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Kernel-based Koopman approximants for control: Flexible sampling, error analysis, and stability

Lea Bold, Friedrich M. Philipp, Manuel Schaller, Karl Worthmann

TL;DR

The paper tackles stability guarantees for data-driven Koopman surrogates of nonlinear dynamics using kernel EDMD. By introducing regularized and proportional error bounds and a flexible macro/micro sampling scheme, it proves that stability properties (and their preservation under surrogate models) transfer between the ground-truth system and the data-driven surrogate, including control-affine cases. The authors develop a two-layer sampling architecture to mitigate kernel conditioning and establish open-loop and closed-loop stability with numerical demonstrations. These results enable end-to-end stability assurances for data-driven controllers derived from surrogate models, with robustness to noise and sampling choices. The framework provides a rigorous foundation for data-driven control design using Koopman-based surrogates in high-dimensional settings.

Abstract

Data-driven techniques for analysis, modeling, and control of complex dynamical systems are on the uptake. Koopman theory provides the theoretical foundation for the popular kernel extended dynamic mode decomposition (kEDMD). In this work, we propose a novel kEDMD scheme to approximate nonlinear control systems accompanied by an in-depth error analysis. Key features are regularization-based robustness and an adroit decomposition into micro and macro grids enabling flexible sampling. But foremost, we prove proportionality, i.e., explicit dependence on the distance to the (controlled) equilibrium, of the derived bound on the full approximation error. Leveraging this key property, we rigorously show that asymptotic stability of the data-driven surrogate (control) system implies asymptotic stability of the original (control) system and vice versa.

Kernel-based Koopman approximants for control: Flexible sampling, error analysis, and stability

TL;DR

The paper tackles stability guarantees for data-driven Koopman surrogates of nonlinear dynamics using kernel EDMD. By introducing regularized and proportional error bounds and a flexible macro/micro sampling scheme, it proves that stability properties (and their preservation under surrogate models) transfer between the ground-truth system and the data-driven surrogate, including control-affine cases. The authors develop a two-layer sampling architecture to mitigate kernel conditioning and establish open-loop and closed-loop stability with numerical demonstrations. These results enable end-to-end stability assurances for data-driven controllers derived from surrogate models, with robustness to noise and sampling choices. The framework provides a rigorous foundation for data-driven control design using Koopman-based surrogates in high-dimensional settings.

Abstract

Data-driven techniques for analysis, modeling, and control of complex dynamical systems are on the uptake. Koopman theory provides the theoretical foundation for the popular kernel extended dynamic mode decomposition (kEDMD). In this work, we propose a novel kEDMD scheme to approximate nonlinear control systems accompanied by an in-depth error analysis. Key features are regularization-based robustness and an adroit decomposition into micro and macro grids enabling flexible sampling. But foremost, we prove proportionality, i.e., explicit dependence on the distance to the (controlled) equilibrium, of the derived bound on the full approximation error. Leveraging this key property, we rigorously show that asymptotic stability of the data-driven surrogate (control) system implies asymptotic stability of the original (control) system and vice versa.

Paper Structure

This paper contains 18 sections, 19 theorems, 85 equations, 5 figures, 1 table, 1 algorithm.

Table of Contents

  1. Introduction
  2. Koopman operator and kernel EDMD
  3. Kernel EDMD (kEDMD)
  4. Approximation error for regularized kEDMD: uniform bounds
  5. Data-driven surrogate dynamics
  6. Data-driven surrogates: Lyapunov stability
  7. Inheritance of stability properties
  8. Proportional bounds and asymptotic stability
  9. Example and numerical simulations
  10. Koopman approximants for control systems
  11. Koopman approximants, flexible sampling, and error bounds
  12. Numerical example: Open-loop error bound
  13. Feedback stabilization
  14. Conclusions and outlook
  15. Results for the alternative surrogate
  16. ...and 3 more sections

Key Result

Theorem 2.4

Let $k\geq 1$, $\lambda \geq 0$ and $F\in C_b^{\lceil \sigma_{n,k} \rceil}(\Omega;\mathbb R^{n})$. Then there are constants $C,h_0>0$ such that for any finite set $\mathcal{X} = \{x_i\}_{i=1}^d \subset\Omega$ of pairwise distinct sample points with $h_{\mathcal{X}}\le h_0$ and for all $f\in\mathcal{ where the RKHS $\mathcal{N}_{\Phi_{n,k}}(\Omega)$ (cf. ex:wend) is generated by Wendland kernels.

Figures (5)

  • Figure 1: Approximation error $\|\widehat{F}(x) - F(x)\|_2$ for system \ref{['eq:Kellett']} with $d \in \{441, 1681, 6561\}$ data points for a uniform (mesh sizes $\delta \in \{0.2, 0.1, 0.05\}$; top ) and a Padua grid (middle ), $d \in \{436, 1654, 6556\}$. Bottom: Error $\|\widehat{F}_\lambda(x) - F(x)\|_2$ with regularization parameter $\lambda = 0.01$.
  • Figure 2: Left: $\| \widehat{F}_\lambda(x(k; x^0_j)) - F(x(k; x^0_j)) \|_2$ along trajectories of \ref{['eq:Kellett']} for $r = 1.9$, $\tilde{r} = 0.95 \sqrt{2}$. Right: Lyapunov decrease $V(x)-V(\widehat{F}(x)) - \alpha_V(\|x\|)$ for system \ref{['eq:Kellett']} with the kEDMD surrogate $\widehat{F}$.
  • Figure 3: Trajectories $x(\cdot;x^0,u)$ of system \ref{['eq: Duffing']} and its kEDMD surrogate $\widehat{x}(\cdot; x^0, u)$ generated with \ref{['alg:cap']} (left ) and their deviation $\| x(k; x^0, u) - \widehat{x}(k; x^0, u)\|_2$, $k \in [1:50]$, in norm w.r.t. a threshold $10^{-{3}}$ (middle ) for initial value $x^0 = (0.1, 0.1)^\top$ for the control sequence $u$ (right ).
  • Figure 4: Approximation error $\max_{j \in [1:20]}\|f(x, u_j) - \widehat{f}(x, u_j)\|$ for the predicted system \ref{['eq: Duffing']} with $d \in \{435, 1653, 6555\}$ cluster points using a Padua-based grid (from left to right ).
  • Figure 5: Left: KEDMD approximants $\widehat{x}(\cdot; x^0, 0)$ (open loop) and $\widehat{x}_\kappa(\cdot; x^0)$ (closed loop) of system \ref{['eq:kanzantzis']}, initial value $x^0 = (0.5, 0.1)^\top$, generated with \ref{['alg:cap']} based on the Padua grid with $d = 435$ cluster points. Deviations $\| x(k; x^0, 0) - \widehat{x}(k; x^0, 0)\|_2$ and $\| x_\kappa(k; x^0) - \widehat{x}_\kappa(k; x^0)\|_2$, $k \in [1:100]$, in open-(mid) and closed-loop (right) control for $d \in \{435, 1653, 6555, 11476\}$.

Theorems & Definitions (45)

  • Remark 2.1: Differential equation
  • Example 2.2: RKHS $\mathbb{H}$ generated by Wendland kernels: $\mathcal{N}_{\Phi_{n,k}}$
  • Remark 2.3
  • Theorem 2.4
  • Lemma 2.5
  • proof
  • proof : Proof of \ref{['thm:error:regularization']}
  • Remark 2.6
  • Remark 2.7
  • Corollary 2.8
  • ...and 35 more