## 4 Theoretical Results  
In this section, we provide the main results of this paper. First, we propose a method based on linear matrix inequality (LMI) to design the event condition (4) in the case that the gain matrices are supposed to be given. Then, we will study the synthesis of the gain matrices. Let the argument variable be \(X_{c}=(x^{\rm T},z^{\rm T})^{\rm T}\). Motivated by [21], we initially use Lemma 1 to deal with the saturation nonlinearity, which shows that if \(\|HX_{c}\|_{\infty}\leq 1\) for a matrix \(H=(H_{1},H_{2})\), \(H_{i}\in\mathbb{R}^{m\times n},i=1,2\), then  
\[Bsat(Kx+Kz+e_{u})\in co\big{\{}BD_{j}^{+}(Kx+Kz+e_{u})+BD_{j}^{-}(H_{1}x+H_{2} z),j\in\mathcal{M}\big{\}}.\]  
Now, consider the state \(X_{c}\) in the set \(\{X_{c}\in\mathbb{R}^{2n}|\ \|HX_{c}\|_{\infty}\leq 1\}\). At this moment, the closed-loop system (5) is translated into the following form:  
\[ \begin{split}\dot{x}&\in co\{(A+BD_{j}^{+}K+BD_{j}^ {-}H_{1})x+(BD_{j}^{+}K+BD_{j}^{-}H_{2})z+BD_{j}^{+}e_{u},j\in\mathcal{M}\},\\ \dot{z}&=(A-LC)z+Ley.\end{split}  \tag{6}\]  
Then, we study the stability of this differential inclusion. Let \(V=x^{\rm T}P_{1}x+z^{\rm T}P_{2}z\) with \(P_{1},P_{2}\) being \(n\)-dimension positive definite matrices. The derivative of \(V\) along the trajectories of Label (6) is  
\[ \begin{split}\dot{V}&\in co\{2x^{\rm T}P_{1}A_{c}x +2x^{\rm T}P_{1}BD_{j}^{+}Kz+2x^{\rm T}P_{1}BD_{j}^{-}H_{2}z+2x^{\rm T}P_{1}BD_ {j}^{+}e_{u}\\ &\qquad+2z^{\rm T}P_{2}(A-LC)z+2z^{\rm T}P_{2}Ley,j\in\mathcal{M }\},\end{split} \]  
with \(A_{c}=A+BD_{j}^{+}K+BD_{j}^{-}H_{1}\). From the fact \(2x^{\rm T}P_{1}BD_{j}^{+}Kz\leq\epsilon_{1}z^{\rm T}z+\frac{1}{\epsilon_{1}}x^ {\rm T}P_{1}BD_{j}^{+}K(P_{1}BD_{j}^{+}K)^{\rm T}x\) for any \(\epsilon_{1}>0\),  
\[ \begin{split}\dot{V}&\in co\{x^{\rm T}(2P_{1}A_{c}+ \frac{1}{\epsilon_{1}}P_{1}BD_{j}^{+}K(P_{1}BD_{j}^{+}K)^{\rm T})x+2x^{\rm T}P_{ 1}BD_{j}^{-}H_{2}z\\ &\qquad+2x^{\rm T}P_{1}BD_{j}^{+}e_{u}+z^{\rm T}(2P_{2}(A-LC)+ \epsilon_{1}I)z+2z^{\rm T}P_{2}Ley,j\in\mathcal{M}\}.\end{split}  \tag{7}\]  
According to Label (7), a sufficient condition for \(\dot{V}<0\) is  
\[X_{c}^{\rm T}\left( \begin{array}{cc}\varphi_{11}&P_{1}BD_{j}^{-}H_{2}\\ *&\varphi_{22}\end{array} \right)X_{c}+2X_{c}^{\rm T}\left( \begin{array}{cc} P_{1}BD_{j}^{+}&0\\ 0&P_{2}L\end{array} \right)E<0,j\in\mathcal{M}, \tag{8}\]  
where \(\varphi_{11}=P_{1}A_{c}+A_{c}^{\rm T}P_{1}+\frac{1}{\epsilon_{1}}P_{1}BD_{j}^ {+}K(P_{1}BD_{j}^{+}K)^{\rm T}\) and \(\varphi_{22}=P_{2}(A-LC)+(A-LC)^{\rm T}P_{2}+\epsilon_{1}I\). Due to the fact that, for any \(\epsilon_{2}>0\),  
\[2X_{c}^{\rm T}\left( \begin{array}{cc}P_{1}BD_{j}^{+}&0\\ 0&P_{2}L\end{array} \right)E\leq\epsilon_{2}E^{\rm T}E+\frac{1}{\epsilon_{2}}X_{ c}^{\rm T}\left( \begin{array}{cc}P_{1}BD_{j}^{+}&0\\ 0&P_{2}L\end{array} \right)\left( \begin{array}{cc}\left(P_{1}BD_{j}^{+}\right) ^{\rm T}&0\\ 0&L^{\rm T}P_{2}^{\rm T}\end{array} \right)X_{c},\]if the following matrix inequality holds for some positive scalar \(\rho_{0}\),  
\[\left( \begin{array}{cc}\psi_{11}+\frac{1}{\epsilon_{2}}(P_{1}BD_{j}^{+})(P_{1} BD_{j}^{+})^{\mathrm{T}}+\varepsilon_{2}\rho_{0}^{2}I&P_{1}BD_{j}^{-}H_{2}\\ *&\varphi_{22}+\varepsilon_{2}\rho_{0}^{2}I+\frac{1}{\epsilon_{2}}P_{2}L(P_{2} L)^{\mathrm{T}}\end{array} \right)<0,j\in\mathcal{M}, \tag{9}\]  
then \(\dot{V}<\epsilon_{2}(-\rho_{0}^{2}X_{c}^{\mathrm{T}}X_{c}+E^{\mathrm{T}}E)\). By a Schur complement, Label (9) is equivalent to  
\[\left( \begin{array}{cccc}\psi_{11}&P_{1}BD_{j}^{-}H_{2}&P_{1}BD_{j}^{+}&0&P_{ 1}BD_{j}^{+}K\\ *&\psi_{22}&0&P_{2}L&0\\ *&*&-\epsilon_{2}I&0&0\\ *&*&*&-\epsilon_{2}I&0\\ *&*&*&*&-\epsilon_{1}I\end{array} \right)<0,j\in\mathcal{M}, \tag{10}\]  
where \(\psi_{11}=P_{1}A_{c}+A_{c}^{\mathrm{T}}P_{1}+\epsilon_{2}\rho_{0}^{2}I\), \(\psi_{22}=P_{2}(A-LC)+(A-LC)^{\mathrm{T}}P_{2}+(\epsilon_{1}+\epsilon_{2}\rho_ {0}^{2})I\) and \(A_{c}=A+BD_{j}^{+}K+BD_{j}^{-}H_{1}\). Hence, at this moment, \(\dot{V}<\epsilon_{2}(-\rho_{0}^{2}X_{c}^{\mathrm{T}}X_{c}+E^{\mathrm{T}}E)\).