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

This paper introduces two new notions of persistence of excitation (PE) in reproducing kernel Hilbert spaces (RKHS) that can be used to establish convergence of function estimates generated by the RKHS embedding method. The two PE conditions are proven to be equivalent provided a type of uniform equicontinuity holds for the composition operator g ı→ g o x, where t ı→ x(t) is the unknown state trajectory. The paper then establishes sufficient conditions for the uniform asymptotic stability (UAS) of the error equations of RKHS embedding in term of these PE conditions. The proof is self-contained and treats the general case. Numerical examples are presented that illustrate qualitatively the convergence of the RKHS embedding method where function estimates converge over the positive limit set, a smooth, regularly embedded submanifold of the state space.