# An Experimental Tuning Approach of Fractional Order Controllers in the Frequency Domain
## 1 Introduction  
When subjected to the task of improving the performance of a physical process, classical control theory tackles the issue using two steps: mathematically modeling the dynamics of the process and tuning a suitable controller for the identified model. Several alternatives have been developed for controller tuning based on information gained from the experimental response of the process. For most processes, transient and steady-state analysis of the process offers enough information to obtain a decent controller. However, there is no guarantee that the obtained controller is the ideal one. Experimental methods such as the well-known [PERSON], Kappa-Tau method, Cohen-Coon, Tyreus-Luyben, Ciancone, and Marlin tune classical proportional-integral-derivative (PID) controllers based on measurements of the different dynamics of the process response [1; 2; 3; 4; 5]. In most cases, the controllers tuned using experimental procedures lead to a closed loop system with poor robustness performance [6]. Robust autotuning has been proposed by [7] where the phase and magnitude margins are specified. The problem of autotuning robustness is also tackled in [8] by imposing the iso-damping property of the closed loop. An increasing trend in controller design integrates the concept of fractional calculus. A rational order for the differentiation and integration extends the classical PID controller to a fractional-order PID. This type of controllers offer increased degrees of freedom and an increased stability as the number of tuned parameters increases [9; 10]. One of the popular methods for tuning fractional-order controllers is by imposing frequency domain constraints such as the gain crossover frequency, phase margin, sensitivity, and complementary sensitivity [11]. Other frequency domain tuning procedures may also impose a flat phase around the gain crossover frequency guaranteeing a certain degree ofrobustness to gain variations [12; 13]. Optimization techniques have also been used to successfully tune fractional-order controllers in the frequency domain [14; 15]. Over the years, fractional calculus proved useful in both identification and control applications. Its numerous advantages have been successfully tested and validated on a manifold of real-life processes such as flexible transmission [16], active suspension [17], buck converters [18], hydraulic actuators [19], irrigation canals [20], pneumatic pressure [21], industrial distributed systems [22], robotic manipulators [23], etc. The autotuning approach has also been applied in the fractional calculus controller design [24; 25; 26; 27]. A fractional-order experimental tuning procedure based on a single experiment has been explored in [28]. The authors present the tuning of a fractional-order PD controller without using an identified model for the process [29]. The process is fed by a sinusoidal input and the amplitude and phase shifts of the output signal are measured. The output signal is then fed to a filter in order to determine the derivative of the signal. From the obtained information regarding the output amplitude, phase and the derivative slope, a fractional-order PD controller is determined. The Ziegler-Nichols method has been extended in order to tune a fractional-order PI controller in [30]. A fractional-order PID controller is tuned using a relay that takes the system at its stability margin in [31]. For the output signal, the amplitude and phase are measured and using the derivative of the phase the fractional-order PID controller is determined. The presented work consists of joining the advantages of optimal controller tuning and experimental procedures. An optimal controller tuning in the frequency domain consists of determining the controller parameters such as the shape of the closed loop satisfies several constraints. A fractional-order controller is more suitable for the task, as in the frequency domain, the first-order integration/differentiation introduces a fixed slope of \(+20/-20\) dB/decade and a fixed phase of \(+90/-90\) degrees.