# Quasi-Consensus of Disturbed Nonlinear Multiagent Systems with Event-Triggered Impulsive Control
## 3 Main Results  
In this section, we derive some sufficient conditions to ensure the quasi-consensus of the considered MASs using ETIC strategy. Additionally, we show that Zeno behavior is excluded. First, the ETM is designed as follows:  
\[ \begin{split}& t_{k}=\min\{t_{k}^{*},t_{k-1}+\tau_{\text{sup}}\}, \\ & t_{k}^{*}=\inf\Big{\{}t>t_{k-1}|E[V(e(t))]\geqslant e^{l_{k}- \lambda(t-t_{k-1})}E[V(e(t_{k-1}))]+vw^{2}\Big{\}},\end{split}  \tag{9}\]  
where \(\tau_{\text{sup}}>0\) is the maximum allowable event-trigger interval to be designed; \(l_{k}>0\), \(k\in\mathbb{N}^{+}\), \(v>1\) and \(\lambda>0\). \(V(e(t))\) is the Lyapunov function to be defined with respect to \(e(t)\). **Remark 1**. : _With the help of impulsive control, information transmission in the designed ETM can only activate at triggered instants. In other words, there is no need for any information transmission during the event-triggered interval, which can effectively reduce the communication cost._  
**Remark 2**. : _Compared with the anti-disturbance ETM in [31], where the effects of disturbances are hidden in some parameters, they are intuitive in our designed ETM. The sensitivity of the ETM (9) to disturbances can be adjusted by the parameter \(v\). Moreover, the measuring error, i.e., \(e(t)-e(t_{k-1})\) for any \(t\in[t_{k-1},t_{k})\), \(k\in\mathbb{N}^{+}\) used in [31] is no longer needed, which can also cut down the communication or computing resources in some sense._  
**Remark 3**. : _The forced impulse generated by the event \(t_{k-1}+\tau_{\rm sup}\) is to ensure the existence of the maximum allowable event-trigger interval. Note that it is important for achieving quasi-consensus of MASs, since one can always derive an upper bound of \(\tau_{\rm sup}\) by the following theorem._  
**Theorem 1**. : _Suppose that Assumptions 1-4 are satisfied. If there exist constants \(\eta_{1},\eta_{2}>0\) such that_  
\[0<\sigma<1,\quad\theta=\frac{\ln\sigma}{\tau_{\rm sup}}+\rho<0, \tag{10}\]  
_where_  
\[\rho =\lambda_{\rm max}(\Lambda)+2\beta\|\|+\eta_{2},\] \[\Lambda =I_{N}\otimes(A+A^{\rm T}+\bar{\lambda}\eta_{1}MM^{\rm T}+\bar{ \lambda}\eta_{1}{}^{-1}Q^{\rm T}Q),\] \[\sigma =\sup_{k\in\mathbb{N}^{+}}\lambda_{\rm max}\Big{(}(\mu_{k}H^{ \rm T}\otimes I_{n}+I_{Nn})(\mu_{k}H\otimes I_{n}+I_{Nn})\Big{)}.\]  
_Then, under the impulsive controller (3), the leader-following MASs consisting of (1) and (2) achieve quasi-consensus with the estimated error bound:_  
\[\chi=\frac{4w^{2}}{|\theta|\sigma\eta_{2}}.\]  
Proof. : Select the following Lyapunov function:  
\[V(e(t))=e^{\rm T}(t)e(t). \tag{11}\]  
The derivative of (11) along (6) can be obtained:  
\[ \begin{split} D^{+}V(e(t)&=e^{\rm T}(t)\dot{e}(t)+ \dot{e}^{\rm T}(t)e(t)\\ &=e^{\rm T}(t)\big{[}(I_{N}\otimes A(t))^{\rm T}+(I_{N}\otimes A (t))|e(t)\\ &\quad+\beta(t)G^{\rm T}(t,e(t))e(t)+\beta(t)e^{\rm T}(t)G(t,e(t)) +e^{\rm T}(t)\widehat{w}(t)+\widehat{w}^{\rm T}(t)e(t)\\ &=e^{\rm T}(t)[I_{N}\otimes(A^{\rm T}+A)]e(t)+\widehat{w}^{\rm T }(t)e(t)\\ &\quad+2e^{\rm T}(t)[I_{N}\otimes\alpha(t)MP(t)Q]e(t)+2\beta(t)G ^{\rm T}(t,e(t))e(t)+e^{\rm T}(t)\widehat{w}(t).\end{split}  \tag{12}\]  
According to Lemma 1 and Assumption 1, we obtain:  
\[ \begin{split}& 2e^{\rm T}(t)[I_{N}\otimes\alpha(t)MP(t)Q]\;e(t)\\ &=2e^{\rm T}(t)[I_{N}\otimes\delta MP(t)Q]\;e(t)+2e^{\rm T}(t)[I_ {N}\otimes(\alpha(t)-\bar{\alpha})MP(t)Q]e(t)\\ &\leqslant e^{\rm T}(t)[I_{N}\otimes\bar{\alpha}(\eta_{1}MM^{\rm T }+\eta_{1}^{-1}Q^{\rm T}Q)]e(t)\\ &\quad+2e^{\rm T}(t)[I_{N}\otimes(\alpha(t)-\bar{\alpha})MP(t)Q] e(t),\end{split}  \tag{13}\]  
and  
\[ \begin{split}& 2\beta(t)G^{\rm T}(t,e(t))e(t)+e^{\rm T}(t) \widehat{w}(t)+\widehat{w}^{\rm T}(t)e(t)\\ &=2\beta G^{\rm T}(t,e(t))e(t)+2(\beta(t)-\bar{\beta})G^{\rm T}(t,e(t))e(t)+e^{\rm T}(t)\widehat{w}(t)+\widehat{w}^{\rm T}(t)e(t)\\ &\leqslant(2\bar{\beta}\|J\|+\eta_{2})e^{\rm T}(t)e(t)+\eta_{2}^ {-1}\widehat{w}^{\rm T}(t)\widehat{w}(t)+2(\beta(t)-\bar{\beta})G^{\rm T}(t,e( t))e(t).\end{split}  \tag{14}\]Combining (12)-(14), we have  
\[