High Order B-Spline Networks and Its Applications to Learning Feedforward Control

This paper presents a derivation of closed-form expression of an arbitrary high order uniform B-spline. This close-form expression is then applied to the stability analysis of a learning feedforward controller (LFFC) using a B-spline network (BSN) in frequency domain. BSN as a promising tool discussed in the existing literature is mainly limited to the 2nd-order B-splines case. Detailed derivation of high order BSN is still missing from the literature. In this paper, we propose to use a special type of matrix called "B-matrix" to facilitate the analysis and computation of high-order BSN for use with the LFFC scheme. Unlike the common recursive definition, the technique we developed is in "piecewise closed form," which is a nice feature especially suitable for the LFFC analysis and design