# Control Strategy of In-Port U-Turn for Ships Based on Arctangent Function Nonlinear Feedback
## 3 Nonlinear Controller Design  
### Design of Controller for U-Turn Operation in Port  
The ship U-turn operation in the port can be divided into two parts. One is the feed-forward link of large angle turning, which can improve the sensitivity of the system and speed up the response of the system [23]. In the process of a ship's U-turn, in order to improve the response speed, the feed-forward control link is necessary. When the ship turns at a large angle without feed-forward control, the system response time will be quite long, which is very unfavorable for ship's that need to make a U-turn in an emergency. Therefore, with the selection and judgment function of the switch, the ship steering angle is divided into seven zones: \(0^{\circ}\)-\(5^{\circ}\), \(5^{\circ}\)-\(15^{\circ}\), \(15^{\circ}\)-\(30^{\circ}\), \(30^{\circ}\)-\(50^{\circ}\), \(50^{\circ}\)-\(90^{\circ}\), \(90^{\circ}\)-\(130^{\circ}\), \(130^{\circ}\)-\(180^{\circ}\). The small angle range in each zone can take all the angle situations into account. The parameters of the feed-forward response link for the seven zones are as follows: \(\frac{508}{35+17}\), \(\frac{1608}{35+17}\), \(\frac{450}{35+17}\), \(\frac{5508}{35+17}\), \(\frac{1500}{35+17}\), \(\frac{3000}{725+17}\), \(\frac{6508}{955+17}\), where \(s\) is the Laplacian operator. According to different turning angles, the corresponding interval can achieve a good control effect. The feed-forward controller is shown in Figure 1.  
The other part mainly adopts a simple and direct robust form of control based on the closed-loop gain-shaping algorithm (CGSA) [24]. It is an engineering algorithm based on the results of the hybrid sensitivity control algorithm of robust control and the closed-loop shaping algorithm. For controller design using the closed-loop gain-shaping algorithm, the third-order model needs to be transformed into a second-order, deeply strict model, as shown in Equation (15): \begin{table}
\begin{tabular}{l r} \hline \hline
**Parameter** & **Value** \\ \hline Length between perpendiculars (m) & 160.93 \\ Ship breadth (m) & 23.17 \\ Design draught (m) & 8.23 \\ Block coefficient & 0.558 \\ Distance from center of gravity to center (m) & 6.8 \\ Ship speed (kn) & 15 \\ Rudder blade area (m\({}^{2}\)) & 30.012 \\ Volume of displacement (m\({}^{3}\)) & 18,541 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Relevant parameters of “Mariner”. Figure 1: Block diagram of feed-forward controller. \[G(s)=\frac{K_{0}}{s(T_{0}s+1)} \tag{15}\]  
\(K_{0}\) and \(T_{0}\) are the ship manipulative indices: see \(K_{0}=K_{0},T_{0}=T_{1}+T_{2}-T_{3}\) in Table 1.  
The bandwidth frequency of the closed-loop system is assumed to be \(1/T_{c}\), according to the second-order CGSA. Thus, Equation (16) can be obtained:  
\[\frac{1}{\left(T_{c}s+1\right)^{2}}=\frac{G(s)K(s)}{1+G(s)K(s)} \tag{16}\]  
Since a ship is a kind of vehicle with a large inertia and it has long adjustment time, the improved simple robust control algorithm presented in [25] is adopted, and a constant, \(\rho\), is added to the proportional coefficient, which improves the dynamic performance of the system. Therefore, the final control law is shown as Equation (17):  
\[K(s)=\frac{1}{K_{0}T_{c}}+\rho+\frac{T_{0}}{K_{0}T_{c}}s \tag{17}\]  
\(K_{0}\) and \(T_{0}\) are the ship manipulative indices; \(T_{c}=4\) s, which makes the system bandwidth frequency less than the high-frequency wave disturbance ((\(0.3\sim 1.25\) rad/s)) and fully suppresses the influence of wave interference. The introduction of a constant can improve the response speed of the closed-loop system, speed up the process of U-turn, reduce the turning time, and make the turning area meet the limited requirements. **Remark 5**: _CGSA obtains the controller parameters directly based on the system characteristics through a frequency domain analysis, omitting the parameter-tuning process of traditional PID controllers. Its control law is similar to that of a PD controller, without introducing an integral term.