Nonparametric adaptive control in native spaces: Finite-dimensional implementations, Part II

Abstract

This two-part work presents a novel theory for model reference adaptive control (MRAC) of deterministic nonlinear ordinary differential equations (ODEs) that contain functional, nonparametric uncertainties that reside in a native space, also called a reproducing kernel Hilbert space (RKHS). As discussed in the first paper of this two-part work, the proposed framework relies on a limiting distributed parameter system (DPS). To allow implementations of this framework in finite dimensions, this paper shows how several techniques developed in parametric MRAC, such as the $\sigma$-modification method, the deadzone modification, adaptive error bounding methods, and projection methods, can be generalized to the proposed nonparametric setting. Some of these techniques assure uniform ultimate boundedness of the trajectory tracking error, while others guarantee its asymptotic convergence to zero. This paper introduces nonparametric metrics of performance that are cast in terms of the functional uncertainty classes in the native space. These performance metrics are relative to the best offline approximation error of the functional uncertainty. All the provided performance bounds are explicit in the dimension of the approximations of the functional uncertainty. Numerical examples show the applicability of the proposed theoretical results.