*Article* **Distributed Dual Closed-Loop Model Predictive Formation Control for Collision-Free Multi-AUV System Subject to Compound Disturbances**

**Mingyao Zhang \*, Zheping Yan, Jiajia Zhou and Lidong Yue**

College of Intelligent Systems Science and Engineering, Harbin Engineering University, Harbin 150001, China; yanzheping@hrbeu.edu.cn (Z.Y.); zhoujiajia@hrbeu.edu.cn (J.Z.); yuelidong@hrbeu.edu.cn (L.Y.) **\*** Correspondence: zhangmingyao@hrbeu.edu.cn

**Abstract:** This paper focuses on the collision-free formation tracking of autonomous underwater vehicles (AUVs) with compound disturbances in complex ocean environments. We propose a novel finite-time extended state observer (FTESO)-based distributed dual closed-loop model predictive control scheme. Initially, a fast FTESO is designed to accurately estimate both model uncertainties and external disturbances for each subsystem. Subsequently, the outer-loop and inner-loop formation controllers are developed by integrating disturbance compensation with distributed model predictive control (DMPC) theory. With full consideration of the input and state constraints, we resolve the local information-based DMPC optimization problem to obtain the control inputs for each AUV, thereby preventing actuator saturation and collisions among AUVs. Moreover, to mitigate the increased computation caused by the control structure, the Laguerre orthogonal function is applied to alleviate the computational burden in time intervals. We also demonstrate the stability of the closed-loop system by applying the terminal state constraint. Finally, based on a connected directed topology, comparative simulations are performed under various control schemes to verify the robustness and superior performance of the proposed scheme.

**Keywords:** multi-AUV system; formation tracking; finite-time extended state observer; distributed model predictive control; Laguerre function
