## 3 Fault-Tolerant Control Design for the Attitude System of HSV
### Fault-Tolerant Control Design for Slow-Loop of HSV
#### 3.1.2 Fault-Tolerant Controller Design under Unexpected Centroid Shift and System Fault  
For the adverse effects caused by complex disturbance \(D_{s}\), as shown in (4), a top-order sliding mode observer is designed in this work to obtain unknown knowledge serving for the basic controller to realize the fault-tolerant control targets. * **High-order sliding mode observer design**. To estimate the complex perturbation \(D_{s}\), a top-order sliding mode observer is designed as follows [21] \[\sigma_{0} =\Omega-z,\dot{z}=f_{s}+g_{s}\omega+\hat{D}_{s},\sigma_{1}=\sigma_{0 }+\theta\sigma_{0}+\zeta\sigma_{0}^{\frac{p}{q}}\] \[\hat{D}_{s} =(\theta+\frac{p}{q}\zeta diag(\sigma_{0}^{\frac{p}{q}-1}))^{-1} (\sigma_{0}+l_{1}\sigma_{1}+l_{2}\text{sgn}(\sigma_{1}))+\hat{\phi}\text{sgn} (\sigma_{1})\] \[\dot{\hat{\phi}} =\vartheta(\sum_{i=1}^{3}{(\theta_{i}+\frac{p}{q}\zeta_{i}diag( \sigma_{0i}^{\frac{p}{q}-1}))|\sigma_{1i}|}),i=1,2,3\] (28) where \(\sigma_{0}\) and \(\sigma_{1}\) are the designed sliding mode surfaces based on the terminal sliding mode trick, and \(z\) denote the state of the designed observer. \(\theta\) and \(\zeta\) are the designed system matrix, satisfying \(\theta=diag\{\theta_{1},\theta_{2},\theta_{3}\},\theta_{i}>0\) and \(\zeta=diag\{\zeta_{1},\zeta_{2},\zeta_{3}\},\zeta_{i}>0\). In addition, \(p\) and \(q\) are odd positive integers, satisfying \(0<p<2q\). Furthermore, \(l_{1}\) and \(l_{2}\) are also the designed positive constants to regulate the estimation effect of \(D_{s}\). Finally, \(\hat{\phi}\) is the estimation of the adaptive parameter \(\phi\). Before analyzing the stability of the disturbance estimate error \(\sigma_{0}\), combined with (4) and (28), the derivative of \(\sigma_{0}\) is deduced as \[\dot{\sigma_{0}}=\hat{\Omega}-\dot{z}=D_{s}-\hat{D}_{s},\ddot{\sigma_{0}}=\hat {D}_{s}-\hat{D}_{s}\] (29) Furthermore, by keeping the exponential reaching law \(\dot{\sigma}_{1}=-l_{1}\sigma_{1}-l_{2}\text{sgn}(\sigma_{1})\) and (29) in mind, \(\dot{\sigma}_{1}\) can be obtained as \[\dot{\sigma}_{1}=-l_{1}\sigma_{1}-l_{2}\text{sgn}(\sigma_{1})+(\theta+\frac{p }{q}\zeta diag(\dot{\sigma}_{0}^{\frac{p}{q}-1}))(\hat{D}_{s}-\hat{\phi}\text {sgn}(\sigma_{1}))\] (30) In what follows, the stability of the designed observer is analyzed through the following Lyapunov function \[V_{\sigma}=\sigma_{1}^{\text{T}}\sigma_{1}\Big{/}2+\ddot{\phi}^{2}\Big{/}2\theta\] (31) where \(\theta\) is a designed parameter, satisfying \(\dot{\theta}>0\), and we define that \(\ddot{\phi}=\phi-\dot{\phi}\). Evaluating the time derivative of \(V_{\sigma}\) along (28) and (30) leads to \[\dot{V}_{\sigma} =\sigma_{1}^{\text{T}}\dot{\sigma}_{1}-\dot{\phi}\dot{\hat{\phi}} /\dot{\theta}\] \[\leq-l_{1}\sigma_{1}^{\text{T}}\sigma_{1}-l_{2}\sum_{i=1}^{3}{ \left|\sigma_{1i}^{\text{T}}\right|}+\sum_{i=1}^{3}{(\theta_{i}+\frac{p}{q} \zeta_{i}diag(\dot{\sigma}_{0i}^{\frac{p}{q}-1}))(\dot{\phi}_{i}|\sigma_{1i}| -\hat{\phi}_{i}|\sigma_{1i}|)}-\dot{\phi}\dot{\phi}/\dot{\theta}\] (32) where \(i=1,2,3\). Furthermore, invoking \(\dot{\hat{\phi}}=\dot{\sigma}(\sum_{i=1}^{3}{(\theta_{i}+\frac{p}{q}\zeta_{i} diag(\dot{\sigma}_{0i}^{\frac{p}{q}-1}))|\sigma_{1i}|})\), (32) can be further deduced as \[\dot{V}_{1}\leq-l_{1}\sigma_{1}^{\text{T}}\sigma_{1}-l_{2}\sum_{i=1}^{3}{ \left|\sigma_{1i}^{\text{T}}\right|}\] (33) The differential term \(\dot{\sigma}_{0}\) in the designed higher-order sliding-mode observer is not directly obtained by using the derivative method. Moreover, according to Lemma 1, applying a higher-order sliding mode differentiator yields the corresponding derivative. Then, the states \(z\) converge to \(\sigma_{0}\) in finite time \(t_{\sigma}\).