# An Experimental Tuning Approach of Fractional Order Controllers in the Frequency Domain
## 4 Controller Tuning
### Fractional Order PI/PD Tuning  
A fractional-order PD controller is first designed to suppress unwanted vibrations in the smart beam. As the controller's transfer function has three parameters that have to be computed: \(k_{p}\), \(k_{d}\), and \(\mu\), three points are computed on the magnitude diagram as follows,  
\[\begin{cases}|G(\omega_{1})|=-51.52\ dB,\quad\omega_{1}=56.55\ rad/s,\\ |G(\omega_{2})|=-41.25\ dB,\quad\omega_{2}=107\ rad/s,\\ |G(\omega_{3})|=-52.55\ dB,\quad\omega_{3}=251.3\ rad/s.\end{cases} \tag{18}\]  
The three frequencies are chosen as follows; \(\omega_{2}\) is close to the resonant frequency of the beam; and \(\omega_{1}\) and \(\omega_{3}\) are chosen to enclose the resonant frequency, and for the beam case they are chosen as the smallest/greatest frequency for which the magnitude of the process can be computed due to the physical limitations of the experimental set-up. The imposed magnitude and phase closed loop values for the chosen points are  
\[\begin{cases}|H_{0}(\omega_{1})|=-12\ dB,\quad\quad\angle(H_{0}(\omega_{1}))=- 0.1\ deg,\\ |H_{0}(\omega_{2})|=-1\ dB,\quad\quad\angle(H_{0}(\omega_{2}))=-60\ deg,\\ |H_{0}(\omega_{3})|=-20\ dB,\quad\quad\angle(H_{0}(\omega_{3}))=-120\ deg.\end{cases} \tag{19}\]  
The constraints are imposed to ensure a lower resonant peak, larger damping, and a stable closed loop system. The graphical representation of the chosen constraints can be seen in Figure 7.  
The optimization routine searches the solution of the minimization problem for \(k_{p}\in(0\ 100)\), \(k_{d}\in(0\ 100)\), and \(\mu\in[-1\ 1]\). The initial points chosen for the parameters are \(k_{p}=15\), \(k_{d}=0.5\), and \(\mu=0.85\). The controller determined that honors the imposed constraints for the closed loop magnitude and phase is a fractional-order PD controller:  
\[C_{FO-PI/PD}=8+0.8888\ s^{0.7683}. \tag{20}\]  
The obtained fractional-order PD controller is approximated to a 6 th-order discrete transfer function using the method in [39], which is based on mapping the time Laplace operator directly into discrete time. The controller is validated in terms of disturbance rejection.  
Figure 7: Frequency domain magnitude constraints for fractional-order proportional integral–proportional derivative (PI/PD) tuning (blue—computed magnitude of the open loop system; red—imposed magnitude for the closed loop system).  
First, a sinusoidal input of the resonant frequency, 14 Hz, and an amplitude of 1V is given as a disturbance and the response of the beam is presented in Figure 8a. The uncompensated response of the beam leads to an amplitude of 0.25 V. The action of the controller is included at t = 25 s, and the fractional PD controller reduces the oscillation amplitude by 80% with a settling time of 2.5 s. The obtained controller is validated on a larger interval by giving a swept sine input with frequencies between 9 and 90 Hz. Two resonant frequencies, representing the first and second flexural modes of the beam, can be visible at 14 Hz and 83 Hz. Figure 8b presents a test performed with the swept sine input for the uncompensated system and two more tests with the control action turned on. It can be seen that the amplitude of the beam is lowered for the entire frequency range.