Notions, Stability, Existence, and Robustness of Limit Cycles in Hybrid Dynamical Systems

TL;DR

This paper investigates the existence, stability, and robustness of limit cycles in hybrid dynamical systems, extending classical stability notions and providing conditions and bounds that ensure their persistence under perturbations.

Contribution

It introduces extended stability notions for hybrid systems, establishes necessary and sufficient conditions for hybrid limit cycles, and analyzes their robustness to perturbations and computational errors.

Findings

Hybrid limit cycles' stability is equivalent to fixed points of the Poincaré map.

Hybrid limit cycles are robust to state noise and model uncertainties.

Computational accuracy of the Poincaré map affects stability verification.

Abstract

This paper deals with existence and robust stability of hybrid limit cycles for a class of hybrid systems given by the combination of continuous dynamics on a flow set and discrete dynamics on a jump set. For this purpose, the notion of Zhukovskii stability, typically stated for continuous-time systems, is extended to the hybrid systems. Necessary conditions, particularly, a condition using a forward invariance notion, for existence of hybrid limit cycles are first presented. In addition, a sufficient condition, related to Zhukovskii stability, for the existence of (or lack of) hybrid limit cycles is established. Furthermore, under mild assumptions, we show that asymptotic stability of such hybrid limit cycles is not only equivalent to asymptotic stability of a fixed point of the associated Poincar\'{e} map but also robust to perturbations. Specifically, robustness to generic perturbations, which capture state noise and unmodeled dynamics, and to inflations of the flow and jump sets are established in terms of $\mathcal{KL}$ bounds. Furthermore, results establishing relationships between the properties of a computed Poincar\'{e} map, which is necessarily affected by computational error, and the actual asymptotic stability properties of a hybrid limit cycle are proposed. In particular, it is shown that asymptotic stability of the exact Poincar\'{e} map is preserved when computed with enough precision. Several examples, including a congestion control system and spiking neurons, are presented to illustrate the notions and results throughout the paper.