Tube-Based Model Predictive Control with Random Fourier Features for Nonlinear Systems

TL;DR

The paper tackles robust MPC for nonlinear systems under constraints by integrating Random Fourier Features (RFF) with tube-based MPC to achieve scalable, deterministic robustness. The core idea is to learn a nonlinear residual on top of a physics-based linear baseline via RFF, yielding a hybrid model with a provable bound $d_{\max}$ on the approximation error and a per-iteration cost of $O(D(n+m))$ independent of training size. The method tightens constraints and propagates tubes using the learned residual, achieving substantially smaller tube sizes and tighter performance bounds while maintaining recursive feasibility through a terminal set and Lyapunov-based design. Demonstrated on a nonlinear bicycle model for autonomous path tracking, the approach yields order-of-magnitude improvements in uncertainty and tracking accuracy, with real-time computational feasibility on standard solvers.

Abstract

This paper presents a computationally efficient approach for robust Model Predictive Control of nonlinear systems by combining Random Fourier Features with tube-based MPC. Tube-based Model Predictive Control provides robust constraint satisfaction under bounded model uncertainties arising from approximation errors and external disturbances. The Random Fourier Features method approximates nonlinear system dynamics by solving a numerically tractable least-squares problem, thereby reducing the approximation error. We develop the integration of RFF-based residual learning with tube MPC and demonstrate its application to an autonomous vehicle path-tracking problem using a nonlinear bicycle model. Compared to the linear baseline, the proposed method reduces the tube size by approximately 50%, leading to less conservative behavior and resulting in around 70% smaller errors in the test scenario. Furthermore, the proposed method achieves real-time performance while maintaining provable robustness guarantees.

Figures (4)

• Figure 1: Tube size evolution showing adaptive uncertainty quantification: RFF-Based (), Traditional ().
• Figure 2: Steering angle command: RFF-Based (), Traditional ().
• Figure 3: Lateral position error ($e_y$): RFF-Based (), Traditional ().
• Figure 4: Heading angle error ($e_{\psi}$) demonstrating reduced orientation deviation: RFF-Based (), Traditional ().