## 3 Main Results
### Consensus Analysis
\right]\\ &=\overline{\varphi}^{\mathrm{T}}(t)\overline{\varphi}(t)+\overline{\psi}^{ \mathrm{T}}(t)(\tilde{M}+\tilde{M}^{\mathrm{T}})\overline{\dot{\overline{\psi} }}(t)+\overline{\xi}^{\mathrm{T}}(t)\overline{\dot{\overline{\psi}}}(t)+( \overline{\psi}^{\mathrm{T}}(t)+\overline{\xi}^{\mathrm{T}}(t))\overline{\dot{ \overline{\xi}}}(t) \tag{15}\]  
By error dynamical system (10), we obtain  
\[\begin{split}\overline{\varphi}(t)^{\mathrm{T}}\overline{\dot{ \overline{\varphi}}}(t)^{\mathrm{T}}=&\sum_{i=1}^{m}\overline{\varphi}_{i}(t) ^{\mathrm{T}}(-\sum_{j=1}^{m}\bar{l}_{ij}(w_{j}f_{j}(x_{j}(t),t)+E_{j}(t)+k_{1 }\overline{\varphi}_{j}(t)+\overline{\varphi}_{j}(t)^{\frac{p}{q}})\\ &-\sum_{j=m+1}^{N}\bar{l}_{ij}w_{j}v_{j}(t)-b_{i}(w_{i}f_{i}(x_{i}(t),t)+E_{ i}(t)+k_{1}\overline{\varphi}_{i}(t)+\overline{\varphi}_{i}(t)^{\frac{p}{q}}))\\ =&-\overline{\varphi}(t)^{\mathrm{T}}((\overline{L}+B)\otimes I_{n})((W \otimes I_{n})F(x(t),t)+E^{1}(t)+k_{1}\overline{\varphi}(t)+\overline{\varphi }(t)^{\frac{p}{q}})\\ &-\overline{\varphi}(t)^{\mathrm{T}}(\overline{L}\otimes I_{n})(W \otimes I_{n})V(t)\\ \overline{\psi}(t)^{\mathrm{T}}((\tilde{M}+\tilde{M}^{\mathrm{T}})\otimes I _{n})\overline{\dot{\overline{\psi}}}(t)+\overline{\xi}(t)^{\mathrm{T}} \overline{\dot{\overline{\psi}}}(t)=& c\overline{\dot{\overline{\psi}}}(t)^{\mathrm{T}}((\tilde{M}+\tilde{M}^{ \mathrm{T}})\otimes I_{n})\overline{\dot{\overline{\xi}}}(t)+\overline{\xi}( t)^{\mathrm{T}}\overline{\dot{\overline{\xi}}}(t) \tag{17}\]\[(\overline{\psi}^{\rm T}(t)+\overline{\xi}^{\rm T}(t))\dot{ \overline{\xi}}(t)= \sum_{i=m+1}^{N}(\overline{\psi}_{i}(t)^{\rm T}+\overline{\xi}_{i}(t)^{ \rm T})(-\sum_{j=m+1}^{N}\tilde{l}_{ij}(w_{j}g_{j}(x_{j}(t),v_{j}(t),t)+E_{j}(t) \tag{18}\] \[+(c+k_{2})(\overline{\psi}_{j}(t)+\overline{\xi}_{j}(t))+( \overline{\psi}_{j}(t)+\overline{\xi}_{j}(t))^{\frac{p}{q}})-b_{i}(w_{i}g_{i}( x_{i}(t),\] \[v_{i}(t),t)+E_{i}(t)+(c+k_{2})(\overline{\psi}_{i}(t)+\overline{ \xi}_{i}(t))+(\overline{\psi}_{i}(t)+\overline{\xi}_{i}(t))^{\frac{p}{q}}))\] \[= -(\overline{\psi}(t)^{\rm T}+\overline{\xi}(t)^{\rm T})(\tilde{M} \otimes I_{n})((W\otimes I_{n})G(x(t),v(t),t)+E^{2}(t)\] \[+(c+k_{2})(\overline{\psi}(t)+\overline{\xi}(t))+(\overline{\psi }(t)+\overline{\xi}(t))^{\frac{p}{q}})\]  
By Assumption 2, Lemma 2, and event-triggered conditions (8),  
\[-\overline{\varphi}(t)^{\rm T}((\overline{L}+B)\otimes I_{n})(( W\otimes I_{n})F(x(t),t)+E^{1}(t)) \tag{19}\] \[\leq \overline{\varphi}(t)^{\rm T}((MM^{\rm T})\otimes I_{n}) \overline{\varphi}(t)+\frac{1}{2}\|(W\otimes I_{n})F(x(t),t)\|^{2}+\frac{1}{2 }\|E^{1}(t)\|^{2}\] \[\leq \overline{\varphi}(t)^{\rm T}((MM^{\rm T})\otimes I_{n}) \overline{\varphi}(t)+\frac{\sigma_{1}}{2}\|(W\otimes I_{n})x(t)-1_{N}\otimes x _{\delta}\|^{2}+\frac{r_{1}}{2}\overline{\varphi}(t)^{\rm T}\overline{\varphi} (t)\]  
