## Appendix A Proof of the Theorems
And then, we obtain  
\[\|\xi_{1}(t)\|\leq\sqrt{\frac{\bar{\omega}}{\underline{\omega}}}\max\biggl{\{} \|\xi_{1}(t_{0})\|e^{-\frac{\lambda(1-\sigma)\ell_{1}}{2\varnothing}(t-t_{0}) },\frac{2}{\lambda\sigma\ell}\|PK(\beta)\|\Pi_{1}+\frac{2}{\lambda\sigma\ell^{4 }}\|P\|M_{1}\biggr{\}} \tag{15}\]  
By substituting condition (7), we have  
\[\|\xi_{1}(t)\|\leq\max\Biggl{\{}\sqrt{\frac{\underline{\omega}}{\underline{ \omega}}}\|\xi_{1}(t_{0})\|e^{-\frac{\lambda(1-\sigma)\ell_{1}}{2\varnothing}(t -t_{0})},\frac{\theta}{\ell^{4}}\Biggr{\}} \tag{16}\]  
We show that  
\[\|\xi_{1}(t_{0})\|e^{-\frac{\lambda t_{1}}{2\overline{\omega}}(t-t_{0})}\leq \frac{\theta}{\ell_{1}^{4}} \tag{17}\]  
holds for any  
\[t\geq t_{0}+\frac{4\bar{\omega}}{\lambda\ell_{1}}\ln\frac{\ell_{1}^{4}\|\xi_{1 }(t_{0})\|}{\theta} \tag{18}\]  
Due to the fact that \(\lim\limits_{\ell_{1}\to\infty}\frac{4\bar{\omega}}{\lambda\ell_{1}}\ln\frac{ \ell_{1}^{4}\|\xi_{1}(t_{0})\|}{\theta}=\lim\limits_{\ell_{1}\to\infty}\frac{4 \bar{\omega}\underline{\theta}}{\lambda\ell_{1}^{4}\|\xi_{1}(t_{0})\|}=0\), we obtain that \(\|\xi_{1}(t_{0})\|\leq\frac{\theta}{\ell_{1}^{4}}\) holds for any \(t\geq t_{0}+T(\ell_{1})\), where \(T(\ell_{1})=\frac{4\bar{\omega}}{\lambda\ell_{1}}\ln\frac{\ell^{4}\|\xi_{1}(t_ {0})\|}{\theta}\). Similarly, the following conclusion is obtained for another two axis: \(\|\xi_{i}(t_{0})\|\leq\frac{\theta}{\ell_{i}^{4}}\) holds for any \(t\geq t_{0}+T(\ell_{i})\), where \(T(\ell_{i})=\frac{4\bar{\omega}}{\lambda\ell_{i}}\ln\frac{\ell_{1}^{4}\|\xi_{i }(t_{0})\|}{\theta}\), here i = 2, 3. Consequently, we obtain that conclusions (8)-(10) hold for any \(t\geq t_{0}+T(\ell)\) by the transformed relationship (17), where \(T(\ell)=\max\{T(\ell_{1}),T(\ell_{2}),T(\ell_{3})\}\). \(\lim\limits_{\ell\to\infty}T(\ell)=0\) holds obviously, and that ends the proof. Proof of Theorem 2.: Firstly, we rewrite (18) as follows:  
\[\dot{\Theta}_{i}(t)=\ell_{i}F(\Theta_{i})+J_{i}\left[ \begin{array}{c}h_{i}(t)- \phi_{i}(t)\\ r_{i}(t_{k})-r_{i}(t)\end{array} \right] \tag{13}\]  
where \(J_{i}=\left[ \begin{array}{cccc}0&\frac{1}{\ell_{i}^{2}}&0&0&0\\ 0&0&\beta_{i1}&\beta_{i2}&\beta_{i3}\end{array} \right]^{T}\). Based on this, we calculate the derivative of \(W_{i}\) as follows:  
\[\frac{dW_{i}}{dt}=\frac{\partial W_{i}}{\partial\Theta_{i}}\left(\ell_{i}F( \Theta_{i})+J_{i}\left[ \begin{array}{c}h_{i}(t)-\phi_{i}(t)\\ r_{i}(t_{k})-r_{i}(t)\end{array} \right]\right) \tag{14}\]  
Substituting conditions (20)-(22) into (14), we have  
\[\dot{W}_{i}(t)\leq -a_{3}\ell_{i}\|\Theta_{i}\|^{2}+a_{4}\|\Theta_{i}\|\left\|I_{i} \left[ \begin{array}{c}h_{i}(t)-\phi_{i}(t)\\ r_{i}(t_{k})-r_{i}(t)\end{array} \right]\right\| \tag{15}\] \[\leq -\frac{a_{3}\ell_{i}}{2}\|\Theta\|^{2}+\frac{{a_{4}}^{2}}{2a_{3} \ell_{i}}\|I_{i}\left[ \begin{array}{c}h_{i}(t)-\phi_{i}(t)\\ r_{i}(t_{k})-r_{i}(t)\end{array} \right]\right\|^{2}\]  
According to the design of the extended state observer and the event-triggered mechanism, it is obvious that inequalities  
\[|r_{i}(t_{k})-r_{i}(t)|\leq\Pi_{i}\leq\frac{1}{\|P_{i}K_{i}(\beta)\|\ell_{i}^ {3}}\left(\frac{\lambda\sigma}{\sqrt{\frac{\varpi}{\varpi}}}\theta-\|P_{i}\| M_{i}\right) \tag{16}\]  
and  
\[\frac{1}{\ell_{i}^{3}}|h_{i}(x)-\phi_{i}|\leq \max\Biggl{\{}\|\xi_{i}(t_{0})\|\sqrt{\frac{\varpi}{\varpi}}e^{- \frac{\lambda(1-\sigma)\ell_{i}}{2\varpi}(t-t_{0})},\frac{\theta}{\ell_{i}^{ 3}}\Biggr{\}} \tag{17}\] \[\leq \|\xi_{i}(t_{0})\|\sqrt{\frac{\varpi}{\varpi}}e^{-\frac{\lambda(1 -\sigma)\ell_{i}}{2\varpi}(t-t_{0})}+\frac{\theta}{\ell_{i}^{3}}\]  
hold. Combining (16) and (17), we have  
\[