## 3 Integral-Type Edge-Event-Triggered Policy  
Here, the control strategy will be introduced. The designed policy on combined relative state measurements is given as follows:  
\[u_{i}(t)=-cKz_{i}(t_{k}^{i}),t\in[t_{k}^{i},t_{k+1}^{i}),\quad i=1,2,\cdots,N, \tag{3}\]  
where \(z_{i}(t)=\sum_{j\in N_{i}}a_{ij}\big{(}x_{i}(t)-x_{j}(t)\big{)}\). Here, \(x_{i}(t)-x_{j}(t)\) with \(j\in N_{i}\) represent the relative states between \(v_{i}\) and all neighboring agents \(v_{j}\) with \(j\in N_{i}\). We introduce the following integral-type triggering condition. Define the state measurement error between the sampled state and the current state by  
\[e_{i}(t)=z_{i}(t_{k}^{i})-z_{i}(t),t\in[t_{k}^{i},t_{k}^{i+1}). \tag{4}\]  
Each triggering instant \(t_{k}^{i}\) is determined by  
\[t_{k+1}^{i}=\inf\Bigl{\{}t>t_{k}^{i}:f_{i}(t)\geq 0\Bigr{\}},\quad i\in\mathbb{N}, \tag{5}\]  
where  
\[f_{i}(t)=\int_{t_{i}}^{t}\|e_{i}(s)\|^{2}\,ds-\int_{t_{i}}^{t}k_{i}\|z_{i}(t_{ k}^{i})\|^{2}+\beta e^{-\alpha s}\,ds. \tag{6}\]  
Here, \(k_{i}\), \(\beta\) and \(\alpha\) are positive constants to be designed. Denote \(R(t)=\{r_{p}(t)\}_{\mathcal{N}\times 1}=(H\otimes I)x(t)\). Under protocol (3), the closed-loop system can be represented by  
\[ \begin{split} R^{\prime}(t)&=(I_{\mathcal{N}} \otimes A)R(t)-c(HH^{T}\otimes BK)R(t)-c(H\otimes BK)e(t)+(H\otimes B_{1}) \Phi(y)\\ y(t)&=(I_{N}\otimes C)x(t),\end{split}  \tag{7}\]  
where \(x(t)=(x_{1}^{T}(t),x_{2}^{T}(t),\cdots,x_{N}^{T}(t))^{T}\), \(e(t)=(e_{1}^{T}(t),e_{2}^{T}(t),\cdots,e_{N}^{T}(t))^{T}\), \(y(t)=(y_{1}^{T}(t),\)\(y_{2}^{T}(t),\cdots,y_{N}^{T}(t))^{T}\). We note that \(z(t)=(E\otimes I_{n})R(t)\), \(E=\{e_{ij}\}\in\mathbb{R}^{N\times\mathcal{N}}\). The entries of matrix \(E\) are defined as \(e_{ij}=1\) if the \(j\)th entry of \(R(t)\) is \(r_{i}(t)-r.(t)\), or \(e_{ij}=-1\) if the \(j\)th entry of \(R(t)\) is \(r.(t)-r_{i}(t)\), or \(e_{ij}=0\) otherwise. ie. \(E=H^{T}\), \(z(t)=(H^{T}\otimes I_{n})R(t)\). According to (1), \(\Phi(y)=(\phi^{T}(y_{1}),\cdots,\phi^{T}(y_{N}))^{T}\) satisfies \([\Phi(y)-(I_{N}\otimes S_{1})y]^{T}[\Phi(y)-(I_{N}\otimes S_{2})y]\leq 0\). Then \(\Phi(\cdot)\) is called sector bounded within the sector \([I_{N}\otimes S_{1},I_{N}\otimes S_{2}]\). The following theorem presents some sufficient conditions for achieving synchronism tracking of system (2) with integral event-triggered information. **Theorem 1**. : _For a given positive real constant \(a\), \(\eta>\lambda_{m}(S_{1}S_{2}+S_{2}S_{1})\), \(\gamma=\frac{2k_{max}}{1-2k_{max}},k_{max}<\frac{1}{2}\), if there exists a symmetric positive definite \(X\in\mathbb{R}^{n\times n}\), a matrix \(Y\in\mathbb{R}^{p\times n}\) and positive real constants \(c\), such that the linear matrix inequality_  
\[\left[ \begin{array}{cccc}\bar{A}+\bar{B}&I_{N}\otimes X&H\otimes X&I_{N} \otimes B_{1}&H\otimes BY\\ I_{N}\otimes X&-\frac{1}{\bar{a}\|\bar{C}\|^{2}}I_{N\times n}&0&0&0\\ H^{T}\otimes X&0&-\frac{1}{\bar{a}T}I_{N\times n}&0&0\\ I_{N}\otimes B_{1}^{T}&0&0&-I_{N\times m}&0\\ H^{T}\otimes Y^{T}B^{T}&0&0&0&-\frac{a}{c^{2}}I_{N}\otimes X\end{array}  \right], \tag{8}\]  
_is negative definite matrix, where \(\bar{A}=I_{N}\otimes(XA^{T}+AX)\), \(\bar{B}=-c[HH^{T}\otimes(BY+Y^{T}B^{T})]\), \(\bar{X}=I_{n}-2X\), then the system (7) with \(K=YX^{-1}\) consensus tracking for sector bounded within the sector \([I_{N}\otimes S_{1},I_{N}\otimes S_{2}]\) under the integral-type edge-event-triggering condition (6). Furthermore, no Zeno behavior occurs for all \(t>t_{0}\)._  
Proof. : Choose the following Lyapunov function:  
\[V(t)=R^{T}(t)(I_{N}\otimes P)R(t), \tag{9}\]  
where \(P\) is a positive-definite matrix with \(P=X^{-1}\). Then, taking the time derivative of \(V(t)\) along the trajectory of (7) yields  
\[