On Path-Complete Lyapunov Functions: Geometry and Comparison

TL;DR

This paper investigates the geometric structure of Path-Complete Lyapunov Functions for switching system stability and introduces a linear programming method to compare their conservativeness.

Contribution

It demonstrates that Path-Complete Lyapunov Functions can be expressed as common Lyapunov functions with minima and maxima, and provides a linear program to compare their conservativeness.

Findings

Path-Complete Lyapunov Functions can be represented as common Lyapunov functions.

A linear programming criterion for comparing Lyapunov functions' conservativeness.

Each criterion implies an invariant set for the system.

Abstract

We study optimization-based criteria for the stability of switching systems, known as Path-Complete Lyapunov Functions, and ask the question "can we decide algorithmically when a criterion is less conservative than another". Our contribution is twofold. First, we show that a Path-Complete Lyapunov Function, which is a multiple Lyapunov function by nature, can always be expressed as a common Lyapunov function taking the form of a combination of minima and maxima of the elementary functions that compose it. Geometrically, our results provide for each Path-Complete criterion an implied invariant set. Second, we provide a linear programming criterion allowing to compare the conservativeness of two arbitrary given Path-Complete Lyapunov functions.