Control Occupation Kernel Regression for Nonlinear Control-Affine Systems

TL;DR

This paper tackles learning unknown high-order control-affine dynamics of the form $\frac{d^s x}{dt^s} = f(x) + g(x) u$ from observed control signals and trajectories by embedding controlled motions in a vector-valued RKHS through higher-order occupation kernels. The proposed Control Occupation Kernel Regression (COKR) converts the infinite-dimensional regularized regression into a finite-dimensional problem via the representer theorem, enabling simultaneous estimation of drift and control-effectiveness components. The paper develops a functional ridge regression (FRR) framework and proves cumulative error bounds, showing that COKR generalizes kernel ridge regression to control-affine settings and inherits favorable convergence properties. Numerical experiments on a Duffing oscillator and a two-link robot manipulator demonstrate robustness to measurement noise, accurate recovery of $f$ and $g$, and practically competitive control performance when used in a computed-torque controller. Overall, the work provides a principled, kernel-based approach for data-driven system identification of complex, nonlinear plants with potential impact on control design and real-time identification.

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.

Figures (6)

• Figure 1: The blue circle marks are the mean of $\left\vert \tilde{f}(x) \right\vert$ and the red square marks are the mean of $\left\vert \tilde{g}(x) \right\vert$ over $x\in[-3,3]$ for different values of $\lambda$ using COKR trained with noisy trajectories.
• Figure 2: The evaluation of $f$ and $\hat{f}$ (left) and $g$ and $\hat{g}$ (right) for the Duffing oscillator with measurement noise. The solid blue lines represent the true values of the functions and the dotted red lines represent the COKR estimates.
• Figure 3: The evaluation of $\left\vert \tilde{f} \right\vert$ (left) and $\left\vert \tilde{g} \right\vert$ (right) for the Duffing oscillator with measurement noise.
• Figure 4: Monte-Carlo results of the max of $\left\vert \tilde{f} \right\vert$ (left) and $\left\vert \tilde{g} \right\vert$ (right) over 1000 trials.
• Figure 5: The relative errors $\frac{\left\Vert \tilde{f}(x) \right\Vert}{\left\Vert f(x) \right\Vert}$ (top), $\frac{\Vert \tilde{g}_1(x) \Vert}{\Vert g_1(x) \Vert}$ (bottom left), and $\frac{\Vert \tilde{g}_2(x) \Vert}{\Vert g_2(x) \Vert}$ (bottom right) evaluated at 100 points, indexed by decreasing distance from the origin.

Theorems & Definitions (23)

• Definition 1
• Proposition 1
• proof
• Proposition 2
• proof
• Proposition 3
• proof
• Proposition 4
• proof
• Proposition 5