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Adaptive Meshing for CPA Lyapunov Function Synthesis

Amy K. Strong, Samuel Akinwande, Leila Bridgeman

TL;DR

This work tackles the computational bottleneck of CPA Lyapunov function synthesis by proposing three mesh-based strategies to reduce the number of simplices required for a valid Lyapunov function. Method 1 online adapts the mesh using slack variables to identify and refine problematic regions via longest-edge bisection, while Method 2 builds a model-informed initial mesh by exploiting Hessian changes through bounding sets and a priori bounds, and Method 3 combines both approaches. Empirical results on 2D and 3D nonlinear systems show substantial reductions in mesh size compared to uniform grids, with Method 1 often delivering the most regular meshes and dramatic improvements in the 3D case (e.g., from about 24,576 simplices in a grid to around 3,431 with Method 1). The findings highlight the practicality of adaptive, model-aware meshing to enable scalable Lyapunov analysis of nonlinear dynamics, and point to future work on lowering recomputation costs and exploring initialization strategies that concentrate density where the dynamics evolve slowly (near $f(x)\approx 0$).

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.

Adaptive Meshing for CPA Lyapunov Function Synthesis

TL;DR

This work tackles the computational bottleneck of CPA Lyapunov function synthesis by proposing three mesh-based strategies to reduce the number of simplices required for a valid Lyapunov function. Method 1 online adapts the mesh using slack variables to identify and refine problematic regions via longest-edge bisection, while Method 2 builds a model-informed initial mesh by exploiting Hessian changes through bounding sets and a priori bounds, and Method 3 combines both approaches. Empirical results on 2D and 3D nonlinear systems show substantial reductions in mesh size compared to uniform grids, with Method 1 often delivering the most regular meshes and dramatic improvements in the 3D case (e.g., from about 24,576 simplices in a grid to around 3,431 with Method 1). The findings highlight the practicality of adaptive, model-aware meshing to enable scalable Lyapunov analysis of nonlinear dynamics, and point to future work on lowering recomputation costs and exploring initialization strategies that concentrate density where the dynamics evolve slowly (near ).

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.

Paper Structure

This paper contains 16 sections, 3 theorems, 9 equations, 2 figures, 2 tables, 1 algorithm.

Table of Contents

  1. INTRODUCTION
  2. Preliminaries
  3. Mesh
  4. Basic Tools
  5. Model-Informed Mesh Generation
  6. CPA Lyapunov Function Synthesis
  7. Motivation and Problem Statement
  8. MAIN RESULTS
  9. Method 1: Online Adaptation
  10. Method 2: Model Informed Triangulation
  11. Method 3: Combined Approach
  12. NUMERICAL EXAMPLES
  13. Two-Dimensional Nonlinear Systems
  14. Three Dimensional Nonlinear System
  15. DISCUSSION
  16. ...and 1 more sections

Key Result

Lemma 1

giesl2014revised Consider the triangulation $\mathcal{T} = \{\sigma_i\}_{i=1}^{m_{\mathcal{T}}},$ where $\sigma_i = \text{co}(\{\mathbf{x}_{i,j}\}_{j=0}^n)$, and a set $\mathbf{W} = \{W_{\mathbf{x}}\}_{\mathbf{x} \in \mathbb{E_{\mathcal{T}}}}\subset\mathbb R ,$ where $W(\mathbf{x}) = W_{\mathbf{x}},

Figures (2)

  • Figure 1: Comparison of the best meshes for system B plotted over the phase plot of the system. Here, \ref{['fig:sysB_grid']} is the best grid mesh, while \ref{['fig:sysB_m1']} shows method 1, \ref{['fig:sysB_m2']} shows method 2, and \ref{['fig:sysB_m3']} shows method 3.
  • Figure 2: Comparison of the best meshes for system C plotted over the phase plot of the system. Here, \ref{['fig:sysC_grid']} is the best grid mesh, while \ref{['fig:sysC_m1']} shows method 1, \ref{['fig:sysC_m2']} shows method 2, and \ref{['fig:sysC_m3']} shows method 3.

Theorems & Definitions (8)

  • Definition II.2
  • Definition II.3
  • Lemma 1
  • Definition II.4
  • Theorem 1
  • Remark 1
  • Corollary 1
  • proof