Persistence of Excitation in Uniformly Embedded Reproducing Kernel Hilbert (RKH) Spaces

Jia Guo, Sai Tej Paruchuri, Andrew J. Kurdila

Published 2020-02-19Version 1

This paper introduces two new notions of the persistence of excitation (PE) in reproducing kernel Hilbert (RKH) spaces that can be used to establish the convergence of function estimates generated by the RKH space embedding method. The equivalence of these two PE conditions is shown to hold if $\mathbb{U}(\bar{S}_1)$ is uniformly equicontinuous, where $\mathbb{U}$ is the Koopman operator and $\bar{S}_1$ is the closed unit sphere in the RKH space. The paper establishes sufficient conditions for the uniform asymptotic stability (UAS) of the error equations of RKH space embedding in terms of these PE conditions. The proof is self-contained, and treats the general case, extending the analysis of special cases studied in the authors' previous work. Numerical examples are presented that illustrate qualitatively the convergence of the RKH space embedding method where function estimates converge over the positive limit set, which is assumed to be a smooth, regularly embedded submanifold of $\mathbb{R}^d$.