Hierarchical stability of nonlinear hybrid systems

TL;DR

This paper proves a hierarchical stability theorem for hybrid dynamical systems, providing conditions for asymptotic stability of nested sets and characterizing basins of attraction, useful for hierarchical control architectures.

Contribution

It introduces a hierarchical stability result for hybrid systems satisfying basic conditions, extending stability analysis to nested sets and basins of attraction.

Findings

Establishes sufficient conditions for uniform asymptotic stability of nested sets.

Shows the basin of attraction equals the largest set with bounded solutions.

Applicable to hierarchical control architectures.

Abstract

In this short note we prove a hierarchical stability result that applies to hybrid dynamical systems satisfying the hybrid basic conditions of (Goebel et al., 2012). In particular, we establish sufficient conditions for uniform asymptotic stability of a compact set based on some hierarchical stability assumptions involving two nested closed sets containing such a compact set. Moreover, mimicking the well known result for cascaded systems, we prove that the basin of attraction of such compact set coincides with the largest set from which all solutions are bounded. The result appears to be useful when applied to several recent works involving hierarchical control architectures.