## 5 Results and Discussions
### Optimization Results  
To evaluate the performance of the two sets of proposed super-helical sliding mode model reference adaptive systems (STSM-MRAS) in optimizing design, simulations were conducted with the motor's initial operating speed set at 1000 r/min and subjected to an initial load torque of 10 N\(\cdot\)m. At 0.5 s into the simulation, an abrupt change occurred, causing the motor speed to transition to 3500 r/min. Figure 10: Motor speed error results. Figure 11 displays the simulated waveforms of motor speed corresponding to the two optimization algorithms when a sudden change in motor speed occurs. Likewise, Figure 9 illustrates the simulated waveforms of motor torque corresponding to the two optimization algorithms under the same conditions. Furthermore, Figure 10 exhibits the Z-curve plots of the standard evaluation of PSSEE complementary to the two optimization algorithms during a sudden change in motor speed. As depicted in Figure 11, the motor's speed overshoot is notably reduced under both optimization algorithms. Specifically, in Figure 11, employing the NSGA-II optimization algorithm, the initial rotational speed overshoot decreases from 138 r/min with the initial STSM algorithm to 124 r/min. Similarly, at 0.5 s, the rotational speed overshoot decreases from 118 r/min to 85 r/min. In Figure 11, utilizing the MOPSO optimization algorithm, the initial rotational speed overshoot decreases from 138 r/min to 112 r/min with the initial STSM algorithm. Correspondingly, at 0.5 s, the rotational speed overshoot diminishes from 118 r/min to 73 r/min. Notably, the time taken to reach a steady state remains consistent across all scenarios. Calculations demonstrate that under the NSGA-II and MOPSO optimization algorithms, the speed overshoot of the permanent magnet synchronous motor decreases by 27.97% and 38.13% during sudden speed changes and by 10.15% and 18.84% during startup, respectively. The torque overshoot measures the degree to which the output torque of the motor surpasses its steady-state value in response to dynamic stimuli. The calculation methodology is delineated in Equation (50), where \(M_{T}\) represents the torque overshoot, \(T_{max}\) denotes the peak torque, and \(T_{s8}\) signifies the steady-state torque. \[\mathbf{M}_{T}=\mathbf{T}_{\text{max}}-T_{ss} \tag{50}\]  
Figure 11: Motor speed results in sudden speed changes. As depicted in Figure 12, the overshooting of the motor torque has significantly decreased through the optimization of both algorithms. In Figure 12a, under the NSGA-II optimization algorithm, the motor torque overshoot decreases from 40.42 N to 37.25 N during motor startup and from 38 N to 33.98 N during the 0.5 s speed mutation. In Figure 12b, employing the MOPSO optimization algorithm, the torque overshoot during motor startup decreases from 40.42 N to 37.15 N, and the motor torque overshoot diminishes from 38 N to 33.95 N at 0.5 s of sudden speed change. Calculations reveal that the torque overshoot of the permanent magnet synchronous motor decreases by 7.8% and 8.1% during sudden speed changes and by 10.57% and 10.66% during startup under the NSGA-II and MOPSO optimization algorithms, respectively. Based on the simulation data and calculations, it is evident that the MOPSO optimization algorithm proves more effective in reducing both speed and torque overshoot compared to the NSGA-II algorithm. However, the performance of the two algorithms remains comparable. Therefore, the Shannon entropy-based PSSEE introduced in this paper can be further utilized to analyze the effect of torque overshoot reduction under both optimization algorithms. This formulation underscores the potential of the PSSEE methodology in providing deeper insights into the performance nuances between different optimization strategies. This statement emphasizes the significance of utilizing the PSSEE methodology for a comprehensive assessment of the torque overshoot reduction effects under various optimization algorithms.