## 5 Simulation Experiments  
### Overshoot Optimization Methods  
Each initial particle generated by the intelligent algorithm contains a total of five parameters: \(K_{p}\), \(K_{i}\), and \(K_{d}\) in the PID controller, as well as \(K_{p}\) and \(K_{d}\) in the PD controller. The intelligent algorithm is integrated with the simulation model of the bridge crane PID control system, enabling simulation analysis of the PID controller model to derive the dynamic performance indicators of the system. Throughout the iteration process, the dynamic performance indicators of the system serve as adaptive fitness values transmitted to the intelligent algorithm program. The adaptive fitness value determines whether the termination condition of the algorithm is met. If the termination condition is satisfied, the iteration concludes, and the optimal solution is returned. Otherwise, the iteration computation continues, repeating the preceding parameter-setting steps until the optimal PID controller parameters are determined.  
The optimal PID gains are determined by minimizing the objective function. There are typically four performance indices used to evaluate system performance [38]: (1) Integral of Squared Error \((ISE)\); (2) Integral of Absolute Error \((IAE)\); (3) Integral of Time multiplied by Squared Error \((ITSE)\); (4) Integral of Time multiplied by Absolute Error \((ITAE)\). Generally, each system aims to reduce error, minimize overshoot, and enhance response speed. In this study, the \(ITAE\) is employed as the performance evaluation criterion, as it enables the system to exhibit rapid response characteristics. The expression for the fitness function is as follows:  
\[ITAE=\int_{0}^{T}t|e(t)|dt \tag{20}\]  
where \(e(t)\) represents the difference between the output and input of the control system, while \(T\) denotes the simulation time. Additionally, considering the necessity of limiting overshoot in crane position, a penalty term is introduced when the overshoot of the crane position controller exceeds 0. Assuming the system comprises two outputs and two controllers, the expression for fitness is as follows:  
\[G=\left\{\begin{array}{l}\int_{0}^{T}\omega_{1}t|e_{1}(t)|+\omega_{2}t|e_{2 }(t)|+\omega_{3}u^{2}(t)dt,e_{1}(t)\leq 0\\ \int_{0}^{T}\omega_{1}t|e_{1}(t)|+\omega_{2}t|e_{2}(t)|+\omega_{3}u^{2}(t)+ \omega_{4}e_{1}(t)dt,e_{1}(t)>0\end{array}\right. \tag{21}\]  
where \(e_{1}(t)\) and \(e_{2}(t)\) represent the errors of the crane position controller and the swing angle suppression controller, respectively. \(G\) denotes the fitness function for all algorithms in the experiment. \(u(t)\) is a control value employed to prevent excessive control margins. \(\omega_{1}\) and \(\omega_{2}\) are weighting factors for adjusting the effect of different errors on the fitness function, typically ranging from 0 to 1. In this study, \(\omega_{1}\) and \(\omega_{2}\) are set to 0.95. \(\omega_{3}\) is another weighting factor, also typically ranging from 0 to 1. In this study, \(\omega_{3}\) is set to 0.05. \(\omega_{4}\) represents the overshoot penalty coefficient. When the error in the trolley position is greater than 0, a larger \(\omega_{4}\) will have a greater effect on the fitness value, and a calibrated value with a larger overshoot will result in a worse fitness value. In this study, \(\omega_{4}\) is set to 100.