Koopman-based Estimation of Lyapunov Functions: Theory on a Reproducing Kernel Hilbert Space

TL;DR

This paper develops a theoretical framework for estimating Lyapunov functions using Koopman operators on reproducing kernel Hilbert spaces, linking spectral properties to stability and providing error bounds for the estimates.

Contribution

It introduces a novel approach to construct Lyapunov functions from Koopman operators on RKHS, establishing existence, uniqueness, and error bounds for the estimates.

Findings

The spectrum of Koopman operator is confined within the unit disk for stable equilibria.

A unique solution to the operator algebraic Lyapunov equation exists.

Error bounds for Lyapunov function estimates are derived and validated numerically.

Abstract

Koopman operator provides a general linear description of nonlinear systems, whose estimation from data (via extended dynamic mode decomposition) has been extensively studied. However, the elusiveness between the Koopman spectrum and the stability of equilibrium point poses a challenge to utilizing the Koopman operator for stability analysis, which further hinders the construction of a universal theory of Koopman-based control. In our prior work, we defined the Koopman operator on a reproducing kernel Hilbert space (RKHS) using a linear--radial product kernel, and proved that the Koopman spectrum is confined in the unit disk of the complex plane when the origin is an asymptotically stable equilibrium point. Building on this fundamental spectrum--stability relation, here we consider the problem of Koopman operator-based Lyapunov function estimation with a given decay rate function. The decay rate function and the Lyapunov function are both specified by positive operators on the RKHS and are related by an operator algebraic Lyapunov equation (ALE), whose solution exists uniquely. The error bound of such a Lyapunov function estimate, obtained via kernel extended dynamic mode decomposition (kEDMD), are established based on statistical learning theory and verified by a numerical study.