## 1 Introduction  
As a deeply coupled nonlinear aircraft, hypersonic vehicles have the advantages of fast speed, strong maneuverability and good penetration. It has important applications in both military and civil fields, attracting many scholars to study its modeling and controller design [1, 2]. The super high speed flight and corresponding complex flight airspace make HSV prone to failure, such as actuator fault, sensor fault, etc. Thus, fault-tolerant control (FTC) systems are central to ensuring HSV safety and achieving its goals. In a fault event on an aircraft, a significant portion of the aircraft's aerodynamic lift surface may become detached when it is damaged [3], resulting in an unexpected shift of the aircraft's center of mass. It is well known that the position of the center of gravity of an aircraft is crucial to the calculation of its control moment [4]. Due to changes in the system, inertia matrix, eccentricity torque and system uncertainty, any unexpected center of gravity movement may put the aircraft in a non-nominal flight state, thus challenging the existing control system to maintain its control performance [3]. A similar situation may arise from fuel consumption, moving mass control [5] as well as the liquid fuel sloshing [6], etc. To ensure the safe flight of the aircraft, adaptive FTC schemes have been developed even in the presence of unexpected centroid shifts, actuator faults and external disturbances. For variants of aircraft centroids, some previous works deliver the following. An aircraft with unexpected centroid shift is modeled in [7], which is derived from the aircraft body coordinate system, which is not convenient to analyze the influence of centroid migration on HSV flow Angle (Angle of attack and Angle of slip). In addition, a hybrid direct-indirect adaptive control method based on neural network is proposed to deal with the damage-induced centroid migration [7; 8]. However, irreversible phenomena may appear in the estimated inertia matrix, leading to singularities in the existing FTC controller resting on the inverse of an inertia matrix. In addition, a linearized aircraft model with centroid shift due to damage is proposed in [9]. Nevertheless, the linearization of the multi-model reference adaptive control makes the system free from the coupling problem caused by centroid migration, leading to a degradation of the control performance. Based on the above studies, the influence of the centroid shift on the HSV motion is mainly reflected in three aspects: (1) the change of inertia matrix of the system, (2) the uncertainty of the system and (3) the eccentric moment, especially in the situation where the inertia matrix appears irreversible. In order to handle the challenge of inertia matrix, RBFNN is adopted in [8] to estimate the inverse of the inertia matrix in order to ensure the normal operation of the controller. To solve this problem, Taylor expansion of the inverse matrix of the inertial matrix is used in [10], but it comes with corresponding Taylor expansion errors and a large number of online computational parameters of the neural network. In addition, Ref. [11] addresses the variation of an inertial matrix by assuming that the minimum and maximum bounds of its eigenvalues are known; however, the mentioned bounds are difficult to measure. With the development of data driven methods [12], a model-free control scheme is developed to handle this challenge, but the training data of the inertial matrix of HSV with centroid-shift is difficult to obtain. Additionally, an observer-based FTC strategy [13], fuzzy approximation [14] and iterative learning [15] are developed to handle this challenge. In summary, how to identify an HSV inertial matrix with uncertainty online and ensure its function in the FTC controller remains elusive. In the past few decades, the fault-tolerant control of actuators, sensors, control surfaces or other failure systems has been extensively studied, and a large number of FTC control strategies have appeared in the literature.