# Discrete-Time Position Control for Autonomous Underwater Vehicle under Noisy Conditions
## 1 Introduction
However, it should be mentioned that all aforementioned controllers [7; 8; 9; 10; 11; 12; 13; 14; 15; 16; 17; 18] do not consider any process and measurement noises. Given that the Global Positioning System (GPS) navigation system does not operate underwater, AUVs estimate their position utilizing acoustic and optical devices that are usually considered to be on-board navigation sensors [21; 22]. Both these sensor types feed the positional AUV controller with velocity information from which the traveled distance is derived. However, especially in the deep-sea, sensor measurements are inevitably affected by noise, influencing the quality of the measurement data and ultimately limiting the precision of the AUV controller. Moreover, the mathematical model of AUV imposes some process noise. Thus, the process and measurement noises incommode the design of a motion AUV controller. To deal with these noises, several estimation approaches have been made. [PERSON] et al. [23] present a Fuzzy Kalman Filter (FKF) to estimate the AUV position, while [24] propose an Ensemble Kalman Filter (EnKF). Compared with FKF, EnKF manages a more accurate position estimation, and additionally, the computational complexity of EnKF is less than that of FKF, saving energy for the AUV platforms evaluated here. [PERSON] et al. [25] introduce a comprehensive scheme that combines a Kalman filter-type navigation for both the cross-track and heading control of AUV. However, current estimation methods become invalid if any uncertain hydrodynamic parameters exist. [PERSON] et al. [26] utilize an advanced control strategy, i.e., the model predictive control (MPC), to improve the trajectory tracking stability of an AUV under external disturbances and noise. [PERSON] et al. [27] propose a high aiming precision method for the unmanned combat air vehicle (UCAV) under the MPC framework. These MPC algorithms can bring a satisfactory control precision. However, exploiting MPC imposes a sizable computational complexity and energy consumption during the mission. In [28], the deterministic artificial intelligence (D.A.I.) is used to eliminate the motion trajectory tracking error for unmanned underwater vehicles in continuous time. However, the performance of D.A.I. is severely degraded by the discretization implementation [29]. In practice, AUV is commonly driven by a propulsion system and controlled by an on-board computer. Given that the computer only processes discrete digital signals, the goal is to design a discrete-time controller. Hence, in this work, the discrete-time position control for AUV under noisy conditions is studied and a discrete-time control law for trajectory tracking is proposed. The contributions of this paper are as follows:  
1. A control law, which is a modified algorithm based on the EnKF that first introduces the TDE into the EnKF. 2. An AUV positional control method that solves the problem of uncertainty in the hydrodynamics parameters, where the traditional EnKF underperforms. 3. Since the proposed solution utilizes the TDE algorithm, it does not need any information about the bounds of uncertain terms of the system hydrodynamics, except the rigid body inertia matrix parameter. 4. Several Matlab/Simulink simulations are performed and the result demonstrates that the proposed discrete-time control law attains better position control precision, compared to the existing control laws, for trajectory tracking. The scenarios include various noise conditions, where the precision of the proposed control law in trajectory tracking is verified and the presented technique is also tested on several sampling times or ensembles. Finally, under the same noise conditions, the performance of the proposed discrete-time control law is tested against that of the conventional linear controller, TDC, and MPC controller. The rest of this paper is organized into four parts as follows. Section 2 introduces the AUV model architecture, while in Section 3 a stability analysis of the proposed discrete-time controller is presented.