# Discrete-Time Position Control for Autonomous Underwater Vehicle under Noisy Conditions
## 1 Introduction  
Autonomous underwater vehicles (AUVs) can perform a variety of deep-sea tasks [1]. These tasks broadly involve marine engineering, military, and marine science fields [2, 3]. Specifically, tasks such as the inspection of underwater pipelines, enemy target tracking, and marine environmental data collection can be performed by controlling AUVs [4, 5]. However, the acquired data quality is highly dependent on the AUV position accuracy [6]. Thus, improving the positional precision of an AUV has recently become a hot research topic. There are two main challenges in achieving an accurate discrete-time position control of an AUV. The first one involves the uncertain hydrodynamic parameters, such as added mass and drag coefficients, and the second, the real-world processing and measurement noise. Due to the uncertainty of system dynamics, common noise filtering algorithms are usually ineffective, and thus the two aforementioned interacting factors impose a low positional accuracy, affecting the precision of AUV control. Over the past years, the literature has offered quite a few control methods, addressing the hydrodynamic uncertainties of an AUV. The existing AUV control methods can be divided into two major classes: adaptive control and robust control techniques. Adaptive control techniques can automatically compensate for unexpected changes related to the order of the model, parameters, and input signals. It is worth noting that adaptive control compensates, even in the absence of the confines of uncertainties. Nevertheless, the shortcoming of adaptive control is its high energy consumption, as the uncertain hydrodynamicparameters to be adapted impose great computational complexity. Given the limited energy resources of an AUV, this deficiency poses a disadvantage for AUV applications. Despite this, adaptive control is still used with [7] a focuse on proportional integral derivative (PID) controllers for AUV. In this work, the PID control algorithm, despite being classic, is proven to be quite effective for AUV control. Based on a set of saturation functions for trajectory tracking of AUV, [PERSON] et al. [8] propose a nonlinear robust PID controller to cope with system uncertainty. In [9], the authors introduce a conventional back-stepping method that applies to AUV platforms for diving control. Sliding mode controllers (SMCs) [10; 11] are designed for AUV control in the presence of uncertain hydrodynamics, with their advantage being their implementation simplicity. Based on a sliding mode control, [PERSON] et al. [12] present a robust adaptive SMC. To track the desired path, [PERSON] et al. [13] propose a robust fuzzy controller for AUV that considers a massive uncertainty of the dynamics. Ref. [14] describes a novel adaptive autopilot, based on \(\mathcal{L}_{1}\) adaptive control theory, that is robust to the system uncertainties. To achieve trajectory tracking control, [PERSON] and [PERSON] [15] propose a robust adaptive tracking controller for AUV. Compared with adaptive control techniques, robust control techniques require fewer computing resources. However, this type of control demands the knowledge of some of the hydrodynamic parameters, which are usually uncertain. To solve this parameter uncertainty problem, [PERSON] et al. successively design a continuous time-delay control law [16] and a discrete time-delay control law [17]. Ref. [18] presents an enhanced time-delay controller (TDC) for the position control of AUV. The enhanced TDC is computationally simple and robust to unmodeled dynamics and disturbances. These time-delay controllers (TDCs) are designed by employing the time-delay estimator (TDE) that is capable of estimating uncertain hydrodynamic parameters, using time-delayed state derivatives and control inputs [19; 20]. The TDCs have two additional advantages, namely, saving computing resources, and having no requirements of prior information about uncertain parameters.