## 1 Introduction  
The autonomous underwater vehicle (AUV) is a highly sophisticated, intricate system that integrates various cutting-edge technologies, such as vehicle control, anomaly detection, information decision-making, and deep-sea navigation. This convergence of advanced technologies enables the AUV to operate efficiently in challenging underwater environments. One significant advantage of the AUV is its ability to cover a larger search area. By utilizing its autonomous capabilities, the AUV can navigate through vast expanses of the ocean floor with precision and speed. This expanded coverage allows for more comprehensive data collection and exploration of previously uncharted territories. Moreover, by employing multiple AUVs in coordinated missions, the fault tolerance of the entire system can be significantly improved. When one AUV encounters technical issues or malfunctions during an operation, other vehicles within the multiple-AUV (multi-AUV) fleet can seamlessly take over its tasks without disrupting overall mission objectives. This redundancy ensures continuous data acquisition and minimizes potential downtime due to equipment failures. Furthermore, advancements in communication systems have enhanced collaboration among multiple AUVs operating together. These vehicles can exchange real-time information regarding their respective positions, collected data, and environmental conditions. Such seamless communication facilitates efficient coordination between individual units within a multi-AUV team while optimizing resource allocation for maximum productivity. In formation control, the leader-follower method of multiple agents is the most common structure. In this method, each follower aircraft is controlled to maintain its velocity and position to a predefined leader aircraft along the trajectory. This structure rapidly reconfigures and extends to new objects, so it can be quickly copied by other multi-AUVs [1]. Therefore, the leader-follower method is widely employed in real-world engineering problems. However, the formation system is too reliant on the leader. One team [2] studied multi-agent systems with input delays and found that parameters were too wide and difficult to adjust in the real marine environment. Others investigated the formation-control problem of leader-follower-structured AUVs through state-prediction estimation in the scenario of an undependable underwater acoustic channel [3]. Ultimately, the team conducted a comparison of the suggested formation-control strategy with the traditional method, and they proved the effectiveness of the former. Targeting the control problem of AUV formation with delays and interruption in the communication, another team [4] signed a feedback linearization controller and the PD control method, and they transformed the strongly coupled nonlinear vehicle model into a second-order model. Another team [5] proposed a basic formation-control system that could minimize the modifications of standardization among an autonomous surface vehicle and AUVs, and they tested it in a sea environment. Furthermore, researchers examined the leader-follower coordination control problem that arises in a continuous-time multi-AUV formation containing two independent topologies and time-varying delays [6,7]. Using the gradient-descent strategy to approximate the communication delay, a time-lag estimator was designed [8]. They found that the time lag of the leader AUV state packets reduced the actual delay, and that the formulation of the estimator led to obstacle avoidance. Additionally, researchers studied the leader-follower consistency problem of a multi-intelligent body system with input delays [2]. Others [9] studied synchronous state information, proposed a leader-follower consensus control protocol with time-varying delays, and performed a controllability analysis using Lyapunov-Krasovskii functionals.