## 4 Fault-Tolerant Controller Design  
### Active FTC without Actuator Saturation  
In this section, an active FTC strategy is proposed for the faulty attitude systems (1) and (11) without actuator saturation. First, a fast terminal sliding model surface is designed as  
\[S=\omega+(k_{1}+k_{h,1}(J_{VS}))\sigma+k_{2}\text{sig}(\sigma)^{p/(q+h(J_{VS}))} \tag{53}\]  
where \(S=[S_{1},S_{2},S_{3}]^{T}\in R^{3}\) is the sliding mode variable; \(k_{1}>0,k_{2}>0\) are two positive scalars; \(k_{h,1}(J_{VS})\), \(h(J_{VS})\) are the variable structure harmonic functions with \(J_{VS}\) as the argument, which are all known scalars, and \(0<(p/(q+h(J_{VS})))<1\); and \(p\), \(q\) are positive odd integers. The function \(\text{sig}(\sigma)p/(q+h(J_{VS}))\) is defined as  
\[\text{sig}(\sigma)^{p/(q+h(J_{VS}))}=\left[ \begin{array}{c}|\phi|^{p/(q+h(J_ {VS}))}\text{sign}(\phi)\\ |\theta|^{p/(q+h(J_{VS}))}\text{sign}(\theta)\\ |\psi|^{p/(q+h(J_{VS}))}\text{sign}(\psi)\end{array} \right] \tag{54}\]  
Taking the derivative of sliding variable \(S\) with respect to time yields  
\[\dot{S}=\dot{\omega}+(k_{1}+k_{h,1}(J_{VS}))\dot{\sigma}+\frac{k_{2}p}{q+h(J _{VS})}\text{diag}(|\sigma|^{[p/(q+h(J_{VS}))]-1})\dot{\sigma} \tag{55}\]  
It was noted that Equation (55) contains a negative fractional term \([p/(q+h(J_{VS}))]^{-1}\), the singularity occurs while \(\sigma_{j}=0\) and its derivative is nonzero. To avoid the singularity problem, the derivative of \(S\) is modified as  
\[\dot{S}=\dot{\omega}+(k_{1}+k_{h,1}(J_{VS}))\dot{\sigma}+k_{2}S^{*}(\sigma) \tag{56}\]  
where  
\[S^{*}(\sigma)=[S_{1}^{*}(\sigma),\ \ S_{2}^{*}(\sigma),\ \ S_{3}^{*}(\sigma)]^{T} \in R^{3\times 1} \tag{57}\]  
with  
\[S_{i}^{*}(\sigma)=\left\{ \begin{array}{cc}\frac{p}{q+h(J_{VS})}|\sigma_{i}| ^{[p/(q+h(J_{VS}))]-1}\dot{\sigma}_{i},&if\ \dot{\sigma}_{i}\
eq 0,\ \ |\sigma_{i}|\geq \varepsilon\\ \frac{p}{q+h(J_{VS})}|\epsilon|^{[p/(q+h(J_{VS}))]-1}\dot{\sigma}_{i},&if\ \dot{ \sigma}_{i}\
eq 0,\ \ |\sigma_{i}|<\varepsilon\\ 0,&if\ \dot{\sigma}_{i}=0\end{array} \right. \tag{58}\]  
where \(S_{i}^{*}(\sigma)\) is the \(i\)th component of \(S^{*}(\sigma)\) and \(\varepsilon\) is a small positive constant. Substituting Equation (11) into Equation (56), it is easily known that  
\[ \begin{array}{c}\dot{S}=-(J+J_{VS})^{-1}\Lambda(J+J_{VS})\omega+(J+J_{VS})^ {-1}DE\tau+(J\\ +J_{VS})^{-1}DF+\rho(t)+(k_{1}+k_{h,1}(J_{VS}))\dot{\sigma}+k_{2}S^{*}(\sigma) \end{array}  \tag{59}\]  
According to the sliding mode control idea, sliding mode dynamics need to match a reaching law. This study selected an improved exponential reaching law as  
\[\dot{S}=-(v_{1}+v_{h,1}(J_{VS}))S-(v_{2}+v_{h,2}\Big{(}J_{VS}))\text{sig}(S)^{ p/(q+h(J_{VS}))} \tag{60}\]  
where  
\[\left\{ \begin{array}{cc}v_{1}=\text{diag}\{v_{11},\ \ \ v_{12},\ \ \ v_{13}\}>0\\ v_{2}=\text{diag}\{v_{21},\ \ \ v_{22},\ \ \ v_{23}\}>0\end{array} \right. \tag{61}\]  
\[\left\{ \begin{array}{cc}v_{h,1}(J_{VS})=\text{diag}\{v_{h,11},\ \ \ v_{h,12},\ \ \ v_{h,13}\}>0\\ v_{h,2}(J_{VS})=\text{diag}\{v_{h,21},\ \ \ v_{h,22},\ \ \ v_{h,23}\}>0\end{array} \right. \tag{62}\]  
Equations (61) and (62) are the diagonal matrices. \(v_{h,11}\), \(v_{h,12}\), \(v_{h,13}\), \(v_{h,21}\), \(v_{h,22}\), and \(v_{h,23}\) are the known harmonic functions with \(J_{VS}\) as the argument. To derive the FTC scheme of this paper, an assumption is introduced in this position. **Assumption 4**. : _There exists two unknown constants \(\gamma\geq 0\) and \(\delta\geq 0\), such that the following inequalities are satisfied_  
\[\|-(J+J_{VS})^{-1}\Lambda(J+J_{VS})\omega\|\leq\gamma \tag{63}\]  
\[\|(k_{1}+k_{h,1}(J_{VS}))\dot{\sigma}+k_{2}S^{*}(\sigma)\|\leq\delta\|\dot{ \sigma}\| \tag{64}\]  
In the following, the third result of this study is given in the form of Theorem 3. **Theorem 3**.