Relationship Between Persistent Excitation Levels and RBF Network Structures, With Application to Performance Analysis of Deterministic Learning

Based on the notion of persistent excitation (PE), a deterministic learning theory is recently proposed for RBF network-based identification of nonlinear systems. In this paper, we study the relationship between the PE levels, the structures of RBF networks and the performance of deterministic learning. Specifically, given a state trajectory generated from a nonlinear dynamical system, we investigate how to construct the RBF networks in order to guarantee sufficient PE levels (especially the level of excitation) for deterministic learning. It is revealed that the PE levels decrease with the density of neural centers, denoted by explicit formulas. As an illustration, these formulas are applied to convergence analysis of deterministic learning. We present exact theoretical conclusions that a finite and definite number of centers can achieve the same performance as global centers. In addition, a tradeoff exists between a relatively high level of excitation and the good approximation capabilities of RBF networks, which indicates that we cannot always obtain better convergence accuracy by increasing the density of centers. These results provide a new perspective for performance analysis of RBF network algorithms based on the notion of PE. Simulation studies are included to illustrate the results.