## 3 Integral-Type Edge-Event-Triggered Policy
\begin{split}\frac{d}{dt}\|e_{i}(t)\|^{2}\leq& 2\|e_{i}(t)\|\|e_{i}^{\prime}(t)\|\\ \leq& 2\|e_{i}(t)\|\left(\|A\|\|e_{i}(t)\|+\Xi_{k}^{i} \right)\\ \leq&(2\|A\|+1)\|e_{i}(t)\|^{2}+(\Xi_{k}^{i})^{2}, \end{split}  \tag{23}\]  
Using the comparison principle, one can further obtain that  
\[ \begin{split}\|e_{i}(t)\|^{2}\leq&(\Xi_{k}^{i})^{2} \int_{t_{k}}^{t}e^{(2\|A\|+1)(t-s)}\,ds\\ \leq&(\Xi_{k}^{i})^{2}(t-t_{k}^{i})e^{(2\|A\|+1)(t-t _{k}^{i})}.\end{split}  \tag{24}\]On the other hand, when triggering condition (6) is satisfied, the next event time \(t_{k+1}^{i}\) is obtained from \(f_{i}(t)=0\). The sufficient condition is obtained to guarantee \(f_{i}(t)\leq 0,t\in[t_{k^{\prime}}^{i},t_{k+1}^{i})\):  
\[ \begin{split}\int_{t_{k}^{i}}^{t}\|e_{i}(s)\|^{2}\,ds\leq& (\Xi_{k}^{i})^{2}\int_{t_{k}^{i}}^{t}(s-t_{k}^{i})e^{(2\|A\|+1)(s-t_{k}^{i})} \,ds\\ \leq&\frac{(\Xi_{k}^{i})^{2}}{2\|A\|+1}(t-t_{k}^{i} )\Big{(}e^{(2\|A\|+1)(t-t_{k}^{i})}-1\Big{)},\end{split}  \tag{25}\]  
Accordingly, with Equation (25), one obtains  
\[ \begin{split}\int_{t_{k}^{i}}^{t}\|e_{i}(s)\|^{2}\,ds\leq& \frac{(\Xi_{k}^{i})^{2}}{2\|A\|+1}(t-t_{k}^{i})\Big{(}e^{(2\|A\|+1)(t-t_{k}^{i} )}-1\Big{)}\\ \leq& k_{i}\|z_{i}(t_{k}^{i})\|^{2}(t-t_{k}^{i})\\ \leq&\int_{t_{k}^{i}}^{t}k_{i}\|z_{i}(t_{k}^{i})\|^ {2}+\beta e^{-\alpha s}\,ds,\end{split}  \tag{26}\]  
Letting \(\tau_{i}=t-t_{k}^{i}\) and combining with the condition (26), one gets  
\[\frac{(\Xi_{k}^{i})^{2}}{2\|A\|+1}\Big{(}e^{(2\|A\|+1)\tau_{i}}-1\Big{)}=k_{i }\|z_{i}(t_{k}^{i})\|^{2}. \tag{27}\]  
In order to show \(\tau_{i}>0\), we proceed by contradiction. Suppose that \(\tau_{i}<0\), the left side of (27) is negative, whereas the right side of (27) is positive. Furthermore, if \(\tau_{i}=0\), the left side of (27) is zero, which is not equal to \(k_{i}\|z_{i}(t_{k}^{i})\|^{2}\). Equation (27) implicitly defines \(\tau_{i}\) as a positive lower bound between any two consecutive event instants. Hence, no Zeno behavior occurs. The proof is completed. **Remark 1**. : _To select the appropriate parameters \(a,\eta>\lambda_{m}(S_{1}S_{2}+S_{2}S_{1})\), \(\gamma=\frac{2k_{max}}{1-2k_{max}}\), \(k_{max}<\frac{1}{2}\) to guarantee the condition (8), we analyze the matrix. Let \(a=c^{2}\) and \(\gamma=\frac{1}{c^{3}}\) i.e., \(k=\frac{1}{2c^{3}+2}<\frac{1}{2}\). It follows that matrix (8) can be represented as_  
\[ \begin{bmatrix}A+B&I_{N}\otimes X&H\otimes X&I_{N}\otimes B_{1}&H\otimes BY \\ I_{N}\otimes X&-\frac{1}{\eta\|C\|^{2}}I_{N\times n}&0&0&0\\ H^{T}\otimes X&0&-cI_{N\times n}&0&0\\ I_{N}\otimes B_{1}^{T}&0&0&-I_{N\times m}&0\\ H^{T}\otimes Y^{T}B^{T}&0&0&0&-I_{N}\otimes X\end{bmatrix} . \tag{28}\]  
_Therefore, only the sufficiently large \(c\) and \(\eta>\lambda_{m}(S_{1}S_{2}+S_{2}S_{1})\) are needed to ensure that the inequality holds._  
**Remark 2**. : _For the triggering condition (6), only the integral of \(\|e_{i}(t)\|^{2}\) from each triggering instant \(t_{k}^{i}\) to the time \(t\), needs to be less than a designed variable in time intervals, and \(\|e_{i}(t)\|^{2}\) does not need to be less than a given variable at any time. That is, the proposed mechanism relaxes the requirement in traditional event-triggered control [13, 14, 15, 16, 17, 19, 20, 21, 22, 23, 25]. Then, we design the following nonintegral-type event condition_  
\[ \begin{split}& t_{k^{\prime}+1}^{i}=\inf\Bigl{\{}t>t_{k^{\prime}} ^{i}:f_{i}^{\prime}(t)\geq 0\Bigr{\}},i\in\mathbb{N},\\ & f_{i}^{\prime}(t)=\|e_{i}(t)\|^{2}-\|z_{i}(t_{k}^{i})\|^{2}- \beta e^{-at}\geq 0.\end{split}  \tag{29}\]  
_where \(\{t_{k^{\prime}}^{i}\}_{k^{\prime}\in\mathbb{Z}_{\geq 0}}\) represents the triggering instant sequence of agent \(v_{i}\). Under the system (2) and control protocol (3), it can achieve synchronization asymptotically, and its minimum inter-event interval is less than Theorem 1._  
**Remark 3**. : _The triggering function (6) has a hybrid form, which contains time-dependent item \(\int_{t_{k}^{i}}^{t}\beta e^{-as}\,ds\) and state-dependent item \(\int_{t_{k}^{i}}^{t}\|z_{i}(t_{k}^{i})\|^{2}\,ds\).