Robust stability of time-varying polytopic systems via parameter-dependent homogeneous Lyapunov functions

This paper deals with robust stability analysis of linear state space systems affected by time-varying uncertainties with bounded variation rate. A new class of parameter-dependent Lyapunov functions is introduced, whose main feature is that the dependence on the uncertain parameters and the state variables are both expressed as polynomial homogeneous forms. This class of Lyapunov functions generalizes those successfully employed in the special cases of unbounded variation rates and time-invariant perturbations. The main result of the paper is a sufficient condition to determine the sought Lyapunov function, which amounts to solving an LMI feasibility problem, derived via a suitable parameterization of polynomial homogeneous forms. Moreover, lower bounds on the maximum variation rate for which robust stability of the system is preserved, are shown to be computable in terms of generalized eigenvalue problems. Numerical examples are provided to illustrate how the proposed approach compares with other techniques available in the literature.