## 3 Main Results
### Consensus Analysis
\begin{split}\dot{V}(t)&\leq-\mathcal{K}_{1}(\overline{ \varphi}(t)^{\mathrm{T}}\overline{\varphi}(t))^{\frac{1}{2}(\frac{p}{q}+1)}- \mathcal{K}_{2}(\frac{2m}{\lambda_{\max}(\Xi)}\zeta(t)^{\mathrm{T}}\Xi\zeta(t ))^{\frac{1}{2}(\frac{p}{q}+1)}\\ &\leq-\mathcal{K}(\overline{\varphi}(t)^{\mathrm{T}}\overline{ \varphi}(t)+\zeta(t)^{\mathrm{T}}\Xi\zeta(t))^{\frac{1}{2}(\frac{p}{q}+1)}\\ &\leq-\mathcal{K}V(t)^{\frac{1}{2}(\frac{p}{q}+1)}\end{split}  \tag{27}\]  
where \(\mathcal{K}=\min\{\mathcal{K}_{1},\mathcal{K}_{2}(\frac{2m}{\lambda_{\max}( \Lambda)})^{\frac{1}{2}(\frac{p}{q}+1)}\}\). Due to \(\mathcal{K}>0\) and \(0<\frac{1}{2}(\frac{p}{q}+1)<1\), based on Lemma 5, one obtains \(V(t)\equiv 0\) for \(t\geq T\), where \(T=\frac{2V(0)^{\frac{1}{2}(n-\frac{p}{q})}}{\mathcal{K}(1-\frac{p}{q})}\). That is, \(\lim_{t\to T}\lVert x_{i}(t)-w_{i}x_{\delta}\rVert=0(i\in I_{N})\), \(\lim_{t\to T}\lVert v_{i}(t)\rVert=0(i\in I_{N-m})\), and \(x_{i}(t)=w_{i}x_{\delta}\), \(v_{i}(t)=0\) for \(t>T\), from Definition 1, the bipartite finite-time consensus of the heterogeneous multi-agent system (1) is realized within the setting time \(T\). The proof is completed. **Remark 5**. : _In the theoretical derivation process using [PERSON]'s stability method, for the sake of convenience, this paper segregated the information of first-order agents from that of second-order agents. Specifically, the error information of all first-order agent was represented by one vector, while that of all second-order agents was encapsulated into another vector. Subsequently, both error vectors were expanded to the same order for consensus._  
Moreover, the following theorem proves that the designed event-triggered protocol does not exhibit a Zeno behavior. In other words, the event-triggered instant has a strictly positive lower bound. **Theorem 2**. : _Consider the heterogeneous multi-agent system (1) under the control input (5), the triggered instant \(t_{k}^{i}\) is decided by the event-triggered mechanism (8). There exists a strictly positive constant which is the lower bound of the event-triggered instant \(t_{k+1}^{i}-t_{k^{\prime}}^{i}\); thus, we can exclude the Zeno behavior._  
Proof. : When \(t\in[t_{k}^{i},t_{k+1}^{i})\), based on the measurement error (9) for first-order agents, one can obtain  
\[ \begin{split} D^{+}(\lVert E_{i}(t)\rVert)\leq& \lVert\dot{E}_{i}(t)\rVert=\lVert k_{1}\overline{\varphi}_{i}(t)+ \frac{p}{q}\overline{\varphi}_{i}(t)^{\frac{p}{q}-1}\overline{\varphi}_{i}(t )\rVert\\ \leq&(k_{1}+\frac{p}{q}|\overline{\varphi}_{i}(t)|^{ \frac{p}{q}-1})\lVert\dot{\overline{\varphi}}_{i}(t)\rVert\\ =&(k_{1}+\frac{p}{q}|\overline{\varphi}_{i}(t)|^{ \frac{p}{q}-1})\rVert\sum_{j=1}^{m}\bar{I}_{ij}(w_{j}f_{j}(x_{j}(t),t)+k_{1} \overline{\varphi}_{j}(t_{k}^{i})+\overline{\varphi}_{j}(t_{k}^{i})^{\frac{p }{q}})\\ &+\sum_{j=m+1}^{N}\bar{I}_{ij}v_{j}(t)+b_{i}(w_{i}f_{j}(x_{i}(t),t )+k_{1}\overline{\varphi}_{i}(t_{k}^{i})+\overline{\varphi}_{i}(t_{k}^{i})^{ \frac{p}{q}})\rVert\\ \leq&(k_{1}+\frac{p}{q}|\overline{\varphi}_{i}(t)|^{ \frac{p}{q}-1})(\lVert\overline{L}+B\rVert\lVert(\mathcal{W}\otimes I_{n})F(x (t),t)\rVert+\lVert\overline{L}\rVert\lVert v(t)\rVert+\lVert\Delta(t_{k}^{i} )\rVert)\\ \leq&(k_{1}+\frac{p}{q}|\overline{\varphi}_{i}(t)|^{ \frac{p}{q}-1})(\sigma_{1}\lVert M\rVert\lVert e(t)\rVert+\lVert\overline{L} \rVert\lVert v(t)\rVert+\lVert\Delta(t_{k}^{i})\rVert)\end{split}  \tag{28}\]  
where \(\lVert\Delta(t_{k}^{i})\rVert=\lVert\sum_{j=1}^{m}\bar{I}_{ij}(k_{1}\overline {\varphi}_{j}(t_{k}^{i})+\overline{\varphi}_{j}(t_{k}^{i})^{\frac{p}{q}})+b_{ i}(k_{1}\overline{\varphi}_{i}(t_{k}^{i})+\overline{\varphi}_{i}(t_{k}^{i})^{ \frac{p}{q}})\rVert\). Let \(e_{i}(t)=x_{i}(t)-x_{\delta}\) denote the consensus error of \(i\)th agent. Since \(V(t)\) is bounded, which is derived in Theorem 1, the error terms \(\varphi(t),\psi(t)\), and \(\xi(t)\) are bounded, which means the consensus error \(e_{i}(t)(i=1,2,\ldots,N)\) and velocity \(v_{i}(t)(i=m+1,m+2,\ldots,N)\) of all agents have the upper bounds \(M_{e}\) and \(M_{v}\), respectively.