Random Features Approximation for Control-Affine Systems

TL;DR

This paper introduces two novel random feature-based methods for modeling control-affine systems, enabling flexible, efficient nonlinear representations suitable for control tasks, demonstrated through simulation on a double pendulum.

Contribution

The paper proposes two new random feature classes that capture control-affine structure with arbitrary complexity, linking them to existing kernels and demonstrating their utility in control applications.

Findings

Methods effectively approximate control-affine dynamics

Simulation shows improved control performance

New kernels enhance expressiveness of nonlinear models

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 Casta\~neda 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.