## 2 Problem Statement
Apart from this, \(F=\{f_{1},\ldots,f_{n}\}^{T}\) is regarded as the actuator bias fault. The relations between fault parameters \(e_{i},f_{i}\), and the actual actuator faults are summarized in Table 1.  
Substituting Equation (10) into Equation (7), the faulty dynamics equation can be described by  
\[ \begin{array}{l c c}\dot{\omega}=-(J+J_{VS})^{-1}\Lambda(J+J_{VS})\omega+(J+ J_{VS})^{-1}DE\tau(t)\\ +(J+J_{VS})^{-1}DF+\rho(t)\end{array}  \tag{11}\]  
In this paper, it was assumed that all of the state variables are measurable. To achieve the objective of this paper, the following Assumptions and Lemma are introduced, which are used in the controller design and the closed-loop stability analysis. **Assumption 1**. : _The desired output trajectory \(\sigma\) and its first-time derivative are continuous and bounded; moreover, there exists a known compact set \(\Pi\) such that \(\big{[}\sigma,\dot{\sigma}\big{]}\in\Pi\)._  
**Assumption 2**. : _The bias fault introduced in the fault model (10) satisfies_  
\[\|F\|\leq f_{0} \tag{12}\]  
_where \(f_{0}\) is a positive constant._  
**Assumption 3**. : _The nonlinear functions \(\Lambda J\omega\) and \(\Lambda J_{VS}\omega\) in the dynamics Equation (11) are locally Lipschitz bounded with respect to a Lipschitz constant \(\xi\), namely_  
\[\|\Lambda(J+J_{VS})\omega-\hat{\Lambda}(J+J_{VS})\dot{\omega}\|\leq\xi\|\dot {\omega}-\omega\|=\xi\|\overline{\omega}\| \tag{13}\] \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Fault Type** & **Loss of Effectiveness Fault** & **Bias Fault** & **Complete Failure** \\ \hline \(e_{i}\) & \(0<e_{i}<l\) & \(l\) & \(0\) \\ \(f_{i}\) & \(0\) & \(f_{i}\
eq 0\) & \(0\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: The reaction wheel fault types of spacecraft. **Lemma 1** ([7]). : _The extended Lyapunov description of finite-time stability with a finite-time convergence is given as_  
\[\dot{V}(x)+\lambda_{1}V(x)+\lambda_{2}V^{\prime}(x)\leq 0,\ \forall t\geq t_{0},\ V (x_{t_{0}})\geq 0 \tag{14}\]  
_Then, for any given \(t_{0}\), the convergence time is given as_  
\[T_{r}\leq t_{0}+\frac{1}{\lambda_{1}(1-r)}\ln\frac{\lambda_{1}V^{1-r}(x_{t_{0}} )+\lambda_{2}}{\lambda_{2}} \tag{15}\]  
_where \(\lambda_{1}\), \(\lambda_{2}\) > 0, and 0 < r< 1._  
**Remark 1**. : _The \(n\) actuators (\(n\) > 3) may suffer from LOE fault or complete failure, but the number of totally failed actuators is no more than \(n-3\), such that \(\mathrm{DE}^{3}\mathrm{D}^{T}\) is a positive definite matrix, which can be viewed as the requirement of controllability of the plant and the existence of a fault-tolerant controller for compensating the effects of actuator faults._  
**Remark 2**. : _Because of increased friction between stator and rotor, inadequate lubrication, marginal failures in bearings, and decreased motor torque, the rate of change of the wheel speed might be less than the nominal value. Consequently, the generated reaction wheel output torque can be less than the torque commanded, and therefore, it is modeled as partial LOE fault. Because of changes in coulomb friction and viscous friction of the bearings caused by aging and lubrication, the wheel may not be capable of holding its speed, which may decelerate the wheel gradually. Therefore, a low-bias torque is produced even when the commanded attitude control torque is zero, and it is modeled as bias fault. If malfunction occurs in the drive motor and power supply, the reaction wheel may decelerate slowly or hold its speed without response to control signals, and it is modeled as complete failure fault._