## 3 Main Results
### Design of Event-Triggered Control Protocol  
To address the bipartite consensus problem of heterogeneous multi-agent systems within a finite time frame, a nonlinear finite-time event-triggered pinning control mechanism was designed. This mechanism was engineered to reduce controller updates, particularly in scenarios where communication bandwidth is constrained. Suppose that the trigger moment sequence of the \(i\)th agent is expressed as \(\{t_{0}^{i},t_{1}^{i},\ldots,t_{k}^{i},\ldots\}\). For convenience, we first define the error term related to neighbor information as follows:  
\[\left\{ \begin{aligned} &\varphi_{i}(t)=\sum_{j=1}^{N}|a_{ij}|( \text{sign}(a_{ij})x_{j}(t)-x_{i}(t))+b_{i}(w_{i}x_{\delta}-x_{i}(t)),& i\in I_{m}\\ &\left\{\begin{aligned} &\psi_{i}(t)=\sum_{j=m+1}^{N}|a_{ij}|( \text{sign}(a_{ij})x_{j}(t)-x_{i}(t))+b_{i}(w_{i}x_{\delta}-x_{i}(t)),& \\ &\xi_{i}(t)=\sum_{j=m+1}^{N}|a_{ij}|(\text{sign}(a_{ij})v_{j}(t)-v_{ i}(t))-b_{i}v_{i}(t),&\end{aligned}\right.& i\in I_{N-m}\end{aligned} \right. \tag{4}\]  
where \(\mathcal{A}=(a_{ij})_{N\times N}\) denotes the adjacency matrix, and \(b_{i}\geq 0\) are the pinning gains. When \(b_{i}>0\), the \(i\)th node is called the pinning node of the network. Then, in order to achieve finite-time consensus within the system (1), the following control protocol is prescribed. \[u_{i}(t)= \begin{cases}k_{1}\varphi_{i}(t_{k}^{i})+\varphi_{i}(t_{k}^{i})^{\frac{ p}{q}},&i\in I_{m}\\ (c+k_{2})(\psi_{i}(t_{k}^{i})+\xi_{i}(t_{k}^{i}))+(\psi_{i}(t_{k}^{i})+\xi_{i}( t_{k}^{i}))^{\frac{p}{q}},&i\in I_{N-m}\end{cases}  \tag{5}\]  
where \(k_{1},k_{2},c>0\) are control gains, and \(p,q\) are both positive odd integers satisfying \(p<q\). **Remark 2**. : _In traditional control methods, the controller can be designed as_  
\[u_{i}(t)= \begin{cases}k_{1}\varphi_{i}(t)+\varphi_{i}(t)^{\frac{p}{q}},&i\in I _{m}\\ (c+k_{2})(\psi_{i}(t)+\xi_{i}(t))+(\psi_{i}(t)+\xi_{i}(t))^{\frac{p}{q}},&i\in I _{N-m}\end{cases} \]  
_The controller must clearly update continuously based on the communication information between the agents. In other words, the controller changes with time \(t\). In this paper, event-triggered control is updated only at the moments determined by the event-triggering conditions within the event sequence\(\{t_{0}^{i},t_{1}^{i},\ldots,t_{k}^{i},\ldots\}\), which significantly saves communication resources._  
Taking into account the bipartite nature of the communication network, and for the sake of expediency in subsequent derivations, we establish \(\overline{\varphi}_{i}(t)=w_{i}\varphi_{i}(t)(i=1,2,\ldots,m)\), \(\overline{\psi}_{i}(t)=w_{i}\psi_{i}(t)\), \(\overline{\xi}_{i}(t)=w_{i}\overline{\xi}_{i}(t)(i=m+1,m+2,\ldots,N)\), where \(w_{i}\) is defined in Lemma 1. Under Assumption 1, through the introduction of the diagonal matrix \(W\), it can be readily inferred that  
\[ \begin{cases}\overline{\varphi}_{i}(t)=\sum_{j=1}^{N}|a_{ij}|(w_{j}x_{j}(t)-w _{i}x_{i}(t))+b_{i}(x_{\delta}-w_{i}x_{i}(t)),&i\in I_{m}\\ \begin{cases}\overline{\psi}_{i}(t)=\sum_{j=m+1}^{N}|a_{ij}|(w_{j}x_{j}(t)-w_{ i}x_{i}(t))+b_{i}(x_{\delta}-w_{i}x_{i}(t)),&\\ \overline{\xi}_{i}(t)=\sum_{j=m+1}^{N}|a_{ij}|(w_{j}v_{j}(t)-w_{i}v_{i}(t))-b_{ i}w_{i}v_{i}(t),&\end{cases}\end{cases}  \tag{6}\]  
Since \(p\) and \(q\) are positive odd integers with \(p<q\), and given any \(w_{i}=\pm 1\), it consequently follows that  
\[w_{i}u_{i}(t)= \begin{cases}k_{1}\overline{\varphi}_{i}(t_{k}^{i})+\overline{ \varphi}_{i}(t_{k}^{i})^{\frac{p}{q}},&i\in I_{m}\\ (c+k_{2})(\overline{\psi}_{i}(t_{k}^{i})+\overline{\xi}_{i}(t_{k}^{i}))+( \overline{\psi}_{i}(t_{k}^{i})+\overline{\xi}_{i}(t_{k}^{i}))^{\frac{p}{q}},& i\in I_{N-m}\end{cases}  \tag{7}\]  
Furthermore, the determination of \(t_{k}^{i}\) is governed by the event-triggered mechanism \(t_{k+1}^{i}=\inf\{t>t_{k}^{i}:h_{i}(t)>0\}\), where \(h_{i}(t)\) is defined as  
\[h_{i}(t)=