## 1 Introduction  
An autonomous underwater vehicle (AUV) is an important piece of equipment in marine scientific research, resource development and marine military forces [1]. The precise motion control of AUVs is helpful to complete the tasks of reconnaissance, underwater measurement and target tracking, and it has received considerable attention from researchers and engineers. Since the 1990s, many advanced control technologies have been adopted for this issue, such as sliding mode control [2], adaptive control [3], neural network control [4], fuzzy-PID [5], predictive formation tracking control [6], etc. The motions of an AUV have six degrees of freedom (DOFs) including surge, sway, heave, roll, pitch and yaw motions (see Figure 1). Among them, surge, sway and heave refer to the longitudinal, sideways and vertical displacements; yaw, roll and pitch describe the rotations around the vertical, longitudinal and transverse axes. The surge motion control of AUVs mainly includes velocity and position tracking control, which plays an important role in many tasks, such as automatic docking, position keeping, precise exploration, reconnaissance and data acquisition [7; 8; 9]. When an AUV is in motion, due to the complex operating environment, current fluctuation and other uncertainties and disturbances, it is difficult to obtain an accurate dynamic model. In [7], [PERSON] et al. developed a method for identifying the decoupled surge motion dynamic parameters of a small AUV. In [10], [PERSON] proposed PI control and PID control methods to design the controller for the accurate position tracking of the underwater vehicle's surge motion. In [11], [PERSON] et al. developed an adaptive controller while ignoring the influence of current. In [12], [PERSON] design a sliding mode variable structure controller for AUV without strict stability analysis. In [13], [PERSON] studied the surge trajectory tracking problem of AUVs under the influence of current and presented an adaptive backstepping sliding mode control scheme. In [9], [PERSON] et al. proposed a sliding-PID method for surge motion control and roll motion control. In [14], [PERSON] et al. proposed an improved adaptive backstepping control algorithm to decrease the too large initial control input. In [15], [PERSON] et al. studied the backstepping control to solve the surge position tracking control problem of AUVs with input limitation. These research results promote the development of AUV control technology. The aforementioned control algorithms can work well while all system states can be accurately measurable. An AUV may be equipped with a compass module to measure its attitude, and angular velocities can be measured by an angular rate sensor. By using the typical positioning systems (including acoustic ultrashort baseline system, GPS, inertial navigation system, etc. ), it is not difficult to obtain the positions of AUVs with very good accuracy. Usually, their translational velocities can be obtained by the Doppler velocity log (DVL). However, in some cases, due to economic and other practical limitations, AUVs may not be equipped with DVLs. Even when an AUV is equipped with a DVL, the DVL can only generate accurate velocity measurements provided that the distance to the seafloor is within a certain boundary, and the measured translational velocities by DVL may not be adequately accurate because of measurement noises [16]. During motion control design without velocity measurement or without accurate velocity measurement, it is necessary to obtain the accurate estimation of velocity for implementing an output feedback controller design. So far, there are mainly two technical means to estimate the velocity signal. A popular approach to velocity estimation entails the use of state observers [17]. Another approach is to use sliding mode differentiators [18]. For uncertain nonlinear systems, the state observer design is very complicated [19]. Even for the ideal situation, high-gain observers can be quite sensitive to measurement noises [18].