# Fractional Order PI-Based Control Applied to the Traction System of an Electric Vehicle (EV)
## 4 Fractional Order Control  
Different variations of PID, including nonlinear approaches, have been reported in the literature. For instance, PID controllers with integral and derivative of fractional order have been published. Those works show an enhancement on the functioning of the controllers [19]. A generalization of the PID controller was proposed by [PERSON] [20; 21], that approach includes an integrator of order \(\lambda\) and a differentiator of order \(\delta\) (\(PI^{\lambda}D^{\delta}\) controller). The transfer function of this controller is shown in (16).  
\[C(s)=\frac{U(s)}{E(s)} =K_{p}+K_{i}s^{-\lambda}+K_{d}s^{\delta} \tag{16}\]  
where  
\(\lambda>0\), real integral operator.  
\(\delta>0\), real differential operator.  
\(K_{p}\), proportional constant.  
\(K_{i}\), integration constant.  
\(K_{d}\), differentiation constant.  
The fractional-order controller comprises three connected parts [22].  
In discrete time the controller is:  
\[C(z)=\frac{U(z)}{E(z)} =K_{p}+\frac{K_{i}}{\left(\omega(z^{-1})\right)^{\lambda}}+K_{d} \Big{(}\omega(z^{-1})\Big{)}^{\delta}. \tag{17}\]  
In Equations (16) and (17), a PID controller in its classic form is conceived, if \(\lambda=1\) and \(\delta=1\). Otherwise, if \(\lambda=1\) and \(K_{d}=0\), then a PI controller is obtained. Actually, these controllers are derived from the fractional order control, which, due to its flexibility, allows to better adjust the dynamic activities of a system [19; 23]. In this work a fractional-order PI was tuned to minimize the integral time-weighted absolute error (ITAE).  
A usual method to implement fractional order parameters is by approximation. An approximation that has been largely studied is to implement the \(S^{q}\) operators, where q is a non-integer, as a Lead-Lag compensators (18) [24].  
\[\frac{V_{o}(t)}{V_{i}(t)}=\frac{1}{S^{q}} =\frac{(1-q)S+(1+q)}{(1+q)S+(1-q)}=A\frac{S+\frac{1}{A}}{S+A}, \tag{18}\]  
where  
\[A=\frac{1-q}{1+q}. \tag{19}\]  
In the case of the derivative factor \(A=\frac{1-q}{1+q}\).  
In order to facilitate the implementation of fractional order derivate and integral operators, a discrete representation by Z transform could be used for the PID (20).  
\[C(z)\frac{U(z)}{E(z)}=K_{p}+K_{i}(\frac{K_{foi}(z-Z_{i})}{z-P_{i}})+K_{d}( \frac{K_{fod}(z-Z_{d})}{z-P_{d}}), \tag{20}\]  
where\(P^{i}\) is the pole of the lag compensator \(Z^{i}\) is the zero of the lag compensator \(P^{d}\) is the pole of the lead compensator \(Z^{d}\) is the zero of the lead compensator  
As all discrete implementation the sample time selected is crucial. If the Equation (20) is developed, the fractional order PID (FO-PID) can be seen as a second order filter, with two real poles and a pair of complex conjugate zeros (21). Therefore, techniques that allow fixing poles and zeros of compensators can be used to tune FO-PID's. For this work, the control parameters were selected using Root Locus and the method proposed by [PERSON] and [PERSON] [25]. The poles and zeros of the FO-PI controller are located depending on the dominant closed-loop pole desired.  
\[\frac{U(z)}{E(z)}=\frac{z^{2}(K_{p}+K_{11}+K_{D1})+z(-K_{p}(P_{t}+P_{d})-K_{11 }(Z_{i}+P_{d})-K_{D1}(Z_{d}+P_{1}))+K_{p}P_{t}P_{d}+K_{11}Z_{i}P_{d}+K_{D1}Z_{d }P_{t}}{(z-P_{t})(z-P_{d})}, \tag{21}\]  
where  
\[K_{I1}=K_{I}K_{foi}\quad\text{and}\quad K_{D1}=K_{D}K_{fod}. \tag{22}\]  
In this work the poles and zeros of the FO-PI are located looking for minimizing the ITAE index.