## 4 Integral-Type Edge-Self-Triggered Policy  
In the event-triggered formulation, it becomes apparent that continuous monitoring of the measurement error norm \(\|e(t)\|\) is required to check condition (6). In the following self-triggered control, this requirement is relaxed. Specifically, the next time \(t_{k+1}^{i}\) at which control law is updated is predetermined at the previous event time \(t_{k}^{i}\) and no state or error measurement is required in between the control updates. Define an increasing positive sequence \(k_{0}^{i}\), \(k_{1}^{i},\cdots,k_{q}^{i}\), taking into account the effects of neighboring agent updates, with \(t_{k}^{i}=t_{k_{0}^{i}}\) and \(t_{k_{q}^{i}}\leq t_{k+1}^{i}\). By the inequality (22), the measurement error \(e_{i}(t)\) satisfies  
\[\|e_{i}(t)\|\leq\mathcal{A}\|e_{i}(t)\|+\Xi_{k_{q}^{i}}^{i}, \tag{30}\]  
where  
\[\Xi_{k_{q}^{i}}^{i}=\Big{\|}-Az_{i}(t_{k}^{i})+cBK\sum_{j\in N_{i}}\left(z_{i }(t_{k}^{i})-z_{j}(t_{k_{q}^{i}})\right)\Big{\|}+\Delta, \tag{31}\]  
and \(\mathcal{A}=\|A\|\). Solving such inequality in \(t\in[t_{k_{q}^{i}},t_{k_{q+1}^{i}}]\), \(q=0,1,\cdots,q-1\), we have  
\[\|e_{i}^{\prime}(t)\|^{2}\leq e^{2\mathcal{A}t}\left(\mathcal{M}_{k_{q}^{i}}^{i}\right)^{2}+2 \mathcal{M}_{k_{q}^{i}}^{i}\Xi_{k_{q}^{i}}^{i}\bigg{(}e^{\mathcal{A}(2t-t_{k_ {q}^{i}})}-e^{\mathcal{A}t}\bigg{)}+\frac{(\Xi_{k_{q}^{i}}^{i})^{2}\bigg{(}e^ {\mathcal{A}(t-t_{k_{q}^{i}})}-1\bigg{)}^{2}}{\mathcal{A}^{2}}, \tag{32}\]  
where  
\[\mathcal{M}_{k_{q}^{i}}^{i}=\sum_{r=0}^{q-1}\frac{\Xi_{k_{r}^{i}}^{i}\Big{(}e^ {-\mathcal{A}_{r}^{i}}-e^{-\mathcal{A}_{r+1}^{i}}\Big{)}}{\mathcal{A}}. \tag{33}\]Here, \(\Xi_{k_{q}^{i}}^{i}\) and \(\mathcal{M}_{k_{q}^{i}}^{i}\) are fix up for any \(t\in[t_{k_{q}^{i}},t_{k_{q+1}^{i}}]\), \(q=0,1,\cdots,\bar{q}-1\). Defined a variable \(\tau_{k}^{i}=\tau_{k}^{i}(t)=t-t_{k}^{i}\), we can now compute  
\[\int_{t_{k}^{i}}^{t_{k}^{i}+\tau_{k}^{i}}\|e_{i}(s)\|^{2}\,ds\leq \frac{(\Xi_{k_{q}^{i}}^{i})^{2}}{\mathcal{A}^{2}}(\tau_{k}^{i})^{2 }+\left(e^{2\mathcal{A}(t_{k}^{i}+\tau_{k}^{i})}-e^{2\mathcal{A}t_{k}^{i}}\right) \tag{34}\] \[\left(\frac{(\mathcal{M}_{k_{q}^{i}}^{i})^{2}}{2\mathcal{A}}+ \frac{\mathcal{M}_{k_{q}^{i}}^{i}\Xi_{k_{q}^{i}}^{i}e^{-\mathcal{A}t_{k_{q}^{i }}}}{\mathcal{A}}+\frac{(\Xi_{k_{q}^{i}}^{i})^{2}e^{-2\mathcal{A}t_{k_{q}^{i }}}}{2\mathcal{A}^{3}}\right)\] \[-\left(e^{\mathcal{A}(t_{k}^{i}+\tau_{k}^{i})}-e^{\mathcal{A}t_{k }^{i}}\right)\left(\frac{2\mathcal{M}_{k_{q}^{i}}^{i}\Xi_{k_{q}^{i}}^{i}}{ \mathcal{A}}+\frac{2(\Xi_{k_{q}^{i}}^{i})^{2}e^{-\mathcal{A}t_{k_{q}^{i}}}}{ \mathcal{A}^{3}}\right),\]  
By (6) and (34), the self-triggered function is as follows:  
\[\tilde{f}(\tau_{k}^{i})= \frac{(\Xi_{k_{q}^{i}}^{i})^{2}}{\mathcal{A}^{2}}(\tau_{k}^{i})^ {2}+\left(e^{2\mathcal{A}(t_{k}^{i}+\tau_{k}^{i})}-e^{2\mathcal{A}t_{k}^{i}} \right)\left(\frac{(\mathcal{M}_{k_{q}^{i}}^{i})^{2}}{2\mathcal{A}}+\frac{ \mathcal{M}_{k_{q}^{i}}^{i}\[PERSON]_{k_{q}^{i}}^{i}e^{-\mathcal{A}t_{k_{q}^{i}}}}{ \mathcal{A}}+\frac{(\[PERSON]_{k_{q}^{i}}^{i})^{2}e^{-2\mathcal{A}t_{k_{q}^{i}}}}{2 \mathcal{A}^{3}}\right) \tag{35}\] \[-\left(e^{\mathcal{A}(t_{k}^{i}+\tau_{k}^{i})}-e^{\mathcal{A}t_{k }^{i}}\right)\left(\frac{2\mathcal{M}_{k_{q}^{i}}^{i}\Xi_{k_{q}^{i}}^{i}}{ \mathcal{A}}+\frac{2(\Xi_{k_{q}^{i}}^{i})^{2}e^{-\mathcal{A}t_{k}^{i}}}{ \mathcal{A}^{3}}\right)-k_{i}\|z_{i}(t_{k}^{i})\|^{2}\tau_{k}^{i}-\frac{\beta }{\alpha}e^{-\alpha t_{k}^{i}}\Big{(}1-e^{-\alpha\tau_{k}^{i}}\Big{)},\]  
For each \(i=1,2,\cdots,N\) the self-triggered ruling defines the next update time as following: if there is a \(\tau_{k}^{i}>0\) such that \(\tilde{f}(\tau_{k}^{i})=0\), the next update time \(t_{k+1}^{i}\) take place at most \(\tau_{k}^{i}\) time units after \(t_{k}^{i}\). Agent \(v_{i}\) also checks the condition (35) when its neighbors updated. Otherwise, if the inequality \(\tilde{f}(\tau_{k}^{i})<0\) holds for all \(\tau_{k}^{i}>0\), then agent \(v_{i}\) waits until the next update of the control law of one of its neighbors to recheck its condition.