# Willems' Fundamental Lemma for Nonlinear Systems with Koopman Linear Embedding

arXiv: https://arxiv.org/abs/2409.16389 · HTML: https://mlsanity.com/paper/2409.16389/

## Key facts

- Cited by 5 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 24 Sep 2024.

## Built on by (most cited first, 5 of 5)

- [Model-free Online Learning for the Kalman Filter: Forgetting Factor and Logarithmic Regret](https://mlsanity.com/paper/2505.08982/) (2025)
- [Koopman operator theory: fundamentals, control, and applications](https://mlsanity.com/paper/2607.01819/) (2026)
- [Data-Driven Prediction and Control of Hammerstein-Wiener Systems with Implicit Gaussian Processes](https://mlsanity.com/paper/2501.15849/) (2025)
- [An Output Feedback Q-learning Algorithm for Optimal Control of Nonlinear Systems with Koopman Linear Embedding](https://mlsanity.com/paper/2603.29858/) (2026)
- [A Behavioral Framework for Data-Driven Modeling of Nonlinear Systems in Vector-Valued Reproducing Kernel Hilbert Spaces](https://mlsanity.com/paper/2605.07052/) (2026)

## Abstract (arXiv)

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.