# Multi-Objective Motion Control Optimization for the Bridge Crane System
## 4 Numerical Simulation  
In this section, the simulation validation and control performance validation are conducted. All simulations are carried out based on Matlab/Simulink, and the MOGA is implemented with its built-in standard code. The crane trajectory is designed with the constraints provided by the TP. The anti-disturbance of the control is solved combining the LQR. The model parameters are shown in Table 1.  
Based on the MOGA, a series of Pareto fronts can be achieved in light of the different requirements after iterations. Since the security of the cargo has a higher priority, the swing optimal solution is considered chiefly in this section, and the optimal control solutions of the Pareto front are shown in Figure 5. The horizontal coordinate and the vertical coordinate denote the ITAEs of the swing angle and the crane's position, respectively. According to the actual application, three solutions, including the time optimization, swing angle minimization, and the trade-off in between, are selected for further analysis, as marked in Figure  
It can be observed that Solution 1 is able to achieve the minimal payload swing. The maximum payload swing is \(1^{\circ}\), and the ITAE of the swing is \(2469.21\), whereas it takes a longer time, i.e., \(4.21\) s, to reach control command. Solution 3 achieves the optimized time regulation, in which the minimum settling time is required to sacrifice the regulation swing. The settling time is \(2.35\) s, and the ITAE of the position control declines by \(83.03\%\). However, the maximum payload swing increases by \(3^{\circ}\) compared with Solution 1. Solution 2 is the trade-off regulation, which means that the time is not at its minimum, and the swing is not yet smallest when compared with solutions 1 and 3. This solution is the most reasonable choice, since that its settling time is reduced by \(1.53\) s compared with Solution 1, and the payload swing decreases by \(2^{\circ}\) compared with Solution 3. Thus, Solution 2 can be selected as the control algorithm for the following study, since it can not only ensure the load swing angle within a small range, it can also control the load to reach the target position with fast speed. By simulation, the parameters based on the LQR and the proposed algorithm are shown in Table 4, respectively. There exists obvious difference among the algorithm parameters, since the crane trajectory is predetermined by the proposed algorithm. From Figure 7, although the TP and LQR can achieve the effective control, the proposed controller can improve the efficiency that the TP algorithm cannot handle. As displayed in Figure 8, the performance indicators of the proposed algorithm are preferably superior to those of the other two control methods under the same swing constraint. Compared with the TP, the settling time shortens to be \(1.35\) s, the ITAE of the swing angle decreases by \(3.24\%\), and the ITAE of the position decreases by \(98.05\%\). Compared with LQR, the settling time reduced by \(1.11\) s, the ITAE of swing decreased by \(12.82\%\), and the ITAE of position fell by \(68.13\%\). Based on the above comparative analysis, we can conclude that the proposed method can achieve the position control rapidly, and meanwhile satisfy the swing constraint requirement. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Parameter** & **LQR** & **Proposed Method** \\ \hline \(K_{I}\) & NA & \(0.02\) \\ \(K_{D}\) & NA & \(169.34\) \\ \([k_{1},k_{2},k_{3},k_{4}]\) & \([1.46,2.56,-3.13,-1.35]\) & \([15.89,13.93,-15.89,-0.95]\) \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Parameters of the linear quadratic regulator (LQR), and the proposed method.