## 5 Simulation Examples  
### Configurations  
In this section, simulation examples are presented to illustrate the effectiveness of the control scheme in fields of marine practice. Based on the LOS-CT guidance system in Section 3.1, the planed route consists of waypoints \(WP_{1}\) (0, 0), \(WP_{2}\) (8000, 8000), \(WP_{3}\) (16,000, 0), \(WP_{4}\) (18,000, 11,000), \(WP_{5}\) (8000, 19,000), and \(WP_{6}\) (\(-\)2000, 11,000), with the unit m. To study the application value of the proposed control schemes on widely merchant ships, the simulation object is a container ship \"SR108\", which is defined in Section 2. As shown in Figure 1, the length is \(L=175\,\)m, together with breadth \(B_{s}=25.4\,\)m, displacement \(D=21,222\,\)tons, mean draft \(d_{m}=8.5\,\)m, block coefficient \(C_{b}=0.559\), propeller revolutions \(P_{r}=70\), metacentric height \(GM=0.3\,\)m, and forward speed \(U_{0}=7.3\,\)m/s. Detailed hydrodynamic force derivatives can be found [2; 3].  
In the simulation experiments, the impact of the wind on the ship course can be divided into two parts: the mean wind and the fluctuating wind. The fluctuating wind is stochastic, which is considered as white noise [30]. In marine practice, the effect of mean wind is taken as leeway drift, which can be expressed by an equivalent rudder angle \(\delta_{ld}\) as follows.  
\[\delta_{ld}=C_{ld}(\frac{S_{w}}{U_{0}})^{2}\sin\alpha_{w} \tag{44}\]  
where \(C_{ld}\) denotes the leeway drift coefficient, \(S_{w}\) denotes the wind speed, and \(\alpha_{w}\) denotes the relative wind angle. The wind force is set to Beaufort No. 6, \(S_{w}\) is about 12 m/s, the set value of \(C_{ld}\) is 0.05, and \(\alpha_{w}\) is 000\({}^{\circ}\). The influence of current on ship's position and speed can be defined as Equation (45), with the current speed \(U_{c}\) 2 km and the current direction \(\alpha_{c}\) 225\({}^{\circ}\).  
\[\left\{\begin{array}{l}\dot{x}=U\sin(\psi)+U_{c}\sin(\alpha_{c})\\ \dot{y}=U\cos(\psi)+U_{c}\cos(\alpha_{c})\end{array}\right. \tag{45}\]  
Figure 6: The Pareto optimal sets distribution for PFA-RRR based on NSGA-II.  
The wave-induced roll motion is produced by Pierson-Moskowitz (PM) spectrum driven by white noise. Supposing \(y_{w}=x_{w2}=\dot{x}_{w1}\), one can obtain the wave-induced rolling as follows  
\[\left[\begin{array}{c}\dot{x}_{w1}\\ \dot{x}_{w2}\end{array}\right]=\left[\begin{array}{cc}0&1\\ -T_{e}^{-2}&-2\xi T_{e}^{-1}\end{array}\right]\left[\begin{array}{c}x_{w1} \\ x_{w2}\end{array}\right]+\left[\begin{array}{c}0\\ K_{w}\end{array}\right]W_{n} \tag{46}\] \[y_{w}=\left[\begin{array}{cc}0&1\end{array}\right]\left[ \begin{array}{c}x_{w1}\\ x_{w2}\end{array}\right]\]  
As shown in Figure 7, the sea state is set to 5, the significant wave height \(H\) is about 2.5 m, the wave encountering frequency \(T_{e}^{-1}\) is 0.8378 rad/s, the damping ratio \(\xi\) is 0.05, and the wave strength factor \(K_{w}\) is \(H/1000\).  
The controllers in Section 3 are optimized by a real-coded NSGA-II, with the population size 100, mutation operator 20, crossover operator 20, and initial searching spaces [0.01 100]. The stopping generation number 100. The optimization results are illustrated by Pareto optimal set distribution in Figure 6, and a parameter set is selected with the smallest \(d\) and \(\phi\), and highest rudder movements \(\delta\).  
Due to the high cost and difficulty of the actual ship verification, the nonlinear ship model in Section 2 is implemented to test the effectiveness of the proposed control schemes. The roll reduction rate is defined as follows  
\[R_{rrr}=\frac{\phi_{intrr}-\phi_{rrr}}{\phi_{intrr}}\times 100\% \tag{47}\]  
where \(\phi_{intrr}\) is the STD value of original wave-induced roll motions, and \(\phi_{rrr}\) is the STD value of roll motions after the actuation of RRR.  
To further study the robustness and effectiveness of the control system in response to different sea conditions, more simulation tests have been done in different sea states with wave-induced maximum roll angle ranging from 10 to 35\({}^{\circ}\).