Random Features Approximation for Control-Affine Systems

TL;DR

The paper introduces two random-feature representations—ADP-RF and AD-RF—that preserve the control-affine structure in nonlinear dynamical systems while allowing flexible state-dependent modelling. By tying these RF bases to the Affine Dot Product (ADP) kernel and a novel Affine Dense (AD) kernel, the authors provide RKHS-backed guarantees and scalable alternatives to kernel methods for data-driven control. They demonstrate utility through a case study using control certificate functions (CCFs) for safety/stability, including both certainty-equivalent and robust formulations, and validate performance with a nonlinear double-pendulum example. The work offers promising avenues for efficient, uncertainty-aware control synthesis in data-driven settings and motivates kernel/RF-tailored kernels for specific control tasks.

Abstract

Modern data-driven control applications call for flexible nonlinear models that are amenable to principled controller synthesis and realtime feedback. Many nonlinear dynamical systems of interest are control affine. We propose two novel classes of nonlinear feature representations which capture control affine structure while allowing for arbitrary complexity in the state dependence. Our methods make use of random features (RF) approximations, inheriting the expressiveness of kernel methods at a lower computational cost. We formalize the representational capabilities of our methods by showing their relationship to the Affine Dot Product (ADP) kernel proposed by Castañeda et al. (2021) and a novel Affine Dense (AD) kernel that we introduce. We further illustrate the utility by presenting a case study of data-driven optimization-based control using control certificate functions (CCF). Simulation experiments on a double pendulum empirically demonstrate the advantages of our methods.

Figures (3)

• Figure 1: Evaluation of models, comparing prediction accuracy on test data (left) and training time on 8859 points (right) for a prediction problem on a double pendulum system. Horizontal lines and markers correspond to kernel methods. Random features are sampled 10 times at varying dimensions; the left panel displays median and quartiles over the trials while the right panel shows the mean. Increasing the feature dimension of each state-dependent basis ${\boldsymbol{\psi}}_i(x)$ results in lower RMSE but longer training time, especially for ADP-RF.
• Figure 2: Left: The value of the Lyapunov function $C({\bm{x}})$ over time for nominal, oracle, and data-driven controllers with initial state $[2,0,0,0]$. Right: Illustration of the pendulum configurations over time. Nominal fails to balance the pendulum; data-driven methods succeed.
• Figure 3: Evaluation of models, comparing prediction accuracy on 100 test data points against average training time on 900 data points for $m=1,10,20$ respectively. Random features are sampled 10 times at matching compound dimensions; the panels display median and quartile RMSE over the trials. Increasing $m$ demonstrates the advantage of AD-RF basis over ADP-RF in RMSE for fixed train-time.

Theorems & Definitions (13)

• definition 1: Control-affine modelling problem
• lemma 1
• definition 2: Affine dot product (ADP) bases
• definition 3: Affine dense (AD) bases
• definition 4: Affine dot product (ADP) kernel
• theorem 1: ADP Approximation
• definition 5: Affine dense (AD) kernel
• theorem 2: AD kernel
• theorem 3: Affine-dense kernel approximation
• proposition 1: Kernel Approximation Errors