Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Willems' Fundamental Lemma for Nonlinear Systems with Koopman Linear Embedding

Xu Shang, Jorge Cortés, Yang Zheng

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.

Willems' Fundamental Lemma for Nonlinear Systems with Koopman Linear Embedding

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.
Paper Structure (21 sections, 9 theorems, 44 equations, 2 figures)

This paper contains 21 sections, 9 theorems, 44 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Preliminaries and Problem Statement
  3. Koopman linear models for nonlinear systems
  4. Willems' Fundamental Lemma
  5. Problem Statement
  6. From Koopman Linear Embeddings to Data-driven Representations
  7. Two behavior spaces
  8. Data-driven representation of nonlinear systems
  9. A special case: Affine systems
  10. Numerical Experiments
  11. Experiment Setup
  12. Prediction and Control Performance
  13. Conclusions
  14. Technical proofs
  15. Proof of \ref{['prop:traj-linear']}
  16. ...and 6 more sections

Key Result

Lemma 1

Consider the LTI system eqn:LTI-system. Assume the Hankel matrix formed by its pre-collected trajectory $H_0 := \textnormal{col}(\mathcal{H}_1(x_{\textnormal{d}, 0:n_\textnormal{d}-L}), \mathcal{H}_L(u_{\textnormal{d}}))$ has full row rank. Then, a length-$L$ input-output data sequence $\textnormal{

Figures (2)

  • Figure 1: Prediction of $x_2$ with a given input $u_\textnormal{F}$. In (a) and (b), the red curve is the predicted trajectory of DD-K with $T_{\textnormal{ini}} = 4$. The orange, blue and green curves in (a) are DD-A, EDMD-K and DNN-K, and the brown and purple curves in (b) are DD-K with initial trajectory of $T_{\textnormal{ini}} = 2$ and $3$.
  • Figure 2: Control performance using different linear representations. (a) Tracking Performance. (b) Realized control costs.

Theorems & Definitions (18)

  • Definition 1: Koopman Linear Embedding
  • Definition 2: Persistently exciting
  • Lemma 1: Willems' fundamental lemma
  • Theorem 1
  • Lemma 2
  • Definition 3
  • Theorem 2
  • Theorem 3
  • Remark 1
  • Theorem 4: Berberich2022
  • ...and 8 more