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A Kernelized Operator Approach to Nonlinear Data-Enabled Predictive Control

Thomas de Jong, Siep Weiland, Mircea Lazar

TL;DR

This work addresses the scalability challenge of nonlinear data-enabled predictive control (DeePC) by introducing a kernelized operator approach based on a product RKHS. By modeling nonlinear dynamics as operators on a product Hilbert space and using a product kernel with Gram matrix $K_{\otimes}=K_u\otimes K_x$, the method learns a nonlinear predictor $G(\mathbf{u})(x)=\Theta^*\mathbf{k}_{\boxtimes}(\mathbf{u},x)$ and formulates a KerODeePC optimization that leverages this operator for multi-step predictions. A key contribution is the computationally efficient KerODeePC (and its reduced KDPC-r) that decouples input and state data, projects the optimization onto a smaller output space, and uses a model-equivalence argument to enable a one-shot solution; this yields large reductions in solve times while maintaining predictive accuracy. Numerical results on a Van der Pol oscillator demonstrate up to ~30x speedups over prior kernelized DeePC approaches and the ability to handle much larger datasets, enabling improved tracking and near-NMPC performance in nonlinear settings. Overall, the approach offers a practical pathway to scalable, data-driven nonlinear control with strong theoretical underpinnings in product RKHS operator learning.

Abstract

This paper considers the design of nonlinear data-enabled predictive control (DeePC) using kernel functions. Compared with existing methods that use kernels to parameterize multi-step predictors for nonlinear DeePC, we adopt a novel, operator-based approach. More specifically, we employ a universal product kernel parameterization of nonlinear systems operators as a prediction mechanism for nonlinear DeePC. We show that by using a product reproducing kernel Hilbert space (RKHS) to learn the system trajectories, big data sets can be handled effectively to construct the corresponding product Gram matrix. Moreover, we show that the structure of the adopted product RKHS representation allows for a computationally efficient DeePC formulation. Compared to existing methods, our approach achieves substantially faster computation times for the same data size. This allows for the use of much larger data sets and enhanced control performance.

A Kernelized Operator Approach to Nonlinear Data-Enabled Predictive Control

TL;DR

This work addresses the scalability challenge of nonlinear data-enabled predictive control (DeePC) by introducing a kernelized operator approach based on a product RKHS. By modeling nonlinear dynamics as operators on a product Hilbert space and using a product kernel with Gram matrix , the method learns a nonlinear predictor and formulates a KerODeePC optimization that leverages this operator for multi-step predictions. A key contribution is the computationally efficient KerODeePC (and its reduced KDPC-r) that decouples input and state data, projects the optimization onto a smaller output space, and uses a model-equivalence argument to enable a one-shot solution; this yields large reductions in solve times while maintaining predictive accuracy. Numerical results on a Van der Pol oscillator demonstrate up to ~30x speedups over prior kernelized DeePC approaches and the ability to handle much larger datasets, enabling improved tracking and near-NMPC performance in nonlinear settings. Overall, the approach offers a practical pathway to scalable, data-driven nonlinear control with strong theoretical underpinnings in product RKHS operator learning.

Abstract

This paper considers the design of nonlinear data-enabled predictive control (DeePC) using kernel functions. Compared with existing methods that use kernels to parameterize multi-step predictors for nonlinear DeePC, we adopt a novel, operator-based approach. More specifically, we employ a universal product kernel parameterization of nonlinear systems operators as a prediction mechanism for nonlinear DeePC. We show that by using a product reproducing kernel Hilbert space (RKHS) to learn the system trajectories, big data sets can be handled effectively to construct the corresponding product Gram matrix. Moreover, we show that the structure of the adopted product RKHS representation allows for a computationally efficient DeePC formulation. Compared to existing methods, our approach achieves substantially faster computation times for the same data size. This allows for the use of much larger data sets and enhanced control performance.

Paper Structure

This paper contains 8 sections, 2 theorems, 34 equations, 4 figures, 2 tables, 2 algorithms.

Table of Contents

  1. INTRODUCTION
  2. Preliminaries
  3. Kernelized Operator Data-Enabled Predictive Control
  4. Kernelized Operator DeePC problem formulation
  5. Data-generation for operator learning
  6. Computationally Efficient KerODeePC
  7. Illustrative example
  8. Conclusions

Key Result

Lemma IV.1

Let $\mathbf{k}_x(x) \neq \mathbf{0}$ for all $x \in \mathcal{X}$, and suppose $K_x$ and $K_u$ are full-rank matrices. Then, the system of equations eq:willems can be equivalently re-written as the reduced system of equations: where $\Omega(x) := \left( I_{T_u} \otimes \frac{\mathbf{k}^\top_{x}(x)}{\| \mathbf{k}^\top_{x}(x) \|_2^2} \right) K_{\otimes} \in \mathbb{R}^{T_u \times T_u T_x}$.

Figures (4)

  • Figure 1: Initial conditions for $T_x = 20$ and $T_x = 200$ generated using Algorithm \ref{['alg:1']}.
  • Figure 2: Comparison of $N=10$ step-ahead predictions (colored lines) from the stacked/product kernel prediction models against the true system trajectory (black line) for various initial conditions.
  • Figure 3: Reference tracking for product kernel, $T=400$ (---), stacked kernel, $T=400$ (---), product kernel, $T=10000$ (---).
  • Figure 4: Reference tracking for product kernel, $T=10000$ (---), compared with NMPC (---).

Theorems & Definitions (9)

  • Definition II.1
  • Definition II.2
  • Remark II.3
  • Remark III.1
  • Lemma IV.1
  • proof
  • Definition IV.2
  • Lemma IV.3
  • proof