## 3 Actuator Fault Detection and Diagnosis Scheme
### Fault Diagnosis Scheme Design  
In this section, a sliding mode-based fault detection observer is designed for the dynamic Equation (7) in the actuator healthy case in order to detect the occurrence of an unknown fault, namely  
\[\hat{\omega}=A_{1}(\omega-\hat{\omega})-(J+J_{VS})^{-1}\hat{\Lambda}(J+J_{VS}) \hat{\omega}+(J+J_{VS})^{-1}D\tau+\overline{\rho}\text{sign}(\widetilde{\omega}) \tag{20}\]  
where \(A_{1}\) is a positive definite matrix given in advance. Define  
\[\widetilde{\omega}=\omega-\hat{\omega} \tag{21}\]  
the following error equation is obtained by subtracting Equation (20) from Equation (7). \[ \begin{array}{l}\dot{\widetilde{\omega}}=-A_{1}\widetilde{\omega}+(J+J_{VS} )^{-1}[\hat{\Lambda}(J+J_{VS})\hat{\omega}-\Lambda(J+J_{VS})\omega]\\ +\rho(t)-\overline{\rho}\text{sigh}(\widetilde{\omega})\end{array}  \tag{22}\]  
In this position, the first result of this study is given in the form of Theorem 1. **Theorem 1**. : _Considering the healthy dynamics Equation (7) of a variable structure spacecraft, a sliding mode-based fault detection observer is designed in Equation (20), for a given positive definite matrix \(A_{1}\), if exists a symmetric positive definite matrix \(P_{1}\), such that inequality (23) holds_  
\[L=[A_{1}-\xi(J+J_{VS})^{-1}]^{T}P_{1}+P_{1}[A_{1}-\xi(J+J_{VS})^{-1}]>0 \tag{23}\]  
_then, a fault detection threshold is designed as follows_  
\[J_{th}=(\frac{\lambda_{\text{max}}(P_{1})}{\lambda_{\text{min}}(P_{1})}e^{-( \lambda_{\text{min}}(L)/\lambda_{\text{max}}(P_{1}))(t-t_{0})}\|\widetilde{ \omega}(t_{0})\|^{2})^{1/2} \tag{24}\]  
_If the \(2\)-norm of the estimation error of \(\omega\) generated by fault detection observer (20) surpasses the detection threshold \(J_{th}\), then at least one actuator is suffering from an unknown fault._  
**Proof of Theorem 1. ** Considering the following Lyapunov candidate function  
\[V_{1}=\frac{1}{2}\widetilde{\omega}^{T}P_{1}\widetilde{\omega} \tag{25}\]  
Taking the derivative of \(V_{1}\) along the trajectory of Equation (22) yields  
\[ \begin{array}{l}\dot{V}_{1}\leq-\frac{1}{2}\widetilde{\omega}^{T}(P_{1}A_{1} +A_{1}^{T}P_{1})\widetilde{\omega}+\xi\widetilde{\omega}^{T}P_{1}(J+J_{VS})^ {-1}\widetilde{\omega}+\widetilde{\omega}^{T}P_{1}(\rho(t)\\ -\overline{\rho}\text{sign}(\widetilde{\omega}))\\ \leq-\frac{1}{2}\widetilde{\omega}^{T}\bigg{\{}P_{1}[A_{1}-\xi(J+J_{VS})^{-1} ]+[A_{1}-\xi(J+J_{VS})^{-1}]^{T}P_{1}\bigg{\}}\widetilde{\omega}\\ +\|P_{1}\|\|\widetilde{\omega}\|(\|\rho(t)\|-\overline{\rho})\\ \leq-\frac{1}{2}\widetilde{\omega}^{T}\bigg{\{}P_{1}[A_{1}-\xi(J+J_{VS})^{-1} ]+[A_{1}-\xi(J+J_{VS})^{-1}]^{T}P_{1}\bigg{\}}\widetilde{\omega}\end{array}  \tag{26}\]  
According to the inequality (23), it can be seen that  
\[\dot{V}_{1}\leq-\frac{1}{2}\lambda_{\text{min}}(L)\|\widetilde{\omega}\|^{2} \leq-\frac{\lambda_{\text{min}}(L)}{\lambda_{\text{max}}(P_{1})}V_{1} \tag{27}\]  
From Equation (27), the following inequality is derived  
\[\|\widetilde{\omega}\|^{2}\leq\frac{\lambda_{\text{max}}(P_{1})}{\lambda_{ \text{min}}(P_{1})}e^{-(\lambda_{\text{min}}(L)/\lambda_{\text{max}}(P_{1}))( t-t_{0})}\|\widetilde{\omega}(t_{0})\|^{2} \tag{28}\]Defining the fault detection residual signal \(r=\widetilde{\omega}\), a so-called threshold \(J_{th}\) is proposed to evaluate the residual signal \(r\), and the following fault detection mechanism is given by  
\[ \begin{array}{l}\left\|r\right\|_{2,T}>J_{th}\Rightarrow a\ fault\ occurs\Rightarrow Alarm\\ \left\|r\right\|_{2,T}\leq J_{th}\Rightarrow no\ fault\end{array}  \tag{29}\]  
where  
\[J_{th}\leq(\frac{\lambda_{\max}(P_{1})}{\lambda_{\min}(P_{1})}e^{-(\lambda_{ \min}(L)/\lambda_{\max}(P_{1}))(t-t_{0})}\|\widetilde{\omega}(t_{0})\|^{2})^{1 /2} \tag{30}\]  
and the residual evaluation function \(\|r\|_{2,T}\) is described as  
\[\|r\|_{2,T}=[\int_{0}^{T}r^{T}(t)r(t)dt]^{1/2} \tag{31}\]  
where \(t\in(0,T]\) is the finite time window.