## 1 Introduction  
Auto-tuning methods are among the most suitable and popular controller design methods for the industry. The reason lies in the elimination of the need for a process model in the tuning of the controller parameters. For PID controllers, several auto-tuning methods have been developed throughout the years. However, this is not the case of fractional order PID controllers. Fractional calculus represents an emerging field of research that has gained increased interest in control applications [1]. The fractional order PID controller is a generalization of the classical one, with the integral and derivative terms modified to accommodate fractional orders. To tune these types of controllers, a couple of methods exist; however, most of these require a process model. The auto-tuning methods for fractional order controllers are in their early stages. Such a method is the so-called phase shaper [2; 3]. This is in fact a traditional PID controller with a twist: It has a fractional order integrator or differentiator \(s^{\mathrm{g}}\), with \(\alpha\)\(\in\)(\(-\)1, 1). The aim of this phase shaper is to achieve closed loop robustness to gain variations. A different approach to auto-tuning of fractional order PIDs is based on a two-step design that first produces a fractional order PI (FO-PI) controller, and then a fractional order PD controller with a filter. A relay test is used in this regard [4]. The design is intended to achieve a certain gain crossover frequency and phase margin, while ensuring the iso-damping property. The procedure is lengthy and is based on maximizing the robustness to plant gain variations. Other auto-tuning methods based on the relay test have been proposed, such as for the design of a FO-PI controller for real-time steam temperature control ([5] new). The relay test is also used in Reference ([6] new) for the auto-tuning of two types of fractional order PI controllers with an iso-damping property for a class of unknown, stable and minimum phase plants. The FO-PI controllers designed by the two methods ensure that the phase Bode plot is flat at a given frequency called the tangential frequency. An iterative procedure is used in this case, since several relay feedback tests are used to identify the plant gain and phase at the tangential frequency. Only simulation results are presented. An iterative procedure combined with the relay test is also used for designing fractional order lead-lag compensators ([7] new). The same relay test has been proposed in Reference ([8] new) for the auto-tuning of a fractional order PID controller for second order plus dead time plants. In this design, firstly, the process dynamics of overdamped, underdamped, and critically damped plants are modelled by using an offline relay-based method, where a maximum of four unknown parameters are determined in the simplest possible way. Although this is a clear limitation of the method, a great variety of plants can be modeled using four parameters only, such as the second order plus dead time processes. A relay with hysteresis is used to obtain the describing function. As performance specifications, the gain margin and phase margin, as well as the iso-damping property are used to tune the five parameters of the controller. A significant advantage of the proposed tuning method is that a priori information of the system's gain and phase crossover frequencies is not required while tuning the controller. The proposed method is validated through simulation studies in a class of process models, and also verified experimentally on a coupled tank system. In Reference ([9] new), experimental validations of an auto-tuning method for fractional order PID controllers, and its implementation on hardware in the loop simulator for real-time control of unknown plants are presented. The application of the auto-tuning procedure is limited to systems with delay and an order greater than one, while the performance specifications refer to the crossover frequency, phase margin and the iso-damping property.