Stability under events for a class of hybrid dynamical systems with continuous and discrete time variables

Abstract

This studies the stability under events for a class of hybrid dynamical systems (HDSs). The HDS is modelled via continuous and discrete time (CDT) variables. It is allowed that an HDS has time-varying, unstable, non-linear, different dwell time, and infinite number of subsystems. The stability under events reflects the effects of switching and impulse type events in HDS. By defining the length of CDT variable and extending the notion of hybrid-event-time in the literature, the concepts on global uniform asymptotic stability (GUAS) and event-GUAS are reasonably defined. By employing multiple Lyapunov-like functions, criteria on GUAS and event-GUAS are established. Moreover, by extending average dwell time (ADT) to hybrid ADT, the relations between GUAS and event-GUAS are derived. And the criteria on global uniform exponential stability under events (event-GUES) are obtained. A method via maximal admissible probabilities of unstable subsystems is proposed to test the conditions of event-GUES. As two special cases, the traditional stability criteria of only continuous or discrete variables are also derived. Finally, three examples are provided to verify the main results.