Willems' Fundamental Lemma for Nonlinear Systems with Koopman Linear Embedding

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.