## 1 Introduction  
In recent years, with the increasing attention paid to the development of marine resources, more and more dynamic positioning ships have been applied in engineering fields such as deep-sea exploration, submarine pipeline construction, and offshore oil drilling [1]. Ocean engineering ships with dynamic positioning systems can maintain a fixed position on the sea surface or track a predetermined motion trajectory in a particular posture under external environmental disturbances such as wind and waves through precise control of the ship's thrusters [2, 3]. Scholars have shown growing interest in the dynamic positioning control of surface boats. Various control algorithms, including fuzzy control, neural network control, backstepping, and hybrid control, have been presented and have shown effective control outcomes [4, 5, 6]. Despite significant research advancements in ship dynamic positioning, both locally and internationally, the study of fault-tolerant control in ship dynamic positioning still needs to be explored. During actual operations, ships' thrusters endure prolonged exposure to severe maritime conditions, leading to inevitable failures. The working nature of dynamic positioning ships makes it difficult to promptly repair or replace faulty thrusters with backup thrusters [7]. Therefore, to improve the system's reliability and ensure the safety of operations, it is necessary to design an effective fault-tolerant control scheme for dynamic positioning ships to prevent position loss even in thruster faults [8]. To achieve better tracking performance, many researchers have combined sliding mode control with different control techniques, such as neural networks, adaptive control, etc., to achieve superior control performance. Aiming at the dynamic safety constraint problem of a dynamic positioning ship under model uncertainty and unknown environmental disturbance, [PERSON] et al. [9] proposed a robust tracking control framework based on a unified barrier function and backstepping control. They designed an observer to observe the total disturbance so that the state of the dynamic positioning ship is always kept within the safety constraint. [PERSON] et al. [10] designed a variable-gain preset performance control scheme to improve the system's transient and steady-state responses. Using the variable-gain technology, the positioning error dynamically adjusts the control gain, and all the system signals are ultimately uniformly bounded to ensure the preset performance. [PERSON] et al. [11] presented a robust adaptive stabilization controller based on observers for the dynamic positioning of ships with time delays. They analyzed the stability of the proposed controller using Lyapunov stability theory and verified the method's effectiveness through simulation experiments. Considering the dynamic positioning problem of fully actuated ships with time-varying disturbances, [PERSON] et al. [12] developed a disturbance observer to observe unknown disturbances and designed a robust adaptive backstepping controller combined with the backstepping controller. The simulation results showed that the proposed control scheme makes all system signals uniformly bounded. However, the control schemes mentioned in the above literature can only achieve asymptotic stability of the system, which cannot guarantee the convergence of the system state within a finite time. [PERSON] et al. [13] proposed a finite-time controller with a fast exponential reaching law for ship dynamic systems subject to environmental disturbances and actuator constraints to ensure that the positioning error converges to zero quickly. To overcome the influence of sampling period variation on the control performance of the system, [PERSON] et al. [14] designed a finite-time discrete sliding mode controller, which can make the tracking error signal of the closed-loop system converge to a small area near zero within a finite time. Considering the actuator failure of the ship, [PERSON] et al.