Series expansion design of variable fractional order integrator and differentiator using logarithm

In this paper, the design problems of variable fractional order integrator and differentiator (FOID) are investigated. First, the transfer function of FOID is obtained by taking fractional power of the transfer function of conventional first order integrator and differentiator. Then, to implement this irrational transfer function, the logarithm and Taylor series expansion are used to get a realizable approximated rational function. The proposed implementation structure is similar to the conventional Farrow structure of fractional sample delay filter. Next, the proposed approach is applied to design fractional rectangular integrator, fractional trapezoidal integrator, fractional Simpson integrator, fractional Al-Alaoui differentiator and fractional maximally flat differentiator. Finally, design examples are demonstrated to illustrate the performance of the proposed design method.