# An Experimental Tuning Approach of Fractional Order Controllers in the Frequency Domain
## 1 Introduction
A fractional-order integration/differentiation offers the flexibility of variable magnitude slope and a variable phase value. Therefore, a fractional controller is more suitable to honor closed loop frequency domain constraints. The tuning of the fractional-order controller is based on an optimization routine that considers the frequency response magnitude values obtained experimentally. The main application to the developed method is vibration mitigation. This specific field has gained more popularity in recent years, especially in combination with fractional-order control. The complexity of control tasks associated to vibration suppression systems requires the usage of controllers based on more advanced mathematical tools, such as the fractional calculus branch. Many fractional-order controller tuning methodologies have been developed and tested with the sole purpose of vibration suppression. A fusion between a fractional-order disturbance observer and PI fuzzy controller is proposed in [32] to reduce vibration amplitude in an industrial servo system. Vibrations of an aeroplane wing modeled as a smart beam are suppressed using fractional-order PID-type controllers in [12; 13; 33; 34; 35]. Structural vibrations of uncertain systems are mitigated using a fractional-order sliding mode controller in [36]. Residual vibrations are tackled in [37] using adaptive-predictive fractional-order control strategies, whereas [38] uses optimal fractional-order controllers for AC motor vibrations. The previously mentioned works use complex fractional-order tuning strategies, based on available models of vibration process' dynamics. The control development is difficult and requires an advanced fractional-order background of the control engineer. In contrast, the experimental fractional-order tuning method presented throughout this study does not require any fractional order background, making the method ideal in industrial applications. From an engineer's point of view, the presented fractional-order controller tuning procedure consists of two steps: The first one implies performing several experiments on the vibration platform, reading the vibration amplitude and experimental approximation of the resonant peak. The second step requires the choice of new magnitude values such as to ensure that the magnitude plot around the magnitude peak is flattened. The ease of application as well as the guaranteed vibration suppression results make the method ideal for industrial scenarios. The novelty of the presented study is asserted in determining optimal fractional-order controller parameters based solely on the experimental response of the process. The method overcomes drawbacks encountered in classical auto-tuning practices by imposing the magnitude and phase of the closed loop system. A certain degree of robustness can also be assured is the phase margin constraints are imposed to satisfy the iso-damping property. The proposed strategy is appropriate to tune fractional-order proportional derivative (PD), proportional integral (PI), and PID controllers with similar tuning effort. The developed methodology is validated experimentally on a vibration unit consisting of a smart beam. The movement of the beam is controlled using piezoelectric actuators placed near the fixed end, whereas the displacement is measured with a strain gauge sensor. The choice of the experimental platform lies in its versatility: the smart beam can be correlated to physical systems such as aeroplane wings, helicopter rotors, cantilever beams, suspended platforms, blades of various heavy-duty machinery, etc. The paper presents the mathematical background of the proposed tuning procedure in Section 2. The method is viable to tune fractional-order PI-, PD-, and PID-type of controllers. Section 3 presents the experimental platform consisting of a smart beam used to prove the efficacy of the proposed method. A fractional-order PD controller is tuned in Section 4, whereas a fractional-order PID is tuned in Section 5.