Koopman-based control of nonlinear systems with closed-loop guarantees

TL;DR

This paper introduces a data-driven Koopman operator framework with error bounds for controlling unknown nonlinear systems, ensuring closed-loop stability and performance guarantees.

Contribution

It extends Koopman-based control with rigorous error bounds and robust control design, enabling guaranteed stabilization and performance in nonlinear systems.

Findings

Guaranteed bounds on Koopman approximation errors.

Successful stabilization in discrete and continuous time.

Demonstrated advantages over existing methods with numerical example.

Abstract

In this paper, we provide a tutorial overview and an extension of a recently developed framework for data-driven control of unknown nonlinear systems with rigorous closed-loop guarantees. The proposed approach relies on the Koopman operator representation of the nonlinear system, for which a bilinear surrogate model is estimated based on data. In contrast to existing Koopman-based estimation procedures, we state guaranteed bounds on the approximation error using the stability- and certificate-oriented extended dynamic mode decomposition (SafEDMD) framework. The resulting surrogate model and the uncertainty bounds allow us to design controllers via robust control theory and sum-of-squares optimization, guaranteeing desirable properties for the closed-loop system. We present results on stabilization both in discrete and continuous time, and we derive a method for controller design with performance objectives. The benefits of the presented framework over established approaches are demonstrated with a numerical example.