# Control Strategy of In-Port U-Turn for Ships Based on Arctangent Function Nonlinear Feedback
## 2 System Design and Implementation
 \tag{12}\]According to the state-space model, ship plane motion is shown as Equation (13):  
\[\left\{ \begin{array}{l}\dot{X}=AX+Bu\\ Y=CX\end{array} \right. \tag{13}\]  
After being converted into the standard form, it is convenient to add the nonlinear force and environment interference to Equation (13). In this paper, the wind and wave model used can be referenced in [22]. The relationship between ship parameters \(T_{1},T_{2},T_{3},T_{0}\), and \(K_{0}\) in Equation (8) and water depth are shown in Table 1; \(h/d\) is the ratio of water depth to ship draft. According to Table 1, it can be seen that there is an imaginary part. These values are related to the ship's operational parameters in the context of a frequency-domain analysis. Specifically, they represent the complex gains of the system at different frequencies, which are essential for understanding the dynamic response of the ship. However, the existence of the imaginary part does not affect the calculation; this is because, when calculating the linear interpolation, the values of \(T_{1}+T_{2}\) and \(T_{1}T_{2}\) in each water depth can be obtained according to the data in the table; then, use the linear interpolation method can be used, which can avoid imaginary parts. **Remark 2**. : _The parameters in Table 1 are dimensionless. The ship maneuverability indices \(K_{0}\) and \(T_{i}\) can be obtained in a dimensionless form using the following formula: \(K_{0}=(V/L)K^{\prime}\), \(T_{i}=(L/V)T_{i}^{\prime}(i=1,2,3)\). Here, \(V\) is the ship speed and \(L\) is the length between the perpendiculars of the ship; \(h\) is the water depth and \(d\) is the ship draft._  
**Remark 3**. : _According to Tables 1 and 2, the values of \(K_{0}\), \(T_{1}\), \(T_{2}\), and \(T_{3}\), corresponding to each water depth, can be obtained by linear interpolation in Matlab. According to the parameters \(K_{0}\) and \(T_{i}\) of the Nomoto model, \(K_{0}\) has a linear relationship with \(V\), and \(T_{i}\) has an inverse linear relationship with \(V\). The mathematical relations are shown as Equation (14):_  
\[\left\{ \begin{array}{l}K=K_{0}(\frac{V}{V_{0}})\\ T=T_{0}(\frac{V}{V_{0}})\end{array} \right. \tag{14}\]  
Therefore, the time constant \(T_{0}=T_{1}+T_{2}-T_{3}\), \(V_{0}\) is the design speed. **Remark 4**. : _In Remarks 1 and 2, the ship maneuverability index is connected with the water depth and ship speed, respectively, which shows that the model is suitable for describing the ship's motion at low speeds in shallow waters._ \begin{table}
\begin{tabular}{l c c c c c} \hline \(h/d\) & **1.21** & **1.50** & **1.93** & **2.50** & \(\infty\) \\ \hline \(T_{1}\) & 0.231 + 0.153\(i\) & 1.072 & 2.678 & 3.244 & 2.749 \\ \(T_{2}\) & 0.231 \(-\) 0.153\(i\) & 0.362 & 0.356 & 0.382 & 0.367 \\ \(T_{3}\) & 0.277 & 0.477 & 0.585 & 0.788 & 0.729 \\ \(K_{0}\) & 0.234 & 0.957 & 2.449 & 2.848 & 2.836 \\ \(T_{1}+T_{2}\) & 0.462 & 1.434 & 3.034 & 3.625 & 3.116 \\ \(T_{1}T_{2}\) & 0.077 & 0.388 & 0.953 & 1.239 & 1.009 \\ \hline \end{tabular}
\end{table} 
Table 1: Parameters under different water depths.