# Multi-Objective Motion Control Optimization for the Bridge Crane System
## 3 Control Design
### Multi-Objective Optimization  
It is critical to determine the control parameters of the designed algorithm, i.e., \(k_{1}\), \(k_{2}\), \(k_{3}\), and \(k_{4}\), and thus, the weighted matrixes \(Q\) and \(R\) need to be optimized. Here, integration of time-weighted absolute value of the error (ITAE) is introduced in order to evaluate the convergence and oscillation of the system, as:  
\[\left\{\begin{array}{c}J_{\theta}=\int_{0}^{\infty}t|e_{\theta}(t)|dt\\ J_{T}=\int_{0}^{\infty}t|e_{T}(t)|dt\end{array}\right. \tag{31}\]  
where \(J_{\theta}\) and \(J_{T}\) denote the ITAEs of the payload swing and the crane position, respectively. Here, the MOGA is applied to find optimal solutions among multiple objective functions. According to the optimal method, a series of points reflecting the control effects can be presented based on the ITAEs of the crane and payload. In addition, the optimization process should be subject to the following constraints:  
\[\begin{split} f_{Pareto}=min(J_{\theta},J_{T})\\ s.t\left\{\begin{array}{c}v_{\max}=min\left(v_{\lim},\frac{S_{ total}}{2\pi\sqrt{L/g}},2\pi a_{\lim}\sqrt{L/g}\right)\\ -\pi/2<\theta(t)<\pi/2\\ a_{\max}\leq min\left(a_{\lim},\frac{\theta\dot{\theta}_{\lim}}{2}\right) \end{array}\right.\end{split} \tag{32}\]  
Here, we employed the MOGA algorithm to calculate the optimization parameters with the criteria of minimizing the ITAE of the crane position and the payload swing. As can be seen in the Figure 4, the whole process will first choose the initial weighted matrixes \(Q\) and \(R\) randomly. Then, the control commands can be generated based on the constraints and the updated ITAE of the crane position, and the payload swing can be calculated based on the fitness function. In the next step, the MOGA is applied to search the optimal parameters through a series of the selection, crossover, and mutation, wherein the roulette wheel selection and the two-point crossover are adopted. For the MOGA algorithm, first, some existing populations are selected to generate the next generation, and this process is known as selection. Some existing populations are considered as elitists, and are directly selected as next generations without any change. The selection criteria is based on the fitness function and their corresponding constraints. During the crossover process, the parent chromosomes are hybridized to generate new offsprings, and meanwhile, some bits of chromosomes are uniformly or randomly changed with a certain possibility. This is the so-called mutation, of which the main function is to avoid falling into a local optimum. By means of these actions, a new chromosome group is generated, which is different from the previous version. The whole action is operated iteratively until the terminal condition is reached, which usually includes exceeding the budget time, reaching the maximum allowable amount, etc. Finally, the Pareto fronts and the optimized parameters can be found. In this paper, the population size, the crossover rate, and the mutation rate of the MOGA algorithm are set to 40, 0.8, and 0.05, respectively, and the number of iterations is set to 50 after repetitive tunning. Furthermore, the constraint \(\theta_{\text{lim}}\) of the payload swing is set to be less than \(4^{\circ}\), the maximum acceleration is defined as 5 m/s\({}^{2}\), the maximum crane velocity is 1.5 m/s, the target position is \(x_{d}=0.4\) m, and the error of two objective functions is limited within \(1~{}\times~{}10^{-4}\).  
Next, the stability analysis is conducted, and the stability proof is given.