State Following (StaF) Kernel Functions for Function Approximation

TL;DR

This paper introduces the State Following (StaF) kernel method for localized function approximation, reducing the number of basis functions needed and maintaining accuracy as the state moves within a compact set, with applications in adaptive dynamic programming.

Contribution

The paper develops a novel StaF kernel approach supported by theoretical bounds and demonstrates its effectiveness in reducing basis functions in adaptive dynamic programming applications.

Findings

Fewer basis functions are needed for accurate local approximation.

The method guarantees stability and near-optimality in control applications.

Simulation results confirm the efficiency of the StaF approach.

Abstract

A function approximation method is developed that aims to approximate a function in a small neighborhood of a state that travels within a compact set. The development is based on the theory of universal reproducing kernel Hilbert spaces over the $n$-dimensional Euclidean space. Several theorems are introduced that 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. Additionally, 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 6.1. To illustrate the advantage, the impact of the StaF method is that for some applications the number of basis functions can be reduced. The StaF method is applied to an adaptive dynamic programming (ADP) application to demonstrate that stability is maintained with a reduced number of basis functions. Simulation results demonstrate the utility of the StaF methodology for the maintenance of accurate function approximation as well as solving an infinite horizon optimal regulation problem through ADP. The results of the simulation indicate that fewer basis functions are required to guarantee stability and approximate optimality than are required when a global approximation approach is used.