Path-Complete Graphs and Common Lyapunov Functions

TL;DR

This paper explores Path-Complete Lyapunov Functions for stability analysis of switching systems, proving their relation to common Lyapunov functions and analyzing their conservativeness compared to other methods.

Contribution

It establishes that Path-Complete Lyapunov Functions imply the existence of a common Lyapunov function and investigates conditions for their conservativeness relative to other functions.

Findings

Satisfiability of Path-Complete Lyapunov criteria implies a common Lyapunov function.

Counterexample showing no equivalence between Path-Complete functions and max-of-quadratics functions.

Tools for comparing the conservativeness of different Lyapunov functions based on graph and algebraic structure.

Abstract

A Path-Complete Lyapunov Function is an algebraic criterion composed of a finite number of functions, called its pieces, and a directed, labeled graph defining Lyapunov inequalities between these pieces. It provides a stability certificate for discrete-time switching systems under arbitrary switching. In this paper, we prove that the satisfiability of such a criterion implies the existence of a Common Lyapunov Function, expressed as the composition of minima and maxima of the pieces of the Path-Complete Lyapunov function. The converse, however, is not true even for discrete-time linear systems: we present such a system where a max-of-2 quadratics Lyapunov function exists while no corresponding Path-Complete Lyapunov function with 2 quadratic pieces exists. In light of this, we investigate when it is possible to decide if a Path-Complete Lyapunov function is less conservative than another. By analyzing the combinatorial and algebraic structure of the graph and the pieces respectively, we provide simple tools to decide when the existence of such a Lyapunov function implies that of another.