## Appendix A Proof of the Theorems
\begin{split}\left\|I_{i}\left[\begin{array}{c}h_{i}(t)-\phi_{ i}(t)\\ r_{i}(t_{k})-r_{i}(t)\end{array}\right]\right\|^{2}&\leq\left(\frac{|h_{i}(t)-\phi_{ i}(t)|}{\ell_{i}^{4}}\right)^{2}+\left[\frac{4\beta_{0}}{\lambda\|P_{i}K_{i}( \beta)\|\ell_{i}^{\frac{3}{3}}}\left(\frac{\lambda\sigma\theta}{2}\sqrt{\frac {\varpi}{\varpi}}-\|P_{i}\|M_{i}\right)\right]^{2}\\ &\leq 2\left(\frac{1}{\ell_{i}}\|\xi_{i}(t_{0})\|\sqrt{\frac{\varpi} {\varpi}}e^{-\frac{\lambda(1-\sigma)\ell_{i}}{2\varpi}(t-t_{0})}+\frac{\delta _{i}\theta}{\ell_{i}^{3}}\right)^{2}\end{split}  \tag{18}\]  
where \(\beta_{0}\) is some positive constant independent of \(\ell_{i}\), and \(\delta_{i}=\max\biggl{\{}\frac{1}{\ell_{i}^{2}}\frac{2\beta_{0}\lambda\sigma} {\lambda\|P_{i}K_{i}(\beta)\|}\sqrt{\frac{\varpi}{\varpi}}\biggr{\}}\). The influence caused by external disturbance and event-triggered sampling is proved to be bounded in (18). After that, we solve \(W_{i}(t)\) from (15), then the following inequality is obtained:  
\[W_{i}(t)\leq e^{-\frac{a_{3}\ell_{i}}{2}(t-t_{0})}V_{i}(t_{0})+\int_{t_{0}}^{t }e^{-\frac{a_{3}\ell_{i}}{2}(t-s)}\frac{{a_{4}}^{2}}{a_{3}\ell_{i}}\Biggl{(} \frac{1}{\ell_{i}}\|\xi_{i}(t_{0})\|\sqrt{\frac{\varpi}{\varpi}}e^{-\frac{ \lambda(1-\sigma)\ell_{i}}{2\varpi}(t-t_{0})}+\frac{\delta_{i}\theta}{\ell_{i} ^{3}}\Biggr{)}^{2}ds \tag{19}\]  
After that, the following bound is obtained:  
\[\limsup_{t\rightarrow\infty}W_{i}(t)\leq\frac{2{a_{4}}^{2}}{a_{3}^{2}\ell_{i} ^{2}}\left(\frac{\delta_{i}\theta}{\ell_{i}^{3}}\right)^{2} \tag{20}\]  
According to condition (20), we obtain  
\[\limsup_{t\rightarrow\infty}\|\Theta_{i}(t)\|\leq\frac{\sqrt{2}a_{4}}{a_{1}a_{ 3}}\frac{\delta_{i}\theta}{\ell_{i}^{4}} \tag{21}\]  
This result concludes the demonstration. **Proof of Theorem 3. ** From the result (A21), we show that  
\[W_{i}(t)\leq V_{i}(t_{0})+\frac{{a_{4}}^{2}}{{a_{3}}}\|I_{i}\|^{2}\bigg{(}\|\xi_{i }(t_{0})\|\sqrt{\frac{\varpi}{\varpi}}+\frac{\delta_{i}\theta}{{\ell_{i}}^{4}} \bigg{)}^{2}\frac{2}{{a_{3}}}\] (A22)  
Here, we define \(R_{i}=V_{i}(t_{0})+\frac{{a_{4}}^{2}}{{a_{3}}}\|I_{i}\|^{2}\Big{(}\|\xi_{i}(t_{0 })\|\sqrt{\frac{\varpi}{\varpi}}+\frac{\delta_{i}\theta}{{\ell_{i}}^{4}}\Big{)} ^{2}\frac{2}{{a_{3}}}\), then we have \(|r_{i}(t)|\leq\frac{\sqrt{R}_{i}}{{a_{1}}}\), \(|\dot{r}_{i}(t)|\leq\frac{\sqrt{R}_{i}}{{a_{1}}}\) for any \(t\geq t_{0}\). After that, we obtain  
\[ \begin{split}\frac{d(r_{i}(t)-r_{i}(t_{k}))^{2}}{dt}=& 2(r_{i}(t)-r_{i}(t_{k}))\dot{r}_{i}(t)\leq 2|r_{i}(t)-r_{i }(t_{k})||\dot{r}_{i}(t)|\\ \leq& 2(|r_{i}(t)|+|r_{i}(t_{k})|)|\dot{r}_{i}(t)| \leq\frac{4R_{i}}{{a_{1}}^{2}}\end{split} \] (A23)  
The minimum sampling interval corresponds to the time required for the driving mechanism to regenerate from 0 to the threshold specified in the driving conditions after the previous trigger event. Consequently, by establishing an upper bound on the derivative of the driving mechanism, we can derive a lower bound on the necessary time interval based on both the threshold and this upper bound. According to the driven mechanism (4)-(6), we obtain the minimum interval between each of the two sampling behaviors:  
\[\Delta t_{\min}=\frac{{a_{1}}^{2}{\Pi_{i}}^{2}}{4R_{i}}\] (A24)  
Therefore, Zeno behavior will not happen based on (A24).