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

TL;DR

The paper addresses functional uncertainty in nonparametric adaptive control and proposes maneuver vRKHS spaces defined via a low-dimensional embedded manifold ${\mathcal M}$ to mitigate the curse of dimensionality. By relating local, manifold-based approximations to global function behavior through trace and extension operators, it derives global pointwise error bounds and refined Sobolev-based interpolation bounds, yielding computational complexity that scales with the manifold dimension $\ell$ as $N(\epsilon) \sim \epsilon^{-\ell/\bar{s}}$. It also demonstrates how smoothed deadzone control laws in this maneuver setting achieve asymptotically optimal tracking guarantees (AAO) with reduced offline approximation error, and it outlines a roadmap for data-driven center management along trajectories. The results bridge deterministic nonparametric control with geometric structure of reference dynamics, offering practical pathways to scalable, reliable adaptive controllers for high-dimensional systems. The work provides a rigorous foundation for maneuver-space design and highlights open questions around dynamic center placement and real-time adaptation of the manifold-based uncertainty classes.

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.

Figures (2)

• Figure 1:
• Figure 2: This figure shows a typical controlled trajectory and centers selected. Here the controller is designed to drive the system to a fixed target reference state $x_r$. More generally, it is desired to choose the controller to drive the trajectory to track some reference trajectory $t\to x_r(t)$. As shown in (a), as time progresses, the state enters and remains in progressively smaller neighborhoods of the target. Figure (b) above illustrates that there is no correlation between centers ordinarily chosen in application of GP methods and the state trajectories, at least in the theoretical proofs of convergence. Figure (c) depicts a family of trajectory-driven centers: they are selected from the positive orbit $\gamma^+(x_0)=\bigcup_{t\geq 0}x(t).$

Theorems & Definitions (26)

• Example 1: Example for Autonomous Underwater Vehicles (AUVs)
• Theorem 1
• Theorem 2
• proof
• Remark 1
• Remark 2
• Theorem 3
• proof
• Theorem 4: The Many-Zeros Theorem, simplified from fuselier2012
• Remark 3