Control Occupation Kernel Regression for Nonlinear Control-Affine Systems

TL;DR

This paper introduces a kernel-based method for approximating nonlinear control-affine systems by embedding trajectories into a vector-valued RKHS, enabling finite-dimensional system identification.

Contribution

It develops a novel control occupation kernel framework that transforms infinite-dimensional system identification into a finite regularized regression problem.

Findings

Effective approximation of nonlinear control systems demonstrated

Simultaneous estimation of drift and control effectiveness components

Finite-dimensional optimization achieved via the representer theorem

Abstract

This manuscript presents an algorithm for obtaining an approximation of a nonlinear high order control affine dynamical system. Controlled trajectories of the system are leveraged as the central unit of information via embedding them in vector-valued reproducing kernel Hilbert space (vvRKHS). The trajectories are embedded as the so-called higher order control occupation kernels which represent an operator on the vvRKHS corresponding to iterated integration after multiplication by a given controller. The solution to the system identification problem is then the unique solution of an infinite dimensional regularized regression problem. The representer theorem is then used to express the solution as finite linear combination of these occupation kernels, which converts an infinite dimensional optimization problem to a finite dimensional optimization problem. The vector valued structure of the Hilbert space allows for simultaneous approximation of the drift and control effectiveness components of the control affine system. Several experiments are performed to demonstrate the effectiveness of the developed approach.