Digital fractional-order differentiator and integrator models based on first-order and higher order operators

Abstract

In this paper, new discretized models for fractional-order differentiator (FOD) (sr) and integrator (FOI) (s-r) using first-order and higher order operators are proposed. The expansions for FOIs of the first-order s-to-z transform proposed by Hsue et al. are obtained by using the Taylor series and continued fraction expansion techniques. Second-order Schneider operator and third-order Al-Alaoui-Schneider-Kaneshige-Groutage (Al-Alaoui-SKG) rule have also been fractionalized to obtain expansions of FODs by using the Taylor series. Specifically, in this paper, Hsue operator based on third-order and fourth-order models of FOI, Schneider operator as well as Al-Alaoui-SKG rule based on third-order, fourth-order, fifth-order and sixth-order models of FOD have been suggested. The stability of the proposed models has been investigated and the unstable ones were stabilized by the pole reflection method. The performance of the proposed models has been compared with that of recent FOD and FOI models based on the Al-Alaoui operator. Performance results using the proposed discrete-time formulations are found to converge to the analytical results of FOD and FOI, in the continuous-time domain. © 2010 John Wiley & Sons, Ltd.