## 6 Bode Plot Based Tuning of PII\({}^{1/2}\)DD\({}^{1/2}\) Control  
In order to assess the possible benefits of replacing a classical PID control with a PII\({}^{1/2}\)DD\({}^{1/2}\), it is necessary to define a tuning criterium. A possible approach is to derive the PII\({}^{1/2}\)DD\({}^{1/2}\) control parameters starting from the Bode plot of a given PID control. In Figure 4 the asymptotic magnitude Bode plot of a generic PID controller with two real negative zeros is represented in blue. Comparing Equations (11) and (15) it is easy to obtain that the two PID zeros are given by the following expression:  
\[z_{1,2}=-\frac{K_{p}\mp\sqrt{K_{p}^{2}-4K_{d}K_{i}}}{2K_{d}} \tag{23}\]  
Equation (23) allows the attainment of the two corner frequencies of the PID magnitude plot of Figure 4, with \(\omega_{c1}=-z_{1}<\omega_{c2}=-z_{2}\). The plot is symmetrical with respect to the frequency \(\omega_{min}=(\omega_{c1},\,\omega_{c2})^{1/2}\), where the amplitude Bode diagram has its minimum \(min_{PID}\) (Figure 4). The asymptotic Bode magnitude plot of a PII\({}^{1/2}\)DD\({}^{1/2}\) controller is characterized by five zones with slopes of \(-20\) dB/dec, \(-10\) dB/dec, \(0\) dB/dec, \(+10\) dB/dec and \(+20\) dB/dec, as discussed in Section 4.4. Figure 4: Derivation of the PII\({}^{1/2}\)DD\({}^{1/2}\) control parameters from the PID control Bode plot: PID (blue); PII\({}^{1/2}\)DD\({}^{1/2}\) with tuning C\({}_{H}\) (red); PII\({}^{1/2}\)DD\({}^{1/2}\) with tuning C\({}_{L}\) (green). Figure 3: Stability regions of fractional order systems with fractional commensurate order \(q=1/2\). A viable tuning criterium (\(C_{H}\)) to derive the P\(\Pi^{1/2}\)DD\({}^{1/2}\) control parameters from the parameters of a given PID is to impose:  
* symmetry of the magnitude plot with respect to \(\omega_{min}\);
* the coincidence of the initial and final asymptotes, with slopes of \(-20\) dB/dec and \(+20\) dB/dec;
* the amplitude of the central zone with null slope. This is shown in Figure 4, where the P\(\Pi^{1/2}\)DD\({}^{1/2}\) Bode plot with tuning \(C_{H}\) is represented in red. Since an equal distance between two frequencies in the logarithmic scale corresponds to an equal ratio between them, the Bode magnitude plot of a P\(\Pi^{1/2}\)DD\({}^{1/2}\) controller is symmetric when the condition expressed by Equation (21) is verified. Observing the blue and red plots of Figure 4, it is possible to note that the P\(\Pi^{1/2}\)DD\({}^{1/2}\) plot with tuning \(C_{H}\) can be obtained from the PID by imposing:  
* the same integral gain of the PID controller,
* the following relations between the corner frequencies:  
\[\omega_{c1}^{\prime}=\frac{\omega_{c1}}{\rho};\;\omega_{c2}^{\prime}=\rho \omega_{c1};\;\omega_{c3}^{\prime}=\frac{\omega_{c2}}{\rho};\;\omega_{c4}^{ \prime}=\rho\omega_{c2} \tag{24}\]  
with \(1<\rho<\rho_{max}=(\omega_{c2}/\omega_{c1})^{1/2}\); for \(\rho=\rho_{max}\), \(\omega^{{}^{\prime}}{{}_{c2}}=\omega^{{}^{\prime}}{{}_{c3}}=\omega_{min}\). Therefore, once the ratio \(\rho\) is selected, the four corner frequencies of the P\(\Pi^{1/2}\)DD\({}^{1/2}\) control can be calculated by Equation (24) and then, considering that \(z_{i}=-(\omega^{{}^{\prime}}{{}_{c1}})^{1/2}\), it is possible to obtain the half-zeros and then the gains \(K_{p}\), \(K_{hi}\), \(K_{d}\), \(K_{hd}\) by Equations (17)-(20). The influence of the parameter \(\rho\) on the controller frequency response will be discussed in Section 7.2.