## 1 Introduction
This control method defines trigger conditions based on system requirements and performance criteria, enabling control decisions and adjustments to occur only when specific events take place rather than at fixed intervals. By doing so, communication and computational burdens are reduced, while system flexibility and efficiency are enhanced. An event-triggered mechanism based on relative information was introduced in [25] to address the consensus issue in stochastic multi-agent systems. [PERSON] et al. proposed a dynamic event-triggered control with an adaptive law to improve control efficiency [26]. For leader-follower and leaderless consensus, corresponding event-triggered mechanisms were designed by [PERSON] [27] and [PERSON] [28], respectively. In traditional communication topologies, the weights of the adjacency matrix are required to be non-negative [23; 25]. However, in certain multi-agent systems, different agents may cooperate to achieve common goals or compete to acquire resources or advantages in various time frames and contexts. As a result, the communication topology of the multi-agent system should be represented as a signed graph, where the weights of the adjacency matrix are positive or negative depending on whether the agents are in a cooperative or competitive relationship [29; 30]. Networks of agent systems are characterized by structurally balanced signed graph topologies, which demonstrate bipartite consensus. The bipartite consensus of multi-agent systems with switching partial couplings and topologies was discussed in [31]. The bipartite consensus of higher-order multi-agent systems was analyzed in [32]. Drawing upon accumulated knowledge and inspiration, this paper endeavors to investigate the finite-time consensus problem in heterogeneous multi-agent systems composed of first-order and second-order agents under a signed directed graph by pinning event-triggered control. The heterogeneous multi-agent system presented in this paper is modeled as a hybrid-order system, comprising both first-order and second-order agents. In contrast to the hybrid-order integral multi-agent system model [6], this formulation incorporates nonlinear terms, which significantly complicate the analytical process but provide a more accurate representation of real-world applications. Moreover, for the heterogeneous system, we develop an appropriate pinning event-triggered control strategy, which not only mitigates data redundancy caused by continuous communication but also ensures that the multi-agent system achieves consensus within a fixed time frame. In contrast, some existingworks only realize an asymptotic consensus, where the time to convergence remains indeterminate [33]. The main contributions are summarized as follows:  
1. The finite-time consensus problem for heterogeneous multi-agent systems with mixed-order dynamics under cooperative-competitive interactions is addressed. To the authors' knowledge, there is scarce literature simultaneously considering the heterogeneity and finite convergence of multi-agent systems. Moreover, this paper achieves finite-time consensus in systems communicated under a signed directed graph. 2. An innovative pinning event-triggered control protocol is designed for the proposed multi-agent system to achieve finite-time consensus. On one hand, pinning control is adopted to constrain those agents with zero in-degree to avoid unnecessary communication; on the other hand, event-triggered control is employed to save communication bandwidth. The control protocol proposed in this paper enhances the convergence performance of the system and prolongs the lifespan of the control devices. 3. The control protocol designed in this paper breaks the communication barrier between first-order agents and second-order agents, enabling communication between hybrid-order agents. **Organization:** In Section 2, the model of multi-systems and control protocol are briefly introduced. In Section 3, the finite-time event-triggered convergent scheme is explored in detail.