## 4 MPTC System Simulation Experiment
### Dynamic Performance  
When the load undergoes a sudden change, the motor speed is set to 600 rad/min, with an initial torque of 2.5 N\(\cdot\)m. After 0.2 s from startup, the torque suddenly increases to 5 N\(\cdot\)m, and then drops back to 2.5 N\(\cdot\)m after 0.4 s. The dynamic waveforms of the IMPTC, IMPTC-I, and IMPTC-II strategies are compared (see Figures 9-11), along with their dynamic performance (see Table 4). The IMPTC-II system has a current harmonic \(THD_{eq}\) of 16.46%, which is 44.01% and 29.21% lower compared to IMPTC and IMPTC-I, respectively. The torque pulsation \(T_{\text{RMSE}}\) of the IMPTC-II system is 15.07%, which is 43.19% and 29.28% lower compared to IMPTC and IMPTC-I, respectively. The magnetic flux pulsation \(\psi_{\text{RMSE}}\) for the IMPTC-II system is 4.24%, which is 6.84% and 2.6% lower compared to IMPTC and IMPTC-I, respectively. Therefore, under dynamic conditions, the control performance of IMPTC-II is superior to both IMPTC-I and IMPTC, while IMPTC-I also performs better than IMPTC.  
To reflect the computation time of the algorithms, the processing time for each segment will be considered by iterating the algorithms 1000 times. The results are shown in Table 5.  
The table shows that there is little difference in the time required for measurement and speed control between IMPTC and IMPTC-I. However, the proposed IMPTC-II requires \(t_{m}\)= 3.20 us for measurement and speed control, _test_ = 6.33 us for estimation and delay compensation, and \(t_{pred}\)= 2.87 us for prediction, demonstrating that IMPTC-II effectively simplifies the computation and improves the system efficiency.  
The parameters of a permanent magnet synchronous motor (PMSM) are ideally considered constant. However, in practice, factors like thermal effects on resistance and environmental disturbances can cause these parameters to vary [31]. Such inaccuracies can lead to model prediction errors, which reduce the quality of the optimal voltage vector determined by the model predictive torque control (MPTC), affecting the system control performance and potentially causing instability. By using an improved extended state  
\begin{table}
\begin{tabular}{c c c c} \hline \hline  & **IMPTC** & **IMPTC-I** & **IMPTC-II** \\ \hline \(THD_{eq}\) & 60.47\% & 45.67\% & 16.46\% \\ \(T_{\text{RMSE}}\) & 58.26\% & 46.03\% & 15.07\% \\ \(\psi_{\text{RMSE}}\) & 14.89\% & 10.65\% & 8.05\% \\ \hline \hline \end{tabular}
\end{table}
Table 4: Evaluation indexes for the dynamic simulation of three control strategies.  
\begin{table}
\begin{tabular}{c c c c} \hline \hline  & **IMPTC** & **IMPTC-I** & **IMPTC-II** \\ \hline \(THD_{eq}\) & 40.68\% & 21.67\% & 12.84\% \\ \(T_{\text{RMSE}}\) & 60.85\% & 40.03\% & 10.75\% \\ \(\psi_{\text{RMSE}}\) & 9.43\% & 7.97\% & 4.24\% \\ \hline \hline \end{tabular}
\end{table}
Table 3: Evaluation indexes of the steady-state simulation of three control strategies.  
observer (ESO) for the parameter compensation, the impact of parameter variations on motor control can be mitigated [32]. For comparison, the same method as used in the dynamic testing was applied [3], setting the resistance to 2R, stator flux linkage to 1.5 \(\varphi\)s, and inductance to 0.5 L [33]. The simulation results are as follows.  
Figure 11: IMPTC–II dynamic performance.  
Figure 10: IMPTC–I dynamic performance.