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Robust Offset-free Kernelized Data-Driven Predictive Control for Nonlinear Systems

Mahmood Mazare, Hossein Ramezani

TL;DR

The paper develops a robust offset-free kernelized data-driven predictive control (KDPC) framework for nonlinear systems. It blends kernel ridge regression in an RKHS with a linearized kernel map to predict future outputs, enabling a standard Quadratic Program (QP) for real-time control, and achieves offset-free tracking by optimizing input increments with a slack term. Theoretical guarantees of recursive feasibility and asymptotic stability are established under incremental stability and kernel approximation assumptions, and the approach is validated on a nonlinear Van der Pol oscillator showing disturbance rejection where a model-based NMPC fails. This work provides a scalable, data-driven alternative to model-based controllers with strong robustness and real-time feasibility for unknown nonlinear dynamics.

Abstract

This paper proposes a novel Kernelized Data-Driven Predictive Control (KDPC) scheme for robust, offset-free tracking of nonlinear systems. Our computationally efficient hybrid approach separates the prediction: (1) kernel ridge regression learns the nonlinear map from past trajectories, and (2) analytical linearization of the kernel map approximates the effect of future inputs. This linearization is key, allowing the controller to be formulated as a standard Quadratic Program (QP) for efficient real-time implementation. Offset-free tracking is inherently achieved by using input increments. We provide theoretical guarantees for recursive feasibility and asymptotic stability. The algorithm is validated on a nonlinear Van der Pol oscillator, where it successfully rejects unmeasured disturbances and eliminates steady-state errors, outperforming a standard model-based controller.

Robust Offset-free Kernelized Data-Driven Predictive Control for Nonlinear Systems

TL;DR

The paper develops a robust offset-free kernelized data-driven predictive control (KDPC) framework for nonlinear systems. It blends kernel ridge regression in an RKHS with a linearized kernel map to predict future outputs, enabling a standard Quadratic Program (QP) for real-time control, and achieves offset-free tracking by optimizing input increments with a slack term. Theoretical guarantees of recursive feasibility and asymptotic stability are established under incremental stability and kernel approximation assumptions, and the approach is validated on a nonlinear Van der Pol oscillator showing disturbance rejection where a model-based NMPC fails. This work provides a scalable, data-driven alternative to model-based controllers with strong robustness and real-time feasibility for unknown nonlinear dynamics.

Abstract

This paper proposes a novel Kernelized Data-Driven Predictive Control (KDPC) scheme for robust, offset-free tracking of nonlinear systems. Our computationally efficient hybrid approach separates the prediction: (1) kernel ridge regression learns the nonlinear map from past trajectories, and (2) analytical linearization of the kernel map approximates the effect of future inputs. This linearization is key, allowing the controller to be formulated as a standard Quadratic Program (QP) for efficient real-time implementation. Offset-free tracking is inherently achieved by using input increments. We provide theoretical guarantees for recursive feasibility and asymptotic stability. The algorithm is validated on a nonlinear Van der Pol oscillator, where it successfully rejects unmeasured disturbances and eliminates steady-state errors, outperforming a standard model-based controller.

Paper Structure

This paper contains 16 sections, 4 theorems, 27 equations, 2 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Problem Formulation and Kernel Setup
  3. Reproducing Kernel Hilbert Spaces (RKHS)
  4. Data Collection and Persistence of Excitation
  5. Kernel Design and Properties
  6. Kernelized Predictor Formulation
  7. Past Data Predictor $\mathbf{P}_1^{\mathrm{K}}$
  8. Future Input Predictor $\mathbf{P}_2^{\mathrm{K}}$
  9. Offset-free Control Formulation
  10. Stability and Feasibility Analysis
  11. Assumptions for Stability Analysis
  12. Lyapunov Stability Analysis
  13. Main Stability Theorems
  14. Numerical Validation and Case Studies
  15. Computational Complexity Analysis
  16. ...and 1 more sections

Key Result

Lemma 1

The optimal cost $J^*(k)$ serves as a Lyapunov function candidate satisfying: for some $\alpha_1, \alpha_2, \alpha_3 > 0$ and $\epsilon_v > 0$ representing approximation errors.

Figures (2)

  • Figure 1: Comparison of the robust offset-free KDPC and standard model-based NMPC in tracking a piecewise reference $y_r$ (dashed black line) under the influence of a piecewise constant input disturbance $d(k)$. The KDPC successfully rejects the disturbance, while the NMPC fails, resulting in a persistent steady-state error.
  • Figure 2: Comparison of the robust offset-free KDPC and standard model-based NMPC in tracking a piecewise reference $y_r$ (dashed black line) under the influence of a piecewise constant output disturbance $d(k)$. The KDPC successfully rejects the disturbance, while the NMPC fails, resulting in a persistent steady-state error.

Theorems & Definitions (9)

  • Definition 1: Persistence of Excitation in RKHS
  • Remark 1
  • Lemma 1: Lyapunov Function Candidate
  • proof
  • Lemma 2: Small Disturbance Bound
  • Theorem 1: Recursive Feasibility
  • proof
  • Theorem 2: Asymptotic Stability
  • proof