## 3 Actuator Fault Detection and Diagnosis Scheme
### Fault Diagnosis Scheme Design
\begin{array}{l}V_{2}=\frac{1}{2}\overline{\omega}^{T}P_{2}\overline{\omega }+\frac{1}{2}(c_{1}+c_{h,1}(J_{VS}))\widetilde{\Phi}^{T}\widetilde{\Phi}+ \frac{1}{2}(c_{2}+c_{h,2}(J_{VS}))\widetilde{F}^{T}\widetilde{F}\\ +\frac{1}{2}(c_{3}+c_{h,3}(J_{VS}))\text{tr}(\widetilde{W}^{T}\widetilde{W})+ \frac{1}{2}(c_{0}+c_{h,0}(J_{VS}))\widetilde{\epsilon}_{0}^{T}\widetilde{ \epsilon}_{0}\end{array}  \tag{46}\]  
Taking the derivative of \(V_{2}\) along the trajectory of Equation (43) yields  
\[ \begin{array}{l}\dot{V}_{2}=\frac{1}{2}(\overline{\omega}^{T}P_{2}\overline{ \omega}+\overline{\omega}^{T}P_{2}\tilde{\omega})+C_{1,JVS}\widetilde{\Phi}^{ T}\widetilde{\Phi}+C_{2,JVS}\widetilde{F}^{T}\widetilde{F}\\ +C_{3,JVS}\text{tr}(\widetilde{W}^{T}\tilde{F})+C_{0,JVS}\widetilde{\epsilon}_ {0}^{T}\widetilde{\epsilon}_{0}+C_{1,JVS}\widetilde{\Phi}^{T}\dot{\Phi}+C_{2,JVS }\widetilde{F}^{T}\dot{F}\\ +C_{3,JVS}\text{tr}(\widetilde{W}^{T}\dot{\Psi})+C_{0,JVS}\widetilde{\epsilon} _{0}^{T}\widetilde{\epsilon}_{0}\\ =\frac{1}{2}\overline{\omega}^{T}\Big{\{}A_{2}^{T}+\xi((J+J_{VS})^{-1})^{T} \Big{\}}P_{2}+P_{2}[A_{2}+\xi(J+J_{VS})^{-1}]\Big{\}}\overline{\omega}\\ +\overline{\omega}^{T}P_{2}(J+J_{VS})^{-1}D\Sigma\widetilde{\Phi}+\overline{ \omega}^{T}P_{2}(J+J_{VS})^{-1}D\widetilde{F}\\ +\overline{\omega}^{T}P_{2}\widetilde{W}^{T}\varphi(t)+\overline{\omega}^{T}P _{2}\widetilde{\epsilon}_{0}+C_{1,JVS}\widetilde{\Phi}^{T}\dot{\Phi}+C_{2,JVS }\widetilde{F}^{T}\dot{F}\\ +C_{3,JVS}\text{tr}(\widetilde{W}^{T}\dot{\Psi})+C_{0,JVS}\widetilde{\epsilon} _{0}^{T}\dot{\xi}_{0}\end{array}  \tag{47}\]  
where  
\[\left\{ \begin{array}{l}C_{1,JVS}=c_{1}+c_{h,1}(J_{VS})\\ C_{2,JVS}=c_{2}+c_{h,2}(J_{VS})\\ C_{3,JVS}=c_{3}+c_{h,3}(J_{VS})\\ C_{0,JVS}=c_{0}+c_{h,0}(J_{VS})\end{array} \right. \tag{48}\]Substituting the adaptive updated laws (35)-(38) into Equation (47), it leads to  
\[ \begin{array}{l}\dot{V}_{2}=\frac{1}{2}\overline{\omega}^{T}Q\overline{\omega}+ \overline{\omega}^{T}P_{2}(J+J_{VS})^{-1}D\Sigma\widehat{\Phi}+\overline{\omega} ^{T}P_{2}(J+J_{VS})^{-1}D\widetilde{F}\\ +\overline{\omega}^{T}P_{2}\widetilde{W}^{T}\varphi(t)+\overline{\omega}^{T}P_{ 2}\widetilde{\xi}_{0}-C_{1,JVS}\widehat{\Phi}^{T}\widehat{\Phi}-\widehat{\Phi }^{T}[(J\\ +J_{VS})^{-1}D\Xi]^{T}P_{2}\overline{\omega}-C_{2,JVS}\widetilde{F}^{T}\hat{F}- \widetilde{F}^{T}[(J+J_{VS})^{-1}D]^{T}P_{2}\overline{\omega}\\ -C_{3,JVS}\text{tr}(\widetilde{W}^{T}\hat{W})-\text{tr}(\widetilde{W}^{T}\varphi (t)\overline{\omega}^{T}P_{2})-C_{0,JVS}\widehat{\xi}_{0}^{T}\xi_{0}- \widetilde{\xi}_{0}^{T}P_{2}\overline{\omega}\\ =\frac{1}{2}\overline{\omega}^{T}Q\overline{\omega}-\frac{C_{1,JVS}}{2} \widehat{\Phi}^{T}\widehat{\Phi}-\frac{C_{2,JVS}}{2}\widetilde{F}^{T}\tilde {F}-\frac{C_{3,JVS}}{2}\text{tr}(\widetilde{W}^{T}\widetilde{W})\\ -\frac{C_{0,JVS}}{2}\widehat{\xi}_{0}^{T}\xi_{0}+\frac{C_{1,JVS}}{2}\Phi^{T} \Phi+\frac{C_{2,JVS}}{2}F^{T}\hat{F}+\frac{C_{3,JVS}}{2}\text{tr}(W^{T}W)\\ +\frac{C_{0,JVS}}{2}\widehat{\epsilon}_{0}^{T}\xi_{0}\end{array}  \tag{49}\]  
According to the inequality (45) described in Theorem 2, the following result is easily obtained  
\[\dot{V}_{2}\leq-\alpha V_{1}+\beta \tag{50}\]  
where  
\[\alpha=\min\{-\lambda_{\min}(Q)/\lambda_{\max}(P_{2}),1\}>0 \tag{51}\]  
\[\beta=\frac{C_{1,JVS}}{2}\|\Phi\|^{2}+\frac{C_{2,JVS}}{2}\|F\|^{2}+\frac{C_{3,JVS}}{2}\|W\|^{2}+\frac{C_{0,JVS}}{2}\|\varepsilon_{0}\|^{2} \tag{52}\]  
According to Lyapunov stability theory, it is easily known from the above inequality that the error dynamics Equation (43) is UUB. This completes the proof. \(\square\)  
Using reachable set computation [27; 28; 29], the robust fault diagnosis was extended to linear uncertain systems. Then, a sliding mode fault detection was designed for the nonlinear attitude systems of a flexible variable structure spacecraft in this study. The sliding mode \(\rho\text{sign}(\widetilde{\omega})\) offsets the uncertainty effects, and therefore, the designed detection threshold is sensitive to unknown faults.