# Model Based Robust Predictive Control of Ship Roll/Yaw Motions with Input Constraints
## 3 Main Result
### Ship Roll/Yaw Robust Model Predict Control with Multiple Input Constraints
**Simulation Case 2: Control performance using different strategies**  
In this case, LQR control law (LQR controller is designed only for the linear model without considering the nonlinear term of the system in nominal model) and \(H_{\infty}\)-SFMPC strategy are used for the ship roll/yaw control in the six-level sea condition at the encounter angle of \(90^{\circ}\). The results are illustrated in Figure 5. It can be seen from (a) and (b) that the control effect of the \(H_{\infty}\)-SFMPC control law proposed is obviously better than that of the traditional LQR control law; it can be seen from (c) and (d) that the difference of fin angle amplitudes using different control methods is not obvious, and the amplitude of rudder angle actions using \(H_{\infty}\)-SFMPC become smaller as a whole, while the rudder actions are more frequent, which is the reason that the \(H_{\infty}\)-SFMPC works by the optimization online and in real time, viz., for a relatively better state response it is necessary to speed up the action rate of the actuators. The statistical values of simulation results are shown in Table 3. \begin{table}
\begin{tabular}{c c c c c c c c} \hline
**S** & \(E(\psi)\) & \(STD(\psi)\) & \(E_{\text{J/3}}(\psi)\) & \(STD(\delta)\) & \(E(\phi)\) & \(STD(\phi)\) & \(E_{\text{1/3}}(\phi)\) & \(STD(\alpha_{\ell})\) \\ \hline
4 & \(2.0\times 10^{-4}\) & 0.0227 & 0.0418 & 0.6725 & 0.0018 & 0.2081 & 0.3833 & 3.6836 \\
5 & \(1.6\times 10^{-4}\) & 0.0431 & 0.0769 & 0.9193 & 0.0047 & 0.4250 & 0.7598 & 6.9192 \\
6 & \(-0.0013\) & 0.0498 & 0.0838 & 0.9720 & \(-0.0213\) & 0.5403 & 0.9238 & 7.7924 \\ \hline \end{tabular} The notation **S** denotes sea codes; the notation **E** denotes average value; the notation **STD** denotes standard deviation; the notation \(E_{\text{J/3}}\) denotes significant value, i.e., the mean value of one-third largest samples.  
\end{table} 
Table 2: Simulation results of the ship roll/yaw motions in different sea states at the encounter angle of \(90^{\circ}\). Figure 4: Simulation curves of the ship roll/yaw motions using \(H_{\infty}\)-SFMPC in different sea states at the encounter angle of \(90^{\circ}\). Figure 5: _Cont._  
**Simulation Case 4: Control performance based on different models (nominal/uncertain model)**  
In this case, the ship roll/yaw control using \(H_{\infty}\)-SFMPC strategy is investigated based on the nominal/uncertain model alternately in the six-level sea state at the encounter angle of \(90^{\circ}\). We assume that hydrodynamic coefficients are perturbed in the proposed model, while the inertia and mass of ship rigid body are constant. The uncertain hydrodynamic coefficient matrices are given such that perturbation coefficient \(e_{i}=1.3-0.6\delta_{i}\), \(\delta_{i}\in N\) (0, 1), i.e., the bounded perturbation value is set as 30%, \(I=1\), 2,..., 21, viz. there are in total 21 hydrodynamic coefficients of \(\mathbf{M^{\prime}}\), \(\mathbf{N^{\prime}}\) and \(\mathbf{G^{\prime}}\) in Equation (2), see Appendix A. The simulation results are shown in Figure 7. It can be seen from (a) and (b) that the ship motion response based on two different models has good similar control effect, and from (c) and (d) it can be seen that the \(H_{\infty}\)-SFMPC also has strong robustness to the parameter uncertainty model when the control inputs based on nominal/uncertain model are not so different. The statistical values of simulation results are shown in Table 4.  
Figure 6: Control performance and the actuator inputs under different design parameters \(Q\) and \(R\). Figure 7: Simulation curves of the ship roll-yaw motions using \(H_{\infty}\)-SFMPC control law based on nominal/uncertain model.