Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Sequentially learning regions of attraction from data

Oumayma Khattabi, Matteo Tacchi-Bénard, Sorin Olaru

TL;DR

This work tackles stability certification for unknown dynamical systems using data-driven Lyapunov analysis. It builds a patchy $PWA$ Lyapunov function $V$ from affine pieces $V_c(x)=g_c^\top x+b_c$ over a tessellation of the state space and iteratively refines the tessellation to certify increasingly large regions of attraction via an SOCP-like optimization with slack variables and a learnability condition. The approach provides a scalable alternative to SDP or neural Lyapunov methods, enabling stability certificates from pointwise evaluations of $f$ within a bounded domain $\mathcal{X}$ and an attractor neighborhood $\mathcal{A}$ around the equilibrium $0$. Applied to a damped pendulum, the method demonstrates convergence of the level sets and certifiable attraction without full model knowledge, highlighting practical data-driven safety guarantees and potential for higher-dimensional extensions.

Abstract

The paper is dedicated to data-driven analysis of dynamical systems. It deals with certifying the basin of attraction of a stable equilibrium for an unknown dynamical system. It is supposed that point-wise evaluation of the right-hand side of the ordinary differential equation governing the system is available for a set of points in the state space. Technically, a Piecewise Affine Lyapunov function will be constructed iteratively using an optimisation-based technique for the effective validation of the certificates. As a main contribution, whenever those certificates are violated locally, a refinement of the domain and the associated tessellation is produced, thus leading to an improvement in the description of the domain of attraction.

Sequentially learning regions of attraction from data

TL;DR

This work tackles stability certification for unknown dynamical systems using data-driven Lyapunov analysis. It builds a patchy Lyapunov function from affine pieces over a tessellation of the state space and iteratively refines the tessellation to certify increasingly large regions of attraction via an SOCP-like optimization with slack variables and a learnability condition. The approach provides a scalable alternative to SDP or neural Lyapunov methods, enabling stability certificates from pointwise evaluations of within a bounded domain and an attractor neighborhood around the equilibrium . Applied to a damped pendulum, the method demonstrates convergence of the level sets and certifiable attraction without full model knowledge, highlighting practical data-driven safety guarantees and potential for higher-dimensional extensions.

Abstract

The paper is dedicated to data-driven analysis of dynamical systems. It deals with certifying the basin of attraction of a stable equilibrium for an unknown dynamical system. It is supposed that point-wise evaluation of the right-hand side of the ordinary differential equation governing the system is available for a set of points in the state space. Technically, a Piecewise Affine Lyapunov function will be constructed iteratively using an optimisation-based technique for the effective validation of the certificates. As a main contribution, whenever those certificates are violated locally, a refinement of the domain and the associated tessellation is produced, thus leading to an improvement in the description of the domain of attraction.
Paper Structure (10 sections, 1 theorem, 10 equations, 7 figures, 1 table, 1 algorithm)

This paper contains 10 sections, 1 theorem, 10 equations, 7 figures, 1 table, 1 algorithm.

Table of Contents

  1. Introduction
  2. Problem Statement
  3. Framework
  4. Data-driven Lyapunov stability analysis
  5. Advantages, challenges and objectives
  6. Proposed Approach
  7. The principle of the iterative construction
  8. Implementation and practical challenges
  9. Numerical Results
  10. Conclusion

Key Result

Theorem 1

Let $\mathcal{X} = \mathcal{X}_0 \supset \mathcal{X}_1 \supset \ldots \supset \mathcal{X}_K$, $\mathcal{A} = \mathcal{A}_0 \supset \mathcal{A}_1 \supset \ldots \supset \mathcal{A}_K$, $\mathcal{D} = \mathcal{D}_0, \ldots, \mathcal{D}_K$ defining ambient sets, prior attracted sets and datasets of ins Then, $\mathbf{L}^{V_0}_{\alpha_0}$ is attracted to $0$ along the unknown dynamics, i.e.

Figures (7)

  • Figure 1: Example case of the simple pendulum
  • Figure 2: $k=0$
  • Figure 3: $k=1$
  • Figure 4: $k=2$
  • Figure 5: $k=3$
  • ...and 2 more figures

Theorems & Definitions (2)

  • Theorem 1
  • proof