## 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
According to (31), if \(\sigma_{1i}\
eq 0,i=1,2,3\) at time \(t_{\sigma}\), the states will arrive at the sliding mode \(\sigma_{1i}=0\) in finite time \(t_{1i}\), namely \[t_{1i}=\frac{1}{I_{1}}[\ln(|\sigma_{1i}(t_{\sigma})|+\frac{I_{2}}{I_{1}})-\ln( \frac{I_{2}}{I_{1}})]\] (34) Examining (34), it is straightforward to show that the sliding mode \(\sigma_{1}=0\) can be reached within the finite time \(t_{a1}=\max(t_{11},t_{12},t_{13})\). 2. When the sliding mode \(\sigma_{1}=0\) is arrived, the convergence time of the system is up to the terminal sliding mode equation \[\hat{D}_{s} =(\theta+\frac{p}{q}\zeta diag(\sigma_{0}^{\frac{p}{q}-1}))^{-1} (\sigma_{0}+l_{1}\sigma_{1}+l_{2}\text{sgn}(\sigma_{1}))+\hat{\phi}\text{sgn} (\sigma_{1})\]0 (35) where \(\sigma_{0i}=0\) denotes a terminal attractor of the system of (35). According to [21], the convergence time \(t_{0i}\) from \(t_{\sigma}+t_{a1}\) when \(\sigma_{0i}(t_{\sigma i}+t_{a1i})\
eq 0\) to \(t_{\sigma}+t_{a1}+t_{0i}\) when \(\sigma_{0i}(t_{\sigma}+t_{a1}+t_{0i})=0\) satisfies \[\hat{D}_{s} =(\theta+\frac{p}{q}\zeta diag(\sigma_{0}^{\frac{p}{q}-1}))^{-1} (\sigma_{0}+l_{1}\sigma_{1}+l_{2}\text{sgn}(\sigma_{1}))+\hat{\phi}\text{sgn} (\sigma_{1})\]1 (36) In a similar way, from the time point \(t_{\sigma}+t_{a1}\) when \(\sigma_{0}(t_{\sigma i}+t_{a1i})\
eq 0\) move to the time point \(t_{\sigma}+t_{a1}+t_{a0}\) when \(\sigma_{0}(t_{\sigma i}+t_{a1i})=0\), the convergence time \(t_{a0}\) is obtained as \(t_{a0}=\max(t_{01},\cdots,t_{03})\). From this, we can obtain that the estimated value \(\hat{D}_{s}\) be approximated by an external perturbation \(D_{s}\) in finite time \(t_{\sigma}=\max(t_{a0},t_{a1})\). Combined with Assumption 3, we can conclude that the disturbance estimation error is bounded and can be defined as \(\left\|D_{s}-\hat{D}_{s}\right\|<\lambda\), where \(\lambda\) is a positive unknown constant. * **Fault-tolerant controller design of slow-loop**. Recall that Assumption 3, together with the estimate of complex disturbance \(\hat{D}_{s}\) from observer (28) and the baseline controller (27), the final control law and adaptive law are synthesized as \[\hat{D}_{s} =(\theta+\frac{p}{q}\zeta diag(\sigma_{0}^{\frac{p}{q}-1}))^{-1} (\sigma_{0}+l_{1}\sigma_{1}+l_{2}\text{sgn}(\sigma_{1}))+\hat{\phi}\text{sgn} (\sigma_{1})\]2 \[\hat{D}_{s} =(\theta+\frac{p}{q}\zeta diag(\sigma_{0}^{\frac{p}{q}-1}))^{-1} (\sigma_{0}+l_{1}\sigma_{1}+l_{2}\text{sgn}(\sigma_{1}))+\hat{\phi}\text{sgn} (\sigma_{1})\]3 (37) where \((g_{s})^{-1}\hat{D}_{s}\) is applied to suppress uncertainties due to unexpected centroid shifts, external perturbations, etc. As for the adaptive term \(\lambda\), it is applied to handle the estimation error of \(D_{s}\), where \(\hat{\lambda}\) represents the estimation of \(\lambda\). To analyze the stability of the observer and the resulting closed-loop system, the following theorem summarizes the main results of this subsection. Theorem 1: _Consider the slow-loop model of HSV describer by (4), with unexpected centroid shift, system fault and disturbance, satisfying Assumptions 1-4, under the designed controller (37) and the observer (28), the slow-loop system of HSV can track expected signals and the whole signals in closed-loop system are bounded._  
Proof: Consider the following Lyapunov function  
\[\hat{D}_{s} =(\theta+\frac{p}{q}\zeta diag(\sigma_{0}^{\frac{p}{q}-1}))^{-1} (\sigma_{0}+l_{1}\sigma_{1}+l_{2}\text{sgn}(\sigma_{1}))+\hat{\phi}\text{sgn} (\sigma_{1})\]4  
where \(P_{s}\) is a positive definite symmetric matrix, and \(V_{\sigma}=\sigma_{1}^{T}\sigma_{1}/2+\hat{\phi}^{2}/2\hat{\vartheta}\) is applied to analyze the estimate error of \(D_{s}\).