Stability of perturbed switched homogeneous systems with delays and uncertainties

This study addresses the stability problem of perturbed switched systems with time‐varying delays and uncertainties. The considered switching signals include five classes and the uncertainties take values on compact sets. It is assumed that the nominal system (perturbation‐free system) is robustly uniformly exponentially stable and that the perturbations satisfy vanishing condition plus one of the following three constraints: being locally Lipschitz at origin, globally Lipschitz, and differentiable at origin. By ‘vanishing’, the authors mean that the perturbation is zero at origin. With these assumptions, it is first revealed that for switched systems being homogeneous of degree one, if the perturbation is sufficiently small, then the perturbed system preserves the stability property of the nominal system, locally or globally, depending on perturbation itself. These results are then applied to switched linear systems with delays, showing that the same conclusion holds for such systems with four types of most frequently encountered uncertainties. Finally, an example is provided to illustrate the proposed theoretical results.