# Quasi-Consensus of Disturbed Nonlinear Multiagent Systems with Event-Triggered Impulsive Control
## 3 Main Results
\begin{split} E[V(e(t))]\leqslant&\sigma^{N(t_{0},t )}e^{\rho(t-t_{0})}E[V(e(t_{0}))]+\eta_{2}^{-1}\int_{t_{0}}^{t}\sigma^{N(s,t)} \bar{w}^{\mathrm{T}}(s)\bar{w}(s)e^{\rho(t-s)}ds\\ \leqslant&\sigma^{\frac{t-t_{0}}{\mathrm{supp}}-1}e^{ \rho(t-t_{0})}E[V(e(t_{0}))]+\eta_{2}^{-1}\int_{t_{0}}^{t}\sigma^{\frac{t-s}{ \mathrm{supp}}-1}\bar{w}^{\mathrm{T}}(s)\bar{w}(s)e^{\rho(t-s)}ds\\ \leqslant&\sigma^{-1}e^{\beta(t-t_{0})}E[V(e(t_{0}) )]+\sigma^{-1}\eta_{2}^{-1}\int_{t_{0}}^{t}\bar{w}^{\mathrm{T}}(s)\bar{w}(s)e^ {\rho(t-s)}ds,\end{split}  \tag{24}\]  
where \(\frac{t-t_{0}}{\tau_{\mathrm{supp}}}-1\leq N(t,t_{0})\) is used and \(N(t,t_{0})\) denotes the number of events in \((t_{0},t]\). Since \(\theta<0\) (see (10)), then  
\[ \begin{split}\limsup_{t\to+\infty}E[V(e(t))]& \leqslant\sigma^{-1}\eta_{2}^{-1}\lim_{t\to+\infty}\int_{t_{0}}^{t}\bar{w}^{ \mathrm{T}}(s)\bar{w}(s)e^{\beta(t-s)}ds\\ &\leqslant\frac{4}{|\theta|}\sigma^{-1}\eta_{2}^{-1}w^{2}:=\chi.\end{split}  \tag{25}\]  
According to Definition 1 and (25), the MASs (1) and (2) realize the quasi-consensus under the ETIC protocol (3), and \(\chi=\frac{4w^{2}}{|\bar{w}|\bar{w}\rho_{2}}\). The proof is completed. Note that the RONs and ROUs are assumed to exist synchronously in the above systems, but this assumption is further relaxed. Suppose the randomly occurring terms of each follower and the leader are asynchronous. In that case, instead of \(\alpha(t)\) and \(\beta(t)\) modeled in the above systems, there exist some \(\alpha_{j}(t)\) and \(\beta_{j}(t)\) for any \(j=0,1,\dots,N\), such that they can be different and independent of each other but satisfy the Bernoulli distribution with \(E[\alpha_{j}(t)]=\bar{\alpha}_{j}\) and \(E[\beta_{j}(t)]=\bar{\beta}_{j}\). Then, we can always set \(\bar{\alpha}=\text{diag}\{\alpha_{1}-\alpha_{0},\alpha_{2}-\alpha_{0},\dots, \alpha_{N}-\alpha_{0}\}\) and \(\bar{\beta}=\text{diag}\{\beta_{1}-\beta_{0},\beta_{2}-\beta_{0},\dots,\beta_ {N}-\beta_{0}\}\). We have slightly abused the notation by using \(\bar{\alpha}\) and \(\bar{\beta}\) to denote both scale-valued and matrix-valued. Clearly, Theorem 1 can still handle this asynchronous case when slightly modified. Next, we exclude Zeno behavior. Theorem 2: _For MASs (1) under control protocol (3), the parameters of ETM (9) satisfy:_  
\[l_{k}-\ln v>0,\quad k\in\mathbb{N}^{+}, \tag{26}\]  
\[\lim_{k\to\infty}\sum_{i=1}^{k}(l_{i}-\ln v)=+\infty, \tag{27}\]then, the interval of event-trigger impulsive control exists clearly, and \(t_{k}\rightarrow+\infty\) as \(k\rightarrow+\infty\). In other words, there is no Zeno behavior for MASs_ (1) _under the event-trigger mechanism_ (9). Proof. : The following proof is divided into three scenarios according to the characteristics of the designed ETM (9). Assume that \(\mathcal{N}:=\{t_{k},k\in\mathbb{N}^{+}\}\) is the generated impulsive sequence by ETM (9). _Scenario 1: \(\mathcal{N}\) completely consists of forced impulses_  
In this scenario, [PERSON] behavior is excluded naturally since \(t_{k}-t_{k-1}\equiv\tau_{\text{sup}},k\in\mathbb{N}^{+}\). _Scenario 2: \(\mathcal{N}\) is completely independent of forced impulses_  
Once any event is triggered, it follows from (9) and (17) that  
\[ \begin{split} E[V(e(t_{k}))]=& e^{l_{k}-\lambda(t_{ k}-t_{k-1})}E[V(e(t_{k-1}))]+vw^{2}\\ \leqslant& e^{\rho(t_{k}-t_{k-1})}E[V(e(t_{k-1}))]+ \eta_{2}^{-1}\int_{t_{k-1}}^{t_{k}}\bar{w}^{\text{T}}(s)\bar{w}(s)e^{\rho(t_{k }-s)}ds.\end{split}  \tag{28}\]  
In this scenario, we further discuss the effects caused by \(\rho\), as shown in the following three cases: \(\rho<0\), \(\rho=0\) and \(\rho>0\). **Case 1: \(\rho<0\). ** It follows from (28) that  
\[e^{l_{k}-\lambda(t_{k}-t_{k-1})}E[V(e(t_{k-1}))]+vw^{2}\leqslant e^{\rho(t_{k}-t_{k-1})}E[V(e(t_{k-1}))]+ \frac{4(t_{k}-t_{k-1})}{\eta_{2}}w^{2}, \tag{29}\]  
which implies that \(\min\{e^{l_{k}-\lambda(t_{k}-t_{k-1})},v\}\leqslant\max\{e^{\rho(t_{k}-t_{k-1 })},\frac{4(t_{k}-t_{k-1})}{\eta_{2}}\}\). **(i) \(e^{\rho(t_{k}-t_{k-1})}\geqslant\frac{4(t_{k}-t_{k-1})}{\eta_{2}}\).