Optimal approximation of analog PID controllers of complex fractional-order

Complex fractional-order (CFO) transfer functions, being more generalized versions of their real-order counterparts, lend greater flexibility to system modeling. Due to the absence of commercial complex-order fractance elements, the implementation of CFO models is challenging. To alleviate this issue, a constrained optimization approach that meets the targeted frequency responses is proposed for the rational approximation of CFO systems. The technique generates stable, {minimum-phase, and real-valued coefficients based approximants}, which are not always feasible for the curve-fitting approach reported in the literature. {Stability and performance studies of the CFO proportional-integral-derivative (CFOPID) controllers for the Podlubny's, the internal model control, and the El-Khazali's forms are considered to demonstrate the feasibility of the proposed technique}. Simulation results highlight that, for a practically reasonable order, all the designs achieve good agreement with the theoretical characteristics. {Performance comparisons with the CFOPID controller approximants determined by the Oustaloup's CFO differentiator based substitution method justify the proposed approach.