## 3 Fault-Tolerant Control Design for the Attitude System of HSV
### Fault-Tolerant Control Design for Fast Loop of HSV
#### 3.2.2 Adaptive Fault-Tolerant Controller without Exact Knowledge of Inertial Matrix  
Due to the unknown effects of system uncertainty \(f(\beta)\), external disturbance \(d_{f}\) and eccentric moment \(\Delta\tau\) deriving from unexpected centroid shifts, actuator failures and external disturbances have a huge impact on the control performance of the designed controller (47). In order to overcome this obstacle, based on the characteristics of system uncertainty (11), eccentric moment (9) and LOE fault of actuator (12), the following adaptive laws are designed as  
* **System uncertainty**. By means of the Jacobi identity of the cross product, \(a\times(b\times c)+b\times(c\times a)+c\times(a\times b)=0\), where \(a,b\) are the vectors. In view of (11), we have \[f(\tilde{\rho}) =m\tilde{\rho}\times\hat{v}_{A}+m\omega\times(\tilde{\rho}\times v _{A})+m\
u_{A}\times(\omega\times\tilde{\rho})-\tilde{\rho}\times W\] \[=m\tilde{\rho}\times\hat{v}_{A}-m\tilde{\rho}\times(v_{A}\times \omega)-\tilde{\rho}\times W=m\tilde{\rho}\times\Phi\] (48) where \(\Phi=\hat{v}_{A}-v_{A}\times\omega-\frac{\Psi}{m}\), remarked as \(\Phi=[\phi_{1}\,\phi_{2}\,\phi_{3}]^{T}\). From this it follows that \[-\tilde{\rho}\times\Phi=\left[ \begin{array}{ccc}\phi_{2}\Delta z-\phi_{3} \Delta y\\ \phi_{3}\Delta x-\phi_{1}\Delta z\\ \phi_{1}\Delta y-\phi_{2}\Delta x\end{array} \right]=\left[ \begin{array}{ccc} 0&-\phi_{3}&\phi_{2}\\ \phi_{3}&0&-\phi_{1}\\ -\phi_{2}&\phi_{1}&0\end{array} \right]\left[ \begin{array}{c}\Delta x\\ \Delta y\\ \Delta z\end{array} \right]=\Phi^{\times}\tilde{\rho}\] (49) Subsequently, its adaptation law was developed as \[\dot{\phi}_{1}=\Psi_{1}({\Phi^{\times}}^{T}\tilde{\omega}-k_{\hat{\phi}_{1} }\hat{\theta}_{1})\] (50) where \(\hat{\theta}_{1}\) is a designed adaptive variable to estimate the unexpected effects of system uncertainties caused by centroid shift. Further, \(\Psi_{1}\in R^{3\times 3}\) represents a designed positive definite symmetric parameter matrix, and \(k_{\hat{\theta}_{1}}\) stands for a constant coefficient, satisfying \(k_{\hat{\theta}_{1}}>0\). * **Eccentric moment**. For clarity, the effect of the eccentricity moment is rewritten as follows. \[\Delta\tau=\left[ \begin{array}{ccc}0&-Z_{a}&Y_{a}\\ Z_{a}&0&-X_{a}\\ -Y_{a}&X_{a}&0\end{array} \right]\left[ \begin{array}{c}\Delta x\\ \Delta y\\ \Delta z\end{array} \right]=\Theta^{\times}\tilde{\rho}\] (51) and the corresponding adaptation law is developed as \[\dot{\theta}_{2}=\Psi_{2}({\Theta^{\times}}^{T}\tilde{\omega}-k_{\dot{\theta }_{2}}\dot{\theta}_{2})\] (52) where \(\dot{\theta}_{2}\) is also an adaptive state to obtain the effect of the eccentricity moment. \(\Psi_{2}\in R^{3\times 3}\) represents a design with positive definite symmetric coefficient matrix, and \(k_{\dot{\theta}_{2}}\) is also a positive parameter. * **LOE fault of actuator and external disturbance**. For actuator faults and external perturbations \(d_{s}\), the complex perturbation \(D_{f}\) is defined as \[D_{f}=d_{f}-\lambda u_{\tau}\] (53) where \(\lambda\) stands for the failure factor of the actuator fault, as shown in (12). Furthermore, combined with Assumption 3, the bounds of \(D_{f}\) are introduced as \(\kappa_{d}\), satisfying \(\left\|D_{f}\right\|\leq\kappa_{d}\), where \(\kappa_{d}>0\). Furthermore, an adaptive estimator is introduced as \[\dot{k}_{d}=P(\tilde{\omega}^{T}\tanh(\tilde{\omega}/\varsigma_{1})-k_{\kappa _{d}}\dot{\kappa}_{d})\] (54) where \(\dot{\kappa}_{d}\) is also utilized to estimate the influence of disturbance and LOE fault of actuator. \(k_{\kappa_{d}}\) and \(\varsigma_{1}\) are both positive constants.