## 3 Numerical Simulations Based on Matlab/Simulink for the Proposed PMSM-SCS Using DTC Strategy
### Numerical Simulation for PMSM-SCS Based on DTC Strategy, Using Constant Flux  
The software implementation carried out in Matlab/Simulink version 2021b for the PMSM-SCS is based on the DTC strategy in the case of constant flux. Thus, for the PI speed controller described in Equation (8), in the case of the sensorless PMSM control system based on the constant-flux DTC strategy, the tuning parameters obtained by empirical methods are \(K_{p}=0.05\) and \(K_{i}=0.91\). Using an EOA-type algorithm, the tuning parameters obtained are \(K_{p}=0.03\) and \(K_{i}=0.66\). The FOMCON toolbox in Matlab allows the tuning of \begin{table}
\begin{tabular}{l c c} \hline \hline
**Parameter** & **Value** & **Unit** \\ \hline Stator resistance (\(R_{s}\)) & 2.875 & \(\Omega\) \\ Inductances on the \(d\)-\(q\) rotating reference frame (\(L_{d}\) and \(L_{d}\)) & 0.0085 & H \\ Combined inertia of rotor and load (\(J\)) & 0.0008 & kg\(\cdot\)m\({}^{2}\) \\ Combined viscous friction of rotor and load (\(B\)) & 0.005 & N\(\cdot\)m\(\cdot\)s/rad \\ Flux induced by the permanent magnets of the rotor in the stator phases (\(\lambda_{0}\)) & 0.175 & Wb \\ PMSM pole pair number (\(n_{P}\)) & 4 & - \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Nominal PMSM parameters of the PMSM-SCS based on the DTC strategy. an FOPI-type controller described by Equation (10), in the case of PMSM-SCS based on a constant-flux DTC strategy. The values obtained are the following: \(K_{p}=6.2\), \(K_{i}=24\), \(\lambda=1.1\), and \(K_{d}=\mu=0\), and the FOPI speed-controller transfer function becomes  
\[H_{FOPI}(s)=\frac{6.2s^{1.1}+24}{s^{1.1}} \tag{53}\]  
For the closed-loop study of the system consisting of the FOPI controller given in Equation (53) and the fixed part given by the PMSM carried out with the FOMCON toolbox from Matlab, the following fractional-order transfer function is obtained:  
\[H_{FOPI\_CL}(s)=\frac{17.71s^{1.1}+68.58}{5.44\cdot 10^{-8}s^{4.2}+2.58\cdot 10^{ -5}s^{3.2}+0.03s^{2.2}+18.47s^{1.1}+68.56} \tag{54}\]  
The stability of the closed-loop system of the PMSM-SCS based on the constant-flux DTC strategy and the PI speed controller is given by satisfying the condition given by \(|\arg(eig(A))|>q\frac{\pi}{2}\)[15], where \(q\in(0,\ 1)\) is the proportional order and \(eig(A)\) is the eigenvalue of the associated matrix \(A\) of the state-space representation for the system defined in Equation (54). The graphical representation is presented in Figure 4a, using the FOMCON toolbox, and the analysis of this figure shows that the system is stable. Figure 4b shows the step-response signal of the closed-loop system when using the FOPI speed controller. Figure 5 depicts the Bode diagram for the closed-loop system consisting of the FOPI controller given by Equation (53) in the case of PMSM-SCS based on the DTC strategy using the FOPI speed controller at constant flux. Thus, Figure 5a presents the Bode magnitude plot, and the Bode phase plot is presented in Figure 5b. According to the Bode stability criterion, it is observed that the system is stable, with a phase stability reserve of about \(90^{\circ}\) and an amplitude stability reserve of about \(20\) dB. These values indicate that the system has a good reserve of stability in both phase and amplitude characteristic elements of a robust control system. Figure 4: Closed-loop stability test for the PMSM-SCS based on the DTC strategy using FOPI speed controller at constant flux: (**a**) Graphical representation for stability analysis of the system; (**b**) Step-response signal of the closed-loop system. Figure 5: Bode diagram graphical representation of the PMSM sensorless control loop based on the DTC strategy using the FOPI speed controller at constant flux: (**a**) Bode magnitude plot; (**b**) Bode phase plot.