Automated Stability Analysis of Piecewise Affine Dynamics Using Vertices

TL;DR

This paper introduces automated methods for stability analysis of piecewise affine systems by adaptively subdividing regions to find valid Lyapunov functions efficiently, reducing computational complexity.

Contribution

It proposes two novel region division techniques and uses Delaunay triangulation to improve Lyapunov function search in PWA systems.

Findings

Fewer regions needed for valid Lyapunov functions.

Reduced computational time for stability analysis.

Effective handling of learned models and MPC controllers.

Abstract

This paper presents an automated algorithm to analyze the stability of piecewise affine (PWA) dynamical systems due to their broad applications. We parametrize the Lyapunov function as a PWA function, with polytopic regions defined by the PWA dynamics. Using this parametrization, Stability conditions can be expressed as linear constraints restricted to polytopes so that the search for a Lyapunov function involves solving a linear program. However, a valid Lyapunov function might not be found given these polytopic regions. A natural response is to increase the size of the parametrization of the Lyapunov function by dividing regions and solving the new linear program. This paper proposes two new methods to divide each polytope into smaller ones. The first approach divides a polytope based on the sign of the derivative of the candidate Lyapunov function, while the second divides it based on the change in the vector field of the PWA dynamical system. In addition, we propose using Delaunay triangulation to achieve automated division of regions and preserve the continuity of the PWA Lyapunov function. Examples involving learned models and explicit MPC controllers demonstrate that the proposed method of dividing regions leads to valid Lyapunov functions with fewer regions than existing methods, reducing the computational time taken for stability analysis