## 5 Simulation Experiments
### Simulation and Analysis  
In this section, the parameters of two PID controllers were set using methods such as ISSA, SSA, PSO, SA, and WOA to validate the optimization effectiveness of the proposed algorithms. A simulation control system for each optimization method was established using the SIMULINK module in MATLAB, and the control performance obtained from each parameter-setting method was compared and analyzed under common conditions. By iteratively manually adjusting the parameters of the PID controller beforehand, the range of variation for each parameter could be approximately determined, thereby significantly reducing the scope of PID parameter optimization. The parameter ranges of the PID controller obtained through repeated testing and operation are shown in Table 4.  
To evaluate the control performance of the designed bridge crane PID control system, the expected position signal of the crane trolley is set to a step signal of 5 m, and the initial swing angle of the crane payload is set to 0\({}^{\circ}\). The simulation time is set to 30 s, and the solver is chosen to be variable step size. The parameter values of the crane are shown in Table 5.  
The improved PID controllers generated by five intelligent algorithms were connected to the bridge crane control system, and the iteration curves are illustrated in Figure 6. The performance index functions of the ISSA, SSA, PSO, SA, and WOA algorithms were recorded as 17.03, 18.64, 37.15, 24.05, and 22.11, respectively. The performance index function value achieved by the ISSA algorithm was the smallest among all algorithms, exhibiting an improvement of 8.64% in search accuracy compared to the SSA algorithm. The iteration count was reduced by 10.77% to 58 iterations. Compared to the PSO, SA, and WOA algorithms, the performance index function values decreased by 20.02, 7.02, and 5.08, respectively, for the ISSA algorithm. These results suggest that the proposed ISSA algorithm holds a certain advantage over other algorithms in terms of iteration count and precision. The values of the PID parameters calculated by the five algorithms are shown in Table 6. \begin{table}
\begin{tabular}{c c c c c} \hline \hline \multirow{2}{*}{**Algorithm**} & \multicolumn{4}{c}{**Parameters**} \\ \cline{2-5}  & \(\mathbf{K_{p}}\) **(PID)** & \(\mathbf{K_{i}}\) **(PID)** & \(\mathbf{K_{d}}\) **(PID)** & \(\mathbf{K_{p}}\) **(PD)** & \(\mathbf{K_{d}}\) **(PD)** \\ \hline ISSA & 9.8435 & 0.0892 & 20.4867 & 0.9452 & 2.0364 \\ SSA & 10.3538 & 0.0878 & 22.9724 & 0.9985 & 6.2987 \\ PSO & 19.9788 & 0.0663 & 49.4362 & 9.9789 & 0.1376 \\ SA & 18.1187 & 0.0654 & 38.5331 & 7.5068 & 0.3206 \\ WOA & 13.1361 & 0.0810 & 29.1224 & 9.9413 & 12.0409 \\ \hline \hline \end{tabular}
\end{table} 
Table 6: Parameters of bridge crane PID controllers obtained by ISSA, SSA, PSO, SA, and WOA algorithms. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \multirow{2}{*}{**Symbols**} & \multirow{2}{*}{**Meaning**} & **ISSA** & **SSA** & **PSO** & **SA** & **WOA** \\ \hline \([r_{\text{min}},r_{\text{max}}]\) & Search range of parameter \(K_{p}\) & [0, 50] & [0, 50] & [0, 50] & [0, 50] & [0, 50] \\ \([r_{\text{min}},r_{\text{max}}]\) & Search range of parameter \(K_{i}\) & [0, 2] & [0, 2] & [0, 2] & [0,  
In the position curves of the trolley, the peak times for the ISSA, SSA, PSO, SA, and WOA algorithms are 5.074 s, 7.904 s, 8.268 s, 12.392 s, and 8.673 s, respectively, while the steady-state times are 4.791 s, 7.379 s, 15.586 s, 12.837 s, and 8.632 s, respectively. Among these algorithms, the ISSA algorithm exhibits the smallest peak time and steady-state times, followed by the SSA and WOA algorithms. Conversely, the PSO algorithm requires the longest time to stabilize after reaching the peak, needing 7.378 s. Furthermore, the ISSA algorithm demonstrates the smallest steady-state error of 0.002, followed by the SSA, WOA, SA, and PSO algorithms, with errors of 0.006, 0.006, 0.048, and 0.050, respectively.