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Data-Enabled Predictive Control for Nonlinear Systems Based on a Koopman Bilinear Realization

Zuxun Xiong, Zhenyi Yuan, Keyan Miao, Han Wang, Jorge Cortes, Antonis Papachristodoulou

TL;DR

This work extends Willems' Fundamental Lemma to nonlinear control-affine systems by leveraging Koopman Bilinear Realization (KBR), enabling direct data-driven control from trajectories without explicit EDMD-based model identification. The authors establish a KBR-based Fundamental Lemma under exact finite-dimensional lifting and develop a Data-Enabled Predictive Control (DeePC) framework that operates on lifted states with regularizers to mitigate finite-KBR errors. They present both KBR-based and KLR-based DeePC formulations and validate them through three nonlinear case studies, showing improved optimality and robustness, particularly when exact finite-dimensional Koopman realizations do not exist. The results indicate a practical, robust data-driven control paradigm for unknown nonlinear dynamics with potential impact on real-time nonlinear control tasks and complex systems.

Abstract

This paper extends the Willems' Fundamental Lemma to nonlinear control-affine systems using the Koopman bilinear realization. This enables us to bypass the Extended Dynamic Mode Decomposition (EDMD)-based system identification step in conventional Koopman-based methods and design controllers for nonlinear systems directly from data. Leveraging this result, we develop a Data-Enabled Predictive Control (DeePC) framework for nonlinear systems with unknown dynamics. A case study demonstrates that our direct data-driven control method achieves improved optimality compared to conventional Koopman-based methods. Furthermore, in examples where an exact Koopman realization with a finite-dimensional lifting function set of the controlled nonlinear system does not exist, our method exhibits advanced robustness to finite Koopman approximation errors compared to existing methods.

Data-Enabled Predictive Control for Nonlinear Systems Based on a Koopman Bilinear Realization

TL;DR

This work extends Willems' Fundamental Lemma to nonlinear control-affine systems by leveraging Koopman Bilinear Realization (KBR), enabling direct data-driven control from trajectories without explicit EDMD-based model identification. The authors establish a KBR-based Fundamental Lemma under exact finite-dimensional lifting and develop a Data-Enabled Predictive Control (DeePC) framework that operates on lifted states with regularizers to mitigate finite-KBR errors. They present both KBR-based and KLR-based DeePC formulations and validate them through three nonlinear case studies, showing improved optimality and robustness, particularly when exact finite-dimensional Koopman realizations do not exist. The results indicate a practical, robust data-driven control paradigm for unknown nonlinear dynamics with potential impact on real-time nonlinear control tasks and complex systems.

Abstract

This paper extends the Willems' Fundamental Lemma to nonlinear control-affine systems using the Koopman bilinear realization. This enables us to bypass the Extended Dynamic Mode Decomposition (EDMD)-based system identification step in conventional Koopman-based methods and design controllers for nonlinear systems directly from data. Leveraging this result, we develop a Data-Enabled Predictive Control (DeePC) framework for nonlinear systems with unknown dynamics. A case study demonstrates that our direct data-driven control method achieves improved optimality compared to conventional Koopman-based methods. Furthermore, in examples where an exact Koopman realization with a finite-dimensional lifting function set of the controlled nonlinear system does not exist, our method exhibits advanced robustness to finite Koopman approximation errors compared to existing methods.
Paper Structure (13 sections, 2 theorems, 25 equations, 4 figures, 2 tables)

This paper contains 13 sections, 2 theorems, 25 equations, 4 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Preliminaries and Problem Formulation
  3. Willems' Fundamental Lemma
  4. Koopman Bi/Linear Realization of Nonlinear Systems
  5. Problem Formulation
  6. Data-Driven Control Method for Nonlinear Systems Based on KBR
  7. KBR-based Fundamental Lemma for Nonlinear System
  8. DeePC for Nonlinear System
  9. Numerical Examples
  10. Nonlinear System with Inexact KLR
  11. Nonlinear System with an Exact KBR
  12. Nonlinear System with Inexact KBR
  13. Conclusions And Future Work

Key Result

Lemma 1

(Willems' Fundamental Lemma RN47RN10): Consider the LTI system eq:linear system, let the input/state data $(u_{[1,T]}^d,x^d_{[1,T]})$ be $L$-persistently exciting for it, then

Figures (4)

  • Figure 1: Prediction comparison between the KLR and the KBR (with different lifting function sets) of system \ref{['eq:inexact KBR system']} after 20 prediction steps (left) and 400 prediction steps (right).
  • Figure 2: Open-loop and closed-loop optimal control trajectories of the Van der Pol oscillator under different control methods.
  • Figure 3: Closed-loop trajectories starting from $x(0)=[-0.8,0.4]$ under different methods after 400 steps (left) and 1000 steps (right).
  • Figure 4: Comparison of state responses under different control methods.

Theorems & Definitions (8)

  • Definition 1
  • Lemma 1
  • Example 1
  • Definition 2
  • Lemma 2
  • proof
  • Remark 1
  • Remark 2