Polytope Lyapunov functions for stable and for stabilizable LSS

TL;DR

This paper introduces a novel method for constructing polytope Lyapunov functions to assess stability and stabilizability of linear switching systems, providing precise Lyapunov exponent bounds in higher dimensions.

Contribution

The paper develops a new discretization-based technique for polytope Lyapunov functions, extending to positive systems with antinorms, and offers theoretical guarantees and practical efficiency.

Findings

Effective stability decision method for high-dimensional LSS

Accurate Lyapunov exponent bounds with linear error estimate

Demonstrated efficiency on systems up to 100 dimensions

Abstract

We present a new approach for constructing polytope Lyapunov functions for continuous-time linear switching systems (LSS). This allows us to decide the stability of LSS and to compute the Lyapunov exponent with a good precision in relatively high dimensions. The same technique is also extended for stabilizability of positive systems by evaluating a polytope concave Lyapunov function ("antinorm") in the cone. The method is based on a suitable discretization of the underlying continuous system and provides both a lower and an upper bound for the Lyapunov exponent. The absolute error in the Lyapunov exponent computation is estimated from above and proved to be linear in the dwell time. The practical efficiency of the new method is demonstrated in several examples and in the list of numerical experiments with randomly generated matrices of dimensions up to $10$ (for general linear systems) and up to $100$ (for positive systems). The development of the method is based on several theoretical results proved in the paper: the existence of monotone invariant norms and antinorms for positively irreducible systems, the equivalence of all contractive norms for stable systems and the linear convergence theorem.