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The stratospheric satellite is regarded as an ideal stratosphere flight platform and is able to accomplish various missions such as surveillance, earth observation, and remote sensing, which requires a robust and effective trajectory tracking control method to support these tasks. A novel observer-based robust finite-time control scheme is proposed to address the trajectory tracking control problem dedicated to a stratospheric satellite in the presence of external disturbance and actuator saturation. Firstly, an extended state observer (ESO) is adopted to observe the unavailable velocity states and unknown disturbances simultaneously, and the estimated data are utilized in the robust control law design. Then, an auxiliary system based on anti-windup compensator is developed to directly compensate for the actuator saturation difference. After that, a backstepping nonsingular fast terminal sliding mode control (BNFTSMC) strategy is designed to track the desired trajectory with high accuracy, fast convergence rate, and finite-time convergence. Then, a stability analysis using Lyapunov-based theory is performed, in which the stabilization of the stratospheric satellite system and finite-time convergence are proven. Furthermore, a number of simulations are conducted further to verify the excellent performance of the designed control strategy.