Autonomous Underwater Vehicles (AUVs) play an essential role in modern ocean exploration, and their speed control systems are fundamental to their efficient operation. Like many other robotic systems, AUVs exhibit multivariable nonlinear dynamics and face various constraints, including state limitations, input constraints, and constraints on the increment input, making controller design challenging and requiring significant effort and time. This paper addresses these challenges by employing a data-driven Koopman operator theory combined with Model Predictive Control (MPC), which takes into account the aforementioned constraints. The proposed approach not only ensures the performance of the AUV under state and input limitations but also considers the variation in incremental input to prevent rapid and potentially damaging changes to the vehicle's operation. Additionally, we develop a platform based on ROS2 and Gazebo to validate the effectiveness of the proposed algorithms, providing new control strategies for underwater vehicles against the complex and dynamic nature of underwater environments.