Reduction Theorems for Hybrid Dynamical Systems

TL;DR

This paper develops reduction theorems for hybrid dynamical systems that relate stability properties of nested sets, providing tools for analyzing stability, attractivity, and detectability in complex hybrid systems.

Contribution

It introduces new reduction theorems for stability and attractivity of nested sets in hybrid systems, including conditions for cascade stability and detectability-based attractivity.

Findings

Reduction theorems for stability and attractivity of nested sets.

Conditions for cascade-connected hybrid system stability.

Hybrid estimator for sinusoidal signal period.

Abstract

This paper presents reduction theorems for stability, attractivity, and asymptotic stability of compact subsets of the state space of a hybrid dynamical system. Given two closed sets $\Gamma_1 \subset \Gamma_2 \subset \Re^n$, with $\Gamma_1$ compact, the theorems presented in this paper give conditions under which a qualitative property of $\Gamma_1$ that holds relative to $\Gamma_2$ (stability, attractivity, or asymptotic stability) can be guaranteed to also hold relative to the state space of the hybrid system. As a consequence of these results, sufficient conditions are presented for the stability of compact sets in cascade-connected hybrid systems. We also present a result for hybrid systems with outputs that converge to zero along solutions. If such a system enjoys a detectability property with respect to a set $\Gamma_1$, then $\Gamma_1$ is globally attractive. The theory of this paper is used to develop a hybrid estimator for the period of oscillation of a sinusoidal signal.