# Edge-Event-Triggered Synchronization for Multi-Agent Systems with Nonlinear Controller Outputs
## 3 Edge-Event-Triggered Policy
\begin{split} 2e(t)^{T}G(\widehat{cr})\leq& b_{2}e(t)^{T}e(t)+ \frac{1}{b_{2}}G(c\widehat{r})^{T}G(c\widehat{r})\\ \leq&\left(b_{2}\sigma_{\max}+\frac{1}{b_{2}}\right) G(c\widehat{r})^{T}G(c\widehat{r})\\ &+\frac{b_{2}\beta_{1}N}{\beta_{2}+\beta_{3}t^{2}}\\ \leq&\left(b_{2}\sigma_{\max}k_{2}^{2}c^{2}+\frac{k _{2}^{2}c^{2}}{b_{2}}\right)\widehat{r}(t)^{T}\widehat{r}(t)\\ &+\frac{b_{2}\beta_{1}N}{\beta_{2}+\beta_{3}t^{2}},\end{split}  \tag{11}\]  
where the final inequality uses that the nonlinear vector function \(G(c\widehat{r})\) satisfies \(G(c\widehat{r})^{2}\leq k_{2}^{2}(c\widehat{r}(t))^{2}\). Noting that \(\sigma_{\max}=max_{i\in\{1,2,\ldots,N\}}\sigma_{i}\). Obtained from the inequality (10), one gets\[-\widehat{r}(t)^{T}\widehat{r}(t)\leq -\frac{1}{1+b_{1}}r(t)^{T}r(t)+\frac{1}{b_{1}}e(t)^{T}e(t)\] \[\leq -\frac{1}{1+b_{1}}r(t)^{T}r(t)+\frac{\sigma_{\max}}{b_{1}}G(c \widehat{r})^{T}G(c\widehat{r})\] \[+\frac{\beta_{1}N}{b_{1}(\beta_{2}+\beta_{3}t^{2})} \tag{12}\] \[\leq -\frac{1}{1+b_{1}}r(t)^{T}r(t)+\frac{\sigma_{\max}k_{2}^{2}c^{2}} {b_{1}}\widehat{r}(t)^{T}\widehat{r}(t)\] \[+\frac{\beta_{1}N}{b_{1}(\beta_{2}+\beta_{3}t^{2})},\]  
Then, we have  
\[-\widehat{r}(t)^{T}\widehat{r}(t)\leq -\frac{b_{1}}{(1+b_{1})(b_{1}+\sigma_{\max}k_{2}^{2}c^{2})}r(t)^{ T}r(t) \tag{13}\] \[+\frac{\beta_{1}N}{(b_{1}+\sigma_{\max}k_{2}^{2}c^{2})(\beta_{2} +\beta_{3}t^{2})}.\]  
The inequality (9) yields  
\[V^{\prime}(t)\leq (aw_{\max}+\frac{\mathcal{L}^{2}}{aw_{\min}})y(t)^{T}(W\otimes I_{ n})y(t) \tag{14}\] \[+(b_{2}\sigma_{\max}k_{2}^{2}c^{2}+\frac{k_{2}^{2}c^{2}}{b_{2}} )\widehat{r}(t)^{T}\widehat{r}(t)\] \[-2\widehat{r}(t)^{T}G(c\widehat{r})+\frac{b_{2}\beta_{1}N}{ \beta_{2}+\beta_{3}t^{2}}\] \[\leq -\frac{1}{1+b_{1}}r(t)^{T}r(t)+\frac{\sigma_{\max}}{b_{1}}G(c \widehat{r})^{T}G(c\widehat{r})\]0 \[\leq -\frac{1}{1+b_{1}}r(t)^{T}r(t)+\frac{\sigma_{\max}}{b_{1}}G(c \widehat{r})^{T}G(c\widehat{r})\]1 \[\leq -\frac{1}{1+b_{1}}r(t)^{T}r(t)+\frac{\sigma_{\max}}{b_{1}}G(c \widehat{r})^{T}G(c\widehat{r})\]2 \[\leq -\frac{1}{1+b_{1}}r(t)^{T}r(t)+\frac{\sigma_{\max}}{b_{1}}G(c \widehat{r})^{T}G(c\widehat{r})\]3 \[\leq -\frac{1}{1+b_{1}}r(t)^{T}r(t)+\frac{\sigma_{\max}}{b_{1}}G(c \widehat{r})^{T}G(c\widehat{r})\]4  
where  
\[ \begin{cases}\mathcal{K}=aw_{\max}+\frac{\mathcal{L}^{2}}{aw_{\min}},\\ S_{1}=\frac{2b_{1}b_{2}k_{1}c-b_{1}b_{2}^{2}\sigma_{\max}k_{2}^{2}c^{2}-b_{1}k _{2}^{2}c^{2}}{b_{2}(1+b_{1})(b_{1}+\sigma_{\max}k_{2}^{2}c^{2})},\\ S_{2}=\frac{1+b_{1}}{b_{1}}S_{1}\end{cases}  \tag{15}\]  
with the nonlinear vector function \(\widehat{r}(t)^{T}G(c\widehat{r})\geq k_{1}c\widehat{r}(t)^{T}\widehat{r}(t)\). Let  
\[S_{3}=\bar{\lambda}_{2}w_{\min}S_{1}-\mathcal{K}. \tag{16}\]The inequality (14) yields  
\[ \begin{split} V^{\prime}(t)\leq&(\mathcal{K}-\bar{ \lambda}_{2}w_{\min}S_{1})y(t)^{T}(W\otimes I_{n})y(t)\\ &+\frac{\beta_{1}N}{\beta_{2}+\beta_{3}t^{2}}S_{2}\\ \leq&-S_{3}V(t)+\frac{\beta_{1}N}{\beta_{2}+\beta_{ 3}t^{2}}S_{2},\end{split}  \tag{17}\]  
Therefore, according to the comparison principle in [56], it follows that \(V(t)\leq\bar{V}(t)\), \(\bar{V}^{\prime}(t)=-S_{3}\bar{V}(t)+\frac{\beta_{1}N}{\beta_{2}+\beta_{3}t^{ 2}}S_{2}\). Here, \(\frac{\beta_{1}N}{\beta_{2}+\beta_{3}t^{2}}S_{2}\in\mathcal{L}^{\infty}\) and \(\lim_{t\to\infty}\frac{\beta_{1}N}{\beta_{2}+\beta_{3}t^{2}}S_{2}\to 0\), then \(\lim_{t\to\infty}\bar{V}^{\prime}(t)=0\)[47]. Therefore, the synchronization is achieved. Next, we show that the considered system does not exhibit Zeno behavior. If there is a lower bound on the inter-event intervals of any agent under the event-triggered condition (6), Zeno-free triggering can be achieved. For each pair of adjacent agents \(v_{i}\) and \(v_{j}\), define an auxiliary vector function \(\mathcal{U}_{p}(t)=\sum_{p\in\mathcal{E}_{i}}h_{ip}w_{p}[g(c\widehat{r_{i}})- g(c\widehat{r_{j}})]\) and \(\Pi_{p}(t)=\mathcal{L}\|\widehat{r}_{i}(t)\|+\|\mathcal{U}_{p}(t)\|\). Here, \(\mathcal{U}_{p}(t)\) and \(\Pi_{p}(t)\) lie between any two consecutive event instants of agent \(v_{i}\) or \(v_{j}\).