# Quasi-Consensus of Disturbed Nonlinear Multiagent Systems with Event-Triggered Impulsive Control
## 3 Main Results
\begin{split} D^{+}V(e(t))\leqslant& e^{\mathsf{T}}(t)[I_{N} \otimes(A+A^{\mathsf{T}})]e(t)+\bar{\alpha}e^{\mathsf{T}}(t)[I_{N}\otimes(\eta_ {1}MM^{\mathsf{T}}+\eta_{1}^{-1}Q^{\mathsf{T}}Q)]e(t)\\ &+2e^{\mathsf{T}}(t)[I_{N}\otimes(\alpha(t)-\bar{\alpha})MP(t)Q]e (t)+2(\beta(t)-\bar{\beta})G^{\mathsf{T}}(t,e(t))e(t)\\ &+(2\bar{\beta}\|I\|+\eta_{2})e^{\mathsf{T}}(t)e(t)+\eta_{2}^{-1 }\bar{w}^{\mathsf{T}}(t)\bar{w}(t)\\ \leqslant&\rho V(e(t))+\eta_{2}^{-1}\bar{w}^{ \mathsf{T}}(t)\bar{w}(t)+2e^{\mathsf{T}}(t)[I_{N}\otimes(\alpha(t)-\bar{\alpha })MP(t)Q]e(t)\\ &+2(\beta(t)-\bar{\beta})G^{\mathsf{T}}(t,e(t))e(t).\end{split}  \tag{15}\]  
When \(t\in[t_{k-1},t_{k}),k\in\mathbb{N}^{+}\), \(V(e(t))\) is continuous, the following equation holds:  
\[E[\dot{V}(e(t))]=D^{+}E[V(e(t))],\]  
which indicates that:  
\[E[\dot{V}(e(t))]\leqslant\rho E[V(e(t))]+\eta_{2}^{-1}\bar{w}^{\mathsf{T}}(t) \bar{w}(t), \tag{16}\]  
where \(\rho=\lambda_{\max}(\Lambda)+2\bar{\beta}\|I\|+\eta_{2}\), and \(\Lambda=I_{N}\otimes(A+A^{\mathsf{T}}+\bar{\alpha}\eta_{1}MM^{\mathsf{T}}+ \bar{\alpha}\eta_{1}^{-1}Q^{\mathsf{T}}Q)\). Then, integrating both sides of (16), for \(t\in[t_{k-1},t_{k})\), one has  
\[E[V(e(t))]\leqslant e^{\rho(t-t_{k-1})}E(V(e(t_{k-1})))+\eta_{2}^{-1}\int_{t_{k-1}}^{t} \bar{w}^{\mathsf{T}}(s)\bar{w}(s)e^{\rho(t-s)}ds. \tag{17}\]  
When \(t=t_{k},k\in\mathbb{N}^{+}\), we have  
\[ \begin{split} E[V(e(t_{k}))]&=E[e^{\mathsf{T}}(t_{k })e(t_{k})]\\ &=E[(\mu_{k}(H\otimes I_{n})+I_{Nn})e(t_{k}^{-})^{\mathsf{T}} \times(\mu_{k}(H\otimes I_{n})+I_{Nn})e(t_{k}^{-})]\\ &=E[e^{\mathsf{T}}(t_{k}^{-})(\mu_{k}(H\otimes I_{n})+I_{Nn})^{ \mathsf{T}}\times(\mu_{k}(H\otimes I_{n})+I_{Nn})e(t_{k}^{-})]\\ &\leqslant\sigma E[V(e(t_{k}^{-}))],\end{split}  \tag{18}\]  
where \(\sigma=\sup_{k\in\mathbb{N}^{+}}\lambda_{\max}\big{(}(\mu_{k}H^{\mathsf{T}} \otimes I_{n}+I_{Nn})(\mu_{k}H\otimes I_{n}+I_{Nn})\big{)}\). In the following, we show the boundedness of the selected Lyapunov function by mathematical induction. For \(t\in[t_{0},t_{1})\), it follows from (18) that:  
\[ \begin{split} E[V(e(t))]\leqslant& e^{\rho(t-t_{0})}E[V(e(t_{0}))]+\eta_{2}^{-1}\int_{t_{0}}^{t} \bar{w}^{\mathsf{T}}(s)\bar{w}(s)e^{\rho(t-s)}ds.\end{split}  \tag{19}\]  
According to (18), for \(t=t_{1}\), we can further obtain the following inequality:  
\[ \begin{split} E[V(e(t_{1}))]\leqslant&\sigma E[V(e(t_ {1}^{-}))]\\ \leqslant&\sigma e^{\rho(t_{1}-t_{0})}E[V(e(t_{0}))]+ \sigma\eta_{2}^{-1}\int_{t_{0}}^{t_{1}}\bar{w}^{\mathsf{T}}(s)\bar{w}(s)e^{ \rho(t_{1}-s)}ds.\end{split}  \tag{20}\]  
Similarly, for \(t\in[t_{1},t_{2})\), we have:  
\[ \begin{split} E[V(e(t))]\leqslant& e^{\rho(t-t_{1})}E[V(e(t_{1}))]+\eta_{2}^{-1}\int_{t_{1}}^{t} \bar{w}^{\mathsf{T}}(s)\bar{w}(s)e^{\rho(t-s)}ds\\ \leqslant&\sigma e^{\rho(t-t_{0})}E[V(e(t_{0}))]+ \sigma\eta_{2}^{-1}\int_{t_{0}}^{t_{1}}\bar{w}^{\mathsf{T}}(s)\bar{w}(s)e^{ \rho(t-s)}ds\\ &+\eta_{2}^{-1}\int_{t_{1}}^{t}\bar{w}^{\mathsf{T}}(s)\bar{w}(s)e ^{\rho(t-s)}ds,\end{split}  \tag{21}\]and for \(t=t_{2}\), we have  
\[ \begin{split} E[V(e(t_{2}))]\leqslant&\sigma E[V(e(t_{2}^ {-}))]\\ \leqslant&\sigma^{2}e^{\rho(t_{2}-t_{0})}E[V(e(t_{0}) )]+\sigma^{2}\eta_{2}^{-1}\int_{t_{0}}^{t_{1}}\bar{w}^{\mathrm{T}}(s)\bar{w}(s )e^{\rho(t_{2}-s)}ds\\ &+\sigma\eta_{2}^{-1}\int_{t_{1}}^{t_{2}}\bar{w}^{\mathrm{T}}(s) \bar{w}(s)e^{\rho(t_{2}-s)}ds.\end{split}  \tag{22}\]  
By iteration calculation, for \(t\in[t_{k-1},t_{k})\), we can finally obtain that  
\[ \begin{split} E[V(e(t))]\leqslant&\sigma^{k-1}e^{ \rho(t-t_{0})}E[V(e(t_{0}))]+\eta_{2}^{-1}\int_{t_{k-1}}^{t}\bar{w}^{\mathrm{T}} (s)\bar{w}(s)e^{\rho(t-s)}ds\\ &+\sum_{i=1}^{k-1}\sigma^{k-i}\eta_{2}^{-1}\int_{t_{i-1}}^{t_{i}} \bar{w}^{\mathrm{T}}(s)\bar{w}(s)e^{\rho(t-s)}ds.\end{split}  \tag{23}\]  
Note that \(\sigma\in(0,1)\), according to Definition 2 and (23), for any \(t\geqslant t_{0}\), the following inequality holds:  
\[