# An Experimental Tuning Approach of Fractional Order Controllers in the Frequency Domain
## 2 Proposed Tuning Method  
The proposed tuning procedure is based on experimental tests without the need of a model for the process. By excluding the model, one can also remove modeling errors and uncertainties, and it also eliminates the need of using a fractional-order model or a high order model in the case of more complex processes. The method has its basis in the Bode magnitude plot which can be created experimentally by feeding the process a series of sine wave signals of a given amplitude \(A_{i}\) and a given frequency \(\omega\). As can be seen in Figure 1, for a sinusoidal input, the output of the process will be a sinusoidal output of the same given frequency \(\omega\), but with a different amplitude \(A_{o}\) and a phase shift \(\tau\). Performing several sine wave tests recreates part of the magnitude frequency diagram through points. The method resides solely on measuring the amplitudes of the input and output signals and computing the corresponding frequency response magnitude values and does not include measurements regarding the time shift, facilitating its ease in real-life applications. The minimum number of points needed resides in the type of the desired controller. Every chosen point of the Bode diagram is part of a system of equations that needs to be solved in order to find the controller parameters, therefore the minimum number of tests needed is equal to the number of controller parameters. After performing the experimental tests and determining several magnitude values at their corresponding frequencies, the process Bode magnitude plot can be recreated. Next, the shape of the closed loop can be imposed by analyzing the actual values of the open loop magnitude. Taking as an example the vibration phenomenon, which exhibits a resonant characteristic, the magnitude at the resonant frequency should be close to 0, as well as the magnitude at the frequencies near the resonance. In real-life situations, this results in a diminished amplitude of oscillation with less visible effects. Figure 1: Sinusoidal input and output. Starting from the following values of the process' magnitude, obtained via the experimental sine tests as described above, for \(n\) different frequencies:  
\[ \begin{cases}|G(\omega_{1})|=magG_{1}\ dB,\\ |G(\omega_{2})|=magG_{2}\ dB,\\...\\ |G(\omega_{n})|=magG_{n}\ dB.\end{cases}  \tag{1}\]  
The desired magnitude and phase of the closed loop at the chosen frequencies are chosen such as  
\[ \begin{cases}|H_{0}(\omega_{1})|=magH_{1}\ dB,&\angle(H_{0}(\omega_{1}))= \textit{phase}H1\ deg,\\ |H_{0}(\omega_{2})|=magH_{2}\ dB,&\angle(H_{0}(\omega_{2}))=\textit{phase}H2\ deg,\\...\\ |H_{0}(\omega_{n})|=magH_{n}\ dB,&\angle(H_{0}(\omega_{n}))=\textit{phase}Hn\ deg.\end{cases}  \tag{2}\]  
Choosing the closed loop design specifications at specific frequencies results in shaping the closed loop magnitude and phase diagrams. The main focus of the specifications is to guarantee the disturbance rejection around the resonant frequency by imposing the magnitude value at the resonant frequency close to 0 dB. The other magnitude values are chosen such that the slope of the magnitude graph is smoother than the uncompensated system implying an increased damping. The phase constraints are chosen such that the close loop system is stable and a certain degree of robustness is obtained through the iso-damping property.