## 6 Simulation Results  
The relevant simulation verifications are carried out in this section by MATLAB/Simulink. The detailed parameters of the selected SRM are listed in Table 1. The DITC method is common in SRM torque control. Therefore, the simulation results of the DITC method, the MPTC method in [31], and the PWM-DITC method in [36] under the same conditions are also given for comparison. The sampling frequency and control frequency are both set at 10 kHz to ensure the consistency of simulation verification and experimental verification. Since the total reference torque signal is calculated by the speed controller, the PI speed controller of the same specification (\(k_{p}=0.14\), \(k_{i}=0.18\)) is used in the verification of the above methods. In the torque controller, the parameters in the MPTC method are determined by the rules in [31]. On the other hand, the cost function in the proposed DPTC method is only related to the current, which avoids the adjustment of parameters and reduces the difficulty of algorithm application. The simulation results of the four methods at 200, 400, and 800 rpm speeds when the load is 5 Nm are shown in Figures 6-8. From the current waveforms, it can be seen that both the DITC and PWM-DITC have a peak current of approximately 6 A. The MPTC and the proposed DPTC methods have smaller current peaks due to their optimization of the phase torque allocation during commutation. However, the MPTC shows a large current ripple, which is caused by a large variation in terminal voltage due to its fixed switching vector (\(-1\), \(0\), \(1\)). On the contrary, the proposed DPTC method makes the change of terminal voltage relatively continuous by calculating the duty cycle, so that the phase current surrounds the reference signal tightly. This advantage is also reflected in the torque. Undoubtedly, in the torque waveforms, the DPTC method has a lower torque ripple than the MPTC method. To quantitatively compare the simulation results of the above methods, three performance indices, namely the peak torque ripple \(T_{r,p}\), the RMS \(T_{r,RMS}\) of the torque ripple, and the copper loss index \(I_{RMS}\), are given. The expressions of \(T_{r,p}\), \(T_{r,RMS}\), and \(I_{RMS}\) are shown in (32)-(34), where \(T_{total}\) is the total output torque, \(T_{av}\) is the average torque, and \(T_{max}\) and \(T_{min}\) are the maximum and minimum values of the torque ripple, respectively. The calculation results are summarized in Table 2.  
\[T_{r,p}=\frac{T_{\max}-T_{\min}}{T_{av}}\times 100\% \tag{32}\]  
\[T_{r,RMS}=\sqrt{\frac{1}{N}\sum_{k=1}^{N}\left(T_{total}(k)-T_{av}\right)^{2}} \tag{33}\]  
\[I_{RMS}=\sqrt{\frac{1}{N}\sum_{p=1}^{3}\sum_{k=1}^{N}i_{p}(k)^{2}R} \tag{34}\] \begin{table}
\begin{tabular}{c c} \hline \hline
**Parameters** & **Value** \\ \hline Phase & 3 phases \\ Stator/rotor poles & 12/8 \\ Rated power & 1.5 KW \\ Rated torque & 9.55 N\(\cdot\)m \\ Speed range of constant torque & 1000-1500 r/min \\ Maximum flux linkage & 0.986 Wb \\ Stator resistance \(R_{p}\) & 0.9 G \\ Moment of inertia \(J_{m}\) & 0.01 kg\(\cdot\)m\({}^{2}\) \\ Friction coefficient \(B_{m}\) & 0.005 N\(\cdot\)m\(\cdot\)s \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Specifications of the SRM. Figure 6: Simulation results of (**a**) the DITC method, (**b**) the FCS MPTC method in [31], (**c**) the PWM-DITC method in [36], and (**d**) the proposed DPTC method at 200 rpm with a load of 5 Nm. Figure 7: Simulation results of (**a**) the DITC method, (**b**) the FCS MPTC method in [31], (**c**) the PWM-DITC method in [36], and (**d**) the proposed DPTC method at 400 rpm with a load of 5 Nm. From Table 2, the DITC method is the worst choice among all the methods with respect to the speeds of 200, 400, and 800 rpm, which is caused by the hysteresis effect of the hysteresis control. In terms of torque ripple, DITC has a relatively high torque ripple index \(T_{r,p}\) of 39.1%, 42.9%, and 58.4%, respectively. Similarly, the \(T_{r,p}\) of MPTC at different speeds are 38.2%, 41.8%, and 53.2%, respectively.