SOS-Convex Lyapunov Functions and Stability of Difference Inclusions

TL;DR

This paper introduces sos-convex Lyapunov functions for stability analysis of difference inclusions, providing algebraic certificates of convexity that can be efficiently computed and are necessary and sufficient for linear systems.

Contribution

It develops the concept of sos-convex Lyapunov functions, proving their universality for linear systems and extending the approach to nonlinear systems with a semidefinite programming method.

Findings

Sos-convex Lyapunov functions are necessary and sufficient for stability of switched linear systems.

Existence of a convex Lyapunov function implies stability in switched nonlinear systems.

The proposed SDP-based method can compute regions of attraction for polynomial systems.

Abstract

We introduce the concept of sos-convex Lyapunov functions for stability analysis of both linear and nonlinear difference inclusions (also known as discrete-time switched systems). These are polynomial Lyapunov functions that have an algebraic certificate of convexity and that can be efficiently found via semidefinite programming. We prove that sos-convex Lyapunov functions are universal (i.e., necessary and sufficient) for stability analysis of switched linear systems. We show via an explicit example however that the minimum degree of a convex polynomial Lyapunov function can be arbitrarily higher than a non-convex polynomial Lyapunov function. In the case of switched nonlinear systems, we prove that existence of a common non-convex Lyapunov function does not imply stability, but existence of a common convex Lyapunov function does. We then provide a semidefinite programming-based procedure for computing a full-dimensional subset of the region of attraction of equilibrium points of switched polynomial systems, under the condition that their linearization be stable. We conclude by showing that our semidefinite program can be extended to search for Lyapunov functions that are pointwise maxima of sos-convex polynomials.