Kernel-based Koopman approximants for control: Flexible sampling, error analysis, and stability

TL;DR

This paper introduces a new kernel-based Koopman approximation method for control systems that offers flexible sampling, robustness, and stability guarantees, supported by detailed error analysis and stability proofs.

Contribution

It proposes a novel kernel extended DMD scheme with regularization and micro-macro grid decomposition, providing explicit error bounds and stability equivalence.

Findings

Explicit error bounds depend on the distance to equilibrium.

Asymptotic stability of the surrogate system implies stability of the original system.

The method enhances robustness and sampling flexibility in data-driven control.

Abstract

Data-driven techniques for analysis, modeling, and control of complex dynamical systems are on the uptake. Koopman theory provides the theoretical foundation for the popular kernel extended dynamic mode decomposition (kEDMD). In this work, we propose a novel kEDMD scheme to approximate nonlinear control systems accompanied by an in-depth error analysis. Key features are regularization-based robustness and an adroit decomposition into micro and macro grids enabling flexible sampling. But foremost, we prove proportionality, i.e., explicit dependence on the distance to the (controlled) equilibrium, of the derived bound on the full approximation error. Leveraging this key property, we rigorously show that asymptotic stability of the data-driven surrogate (control) system implies asymptotic stability of the original (control) system and vice versa.