Willems' Fundamental Lemma for Nonlinear Systems with Koopman Linear Embedding

TL;DR

This paper extends Willems' fundamental lemma to nonlinear systems with Koopman linear embedding, enabling accurate data-driven modeling without lifting functions by leveraging rich trajectory data.

Contribution

It establishes a theoretical link between nonlinear systems and their Koopman embeddings, enabling a new data-driven modeling approach that avoids lifting function biases.

Findings

Trajectory space of Koopman embedding is linearly generated by nonlinear system trajectories.

Rich and longer trajectories improve model accuracy.

Bypasses the need for lifting functions in data-driven modeling.

Abstract

Koopman operator theory and Willems' fundamental lemma both can provide (approximated) data-driven linear representation for nonlinear systems. However, choosing lifting functions for the Koopman operator is challenging, and the quality of the data-driven model from Willems' fundamental lemma has no guarantee for general nonlinear systems. In this paper, we extend Willems' fundamental lemma for a class of nonlinear systems that admit a Koopman linear embedding. We first characterize the relationship between the trajectory space of a nonlinear system and that of its Koopman linear embedding. We then prove that the trajectory space of Koopman linear embedding can be formed by a linear combination of rich-enough trajectories from the nonlinear system. Combining these two results leads to a data-driven representation of the nonlinear system, which bypasses the need for the lifting functions and thus eliminates the associated bias errors. Our results illustrate that both the width (more trajectories) and depth (longer trajectories) of the trajectory library are important to ensure the accuracy of the data-driven model.