On Robust Stability of Switched Systems in the Context of Filippov Solutions

TL;DR

This paper investigates the stability of nonlinear switched systems using Filippov solutions, providing a Lyapunov-based stability theorem and extending results to uncertain and polynomial systems with sum of squares methods.

Contribution

It introduces a Lyapunov-like stability criterion for Filippov solutions and extends stability analysis to uncertain and polynomial switched systems.

Findings

Stability conditions for Filippov solutions are established.

Sum of squares programming can verify stability of polynomial systems.

Results encompass systems with sliding modes and infinite switching.

Abstract

The stability problem of a class of nonlinear switched systems defined on compact sets with state-dependent switching is considered. Instead of the Caratheodory solutions, the general Filippov solutions are studied. This encapsulates solutions with infinite switching in finite time and sliding modes in the neighborhood of the switching surfaces. In this regard, a Lyapunov-like stability theorem, based on the theory of differential inclusions, is formulated. Additionally, the results are extended to switched systems with simplical uncertainty. It is also demonstrated that, for the special case of polynomial switched systems defined on semi-algebraic sets, stability analysis can be checked based on sum of squares programming techniques.