## 4 Theoretical Results
\begin{array}{cccccccc}1&\tilde{h}_{l}^{+}&\tilde{h}_{l}^ {-}\\ *&Q_{1}&0\\ *&*&Q_{2}\\ \end{array} \right)\geq 0,l=1,\ldots,m, \tag{20}\]  
_where \(\bar{\varphi}_{11}=AQ_{1}+BD_{j}^{+}KQ_{1}+BD_{j}^{-}\bar{H}_{1}+(AQ_{1}+BD_{ j}^{+}KQ_{1}+BD_{j}^{-}\bar{H}_{1})^{\mathrm{T}}\), and \(\bar{\varphi}_{22}=(A-LC)Q_{2}+Q_{2}(A-LC)^{\mathrm{T}}\). The LMI variables are positive definite matrices \(Q_{1}\),\(Q_{2}\), and matrices \(\bar{H}_{1}\), \(\bar{H}_{2}\). \(\tilde{h}_{l}^{+}\) and \(\bar{h}_{l}^{-}\) are the \(l^{\mathrm{th}}\) row of \(\bar{H}_{1}\) and \(\bar{H}_{2}\), respectively. Moreover, the solutions to Label (18) are \(\{Q_{1}^{-1},Q_{2}^{-1},\bar{H}_{1}Q_{1}^{-1},\bar{H}_{2}Q_{2}^{-1}\}\)._  
Proof of Theorem 2.: Using the Schur complement, Label (20) is equivalent to  
\[\left( \begin{array}{cccccccc}\bar{\varphi}_{11}+\varepsilon_{2}\rho_{0}^{2}Q _{1}^{2}&BD_{j}^{-}\bar{H}_{2}&BD_{j}^{+}&0&BD_{j}^{+}K\\ *&\bar{\varphi}_{22}+(\varepsilon_{2}\rho_{0}^{2}+\varepsilon_{1})Q_{2}^{2}&0& L&0\\ *&*&-\varepsilon_{2}I&0&0\\ *&*&*&-\varepsilon_{2}I&0\\ *&*&*&*&-I\\ \end{array} \right)<0.\]  
Pre- and post-multiplying both sides of the above inequality by \(\mathit{diag}\{Q_{1}^{-1},Q_{2}^{-1},I,I,I\}\), it is transformed to  
\[\left( \begin{array}{cccccccc}Q_{1}^{-1}\bar{\varphi}_{11}Q_{1}^{-1}+\rho_{0 }^{2}I&Q_{1}^{-1}BD_{j}^{-}\bar{H}_{2}Q_{2}^{-1}&Q_{1}^{-1}BD_{j}^{+}&0&Q_{1}^{ -1}BD_{j}^{+}K\\ *&Q_{2}^{-1}\bar{\varphi}_{22}Q_{2}^{-1}+(\varepsilon_{2}\rho_{0}^{2}+ \varepsilon_{1})I&0&Q_{2}^{-1}L&0\\ *&*&-I&0&0\\ *&*&*&-I&0\\ *&*&*&*&-\varepsilon_{1}I\\ \end{array} \right)<0. \tag{21}\]  
By denoting \(P_{1}=Q_{1}^{-1}\), \(P_{2}=Q_{2}^{-1}\), \(H_{1}=\bar{H}_{1}Q_{1}^{-1}\) and \(H_{2}=\bar{H}_{2}Q_{2}^{-1}\), Label (21) is the same as Label (10) obviously. By a similar process, it is proved that Label (20) is equivalent to Label (12) with \(h_{l}^{+}=\bar{h}_{l}^{+}Q_{1}^{-1}\) and \(h_{l}^{-}=\bar{h}_{l}^{-}Q_{2}^{-1}\). Therefore, the proof is completed. For given gain matrices \(K\) and \(L\), if the LMIs (10) and (12) are feasible, one can design the parameters in event condition (4) according to Theorem 1 and give the largest ellipsoid by solving the LMI optimization problem in Theorem 2. Hence, in the following, we focus attention on how to find such gain matrices, i.e., a controller synthesis jointly with the parameters in the event condition. **Theorem 3**. : _For the closed-loop system (5), if there exist positive definite matrices \(Q_{1}\), \(P_{2}\), matrices \(Y_{k}\)\(Y_{L}\), \(\bar{h}_{l}^{+}\), and \(h_{l}^{-},l=1,\ldots,m\), and a positive scalar \(\rho_{1}\) such that the following LMIs hold for some \(\epsilon_{1},\epsilon_{2}>0\):_  
\[\left( \begin{array}{cccccc}\Lambda_{11}&BD_{j}^{-}H_{2}&BD_{j}^{+}&0&BD_{j}^ {+}Y_{K}&\sqrt{\epsilon_{2}}Q_{1}&0\\ *&\Lambda_{22}&0&Y_{L}&0&0&\sqrt{\epsilon_{2}}I\\ *&*&-\epsilon_{2}I&0&0&0&0\\ *&*&*&-\epsilon_{2}I&0&0&0\\ *&*&*&*&-2\epsilon_{1}Q_{1}+\epsilon_{1}I&0&0\\ *&*&*&*&*&-\rho_{1}I&0\\ *&*&*&*&*&-\rho_{1}I\end{array} \right)<0,j\in\mathcal{M}, \tag{22}\]  
\[\left( \begin{array}{cccccc}1&\bar{h}_{l}^{+}&h_{l}^{-}\\ *&\Omega_{1}&0\\ *&\pi&P_{2}\end{array} \right)\geq 0,l=1,\ldots,m, \tag{23}\]  
_where \(\Lambda_{11}=AQ_{1}+BD_{j}^{+}Y_{K}+BD_{j}^{-}\bar{H}_{1}+(AQ_{1}+BD_{j}^{+}Y_ {K}+BD_{j}^{-}\bar{H}_{1})^{\mathrm{T}}\) and \(\Lambda_{22}=P_{2}A-Y_{L}C+(P_{2}A-Y_{L}C)^{\mathrm{T}}+\epsilon_{1}I\). \(\bar{h}_{l}^{+}\) and \(h_{l}^{-}\) are, respectively, the \(l^{\mathrm{th}}\) row of \(\bar{H}_{1}\) and \(H_{2}\). Then, \(\{Q_{1}^{-1},P_{2},\rho_{1}^{-1},\bar{h}_{l}^{+}Q_{1}^{-1},h_{l}^{-}\}\) are the solutions to the LMIs (10) and (12) with \(K=Y_{K}Q_{1}^{-1}\) and \(L=P_{2}^{-1}Y_{L}\)._  
Proof of Theorem 3.: Due to the Schur complement and the fact that \(-Q_{1}^{2}\leq-2Q_{1}+I\), Label (22) yields  
\[\left(