## 2 Problem Formulation and Preliminaries  
### Mathematical Model  
This article focuses on adaptive fault-tolerant control of ship dynamic positioning. Therefore, based on [PERSON]'s proposed dynamic positioning ship model Fossen (2016), a simplified three-degree-of-freedom kinematic and dynamic model is established.  
\[\left\{\begin{array}{l}\dot{\eta}=J(\psi)\
u\\ M\dot{\
u}=-D(\
u)\
u+\tau+d\end{array}\right. \tag{1}\]  
where \(\eta=[x,y,\psi]^{T}\) denotes the position and heading angle of the ship vessel in the earth coordinate system; \(\
u=[u,v,r]^{T}\) represents the ship's surge velocity, sway velocity, and yaw angular velocity in the body-fixed coordinate system; \(d\) denotes the external disturbance; and \(\tau=[\tau_{u},\tau_{v},\tau_{r}]^{T}\) represents the control force and torque. \(J(\psi)\in R^{3\times 3}\) represents the transfer matrix from the body-fixed coordinate to the earth coordinate, with the specific form as follows:  
\[J(\psi)=\begin{bmatrix}\cos(\psi)&-\sin(\psi)&0\\ \sin(\psi)&\cos(\psi)&0\\ 0&0&1\end{bmatrix} \tag{2}\]  
The following properties are satisfied:  
\[\left\{\begin{array}{l}J^{T}(\psi)J(\psi)=I_{3\times 3},\|J(\psi)\|=1\\ \dot{J}(\psi)=J(\psi)S(r)\\ J^{T}(\psi)S(r)J(\psi)=J(\psi)S(r)J^{T}(\psi)=S(r)\end{array}\right. \tag{3}\]  
The nominal inertia matrix \(\overline{M}\in R^{3\times 3}\) and the nominal hydrodynamic damping matrix \(\overline{D}(v)\in R^{3\times 3}\) are as follows:  
\[\overline{M}=\begin{bmatrix}m_{11}&0&0\\ 0&m_{22}&m_{23}\\ 0&m_{32}&m_{33}\end{bmatrix} \tag{4}\]  
\[\overline{D}(\
u)=\begin{bmatrix}d_{11}(\
u)&0&0\\ 0&d_{22}(\
u)&d_{23}(\
u)\\ 0&d_{32}(\
u)&d_{33}(\
u)\end{bmatrix} \tag{5}\]  
where \(m_{11}=m-X_{\dot{u}}\), \(m_{22}=m-Y_{\dot{v}}\), \(m_{23}=m_{xg}-Y_{\dot{v}}\), \(m_{32}=m_{xg}-N_{\dot{v}}\), \(m_{33}=I_{z}-N_{\dot{r}},c_{13}(\
u)=-m_{11}\
u-m_{23}r\), \(c_{23}(\
u)=-m_{11}u,d_{11}(\
u)=-X_{u}-X_{|u|}u|-X_{uuuu}|u|^{2},d_{22}(\
u) =-Y_{\dot{v}}-Y_{|v|v}|v|\), \(d_{23}(\
u)=-Y_{r}-Y_{|v|r}|v|-Y_{|r|r}|\), \(d_{32}(\
u)=-N_{\dot{v}}-N_{|v|\dot{v}}|v|-N_{|r|v}|r|\), \(d_{33}(\
u)=-N_{r}-N_{|v|r}|v|-N_{|r|r}|r|\), and \(Y_{\dot{r}}=N_{\dot{v}}\). \(m\) denotes the mass of the ship, and \(I_{z}\) represents the inertia moment of the ship.  
In practical applications, it is difficult to obtain an accurate model of the ship. Therefore, based on the nominal model and model uncertainty, the following descriptions are given D. [PERSON] (2016):  
\[\begin{array}{l}M=\overline{M}+\Delta M\\ D(\
u)=\overline{D}(\
u)+\Delta D(\
u)\end{array} \tag{6}\]where \(\overline{M}\) and \(\overline{D}(\
u)\) represent the matrices of the nominal model, and \(\Delta M\) and \(\Delta D(\
u)\) denote the matrices of the unmodeled dynamics. Therefore, Equation (1) can be rewritten as:  
\[\left\{\begin{array}{l}\dot{\eta}=J(\psi)\
u\\ \overline{M}\
u=-\overline{D}(\
u)\
u+\tau+d+f\end{array}\right. \tag{7}\]  
where \(f=-\Delta M\dot{\
u}-\Delta D(\
u)\
u\).  
In addition, considering the thruster faults, the actual control force and torque can be expressed as [19]:  
\[\tau=\vartheta\tau_{c}+\delta \tag{8}\]  
where \(\vartheta=diag\{\vartheta_{u},\vartheta_{v},\vartheta_{r}\}\) represents the efficiency matrix of thrusters, \(0<\vartheta_{j}\leq 1,j=u,v,r\). If \(\vartheta_{j}=1\), it indicates that the thruster is working normally; if \(0<\vartheta_{j}<1\), it means that the thruster has a partial fault. \(\delta\in R^{3}\) is the bias fault. \(\tau_{c}\) denotes the command control vector calculated by the controller.  
Based on the above analysis, the mathematical model of the ship, considering thruster faults and input saturation constraints, can be represented as follows:  
\[\left\{\begin{array}{l}\dot{\eta}=J(\psi)\
u\\ \overline{M}\
u=-\overline{D}(\
u)\
u+\vartheta\tau_{c}+\Omega\end{array}\right. \tag{9}\]  
where \(\Omega=d+f+\delta\) represents the lumped uncertainties.