Contraction Analysis of Filippov Solutions in Multi-Modal Piecewise Smooth Systems

TL;DR

This paper establishes conditions for contraction in Filippov solutions of multi-modal piecewise smooth systems, aiding in analyzing their long-term behavior such as convergence to equilibria or limit cycles.

Contribution

It extends contraction analysis from bimodal systems to multi-modal systems with multiple and intersecting switching manifolds, providing new theoretical conditions.

Findings

Conditions for contraction in multi-modal PWS systems

Validation through numerical examples

Extension to intersecting switching manifolds

Abstract

This paper provides conditions to ensure contractive behavior of Filippov solutions generated by multi-modal piecewise smooth (PWS) systems. These conditions are instrumental in analyzing the asymptotic behavior of PWS systems, such as convergence towards an equilibrium point or a limit cycle. The work is motivated by a known principle for contraction analysis of bimodal PWS systems which ensures that the flow dynamics of each mode and the sliding dynamics on the switching manifold are contracting. This approach is extended first to PWS systems with multiple non-intersecting switching manifolds in Rn, and then to two intersecting switching manifolds in R2. Numerical examples are provided to validate the theoretical findings, along with a discussion on extensions to more general intersecting switching manifolds in Rn.