Data-driven Kernel-based Predictive Control with Stability and Robustness Guarantees

Technical Summary: Data-driven Kernel-based Predictive Control with Stability and Robustness Guarantees

Problem Formulation
The paper addresses the control of unknown nonlinear discrete-time systems using only input-output data. While Model Predictive Control (MPC) is effective for handling constraints and nonlinearities, its reliance on accurate parametric models limits its application to complex, high-dimensional systems where modeling is costly or infeasible. Existing data-driven predictive control (DDPC) methods, largely based on Willems' fundamental lemma, are primarily restricted to linear time-invariant (LTI) systems. Although recent extensions attempt to handle nonlinearities via local linearization, neural networks, or kernel methods, these approaches often lack rigorous theoretical guarantees regarding recursive feasibility and closed-loop stability, particularly when terminal sets and costs are omitted. The core challenge is to develop a data-driven control framework for unknown nonlinear systems that provides formal stability guarantees without requiring explicit system identification or terminal ingredients.

Methodology
The proposed framework, termed Data-driven Kernel-based Predictive Control (DDKPC), constructs a multi-step predictor directly from historical input-output trajectories using kernel methods and the representer theorem.

1. Kernel-based Multi-step Predictor: The authors assume the system's input-output behavior can be represented in a Reproducing Kernel Hilbert Space (RKHS). Instead of learning a state-space model, they approximate the unknown nonlinear mapping from initial conditions and future inputs to future outputs. This is achieved via a kernel-based estimator fitted to offline data. The predictor implicitly captures nonlinear dynamics, and the representer theorem ensures the estimator takes a finite-dimensional linear form in the feature space, allowing for explicit error bounds.
2. Nominal Control Scheme: For noise-free data, the control problem is formulated as an optimization that minimizes a quadratic stage cost subject to an implicit equality constraint derived from the kernel predictor. To handle the inevitable mismatch between the kernel approximation and the true system, a slack variable is introduced into the constraints, penalized in the objective function. Theoretical analysis proves that if the prediction horizon is sufficiently long and the kernel representation error is sufficiently small, the scheme guarantees recursive feasibility and practical closed-loop stability. Crucially, these guarantees are achieved without conventional terminal sets or terminal costs.
3. Computational Relaxation: Recognizing that the implicit kernel constraint renders the optimization problem nonconvex, the authors introduce a penalty relaxation formulation. This converts the hard equality constraint into a quadratic penalty term within the objective function, enabling the use of efficient first-order optimization methods (e.g., projected gradient descent) for real-time implementation.
4. Robustness to Noise: To address measurement noise in both offline data collection and online operation, the framework aggregates the kernel representation error and the bounded measurement noise into a unified uncertainty bound. The control formulation is modified to rescale regularization weights based on this bound, ensuring robust stability.
5. Extension to Time-Varying Systems: For slowly time-varying nonlinear systems, where a fixed offline dictionary becomes obsolete, the authors propose an online extension. This involves maintaining a fixed-budget online dictionary managed by the Approximate Linear Dependency (ALD) criterion. New data points are admitted based on their informational novelty, and redundant points are pruned. The kernel predictor is reconstructed periodically from this evolving dictionary. Theoretical analysis demonstrates that recursive feasibility and practical stability are preserved under bounded rates of system variation and uniform online prediction errors.

Key Contributions
The paper makes three primary contributions:

• Nominal DDKPC Framework: It establishes a DDKPC framework for unknown nonlinear systems that provides provable recursive feasibility and practical stability without relying on terminal ingredients. It utilizes a penalty relaxation strategy to facilitate efficient real-time computation.
• Robustness Extension: It systematically extends the theoretical guarantees to accommodate measurement noise by aggregating data uncertainty and representation mismatch into a unified bound, ensuring stability in noisy environments.
• Time-Varying Adaptation: It extends the framework to slowly time-varying nonlinear systems via an ALD-managed online dictionary and periodic predictor updates. It proves that recursive feasibility and practical stability are retained under bounded-rate evolution and uniform online prediction-error conditions.

Results
The effectiveness of the proposed approach is validated through numerical examples:

• One-dimensional Vehicle Benchmark: Simulations on a nonlinear vehicle model demonstrate that the kernel-based DDKPC successfully stabilizes the system and tracks references, whereas standard linear DDPC fails. The results show that Gaussian kernels outperform polynomial kernels in steady-state accuracy. Furthermore, the use of first-order solvers (FastGD) achieves closed-loop performance comparable to general-purpose nonlinear programming (NLP) solvers but with significantly reduced computational time. The online DDKPC with periodic dictionary updates effectively mitigates tracking bias caused by slowly time-varying parameters, outperforming static offline designs.
• Quadruped Robot Simulation: A PyBullet simulation of an A1 quadruped robot carrying an off-center payload illustrates the method's applicability to contact-rich, nonlinear locomotion systems. The DDKPC acts as a residual velocity controller, compensating for payload-induced dynamics. Results show significant reductions in tracking Root Mean Square Error (RMSE) compared to a nominal whole-body controller. The online DDKPC variant adapts more effectively to sudden payload changes than the static offline version.

Significance
The paper claims to bridge the gap between theoretical data-driven control and practical application for nonlinear systems. By leveraging the representer theorem, the method circumvents the need for explicit parametric modeling while providing formal stability guarantees that are often absent in existing kernel- or neural-network-based approaches. The work addresses critical implementation hurdles, including the computational burden of nonconvex optimization and the presence of measurement noise, and extends the theory to time-varying scenarios. The authors emphasize that their approach offers a systematic way to synthesize stabilizing receding-horizon controllers purely from input-output data for a broad class of nonlinear systems, without requiring full state measurements or complex terminal ingredients.