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Koopman Data-Driven Predictive Control with Robust Stability and Recursive Feasibility Guarantees

Thomas de Jong, Valentina Breschi, Maarten Schoukens, Mircea Lazar

TL;DR

This paper develops a Koopman data-driven predictive control framework to stabilize nonlinear systems using a linear-in-control lifted model. It introduces a multi-step predictor learned directly from input-output data via a nonlinear observable learning step and a subsequent least-squares refinement, enabling a predictive controller based on Koopman dynamics. To guarantee recursive feasibility and stability despite model incompleteness, the authors employ an interpolated initial state and a novel regularization term that yields input-to-state stability with respect to prediction error. The approach is validated on two nonlinear benchmarks (cart-spring–damper and pendulum), showing high prediction accuracy and competitive performance with NMPC under constraints. The work highlights opportunities for extending to noisy scenarios and fully leveraging the complete Koopman formulation for improved robustness and accuracy.

Abstract

In this paper, we consider the design of data-driven predictive controllers for nonlinear systems from input-output data via linear-in-control input Koopman lifted models. Instead of identifying and simulating a Koopman model to predict future outputs, we design a subspace predictive controller in the Koopman space. This allows us to learn the observables minimizing the multi-step output prediction error of the Koopman subspace predictor, preventing the propagation of prediction errors. To avoid losing feasibility of our predictive control scheme due to prediction errors, we compute a terminal cost and terminal set in the Koopman space and we obtain recursive feasibility guarantees through an interpolated initial state. As a third contribution, we introduce a novel regularization cost yielding input-to-state stability guarantees with respect to the prediction error for the resulting closed-loop system. The performance of the developed Koopman data-driven predictive control methodology is illustrated on a nonlinear benchmark example from the literature.

Koopman Data-Driven Predictive Control with Robust Stability and Recursive Feasibility Guarantees

TL;DR

This paper develops a Koopman data-driven predictive control framework to stabilize nonlinear systems using a linear-in-control lifted model. It introduces a multi-step predictor learned directly from input-output data via a nonlinear observable learning step and a subsequent least-squares refinement, enabling a predictive controller based on Koopman dynamics. To guarantee recursive feasibility and stability despite model incompleteness, the authors employ an interpolated initial state and a novel regularization term that yields input-to-state stability with respect to prediction error. The approach is validated on two nonlinear benchmarks (cart-spring–damper and pendulum), showing high prediction accuracy and competitive performance with NMPC under constraints. The work highlights opportunities for extending to noisy scenarios and fully leveraging the complete Koopman formulation for improved robustness and accuracy.

Abstract

In this paper, we consider the design of data-driven predictive controllers for nonlinear systems from input-output data via linear-in-control input Koopman lifted models. Instead of identifying and simulating a Koopman model to predict future outputs, we design a subspace predictive controller in the Koopman space. This allows us to learn the observables minimizing the multi-step output prediction error of the Koopman subspace predictor, preventing the propagation of prediction errors. To avoid losing feasibility of our predictive control scheme due to prediction errors, we compute a terminal cost and terminal set in the Koopman space and we obtain recursive feasibility guarantees through an interpolated initial state. As a third contribution, we introduce a novel regularization cost yielding input-to-state stability guarantees with respect to the prediction error for the resulting closed-loop system. The performance of the developed Koopman data-driven predictive control methodology is illustrated on a nonlinear benchmark example from the literature.
Paper Structure (13 sections, 3 theorems, 40 equations, 6 figures, 4 tables)

This paper contains 13 sections, 3 theorems, 40 equations, 6 figures, 4 tables.

Table of Contents

  1. INTRODUCTION
  2. Preliminaries & problem statement
  3. Problem Statement
  4. Multi-step learning of the Koopman observables and prediction matrices
  5. Koopman data-driven predictive control with interpolated initial state
  6. Properties of proposed scheme
  7. Illustrative Example
  8. 2D cart-spring–damper
  9. Pendulum
  10. Conclusions and Future Works
  11. Future Works
  12. Computation of Terminal Ingredients
  13. Identification data

Key Result

Theorem IV.2

Suppose that Assumption assm:TerminalCost_and_Set holds. At any time $k\in\mathbb{N}$, given $\mathbf{x}_\text{ini}(k)$ and $z_{1|k-1}^\ast$, suppose that eq:3:1 is feasible. Then the problem in eq:3:1 is feasible at time $k+1$ for $\mathbf{x}_\text{ini}(k+1)$ and $z_{1|k}^\ast$.

Figures (6)

  • Figure 1: Simulations chart: position $x_1$, control input $u$, interpolation variable $\xi$ and observables $\varphi_i$.
  • Figure 2: Simulations chart: interpolation variable $\xi$, observables $\varphi_i$ and Koopman model error $e$.
  • Figure 3: Simulations pendulum: angle $x_2$, control input $u$, interpolation variable $\xi$ and observables $\varphi_i$.
  • Figure 4: Simulations pendulum: interpolation variable $\xi$, observables $\varphi_i$ and Koopman model error $e$.
  • Figure 5: Input-output data for 2D chart-spring damper.
  • ...and 1 more figures

Theorems & Definitions (12)

  • Definition II.3
  • Remark II.4
  • Remark II.5
  • Theorem IV.2
  • proof
  • Remark IV.3
  • Lemma IV.4
  • proof
  • Theorem IV.5
  • proof
  • ...and 2 more