# 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  
For the ship roll/yaw controller design, the discrete-time ship motion state prediction model is carried out according to \(H_{\infty}\)-type cost function, the minimum value of cost function is treated as inequality constraint and the multiple input constraints are considered, including the saturation limitation of the rudder/fin stabilizers and the delay constraint of the fin stabilizers. With the help of Matlab toolbox, the control law gain \(K\) at each time instant is obtained. In addition, considering that there is extreme wave interference when the ship is sailing in the ocean, the feasible region of online solution may exceed, making the control inputs lose efficacy further, resulting in being out of control. To resolve the issue, the maximal feasible region can be calculated offline in advance. In cases where the system states lie outside this region, a linear quadratic regulator (LQR) is employed to improve the safety of the ship. As a summary of this section, the \(H_{\infty}\)-type MPC approach is integrated with the prediction method by considering the fin angles delay constraint. Note that the control method employed in this paper is inherently developed for linear state space models. In fact, the nonlinearity is not strong for a complete ship control system model under mild conditions. Hence, it is reasonable to consider the model mismatches in the disturbance term when a linear model is employed for prediction. A byproduct of this treatment is that less computational resources are required, which facilitates real-time implementation compared with a nonlinear model based predictive control method. It can be seen that the robust \(H_{\infty}\)-type model predictive control strategy with multiple input constraints (\(H_{\infty}\)-SFMPC) is constructed in this paper, as is shown in Figure 1.  
In cases where the order of AR model is \(p=5\), the fitting model has high accuracy [37]. Generally, the delay time of fin stabilizer actuator is more than 1 s. It is assumed that the delay time is 1.2 s, which can compensate the dynamic delay of fin stabilizers. In the simulations, the parameter identification accuracy and stability can be considered, and the forgetting factor is set \(\lambda=0.975\) in this paper. The predicted roll angles can be used to participate in the subsequent controller design. **Simulation Case 1: Control performance of \(H_{\infty}\)-SFMPC strategy based on nominal model in different sea states**  
In this case, the robustness of the proposed approach in different sea states is studied. The sea state codes are selected as four, five and six levels, and the encounter angle is set as \(90^{\circ}\). The simulation results are shown in Figure 4, where Figure 4 shows the time history curves of the ship yaw angles in these levels of sea conditions; (b) represents the time history curves of the ship roll angles; (c) represents the time history curves of the fin angle inputs; (d) represents the time history curves of the rudder angle inputs. From the results in Figure 4, it can be seen that the \(H_{\infty}\)-SFMPC strategy can satisfy control requirements for roll and yaw angles. In addition, the larger the sea state level is, the greater the response of roll angles and yaw angles is, and the input effect of the controller is also increased. In the six-level sea state, there exist the larger fin angles and rudder angles relative to other low level sea states. The statistical values of simulation results are shown in Table 2. It clearly shows that the \(H_{\infty}\)-SFMPC strategy has considerable control effect and robustness in different sea states. Figure 4: _Cont._  
Figure 3: Time history curve of the ship rolling disturbance moment.