Realization and comparative analysis of Fractional Order Controllers for different discretization methods

There is a great variety of applications for the use of fractional operators, with a special case in fractional control systems. Different methods of realizing the fractional operators of the FOPID controllers offer a variety of results. This article presents some methodologies for realizing fractional order controllers using MATLAB toolboxes. Three methodologies are developed to obtain the integer approximations and each integer transfer function is discretized with three methods (Tustin, Backward, FOH), in the same way three direct discretization methods are used (PSE, CFE/Al-Alaoui, CFE/Tustin) for the integrator and differentiator of a FOPID controller. To show the usefulness of the integer approximation and discretization of the proposed schemes, test simulations are carried out for a DC motor speed control system in closed loop with the controllers made with the proposed schemes and each proposed method is compared. Finally, it is concluded that the direct discretization with the CFE schemes of order 6, with Al-Alaoui and Tustin operators, with a sampling time of Ts = 0.001 seconds, present better control results in continuous time. Generating a fast response in the output signal of the control system.