## 2 Analysis of Mathematical Model  
To design control system for roll stabilization, it is necessary to incorporate roll motion to the horizontal plane model, i.e. restoring moment due to gravity and buoyancy must be included. The resulting maneuvering models including roll (surge, sway, yaw and roll) have been concluded in [4; 10]. Thereinto, a four degrees of freedom (DOF) nonlinear model of a high speed container ship was proposed by [PERSON] and [PERSON] [24; 25], just as shown in Figure 1.  
\[ \begin{array}{l}(m+m_{x})\dot{u}-(m+m_{y})vr=X+\tau_{X}\\ (m+m_{y})\dot{v}+(m+m_{x})ur+m_{y}\kappa_{y}\dot{r}-m_{y}l_{y}\dot{p}=Y+\tau_{Y }\\ (I_{z}+I_{x})\dot{r}+m_{y}\alpha_{y}\dot{v}=N-Y\tau_{X}+\tau_{N}\\ (I_{x}+J_{x})\dot{p}-m_{y}l_{y}\dot{v}-m_{x}l_{x}ur=J-W_{\Delta}GM\phi+\tau_{J }\end{array}  \tag{1}\]  
where \(m\) denotes ship mass, and \(I_{x}\), \(I_{z}\) denote the moments of inertia about the \(x\) and \(z\) axes, respectively. \(m_{x}\), \(m_{y}\) and \(J_{x}\), \(J_{z}\) denote the added mass and added moments of inertia. \(\alpha_{y}\) is the \(x\)-coordinate of the center of \(m_{y}\), while \(l_{x}\) and \(l_{y}\) are the \(z\)-coordinates of the centers of \(m_{x}\) and \(m_{y}\) respectively. \(x_{G}\) is the distance of gravity center in front of the midship. \((\psi,\phi)\) are the yaw and roll angle in the earth-fixed coordinate frame. \((u,v,r,p)\) denote surge, sway, yaw and roll velocities. \((\tau_{X},\tau_{Y},\tau_{N},\tau_{J})\) are recognized as control inputs. \(W_{\Delta}\) is the ship displacement weight and \(GM\) is the transverse metacentric height. \((X,Y,N,J)\) denote the hydrodynamic forces and moments, which are defined by third-order Taylor series expressions with small coefficients neglected. Figure 1: Ship’s body fixed and earth fixed coordinate frames. Defining the state vector as \(x:=\left[v,p,r,\phi,\psi\right]^{\text{T}}\), the linearized model of Equation (1) can be written in state-space form as follows  
\[y=\underbrace{\left[ \begin{array}{ccccc}a_{11}&a_{12}&a_{13}&a_{14}&0\\ a_{21}&a_{22}&a_{23}&a_{24}&0\\ a_{31}&a_{32}&a_{33}&a_{34}&0\\ 0&1&0&0&0\\ 0&0&1&0&0\\ \end{array} \right]}_{\text{C}}x+\underbrace{\left[ \begin{array}{ccccc}b_{11 }&b_{12}&\cdots&b_{1r}\\ b_{21}&b_{22}&\cdots&b_{2r}\\ b_{31}&b_{32}&\cdots&b_{3r}\\ 0&0&\cdots&0\\ 0&0&\cdots&0\\ \end{array} \right]}_{\text{C}}u \tag{2}\]  
**Assumption 1**. : _The dynamics between yaw and roll are decoupled in frequency domain. That is to say, the yaw motion caused by rolling is negligible, and vice versa._  
**Remark 1**. : _Assumption 1 is a common precondition for RRR control in the existing literature. The roll motion and related RRR rudder motion are of higher frequency, compared to yaw motion and regular course control rudder motion, which are typical inertial motions in lower frequency domain. Consequently, the course and RRR controllers can be designed separately in accordance with the additive principle._  
Considering Assumption 1 and the container ship with one rudder, \(u=\delta\) and \(b=\left[ \begin{array}{ccccc}b_{11}&b_{12}&b_{13}&0&0\\ \end{array} \right]^{\text{T}}\), the transfer functions can be derived as follows. \[G_{\psi}(s)=\frac{K_{\psi}(T_{3}s+1)}{s(T_{1}s+1)(T_{2}s+1)} \tag{3}\]  
\[G_{\phi}(s)=\frac{K_{\phi}\omega_{\phi}^{2}(-T_{5}s+1)}{(T_{4}s+1)(s^{2}+2\xi \omega_{\phi}s+\omega_{\phi}^{2})} \tag{4}\]  
where \(T_{i},i=1,2,3,4\) are time constants, \(K_{\phi}\) and \(K_{\psi}\) are speed-related gains in rolling and yawing, \(\xi\) is the rolling damping coefficient, and \(\omega_{\phi}\) is the rolling natural frequency. **Remark 2**. : _The parameters of Equations (3) and (4) are of typical navigation significance.