## 2 Proposed PMSM-SCS Based on DTC Strategy
### PMSM-SCS Based on DTC Strategy Using Constant Flux
#### 2.1.1 Fractional-Order Speed Controllers for the PMSM-SCS Based on DTC Strategy  
For the implementation of the FO-type controllers proposed in Figure 1 (FOPI, TID and FO-Lead-Lag speed controllers), some notions of fractional integration and differentiation are briefly presented. Thus, the non-integer order operator applied for integration and differentiation is denoted by \({}_{a}D_{t}^{\alpha}\), where \(\alpha\) is the fractional order, and the limits of the interval to which the operator is applied are denoted by \(a\) and \(t\). The operator \({}_{a}D_{t}^{\alpha}\) is shown in the following equation [15; 16; 41]:  
\[{}_{a}D_{t}^{\alpha}=\left\{ \begin{array}{c}\frac{d^{\alpha}}{dt^{\alpha}} \operatorname{Re}(\alpha)>0\\ 1\operatorname{Re}(\alpha)=0\\ \int_{a}^{t}\left(dt\right)^{-\alpha}\operatorname{Re}(\alpha)<0\end{array} \right. \tag{5}\]  
The Riemann-Liouville definition of the operator \({}_{a}D_{t}^{\alpha}\) is given by the next equation [15; 16]:  
\[{}_{a}D_{t}^{\alpha}f(t)=\frac{1}{\Gamma(m-\alpha)}\bigg{(}\frac{d}{dt}\bigg{)} ^{m}\int\limits_{a}^{t}\frac{f(\tau)}{\left(t-\tau\right)^{\alpha-m+1}}d\tau \tag{6}\]  
where \(m-1<\alpha<m,\ m\in N\); \(\Gamma(\cdot)\)--[PERSON]'s gamma function. The Grunwald-Letnikov definition of the operator \({}_{a}D_{t}^{\alpha}\) is useful in practical applications of numerical representation [15; 16]:  
\[{}_{a}D_{t}^{\alpha}f(t)=\lim_{h\to 0}\frac{1}{h^{\alpha}}\sum_{j=0}^{ \left[\frac{t-h}{h}\right]}(-1)^{j}\bigg{(}\overset{\alpha}{j}\bigg{)}f(t-jh) \tag{7}\]  
where operator \(\left[\cdot\right]\) provides the integer part. Note that the application of these definitions to the system presented in this article is under initial zero conditions [16; 17], and the two definitions in Equations (6) and (7) become equivalent. Starting with PID control law, after applying the Laplace transform in the complex domain \(s\), the following equation can be obtained [15]:  
\[U(s)=K_{p}E(s)+K_{i}\frac{1}{s}E(s)+K_{d}sE(s) \tag{8}\]  
From Equation (8), which represents the output of the controller in the complex domain \(s\) in the integer case, FO-type controllers can be further described in the fractional case, using the notions of fractional computation described above. * FO-PI Speed Controller for PMSM-SCS Based on DTC Strategy  
The most commonly used FO controllers are of the \(PI^{\lambda}D^{\mu}\) type. The description equation in the time domain is expressed as follows [15]:  
\[u(t)=K_{p}e(t)+K_{i}D^{-\lambda}e(t)+K_{d}D^{\mu}e(t) \tag{9}\]  
where \(e(t)\)--signal error. Using the Laplace transform for the Equation (9) with zero initial conditions gives the next equation, which represents the FO-PI speed controller for the PMSM-SCS based on DTC strategy [15]:  
\[G_{c}(s)=K_{p}+\frac{K_{i}}{s^{\lambda}}+K_{d}s^{\mu} \tag{10}\]  
where the proportional factor is noted with \(K_{p}\); the integral factor is noted with \(K_{i}\); the integrator order (positive) is noted with \(\lambda\); the differential factor is denoted by \(K_{d}\); and the differentiator order is denoted by \(\mu\). For \(\lambda=\mu=1\), the result is an integer-order PID controller. * TID Speed Controller for PMSM-SCS Based on DTC StrategyAnother fractional controller similar to the PI\({}^{\Lambda}\)D\({}^{\mu}\) controller is the TID controller, which can be expressed by the following transfer function for the PMSM-SCS, based on DTC strategy [15]:  
\[G_{c}(s)=\frac{K_{t}}{s^{1/n}}+\frac{K_{i}}{s}+K_{d}s \tag{11}\]  
where the slope gain is denoted by \(K_{i}\); the integration order of the slope term is noted with \(n\); the gain of the integrator term is denoted by \(K_{i}\); and the gain of the derivator term is denoted by \(K_{d}\).