## 5 Simulation  
### Design of Event Conditions and Controllers  
In this subsection, we provide numerical simulations to illustrate the feasibility of Theorems 1 and 3. Consider the following plant borrowed from [25] with  
\[A=\left( \begin{array}{cccc}0&1&0&0\\ -48.6&-1.25&48.6&0\\ 0&0&0&1\\ 19.5&0&-19.5&0\end{array} \right),B=\left( \begin{array}{c}0\\ 21.6\\ 0\\ 0\end{array} \right),C=\left( \begin{array}{cccc}1&0&0&0\\ 0&1&0&0\end{array} \right). \tag{26}\]  
Hence, we suppose that there are two sensor nodes, \(y_{1}\) and \(y_{2}\), and one actuator node, \(u_{1}\), in the plant. The initial states of the plant and the observer are, respectively, selected as  
\[x^{\rm T}(0)=\left( \begin{array}{cccc}0&-4.5&-0.05&1.5\end{array} \right); \tilde{x}^{\rm T}(0)=\left( \begin{array}{cccc}-0.05&-5.4&-0.25&1.8\end{array}  \right).\]  
Then, by solving the LMIs in Theorem 3 with \(\varepsilon_{1}=8\) and \(\varepsilon_{2}=17\), a group of feasible solutions are \(\rho_{0}=0.08\), the gain matrices  
\[K=\left( \begin{array}{cccc}-0.0962&-0.3757&-0.3768&-0.2257\end{array} \right), L^{\rm T}=\left( \begin{array}{cccc}8.5835&3.9595&9.1535&-1.1554\\ 1.9763&45.9760&8.6555&-14.5631\end{array} \right),\]  
and the ellipsoid \(\Gamma(P_{1},P_{2})\) with  
\[P_{1}=\left( \begin{array}{cccc}1.1606&0.0813&-0.8561&0.1488\\ 0.0813&0.0406&-0.0401&0.0325\\ -0.8561&-0.0401&0.9802&-0.0607\\ 0.1488&0.0325&-0.0607&0.1063\end{array} \right),\]\[P_{2}=\left( \begin{array}{cccc}9.4717&3.1637&-8.4906&6.2623\\ 3.1637&2.2400&-3.3975&4.6866\\ -8.4906&-3.3975&9.4641&-6.3464\\ 6.2623&4.6866&-6.3464&11.9925\end{array} \right).\]  
By simple calculation, the initial states belong to the above ellipsoid. According to Theorem 1 with \(\theta=0.9375\), the decentralized integral-based event conditions of the sensor and actuator nodes are designed as  
\[t_{j_{1}+1}^{1}=\inf_{t}\{t\geq t_{j_{1}}^{1}+3.5000\times 10^{-4}|\int_{t_{j_{1} }^{1}}^{t}\|e_{u_{1}}\|^{2}\mathrm{d}\omega>0.0025\int_{t_{j_{1}}^{1}}^{t}\|u_ {1}\|^{2}\mathrm{d}\omega\},\]  
\[t_{k_{1}+1}^{1}=\inf_{t}\{t\geq t_{k_{1}}^{1}+3.0000\times 10^{-4}|\int_{t_{k_{1} }^{1}}^{t}\|e_{y_{1}}\|^{2}\mathrm{d}\omega>0.0020\int_{t_{k_{1}}^{1}}^{t}\|y_ {1}\|^{2}\mathrm{d}\omega\},\]  
\[t_{k_{2}+1}^{2}=\inf_{t}\{t\geq t_{k_{2}}^{2}+3.0000\times 10^{-4}|\int_{t_{k_{2} }^{2}}^{t}\|e_{y_{2}}\|^{2}\mathrm{d}\omega>0.0020\int_{t_{k_{2}}^{2}}^{t}\|y_ {2}\|^{2}\mathrm{d}\omega\}.\]  
Figure 2 shows the convergence of the plant state and the observer error at the origin. The top row of Figure 3 provides the input signal trajectory, which is saturated at the initial stage of the control process. The bottom row of Figures 3 and 4 show the evolutions of the inter-event times of the actuator and sensor nodes. Table 1 provides the sampling numbers of the nodes \(u_{1},y_{1}\) and \(y_{2}\) of the proposed scheme. By simple calculations, the average inter-event times are 0.0044 s, 0.0119 s and 0.0048 s for, respectively, \(u_{1}\), \(y_{1}\) and \(y_{2}\). All of them are quite larger than \(\tau_{u}^{1}\), \(\tau_{y}^{1}\) and \(\tau_{y}^{2}\). Hence, compared to the time-triggered manner, event-triggered control can save more communication resources without loss of stability. Meanwhile, we also make a comparison among the proposed integral-based event-triggered control scheme and some existing ones, i.e., the centralized relative event-triggered control in [21] and the decentralized one in [25]. The evolutions of the inter-event times are also depicted in Figures 3 and 4. Obviously, almost all of the inter-event times decided by (4) are larger than those by the decentralized relative event condition, like the analysis in Remark 4. The sampling numbers for the two schemes are given in Table 1. Although the centralized event-triggered control scheme contradicts the decentralized configuration in Figure 1, comparing it with our scheme still contributes to illustrating the advantages of integral-based event condition.