**1. Introduction**

With the further exploration of the ocean, Autonomous Underwater Vehicles (AUVs) have started to play an important role in various marine activities, and AUVs are commonly used in tasks such as marine ecosystem detection, underwater inspection and surveillance, and subsea pipeline laying [1–3]. As the complexity of AUV missions increases, the operating environment of AUVs will become more and more complex. Due to constraints, such as the limited energy carried by them, AUVs start to look overwhelmed when facing some more demanding tasks. Therefore, multi-AUV collaboration, information sharing, and joint mission accomplishment have become the new direction of AUV development today. Multi-AUV collaboration can accomplish difficult tasks faster and better for single AUVs, especially in data acquisition [4], target search [5–7] and path planning [8,9], etc. Therefore, multi-AUV collaborative operation is the future development trend of AUVs to deal with complex problems in complex environments.

In the actual application, the multi-AUV formation will inevitably be affected by the actual environment, there will be a communication delay when multi-AUVs communicate with each other, and it takes some time to fuse and calculate the information of each sensor, so the real-time information sharing between multi-AUVs cannot be achieved in the actual application and the multi-AUV formation control will produce large control errors [10,11]. Therefore, the study of multi-AUV formation control under communication delay is helpful to apply the theory to practice and promote the development of multi-AUV formation technology.

**Citation:** Li, J.; Tian, Z.; Zhang, G.; Li, W. Multi-AUV Formation Predictive Control Based on CNN-LSTM under Communication Constraints. *J. Mar. Sci. Eng.* **2023**, *11*, 873. https:// doi.org/10.3390/jmse11040873

Academic Editor: Rafael Morales

Received: 27 March 2023 Revised: 17 April 2023 Accepted: 18 April 2023 Published: 20 April 2023

**Copyright:** © 2023 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/).

For the multi-AUV formation control problem, different authors have proposed different solutions. Kang [12] used fuzzy control theory to coordinate the behavior of multiple AUV members, and the fuzzy control scheme inputs for the leader AUV in a multi-AUV formation were the yaw angle during obstacle avoidance and the yaw angle during target finding maneuvers, and the fuzzy control scheme for the follower consisted of the yaw angle deviations during obstacle avoidance and formation keeping. Borhaug [13] proposed a time-varying smooth feedback control law for multiple non-complete AUVs to maintain formation. An integral backstepping method was used to cooperatively park the follower AUV in its desired docking position and orientation relative to the leader, and the above control law was applied to a real AUV formation system to investigate the implementation problem and singularity avoidance problem of the physical AUV system. Ding [14] proposed a multi-AUV 3D formation control and obstacle avoidance method based on backstepping control and a bio-inspired neural network model. The followers track the virtual AUVs, during which the backstepping control method is guided to achieve 3D underwater formation control. The formation of AUVs was transformed using a bio-neural network model in order to avoid obstacles and pass through the area of obstacles. For the problem of leader failure in multi-AUV leader-following formations, Juan [15] proposed a solution to the problem of leader failure in multi-AUV leader-following formations by using the Hungarian algorithm to reconstruct the failed formation with the lowest cost. The Hungarian algorithm was improved to solve the nonstandard assignment problem. To address the issue of increased leader communication pressure after formation reconstruction, an event trigger mechanism was applied to reduce unnecessary communication. The efficiency of the event trigger mechanism was improved by increasing the event trigger condition of the sampling error threshold. Zheping [10] considered the presence of bounded communication delay and non-convex control input constraints in multi-AUV formation under weak communication conditions. They proposed a formation consistency constrained controller algorithm for discrete-time leaderless multi-AUV systems with dual independent communication topologies by introducing a constraint operator. For the problem of hydroacoustic communication constraints between multiple AUVs, Yuepeng [16] proposed a consensus control algorithm for multi-AUVs combined with the leader-following method under communication time delay, using graph theory to describe the communication topology of multi-AUVs and introducing a hybrid communication topology to accommodate large formation control. The consensus theory was combined with the leader-following method to construct distributed control laws. Suryendu [17] designed a time-lag estimator based on the gradient descent method to estimate the communication delay, and the actual delay was significantly reduced because the time tagging of the leader AUV state packets was avoided in the formulation of the estimator. Shibin [18] investigated the leader-following consistency problem for a multi-intelligent body system with input delays. A distributed state observer was designed to estimate the states of neighbors using the output information between neighboring intelligences, and a consistency algorithm was proposed using the estimated state information. Sufficient conditions for stability were constructed using Lyapunov theory and solved by a set of linear matrix inequalities with iterative parameters.

Based on the above research results, this paper proposes a formation control method combining CNN-LSTM prediction and backstepping sliding mode control. The specific contributions of this paper are summarized as follows:

1. A multi-AUV formation control method combining CNN-LSTM prediction and backstepping sliding mode control is proposed, the stability of the control method is demonstrated, and the effectiveness of the control method is verified by simulation.

2. Combining the advantages of CNN feature extraction, filtering noise and LSTM temporal memory, a CNN-LSTM prediction model is built for predicting the state information of navigators.

3. Applying the feedback linearization method, the AUV nonlinear model is transformed into a second-order integral model, and the controller is designed by combining

the backstepping method and sliding mode control, which improves the robustness of the controller.

### **2. AUV Nonlinear Model Building and Feedback Linearization**
