New tuning design schemes of fractional complex-order PI controller

This manuscript presents two systematic design procedures, to tune parameters of a fractional complex-order PI (FCO-PI) controller in the form of $$\mathrm{PI}^{a+ib}$$PIa+ib. The $$\mathrm{PI}^{a+ib}$$PIa+ib controller uses extra parameter(s) than the conventional fractional- and/or integer-order PI controllers. Therefore, more specifications can be achieved. These are investigated in two different approaches through comparative studies. The proposed design procedures are based on realizing some frequency domain restrictions. These are eventually stated in terms of $$M_{s}$$Ms and $$M_{p}$$Mp constraints, developing integral gain optimization tuning method. In this method, optimized amount of parameters are assessed based on minimizing the integral error indices with a constraint on the maximum sensitivity functions. In this aim, tuning of parameters of fractional complex-order controller via M constraint integral gain optimization (FC-MIGO) algorithm is innovatively defined and then the so-called FC-MIGO rule is proposed by applying FC-MIGO algorithm on a test batch. Comprehensive simulations illustrate how systematic and significant the proposed algorithms are. Capability of the design procedure will be also investigated on a PEM fuel cell as a case study.