# Edge-Event-Triggered Synchronization for Multi-Agent Systems with Nonlinear Controller Outputs
## 4 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 algorithm, 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 between the control update instants. 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 (20), the measurement error \(e_{i}(t)\) satisfies  
\[\|e_{i}(t)\|^{2}\leq\frac{\Pi_{k}^{p}\left(e^{(2\mathcal{L}+\beta)(t-t_{k}^{i} )}-1\right)}{\beta(2\mathcal{L}+\beta)}, \tag{23}\]  
where  
\[\begin{split}\mathcal{U}_{p}(t)&=\sum_{p\in\hat{ \mathcal{E}}_{i}}h_{ip}w_{p}\left(g(\hat{c}\widehat{r}_{i})-g(\hat{c}\widehat{ r}_{j})\right),\\ \mathcal{U}_{k}^{p}&=\max_{t\in[t_{k}^{i},t_{k+1}^{ i})}\|\mathcal{U}_{p}(t)\|,\end{split} \tag{24}\]  
and  
\[\begin{split}\Pi_{p}(t)=&\mathcal{L}\|\widehat{r}_{ i}(t)\|+\|\mathcal{U}_{p}(t)\|,\\ \Pi_{k}^{p}&=\max_{t\in[t_{k}^{i},t_{k+1}^{i})}\| \Pi_{p}(t)\|.\end{split} \tag{25}\]  
By (6) and (23), the self-triggered function is as follows,  
\[\begin{split}\tilde{f}(\tau_{k}^{i})=&\frac{\Pi_{k} ^{p}\left(e^{(2\mathcal{L}+\beta)\tau_{k}^{i}}-1\right)}{\beta(2\mathcal{L}+ \beta)}-\sigma_{i}\|g(\hat{c}\widehat{r}_{i})\|^{2}\\ &-\frac{\beta_{1}}{\beta_{2}+\beta_{3}(t_{k}^{i}+\tau_{k}^{i})^{ 2}},\end{split} \tag{26}\]  
For each \(i=1,2,\cdots,N\), the self-triggered rule defines the next update time as follows. 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 \(t_{k}^{i}\). Moreover, agent \(v_{i}\) also checks the condition (26) 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.  
According to the above analysis, we give Algorithm 1 and Theorem 2 as follows.  
**Theorem 2**.: _Under the assumptions and conditions in Theorem 1, the system (1) with the control law (3) and Algorithm 1 can achieve synchronization for any initial condition. Furthermore, Zeno-free triggering can be achieved for all agents._  
**Remark 2**.: _The proposed edge-event- and edge-self-triggered policies are suitable for different scenarios with different system resources. The proposed event-triggered policy employs sensors to continuously measure edge state information to generate measurement errors and does not require communication resources, for instance, each individual employs LiDAR, ultrasonic rangefinder, and sonar to measure the relative position and relative speed of all neighbors. However, the proposed self-triggered policy can avoid continuous monitoring of these sensors, but it requires communication networks for information exchange. Therefore, these two policies include a trade-off between the scheduling of sensor resources and communication resources, which can be suitable for the scenarios with different levels of resource utilization._  
**Remark 3**.: _Compared with the previous results on self-triggered control [34; 39; 41; 42; 46; 47; 52], the interactive control input information of our algorithm contains the considered nonlinear factors, which include the influence of the logarithmic quantizer on the interactive information. This setting can save the communication bandwidth. This is also the theoretical expansion of the distributed self-triggered mechanism in these scenarios._