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 [1] and Hsue operator [2], second order Schneider operator [3] and third order Al-Alaoui - Schneider-Kaneshige-Groutage (ALSKG) rule [4, 5] 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.