Stability conditions of switched nonlinear systems with unstable subsystems and destabilizing switching behaviors

Abstract In this paper, the global uniform exponential stability (GUES) for a class of switched nonlinear systems is discussed in both continuous-time and discrete-time contexts. Different from multiple Lyapunov function and average dwell time in the previous work, a general multiple Lyapunov-like function and a modified admissible edge-dependent average dwell time switching scheme based on a directed graph are implemented in this paper. By further allowing the Lyapunov-like function to increase and decrease during the running time of active subsystems, and permitting the admissible transition edge-dependent weights (ATEDWs) to be greater or less than one, the extended stability results for switched systems in a nonlinear situation are first derived. Given a new amalgamation condition between the increasing ratio and decreasing ratio of the Lyapunov-like function at switching instants, the minimal admissible edge-dependent average dwell time (AEDADT) for admissible switching signals are is obtained. By contrasting with the results available on the subject, we allow that all the switching behaviors of the switching systems at switching instants are destabilizing, and the transition weights of admissible transition edge may be less than one1. More especially, even if some of the subsystems are not stable and all the switching behaviors of are destabilizing, the stability property of the switched system can still be preserved. Finally, two numerical examples are illustrated to show the validity of the proposed results.