## 1 Introduction
Expanding the research on the leader-follower consensus, one team [10] evaluated the issues of the immeasurable states of intelligent agents and the inherent time delays in the signal-transmission process. They subsequently created an unknown input observer to estimate time delays and established a consensus control protocol for multi-agent systems according to the observer's estimated values. Moreover, one team [11] examined a novel optimal control method that can achieve the consensus of multiple AUVs and avoid obstacles with minimal control effort. Another team [12] focused on collision-free formation-tracking AUVs under the influence of compound disturbances in intricate ocean environments. To that end, they proposed a novel finite-time extended state observer (FTESO)-based distributed dual closed-loop model-predictive control scheme. In [13], the authors designed a control protocol based on a distributed robust model-predictive control to better coordinate cooperation between AUVs. In [14], the researchers established a robust model-predictive control scheme based on the active disturbance-rejection control strategy for the Untracking task. Of course, extending the use of these systems to multi-AUV systems can be challenging, as it involves orchestrating the control behavior of each subsystem and making sure that the local MPC (Model Predictive Control) optimization problem maintains its closed-loop stability under system constraints. In [15], the authors presented an unprecedented solution to the issue posed by the need to control the nonplanar multi-AUV formation reconfiguration. The proposed method utilizes the graph rigidity and affine transformation (GR-AT) approach, which can tackle this complex problem in a unique manner. In [16], the researchers established the distance rigid graph-based formation-control strategy for vehicles modeled with Euler-Lagrange-like (E-L) equations. The approach differs from linear systems due to the nonholonomic constraint in the former method. Furthermore, researchers [17] suggested a time-varying strategy based on optimization for networked uncertain Euler-Lagrange mobile agents. This strategy adaptively and flexibly considers objective functions that can change in real time, enabling its versatile application. Others [18] investigated the distance-based formation-control problem for multi-agent systems in arbitrary dimensional space. The agents are treated as point masses with dynamics governed by single and double integrators. In addressing rigid graph-based formation acquisition control problems, one team [19]focused on a class of double-integrator multiagent systems operating in two independent layers. This research illuminated how to acquire formations efficiently within such systems. Also, the work presented in [20] studied the formation coordination control of discrete-time distributed leaderless multi-AUV systems with two independent position-velocity communication topology and control inputs on a nonconvex set. Its focus was how to effectively coordinate multiple AUVs without relying on centralized leadership or global information exchange. In [21], the authors addressed the formation-control problem to deploy a group of AUVs by considering vehicle nonholonomic constraints and obstacle avoidance in the environment. In [22], the authors suggested a fixed-time-based method involving a multi-AUV leader-follower formation-control, achieving the finite-time convergence of formation-tracking error. In [23], the researchers created a two-loop controller to regulate multi-AUV formation, as well as a neural-network based controller and a conditional integrator to tackle model uncertainty and disturbances. In [24], the researchers proposed a new distributed control protocol for solving the guaranteed cost formation tracking control of multi-AUVs with switching topology communication. The protocol realizes formation tracking control of a multi-AUV system, and considers the characteristics of switching topology communication between individuals.