## 2 Problem Statement  
In this section, the Euler-angles moment equations are adopted to describe the attitude systems of spacecraft, and the equations of motion in terms of kinematics are given by [7]:  
\[\left\{ \begin{array}{l}\dot{\phi}=\omega_{1}-\tan\theta(\omega_{2}\cos\phi- \omega_{3}\sin\phi)\\ \dot{\theta}=\omega_{2}\sin\phi+\omega_{3}\cos\phi\\ \dot{\psi}=(\omega_{2}\cos\phi-\omega_{3}\sin\phi)/\cos\theta\end{array} \right. \tag{1}\]  
where \(\phi,\theta\), and \(\psi\) are roll angle, pitch angle, and yaw angle, respectively. \(\omega_{1}\), \(\omega_{2}\), and \(\omega_{3}\) denote the angular velocities with respect to the body-fixed frame. For simplicity, the kinematic Equation (1) can be rewritten in the following form  
\[\dot{\sigma}=G(\sigma)\omega \tag{2}\]  
where  
\[\sigma=[\phi,\ \theta,\ \psi]^{T},\ \omega=[\omega_{1},\ \omega_{2},\ \omega_{3}]^{T} \tag{3}\]  
and  
\[G(\sigma)=\frac{1}{\cos\theta}\left[ \begin{array}{ccc}\cos\theta&-\cos\phi \sin\theta&\sin\phi\sin\theta\\ 0&\sin\phi\cos\theta&\cos\phi\cos\theta\\ 0&\cos\phi&-\sin\phi\end{array} \right] \tag{4}\]  
According to reference [7], the dynamic equation of a flexible variable structure spacecraft is modeled as  
\[(J+J_{VS}+\Delta J)\dot{\omega}=-\Lambda(J+J_{VS}+\Delta J)\omega+Du(t)+d(t) \tag{5}\]  
where \(J\in\mathbb{R}^{3\times 3}\) is the symmetric inertia matrix, \(J_{VS}\in\mathbb{R}^{3\times 3}\) is the added value of inertia matrix generated by changing the mechanical structure of the spacecraft, that is, the variable structure parameter, \(\Delta J\in\mathbb{R}^{3\times 3}\) is the norm-bounded inertia uncertainty, \(u(t)\in\mathbb{R}^{n}\) is the control torque generated by \(n\) reaction wheels, \(D\in\mathbb{R}^{3\times n}\) is the reaction wheel distribution matrix, and \(n\) is the number of reaction wheels. \(d(t)\in\mathbb{R}^{3}\) is the bounded external disturbance, and \(\Lambda\) is a skew-symmetric matrix and has the following form  
\[\Lambda=\left[ \begin{array}{ccc}0&-\omega_{3}&\omega_{2}\\ \omega_{3}&0&-\omega_{1}\\ -\omega_{2}&\omega_{1}&0\end{array} \right] \tag{6}\]  
After some manipulations, Equation (5) can be transformed into the following form  
\[\dot{\omega}=-(J+J_{VS})^{-1}\Lambda(J+J_{VS})\omega+(J+J_{VS})^{-1}Du(t)+ \rho(t) \tag{7}\]  
where  
\[\rho(t)=-(J+J_{VS})^{-1}\Delta J\dot{\omega}-(J+J_{VS})^{-1}\Delta J\omega+(J +J_{VS})^{-1}d(t) \tag{8}\]represents the generalized perturbation caused by disturbance and inertia uncertainty, which is unknown and satisfies  
\[\|\rho(t)\|\leq\overline{p} \tag{9}\]  
All the environmental disturbances due to solar radiation pressure, gravitation, magnetic forces, and aerodynamic drag can be assumed to be bounded, and thus it is reasonable. The actuator considered in this study is the reaction wheel, which is usually composed of a flywheel and an electric motor. If the speed of the flywheel is changed, the flexible variable structure spacecraft starts to rotate in reverse by keeping the angular momentum. Several components of the reaction wheel may cause fault, such as electronics, motor, and power supply; the reaction wheels are susceptible to the following three types of faults: LOE, increased bias torque, and complete failure. It was noted that the three different types of actuator faults described above may happen in the attitude systems of a flexible variable structure spacecraft, and the actual control torques generated by the faulty actuators could be modeled as the following form  
\[u(t)=E\tau(t)+F \tag{10}\]  
where \(\tau(t)\in\mathbb{R}^{n}\) is the desired torque of the reaction wheel produced by the attitude controller. \(E=\text{diag}\{e_{1},\ldots,e_{n}\}\) is the effectiveness matrix of the reaction wheel and satisfies \(0\leq e_{i}\leq 1\) (\(i=1,2,\ldots,n\)). If \(e_{i}=1\), the \(i\)th actuator works normally, \(0<e_{i}<1\) indicates that the \(i\)th actuator produces a reduced effectiveness, and \(e_{i}=0\) implies that the \(i\)th actuator undergoes a complete failure.