## 3 The PII\({}^{1/2}\)DD\({}^{1/2}\) Control Scheme
### \(Pl^{1/2}Dd^{1/2}\) Frequency Response  
The \(Pl^{1/2}DD^{1/2}\) controller is a commensurate-order system, with order \(q=1/2\), and this represents an advantage with respect to the \(Pl^{\lambda}D^{\mu}\). First of all, if this controller is used in combination with an IO plant, the closed-loop transfer function has also commensurate order \(q=1/2\); therefore, the [PERSON]'s stability theorem [48] can be applied and the roots location in the complex plane gives relevant information about the system behaviour, as it will be discussed in Section 5. The transfer function (13), applying the factorization (14), can be rewritten as follows:  
\[G_{c,PII^{1/2}DD^{1/2}}(s)=\frac{\prod\limits_{i=1}^{4}\left(s^{1/2}-z_{i} \right)}{s}=\frac{K_{i}}{s}\prod\limits_{i=1}^{4}\left(1-\frac{s^{1/2}}{z_{i} }\right) \tag{16}\]  
The asymptotic Bode plot is characterized by an initial magnitude slope of \(-20\) dB/dec and an initial phase of \(-\pi/2\) rad at low frequencies; after each corner frequency \(\omega_{c,i}=1z_{i}\mid^{2}\) the magnitude slope increases by \(10\) dB/dec, therefore at frequencies over the highest corner frequency the magnitude slope tends towards \(+20\) dB/dec; the phase increases by \(\pi/4\) rad after each corner frequency, and tends towards \(\pi/2\) at frequencies over the highest corner frequency. Expanding the product of Equation (16), it is possible to find a relation between the half-zeros \(z_{i}\) and the controller gains:  
\[K_{hi}=K_{i}\left(-\frac{1}{z_{1}}-\frac{1}{z_{2}}-\frac{1}{z_{3}}-\frac{1}{z _{4}}\right) \tag{17}\]  
\[K_{p}=K_{i}\left(\frac{1}{z_{1}z_{2}}+\frac{1}{z_{1}z_{3}}+\frac{1}{z_{1}z_{4} }+\frac{1}{z_{2}z_{3}}+\frac{1}{z_{2}z_{4}}+\frac{1}{z_{3}z_{4}}\right) \tag{18}\]  
\[K_{hd}=K_{i}\left(-\frac{1}{z_{1}z_{2}z_{3}}-\frac{1}{z_{1}z_{2}z_{4}}-\frac{1 }{z_{1}z_{3}z_{4}}-\frac{1}{z_{2}z_{3}z_{4}}\right) \tag{19}\]  
\[K_{d}=\frac{K_{i}}{z_{1}z_{2}z_{3}z_{4}} \tag{20}\]  
Using Equations (17) to (20) it is possible to select \(K_{i}\) and \(\omega_{c,i}\), \(i=1\dots 4\), and then to obtain the remaining four gains. Figure 2 shows in red the \(Pl^{1/2}DD^{1/2}\) controller frequency response for \(K_{i}=1.6\), \(\omega_{c,1}=1\) rad/s, \(\omega_{c,2}=10\) rad/s, \(\omega_{c,3}=10^{3}\) rad/s, \(\omega_{c,4}=10^{4}\) rad/s. Assuming these example parameters, the two central corner frequencies correspond to the two corner frequencies of the PID considered in Section 4.2 (blue); therefore, the two controllers have the same central range (\(10\div 1000\) rad/s) with a constant asymptotic magnitude plot. Let us note that the PID frequency response is symmetrical with respect to \(10^{2}\) rad/s, since its two corner frequencies are placed at \(10^{1}\) rad/s and \(10^{3}\) rad/s; furthermore, the PII\({}^{1/2}\)DD\({}^{1/2}\) frequency response is symmetrical with respect to \(10^{2}\) rad/s, since its four corner frequencies are placed symmetrically with respect to this value in logarithmic scale. In general, while the PID frequency response is always symmetric with respect to the middle frequency in logarithmic scale between the two zeros, the PII\({}^{\lambda}\)D\({}^{\mu}\) frequency response is symmetric only if \(\lambda=\mu\); as regards the PII\({}^{1/2}\)DD\({}^{1/2}\) frequency response, it is evident that it is symmetric if the four corner frequencies are symmetrically placed in logarithmic scale; this condition is verified if:  
\[\frac{\omega_{c,2}}{\omega_{c,1}}=\frac{\omega_{c,4}}{\omega_{c,3}} \tag{21}\]  
Stability of Closed-Loop Systems with Integer-Order Plant and PII\({}^{1/2}\)DD\({}^{1/2}\) Control  
If the closed-loop control scheme of Figure 1 is implemented adopting a PII\({}^{1/2}\)DD\({}^{1/2}\) controller in combination with an IO plant, it is easy to verify that the closed-loop transfer function has commensurate order \(q=1/2\).