## 4 Fault-Tolerant Controller Design
### Active Anti-Saturation FTC  
To address the problem of actuator saturation, the other active FTC scheme is further presented in this section. Consider the dynamics Equation (7) of a flexible variable structure spacecraft under actuator saturation, which is described by  
\[\dot{\omega}=-(J+J_{VS})^{-1}\Lambda(J+J_{VS})\omega+(J+J_{VS})^{-1}D\text{sat}( \tau)+\rho(t) \tag{84}\]  
in which sat(\(\tau\)) is given by  
\[\text{sat}(\tau_{i})=\left\{ \begin{array}{ll}\tau_{i},&if\ |\tau_{i}|<\tau_{ \text{max}}\\ \tau_{\text{max}}\text{sign}(\tau_{i}),&otherwise\end{array} \right. \tag{85}\]  
where \(t\) is the designed control input and \(t_{max}\) is the upper boundary of saturation nonlinearity. Here, a nonlinear saturation function is introduced to handle the saturation input  
\[\text{sat}(\tau)=\chi(\tau)\tau \tag{86}\]  
where \(\chi(\tau)=\text{diag}[\chi_{1}(\tau_{1}),\ \ldots,\ \chi_{n}(\tau_{n})]\) represents the saturation characteristic of the actuators with  
\[\chi_{i}(\tau_{i})=\left\{ \begin{array}{ll}1,&if\ \ |\tau_{i}|\leq\tau_{ \text{max}}\\ \frac{\tau_{\text{max}}}{\tau_{i}}\text{sign}(\tau_{i}),&otherwise\end{array} \right. \tag{87}\]  
It was assumed that \(\chi_{i}(\tau_{i})\) satisfies the following constraint  
\[0<\mu\leq\min(\chi_{i}(\tau_{i}))\leq 1 \tag{88}\]  
By referring to the fault modeling method developed in the study by [21], the faulty dynamics equation in the actuator saturation case is written as  
\[ \begin{array}{l}\dot{\omega}=-(J+J_{VS})^{-1}\Lambda(J+J_{VS})\omega+(J+J_{ VS})^{-1}D(E\text{sat}(\tau)\\ +F)+\rho(t)\end{array}  \tag{89}\]  
By designing the same fast terminal sliding model surface  
\[S=\omega+(k_{1}+k_{h,1}(J_{VS}))\sigma+k_{2}\text{sig}(\sigma)^{p/(q+h(J_{VS}))} \tag{90}\]  
the derivative of \(S\) is given by  
\[ \begin{array}{l}\dot{S}=-(J+J_{VS})^{-1}\Lambda(J+J_{VS})\omega+(J+J_{VS})^ {-1}D(E\text{sat}(\tau)+F)\\ +\rho(t)+(k_{1}+k_{h,1}(J_{VS}))\dot{\sigma}+k_{2}S^{*}(\sigma)\end{array}  \tag{91}\]  
Based on the above content, the fourth result of this study is given in the form of Theorem 4. **Theorem 4**. : _Considering the faulty attitude systems (1) and (89) in the actuator saturation case, a fast terminal sliding model-based active FTC scheme is designed as_  
\[\tau=-\hat{E}^{2}D^{T}[(J+J_{VS})^{-1}D\hat{E}^{3}D^{T}]^{-1}(\tau_{1}+\tau_{2}) \tag{92}\]  
_with_  
\[\tau_{1}=(v_{1}+v_{h,1}(J_{VS}))S+(v_{2}+v_{h,2}\Big{(}J_{VS}))\text{sig}(S)^{ p/(q+h(J_{VS}))} \tag{93}\]  
\[\tau_{2}=\frac{S}{\|S\|+\varsigma}\hat{\pi}[\hat{\gamma}+\hat{\delta}\|\hat{ \sigma}\|+\|(J+J_{VS})^{-1}D\hat{F}\|+\|\hat{W}^{T}\varphi(t)+\hat{\epsilon}_{ 0}\|] \tag{94}\]_The adaptive parameter updated algorithms are given by_  
\[\dot{\gamma}=(c_{6}+c_{h,6}(J_{VS}))\|S\| \tag{95}\]  
\[\dot{\delta}=(c_{7}+c_{h,7}(J_{VS}))\|S\|\|\dot{\sigma}\| \tag{96}\]  
\[\dot{\mu}=(c_{8}+c_{h,8}(J_{VS}))\dot{\eta}^{3}\|S\|(\dot{\gamma}+\delta\|\dot{ \sigma}\|+\|(J+J_{VS})^{-1}D\dot{F}\| \tag{97}\]  
_where \(\zeta>0\) is a small positive constant, \(c_{i}>0\) (\(i=6\), \(7\), \(8\)) are positive scalars, and \(c_{h,i}\) are the variable structure harmonic functions, all of which can be preset. Then, the state trajectory of the faulty attitude systems can converge in finite time._  
**Proof of Theorem 4. ** Defining  
\[\widetilde{\mu}=\tilde{\mu}^{-1}-\mu \tag{98}\]  
a Lyapunov candidate function was selected as  
\[ \begin{array}{l}V_{4}=\frac{1}{2}S^{T}S+\frac{1}{2(c_{6}+c_{h,6}(J_{VS}))} \widetilde{\gamma}^{2}+\frac{1}{2(c_{7}+c_{h,7}(J_{VS}))}\widetilde{\delta}^ {2}\\ +\frac{1}{2(c_{8}+c_{h,8}(J_{VS}))}\tilde{\mu}^{2}+V_{2}\end{array}  \tag{99}\]  
Set  
\[\left\{ \begin{array}{l}C_{6,JVS}=c_{6}+c_{h,6}(J_{VS})\\ C_{7,JVS}=c_{7}+c_{h,7}(J_{VS})\\ C_{8,JVS}=c_{8}+c_{h,8}(J_{VS})\end{array} \right. \tag{100}\]  
and taking the derivative of \(V_{4}\) along the trajectory of Equation (91) yields  
\[