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Kernel EDMD for data-driven nonlinear Koopman MPC with stability guarantees

Lea Bold, Manuel Schaller, Irene Schimperna, Karl Worthmann

TL;DR

The paper addresses stability in data-driven nonlinear MPC by using kernel EDMD (kEDMD) to approximate the Koopman operator within a reproducing kernel Hilbert space. It leverages RKHS-based finite-data error bounds to construct a data-driven surrogate $F^\varepsilon$ and proves practical asymptotic stability of the origin for the MPC closed loop without stabilizing terminal conditions, relying on cost controllability and a relaxed dynamic programming inequality. A key theoretical advance is achieving stability guarantees without restrictive invariance assumptions on the dictionary, while explicitly characterizing how approximation error $\varepsilon$ (and thus the fill distance $h_{\mathcal{X}}$) affects the convergence neighborhood. Numerical experiments on a discrete nonlinear oscillator illustrate PAS and show how the neighborhood size improves as the data grid becomes denser or the horizon increases, validating the theoretical results.

Abstract

Extended dynamic mode decomposition (EDMD) is a popular data-driven method to predict the action of the Koopman operator, i.e., the evolution of an observable function along the flow of a dynamical system. In this paper, we leverage a recently-introduced kernel EDMD method for control systems for data-driven model predictive control. Building upon pointwise error bounds proportional in the state, we rigorously show practical asymptotic stability of the origin w.r.t. the MPC closed loop without stabilizing terminal conditions. The key novelty is that we avoid restrictive invariance conditions. Last, we verify our findings by numerical simulations.

Kernel EDMD for data-driven nonlinear Koopman MPC with stability guarantees

TL;DR

The paper addresses stability in data-driven nonlinear MPC by using kernel EDMD (kEDMD) to approximate the Koopman operator within a reproducing kernel Hilbert space. It leverages RKHS-based finite-data error bounds to construct a data-driven surrogate and proves practical asymptotic stability of the origin for the MPC closed loop without stabilizing terminal conditions, relying on cost controllability and a relaxed dynamic programming inequality. A key theoretical advance is achieving stability guarantees without restrictive invariance assumptions on the dictionary, while explicitly characterizing how approximation error (and thus the fill distance ) affects the convergence neighborhood. Numerical experiments on a discrete nonlinear oscillator illustrate PAS and show how the neighborhood size improves as the data grid becomes denser or the horizon increases, validating the theoretical results.

Abstract

Extended dynamic mode decomposition (EDMD) is a popular data-driven method to predict the action of the Koopman operator, i.e., the evolution of an observable function along the flow of a dynamical system. In this paper, we leverage a recently-introduced kernel EDMD method for control systems for data-driven model predictive control. Building upon pointwise error bounds proportional in the state, we rigorously show practical asymptotic stability of the origin w.r.t. the MPC closed loop without stabilizing terminal conditions. The key novelty is that we avoid restrictive invariance conditions. Last, we verify our findings by numerical simulations.
Paper Structure (6 sections, 5 theorems, 49 equations, 2 figures, 1 algorithm)

This paper contains 6 sections, 5 theorems, 49 equations, 2 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Kernel EDMD and Control
  3. MPC via kernel EDMD
  4. Stability of kEDMD-MPC
  5. Numerical simulations
  6. Conclusions

Key Result

Theorem 1

Let $\mathbb{H}$ be the RKHS on $\Omega$ generated by the Wendland kernels with smoothness degree $k \in \mathbb N$. Further, let $\mathcal{X} :=\{ x_i \mid i \in [1:d] \} \subset \Omega$ be a set of finitely-many pairwise-distinct data points and the right-hand side ${F \in C^p(\overline{\Omega},\m holds for all fill distances $h_{\mathcal{X}}$ satisfying $h_\mathcal{X} \leq h_0$.

Figures (2)

  • Figure 1: System \ref{['eq:van der Pol']}: Error $\|x(k)\|$ of the kEDMD-MPC closed-loop with horizon $N = 10$ for vaying $d$ and $\varepsilon_c$.
  • Figure 2: System \ref{['eq:van der Pol']}: Error $\|x(k)\|$ of the kEDMD-MPC closed-loop with horizon $N = 20$ with varying $\varepsilon_c$.

Theorems & Definitions (10)

  • Remark 1: Wendland kernels
  • Theorem 1
  • Theorem 2: Approximation error
  • proof
  • Remark 2
  • Definition 1
  • Definition 2
  • Lemma 1: Uniform continuity of surrogate $F^\varepsilon$
  • Proposition 1: Cost controllability
  • Theorem 3