Functional Uncertainty Classes, Nonparametric Adaptive Contro Functional Uncertainty Classes for Nonparametric Adaptive Control: the Curse of Dimensionality

TL;DR

This paper introduces a new class of vector-valued reproducing kernel Hilbert spaces tailored for nonparametric adaptive control, aiming to mitigate the curse of dimensionality in high-dimensional systems.

Contribution

It develops maneuver vRKHS based on Riemannian manifolds to better represent functional uncertainty in nonparametric adaptive control.

Findings

New vRKHS framework for functional uncertainty

Addresses curse of dimensionality in high-dimensional control

Applicable to systems with manifold-structured dynamics

Abstract

This paper derives a new class of vector-valued reproducing kernel Hilbert spaces (vRKHS) defined in terms of operator-valued kernels for the representation of functional uncertainty arising in nonparametric adaptive control methods. These are referred to as maneuver or trajectory vRKHS KM in the paper, and they are introduced to address the curse of dimensionality that can arise for some types of nonparametric adaptive control strategies. The maneuver vRKHSs are derived based on the structure of a compact, l-dimensional, smooth Riemannian manifold M that is regularly embedded in the state space X = Rn, where M is assumed to approximately support the ultimate dynamics of the reference system to be tracked.