## 1 Introduction  
A multi-agent system refers to the mutual cooperation and coordination among multiple individual agents to collectively accomplish complex tasks. Currently, multi-agent systems are widely applied in various fields such as aircraft formation, sensor networks, parallel computing, and network resource allocation [1; 2; 3; 4]. Researching heterogeneous multi-agent systems is of paramount significance in bolstering system robustness and adaptability, as well as optimizing task performance [5]. Scholars have conducted in-depth research on multi-agent systems, yet the aspect of heterogeneity within the system, wherein agents possess distinct characteristics, remains largely overlooked. Heterogeneous systems typically involve different types of agents or control strategies. Effectively addressing the consensus problem in these systems can enhance resource utilization efficiency, optimize the collaboration process, and reduce redundant operations. [PERSON] and [PERSON] sought to investigate the consensus issue of heterogeneous multi-agent system formed by second-order linear and nonlinear agents [6]. [PERSON] et al. considered the consensus problem of heterogenous multi-agent systems composed of two different dynamics: first-order and second-order integrator agents [7]. Nevertheless, there is ample scope for further exploration in the field of heterogeneous systems  
In multi-agent systems, achieving consensus is a critical research focus for coordinating control [8; 9; 10; 11]. This involves individuals adjusting and optimizing their behavior through mutual communication, leading to eventual convergence to a unified state. Multi-agent systems exhibit two convergence modes [12], namely, asymptotic convergence and finite-time convergence. Finite-time mode involves each agent in the system reaching its target state or achieving the desired performance within a finite number of steps or time, thereby meeting engineering requirements effectively. Finite-time consensus can significantly enhance the responsiveness and coordination of multi-agent systems within a specific time frame, ensuring that the system can quickly adapt to dynamic environments or unforeseen events. This is crucial for many applications, such as drone formations and autonomous driving. Due to its higher level of convergence accuracy, speed, and robust stability, finite-time convergence has attracted significant interest [13; 14; 15]. For instance, researchers explored finite-time bipartite consensus for multi-agent systems under random switching topologies [16]. Sliding-mode control was utilized to attain finite-time tracking for multi-agent systems encountering false data injection attacks [17]. However, in practical applications, multi-agent systems frequently face challenges in autonomously reaching consensus states. Therefore, exerting control becomes essential to ensure the attainment of consensus. Numerous control strategies have been developed in existing research to enable multi-agent systems to achieve consensus. These strategies include adaptive control [10; 18], optimal control [19; 20], containment control [21], observer-based control [22], among others. It is important to recognize that not all agents need to directly participate in the control process, as certain agents may have minimal influence on the overall system dynamics. Pinning control methods involve selecting specific agents for control to reduce data redundancy and enhance system efficiency. In the study by [23], achieving group consensus only required at least one agent in the system to be pinned. [PERSON] proposed an edge-pinning-based synchronous update strategy that adjusted partial coupling weights between neighboring agents instead of all, offering effective reductions in estimation complexity and resource usage [24]. Event-triggered control is another advantageous approach for maximizing the use of network resources.