Adaptive Meshing for CPA Lyapunov Function Synthesis

TL;DR

This paper investigates adaptive meshing techniques to improve the efficiency of CPA Lyapunov function synthesis for nonlinear systems, reducing computational costs while maintaining effectiveness.

Contribution

It introduces and compares three adaptive meshing methods, including model-based and combined approaches, to enhance CPA Lyapunov synthesis efficiency.

Findings

Adaptive meshing reduces computational effort.

Model-based meshing improves mesh quality.

Combined methods outperform individual approaches.

Abstract

Continuous piecewise affine (CPA) Lyapunov function synthesis is one method to perform Lyapunov stability analysis for nonlinear systems. This method first generates a mesh over the region of interest in the system's state space and then solves a linear program (LP), which enforces constraints on each vertex of the mesh, to synthesize a Lyapunov function. Finer meshes broaden the class of Lyapunov function candidates, but CPA function synthesis is more computationally expensive for finer meshes -- particularly so in higher dimensional systems. This paper explores methods to mesh the region of interest more efficiently so that a Lyapunov function can be synthesized using less computational effort. Three methods are explored -- adaptive meshing, meshing using knowledge of the system model, and a combination of the two. Numerical examples for two and three dimensional nonlinear dynamical systems are used to compare the efficacy of the three methods.