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Error bounds on analytic Koopman-based Lyapunov functions

François-Grégoire Bierwart, Alexandre Mauroy

Abstract

The Koopman operator provides an infinite-dimensional linear description of nonlinear dynamical systems that can be leveraged in the context of stability analysis. In particular, Lyapunov functions can be obtained in a systematic way via the eigenfunctions of the Koopman operator. However, these eigenfunctions are computed from finite-dimensional approximations, resulting in approximated Lyapunov functions that must be validated. In this paper, we provide theoretical error bounds on the approximation of the eigenfunctions of the Koopman operator in the case of analytic vector field and finite-dimensional approximation in polynomial subspaces. We leverage these results to assess the validity of Koopman-based Lyapunov functions and obtain an optimization-free inner approximation of the region of attraction of an equilibrium.

Error bounds on analytic Koopman-based Lyapunov functions

Abstract

The Koopman operator provides an infinite-dimensional linear description of nonlinear dynamical systems that can be leveraged in the context of stability analysis. In particular, Lyapunov functions can be obtained in a systematic way via the eigenfunctions of the Koopman operator. However, these eigenfunctions are computed from finite-dimensional approximations, resulting in approximated Lyapunov functions that must be validated. In this paper, we provide theoretical error bounds on the approximation of the eigenfunctions of the Koopman operator in the case of analytic vector field and finite-dimensional approximation in polynomial subspaces. We leverage these results to assess the validity of Koopman-based Lyapunov functions and obtain an optimization-free inner approximation of the region of attraction of an equilibrium.

Paper Structure

This paper contains 14 sections, 7 theorems, 39 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Koopman operator framework for stability analysis
  4. Approximation of the Lyapunov function
  5. Error bounds on the approximation of the eigenfunctions
  6. Error bounds on the Lyapunov function and application to stability
  7. Error bound on $|V-\widetilde{V}|$
  8. Error bound on $|\dot{V}-\dot{\widetilde{V}}|$
  9. Estimation of the region of attraction
  10. Alternative result with a surrogate system
  11. Numerical Application
  12. Example 1
  13. Example 2
  14. Conclusions and Perspectives

Key Result

Proposition 1

Let $\phi_{\lambda}$ be an analytic function over $\mathbb{D}^n(S)$ such that $\sup_{x\in\overline{\mathbb{D}^n(S)}}|\phi_{\lambda}(x)| \leq M$. Then, for any $x\in \mathbb{D}^n(R)$, where $0<R<S$.

Figures (2)

  • Figure 3: \ref{['Fig1a']}Light green: The convergence of the sequence $|c_k|^{1/k}$ to the value $2$ indicates that $\phi_{\lambda}$ is analytic over $\mathbb{D}^1(1/2)$. Dark green: The sequence $(|P_k\phi_{\lambda}|)_k$ is upper bounded by the value $5$. \ref{['Fig1b']} The decreasing sequence $|c_k|S^k$ allows to set $\max_{k>70}|c_{k}|S^{k} < 4\times10^{-4}$. \ref{['Fig1c']} An inner approximation of the ROA is computed with $R = 0.39$ and with the bounds obtained from Proposition \ref{['thm:error_phi1']} (light green) and Proposition \ref{['thm:error_phi2']} (dark green).
  • Figure 4: \ref{['Fig2a']} The sequence $(|P_k\phi_{\lambda}|)_k$ is upper bounded by the value $2$. \ref{['Fig2b']} The decreasing sequence of the nonzero coefficients $\max_{|k|=i}S^i|c_k|$ allows to set $\max_{|k|>70}|c_{k}|S^{k} < 3\times10^{-4}$. \ref{['Fig2c']} An inner approximation of the ROA is computed with $R = 0.385$ and with the bounds obtained from Proposition \ref{['thm:error_phi1']} (light green) and Proposition \ref{['thm:error_phi2']} (dark green). The dotted black curve is the largest level set $\widetilde{V}=\gamma_2$ inside $\mathbb{D}^n(R)$ (red box) while the full black curve is the true ROA of the system.

Theorems & Definitions (16)

  • Definition 1
  • Proposition 1: c13
  • Proposition 2
  • proof
  • Proposition 3
  • proof
  • Remark 1: Estimation of the domain of analyticity
  • Remark 2
  • Proposition 4
  • proof
  • ...and 6 more