## 1 Introduction
The auto-tuning procedure consists of two phases: The first one is dedicated to the identification of the process at the desired crossover frequency and the second one to determine all the parameters of the fractional order PID controllers. Experimental validations are also presented in Reference ([10] new), where a model-free tuning method of an FO-PI controller and its application for the speed regulation of a permanent magnet synchronous motor (PMSM) is described. Here, virtual reference feedback tuning is used to determine the parameters of the FO-PI controller. Under the lack of accurate models, the proposed model-free method depends only on the measured input-output data of the closed-loop PMSM servo system. Also, Bode's ideal transfer function is incorporated into the virtual reference feedback tuning with consideration of the systematic fractional dynamics. Optimal performance constraints are taken into consideration by the proposed method, such as stability requirements, sensitivity criteria, frequency-domain and time-domain characteristics. Based on these, a comprehensive optimization problem is derived and solved, such that the robustness and disturbance rejection ability of the resulting FO-PI controller are enhanced to achieve optimal performance. In Reference ([11] new), the problem of fractional modeling and the control of an industrial selective compliant assembly robot arm (SCARA) is presented. Induction motors are used to drive each joint of the robot manipulator and for each motor a fractional model is formulated. For comparison purposes, both integer order and fractional order controllers are designed. The tuning procedure is based on a Particle swarm optimization algorithm, with the ITAE index used as a cost function to be minimized. Only numerical simulations are included to show the effectiveness of the fractional order controllers. The same performance specification set (phase margin, gain crossover frequency and iso-damping) is considered in the design of fractional order controllers based on a combination of the Ziegler-Nichols tuning procedure and the Astrom-Hagglund method [12]. First, the proportional and integral gains of the fractional order controller are computed based on the Ziegler-Nichols approach, then the derivative gain is obtained using the Astrom-Hagglund method [13; 14]. An optimization procedure is required to solve the resulting nonlinear equations. The core idea is that, in case of a better step response of the closed loop system, an optimization model is used, with the previously computed controller parameters taken as initial values. The result of the optimization consists in new values for the controller parameters, considered to be optimal. The same three frequency domain specifications are considered also in Reference [15]. The auto-tuning method can be used to determine either fractional order PI or PD controllers. In this case, a simple sine test is used to determine the required process information for the tuning of the fractional order controller, then, much like in most of the fractional order controller design methods, either a graphical approach or an optimization routine is required to solve the resulting system of nonlinear equations. A novel auto-tuning approach for fractional order PI controllers, the FO-KC method, is presented in this paper. The idea comes as an extension of a previously designed autotuning method for integer order PID controllers, namely the KC autotuner [16]. The tuning of fractional order PI controllers using the FO-KC auto-tuning method is similar to the classical tuning methods in the sense that it determines the three controller parameters using three performance specifications: the gain crossover frequency \(\omega_{c}\), the phase margin \(PM\) and the iso-damping property. These three performance specifications are quite frequently used as design specifications and lead to a system of nonlinear equations that need to be solved to estimate the FO-PI parameters. However, this can be time consuming.