## 3 The PII\({}^{1/2}\)DD\({}^{1/2}\) Control Scheme  
To introduce the proposed PII\({}^{1/2}\)DD\({}^{1/2}\) control, let us consider a second-order plant (Figure 1). Many mechatronic systems in which friction can be considered viscous can be suitably modelled by a second-order linear system. In the following a rotor with inertia \(J\) and viscous coefficient \(B\) driven by a torque \(M\) commanded by the controller will be considered.  
Figure 1: Closed-loop system with a second-order plant.  
The open-loop plant dynamics is expressed by the following differential equation:  
\[J\frac{d^{2}}{dt^{2}}\theta+B\frac{d}{dt}\theta=M(e_{\theta}) \tag{7}\]  
where \(M\) is the controller output, calculated as a function of the error \(e_{\theta}\), difference between the set-point angle \(\theta_{r}\) and the current angle \(\theta\).  
In the following, for the closed-loop control scheme of Figure 1, three control laws will be considered: the integer-order PID, the fractional-order \(\Pi^{A}\mathrm{D}^{\mu}\) and the proposed \(\Pi\mathrm{I}^{1/2}\mathrm{DD}^{1/2}\).  
The classical PID control law is based on the well-known proportional, integral and derivative gains \(K_{p}\), \(K_{i}\), \(K_{d}\):  
\[M(e_{\theta})=\Big{(}K_{p}+K_{i}D^{-1}+K_{d}D^{1}\Big{)}e_{\theta} \tag{8}\]  
In case of \(\Pi^{A}\mathrm{D}^{\mu}\), the control law is given by:  
\[M(e_{\theta})=\Big{(}K_{p}+K_{fi}D^{-\lambda}+K_{fd}D^{\mu}\Big{)}e_{\theta} \tag{9}\]  
where \(K_{p}\), \(K_{fi}\) and \(K_{fd}\) are the proportional, fractional-order integral and fractional-order derivative gains, \(\lambda\) is the fractional integral order and \(\mu\) is the fractional derivative order [10].  
Similarly to the \(\mathrm{PDD}^{1/2}\) concept, in which the half-derivative term is added to the derivative one instead of replacing it with an FO derivative term, in the proposed \(\Pi\mathrm{I}^{1/2}\mathrm{DD}^{1/2}\) control the half-derivative and the half-integral terms are added to the PID; therefore, the \(\Pi\mathrm{I}^{1/2}\mathrm{DD}^{1/2}\) control law is:  
\[M(e_{\theta})=\Big{(}K_{p}+K_{i}D^{-1}+K_{hi}D^{-1/2}+K_{d}D^{1}+K_{hd}D^{1/2} \Big{)}e_{\theta} \tag{10}\]  
where \(K_{hd}\) is the half-derivative gain and \(K_{hi}\) is the half-integral gain.  
Applying a Laplace transform to Equations (8) to (10) with null initial condition, the transfer functions of the three controllers can be expressed by:  
\[G_{c,PID}(s)=K_{p}+\frac{K_{i}}{s}+K_{d}s=K_{i}\frac{\big{(}1+\big{(}K_{p}/K_{ i}\big{)}s+(K_{d}/K_{i})s^{2}\big{)}}{s} \tag{11}\]  
\[G_{c,P\mathrm{I}^{1/2}D^{\mu}}(s)=K_{p}+\frac{K_{fi}}{s^{\lambda}}+K_{fd}s^{ \mu}=K_{fi}\frac{\Big{(}1+\big{(}K_{p}/K_{fi}\big{)}s^{\lambda}+\big{(}K_{fd}/ K_{fi}\big{)}s^{\lambda+\mu}\Big{)}}{s^{\lambda}} \tag{12}\]  
\[G_{c,P\mathrm{I}\mathrm{I}^{1/2}DD^{1/2}}(s)=K_{p}+\frac{K_{i}}{s}+\frac{K_{ hi}}{s^{1/2}}+K_{d}s+K_{hd}s^{1/2}= \tag{13}\]  
Let us note that while the PID has three degrees of freedom for tuning (the three gains), both the FO controls have five degrees of freedom for tuning: three gains and two orders for \(\Pi^{A}\mathrm{D}^{\mu}\), five gains for \(\Pi\mathrm{I}^{1/2}\mathrm{DD}^{1/2}\).  
Frequency Domain Response of PID, \(\Pi^{A}\mathrm{D}^{\mu}\) and \(\Pi\mathrm{I}^{1/2}\mathrm{DD}^{1/2}\) Controllers