# Output Feedback Tracking Control for Vessel with Collision-Avoidance and Performance Constraints
## 2 Problem Formulation  
As shown in Figure 1, the vessel's mathematical model is  
\[\dot{\eta}= R(\psi)\
u \tag{1}\] \[M\dot{\psi}+C\
u+D\
u= \tau+d, \tag{2}\]  
where \(\eta=[x,y,\psi]^{\mathrm{T}}\) is the vector containing position \((x,y)\) and yaw angle \(\psi\), which is expressed in earth-fixed frame \(X_{E}O_{E}Y_{E}\). \(\
u=[u,v,r]^{\mathrm{T}}\) is the velocity vector expressed in the body-fixed frame \(X_{B}O_{B}Y_{B}\), \(u\) and \(v\) are surge and sway, and \(r\) is the yaw velocity. \(M\in\mathbb{R}^{3\times 3}\) is the positive definite inertia matrix including added mass. \(C\in\mathbb{R}^{3\times 3}\) is the skew-symmetric matrix of Coriolis and the centripetal term. \(D\in\mathbb{R}^{3\times 3}\) is the positive definite damping matrix. \(\tau=[\tau_{u},\tau_{v},\tau_{r}]^{\mathrm{T}}\) is the input signal of the vessel control system, which is generated using the thruster system. \(d\) values are the external environmental disturbances. \(R\) denotes the rotation matrix, defined as  
\[R(\psi)=\left[\begin{array}{ccc}\cos(\psi)&-\sin(\psi)&0\\ \sin(\psi)&\cos(\psi)&0\\ 0&0&1\end{array}\right].\]  
In practice, owing to the thruster system's physical limitations, control inputs are limited. And the input limitation is modeled as  
\[\tau=\left\{\begin{array}{ccc}\tau_{\max}&&\tau_{c}>\tau_{\max}\\ \tau_{c}&&\tau_{\min}\leq\tau_{c}\leq\tau_{\max}\\ \tau_{\min}&&\tau_{c}<\tau_{\min}\end{array}\right.\]  
Figure 1: Earth-fixed frame and body-fixed frame.  
\[A=\left[\begin{array}{ccc}-K_{o1}&R(\psi)&0_{3}\\ -K_{o2}R^{T}(\psi)&0_{3}&I_{3}\\ -K_{o3}R^{T}(\psi)&0_{3}&0_{3}\end{array}\right], \tag{12}\]  
\[B=\left[\begin{array}{c}0_{3}\\ 0_{3}\\ I_{3}\end{array}\right], \tag{13}\]  
\[C_{oi}=\left[\begin{array}{ccc}I_{3}&0_{3}&0_{3}\end{array}\right], \tag{14}\]  
and \(\phi=-\zeta\). \(0_{3}\) represents a \(3\times 3\) dimensional zero matrix. \(I_{3}\) represents a \(3\times 3\) dimensional identity matrix.  
To make the stability analysis of the ESO convenient, a transformation \(\chi=TX\) with \(T=\text{diag}\{R^{T}(\psi),\,I_{3},I_{3}\}\) is introduced to remove the nonlinear term \(R(\psi)\). We have  
\[\dot{\chi}=(A_{0}+rS_{T})\chi+B\phi, \tag{15}\]  
where \(S_{T}=\text{diag}\{S_{1},0_{3},0_{3}\}\), \(\dot{\phi}\) denotes a positive constant, and we make the assumption that \(\|\phi\|\leq\ddot{\phi}\) according to Assumption 2.  
\[S_{1}=\left[\begin{array}{ccc}0&1&0\\ -1&0&0\\ 0&0&0\end{array}\right], \tag{16}\]  
\[A_{0}=\left[\begin{array}{ccc}-K_{o1}&I_{3}&0_{3}\\ -K_{o2}&0_{3}&I_{3}\\ -K_{o3}&0_{3}&0_{3}\end{array}\right]. \tag{17}\]  
**Lemma 1**.: _The estimation error \(X\) is bounded; if there are symmetric definite positive matrices \(Q,P\in\mathbb{R}^{9\times 9}\) results in_  
\[A_{0}^{T}P +PA_{0}+PBB^{T}P+Q\] \[+r(S_{T}^{T}P+PS_{T})\leq 0, \tag{18}\] \[A_{0}^{T}P +PA_{0}+PBB^{T}P+Q\] \[-r(S_{T}^{T}P+PS_{T})\leq 0, \tag{19}\]  
\(\tilde{r}\in\mathbb{R}\) _is the upper bound of \(r\)._  
Proof.: Chose a Lyapunov function as follows  
\[V_{o}=\frac{1}{2}\chi^{T}P\chi. \tag{20}\]  
Differentiating \(V_{o}\) with respect to time,  
\[V_{o}= \frac{1}{2}\chi^{T}(PA_{0}+A_{0}^{T}P+rPS_{T}+rS_{T}^{T}P)\chi+ \chi^{T}PB\phi\] \[\leq \frac{1}{2}\chi^{T}(PA_{0}+A_{0}^{T}P+rPS_{T}+rS_{T}^{T}P+PBB^{T} P)\chi+\frac{1}{2}\phi^{T}\phi\] \[\leq-c_{o_{1}}V_{o}+c_{o_{2}}, \tag{21}\]  
where \(c_{o_{1}}=\frac{\lambda_{\min}(Q)}{\lambda_{\max}(P)}\), \(c_{o_{2}}=\frac{1}{2}\ddot{\phi}^{2}\). Since \(c_{o_{2}}\) is symmetric definite positive, the state \(\chi_{i}\) is bounded using \(\|T^{-1}\|=1\) and \(X=T^{-1}\chi\). The estimation error signal \(X\) is bounded.