Design of a Complex fractional Order PID controller for a First Order Plus Time Delay system

Complex fractional Order PID (COPID) controller is an extension to the Real fractional Order PID (ROPID) controller by extending the orders of differentiation and integration to include complex numbers, i.e., two extra parameters (the imaginary parts of the orders of the differentiator and the integrator) are introduced into the formula of the controller. The purpose is to overcome the limitation stemmed from restricting the parameters of the ROPID controller to belong to certain intervals, where this limitation results in a control system that does not satisfy the required design specification accurately. In this paper, analysis and design of COPID controller is presented, and for comparison purposes, both ROPID and COPID controllers are designed for a low pressure flowing water circuit, which is a First Order Plus Time Delay (FOPTD) system. The design specifications are given in frequency domain, which are gain crossover frequency, phase margin, and robustness against gain variation. The design specifications are taken as two cases, simple an rigorous, where the latter is considered to demonstrate the superiority of the COPID controller over the ROPID controller to achieve hard specifications. Although the design of the COPID controller is more complex than that of the ROPID controller, the first achieves the required design specification more accurately.