# Concise Robust Control of the Nonlinear Ship in Course-Keeping Control
## 4 Design of a Nonlinear Robust Controller for Ship Course Control
### Nominal Model for Control Law Design  
At present, the nonlinear mathematical model of ship motion control is commonly used. The model has a simple form and few parameters, which can also basically reflect the nonlinear characteristics of the ship turning. However, it can only reveal the relationship between rudder angle and yawing, which cannot reflect the influence of the ship's velocity on yawing, hence limiting its application range. In practical application, ship motion attitude is often considered in the design of ship motion controllers, so it is necessary to study the design of a ship controller with multiple degrees of freedom. In other words, the longitudinal and the lateral velocity of the ship should be considered at least. The transformation of the terms \(f_{3}\) in the Abkowitz model as mentioned in Equation (6) is carried out, and the terms include \(\delta\), and higher-order terms including \(N_{\delta\delta\delta}\delta^{3}\), \(N_{\delta\delta\delta}\delta\delta^{2}\), \(N_{r\delta\delta}\delta^{2}\) are ignored in the control law design, as they exert a minor influence on the final result, the parameter could be found in Table A1 in the Appendix A. Assume that:  
\[f_{3}(u,v,r,\delta)=N_{1}+N_{2}\delta \tag{13}\]  
Among them:  
\[\left\{ \begin{array}{rl}N_{1}=N_{0}&+N_{u}\Delta u+N_{uu}\Delta u^{2}+N_{uu} \Delta u^{3}+N_{v}v\\ &+N_{r}r+N_{vv\delta}\Delta v^{3}+N_{rrr}\Delta v^{3}+N_{vvr}v^{2}\\ &+N_{rrr}r^{2}+N_{vv}\Delta u+N_{nn}r\Delta u-mx_{G}r\Delta u\\ &+N_{uu}v\Delta u^{2}+N_{nn}r\Delta u^{2}\\ &N_{2}=N_{\delta}&+N_{vv\delta}v^{2}+N_{rr\delta}r^{2}+N_{rr\delta}vr+N_{ \delta u}\Delta u+N_{\delta uu}\Delta u^{2}\end{array} \right. \tag{14}\]  
\(N_{1}\) and \(N_{2}\) are nonlinear terms related to ship motion. In 1965, [PERSON] carried out comprehensive research on \"mariner\" by Planar Motion Mechanism (PMM); the hydrodynamics of the vessel were obtained. The result had good conformity with the sea trial results in ship steady-turning and zigzag experiments. The parameters of the mariner are presented in Table 1.  
With the purpose of verifying the influence on the simplification of \(\delta\) and related terms, the steady-turning and the zigzag tests are conducted for the nominal model as described by Equation (13). Meanwhile, the simulation result is compared with the result from [PERSON] as well as the sea trial results. As illustrated in Figure 3, the result of 2 the \(0^{\circ}/20^{\circ}\) zigzag test is plotted. This refers to the results from [PERSON], our simulation results, and sea trial results, respectively. It can be observed that the turning rate starts to increase under the rudder angle. The rudder starts to revert once the course reaches its target value. The course continues to increase due to the huge inertial of the ship even with the rudder reverted, but the course starts to revert until the turning rate decrease to zero. As indicated by the fact that our nominal model agrees with the result from [PERSON] and the sea trial results, the nominal model is close to the original [PERSON] model even if the \(\delta\)-related higher-order terms are ignored, as shown in Equation (13). The steady-state turning test is carried out at a rudder angle of \(5^{\circ}\), \(10^{\circ}\), \(20^{\circ}\) for both starboard and port side, and the results are shown in Figure 4. It is clear that the turning process includes the transition stage and steady-turning stage. In the transition stage, the ship starts yawing under the rudder angle applied; then, the turning rate reaches the maximum. After this, the turning rate begins to decrease slightly; shortly after, the turning rate becomes steady at around 200 s, and the ship begins a steady-state turning process.