## 3 Event-Triggered Extended-State Observer
The first event-triggering time is \(t_{1}=0\)._  
To facilitate controller design and stability analysis, \(\bar{\xi}_{i}=[\bar{\xi}_{i,1},\bar{\xi}_{i,2},\bar{\xi}_{i,3}]^{T}\) (16) can be written as  
\[\hat{\mu}_{i} =\xi_{i,1}+\hat{\upsilon}_{i,1}+\frac{\tau_{i,uc}-\varpi_{i,u}}{ m_{i,11}}, \tag{21}\] \[\hat{\upsilon}_{i} =\xi_{i,2}+\hat{\upsilon}_{i,2},\] (22) \[\hat{\mu}_{i} =\xi_{i,3}+\hat{\upsilon}_{i,3}+\frac{\tau_{i,rc}-\varpi_{i,r}}{ m_{i,33}} \tag{23}\]  
with  
\[\xi_{i,1} = -\frac{3}{\epsilon_{i}^{2}}[\cos(\psi_{i})(\hat{x}_{i}-\bar{\xi}_ {i,1})+\sin(\psi_{i})(\hat{y}_{i}-\bar{\xi}_{i,2})], \tag{24}\] \[\xi_{i,2} = -\frac{3}{\epsilon_{i}^{2}}[-\sin(\psi_{i})(\hat{x}_{i}-\bar{\xi} _{i,1})+\cos(\psi_{i})(\hat{y}_{i}-\bar{\xi}_{i,2})],\] (25) \[\bar{\zeta}_{i,3} = -\frac{3}{\epsilon_{i}^{2}}(\hat{y}_{i}-\bar{\xi}_{i,3}), \tag{26}\]  
Define \(\bar{x}_{i}=\hat{x}_{i}-x_{i}\), \(\hat{y}_{i}=\hat{y}_{i}-y_{i}\), \(\bar{\psi}_{i}=\hat{\psi}_{i}-\psi_{i}\), \(\bar{u}_{i}=\hat{u}_{i}-u_{i}\), \(\bar{v}_{i}=\hat{v}_{i}-v_{i}\), \(\bar{r}_{i}=\hat{r}_{i}-r_{i}\),  
\[\bar{\eta}_{i} =[\bar{x}_{i},\bar{y}_{i},\bar{\psi}_{i}]^{T},\;e_{i,1}(t)=[e_{11},e _{12},e_{13}]^{T}=\frac{\bar{\eta}_{i}(e_{i}t)}{\epsilon_{i}^{2}}, \tag{27}\] \[\hat{\upsilon}_{i} =[\hat{u}_{i},\hat{\upsilon}_{i},\bar{r}_{i}]^{T},\;e_{i,2}(t)=\frac {\bar{v}_{i}(e_{i}t)}{\epsilon_{i}},\] (28) \[\tilde{\upsilon}_{i} =\hat{\upsilon}_{i}-\bar{\upsilon}_{i},\;e_{i,3}(t)=\tilde{ \upsilon}_{i}(e_{i}t). \tag{29}\]It follows that  
\[\hat{\mu}_{i} =\xi_{i,1}+\hat{\upsilon}_{i,1}+\frac{\tau_{i,uc}-\varpi_{i,u}}{ m_{i,11}}, \tag{21}\]0 \[\hat{\mu}_{i} =\xi_{i,1}+\hat{\upsilon}_{i,1}+\frac{\tau_{i,uc}-\varpi_{i,u}}{ m_{i,11}}, \tag{21}\]1 (31) \[\hat{\mu}_{i} =\xi_{i,1}+\hat{\upsilon}_{i,1}+\frac{\tau_{i,uc}-\varpi_{i,u}}{ m_{i,11}}, \tag{21}\]2 \[\hat{\mu}_{i} =\xi_{i,1}+\hat{\upsilon}_{i,1}+\frac{\tau_{i,uc}-\varpi_{i,u}}{ m_{i,11}}, \tag{21}\]3  
**Theorem 1**. : _Consider the USV models (7) and (8), the ETESO (15)-(17) and the event-triggering condition (19). If Assumptions 1 and 2 are satisfied, there exist \(e_{i}>0\) and \(\iota>0\) such that for any \(k>0\),_  
\[\hat{\mu}_{i} =\xi_{i,1}+\hat{\upsilon}_{i,1}+\frac{\tau_{i,uc}-\varpi_{i,u}}{ m_{i,11}}, \tag{21}\]4  
Proof. : Along with (18) and (19), the event-triggering mechanism can be described as  
\[\xi_{i}(t)= \left\{ \begin{array}{ll}\eta_{i}(t_{k}),&\text{if }\|\eta_{i}(t_{k})- \eta_{i}(t)\|<c_{q}e_{i}^{2}\\ \eta_{i}(t),&\text{otherwise.}\end{array} \right. \tag{34}\]  
When \(t\in[t_{k},t_{k+1})\), the sample error (\(\eta_{i}(t_{k})-\eta_{i}(t)\)) becomes  
\[\|\eta_{i}(t_{k})-\eta_{i}(t)\| = \|\int_{t_{k}}^{t}\dot{\eta}_{i}(i)d\| \tag{35}\] \[\leq\int_{t_{k}}^{t}\|R_{i}(\psi_{i}(t))\
u_{i}\|d\iota.\]  
Assumption 2 implies that  
\[\|\eta_{i}(t_{k})-\eta_{i}(t)\|\leq(t-t_{k})\
u_{i}^{*}. \tag{36}\]  
There exist \(\iota=e_{i}^{2}c_{q}/\
u^{*}\) and \(\iota=t-t_{k}\) such that \(\|\eta_{i}(t_{k})-\eta_{i}(t)\|\leq c_{q}e_{i}^{2}\). It is worth noting that \(t\in[t_{k},t_{k+1})\), leading to the conclusion that \(\min\{t_{k+1}-t_{k}\}\geq\iota\). Theorem 1 shows that [PERSON] behavior can be avoided by using the proposed observer. Next, the error dynamics of the proposed ETESO are investigated. To facilitate stability analysis, the following definitions are posited:\(E_{i}(t)=[e_{i,1}(t)^{T},e_{i,2}(t)^{T},e_{i,3}(t)^{T}]^{T}\in\mathbb{R}^{9}\). The error dynamics (Equations (30)-(32)) can be expressed as  
\[\dot{E}_{i}(t)=A_{i}E_{i}(t)-B_{i}e_{i}\dot{\
u}_{i}(t)-H_{i}D_{i}q(t), \tag{37}\]  
\[A_{i}=\left[ \begin{array}{ccc}-3I_{3}&R_{i}(\psi_{i}(t))&0_{3}\\ -3R_{i}^{T}(\psi_{i}(t))&0_{3}&I_{3}\\ -R_{i}^{T}(\psi_{i}(t))&0_{3}&0_{3}\end{array} \right], \tag{38}\]  
\[B_{i}=\left[ \begin{array}{c}0_{3}\\ 0_{3}\\ I_{3}\end{array} \right], \tag{39}\]  
\[D_{i}=\left[ \begin{array}{c}3_{3}\\ 3_{3}\\ I_{3}\end{array} \right], \tag{40}\]  
and \(H_{i}=\text{diag}\{I_{3},R_{i}^{T}(\psi_{i}(t)),R_{i}^{T}(\psi_{i}(t))\}\).