# Quasi-Consensus of Disturbed Nonlinear Multiagent Systems with Event-Triggered Impulsive Control
## 4 Numerical Examples  
This section verifies the main results through a simulation example. Consider the leader-following MASs, which consist of one leader and four followers; their topology is shown as Figure 1. According to Figure 1, we have:  
\[L= \begin{bmatrix}2&-1&-1&0\\ -1&2&0&-1\\ -1&0&1&0\\ 0&-1&0&1\end{bmatrix} ,\quad B= \begin{bmatrix}1&&&\\ &0&&&\\ &&1&&\\ &&&0\end{bmatrix} ,\quad H= \begin{bmatrix}3&-1&-1&0\\ -1&2&0&-1\\ -1&0&2&0\\ 0&-1&0&1\end{bmatrix} .\]It is assumed that the MASs model is described as (1) and (2). Moreover, let \(f(t,x_{j}(t))=0.25\tanh(x_{j}(t))\), and thus \(\|f\|=0.25\). The controller \(u_{i}(t)\) is designed as (3). Choosing \(E[\alpha(t)]=E[\beta(t)]=0.5\). \(A(t)=A+\alpha(t)MP(t)Q\), where \(M=Q=I_{3}\), and  
\[A= \begin{bmatrix}-1.87&0&0\\ 1&-1&1\\ 0&-0.82&0\end{bmatrix} ,\quad P(t)= \begin{bmatrix}0.2\cos(t)&0&0\\ 0&-0.5\sin(t)&0\\ 0&0&0.4\sin(t)\end{bmatrix} .\]  
In addition, setting  
\[ \begin{bmatrix}w_{0}(t),w_{1}(t),&w_{2}(t),w_{3}(t),w_{4}(t)\end{bmatrix} \\ = \begin{bmatrix}0.5&0.5&0.1&0.3&0.5\\ -0.1&0.1&-0.2&0.5&-0.2\\ 0.3&0.4&0.3&0.4&0.3\end{bmatrix} \times\sin(t),\]  
and we have \(w=0.8\). Let the initial value of the leader and four followers be as follows:  
\[ \begin{bmatrix}\varphi_{0},\varphi_{1,}\varphi_{2},\varphi_{3}, \varphi_{4}\end{bmatrix} \\ = \begin{bmatrix}1.5&0.5&-0.5&-1&1\\ -1&2&-1&-2&-1.5\\ 2&-2.5&2.5&2&1\end{bmatrix} ,\]  
Selecting \(\eta_{1}=\eta_{2}=1\) and \(\mu_{k}=0.1\), one can obtain that \(\rho=3.8177,\sigma=0.1658\) and \(\tau_{\rm sup}<0.0881\). Then, we choose \(\tau_{\rm sup}=0.05\), and \(\theta=-14.1537<0\). Hence, the conditions in Theorem 1 are satisfied, and the quasi-consensus can be achieved with the error bound \(\chi~{}=~{}1.0446\). Moreover, setting \(l_{k}\equiv 1.2\) and \(v=1.2\), one can check that conditions in Theorem 2 are also met, and Zeno behavior can be excluded. Let \(\lambda=1\), the trajectories of error states and event-triggered instants under ETM (9) are shown as Figure 1(a),1(b), respectively. Figure 1: Communication topology of the system. To highlight the superiority of the obtained results in this paper, the results (Theorem 2) in [31] are considered, where their ETM is designed as follows (the fixed maximum allowable event-trigger interval \(\rho_{k}=\rho_{\text{sup}}\) case is considered):  
\[ \begin{split} t_{k}&=\min\{t_{k}^{*},t_{k-1}+\rho_{ \text{sup}}\},\\ t_{k}^{*}&=\inf\Big{\{}t>t_{k-1}|\|\eta(t)\|\geqslant \alpha_{1}\|e(t_{k-1})\|+\bar{\lambda}e^{-\alpha_{2}(t-t_{k-1})}\Big{\}},\end{split}  \tag{33}\]  
where \(\eta(t)=e(t)-e(t_{k-1}),k\in\mathbb{N}^{+}\) and \(\alpha_{1},\alpha_{2},\bar{\lambda}>0\). Neglecting the delay considered in [31], a set of feasible solutions for LMIs in (Theorem 2 in [31]) are obtained by setting \(\varepsilon_{1}=1,\beta_{1}^{\prime}=2\) and \(\bar{\mu}=0.8\). Then, one has \(\beta_{2}\geq 1.3934\) and \(\rho_{\text{sup}}<0.0595\) with \(\beta_{2}=1.4\). Choose \(\rho_{\text{sup}}=0.05,\alpha_{1}=0.5,\alpha_{2}=1,\bar{\lambda}=e^{1.5}\) and \(K=0.1t_{4}\); it is calculated that the error bound under (33) is \(\chi=1.6378\) with \(\bar{w}=1.2\) and \(\eta=1.0623\). With the parameters above, the trajectories of error states and event-triggered instants under ETM (9) are shown as Figure 2(a),b, respectively. Compared with the performance of both ETM (9) and (33), one can find that the error bound \(\chi\) is smaller, and the triggered instants are fewer under ETM (9) than those under ETM (33). Hence, less conservative results can be obtained with the ETM designed in this paper. Figure 3: Trajectories of error states and event-triggered instants under ETM (33). (**a**) Trajectories of error states; (**b**) event-triggered instants. Figure 2: Trajectories of error states and event-triggered instants under ETM (9). (**a**) Trajectories of error states; (**b**) event-triggered instants. Setting \(\lambda=2\) and \(\lambda=0.1\), the event-triggered instants of the system are shown as Figure 4a,b, respectively.