# Model Based Robust Predictive Control of Ship Roll/Yaw Motions with Input Constraints
## 2 Problem Formulation  
### Ship Lateral Motion Model Control with Rudder-Fin Joint System  
It is known that the common 4 degrees of freedom (4 DOF) maneuvering model includes surge, sway, roll and yaw, which is a relatively comprehensive nonlinear model proposed by [PERSON] and [PERSON] (1981, 1982) [28; 29]. \[\left\{ \begin{array}{l}(m-X_{u})u=(m-Y_{y})vr+X_{\rm H}+X_{\rm P}+X_{\rm R}+X_ {\rm F}+X_{\rm env}\\ (m-Y_{y})v-N_{y}r+K_{\rm P}v=-(m-X_{u})ur+Y_{\rm H}+Y_{\rm P}+Y_{\rm R}+Y_{\rm F }+Y_{\rm env}\\ (I_{\rm x}+I_{\rm x})p+K_{\rm v}v=-X_{u}l_{\rm x}ur-W\cdot\overline{GM}\cdot \phi+K_{\rm H}+K_{\rm P}+K_{\rm R}+K_{\rm F}+K_{\rm env}\\ (I_{\rm z}+I_{\rm z})r-N_{\rm p}v=N_{\rm H}+N_{\rm P}+N_{\rm R}+N_{\rm F}+N_{ \rm env}\end{array} \right., \tag{1}\]  
where \(-X_{u}=A_{11}(0)\), \(-Y_{y}=A_{22}(0)\), \(J_{\rm x}=A_{44}(\omega_{\rm roll})\) and \(J_{\rm z}=A_{66}(0)\) denote the added mass and added moments of inertia. Subscript H denotes the viscous hydrodynamic force in terms of a third Taylor series expansion in which small coefficients are neglected, subscript P denotes the thrusting force due to the propeller, subscript R denotes the force due to the rudder, subscript F denotes the force due to the fin stabilizers, and subscript env denotes the forces due to environment, such as wind and waves [28]. Equation (1) can be decoupled from the sway, roll and yaw modes. To facilitate understanding of ship motion system, here the control object is formulated still in the continuous-time system form, which takes the matrix form Equation (2), viz., the following three DOF nonlinear ship lateral motion models with wave disturbances at a constant speed. \[\mathbf{M^{\prime}v^{\prime}}+\mathbf{N^{\prime}v^{\prime}}+\mathbf{G^{\prime}\eta^{\prime }}=\mathbf{b^{\prime}u}+\mathbf{F^{\prime}_{\rm w}}+\mathbf{a^{\prime}}(\mathbf{x}), \tag{2}\]  
where the superscript \(\mathbf{\prime}\) denotes dimensionless quantities, subscript \(\rm w\) denotes wave disturbances, \(\mathbf{v^{\prime}}=\left[ \begin{array}{cc}v^{\prime}&p^{\prime}&r^{\prime}\end{array}  \right]^{T}\) is the vector of velocities, \(\mathbf{\eta}=\left[ \begin{array}{cc}y^{\prime}&\phi&\psi\end{array} \right]^{T}\) is the vector of position/Euler angles, \(\mathbf{u}=\left[ \begin{array}{cc}\alpha&\delta\end{array} \right]^{T}\) is the vector of control inputs, \(\mathbf{b}=\left[ \begin{array}{ccc}Y^{\prime}_{\alpha}&K^{\prime}_{\alpha}&N^{ \prime}_{\alpha}\\ Y^{\prime}_{\delta}&K^{\prime}_{\delta}&N^{\prime}_{\delta}\end{array} \right]^ {T}\) is the vector of control coefficients, \(\mathbf{F^{\prime}_{\rm w}}=\left[ \begin{array}{cc}Y^{\prime}_{\rm w}&K^{\prime}_ {\rm w}&N^{\prime}_{\rm w}\end{array} \right]^{T}\) is the vector of the wave force and moments and \(\mathbf{a^{\prime}}(x)=\left[ \begin{array}{cc}\alpha^{\prime}_{\rm y}(x)&\alpha ^{\prime}_{\rm K}(x)&\alpha^{\prime}_{\rm N}(x)\end{array} \right]^{T}\) is the combination of system nonlinear terms. The inertia matrix \(\mathbf{M^{\prime}}\) is defined as  
\[\mathbf{M^{\prime}}=\left[ \begin{array}{ccc}m^{\prime}-Y^{\prime}_{v}&-m^{ \prime}z^{\prime}_{g}+Y^{\prime}_{p}&m^{\prime}x^{\prime}_{g}-Y^{\prime}_{r} \\ -m^{\prime}z^{\prime}_{g}-K^{\prime}_{v}&I^{\prime}_{x}-K^{\prime}_{p}&-I^{ \prime}_{xz}-K^{\prime}_{r}\\ m^{\prime}x^{\prime}_{g}-N^{\prime}_{v}&-I^{\prime}_{xz}-N^{\prime}_{p}&I^{ \prime}_{z}-N^{\prime}_{r}\end{array} \right], \tag{3}\]  
the expression for \(\mathbf{N^{\prime}}\) is obtained by local linearization of Coriolis/centripetal and damping coefficient matrix, which gives  
\[\mathbf{N^{\prime}}=\left[ \begin{array}{ccc}-Y^{\prime}_{v}&-Y^{\prime}_{p}&m^{ \prime}u^{\prime}_{0}-Y^{\prime}_{r}\\ -K^{\prime}_{v}&-K^{\prime}_{p}&-m^{\prime}z^{\prime}_{g}u^{\prime}_{0}-K^{ \prime}_{r}\\ -N^{\prime}_{v}&-N^{\prime}_{p}&m^{\prime}x^{\prime}_{g}u^{\prime}_{0}-N^{ \prime}_{r}\end{array} \right], \tag{4}\]  
and the linear restoring forces and moments for the ship can be written by  
\[\mathbf{G^{\prime}}=\left[