# Model Based Robust Predictive Control of Ship Roll/Yaw Motions with Input Constraints
## 3 Main Result  
### Design of Robust Model Predictive Controller for Rolllyaw with Saturation Constraints  
The optimal problem of infinite-horizon MPC is given by  
\[\min_{\mathbf{u}(k)}\ \ \max_{D(k)}\ \ \ J_{\infty}(k), \tag{10}\]  
Equation (13) is a min-max optimal problem, i.e., maximize the cost function with respect to the disturbance over the prediction horizon, and then minimize it by the optimal control input sequence [23]. Although this optimization problem is convex, it is not computationally feasible. In order to simplify the solution, we may derive the upper bound of the performance index, and then use the state feedback control law to minimize the upper bound. Hence, a quadratic function \(L(k)=\mathbf{x}^{T}\mathbf{U}\mathbf{x}\) is designed such that  
\[L(\mathbf{x}(k+i+1\big{|}k))-L(\mathbf{x}(k+i\big{|}k))\leq-\Delta(k+i\big{|}k). \tag{11}\]  
Based on the above inequality, one has  
\[\sum_{i=0}^{\infty}\Delta(\mathbf{x}(k+i\big{|}k),\mathbf{u}(k+i\big{|}k),\mathbf{d}(k+i \big{|}k))\leq L(\mathbf{x}(k))\,. \tag{12}\]  
Thus \(L(\mathbf{x}(k))\) is the upper bound of the performance index. Now that we have the upper bound of the cost function, we need to consider the following state feedback control law,  
\[\mathbf{u}(k+i\big{|}k)=\mathbf{K}\mathbf{x}(k+i\big{|}k),i\geq 0. \tag{13}\]  
Substituting Equation (13) into Equation (11), then Equation (11) can be written in an inequality form  
\[\|\,(\mathbf{A}+\mathbf{B}\mathbf{K})\mathbf{x}(k+i\big{|}k)+\mathbf{d}(k+i\big{|}k)\,\|_{\mathbf{U}}^ {2}-\|\mathbf{x}(k+i\big{|}k)\,\|_{\mathbf{U}}^{2}\leq-\|\mathbf{x}(k+i\big{|}k)\,\|_{(\bm {Q}+\mathbf{K}^{T}\mathbf{R}\mathbf{K})}^{2}+\lambda\|\mathbf{d}\|^{2}. \tag{14}\]  
In this way, infinite-horizon MPC optimization problem is transformed into: at every moment \(k\), find the state feedback control law to minimize \(L(\mathbf{x}(k))\), but only implement Equation (14); at the next moment \(k+1\), repeat the same optimization problem according to the new measurement value \(\mathbf{x}(k+1)\) and get the new \(\mathbf{K}\). Define a scalar \(\delta\), which follows that  
\[\min\ \delta \tag{15}\] \[s.t.\ L(\mathbf{x}(k))\leq\delta\]  
Then minimization of the upper bound of the cost function is approximated as minimization of \(\delta\). Define \(\mathbf{V}:=\mathbf{U}^{-1}\delta\), by resorting to the Schur complement, Equation (15) is equivalent to  
\[\left[ \begin{array}{cc}1&\mathbf{x}^{T}(k)\\ \mathbf{x}(k)&\mathbf{V}\end{array} \right]\geq 0. \tag{16}\]  
Equation (14) is equivalent to  
\[ \begin{array}{l}\mathbf{x}^{T}(k+i\big{|}k)(\|(\mathbf{A}+\mathbf{B}\mathbf{K})\|_{\mathbf{U}}^ {2}-\mathbf{U}+(\mathbf{Q}+\mathbf{K}^{T}\mathbf{R}\mathbf{K}))\mathbf{x}(k+i\big{|}k)\\ +\mathbf{x}^{T}(k+i\big{|}k)(\mathbf{A}+\mathbf{B}\mathbf{K})^{T}\mathbf{U}\mathbf{d}+d^{T}\mathbf{U}(\mathbf{A} +\mathbf{B}\mathbf{K})\mathbf{x}(k+i\big{|}k)+\mathbf{d}^{T}(\mathbf{U}-\lambda\mathbf{I})\mathbf{d}\leq 0 \end{array}  \tag{17}\]Letting \(\prod_{11}=\left\|(A+BK)\right\|_{\mathbf{U}}^{2}-\mathbf{U}+(Q+K^{T}\mathbf{R}K)\), \(\prod_{12}=(A+BK)^{T}\mathbf{U}\), \(\prod_{21}=\mathbf{U}(A+BK)\), \(\prod_{22}=\mathbf{U}-\lambda\mathbf{I}\), then Equation (17) is expressed as  
\[\left[ \begin{array}{cc}x^{T}(k+i|k)&\mathbf{d}^{T}(k+i|k)\\ \end{array} \right]\prod\left[ \begin{array}{cc}\mathbf{x}(k+i|k)\\ \mathbf{d}(k+i|k)\\ \end{array} \right]\leq 0. \tag{18}\]  
where \(\prod=\left[ \begin{array}{cc}\prod_{11}&\prod_{12}\\ \prod_{21}&\prod_{22}\\ \end{array} \right]\). By resorting to the Schur complement, a sufficient condition can be derived for the inequality (18) as follows  
\[\left[ \begin{array}{ccccc}-\mathbf{V}&*&*&*&*\\ 0&-\alpha\mathbf{I}&*&*&*\\ \mathbf{AV}+\mathbf{BY}&\delta\mathbf{I}&-\mathbf{V}&*&*\\ \mathbf{QV}&0&0&-\delta\mathbf{Q}&*\\ \mathbf{RY}&0&0&0&-\delta R\\ \end{array} \right]\leq 0. \tag{19}\]  
where \(\mathbf{Y}:=\mathbf{K}\mathbf{V}\) and \(\sigma:=\lambda\delta\). The details of the derivation of the above linear matrix inequality (LMI) is shown below. \[