## 2 Preliminaries and Problem Formulation
### Extended State Observer  
In this section, an ESO is used for estimating total disturbances consisting of the unknown term of the system matrix \(C_{i}(v_{i})\), \(D_{i}(v_{i})\) and environmental disturbances \(d_{i}\)[25]. The \(i\)th USV dynamic (1) is rewritten as  
\[ \begin{split}\dot{\eta}_{i}&=J_{i}(\psi_{i})v_{i} \\ \dot{v}_{i}&=\zeta_{i}+M_{i}^{-1}\tau_{i},\end{split}  \tag{5}\]  
where \(\zeta_{i}=[\zeta_{i,1},\zeta_{i,2},\zeta_{i,3}]^{T}\in\mathbb{R}^{3}\) is total disturbances; it is a state vector expressed as  
\(\zeta_{i}=M_{i}^{-1}(-C_{i}(v_{i})-D_{i}(v_{i})v_{i}+d_{i})\). The following assumption is made during ESO design. **Assumption 5**. : _There exists a positive constant \(\zeta_{i}^{*}\) satisfying \(||\dot{\zeta}_{i}||\leq\zeta_{i}^{*}\)._  
**Remark 1**. : _Note that \(\zeta_{i}=M_{i}^{-1}(-C_{i}(v_{i})-D_{i}(v_{i})v_{i}+d_{i})\). The velocity \(v_{i}\) is bounded, and control inputs to drive USVs are bounded, and thus the derivative of \(v_{i}\) is bounded. According to Assumption 3 and the disturbances \(d_{i}\) are bounded, the derivative of \(d_{i}\) is bounded. Then, Assumption 5 is reasonable._  
An ESO is used for estimating the total disturbances as follows:  
\[ \begin{split}\dot{\eta}_{i}&=-K_{i,\rho 1}(\hat{\eta}_{i}-\eta_{i})+J_{i}(\psi_{i})\hat{\eta}_{i},\\ \dot{v}_{i}&=-K_{i,\rho 2}I_{i}^{T}(\psi_{i})(\hat{ \eta}_{i}-\eta_{i})+\hat{\zeta}_{i}+M_{i}^{-1}\tau_{i},\\ \dot{\xi}_{i}&=-K_{i,\rho 3}I_{i}^{T}(\psi_{i})(\hat{ \eta}_{i}-\eta_{i}),\end{split}  \tag{6}\]where \(\hat{\eta}_{i}=[\hat{x}_{i},\hat{y}_{i},\hat{y}_{i}]^{T}\in\mathbb{R}^{3}\), \(\hat{v}_{i}=[\hat{u}_{i},\hat{v}_{i},\hat{r}_{i}]^{T}\in\mathbb{R}^{3}\), \(\hat{\xi}=[\hat{\xi}_{i,1},\hat{\xi}_{i,2},\hat{\xi}_{i,3}]^{T}\in\mathbb{R}^{3}\), \(\hat{x}_{i},\hat{y}_{i}\), \(\hat{y}_{i}\), \(\hat{u}_{i}\), \(\hat{v}_{i}\), \(\hat{r}_{i}\), \(\hat{\xi}_{i,1}\), \(\hat{\xi}_{i,2}\), \(\hat{\xi}_{i,3}\) are the estimates of \(x_{i}\), \(y_{i}\), \(\psi_{i}\), \(u_{i}\), \(v_{i}\), \(r_{i}\), \(\zeta_{i,1}\), \(\zeta_{i,2}\) and \(\zeta_{i,3}\). \(K_{i,01}\in\mathbb{R}^{3\times 3}\), \(K_{i,02}\in\mathbb{R}^{3\times 3}\) and \(K_{i,03}\in\mathbb{R}^{3\times 3}\) are gain matrices. From (5) and (6), the error dynamics of the observer can be expressed as  
\[\dot{\hat{\eta}}_{i} =-K_{i,01}\tilde{\eta}+I_{i}(\psi_{i})\tilde{v}_{i},\] \[\dot{\hat{v}}_{i} =-K_{i,02}I_{i}^{T}(\psi_{i})\tilde{\eta}+\tilde{\zeta}_{i},\] \[\dot{\hat{\xi}}_{i} =-K_{i,03}I_{i}^{T}(\psi_{i})\tilde{\eta}-\dot{\zeta}_{i}, \tag{7}\]  
where \(\tilde{\eta}=\hat{\eta}_{i}-\eta_{i}\), \(\tilde{v}=\hat{v}_{i}-v_{i}\) and \(\tilde{\xi}_{i}=\hat{\xi}_{i}-\zeta_{i}\) are estimation errors. (7) can be rewritten as  
\[\dot{\hat{X}}_{i}=-A_{i}\hat{X}_{i}-B_{i}\hat{\xi}_{i}, \tag{8}\]  
where \(\hat{X}_{i}=[\hat{\eta}_{i}^{T},\hat{\eta}_{i}^{T},\hat{\xi}_{i}^{T}]^{T}\in \mathbb{R}^{9\times 1}\),  
\[A_{i}=\left[ \begin{array}{cc}K_{i,01}&-J_{i}(\psi_{i})&0_{3\times 3}\\ K_{i,02}I_{i}^{T}(\psi_{i})&0_{3\times 3}&-I_{3}\\ K_{i,03}I_{i}^{T}(\psi_{i})&0_{3\times 3}&0_{3\times 3}\end{array} \right] \tag{9}\]  
and  
\[B_{i}=\left[ \begin{array}{c}0_{3\times 3}\\ 0_{3\times 3}\\ I_{3}\end{array} \right]. \tag{10}\]  
**Theorem 1**. : _Consider the system (5) under Assumptions 2-5, the proposed observer (6) guarantees estimation error is bounded._  
Proof.