## 5 Simulations  
### FO PI\({}^{\lambda}\)D\({}^{\mu}\) Controller  
The research object used the actual ship data of COSCO's large container vessel of 5446 TEU, whose parameters are shown in Table 2. Let the ship's rated speed be \(V\) = 19.8 knots = 10.186 m/s (1 knot = 1.852 km/s) and the ship's nonlinear course model adopt the Norrbin nonlinear model. Since the ship could be regarded as a large mass object in movement, the ship's course control could be regarded as a kind of slow turning movement when the steering was not very frequent, so Equation (3) could be simplified. The simplified equation [37] is shown here:  
\[T\dot{\psi}+\psi=K\delta \tag{36}\]  
According to the calculation, the model coefficients \(K\) = 0.1955 and \(T\) = 255.8837 were obtained. \(A_{m}\) = 1.3, \(\mathcal{O}_{m}\) = \(\pi\)/4, \(\lambda\) = 0.1, and \(\mu\) = 0.85 for the FO PI\({}^{\lambda}\)D\({}^{\mu}\) controller were determined by the system characteristics and practical experience. Then, \(K_{p}\) = 1.4516, \(K_{i}\) = 1.1355, and \(K_{d}\) = 95.0636 were calculated from Equations (22)-(26), respectively. The fitting frequency of the modified Oustaloup filtering method (\(\omega_{b}\), \(\omega_{h}\)) chose [0.001, 1000] and the order \(N\) = 5. According to the design method of the _PID_ ship autopilot [37], the \(K_{p}\), \(K_{i}\), and \(K_{d}\) are shown here:  
\[K_{p}=\frac{T\omega_{n}^{2}}{K},K_{i}=\frac{T\omega_{n}^{3}}{10K},K_{d}=\frac {2T\epsilon\omega_{n}-1}{K} \tag{37}\]  
where \(\omega_{n}\) is the system's natural frequency and \(\epsilon\) is the system's relative attenuation coefficient. Usually, one would have 0.8 \(\leq\)\(\epsilon\)\(\leq\) 1.0, \(\omega_{n}\) = 0.06, and \(\epsilon\) = 0.9. From Equation (37), we could obtain \(K_{p}\) = 4.712, \(K_{i}\) = 0.028, and \(K_{d}\) = 136.243. The FO PI\({}^{\lambda}\)D\({}^{\mu}\) controller was applied to the ship's course, where the maximum rudder angle was restricted from \(-35^{\circ}\) to \(+35^{\circ}\), and the desired course angle was \(30^{\circ}\) when time was 0-1000 s. The FO PI\({}^{\lambda}\)D\({}^{\mu}\) controller and PID controller were compared. The comparison results can be seen in Figures 9-11 and Table 3.  
In the next step, we set the ship's course angle to change in the following order: (1) 0 \(\leq\)\(t\)\(\leq\) 1000 while setting the desired course angle to \(\psi_{r}(t)\) = 20\({}^{\circ}\) and (2) 1000 \(\leq\)\(t\)\(\leq\) 2000, while setting the desired course angle to \(\psi_{r}(t)\) = 30\({}^{\circ}\). The course-tracking curve and rudder angle curve with the FO PI\({}^{\lambda}\)D\({}^{\mu}\) controller and the PID controller are shown in Figures 12 and 13.  
\begin{table}
\begin{tabular}{c c c c} \hline
**Structure** & **Value** & **Structure** & **Value** \\ \hline length of ship & 280 m & length of two columns & 267 m \\ width of ship & 39.8 m & distance of gravity center & 2.64 m \\ area of Rudder & 61.0 m\({}^{2}\) & square coefficient & 0.67 \\ no-load weight & 3.5453 M tons & full load weight & 6.5531 M tons \\ draft & 12.532 m & draft with fully load & 14.023 m \\ \hline \end{tabular}
\end{table}
Table 2: Structure data of 5446 TEU container vessel.  
Figure 11: Steady state error curve with FO PI\({}^{\lambda}\)D\({}^{\mu}\) controller and the PID controller.  
Figure 10: Rudder angle curve with FO PI\({}^{\lambda}\)D\({}^{\mu}\) controller and the PID controller.  
Figure 9: Course-keeping curve with FO PI\({}^{\lambda}\)D\({}^{\mu}\) controller and the PID controller.  
By comparing the simulation experiments, it can be seen that the FO PI\({}^{\lambda}\)D\({}^{\mu}\) controller had a significantly better control effect than the PID automatic rudder controller proposed in [37]. The FO PI\({}^{\lambda}\)D\({}^{\mu}\) controller had the advantages of a small overshoot, short adjustment time, and accurate control.