Analytical approach in designing PID controller for complex fractional order transfer function

The study focuses on the fractional complex order plant model, which has gained popularity in applied mathematics, physics, and control systems. A significant contribution of this research lies in discussing the physical phenomena associated with complex plant models and their impact on system stability and robustness. The main purpose of the method presented in this paper is to tune the controller parameters to ensure the stability and robustness of the system. There are methods presented in the literature for this purpose. One of these methods is to keep the phase curve in the system frequency response flat within a certain range. However, this process is based on equating the derivative of the phase value to zero at a certain frequency and adds great mathematical complexity to the calculations. In this study, reliable analytical formulas are presented for the same purpose using a graphical approach. Since the fractional complex order plant model represents the most general mathematical form, it enables easy creation of other plant models, including integer order and fractional order plant. The reason why this plant is chosen is that this structure can be named as the universal plant, which all other structures can be built by making little variations. For instance, a transfer function having integer, real and/or complex number coefficients and/or orders can be obtained by proper determination of the parameters of the universal plant. A time delay can also be added towards researcher's desire. The main inspiration comes from studying on an inclusive plant. The method in this paper intends to tune the well-known classical Proportional Integral Derivative controller. Thus, effectiveness of the integer order controller on various plants will be shown. This approach provides analytical calculation equations for the physical modifications of plants with integer, fractional, and/or complex coefficients and/or orders. The effectiveness of the method is demonstrated visually with different examples that include these different possible situations. The results observed in the changes of the parameters in the transfer functions were also examined. Thus, pros and cons of the variations of integer, fractional, and complex numbers on system parameters have been shown.