Sufficient Conditions for Parameter Convergence Over Embedded Manifolds Using Kernel Techniques

Abstract

The persistence of excitation (PE) condition is sufficient to ensure parameter convergence in adaptive estimation problems. Recent results on adaptive estimation in reproducing kernel Hilbert spaces (RKHS) introduce PE conditions for RKHS. This paper presents sufficient conditions for PE for the particular class of uniformly embedded reproducing kernel Hilbert spaces (RKHS) defined over smooth Riemannian manifolds. This paper also studies the implications of the sufficient condition in the case when the RKHS is finite or infinite-dimensional. When the RKHS is finite-dimensional, the sufficient condition implies parameter convergence as in the conventional analysis. On the other hand, when the RKHS is infinite-dimensional, the same condition implies that the function estimate error is ultimately bounded by a constant that depends on the approximation error in the infinite-dimensional RKHS. We illustrate the effectiveness of the sufficient condition in a practical example.

Kernel center adaptation in the reproducing kernel Hilbert space embedding method

“…Theorem 5 (6,31). Suppose the RKHS is generated by a strictly positive definite kernel and the hypotheses of Reference 31 hold.…”

“…Notice that this ultimate bound actually implies a stronger result than the one in the conventional analysis in Euclidean space: here, the ultimate bound is explicit in terms of the approximation space error. 31…”

“…However, there is a much more practical and intuitive way for selecting the kernel centers in the set when the RKHS is generated by a strictly positive definite kernel. For the precise hypothesis of the theorem below, the reader should see Reference 31: the context of the following theorem is rather detailed.…”

“…Studies on PE in RKHS show that only the kernel centers regularly visited by the state trajectory are persistently excited. 6,31 Such a relationship makes it easier to ensure PE in RKHS for a certain class of dynamical systems. Furthermore, recent results have shown that the persistently exciting sets are contained in the positive limit sets for a particular class of RKHS adaptive estimators.…”

“…However, in the RKHS embedding method, the modified PE conditions, studied in References 29,30, are directly related to the kernel center positions in the state‐space. Studies on PE in RKHS show that only the kernel centers regularly visited by the state trajectory are persistently excited 6,31 . Such a relationship makes it easier to ensure PE in RKHS for a certain class of dynamical systems.…”