Nonpathological ISS-Lyapunov Functions for Interconnected Differential Inclusions

TL;DR

This paper develops less conservative ISS stability conditions for interconnected systems modeled by differential inclusions using non-smooth Lyapunov functions and generalized derivatives, with applications to switched systems.

Contribution

It introduces new ISS conditions for differential inclusions using locally Lipschitz Lyapunov functions and Lie generalized derivatives, improving upon traditional Clarke derivative methods.

Findings

Provided sufficient ISS conditions for differential inclusions.

Applied results to state-dependent switched systems.

Demonstrated an observer-based controller for nonlinear switched systems.

Abstract

This article concerns robustness analysis for interconnections of two dynamical systems (described by upper semicontinuous differential inclusions) using a generalized notion of derivatives associated with locally Lipschitz Lyapunov functions obtained from a finite family of differentiable functions. We first provide sufficient conditions for input-to-state stability (ISS) for differential inclusions, using a class of non-smooth (but locally Lipschitz) candidate Lyapunov functions and the concept of Lie generalized derivative. In general our conditions are less conservative than the more common Clarke derivative based conditions. We apply our result to state-dependent switched systems, and to the interconnection of two differential inclusions. As an example, we propose an observer-based controller for certain nonlinear two-mode state-dependent switched systems.