## 5 Results and Discussion
In terms of motor-side and load-side inertias (\(J_{1n}\) and \(J_{2n}\)) and shaft stiffness (\(K_{n}\)), the HPSO-QN MLSM method achieves the lowest error, with APE values ranging from 3.53% to 15.05%. Additionally, it performs well for \(B_{2n}\) with an APE of 9.76%. These results highlight the effectiveness of the HPSO-QN method in accurately identifying mechanical parameters, providing more reliable results compared to other methods. However, it is important to note that there is still a significant discrepancy between the estimated and actual values for the friction coefficients. This discrepancy may be attributed to factors such as measurement noise and the potential variability of these coefficients over time. Despite this limitation, the HPSO-QN method yields lower errors for the friction coefficients compared to the SPSO method. The convergence of the proposed methods is demonstrated in Figure 4, showing the \(Fcost\) versus the number of iterations for each algorithm. The plot highlights the effectiveness of the implemented switching criteria and particle selection in the hybrid method. Compared to the SPSO algorithm, which often gets trapped in the local optima, the HPSO-QN methods demonstrate faster convergence to a low-cost function in the initial stages and continue to search within that region. Moreover, the SPSO algorithm demonstrates a slower convergence speed when compared to the HPSO-QN methods. \begin{table}
\begin{tabular}{l c c c c c c c} \hline \hline
**PSO Methods** & \(J_{1n}\) **(\%)** & \(J_{2n}\) **(\%)** & \(K_{n}\) **(\%)** & \(C_{1n}\) **(\%)** & \(C_{2n}\) **(\%)** & \(B_{1n}\) **(\%)** & \(B_{2n}\) **(\%)** \\ \hline SPSO & 42.95 & 69.13 & 52.13 & 48.03 & 5.73 & 25.02 & 53.21 \\ HPSO-QN SM & 20.40 & 43.62 & 28.98 & 30.59 & 0.18 & 2.11 & 37.93 \\ HPSO-QN SLSM & 3.68 & 27.95 & 9.92 & 4.93 & 16.82 & 24.10 & 24.50 \\ HPSO-QN MLSM & 6.46 & 15.05 & 3.53 & 5.59 & 22.74 & 18.86 & 9.76 \\ \hline \hline \end{tabular}
\end{table} 
Table 6: Performance comparison of the proposed method: absolute percentage error comparison of estimated and actual parameters. Figure 5 compares the _Fcost_ of solutions obtained using the Hybrid PSO-QN MLSM optimization process for both exploration and exploitation processes. This graph demonstrates the effect of both methods at each iteration. It is evident that the QN method aids in escaping from the local optima and finding superior solutions during both exploration and exploitation phases, while the PSO method tends to remain stable in the optimum for a period of iterations. The accuracy of the estimated speeds was evaluated using performance measures, such as the \(IAE\), \(ISE\), and \(ITAE\) metrics as shown in Figure 6. These metrics provide a quantitative measure of the difference between simulated and experimental speed responses. The \(IAE\), \(ISE\), and \(ITAE\) are defined as  
\[IAE=\int_{T_{0}}^{T_{f}}|e(t)|dt \tag{25}\]  
Figure 4: Cost function (_Fcost_) analysis of the proposed methods with the best performance. Figure 5: Comparison of _Fcost_ values obtained using the HPSO-QN MLSM optimization process. \[ISE=\int_{T_{0}}^{T_{f}}e^{2}(t)dt \tag{26}\]  
\[ITAE=\int_{T_{0}}^{T_{f}}t|e(t)|dt \tag{27}\]  
The error \(e(t)\) between the estimated and measured motor speeds is given by \(\Omega_{1}(t)-\Omega_{2}(t)\), and the error between the estimated and measured load speeds is \(\Omega_{2}(t)-\Omega_{2}(t)\). The performance of the estimated model in accurately capturing the speed responses is indicated by smaller values of these measures. The plot indicates that the hybrid PSO-QN MLSM method achieved lower error values compared to SPSO. This demonstrates a higher degree of precision and a significant reduction in error using the hybrid PSO-QN MLSM method. The comparison highlights the accuracy and effectiveness of the hybrid PSO algorithm in estimating the mechanical parameters of the 2 MM system.