## 1 Introduction  
The improvement of marine resources exploitation technology has become important for the development of the marine industry. The design and development of new types of marine engineering equipment to enhance the technological level of the ocean engineering industry is of considerable significance. Because of the complexity of the marine environment, underwater vehicles have become an important and valuable tool for the exploration and development of marine resources. Unmanned underwater vehicles, including remotely operated vehicles (ROVs), autonomous underwater vehicles (AUVs), and underwater gliders, have been extensively and increasingly used in marine environments for exploration, inspection, and engineering operations [1]. They can provide safe and effective access to deep sea and hadal environments without physically entering them. With advantages like functional diversity, strong operational ability, and a high safety factor, ROVs have been widely used and have become important technical equipment in areas such as ocean resource development, exploration, marine scientific research, and underwater engineering [2]. To guarantee the operation quality, high precision, and safety of ROVs, a high-performance controller for the trajectory tracking or station-keeping of ROVs is required. In practice, there are a number of technical challenges in the control of an underwater vehicle, such as model uncertainties and unknown external disturbances [3; 4; 5; 6; 7]. The model uncertainties of ROVs are usually caused by inaccurate hydrodynamic coefficients, which are obtained using towing tank experiments or computational fluid dynamics methods. The unknown disturbances in practical oceanic environments often include currents, waves, and tides. For the control design of ROVs, the external disturbing force caused by the cable that connects with the support ship should also be considered. The currentsand the umbilical cable simultaneously affect the ROV motions as the umbilical cable is connected to the ROV. In many works on the control of underwater dynamic systems, the state vector is assumed to be measured. The quality of feedback signals from the ROV sensor system plays an important role in the control performance of the vehicle as the signals affected by noises can debate the control quality and even lead to system instability [8]. However, in practice, it is difficult or impossible to measure all of the state variables of an ROV system with sensors for technical or economic reasons. For example, cable disturbing force cannot be easily obtained with normal sensors, and there is not enough money to buy high-precision but expensive sensors. Hence, effective state observers for ROVs need to be developed to provide accurate and robust signals. Moreover, it is necessary and is of considerable significance to design an observer to fully know the state variables of the system. The state observers for vehicles have been studied by a number of researchers in the past [9; 10; 11; 12]. The use of a complementary observer is one of the most popular techniques for sensor combination. The algorithm allows for straightforward implementation without requiring high computational resources and is suitable for small and low-cost autonomous vehicles with limited onboard power. The linear observer has been solved by [PERSON] and [PERSON], but the nonlinear case is still an active domain of research. [PERSON] et al. developed the high-gain observer approach, which is closely related to the triangular structure and is derived from the uniform observability of nonlinear systems [13]. [PERSON] et al. proposed a higher-order sliding-mode observer to estimate precisely the observable states and asymptotically the unobservable ones in a multi-input-multi-output nonlinear system with unknown inputs and stable internal dynamics [14].