## 4 Theoretical Results
\begin{array}{cccccc}\Lambda_{11}+\epsilon_{2}\rho_{1}^{-1}Q_{1}^{2}& BD_{j}^{-}H_{2}&BD_{j}^{+}&0&BD_{j}^{+}Y_{k}\\ *&\Lambda_{22}+\epsilon_{2}\rho_{1}^{-1}I&0&Y_{L}&0\\ *&*&-\epsilon_{2}I&0&0\\ *&*&*&-\epsilon_{2}I&0\\ *&*&*&-\epsilon_{1}Q_{1}^{2}\end{array} \right)<0. \tag{24}\]  
Pre- and post-multiplying both sides of Label (24) by \(diag\{Q_{1}^{-1},I,I,I,Q_{1}^{-1}\}\) and substituting \(Y_{K}Q_{1}^{-1}=K\) and \(Y_{L}=P_{2}L\) into Label (24), one has that Label (24) is equivalent to  
\[\left( \begin{array}{cccccc}\bar{\Lambda}_{11}&Q_{1}^{-1}BD_{j}^{-}H_{2}&Q_{1 }^{-1}BD_{j}^{+}&0&Q_{1}^{-1}BD_{j}^{+}K\\ *&\bar{\Lambda}_{22}&0&P_{2}L&0\\ *&*&-\epsilon_{2}I&0&0\\ *&*&*&-\epsilon_{2}I&0\\ *&*&*&-\epsilon_{1}I\end{array} \right)<0, \tag{25}\]  
where \(\bar{\Lambda}_{11}=Q_{1}^{-1}A+Q_{1}^{-1}BD_{l}^{+}K+Q_{1}^{-1}BD_{l}^{-} \bar{H}_{1}Q_{1}^{-1}+(Q_{1}^{-1}A+Q_{1}^{-1}BD_{l}^{+}K+Q_{1}^{-1}BD_{l}^{-} \bar{H}_{1}Q_{1}^{-1})^{\mathrm{T}}+\epsilon_{2}\rho_{1}^{-1}I\), and \(\bar{\Lambda}_{22}=P_{2}(A-LC)+(A-LC)^{\mathrm{T}}P_{2}+(\epsilon_{1}+\epsilon _{2}\rho_{1}^{-1})I\). By denoting \(P_{1}=Q_{1}^{-1}\), \(H_{1}=\bar{H}_{1}Q_{1}^{-1}\) and \(\rho_{0}^{2}=\rho_{1}^{-1}\), Label (25) is the same as Label (10). This shows that Label (22) is a sufficient condition for Label (10). Similarly, by pre- and post-multiplying both sides of Label (25) by \(diag\{1,Q_{1}^{-1},1\}\), Label (23) is equivalent to Label (12). Therefore, the proof is completed. **Remark 3**. : _In Label (10), scalars \(\epsilon_{1},\epsilon_{2}>0\) are two free parameters for improving the feasibility of the LMI. To make the LMI feasible, properly large \(\epsilon_{1}\) and \(\epsilon_{2}\) are expected. On the one hand, \(\epsilon_{1}\) has a counter effect on the feasibility. In fact, as shown in the last column, for a fixed \(P_{1}\), \(\epsilon_{1}\) is expected to be large enough, which, however, may yield a large \(P_{2}\). On the other hand, for a large \(P_{2}\), as shown in the forth column, a large \(\epsilon_{2}\) is expected as well. To eliminate the effect of large \(\epsilon_{2}\), as shown in \(\psi_{11}\), \(\rho_{0}\) is required to be small enough, which can always be satisfied. Therefore, to make Label (10) feasible, large \(\epsilon_{1}\) and \(\epsilon_{2}\) are expected, although too large \(\epsilon_{1}\) and \(\epsilon_{2}\) lead \(\rho_{0}\) to be very small. Since a small \(\rho_{0}\) would make the event condition more easy to be triggered, small \(\epsilon_{1},\epsilon_{2}\) are expected to improve the sampling performance. To balance the feasibility of LMIs and the samplingperformance, one may follow a two-step procedure for choosing the proper free parameters. First, one should select large enough \(\varepsilon_{1},\varepsilon_{2}\) such that the LMIs are feasible. Second, to obtain a large \(\rho_{0}\), one should gradually decrease the values of the free parameters until the obtained \(\rho_{0}\) is satisfactory or the LMIs are not feasible. The above analysis is also applicable to the LMIs in Theorems 2 and 3._  
**Remark 4**. : _In Ref. [25], the decentralised relative event condition is employed, i.e.,_  
\[t_{j_{k}+1}^{s}=\inf_{t}\{t\geq t_{j_{k}}^{s}+\tau_{u}^{s}\,\|\,\|e_{u_{k}}\|> \sigma_{u}^{s}\,\|u_{s}\|\},s=1,\ldots,p,\]  
\[t_{k_{l}+1}^{l}=\inf_{t}\{t\geq t_{k_{l}}^{l}+\tau_{y}^{l}\,\|\,\|e_{y_{l}}\|> \sigma_{y}^{l}\,\|y_{l}\|\},l=1,\ldots,r.\]  
_Similar to the analysis in [15] (Theorem 2), the integral-based event condition (4) is better than the relative one to some degree._  
_For example, we consider the node \(u_{1}\). For the same triggering input \(u_{1}(t_{k_{1}}^{1})\), the next triggering instant \(t_{k_{1}+1}^{1}\) decided by the relative event condition cannot be larger than that decided by Label (4). In fact, when \(t_{k_{1}+1}^{1}\) determined by the relative event condition is \(\tau_{u}^{1}\), the statement is obvious.