Hybrid Systems With Delayed Jumps: Asymptotic Stability via Robustness and Lyapunov Conditions

Abstract

Hybrid systems subject to delayed jumps form a class of dynamical systems with broad applications. This article develops sufficient conditions for robust asymptotic stability of hybrid systems in the presence of delayed jumps. More precisely, given a delay-free hybrid system, we introduce a higher order delayed system parametrized by the length of delays. We show that when the delay parameter is set to zero, the higher order model captures the solutions of the delay-free system. Under mild conditions, it is shown that when the delay-free system has an asymptotically stable compact set, for small enough delays, solutions of the delayed system converge to a neighborhood of a set of interest related to the aforementioned compact set. Then, Lyapunov functions for the delay-free system are used to develop sufficient conditions for asymptotic stability in the presence of delays. Unlike prior work in the literature, the results pertaining to these notions of stability hold for systems possessing Zeno solutions, with time-varying delays. Importantly, the required conditions are expressed in finite-dimensional space, and depend primarily on the data of the delay-free system. The practical stability result is validated numerically through the hybrid system model of a controlled boost converter circuit with state-triggered switches and Zeno solutions. The higher order model and the derived Lyapunov conditions are utilized to obtain quantitative bounds on maximum allowable delays for switched systems and sampled-data control.