Parameter-dependent homogeneous Lyapunov functions for robust stability of linear time-varying systems

In this paper robust stability of linear state space models with respect to time-varying uncertainties with bounded variation rates is considered. A new class of parameter-dependent Lyapunov functions to establish stability of a polytope of matrices in presence of a polytopic bound on the variation rate of the uncertain parameters is introduced, i.e., the class of homogeneously parameter-dependent homogeneous Lyapunov functions (HPD-HLFs). Such a class, where the dependence on the uncertain parameter vector and the state vector are both expressed as polynomial homogeneous forms, generalizes those successfully employed in the special cases of unbounded variation rates or time-invariant uncertainties. The main result of the paper is a sufficient condition to determine the sought HPD-HLF, which amounts to solving a set of linear matrix inequalities (LMIs) derived via a suitable parameterization of polynomial homogeneous forms. Also, lower bounds for the maximum scaling factor of the variation rates polytope for which the stability of the system is preserved, are shown to be computable in terms of generalized eigenvalue problems (GEVPs). Several numerical examples are provided to show the effectiveness of the proposed approach.