# Multi-Objective Motion Control Optimization for the Bridge Crane System
## 1 Introduction  
Nowadays, the development of port transportation brings with it increasing demands of cargo movement. The bridge crane system is widely applied in order to move the cargo with less cost compared to other transit systems [1]. For the crane system, it is imperative to design an effective controller that can achieve fast and safe cargo movement. For the cargo movement control, in addition to satisfying the target position, some constraints and disturbances need to be properly dealt with, including maximum moving speed and distance, maximum swing angle, etc. Focusing on these practical challenges, the designed controller should satisfy the demands, including fast response, high robustness, and stability, as well as strong anti-disturbance. Among them, the most important task is to regulate the crane to the desired position, and meanwhile to suppress the payload swing. In terms of this, the traditional control methods can be divided into two categories, i.e., open loop and closed loop methods. For open loop control methods, the control commands are determined in advance according to the requirements before operation. The popular methods include input shaping control [2; 3], trajectory planning (TP) control [4; 5], etc. The input shaping control is imposed in order to produce a series of pulses according to the ratio between the system frequency and the damping; then, the convolution between the pulse and the reference trajectory is conducted in order to generate the control command. This method can guarantee the control performance for the crane position when the cable length changes, whereas the residual swing of the payload cannot be eliminated. By TP,the crane trajectory can be designed based on the nonlinear coupling relationship between the crane position and the swing angle, thereby ensuring the cargo's safety when moving and suppressing the residual oscillation. Both methods can satisfy the controlling target. However, when the system is under external disturbance, such as wind and human touch, the payload oscillation can be potentially triggered, and yet cannot be eliminated effectively based on the open loop methods. In order to overcome the influence of the external disturbance, various closed loop methods are proposed, which can be divided into adaptive control [6; 7; 8], fuzzy logic-based control [9; 10], genetic algorithms (GA) [11; 12], feedback linearization control [13; 14], linear quadratic regulator (LQR) control [15; 16; 17], proportional-integral-differential (PID) control [18; 19; 20], etc. In [21], [PERSON] and [PERSON] considered the crane velocity, the position error, and the swing angle totally in a linearized manner, and an adaptive sliding mode controller is designed. However, the convergence and boundedness proof of the position error and swing angle are not given. Based on linearization, the PID control algorithm, regulated by the fuzzy logic algorithm, is employed in order to improve the crane's transient performance and robustness. Since the GA is capable of finding the optimal solutions, a nonlinear control algorithm is proposed based on a real-time GA in order to solve the difficulty of regulating the control gain [12; 22; 23; 24]. In this manner, the whole control performance of the crane system is improved. Actually, it is difficult for the GA algorithm to ensure the stability strictly at the equilibrium point theoretically, and the controller design is relatively complex, bringing inconvenience for practical application. For sake of reducing the controller complexity and easily applying it in actual operation, the feedback linearization control algorithm is introduced based on the simplified crane model, which divides the crane motion and the payload swing into two subsystems [25; 26; 27; 28]. A nonlinear coupling control method is designed to ensure that the crane position and the payload swing can finally converge to specified ranges. Based on the model linearization, the LQR method is harnessed to achieve the control using the feedback gain of the linear systems [29].