Dynamic Mode Decomposition of Control-Affine Nonlinear Systems using Discrete Control Liouville Operators

TL;DR

This paper introduces a novel linear operator representation for discrete-time control-affine nonlinear systems, enabling improved prediction of system behavior under feedback control using data-driven methods.

Contribution

It extends dynamic mode decomposition to control-affine nonlinear systems, allowing for predictive analysis with recorded system snapshots and feedback laws.

Findings

Successfully predicted controlled Duffing oscillator responses.

Demonstrated advantages over existing techniques.

Validated method with numerical experiments.

Abstract

Representation of nonlinear dynamical systems as infinite-dimensional linear operators over Hilbert spaces enables analysis of nonlinear systems via pseudo-spectral operator analysis. In this paper, we provide a novel representation for discrete-time control-affine nonlinear dynamical systems as linear operators acting on a Hilbert space. We also demonstrate that this representation can be used to predict the behavior of the closed-loop system given a known feedback law using recorded snapshots of the system state resulting from arbitrary, potentially open-loop control inputs. We thereby extend the predictive capabilities of dynamic mode decomposition to discrete-time nonlinear systems that are affine in control. We validate the method using two numerical experiments by predicting the response of a controlled Duffing oscillator to a known feedback law, as well as demonstrating the advantage of the developed method relative to existing techniques in the literature.