# Control Strategy of In-Port U-Turn for Ships Based on Arctangent Function Nonlinear Feedback
## 2 System Design and Implementation
The third-order Nomoto model has been widely applied and verified in the field of ship maneuverability research, accumulating extensive experience and data. Research findings based on this model are highly reliable and credible, meaning that they can be easily accepted and applied in the industry. Therefore, this paper selected the third-order Nomoto model for the simulation experiments, and the third-order Nomoto model is adopted as Equation (8):  
\[G_{1}(s)=\frac{K_{0}(T_{3}s+1)}{s(T_{1}s+1)(T_{2}s+1)} \tag{8}\]  
where \(K_{0}\) is called the ship-turning ability index, which is the static gain from \(\delta\) to \(\psi\). The larger the \(K_{0}\) is, the better the ship's turning ability will be. \(T_{1}\), \(T_{2}\), and \(T_{3}\) are called the ship turning lag indices; the smaller \(T_{1}\) and \(T_{2}\) are, the larger \(T_{3}\) will be, and the better the ship's following ability will be. Small \(T_{1}\) and \(T_{2}\) mean that the transient motion decays to stable rotation quickly, while a large \(T_{3}\) mainly increases the angular acceleration of initial rotation [20]. In order to add nonlinear force and wind and wave interference, the form of transfer function must be transformed into three degrees of freedom for the state-space model [21]. Using the symbolic toolbox in Matlab, the third-order Nomoto model can be transformed into a three-degrees-of-freedom state-space model. The transformed model parameter matrices \(A\), \(B\), and \(C\) are shown as Equation (9). \[ \begin{split} A&=\left[\begin{array}{ccc}-(T_{1}+T_{ 2})/(T_{1}T_{2})&-1/(T_{1}T_{2})&0\\ 1&0&0\\ 0&1&0\end{array}\right]\\ B&=\left[\begin{array}{c}1\\ 0\\ 0\end{array}\right],C=\left[\begin{array}{cc}0&K_{0}T_{3}/(T_{1}T_{2})&K_{0} /(T_{1}T_{2})\end{array}\right]\end{split}  \tag{9}\]  
**Remark 1**. : _When transforming the third-order Nomoto model into a state-space model, certain assumptions are typically made to simplify the model and make it more suitable for control design and analysis:_  
1. _Higher-order nonlinear hydrodynamic effects in the model, such as wave-making resistance and the nonlinear variation of viscous resistance, are neglected. These higher-order terms have a relatively minor impact on ship motion under small-angle and low-speed conditions, and can be reasonably ignored to simplify the model._
2. _It is assumed that the maneuvering parameters of the ship (such as_ \(K_{0}\)_,_ \(T_{1}\)_,_ \(T_{2}\)_, and_ \(T_{3}\)_) remain constant during the operation. These parameters mainly depend on the ship's design and underwater shape, and can be considered constants over short periods, thereby simplifying the dynamic characteristics of the model._
3. _It is assumed that dynamic disturbances change relatively slowly and can be treated as quasi-static processes. This allows the use of static gains in control design to compensate for these disturbances._  
However, this form of model is different from the standard form; it cannot add the interference items in directly. In order to convert it into the standard form, according to [PERSON]'s norm and balance realization definition, the transformation matrix can be obtained using Equation (10). \[T_{p}=\left[ \begin{array}{ccc}-T_{1}T_{2}/(K_{0}T_{3}T_{3})&T_{1}T_{2}/(K_{0 }T_{3})&0\\ T_{1}T_{2}/(K_{0}T_{3})&0&0\\ -T_{1}T_{2}/K_{0}&0&T_{1}T_{2}/K_{0}\end{array} \right] \tag{10}\]  
The transformed \(A^{\prime}\), \(B^{\prime}\), and \(C^{\prime}\) are, respectively, shown as Equation (11):  
\[A^{\prime}=\left[ \begin{array}{ccc}-1/T_{3}&1&0\\ A_{21}&A_{22}&0\\ 0&1&0\end{array} \right],B^{\prime}=\left[ \begin{array}{ccc}0\\ K_{0}T_{3}/(T_{1}T_{2})\\ 0\end{array} \right],C^{\prime}=\left[ \begin{array}{ccc}0&0&1\end{array} \right] \tag{11}\]  
Furthermore, \(A_{21}\), \(A_{22}\) is shown as Equation (12):  
\[ \begin{split}& A_{21}=-(-K_{0}T_{3}(T_{1}+T_{2})/(T_{1}T_{2}T_{1}T_{2})+K_{0}/(T_{1}T_{2}))T_{1}T_{2}/(K_{0 }T_{3}T_{3})-1/(T_{1}T_{2})\\ & A_{22}=(-K_{0}T_{3}(T_{1}+T_{2})/(T_{1}T_{2}T_{1}T_{2})+K_{0}/(T_{1}T_{2}))T_{ 1}T_{2}/(K_{0}T_{3})\end{split}