## 3 Main Results
### Consensus Analysis
In the same way, if there exists a positive constant \(0<\omega_{2}<1\) such that \(\gamma_{2i}\|\overline{\psi}_{i}(t)+\overline{\xi}_{i}(t)\|\geq\omega_{2}>0\) holds, one can obtain  
\[D^{+}(\|E_{i}(t)\|)\leq(k_{1}+\frac{p}{q}(\frac{\omega_{1}}{\gamma_{1i}})^{ \frac{p}{q}-1})(\sigma_{1}N\|M\|M_{e}+N\|\overline{L}\|M_{v}+\|\Delta(t_{k}^{ i})\|) \tag{30}\]5  
Accordingly,  
\[D^{+}(\|E_{i}(t)\|)\leq(k_{1}+\frac{p}{q}(\frac{\omega_{1}}{\gamma_{1i}})^{ \frac{p}{q}-1})(\sigma_{1}N\|M\|M_{e}+N\|\overline{L}\|M_{v}+\|\Delta(t_{k}^{ i})\|) \tag{30}\]6  
where \(\tilde{\Phi}_{i}=(c+k_{2}+\frac{p}{q}(\frac{\omega_{2}}{\gamma_{2i}})^{\frac {p}{q}-1})(2(\sum_{j=m+1}^{N}|a_{ij}|+b_{i})M_{v}+\sigma_{2}\|\tilde{M}\|(N-m )M_{v}+\|\tilde{\Delta}(t_{k}^{i})\|)\), and at the trigger time \(t_{k+1}^{i}\), we also have \(\omega_{2}\leq\gamma_{2i}\|\overline{\psi}_{i}(t_{k+1}^{i})+\overline{\xi}_{i }(t_{k+1}^{i})\|<\|E_{i}(t_{k+1}^{i})\|\leq\tilde{\Phi}_{i}(t_{k+1}^{i}-t_{k} ^{i})\), which means the inter-event interval \(\tau_{k}^{i}=t_{k+1}^{i}-t_{k}^{i}>\frac{\omega_{2}}{\Phi_{i}}>0\). The proof is completed. **Remark 6**: _Based on Lyapunov stability theory and Lemma 5, Theorem 1 proves that when the system parameters satisfy conditions (11)-(13), multi-agent system (1) can achieve consensus within a finite-time under the influence of Controller (5). Utilizing the comparison principle and event-triggering conditions, Theorem 2 demonstrates that the event intervals of our designed event-triggering Controller (5) are greater than a positive number, thereby preventing the occurrence of any Zeno behavior, which means that an infinite number of events will not occur within a finite-time interval._  
**Remark 7**: _The control protocol designed in this paper is notably flexible, as the update frequency of control protocol (5) can be tailored according to the threshold settings of the control protocol (8). Specifically, controllers are updated only when predefined events occur, remaining unchanged otherwise. This effectively minimizes unnecessary overhead for controllers, conserving communication bandwidth and enhancing the efficiency of multi-agent systems._  
**Remark 8**: _The controllers designed in this paper overcome the communication barrier between first-order and second-order multi-agent systems, enabling first-order agents to receive information from second-order agents. Although [33] analyzed the consensus of hybrid-order heterogeneous multi-agent systems, their approach treated first-order and second-order agents in separate groups, failing to address the issue of communication between hybrid-order agents._  
**Remark 9**: _Notably, if the control protocol is not properly designed, the Zeno behavior is highly likely to occur. The theoretical analysis of Theorem 2 proves that the proposed control protocol guarantees a positive inter-event interval, effectively preventing the occurrence of any Zeno behavior._  
**Remark 10**: _Compared with the consensus results in [33; 38], this paper possesses the following advantages: (1) This paper delves into a more intricate signed directed topology for consensus analysis. (2) This paper provides finite-time convergence results for hybrid-order heterogeneous multi-agent systems, which are more practically relevant._