Optimal approximation of analog PID controllers of complex fractional-order

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