# Concise Robust Control of the Nonlinear Ship in Course-Keeping Control
## 5 Simulation Experiments and Result Analysis
### Course-Tracking Experiment and Analysis  
Figure 8 is the result of dynamically tracking course change with the target course defined by \(10\)sin(\(2\pi/100\)). It can be seen from the results that the PID control and CROB control based on backstepping can effectively track the target course. At approximately 30 s, the course has been tracked after a short adjustment. Moreover, the rudder angle also shows a certain periodic law compared with the periodic course. However, due to the large inertia of the ship, there is a certain delay between the maximum rudder angle and the maximum course. Hence, the influence of the target course derivative is ignored in the course control law design, as described in Equation (21), but the effective tracking of the dynamic change course can still be achieved for a certain period (200 s). Figure 6: Comparison of CROB and PID based on backstepping. Figure 7: Rudder angle comparison of CROB and PID based on backstepping (the first 500 s). Nonlinear Ship Robust Course-Keeping Control Experiment Results and Analysis in Case of Wind and Wave Disturbance  
Under the wind interference with Beaufort scale 6, the control result of ship course-keeping is described in Figures 9 and 10, and the details of wind and wave are shown in reference [24; 25]. The target course is set at \(30^{\circ}\). The maximum rudder angles are \(35^{\circ}\), \(32.4^{\circ}\), \(23.4^{\circ}\), and \(35^{\circ}\), respectively. Compared with PID control, the maximum rudder angle is decreased by \(7.4\%\) and \(33.2\%\) when the CROB method with a door-closing slope of \(-40\) dB and \(-60\) dB, respectively, is used. Therefore, it can be observed that an increase in the door-closing slope can effectively reduce the maximum rudder angle. In the stable condition (500 s-1000 s), the maximum rudder angle, average rudder angle, rudder angle variance, and the time to the target heading of different control methods  
Figure 8: Comparison of dynamic course tracking. Figure 10: Rudder angle comparison under concise robust control and PID in case of wind and wave disturbance. Figure 9: Concise robust control in case of wind–wave disturbance (scale 6). are shown in Table 2, which shows that: 1 the applied average rudder angle is \(2.95\lx@math@degree\), which is consistent with the reference [24,25] with a \(3\lx@math@degree\) rudder angle; 2 with the increase in the door-closing slope, the concise robust control algorithm can reduce the maximum rudder angle and rudder angle variance under wind-wave interference, achieving the goal of reducing the rudder wear and energy consumption. However, in practice, a larger the door-closing slope does not yield better results. For example, the maximum rudder angle is \(5.68\lx@math@degree\), \(3.61\lx@math@degree\), and \(3.20\lx@math@degree\) in the concise robust control algorithm, while the relative maximum rudder angle decreased by \(36\%\) and \(11\%\) for \(-40\) dB and \(-60\) dB door-closing slopes, respectively, as shown in Table 2; the rudder angle variance is decreased by \(61.4\) times and \(3.6\) times, while the time to the target heading basically changes linearly (\(110\) s, \(209\) s, \(303\) s). In other words, when the door-closing slope is increased from \(-20\) dB to \(-40\) dB, the maximum rudder angle is decreased by \(36\%\), and the rudder angle variance is reduced by \(61.4\) times; correspondingly, the maximum rudder angle is decreased by \(11\%\) when the closing slope is increased from \(-40\) dB to \(-60\) dB, and the rudder angle variance is only decreased by \(3.6\) times. Therefore, the cost performance of the \(-40\) dB door-closing slope is higher than \(-60\) dB. Compared with the PID method, the maximum rudder angle and rudder angle variance of \(-40\) dB and \(-60\) dB door-closing slopes are reduced by \(20.9\%\) and \(34\) times and by \(29.8\%\) and \(160\) times, respectively.