First and higher order operator based fractional order differentiator and integrator models

In this paper, new discretized models of fractional order differentiator (FOD) (s^r) and fractional order integrator (FOI) (s^-r) based on first and higher order operators are proposed. Specifically in this work, one-third (s^±1/3) and one-fourth (s^±1/4) order differentiator and integrator models based on first order Al-Alaoui and Hsue operator, second order Schneider operator and third order Al-Alaoui - Schneider-Kaneshige-Groutage (ALSKG) rule have been derived. The stability of the proposed models has been investigated and the unstable ones stabilized by the pole reflection method. Performance results using the proposed discrete-time formulations are found to converge to the analytical results of fractional order differentiator and integrator, in the continuous-time domain. MATLAB simulation results show that the responses of the fractional differentiators and integrators match with the results of the theoretical results of the continuous-time domain fractional differentiators and integrators.