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Data-driven Model Predictive Control: Asymptotic Stability despite Approximation Errors exemplified in the Koopman framework

Irene Schimperna, Karl Worthmann, Manuel Schaller, Lea Bold, Lalo Magni

TL;DR

The paper addresses stability for nonlinear MPC when the prediction model is a data-driven surrogate. It develops a general framework showing that, under proportional error bounds and a sufficiently long horizon, the MPC closed-loop stabilizes the origin asymptotically even without terminal conditions, provided cost controllability holds for the nominal system. It verifies the necessary assumptions for data-driven surrogates using kernel EDMD within an RKHS, deriving finite-data error bounds and a learning approach with flexible sampling; this enables constructing surrogate dynamics that satisfy the required Lipschitz and error properties. Numerical simulations on the Van der Pol oscillator and a four-tank process demonstrate that PI-kEDMD-based MPC achieves true asymptotic stability, while standard kEDMD yields only practical stability, highlighting the practical relevance of the proposed method and its independence from the Koopman formulation.

Abstract

In this paper, we analyze stability of nonlinear model predictive control (MPC) using data-driven surrogate models in the optimization step. First, we establish asymptotic stability of the origin, a controlled steady state, w.r.t. the MPC closed loop without stabilizing terminal conditions for sufficiently long prediction horizons. To this end, we prove that cost controllability of the original system is preserved if sufficiently accurate proportional bounds on the approximation error hold. Here, proportional refers to state and control. The proportionality of the error bounds is a key element to derive asymptotic stability in presence of modeling errors and not only practical asymptotic stability. Second, we exemplarily verify the imposed assumptions for data-driven surrogates generated with kernel extended dynamic mode decomposition based on Koopman operator theory. Hereby, we do not impose invariance assumptions on finite dictionaries, but rather derive all conditions under non-restrictive conditions. Finally, we demonstrate our findings with numerical simulations.

Data-driven Model Predictive Control: Asymptotic Stability despite Approximation Errors exemplified in the Koopman framework

TL;DR

The paper addresses stability for nonlinear MPC when the prediction model is a data-driven surrogate. It develops a general framework showing that, under proportional error bounds and a sufficiently long horizon, the MPC closed-loop stabilizes the origin asymptotically even without terminal conditions, provided cost controllability holds for the nominal system. It verifies the necessary assumptions for data-driven surrogates using kernel EDMD within an RKHS, deriving finite-data error bounds and a learning approach with flexible sampling; this enables constructing surrogate dynamics that satisfy the required Lipschitz and error properties. Numerical simulations on the Van der Pol oscillator and a four-tank process demonstrate that PI-kEDMD-based MPC achieves true asymptotic stability, while standard kEDMD yields only practical stability, highlighting the practical relevance of the proposed method and its independence from the Koopman formulation.

Abstract

In this paper, we analyze stability of nonlinear model predictive control (MPC) using data-driven surrogate models in the optimization step. First, we establish asymptotic stability of the origin, a controlled steady state, w.r.t. the MPC closed loop without stabilizing terminal conditions for sufficiently long prediction horizons. To this end, we prove that cost controllability of the original system is preserved if sufficiently accurate proportional bounds on the approximation error hold. Here, proportional refers to state and control. The proportionality of the error bounds is a key element to derive asymptotic stability in presence of modeling errors and not only practical asymptotic stability. Second, we exemplarily verify the imposed assumptions for data-driven surrogates generated with kernel extended dynamic mode decomposition based on Koopman operator theory. Hereby, we do not impose invariance assumptions on finite dictionaries, but rather derive all conditions under non-restrictive conditions. Finally, we demonstrate our findings with numerical simulations.
Paper Structure (11 sections, 8 theorems, 87 equations, 3 figures, 2 tables, 1 algorithm)

This paper contains 11 sections, 8 theorems, 87 equations, 3 figures, 2 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Data-driven MPC formulation
  3. Asymptotic stability of MPC
  4. Data-driven models via kernel EDMD
  5. Koopman operator and EDMD
  6. Extension of kernel EDMD to control systems
  7. Relation to kernel approximation of control systems
  8. Numerical simulations
  9. Van der Pol
  10. Four-tanks process
  11. Conclusions and outlook

Key Result

Proposition 1

Suppose that system eq:system is cost controllable on a set $S$. Further, let $\bar{N} \in \mathbb{N}$ be given and the Assumptions ass:error_bound and ass:lipschitz hold on a set $\Omega$ satisfying $S \oplus \mathcal{B}_{r(\bar{N}) \eta^\varepsilon}(0) \subseteq \Omega$ with $r(\bar{N}) := \sum_{i Moreover, $\lim_{\varepsilon \searrow 0} B^\varepsilon_{N} = B_{N}$ for all $N \in [1:\bar{N}]$. Th

Figures (3)

  • Figure 1: Van der Pol: Error $\|x(k)\|$ for the kEDMD-based MPC closed loop with different horizons and number of clusters.
  • Figure 2: Van der Pol: trajectories of the kEDMD-based MPC closed loop with different horizons and number of clusters.
  • Figure 3: Four-tank: Error $\|x(k) - \bar{x}\|$ for the kEDMD-based MPC closed loop with horizon $N=10$.

Theorems & Definitions (12)

  • Definition 1: Admissible control sequences
  • Definition 2: Cost controllability on a set $S$
  • Proposition 1
  • Theorem 1
  • Corollary 1: Cost controllability of the surrogate model
  • Remark 1
  • Theorem 2
  • Theorem 3: Uniform approximation error
  • Lemma 1: Lipschitz continuity of surrogate $f^\varepsilon$
  • Proposition 2: Proportional error bound
  • ...and 2 more