## 4 Simulation Study  
To verify the effectiveness of this paper's theory in error compensation and weakening overshoot, a control simulation model was built in MATLAB/Simulink R2022a for a simulation analysis. The relevant SPMSM parameters are listed in Table 1. Because the motor parameters cannot be changed artificially in practice, the mismatches in the parameters of the actual motor caused by external influences were simulated by changing the predictive model motor parameters. Further, the DPCC method used in the simulations and in the experiment was one-step delay compensation DPCC. The conditions of the simulation were set as follows: the control frequencies of the current and speed loop were 10 kHz and 1 kHz, respectively. The DC voltage source was set to 60 V, and the weight factors were \(x=y=0.5\). In order to demonstrate the effects of adding weight factors under a mismatch of the inductance parameters, a simulation of the inductance mismatch (\(L_{0}=2L_{s}\)) of the predictive model was set under the following conditions: the rotational speed was 450 rpm and the load torque increased from 0 to 1 Nm at 0.2 s and decreased from 1 to 0 Nm at 0.4 s when the weight factors were \(x=y=0.5\). Figure 3a shows that the \(dq\)-axis feedback current under DPCC undergoes obvious fluctuations, while Figure 3b shows that the \(dq\)-axis feedback current does not undergo obvious fluctuations after weight factors are added. This shows that the impact generated by the inductance mismatch is suppressed, but it also shows the generation of certain steady-state errors. Simulation tests of the current loop were carried out to demonstrate that the weakened integral sliding mode structure improves the integral saturation and reduces the overshoot. The simulation conditions were such that the \(q\)-axis reference current increased from 0 to 3 A at 0.1 s. Figures 4-6 represent the \(q\)-axis feedback and the reference currents when the three parameters of the model, namely inductance, resistance, and flux linkage, were not mismatched as well as when they were simultaneously mismatched. Figures 4a, 5a and 6a indicate the \(q\)-axis feedback and the reference currents of the normal integral sliding mode optimization combined with the weight factors, and Figures 4b, 5b and 6b indicate the \(q\)-axis feedback \begin{table}
\begin{tabular}{c c} \hline \hline
**Parameter** & **Value** \\ \hline Rated voltage (V) & 220 \\ Rated speed (rpm) & 3000 \\ Rated torque (Nm) & 2.4 \\ Number of pole pairs & 4 \\ \(dq\)-axis inductance (mH) & 5 \\ Stator resistance (\(\Omega\)) & 1.08 \\ Flux linkage (Wb) & 0.0819 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: SPMSM parameters. Figure 3: Comparison of \(dq\)-axis current simulations of two methods at 450 rpm with torque variation (1 Nm) in case of inductance mismatch (\(L_{0}=2L_{s}\)): (**a**) DPCC; (**b**) DPCC with weight factors. and the reference currents of the weakened integral sliding mode optimization combined with the weight factors. Moreover, methods (a) and (b) differ from each other only in the construction of the sliding mode surface. It can be seen in (a) that current overshooting occurs during current variation, while it is clear that in (b) the weakened integral sliding mode structure suppresses this phenomenon well. Figures 7-9 represent comparisons of simulations about \(dq\)-axis currents between DPCC and DPCC optimized with weakened integral sliding mode compensation combined  
Figure 4: Comparison of \(q\)- axis current simulations of two methods with current variations (0 A to 3 A) and without parameter mismatches (\(R_{0}=R_{s}\), \(L_{0}=L_{s}\), and \(\psi_{f0}=\psi_{f}\)): (**a**) DPCC with normal integral sliding mode compensation combined with weight factors; (**b**) DPCC with weakened integral sliding mode compensation combined with weight factors.