Robust data-driven control for nonlinear systems using the Koopman operator

TL;DR

This paper introduces a data-driven control method for nonlinear systems using the Koopman operator, enabling robust local stability guarantees through finite-dimensional approximations and linear matrix inequalities.

Contribution

It develops a novel control design approach for nonlinear systems based on Koopman operator theory, including finite-dimensional lifting and stability guarantees.

Findings

Successfully applied to Van der Pol oscillator

Guarantees robust local stability under finite-gain bounds

Provides a linear fractional representation for control design

Abstract

Data-driven analysis and control of dynamical systems have gained a lot of interest in recent years. While the class of linear systems is well studied, theoretical results for nonlinear systems are still rare. In this paper, we present a data-driven controller design method for discrete-time control-affine nonlinear systems. Our approach relies on the Koopman operator, which is a linear but infinite-dimensional operator lifting the nonlinear system to a higher-dimensional space. Particularly, we derive a linear fractional representation of a lifted bilinear system representation based on measured data. Further, we restrict the lifting to finite dimensions, but account for the truncation error using a finite-gain argument. We derive a linear matrix inequality based design procedure to guarantee robust local stability for the resulting bilinear system for all error terms satisfying the finite-gain bound and, thus, also for the underlying nonlinear system. Finally, we apply the developed design method to the nonlinear Van der Pol oscillator.