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Data-driven MPC with terminal conditions in the Koopman framework

Karl Worthmann, Robin Strässer, Manuel Schaller, Julian Berberich, Frank Allgöwer

TL;DR

The paper presents a data-driven MPC framework that uses SafEDMD to build a bilinear surrogate of nonlinear dynamics in lifted coordinates and designs terminal regions and costs to guarantee recursive feasibility and practical stability. By leveraging proportional finite-data error bounds, it constructs terminal conditions that ensure Lyapunov decrease within the MPC loop, with a dual-mode extension offering exponential stability. The approach is validated on an inverted pendulum, showing robust stabilization and improved performance over EDMDc-based linear Koopman models. This work provides a systematic, data-driven path to stabilization guarantees in nonlinear MPC using Koopman-based surrogates and SafEDMD, with clear guidance for terminal design and robustness analysis.

Abstract

We investigate nonlinear model predictive control (MPC) with terminal conditions in the Koopman framework using extended dynamic mode decomposition (EDMD) to generate a data-based surrogate model for prediction and optimization. We rigorously show recursive feasibility and prove practical asymptotic stability w.r.t. the approximation accuracy. To this end, finite-data error bounds are employed. The construction of the terminal conditions is based on recently derived proportional error bounds to ensure the required Lyapunov decrease. Finally, we illustrate the effectiveness of the proposed data-driven predictive controller including the design procedure to construct the terminal region and controller.

Data-driven MPC with terminal conditions in the Koopman framework

TL;DR

The paper presents a data-driven MPC framework that uses SafEDMD to build a bilinear surrogate of nonlinear dynamics in lifted coordinates and designs terminal regions and costs to guarantee recursive feasibility and practical stability. By leveraging proportional finite-data error bounds, it constructs terminal conditions that ensure Lyapunov decrease within the MPC loop, with a dual-mode extension offering exponential stability. The approach is validated on an inverted pendulum, showing robust stabilization and improved performance over EDMDc-based linear Koopman models. This work provides a systematic, data-driven path to stabilization guarantees in nonlinear MPC using Koopman-based surrogates and SafEDMD, with clear guidance for terminal design and robustness analysis.

Abstract

We investigate nonlinear model predictive control (MPC) with terminal conditions in the Koopman framework using extended dynamic mode decomposition (EDMD) to generate a data-based surrogate model for prediction and optimization. We rigorously show recursive feasibility and prove practical asymptotic stability w.r.t. the approximation accuracy. To this end, finite-data error bounds are employed. The construction of the terminal conditions is based on recently derived proportional error bounds to ensure the required Lyapunov decrease. Finally, we illustrate the effectiveness of the proposed data-driven predictive controller including the design procedure to construct the terminal region and controller.
Paper Structure (8 sections, 1 theorem, 33 equations, 2 figures, 1 algorithm)

This paper contains 8 sections, 1 theorem, 33 equations, 2 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. The Koopman operator and SafEDMD
  3. EDMD-based MPC with terminal conditions
  4. Analysis of EDMD-based MPC and construction of the terminal conditions
  5. MPC closed-loop analysis
  6. Construction of the terminal conditions using SafEDMD
  7. Simulation
  8. Conclusions and outlook

Key Result

Theorem IV.2

Let Assumptions ass:invariant-dictionary and ass:terminal:condition hold. Then, the MPC closed loop is recursively feasible and the origin is practically asymptotically stable w.r.t. the approximation error on the set $\mathbb{X}$ in the sense of Definition def:stability:practical.

Figures (2)

  • Figure 1: Closed-loop results of the proposed MPC controller () with input constraints (), where we switch to the terminal control law after $t=10$. MPC results for a linear Koopman surrogate model () are included for comparison.
  • Figure 2: Closed-loop results using $\Phi_1$ and the proposed MPC controller () compared to L-MPC ().

Theorems & Definitions (6)

  • Definition III.1: Admissibility
  • Definition III.2: EDMD admissibility
  • Definition III.4: Practical asymptotic stability
  • Theorem IV.2
  • proof
  • Remark IV.3: Dual-mode MPC