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Least-Squares Multi-Step Koopman Operator Learning for Model Predictive Control

Liang Wu, Wallace Gian Yion Tan, Leqi Zhou, Richard D. Braatz, Jan Drgona

TL;DR

This work addresses long-horizon control of nonlinear systems by integrating Koopman-based MPC with a novel multi-step EDMD learning framework. Instead of identifying lifted system matrices and condensing them post hoc, the authors directly learn the condensed horizon-mapping matrices $(oldsymbol{E}, oldsymbol{F})$ in a convex least-squares problem, enabling parallel computation and automatic dictionary pruning. Theoretical analysis shows that multi-step EDMD avoids error accumulation inherent in one-step EDMD, with non-asymptotic bounds depending on the target multi-step map and data budget. Numerical experiments on the Van der Pol and Duffing oscillators demonstrate improved long-horizon prediction accuracy and robust closed-loop performance, illustrating the practical impact for real-time MPC of nonlinear dynamics.

Abstract

MPC is widely used in real-time applications, but practical implementations are typically restricted to convex QP formulations to ensure fast and certified execution. Koopman-based MPC enables QP-based control of nonlinear systems by lifting the dynamics to a higher-dimensional linear representation. However, existing approaches rely on single-step EDMD. Consequently, prediction errors may accumulate over long horizons when the EDMD operator is applied recursively. Moreover, the multi-step prediction loss is nonconvex with respect to the single-step EDMD operator, making long-horizon model identification particularly challenging. This paper proposes a multi-step EDMD framework that directly learns the condensed multi-step state-control mapping required for Koopman-MPC, thereby bypassing explicit identification of the lifted system matrices and subsequent model condensation. The resulting identification problem admits a convex least-squares formulation. We further show that the problem decomposes across prediction horizons and state coordinates, enabling parallel computation and row-wise $\ell_1$-regularization for automatic dictionary pruning. A non-asymptotic finite-sample analysis demonstrates that, unlike one-step EDMD, the proposed method avoids error compounding and yields error bounds that depend only on the target multi-step mapping. Numerical examples validate improved long-horizon prediction accuracy and closed-loop performance.

Least-Squares Multi-Step Koopman Operator Learning for Model Predictive Control

TL;DR

This work addresses long-horizon control of nonlinear systems by integrating Koopman-based MPC with a novel multi-step EDMD learning framework. Instead of identifying lifted system matrices and condensing them post hoc, the authors directly learn the condensed horizon-mapping matrices in a convex least-squares problem, enabling parallel computation and automatic dictionary pruning. Theoretical analysis shows that multi-step EDMD avoids error accumulation inherent in one-step EDMD, with non-asymptotic bounds depending on the target multi-step map and data budget. Numerical experiments on the Van der Pol and Duffing oscillators demonstrate improved long-horizon prediction accuracy and robust closed-loop performance, illustrating the practical impact for real-time MPC of nonlinear dynamics.

Abstract

MPC is widely used in real-time applications, but practical implementations are typically restricted to convex QP formulations to ensure fast and certified execution. Koopman-based MPC enables QP-based control of nonlinear systems by lifting the dynamics to a higher-dimensional linear representation. However, existing approaches rely on single-step EDMD. Consequently, prediction errors may accumulate over long horizons when the EDMD operator is applied recursively. Moreover, the multi-step prediction loss is nonconvex with respect to the single-step EDMD operator, making long-horizon model identification particularly challenging. This paper proposes a multi-step EDMD framework that directly learns the condensed multi-step state-control mapping required for Koopman-MPC, thereby bypassing explicit identification of the lifted system matrices and subsequent model condensation. The resulting identification problem admits a convex least-squares formulation. We further show that the problem decomposes across prediction horizons and state coordinates, enabling parallel computation and row-wise -regularization for automatic dictionary pruning. A non-asymptotic finite-sample analysis demonstrates that, unlike one-step EDMD, the proposed method avoids error compounding and yields error bounds that depend only on the target multi-step mapping. Numerical examples validate improved long-horizon prediction accuracy and closed-loop performance.
Paper Structure (35 sections, 5 theorems, 101 equations, 5 figures, 1 algorithm)

This paper contains 35 sections, 5 theorems, 101 equations, 5 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Related work
  3. Contributions
  4. Notations
  5. Problem Formulation and Preliminaries
  6. Koopman operator and Extended Dynamic Mode Decomposition
  7. Condensed Koopman-MPC formulation
  8. Multi-step Extended Dynamic Mode Decomposition
  9. Decomposition at the prediction horizons level and state coordinates level
  10. Theoretical Analysis
  11. Analysis framework and assumptions
  12. Koopman approximations for EDMD
  13. Single-step EDMD
  14. Polynomial chaos expansions as dictionary observables
  15. Regression lemma
  16. ...and 20 more sections

Key Result

Lemma 1

For any $g \in H^s_\mu(\Omega)$, there exists a constant $C_{s,n_x}>0$ only depending on $s >0$ and $n_x$ such that, for any $0 \leq q \leq \lfloor s/2\rfloor$, where and $\left\lVert g\right\rVert_{H^q_\mu(\Omega)}$ denotes the weighted Sobolev norm

Figures (5)

  • Figure 1: Top: Prediction MSE comparison across different models for the van der Pol oscillator. Bottom: Example $x_2$ trajectory prediction which one-step EDMD, which diverges over longer horizons.
  • Figure 2: Closed-loop state (top) and control (bottom) trajectories for the van der Pol oscillator.
  • Figure 3: Prediction MSE comparison for the Duffing oscillator across different models.
  • Figure 4: Closed-loop state (top) and control (bottom) trajectories for the oscillator.
  • Figure 5: Closed-loop phase trajectories of the Duffing oscillator with separatrix. The separatrix separates the two basins of attraction of the equilibria (black dots). Multi-step MPC and its pruned variant converge to the origin, while single-step MPC crosses the separatrix, leading to poor closed-loop behavior.

Theorems & Definitions (15)

  • Remark 1
  • Remark 2
  • Remark 3
  • Lemma 1
  • Proof 1
  • Remark 4
  • Lemma 2: Regression Error
  • Proof 2
  • Theorem 1
  • Proof 3
  • ...and 5 more