## 3 Main Results
### Consensus Analysis  
Based on the proposed control protocol and event-triggering conditions, the following theorem is established to ensure the finite-time consensus of heterogeneous nonlinear multi-agent systems under a bipartite topology. **Theorem 1**. : _Assumptions 1 and 2 hold. The heterogeneous multi-agent systems (1) under the control protocol (5) with the event-triggered mechanism (8) will achieve the finite-time consensusif there exist positive constants \(\mu_{\min}\), \(\bar{\mu}_{\min}\), \(c\), \(\sigma_{1}\), \(\sigma_{2}\), \(\gamma_{1}\), \(\gamma_{2}\), and \(\mathcal{K}\) such that the following conditions hold:_  
\[\mu_{\min}k_{1} >\alpha+\frac{\sigma_{1}}{\lambda_{\min}(M^{\mathrm{T}}I_{m}M)}+ \gamma_{1} \tag{11}\] \[\bar{\mu}_{\min}k_{2} >(\varepsilon+1)\lambda_{\max}(\tilde{M}\tilde{M}^{\mathrm{T}})+ \gamma_{2}\] (12) \[\bar{\mu}_{\min} >\frac{2}{c}+\frac{1}{c\lambda_{\min}(\tilde{M}^{\mathrm{T}} \tilde{M})} \tag{13}\]  
_where \(\alpha=2\lambda_{\max}(MM^{\mathrm{T}})+\lambda_{\max}(LL^{\mathrm{T}})\) and \(\varepsilon=\sigma_{2}/\lambda_{\min}(\tilde{M}^{\mathrm{T}}\tilde{M})\). Moreover, the bipartite consensus of MASs (1) is achieved in a finite time \(T=\frac{V^{1-\frac{1}{2}(\frac{\sigma}{2}+1)}(t_{0})}{\mathcal{K}(1-\frac{1}{ 2}(\frac{\sigma}{2}+1))}\)._  
Proof. : Construct the Lyapunov function as  
\[V(t)=\frac{1}{2}\left[ \begin{array}{c}\overline{\varphi}(t)\\ \overline{\psi}(t)\\ \overline{\xi}(t)\end{array} \right]^{\mathrm{T}}\left[ \begin{array}{cc}I_{Nn }&0\\ 0&\Xi\end{array} \right]\left[ \begin{array}{c}\overline{\varphi}(t)\\ \overline{\psi}(t)\\ \overline{\xi}(t)\end{array} \right], \tag{14}\]  
where \(\Xi=\left[ \begin{array}{cc}c(\tilde{M}+\tilde{M}^{\mathrm{T}})\otimes I_{n }&I_{Nn}\\ I_{Nn}&I_{Nn}\end{array} \right]\), \(\tilde{M}=\left[ \begin{array}{cc}I_{m}&0\\ 0&L_{2}+B_{2}\end{array} \right]\), \(L_{2}\) denotes the Laplacian matrix of the unsigned communication network for all second-order multi-agent system, and \(B_{2}=\mathrm{diag}(b_{m+1},b_{m+2},\ldots,b_{N})\). Based on the definition of \(\tilde{M}\), all eigenvalues of \(\tilde{M}\) are positive, which means that \(\tilde{M}+\tilde{M}^{\mathrm{T}}\) is a positive definite matrix. Therefore, there exists an invertible matrix \(\Gamma\) such that \(\Gamma(\tilde{M}+\tilde{M}^{\mathrm{T}})\Gamma^{-1}=\Lambda\), where \(\Lambda=\mathrm{diag}\{\bar{\mu}_{1},\bar{\mu}_{2},\ldots,\bar{\mu}_{N}\}\), \(\bar{\mu}_{i}\) is the \(i\)th eigenvalue of \(\tilde{M}+\tilde{M}^{\mathrm{T}}\), and \(\bar{\mu}_{\min}=\min_{i=1,2,\ldots,N}\{\bar{\mu}_{i}\}\). It is known from Equation (13) in Theorem 1 that \(\bar{\mu}_{\min}>\frac{1}{2}\); thus, \(c(\tilde{M}+\tilde{M}^{\mathrm{T}})\otimes I_{n}-I_{Nn}>0\), and \(\Xi\) is a positive definite matrix from the Shur complement theorem. Then, we can obtain \(V(t)>0\). The time derivative of \(V(t)\) is calculated as follows:  
\[\begin{split}\dot{V}(t)&=\left[ \begin{array}{c} \overline{\varphi}(t)\\ \overline{\psi}(t)\\ \overline{\xi}(t)\end{array} \right]^{\mathrm{T}}\left[ \begin{array}{cc}I_{ Nn}&0&0\\ 0&c(\tilde{M}+\tilde{M}^{\mathrm{T}})\otimes I_{n}&I_{Nn}\\ 0&I_{Nn}&I_{Nn}\\ \end{array} \right]\left[ \begin{array}{c}\frac{\overline{\varphi}(t)}{ \overline{\psi}(t)}\\ \overline{\xi}(t)\end{array}