\[-(\overline{\varphi}(t)^{\rm T}+\overline{\xi}(t)^{\rm T})( \tilde{M}\otimes I_{n})((W\otimes I_{n})G(x(t),v(t),t)+E^{2}(t)) \tag{20}\] \[(\overline{\psi}^{\rm T}(t)+\overline{\xi}^{\rm T}(t))\dot{ \overline{\xi}}(t)= \sum_{i=m+1}^{N}(\overline{\psi}_{i}(t)^{\rm T}+\overline{\xi}_{i}(t)^{ \rm T})(-\sum_{j=m+1}^{N}\tilde{l}_{ij}(w_{j}g_{j}(x_{j}(t),v_{j}(t),t)+E_{j}(t) \tag{18}\]0  
\[(\overline{\psi}^{\rm T}(t)+\overline{\xi}^{\rm T}(t))\dot{ \overline{\xi}}(t)= \sum_{i=m+1}^{N}(\overline{\psi}_{i}(t)^{\rm T}+\overline{\xi}_{i}(t)^{ \rm T})(-\sum_{j=m+1}^{N}\tilde{l}_{ij}(w_{j}g_{j}(x_{j}(t),v_{j}(t),t)+E_{j}(t) \tag{18}\]1  
According to (19) and (20), we obtain  
\[(\overline{\psi}^{\rm T}(t)+\overline{\xi}^{\rm T}(t))\dot{ \overline{\xi}}(t)= \sum_{i=m+1}^{N}(\overline{\psi}_{i}(t)^{\rm T}+\overline{\xi}_{i}(t)^{ \rm T})(-\sum_{j=m+1}^{N}\tilde{l}_{ij}(w_{j}g_{j}(x_{j}(t),v_{j}(t),t)+E_{j}(t) \tag{18}\]2 \[(\overline{\psi}^{\rm T}(t)+\overline{\xi}^{\rm T}(t))\dot{ \overline{\xi}}(t)= \sum_{i=m+1}^{N}(\overline{\psi}_{i}(t)^{\rm T}+\overline{\xi}_{i}(t)^{ \rm T})(-\sum_{j=m+1}^{N}\tilde{l}_{ij}(w_{j}g_{j}(x_{j}(t),v_{j}(t),t)+E_{j}(t) \tag{18}\]3 \[(\overline{\psi}^{\rm T}(t)+\overline{\xi}^{\rm T}(t))\dot{ \overline{\xi}}(t)= \sum_{i=m+1}^{N}(\overline{\psi}_{i}(t)^{\rm T}+\overline{\xi}_{i}(t)^{ \rm T})(-\sum_{j=m+1}^{N}\tilde{l}_{ij}(w_{j}g_{j}(x_{j}(t),v_{j}(t),t)+E_{j}(t) \tag{18}\]4 \[(\overline{\psi}^{\rm T}(t)+\overline{\xi}^{\rm T}(t))\dot{ \overline{\xi}}(t)= \sum_{i=m+1}^{N}(\overline{\psi}_{i}(t)^{\rm T}+\overline{\xi}_{i}(t)^{ \rm T})(-\sum_{j=m+1}^{N}\tilde{l}_{ij}(w_{j}g_{j}(x_{j}(t),v_{j}(t),t)+E_{j}(t) \tag{18}\]5 \[(\overline{\psi}^{\rm T}(t)+\overline{\xi}^{\rm T}(t))\dot{ \overline{\xi}}(t)= \sum_{i=m+1}^{N}(\overline{\psi}_{i}(t)^{\rm T}+\overline{\xi}_{i}(t)^{ \rm T})(-\sum_{j=m+1}^{N}\tilde{l}_{ij}(w_{j}g_{j}(x_{j}(t),v_{j}(t),t)+E_{j}(t) \tag{18}\]6  
\[(\overline{\psi}^{\rm T}(t)+\overline{\xi}^{\rm T}(t))\dot{ \overline{\xi}}(t)= \sum_{i=m+1}^{N}(\overline{\psi}_{i}(t)^{\rm T}+\overline{\xi}_{i}(t)^{ \rm T})(-\sum_{j=m+1}^{N}\tilde{l}_{ij}(w_{j}g_{j}(x_{j}(t),v_{j}(t),t)+E_{j}(t) \tag{18}\]7 \[(\overline{\psi}^{\rm T}(t)+\overline{\xi}^{\rm T}(t))\dot{ \overline{\xi}}(t)= \sum_{i=m+1}^{N}(\overline{\psi}_{i}(t)^{\rm T}+\overline{\xi}_{i}(t)^{ \rm T})(-\sum_{j=m+1}^{N}\tilde{l}_{ij}(w_{j}g_{j}(x_{j}(t),v_{j}(t),t)+E_{j}(t) \tag{18}\]8 \[(\overline{\psi}^{\rm T}(t)+\overline{\xi}^{\rm T}(t))\dot{ \overline{\xi}}(t)= \sum_{i=m+1}^{N}(\overline{\psi}_{i}(t)^{\rm T}+\overline{\xi}_{i}(t)^{ \rm T})(-\sum_{j=m+1}^{N}\tilde{l}_{ij}(w_{j}g_{j}(x_{j}(t),v_{j}(t),t)+E_{j}(t) \tag{18}\]9 \[+(c+k_{2})(\overline{\psi}_{j}(t)+\overline{\xi}_{j}(t))+( \overline{\psi}_{j}(t)+\overline{\xi}_{j}(t))^{\frac{p}{q}})-b_{i}(w_{i}g_{i}( x_{i}(t),\]0 \[+(c+k_{2})(\overline{\psi}_{j}(t)+\overline{\xi}_{j}(t))+( \overline{\psi}_{j}(t)+\overline{\xi}_{j}(t))^{\frac{p}{q}})-b_{i}(w_{i}g_{i}( x_{i}(t),\]1where \(\|e(t)\|=\|(W\otimes I_{n})x(t)-1_{N}\otimes x_{\delta}\|\) denotes the consensus error, and \(\mu_{\min}\) and \(\check{\mu}_{\min}\) are the non-zero eigenvalues of matrices \(M+M^{\rm T}\) and \(\tilde{M}+\tilde{M}^{\rm T}\), respectively.