## 5 Results and Discussion  
To evaluate the effectiveness of the proposed hybrid PSO methods, we conducted a series of simulation runs in this study. Four different algorithms were employed:PSO, HPSO-QN SM, HPSO-QN SLSM, and HPSO-QN MLSM. Each algorithm was tested independently five times, with randomized initial positions. The proposed algorithms utilized a particle count (\(N_{PSO}\)) of 50 and a maximum iteration count (\(k_{max}\)) of 300. The choice of \begin{table}
\begin{tabular}{c c c} \hline \hline
**Parameter** & **SI Units** & **Normalized Units** \\ \hline Motor Inertia (\(J_{1}\)) & \(1.5602\times 10^{-3}\) kg\(\cdot\)m\({}^{2}\) & 0.8713 s \\ Load Inertia (\(J_{2}\)) & \(1.3966\times 10^{-3}\) kg\(\cdot\)m\({}^{2}\) & 0.7799 s \\ Shaft stiffness (\(K\)) & \(19.4840\) N\(\cdot\)m/rad & 10.8819233 p.u. \\ Motor Coulomb Friction (\(C_{1}\)) & 0.0182 N\(\cdot\)m & 0.0304 p.u. \\ Load Coulomb Friction (\(C_{2}\)) & 0.0162 N\(\cdot\)m & 0.0271 p.u. \\ Motor Viscous Friction (\(B_{1}\)) & \(3.7167\times 10^{-3}\) N\(\cdot\)m/rad\(\cdot\)s & 2.0759 p.u. \\ Load Viscous Friction (\(B_{2}\)) & \(3.5530\times 10^{-3}\) N\(\cdot\)m/rad\(\cdot\)s & 1.9844 p.u. \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Benchmark parameters of the two-mass model (2 MM). Figure 3: Block diagram illustrating the overall system configuration of the two-mass model (2 MM). 300 iterations as the maximum value was determined by monitoring the convergence of the algorithm, specifically through the _Fcost_ plot analysis. To define the search space bounds, the following upper and lower bounds for the position and velocity vectors were assigned:  
\[ \begin{split}\vec{x}_{\min}(k)&=\begin{bmatrix}0&0&0&0&0&0 &0\\ \vec{x}_{\max}(k)&=\begin{bmatrix}4&4&40000&0.5&0.5&5&5\end{bmatrix}\end{split}  \tag{22}\]  
These bounds correspond to the following parameters:  
\[\vec{x}(k)= \begin{bmatrix}J_{1n}&J_{2n}&K_{n}&C_{1n}&C_{2n}&B_{1n}&B_{2n} \end{bmatrix}  \tag{23}\]  
To assess the effectiveness of the switching criteria for the local search methods (HPSO-QN SLSM and HPSO-QN MLSM), we evaluated different values of \(F_{th}\): specifically, 5%, 10%, and 20%. Our findings showed that HPSO-QN SLSM achieved the lowest cost function and standard deviation of the cost function when \(F_{th}\) was set to 5%. Furthermore, we investigated the impact of different values of \(B\) in the HPSO-QN MLSM method. We tested two scenarios: selecting either the 5 or 10 best particles for the QN local search with \(B\) set to 5% and 10%, respectively. After evaluating the results, we observed that the lowest standard deviation of the cost function and parameters, as well as the lowest absolute percentage error (APE) of the parameters, were achieved when \(F_{th}\) was set to 5% and \(B\) was set to 5%. The results of the lowest cost function (_Fcost_) using \(F_{th}=5\%\) and \(BP=5\%\) are presented in Table 2.  
Based on these results and compared with the benchmark parameters in Table 1, the proposed HPSO-QN methods demonstrated their effectiveness in estimating the mechanical parameters of the 2 MM system. The HPSO-QN MLSM method achieved the lowest cost function value of \(2.32\times 10^{-6}\) with a high degree of speed response accuracy. Table 3 compares the standard deviation values of the identified parameters using four different PSO methods. The HPSO-QN MLSM method consistently exhibits the lowest standard deviation values, highlighting its robustness. In contrast, the SPSO method demonstrates relatively higher standard deviation values compared to the HPSO-QN methods.