## 1 Introduction  
In recent years, the consensus problem has gained significant attention within the field of multi-agent systems, owing to its extensive applications across various domains, including formation control [1; 2; 3], flocking behavior [4; 5], and collaborative decision-making processes [6; 7]. The primary objective in addressing the consensus problem is to develop an efficient control protocol that ensures all agents converge on a collectively agreed-upon value through iterative local information exchanges with their neighboring agents [8; 9; 10; 11; 12]. Generally, there are two approaches for multi-agent systems to achieve consensus: leader-following and leaderless consensus. The leader-following approach allows all followers to converge to the trajectory of a designated leader, thereby utilizing the leader's state as the target for control. This method facilitates the achievement of prescribed behaviors among all agents and is widely applied in practical scenarios. The exchange of information between agents occurs through wireless networks, which are inherently vulnerable to cyber-attacks. This vulnerability is particularly concerning in the context of information exchange, as cyber-attacks can manifest in various forms, including false data injection attacks [13; 14] and repay attacks [15; 16]. Notably, denial-of-service (DoS) attacks are a prevalent type of cyber-attack that can cause significant disruptions in information transmission across networks. Consequently, considerable research efforts have been directed towards establishing consensus in networked systems under the threat of DoS attacks, with the goal of maintaining operational integrity and ensuring communication reliability in these challenging environments [17; 18; 19]. To achieve consensus in multi-agent systems, various strategies such as containment control [20; 21], tracking control [22; 23; 24], and observer-based control mechanisms [25; 26]have been explored and implemented. Each of these approaches plays a distinct role in advancing the overarching goal of attaining systemic consensus within multi-agent environments. However, these control methods are predominantly continuous, which poses practical challenges in real-world applications. Consequently, addressing the constraints imposed by limited resources emerges as a critical issue that necessitates resolution. A promising strategy that offers a resource-efficient alternative is event-triggered control. This methodology predicates the updating and transmission of control signals based on specific events, defined by the violation of pre-established event-triggered thresholds. By employing a selective update mechanism, the frequency of control signal adjustments is inherently reduced, thereby optimizing resource utilization and significantly extending the operational lifespan of network components. In recent years, there has been a notable increase in scholarly investigations focused on the development of event-triggered control protocols, particularly regarding their applicability to systems characterized by both linear [27; 28; 29; 30] and nonlinear [31; 32] dynamics. These studies, tailored to various multi-agent systems, utilize system state errors to design the event-triggered function, demonstrating effective resource conservation. Pinning control is a technique employed in multi-agent systems to tackle the challenges associated with implementing controllers across all nodes, particularly in the context of resource constraints. This method involves applying local feedback to a carefully selected subset of nodes, thereby reducing the number of controlled nodes while still achieving the desired consensus dynamics. Extensive research has been dedicated to refining consensus protocols for multi-agent systems, building upon the foundational concept of pinning control. For example, ref. [33] investigated the consensus of switching multi-agent systems through the application of pinning control, while ref.