## 1 Introduction
However, in all of the above methods, the observation error converges to zero in an asymptotically convergent manner, which only guarantees the convergence of the observation error at infinite time. With the further development of stabilization control theory, the theory of \"finite time control\" has gradually emerged. [PERSON] et al. [25] designed a finite-time sliding mode observer to estimate the disturbance. Inspired by the multivariate super-twisting algorithm, [PERSON] et al. [26] designed a finite-time disturbance observer (FDO) to estimate the system disturbance, which improves the robustness of the controller. [PERSON] et al. [27] also took actuator faults as disturbance into account and proposed an FDO to achieve an accurate estimation of disturbance and actuator faults. In the above studies, the output of the observer is prone to a chattering phenomenon due to the presence of sign function terms in the observer. To address this problem, studies [28; 29; 30; 31] investigated an FDO based on the homogeneous theory. [PERSON] et al. [31] designed a continuous FDO based on the homogeneous theory and the saturation function method to estimate the disturbance, but when the sampling time becomes longer, this FDO could not achieve a stable output of the observed signals. In addition, the saturation constraint of the actuator is also a non-negligible problem. In practical systems, the output signal of the actuator frequently experiences amplitude saturation, leading to a situation where the controller struggles to promptly regulate the system state. Concerning the saturation issue, anti-saturation control methods are increasingly categorized into two main groups: active anti-saturation methods [32; 33; 34; 35] and passive anti-saturation methods [36; 37; 38; 39]. [PERSON] et al. [36] proposed a nonlinear controller based on nested saturation for the rotational speed saturation problem of the quadrotor UAV. Based on this, [PERSON] et al. [37] considered a more complex quadrotor UAV model, and the anti-saturation auxiliary system prevents the saturation phenomenon by adaptively adjusting the system error. Different from the study [33], [PERSON] et al. [38] introduced the output of the anti-saturation auxiliary system into the control volume, which also achieved effective control. [PERSON] et al. [39] designed an anti-saturation auxiliary system with non-singularity for the saturation problem in spacecraft attitude control, which achieved the fast convergence of the auxiliary system state. However, there are fewer studies on finite-time anti-saturation control. Inspired by the above work, a finite-time disturbance observer and an anti-saturation auxiliary control method (FDOACM) are proposed in this paper with the following main contributions:  
* The assumptions based on small-angle flight are eliminated, so that the rate of change of the Euler angle and the angular velocity of the airframe satisfy the nonlinear relationship in an actual flight, and the total disturbance of the system and the actuator saturation are taken into account. The established model better reflects real flight conditions. * For the saturation problem of the actuator, an anti-saturation auxiliary control system is designed. In the presence of saturation, this anti-saturation auxiliary control system promptly guides the controller's output moment out of the saturation range by correcting attitude feedback errors. This ensures finite-time convergence of attitude errors in anti-saturation scenarios. * Targeting the overall disturbance experienced during flight, a disturbance observer rooted in finite-time stability theory is crafted to precisely estimate disturbances. IN contrast with the existing literature-based observers [31; 40], the observer proposed in this study not only achieves finite-time convergence of observation errors but also guarantees the continuity of observer output results.