## 7 Case Study: Position Control of a Rotor by PII\({}^{1/2}\)DD\({}^{1/2}\) Control  
### Comparison of PID and PII\({}^{1/2}\)DD\({}^{1/2}\) Control in Frequency Domain and Time Domain  
Let us consider the position control of a rotor with inertia \(J\) and viscous coefficient \(B\), according to the closed-loop scheme of Section 3. The numerical values of \(J=1.04\,\times\,10^{-3}\) kg\(\cdot\)m\({}^{2}\) and \(B=1.45\,\times\,10^{-3}\) Nms/rad are related to the test bench that will be used in the experimental tests (Section 8). Let us start from the following PID control gains: \(K_{p}=0.25\) Nm/rad, \(K_{i}=0.005\) Nm/rad-s, \(K_{d}=0.035\) Nms/rad, which provide a closed-loop stable behavior, with a phase margin of \(80.5^{\circ}\) and a bandwidth of \(34\) rad/s. The two zeros of the PID transfer function can be calculated by Equation (23) and their opposites are the corner frequencies \(\omega_{c1}=2.01\,\times\,10^{-2}\) rad/s and \(\omega_{c2}=7.12\) rad/s; according to the considerations developed in the next section, we choose \(\rho=4\), and consequently the four corner frequencies of the PII\({}^{1/2}\)DD\({}^{1/2}\) control can be calculated by equations (24): \(\omega^{\prime}{}_{c1}=5\,\times\,10^{-3}\) rad/s, \(\omega^{\prime}{}_{c2}=8.04\,\times\,10^{-2}\) rad/s, \(\omega^{\prime}{}_{c3}=\)1.78 rad/s, \(\omega^{\prime}{}_{c4}=28.5\) rad/s, and are equal for the two tuning criteria \(C_{H}\) and \(C_{L}\). The gains of the controllers for the two tuning criteria have been calculated by the procedure discussed in Section 6 and are collected in Table 1.  
Figures 5-7 compare the three controllers: PID (blue), PII\({}^{1/2}\)DD\({}^{1/2}\) with tuning \(C_{H}\) (PII\({}^{1/2}\)DD\({}^{1/2}\)\({}_{H}\), red), PII\({}^{1/2}\)DD\({}^{1/2}\) with tuning \(C_{L}\) (PII\({}^{1/2}\)DD\({}^{1/2}\)\({}_{L}\), green). The Bode plots of the controllers are represented in Figure 5. All the plots of the controller as symmetrical with respect to the frequency \(\omega_{min}=3.78\,\times\,10^{-1}\) rad/s. The two PII\({}^{1/2}\)DD\({}^{1/2}\) controllers have the same phase plots since their transfer functions are only shifted in magnitude. Figure 6 shows the Bode plot of the closed-loop system; both the PII\({}^{1/2}\)DD\({}^{1/2}\) controllers exhibit a larger bandwidth, which is correlated to an improved readiness in the time domain, as shown by the step response (Figure 7). This is well understandable for the PII\({}^{1/2}\)DD\({}^{1/2}\)\({}_{H}\), which has a higher gain at all the frequencies (Figure 5), but not so obvious for the PII\({}^{1/2}\)DD\({}^{1/2}\)\({}_{L}\). Figure 7: Comparison of the unit step responses of the closed-loop systems: PID (blue); PII\({}^{1/2}\)DD\({}^{1/2}\)\({}_{H}\) (red); PII\({}^{1/2}\)DD\({}^{1/2}\)\({}_{L}\) (green). Influence of the Ratio \(\rho\) on the PII\({}^{1/2}\)DD\({}^{1/2}\) Controller Frequency Response  
In Section 7.1, a ratio \(\rho=4\) was chosen to evaluate the four corner frequencies from Equation (24). Figure 8 shows the influence of this ratio on the controller response frequency for the case study of Section 7.1. This figure shows the Bode plots of the PID controller and of the PII\({}^{1/2}\)DD\({}^{1/2}\)\({}_{H}\) controllers with \(\rho=1\), \(4\), \(10\), \(50\), \(200\), \(\rho_{max}=355.1\). The PII\({}^{1/2}\)DD\({}^{1/2}\)\({}_{L}\) controllers are not represented since the phase plot is the same and the magnitude plot is simply shifted to have the same minimum of the PID controller.