A Piecewise Smooth Control-Lyapunov Function Framework for Switching Stabilization

TL;DR

This paper introduces a unified piecewise smooth control-Lyapunov function framework for stabilizing switched nonlinear systems, effectively handling nonsmooth surfaces and sliding motions, and improving upon existing methods.

Contribution

It develops a systematic PSCLF-based approach for designing stabilizing switching laws, encompassing many existing results as special cases.

Findings

Derived sufficient stability conditions for Filippov solutions.

Unified framework includes linear and nonlinear switching stabilization.

Numerical example demonstrates improved stabilization results.

Abstract

This paper studies switching stabilization problems for general switched nonlinear systems. A piecewise smooth control-Lyapunov function (PSCLF) approach is proposed and a constructive way to design a stabilizing switching law is developed. The switching law is constructed via the directional derivatives of the PSCLF with a careful discussion on various technical issues that may occur on the nonsmooth surfaces. Sufficient conditions are derived to ensure stability of the closed-loop Filippov solutions including possible sliding motions. The proposed PSCLF approach contains many existing results as special cases and provides a unified framework to study nonlinear switching stabilization problems with a systematic consideration of sliding motions. Applications of the framework to switched linear systems with quadratic and piecewise quadratic control-Lyapunov functions are discussed and results stronger than the existing methods in the literature are obtained. Application to stabilization of switched nonlinear systems is illustrated through an numerical example.