Global results on reset-induced periodic trajectories of planar systems

TL;DR

This paper investigates the conditions under which reset feedback induces stable periodic trajectories in planar hybrid systems, using hybrid Lyapunov methods to analyze stability and robustness.

Contribution

It introduces a hybrid framework to analyze reset-induced periodic orbits in planar systems, highlighting energy balance and stability conditions.

Findings

Periodic orbits arise from energy balance between flows and resets.

Hybrid Lyapunov functions ensure robustness of stability.

Extensions to mechanical systems are discussed.

Abstract

We study the existence of asymptotically stable periodic trajectories induced by reset feedback. The analysis is developed for a planar system. Casting the problem into the hybrid setting, we show that a periodic orbit arises from the balance between the energy dissipated during flows and the energy restored by resets, at jumps. The stability of the periodic orbit is studied with hybrid Lyapunov tools. The satisfaction of the so-called hybrid basic conditions ensures the robustness of the asymptotic stability. Extensions of the approach to more general mechanical systems are discussed.