 0 (< 0) is positive (negative) definite. For a matrix A ∈ R with rank(A) = r < n,A ∈ R is a matrix such that AA = 0 and (A)A > 0. R+ denotes the set of nonnegative integers. C denotes the complex plain. He{AB} = AB +BA . 2 PROBLEM FORMULATION AND PRELIMINARIES