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Stability of data-driven Koopman MPC with terminal conditions

Irene Schimperna, Lea Bold, Johannes Köhler, Karl Worthmann, Lalo Magni

TL;DR

This work addresses reliable stabilization of nonlinear systems with data-driven MPC when the prediction model is a surrogate. It proves that, under sufficiently small proportional error bounds for the surrogate and with appropriately designed terminal costs and terminal sets, MPC with terminal conditions achieves recursive feasibility and asymptotic stability of the true plant. The authors show how kernel EDMD (kEDMD) surrogates in the Koopman framework can satisfy these error bounds, and they provide a constructive probe using PI-kEDMD to enforce proportionality at the equilibrium. A Lyapunov-based analysis demonstrates exponential stability of the closed-loop, with a design that decouples stability guarantees from horizon length. A numerical example on the controlled van der Pol oscillator illustrates the practical benefits of using PI-kEDMD over standard approaches, validating the theoretical results and highlighting potential conservatism in constraint tightening that could be mitigated by future work.

Abstract

This paper derives conditions under which Model Predictive Control (MPC) with terminal conditions, using a data-driven surrogate model as a prediction model, asymptotically stabilizes the plant despite approximation errors. In particular, we prove recursive feasibility and asymptotic stability if a proportional error bound holds, where proportional means that the bound is linear in the norm of the state and the input. For a broad class of nonlinear systems, this condition can be satisfied using data-driven surrogate models generated by kernel Extended Dynamic Mode Decomposition (kEDMD) using the Koopman operator. Last, the applicability of the proposed framework is demonstrated in a numerical case study.

Stability of data-driven Koopman MPC with terminal conditions

TL;DR

This work addresses reliable stabilization of nonlinear systems with data-driven MPC when the prediction model is a surrogate. It proves that, under sufficiently small proportional error bounds for the surrogate and with appropriately designed terminal costs and terminal sets, MPC with terminal conditions achieves recursive feasibility and asymptotic stability of the true plant. The authors show how kernel EDMD (kEDMD) surrogates in the Koopman framework can satisfy these error bounds, and they provide a constructive probe using PI-kEDMD to enforce proportionality at the equilibrium. A Lyapunov-based analysis demonstrates exponential stability of the closed-loop, with a design that decouples stability guarantees from horizon length. A numerical example on the controlled van der Pol oscillator illustrates the practical benefits of using PI-kEDMD over standard approaches, validating the theoretical results and highlighting potential conservatism in constraint tightening that could be mitigated by future work.

Abstract

This paper derives conditions under which Model Predictive Control (MPC) with terminal conditions, using a data-driven surrogate model as a prediction model, asymptotically stabilizes the plant despite approximation errors. In particular, we prove recursive feasibility and asymptotic stability if a proportional error bound holds, where proportional means that the bound is linear in the norm of the state and the input. For a broad class of nonlinear systems, this condition can be satisfied using data-driven surrogate models generated by kernel Extended Dynamic Mode Decomposition (kEDMD) using the Koopman operator. Last, the applicability of the proposed framework is demonstrated in a numerical case study.

Paper Structure

This paper contains 6 sections, 2 theorems, 53 equations, 1 figure, 1 algorithm.

Table of Contents

  1. Introduction
  2. Setup and MPC with terminal conditions
  3. Recursive feasibility & asymptotic stability
  4. Kernel Extended Dynamic Mode Decomposition
  5. Numerical example
  6. Conclusion

Key Result

Theorem 1

Let Assumptions ass:error_bounds and ass:terminal hold for a sufficiently small $\varepsilon > 0$ and assume initial feasibility, i.e., $\mathcal{U}^\varepsilon_N(x_{\mu_N^\varepsilon}(0)) \neq \emptyset$. Then, the optimization problem of Algorithm alg:MPC is feasible for all $k \in \mathbb{Z}_{\ge

Figures (1)

  • Figure 1: Van der Pol \ref{['eq:sys:vdp']}: Error $\|x(k)\|$ of the kEDMD-based MPC with different numbers of clusters.

Theorems & Definitions (4)

  • Definition 1: $\varepsilon$-admissibility
  • Theorem 1
  • proof
  • Theorem 2: Approximation error; Proposition 2 in schimperna2025data