## 4 Fault-Tolerant Controller Design
### Active FTC without Actuator Saturation
: _Considering the faulty attitude systems (1) and (11) of a flexible variable structure spacecraft without actuator saturation, a fast terminal sliding mode-used active FTC scheme is given by_  
\[\tau=-\hat{E}^{2}D^{\mathsf{T}}[(J+J_{VS})^{-1}D\hat{E}^{3}D^{\mathsf{T}}]^{-1}( \tau_{1}+\tau_{2}) \tag{65}\]  
_with_  
\[\tau_{1}=(v_{1}+v_{h,1}(J_{VS}))S+(v_{2}+v_{h,2}(J_{VS}))\mathrm{sig}(S)^{p/( q+h(J_{VS}))} \tag{66}\]  
\[\tau_{2}=\frac{S}{\|S\|+\xi}(\delta\|\dot{\sigma}\|+\hat{\gamma}+\|(J+J_{VS})^ {-1}D\hat{F}\|+\|\hat{W}^{\mathsf{T}}\varphi(t)+\xi_{0}\|) \tag{67}\]  
\[\dot{\hat{\gamma}}=(c_{4}+c_{h,4}(J_{VS}))\|S\| \tag{68}\]  
\[\dot{\delta}=(c_{5}+c_{h,5}(J_{VS}))\|S\|\|\dot{\sigma}\| \tag{69}\]  
_where \(\xi>0\) is a small positive scalar, \(c_{i}>0(i=\boldsymbol{4},\,5)\) are positive scalars, and \(c_{h,i}\) are the variable structure harmonic functions; additionally, these parameters can be preset. Then, the state trajectory of the faulty attitude systems can converge to a neighborhood of the origin in finite time._  
Proof of Theorem 3.: Let  
\[\left\{ \begin{array}{l}\widetilde{\delta}=\delta-\delta\\ \widetilde{\gamma}=\hat{\gamma}-\gamma\end{array} \right. \tag{70}\]  
a Lyapunov candidate function was selected as  
\[V_{3}=\frac{1}{2}S^{\mathsf{T}}S+\frac{1}{2(c_{4}+c_{h,4}(J_{VS}))}\widetilde {\gamma}^{2}+\frac{1}{2(c_{5}+c_{h,5}(J_{VS}))}\widetilde{\delta}^{2}+V_{2} \tag{71}\]  
Taking the derivative of \(V_{3}\) along the trajectory of Equation (59) yields  
\[\dot{V}_{3}=S^{\mathsf{T}}[-(J+J_{VS})^{-1}\Lambda(J+J_{VS}) \omega+(J+J_{VS})^{-1}D\varepsilon\tau+(J\] \[+J_{VS})^{-1}D\varepsilon\tau+(J\] \[+J_{VS})^{-1}D\varepsilon+\rho(t)+(k_{1}+k_{h,1}(J_{VS}))\dot{ \sigma}+k_{2}S^{*}(\sigma)]+\frac{\widetilde{\gamma}\dot{\gamma}}{c_{4,J_{VS}}} \tag{72}\] \[+\frac{\widetilde{\delta}\widetilde{\delta}}{\mathrm{C}_{5,J_{VS} }}+\dot{V}_{2}\]  
where  
\[\left\{ \begin{array}{l}C_{4,JVS}=c_{4}+c_{h,4}(J_{VS})\\ C_{5,JVS}=c_{5}+c_{h,5}(J_{VS})\end{array} \right. \tag{73}\]  
Substituting Equation (59) into the above equation, it can be seen that  
\[\dot{V}_{3}\leq\|S\|(\|(k_{1}+k_{h,1}(J_{VS}))\dot{\sigma}+k_{2}S ^{*}(\sigma)\|-\delta\|\dot{\sigma}\|)+\widetilde{\gamma}\|S\|+\widetilde{ \delta}\|S\|\|\dot{\sigma}\|\] \[+\|S\|(\|-(J+J_{VS})^{-1}\Lambda(J+J_{VS})\omega\|-\hat{\gamma})+S ^{\mathsf{T}}[-(v_{1}+v_{h,1}(J_{VS}))S\] \[-(v_{2}+v_{h,2}\big{(}J_{VS}))\mathrm{sig}(S)^{p/(q+h(J_{VS}))}]- \alpha V_{2}+\beta\] \[\leq S^{\mathsf{T}}[-(v_{1}+v_{h,1}(J_{VS}))S-(v_{2}+v_{h,2} \big{(}J_{VS}))\mathrm{sig}(S)^{p/(q+h(J_{VS}))}]+\beta\] \[\leq-S^{\mathsf{T}}(v_{1}+v_{h,1}(J_{VS}))S-2\big{[}^{(p/(q+h(J_{ VS}))+1)+1/2}[\tfrac{1}{2}S^{\mathsf{T}}(v_{2}\] \[+v_{h,2}\big{(}J_{VS}))S]^{\lfloor\{p/(q+h(J_{VS}))\}\rfloor+1\}/2+\beta\] \[\leq-2\lambda_{\min}(v_{1}+v_{h,1}  
where  
\[\Omega=(1/2\text{C}_{4 JVS})\tilde{\gamma}^{2}(t)+(1/2\text{C}_{5 JVS})\tilde{ \delta}^{2}(t)+V_{2} \tag{75}\]  
and  
\[\left\{ \begin{array}{l}(\Omega/V_{3})<1\\ (\Omega/V_{3})^{\{[p/(q+h(I_{VS}))]+1\}/2}<1\end{array} \right. \tag{76}\]  
Define  
\[\vartheta_{1}=2\lambda_{\min}(v_{1}+v_{h,1}(I_{VS}))(1-(\Omega/V_{3})) \tag{77}\]  
\[\vartheta_{2}=2^{\{[p/(q+h(I_{VS}))]+1\}/2}\lambda_{\min}(v_{2}+v_{h,2}(I_{VS} ))[1-(\Omega/V_{3})^{\{[p/(q+h(I_{VS}))]+1\}/2}] \tag{78}\]  
then, we have  
\[\dot{V}_{3}\leq-\vartheta_{1}V_{3}-\vartheta_{2}V_{3}^{\{[p/(q+h(I_{VS}))]+1 \}/2}+\beta \tag{79}\]  
If \(V_{3}>\beta/\vartheta_{1}\), the inequality (79) is further rewritten as  
\[\dot{V}_{3}\leq-(\vartheta_{1}-\beta/V_{3})V_{3}-\vartheta_{2}V_{3}^{\{[p/( q+h(I_{VS}))]+1\}/2} \tag{80}\]  
According to Lemma 1, the convergence time satisfies the following inequality  
\[T_{5}\leq t_{0}+\frac{1}{(\vartheta_{1}-\beta/V_{3})(1-\frac{p}{q+h(I_{VS})} )}\ln\left\{\frac{(\vartheta_{1}-\beta/V_{3})V_{3}(t_{0})^{[1-p/(q+h(I_{VS}))] /2}}{\vartheta_{2}}+1\right\} \tag{81}\]  
where \(t_{0}\) is the initial time and \(V_{3}(t_{0})\) is the initial value.