# Edge-Event-Triggered Synchronization for Multi-Agent Systems with Nonlinear Controller Outputs
## 3 Edge-Event-Triggered Policy  
We now introduce the following event-triggered condition. For \(i=1,2,\cdots,N\), denote \(r_{i}(t)=\sum_{p\in\mathcal{E}_{i}}h_{ip}w_{p}y_{p}(t)\) and \(\widehat{r}_{i}(i_{k}^{i})=\widehat{r}_{i}(t)=\sum_{p\in\mathcal{E}_{i}}h_{ip }w_{p}\widehat{y}_{p}\). Define the state measurement error by  
\[e_{i}(t)=\widehat{r}_{i}(t)-r_{i}(t),t\in[t_{k}^{i},t_{k}^{i+1}). \tag{4}\]  
Each triggering instant \(t_{k}^{i}\) is determined by  
\[t_{k+1}^{i}=\inf\left\{t>t_{k}^{i}:\bar{f}_{i}(t)\geq 0\right\},i\in\mathbb{N}, \tag{5}\]  
where  
\[\bar{f}_{i}(t)=\|e_{i}(t)\|^{2}-\sigma_{i}\|g(c\widehat{r_{i}}(t_{k}^{i}))\|^ {2}-\frac{\beta_{1}}{\beta_{2}+\beta_{3}t^{2}}. \tag{6}\]  
Here, \(\sigma_{i}\) and \(\beta_{i}\), \(i=1,2,\cdots,N\) are positive constants to be designed. Denote \(\widetilde{f}(y_{p})=f(x_{i})-f(x_{j})\), the edge state vector \(y(t)\) satisfies  
\[\dot{y}(t)= F(y)-c(H\otimes I_{n})G(c\widehat{r}) \tag{7}\] \[= F(y)-c(H\otimes I_{n})G(c(r+e)),\]  
where \(x(t)=[x_{1}^{T}(t)\ x_{2}^{T}(t)\cdots x_{N}^{T}(t)]^{T}\), \(y(t)=[y_{1}^{T}(t)\ y_{2}^{T}(t)\cdots y_{m}^{T}(t)]^{T}\), \(e(t)=[e_{1}^{T}(t)\ e_{2}^{T}(t)\cdots e_{N}^{T}(t)]^{T}\), \(F(y)=[\widetilde{f}(y_{1})^{T}\ \widetilde{f}(y_{2})^{T}\cdots\widetilde{f}(y_{m})^{T}]^{T}\), and \(G(\widehat{r})=[g(c\widehat{r}_{1})^{T}\ g(c\widehat{r}_{2})^{T}\cdots g(c \widehat{r}_{m})^{T}]^{T}\). Here, \(r(t)=(H^{T}W\otimes I_{n})y(t)\), where \(r(t)=[r_{1}(t)^{T}\ r_{2}(t)^{T}\cdots r_{N}(t)^{T}]^{T}\). The following theorem presents some sufficient conditions for achieving synchronism tracking of system (1) with event-triggered information. **Theorem 1**. : _Suppose that the underlying topology \(\mathcal{G}\) of the Lipschitz nonlinear multi-agent system (1) with control input (3) and triggering condition (6) is connected. If the nonlinear function \(g(\cdot)\) satisfies properties P1-P3 and there exist positive scalars \(a\), \(b_{1}\), \(b_{2}\), \(c\), \(\beta_{1}\), \(\beta_{2}\), \(\beta_{3}\), \(k_{1}\), \(k_{2}\), and \(\sigma_{i}\), \(i=1,2,\cdots,N\), such that the defined parameter \(S_{3}>0\) in (16). Then, the system (1) achieves state synchronization asymptotically for any initial condition. Moreover, all triggering time intervals are lower bounded by a positive scalar such that all agents achieve Zeno-free triggering._  
Choose the following Lyapunov function. \[V(t)=y(t)^{T}(W\otimes I_{n})y(t), \tag{8}\]  
Taking the time derivative of \(V(t)\) along the trajectory of (7), which yields  
\[ \begin{split}\frac{dV(t)}{dt}=& 2y(t)^{T}(W\otimes I_{n})F(y)- 2y(t)^{T}(WH\otimes I_{n})G(\widehat{cr})\\ \leq& aw_{\max}y(t)^{T}(W\otimes I_{n})y(t)+\frac{1 }{a}F(y)^{T}F(y)\\ &-2[(H^{T}W\otimes I_{n})y(t)]^{T}G(\widehat{cr})\\ \leq&\left(aw_{\max}+\frac{\mathcal{L}^{2}}{aw_{ \min}}\right)y(t)^{T}(W\otimes I_{n})y(t)\\ &-2\widehat{r}(t)^{T}G(\widehat{cr})+2e(t)^{T}G(\widehat{cr}), \end{split}  \tag{9}\]  
where we use the fact that \(2y(t)^{T}(W\otimes I_{n})F(y)\leq ay(t)^{T}(W^{2}\otimes I_{n})y(t)+\frac{1}{ a}F(y)^{T}F(y)\) for any positive real constant \(a\) to derive the first inequality, and the second one takes advantage of the Lipschitz condition of the function \(f(y)\). However, \(a\) has to be fixed due to the \(S_{1}\). Furthermore, the triangle inequality \(\xi^{T}\zeta\leq\frac{b}{2}\xi^{T}\xi+\frac{1}{2b}\xi^{T}\zeta\) holds for any real vectors \(\xi\), \(\zeta\) and positive scalar \(b\), which implies that  
\[ \begin{split} r(t)^{T}r(t)=&[\widehat{r}(t)-e(t)]^ {T}[\widehat{r}(t)-e(t)]\\ \leq&(1+b_{1})\widehat{r}(t)^{T}\widehat{r}(t)+(1+ \frac{1}{b_{1}})e(t)^{T}e(t),\end{split}  \tag{10}\]  
and  
\[