# An Experimental Tuning Approach of Fractional Order Controllers in the Frequency Domain
## 4 Controller Tuning
### Fractional-Order PID Tuning  
In the previous section, the experimental method was successfully used to tune a fractional-order PD controller. The obtained controller proved useful in the case of disturbance rejection making the  
Figure 8: Experimental disturbance rejection of the closed loop system with the fractional-order PD controller. controller suitable for the chosen process. However, to prove the efficacy of the proposed method, a more complex fractional-order PID controller is tuned. The transfer function of the fractional PID from (13) has five parameters: \(k_{p}\), \(k_{i}\), \(k_{d}\), \(\lambda\), and \(\mu\). Therefore, a minimum number of five points must be chosen from the magnitude plot. \[ \begin{cases}|G(\omega_{1})|=-51.52\ dB,&\omega_{1}=56.55\ rad/s,\\ |G(\omega_{2})|=-38.87\ dB,&\omega_{2}=78.54 rad/s,\\ |G(\omega_{3})|=-12.56\ dB,&\omega_{3}=87.96\ rad/s,\\ |G(\omega_{4})|=-41.25\ dB,&\omega_{4}=107\ rad/s,\\ |G(\omega_{5})|=-52.55\ dB,&\omega_{5}=251.3\ rad/s.\end{cases}  \tag{21}\]  
The frequencies from Equation (17) are chosen as follows; \(\omega_{3}\) is the resonant frequency of the beam as can be seen in Figure 6; \(\omega_{1}\) and \(\omega_{5}\) are lower and upper bounds of the measured frequencies interval, respectively; and \(\omega_{2}\) and \(\omega_{4}\) are evenly distributed frequencies between the resonant peak frequency and the upper and lower bounds, respectively. The closed loop magnitude and phase are imposed as  
\[ \begin{cases}|H_{0}(\omega_{1})|=-12\ dB,&\angle(H_{0}(\omega_{1}))=-0.1\ deg,\\ |H_{0}(\omega_{2})|=-2\ dB,&\angle(H_{0}(\omega_{2}))=-60\ deg,\\ |H_{0}(\omega_{3})|=-1\ dB,&\angle(H_{0}(\omega_{3}))=-120\ deg,\\ |H_{0}(\omega_{4})|=-4\ dB,&\angle(H_{0}(\omega_{4}))=-120\ deg,\\ |H_{0}(\omega_{5})|=-20\ dB,&\angle(H_{0}(\omega_{5}))=-120\ deg.\end{cases}  \tag{22}\]  
Figure 9 presents the chosen values for the magnitude. The magnitude values are chosen in order to reduce the amplitude of the closed loop system at the resonant frequency, therefore ensuring a lower damping. The phase values are chosen such that the closed loop system is stable. The advantage of using a fractional-order PID controller over a fractional-order PD is that robustness can also be a design specification through the iso-damping property. The phase constraints are imposed to be \(\angle H_{0}(\omega_{3})=\angle H_{0}(\omega_{4})=\angle H_{0}(\omega_{5})=-12 0\deg\), resulting in a constant phase margin \(\phi_{m}=60\deg\) between the frequencies belonging to the interval \([\omega_{3},\omega_{5}]\), ensuring a certain robustness to small gain variations. Initial points have been given for the optimization routine consisting of \(k_{p}=15\), \(k_{d}=0.05\), \(k_{i}=0.05\), \(\mu=0.85\) and \(\lambda=0.15\). The obtained fractional-order PID controller by performing the minimization for \(k_{p}\in(0,100)\), \(k_{d}\in(0,100)\), \(k_{i}\in(0,100)\), \(\mu\in[0,1]\), and \(\lambda\in[0,1]\) is given by  
\[C_{FO-PID}=15.4506+0.8278\mathrm{s}^{0.8220}+\frac{0.1}{s^{0.1049}}. \tag{23}\]  
The obtained fractional-order PID controller has been approximated using the same complexity, \(6^{th}\) order, and the same settling time, \(T_{s}=0.005\,s\), as the fractional-order PD controller. Implementing the PID controller on the experimental unit successfully diminishes the effect of the disturbance, as can be seen in Figure 10a,b. For a sinusoidal disturbance of 14 Hz and 1V the oscillation amplitude is reduced by 80% and the settling time is around 3 s. The same tests are performed as in the case of the fractional-order PD controller, but the results are different because the two controllers were tuned with different closed loop magnitude and phase constraints. The purpose of tuning the PID controller is not to outperform the fractional PD, but to prove that the method is applicable to different controllers. The fractional-order PD and PID are different controllers tuned using the same method and validated on the same equipment, under the same conditions.