Stability Region Design of Fractional Complex Order $PI^{\lambda+\mu \mathrm{i}}$ Controller Using D Segmentation

The fractional complex order PI(FCO-PI) controller is a generalized fractional-order PI(FO-PI) controller, which can make the system have more complex structure and more flexible control strategy because of its additional adjustable complex-order parameters. However, due to the existence of complex order, the design of complex order controller is more difficult than that of fractional order controller. In order to solve this problem, the D-partition method is extended to the complex order control system. For a typical class of fractional order systems with time delay, the D-partition method with real root boundary (RRB) and complex root boundary (CRB) conditions is used to obtain the stable region of the FCO-PI controller. Finally, through the analysis of the simulation results, the parameter range of the complex controller and the coupling effect between the parameters are given. At the same time, the simulation example shows that the designed fractional complex order controller can make the system asymptotically stable according to the parameters in the stable region given in this paper.