The State Following Approximation Method

A function approximation method is developed which aims to approximate a function in a small neighborhood of a state that travels within a compact set. The method provides a novel approximation strategy for the efficient approximation of nonlinear functions for real-time simulations and experiments. The development is based on the theory of universal reproducing kernel Hilbert spaces over the n-dimensional Euclidean space. Several theorems are introduced which support the development of this state following (StaF) method. In particular, it is shown that there is a bound on the number of kernel functions required for the maintenance of an accurate function approximation as a state moves through a compact set. In addition, a weight update law, based on gradient descent, is introduced where arbitrarily close accuracy can be achieved provided the weight update law is iterated at a sufficient frequency, as detailed in Theorem 4. An experience-based approximation method is presented which utilizes the samples of the estimations of the ideal weights to generate a global approximation of a function. The experience-based approximation interpolates the samples of the weight estimates using radial basis functions. To illustrate the StaF method, the method is utilized for derivative estimation, function approximation, and is applied to an adaptive dynamic programming problem where it is demonstrated that the stability is maintained with a reduced number of basis functions.