## 4 Fault-Tolerant Controller Design
### Active Anti-Saturation FTC
\begin{array}{l}\dot{V}_{4}=S^{T}[-(J+J_{VS})^{-1}\Lambda(J+J_{VS})\omega+( J+J_{VS})^{-1}DEs{\rm sat}(\tau)\\ +(J+J_{VS})^{-1}DF+\rho(t)+(k_{1}+k_{h,1}(J_{VS}))\dot{\sigma}+k_{2}S^{*}( \sigma)]\\ +\frac{\widetilde{\gamma}\dot{\gamma}}{c_{6,JVS}}+\frac{\widetilde{\delta} \dot{\delta}}{C_{7,JVS}}-\frac{\widetilde{\mu}^{2}-\hat{\mu}^{2}}{C_{8,JVS}}+ \dot{V}_{2}\end{array}  \tag{101}\]  
Substituting Equation (92) into \(\dot{V}_{4}\), it is easily known that  
\[ \begin{array}{l}\dot{V}_{4}\leq\|S\|[\|-(J+J_{VS})^{-1}\Lambda(J+J_{VS}) \omega\|-\hat{\gamma}-\delta\|\dot{\sigma}]\\ -\|([PERSON]+J_{VS})^{-1}DF\|-\|\dot{W}^{T}\varphi(t)+\hat{\varepsilon}_{0}\|+\|([PERSON]+J _{VS})^{-1}DF\|+\|\rho(t)\|\\ +\|(k_{1}+k_{h,1}(J_{VS}))\dot{\sigma}+k_{2}S^{*}(\sigma)\|+\widetilde{\gamma }\|S\|+\delta\|S\|\|\dot{\sigma}\|+\mu S^{T}[-(v_{1}\\ +v_{h,1}([PERSON]_{VS}))S-(v_{2}+v_{h,2}(J_{VS})){\rm sig}(S)^{p/(q+h(J_{VS}))}]+\dot {V}_{2}\\ \leq\|S\|\left[(\|-(J+J_{VS})^{-1}\Lambda(J+J_{VS})\omega\|-\gamma)+(\|\rho(t )\|-\|\dot{W}^{T}\varphi(t)+\hat{\varepsilon}_{0}\|)\right.\\ +(\|(k_{1}+k_{h,1}(J_{VS}))(J+J_{VS})\dot{\sigma}+k_{2}(J+J_{VS})S^{*}(\sigma) \|-\delta\|\dot{\sigma}\|)\\ +(\|(J+J_{VS})^{-1}DF\|-\|([PERSON]+[PERSON]_{VS})^{-1}DF\|)]+\mu S^{T}[-(v_{1}+v_{h,1}([PERSON]_{VS} ))S\\ -(v_{2}+v_{h,2}([PERSON]_{VS})){\rm sig}(S)^{p/(q+h([PERSON]_{VS}))}]-\alpha V_{2}+\beta\\ \leq\mu S^{T}[-(v_{1}+v_{h,1}([PERSON]_{VS}))S-(v_{2}+v_{h,2}([PERSON]_{VS})){\rm sig}(S)^{p /(q+h([PERSON]_{VS}))}]+\beta\\ \leq-2\mu\{\hat{\mu}^{3}S^{T}(v_{1}+v_{h,1}([PERSON]_{VS}))S\}-2\{[p/(q+h([PERSON]_{VS}))]+1 \}/2\mu[\frac{1}{2}S^{T}(v_{2}\\ +v_{h,2}([PERSON]_{VS}))S]^{[p/(q+h([PERSON]_{VS}))]+1\}/2+\beta\\ \leq-2\mu\lambda_{\min}(v_{1}+v_{h,1}([PERSON]_{VS}))(1-\frac{\Theta}{V_{4}})V_{4}-2 \{[p/(q+h([PERSON]_{VS}))]+1\}/2\mu\lambda_{\min}(v_{2}\\ +v_{h,2}([PERSON]_{VS}))(1-(\frac{\Theta}{V_{4}})^{\{[p/(q+h(J_{VS}))]+1\}/2\})V_{4} ^{\{[p/(q+h(J_{VS}))]+1\}/2+\beta\end{array}  \tag{102}\]  
where  
\[\Theta=\frac{1}{2C_{6,JVS}}\widetilde{\gamma}^{2}(t)+\frac{1}{2C_{7,JVS}} \widetilde{\delta}^{2}(t)+\frac{1}{2C_{8,JVS}}\widetilde{\mu}^{2}(t)+V_{2} \tag{103}\]  
and  
\[\left\{ \begin{array}{l}\frac{\Theta}{V_{4}}<1\\ (\frac{\Theta}{V_{4}})^{\{[p/(q+h(J_{VS}))]+1\}/2}<1\end{array} \right. \tag{104}\]Define  
\[\theta_{3}=2\mu\lambda_{\min}(v_{1}+v_{h,1}(J_{VS}))(1-(\frac{\Theta}{V_{4}})) \tag{105}\]  
\[\theta_{4}=2^{\{[p/(q+h(J_{VS}))]+1\}/2}\mu\lambda_{\min}(v_{2}+v_{h,2}(J_{VS}))( 1-(\frac{\Theta}{V_{4}})^{\{[p/(q+h(J_{VS}))]+1\}/2}) \tag{106}\]  
then, we have  
\[\dot{V}_{4}\leq-\theta_{3}V_{4}-\theta_{4}V_{4}^{\{[p/(q+h(J_{VS}))]+1\}/2}+\beta \tag{107}\]  
By referring to the proof process of Theorem 3, all the state trajectory of the considered closed-loop attitude systems can converge to a neighborhood of the origin in finite time. This completes the proof. The unknown \(P_{1}\) and \(P_{2}\) could be obtained by solving (23) and (45) using the linear matrix inequality toolbox in MATLAB. Hence, the fault detection threshold \(J_{th}\) and the adaptive fault parameter updated algorithms could be accomplished online and the final FTC (64) or (91) was successfully reconstructed to the faulty attitude systems. The whole active FTC scheme in this study had an acceptable computational burden and could be easily applied. \(\square\)  
**Remark 5**. : _In this study, the variable structure control has one more \(J_{VS}\) including the harmonic functions with \(J_{VS}\) as the independent variable than the non-variable structure control. When \(J_{VS}\) = 0, the harmonic functions in the controller are 0. This shows that the control method without the variable structure is the same as the control method after all harmonic functions are removed.