RKHS method for computing Koopman-based Lyapunov functions

TL;DR

This paper introduces a kernel-based method to accurately compute Koopman eigenfunctions for stability analysis of nonlinear systems, especially high-dimensional ones, ensuring spectrum preservation and reliable region of attraction estimation.

Contribution

It develops a novel RKHS-based approach that preserves the Jacobian spectrum in Koopman eigenfunction approximation, improving stability analysis accuracy.

Findings

Kernel method effectively approximates Koopman eigenfunctions.

Spectrum preservation enhances stability analysis reliability.

Scenario-based optimization estimates the region of attraction.

Abstract

The Koopman operator is a powerful approach to global stability analysis of nonlinear systems, which provides a systematic procedure for Lyapunov function design. In this framework, Lyapunov functions are obtained through the eigenfunctions of the Koopman operator associated with the eigenvalues of the Jacobian matrix at the equilibrium. In practice, the eigenfunctions are approximated via a finite-dimensional representation of the operator, and there is no guarantee that the approximated spectrum accurately matches the true one. In this paper, we develop a kernel-based method to compute Koopman eigenfunctions and preserve the spectrum of the Jacobian matrix. This approach is suitable for stability analysis of high-dimensional systems thanks to the kernel trick. Moreover, the Lyapunov function candidate is validated through a scenario-based optimization technique that provides a reliable estimation of the region of attraction of the system.