## 5 Simulation Experiments
### Robustness Simulation Experiments  
In practical operating scenarios, certain parameters of a bridge crane may incur errors due to measurement methods and environmental variations. Therefore, to evaluate the adaptability and stability of the PID gains obtained by each optimization technique, simulations of the crane control systems using the five algorithms were conducted under different operating conditions. The parameters corresponding to each of the five scenarios are listed in Table 8.  
Figures 9-18 illustrate the trolley position and payload swing obtained by the five algorithms under five different working conditions. The values of the performance index of the five algorithms under these conditions are presented in Table 9. In the position curves under conditions 1 to 5, the ISSA algorithm exhibits the smallest peak time among the four algorithms, with values of 5.014, 9.147, 8.247, 6.070, and 8.147, respectively. Under conditions 2 and 4, all five algorithms demonstrate no overshoot. However, in conditions 1  
Figure 8: Payload swing angle suppression effects of five algorithms. and 5, the PSO algorithm experiences overshoot, measuring 0.46% and 0.72%, respectively. In condition 3, four algorithms exhibit overshoot, with PSO having the highest value of 0.7%. Across all five scenarios, ISSA showcases shorter settling times and lower steady-state errors compared to the other four algorithms, with values of 7.101 and 0.002, 7.715 and 0.002, 7.469 and 0.005, and 7.136 and 0.002, respectively. In the payload swing angle curves for the five scenarios, ISSA consistently demonstrates  
Figure 11: Trolley position control effects for five algorithms under working condition 2. Figure 12: Payload swing angle suppression effects for five algorithms under working condition 2. Figure 13: Trolley position control effects for five algorithms under working condition 3. Figure 16: Payload swing angle suppression effects for five algorithms under working condition 4. Figure 14: Payload swing angle suppression effects for five algorithms under working condition 3. Figure 15: Trolley position control effects for five algorithms under working condition 4. These findings suggest that, following the integration of overshoot penalty terms into the fitness function, the occurrence of overshoot in the position curves across the five scenarios is notably reduced for all five algorithms, with overshoot values consistently below 1%. This indicates that the devised fitness function effectively suppresses overshoot. Furthermore, in comparison to the SSA, PSO, SA, and WOA algorithms, the PID control optimized by the ISSA algorithm demonstrates enhanced adaptability and stability, enabling effective control performance in bridge crane control systems across diverse operational scenarios. Figure 17: Trolley position control effects for five algorithms under working condition 5. \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline \multirow{2}{*}{\begin{tabular}{c} **Working** \\ **Condition** \\ \end{tabular} } & \multirow{2}{*}{\
\begin{tabular}{c} **Improved PID** \\ **Controller** \\ \end{tabular} } & \multirow{2}{*}{
\begin{tabular}{c} **Peak Time** \\ **(s)** \\ \end{tabular} } & \multicolumn{2}{c}{**Trolley Position**} & \multicolumn{2}{c}{**Payload Swing Angle**} \\ \cline{3-8}  & & & **Overshoot** & **Steady-State** & **Steady-State** & **Max Angle** & **Steady State** \\ \hline \multirow{4}{*}{\(1\)} & PSO & 8.206 & 0.46\% & 15.737 & 4.947 & 0.265 & 26.284 \\  & SA & 12.422 & 0 & 15.368 & 4.948 & 0.217 & 25.239 \\  & WOA & 8.697 & 0 & 8.632 & 4.993 & 0.187 & 12.940 \\  & SSA & 8.902 & 0 & 7.715 & 4.998 & 0.160 & 11.254 \\  & ISSA & 5.014 & 0 & 7.101 & 4.998 & 0.156 & 10.836 \\ \hline \multirow{4}{*}{\(2\)} & PSO & 8.247 & 0 & 15.778 & 4.943 & 0.266 & 28.701 \\  & SA & 12.503 & 0 & 15.368 & 4.956 & 0.218 & 22.766 \\  & WOA & 8.861 & 0 & 7.879 & 4.994 & 0.188 & 15.149 \\  & SSA & 9.229 & 0 & 7.960 & 4.998 & 0.161 & 11.149 \\  & ISSA & 9.147 & 0 & 7.715 & 4.998 & 0.157 & 11.030 \\ \hline \hline \end{tabular}
\end{table}