Dynamics of discrete-time homogeneous switched systems with delays and perturbations

Addressed in this paper is the dynamics of discrete-time homogeneous switched systems subject to perturbations. The nominal systems (perturbation-free systems) are with time-varying delays and uncertainties taking values on a compact set and are supposed to be robustly uniformly exponentially stable. Three types of vanishing perturbations are considered: Perturbations being locally Lipschitz at origin, globally Lipschitz, and differentiable at zero with differentiation zero. By “vanishing”, we mean that the perturbation is zero if the state is zero. It is shown that, with these assumptions, the perturbed systems are still robustly uniformly exponentially stable provided that perturbations are small enough, and that the stable property may hold locally or globally, depending on the property of perturbations. These results are then applied to discrete-time linear switched systems with delays and uncertainties (including four types of most frequently investigated structured uncertainties). Finally, an example is provided to illustrate the presented theoretical results.