## 5 Simulation  
This section showcases the simulation results that validate the effectiveness of the proposed control strategy. Detailed simulation parameters for the model, observer, and controllers are provided in Tables 1-3, respectively. The parameters in Tables 2 and 3 were obtained through multiple rounds of tuning and experience. These tables list the specific values and configurations employed in the simulations, ensuring that the results are clear and reproducible. The letter A, B, C represent the enlarged part and correspond to the corresponding figures. Figure 2 illustrates a comparison of the time taken by the CSTRL, NSMRL, and FTSTRL systems to achieve speed regulation at a target of 1000 rpm. This comparison highlights the relative performance and efficiency of each control strategy. As shown, the FTSTRL reached the desired speed of 1000 rpm in just 0.02 s, whereas the CSTRL and NSMRL took 0.05 and 0.048 s, respectively. This indicates that the FTSTRL outperformed both the CSTRL and NSMRL. Figure 3 illustrates the corresponding variations in current, reflecting the changes in control inputs. Moreover, Figure 4 presents an in-depthcomparative and quantitative analysis of the speed regulation simulation results. This analysis employs key performance metrics such as ISE, ITSE, IAE, and ITAE. By leveraging these metrics, we can rigorously evaluate each control strategy's performance, focusing on its precision, response time, and overall effectiveness. Figure 3: Comparison between CSTRL, NSMRL, and FTSTRL for control input of speed regulation. Figure 2: Comparison between CSTRL, NSMRL, and FTSTRL for speed regulation. **Remark 2**. : _To guarantee a fair comparison, all three controllers are implemented with the same ESDO and identical parameter settings. The exact parameters used for the simulations are comprehensively listed in Table 2. This uniformity in the setup ensures that any observed differences in performance can be attributed solely to the control strategies themselves, rather than variations in the observer or parameter configurations._ \begin{table}
\begin{tabular}{c c c} \hline \hline
**Parameters** & **Value** & **Unit** \\ \hline Resistance \(R\) & 2.875 & \([\Omega]\) \\ Inductance \(L_{s}\) & 8.5 & \([\text{mH}]\) \\ Flux linkage \(\psi_{f}\) & 0.175 & \([\text{Wb}]\) \\ Inertia \(J\) & 0.0003 & \([\text{kg}\cdot\text{m}^{2}]\) \\ Damping coefficient \(B\) & 0.0008 & \([\text{N}\cdot\text{m}\cdot\text{s}/\text{r}]\) \\ Pole pairs \(p\) & 4 & \\ Switching frequency & 20,000 & \([\text{Hz}]\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Parameter values of the PMSM [31]. \begin{table}
\begin{tabular}{c c} \hline \hline
**Reaching Laws** & **Value** \\ \hline CSTRL & \(k_{1}=0.28,k_{2}=1\) \\ NSMRL & \(\kappa=0.1,\chi=0.03,\zeta=0.01,\rho=0.01\) \\ FTSTRL & \(k_{1}=0.28,k_{2}=1,k_{3}=0.03,k_{4}=0.1,p=3,q=1\) \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Parameter values of the 3 reaching laws. Figure 4: Performance indexes such as ISE, IAE, ITAE, and ITSE for speed regulation. **Remark 3**. : _The parameter values for the same parameters are identical and are listed in Table 3 along with their corresponding values._  
Figure 5 illustrates the speed tracking performance of the CSTRL, NSMRL, and FTSTRL. The desired speed undergoes several changes: from 0 to 200 rpm at 0.01 s, from 200 to 900 rpm at 0.11 s, from 1000 to \(-200\) rpm at 0.22 s, and from \(-200\) to 800 rpm at 0.42 s. The results clearly show that the FTSTRL achieves superior tracking performance compared to the CSTRL and NSMRL. Figure 6 depicts the corresponding current variations, which reflect the control input. Additionally, Figure 7 presents a detailed comparative and quantitative analysis of the simulation results for speed tracking, utilizing the metrics of ISE, ITSE, IAE, and ITAE.