## 1 Introduction  
With the growing demand for unmanned underwater operations, autonomous underwater vehicles (AUVs) have been widely utilized in fields such as underwater exploration, environmental monitoring, energy extraction, and marine research [1, 2, 3]. However, the limitations of traditional single-body AUVs have become increasingly evident in deep-sea or complex environments, highlighting the urgent need for multi-body collaborative systems with greater payload capacity and higher adaptability. As an innovative platform, the Multiple AUVs in Tandem Connection (MATC) system utilizes a modular design and distributed task allocation, enabling the system to flexibly adjust the number of units and their functional distribution based on mission requirements [4]. This approach offers greater flexibility and scalability compared to traditional AUVs, making it well-suited for challenging underwater operations. The MATC system comprises multiple interconnected AUV units, each independently capable of carrying sensors, payloads, or other equipment required for specific tasks, thereby enhancing flexibility and payload capacity. Additionally, tandem underwater vehicles offer significant advantages in terms of system fault tolerance [5], maneuverability [6; 7], as well as navigation and positioning [8]. Compared to traditional single-body AUVs, the MATC system demonstrates superior task adaptability, allowing for expansion and adjustments to suit various application scenarios. For example, in underwater exploration or environmental monitoring missions, the arrangement and operational modes of the AUVs can be customized to meet specific mission requirements and environmental conditions. Due to the interconnected nature of multiple AUV units, the MATC system exhibits the characteristics of a distributed thrust system in its design [9]. The propellers of each AUV unit, typically located on both sides, work together to propel the entire system forward. Furthermore, the MATC system often incorporates flexible components, such as extendable springs, to minimize rigid collisions between units. Accurately controlling the relative displacement between units and ensuring stable straight-line motion of the entire system are critical for the successful execution of MATC system missions. However, due to the distributed nature of the system's propulsion, the relative motion and coupled dynamic effects between units make both overall speed control and inter-unit distance management particularly complex. The relative displacement between units not only impacts the system's stability but also increases the risk of collisions and deviations from the planned trajectory, especially in high-speed or complex underwater environments. Ensuring steady straight-line motion thus becomes a significant challenge. Moreover, as the number of units' increases, conventional thrust allocation methods lack the real-time efficiency needed for large-scale systems, highlighting the necessity of improving the efficiency of online thrust allocation methods. While advancements in thrust allocation methods for distributed propulsion platforms, such as land trains, have been well-documented, to the best of our knowledge, no existing literature directly addresses the specific challenges of thrust allocation in multi-AUV systems with varying unit numbers and dynamic constraints. Land trains provide a representative example of distributed propulsion platforms. With advancements in control theory, thrust allocation methods for land trains have evolved over time. These methods primarily include optimal control [10], fuzzy control [11], adaptive control [12], and data-driven control [13]. For instance, [PERSON] et al. developed a cooperative model predictive control (CMPC) strategy incorporating a multi-objective rolling optimization scheme [14]. Key optimization objectives included maximizing overall line capacity, minimizing energy consumption, and enhancing ride comfort for train platoons operating within the railway section.