Computing the domain of attraction of switching systems subject to non-convex constraints

TL;DR

This paper presents a method to compute the domain of attraction for stable switching systems with non-convex polynomial constraints by transforming the problem into a linear or semidefinite programming framework using Veronese lifting.

Contribution

It introduces a novel approach that leverages Veronese lifting and reachability analysis to exactly compute the maximal invariant set for constrained switching systems with non-convex polynomial constraints.

Findings

The method successfully computes the domain of attraction in numerical examples.

The approach transforms non-convex constraints into polyhedral sets via Veronese lifting.

Exact computation is achieved through linear or semidefinite programming.

Abstract

We characterize and compute the maximal admissible positively invariant set for asymptotically stable constrained switching linear systems. Motivated by practical problems found, e.g., in obstacle avoidance, power electronics and nonlinear switching systems, in our setting the constraint set is formed by a finite number of polynomial inequalities. First, we observe that the so-called Veronese lifting allows to represent the constraint set as a polyhedral set. Next, by exploiting the fact that the lifted system dynamics remains linear, we establish a method based on reachability computations to characterize and compute the maximal admissible invariant set, which coincides with the domain of attraction when the system is asymptotically stable. After developing the necessary theoretical background, we propose algorithmic procedures for its exact computation, based on linear or semidefinite programs. The approach is illustrated in several numerical examples.