## 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
Furthermore, by defining that \(V_{f}(t_{0})e^{-\mathrm{H}(t-t_{0})}+\frac{\mathrm{M}}{\mathrm{H}}(1-e^{-\mathrm{H} (t-t_{0})})=\mathrm{N}(t)\), combined with the definition of \(V_{f}\) in (56), the boundedness of attitude angle rate error can be obtained as  
\[\dot{k}_{d}=P(\tilde{\omega}^{T}\tanh(\tilde{\omega}/\varsigma_{1})-k_{\kappa _{d}}\dot{\kappa}_{d})\]3where \(\lambda_{\min}(J)\) is the minimum eigenvalue of inertia matrix \(J\). Moreover, note that \(\mathrm{N}(t)\to\mathrm{M}/\mathrm{H}\) as \(t\to\infty\). As a result, \(|\tilde{\omega}_{i}(t)|\leq\sqrt{\mathrm{N}(t)/\lambda_{\min}(J)},i=1,2,3\). In addition, to guarantee the boundedness of the system parameters, the adaptive update law is modified by the parameter projection algorithm. The compact sets \(\Pi_{\underline{\Sigma}}\), \(\Pi_{\underline{\Phi}_{1}}\), \(\Pi_{\underline{\Phi}_{2}}\) and \(\Pi_{\underline{\delta}_{f}}\) are defined to restrain the above adaptive laws and the details are shown as follows [19]  
\[\dot{k}_{d}=P(\tilde{\omega}^{T}\tanh(\tilde{\omega}/\varsigma_{1})-k_{\kappa _{d}}\dot{\kappa}_{d})\]4 \[\dot{k}_{d}=P(\tilde{\omega}^{T}\tanh(\tilde{\omega}/\varsigma_{1})-k_{\kappa _{d}}\dot{\kappa}_{d})\]5  
where \(\underline{\Xi}\), \(\underline{\theta}_{1}\), \(\underline{\theta}_{2}\), \(\underline{\kappa}_{d}\) and \(\underline{\Xi}\), \(\overline{\theta_{1}}\), \(\overline{\theta_{2}}\), \(\overline{\kappa_{d}}\) are the lower and upper bounds of adaptive laws. Then, the adaptive laws of \(\underline{\hat{\Xi}}\), \(\hat{\theta_{1}}\), \(\hat{\theta_{2}}\) and \(\hat{\kappa_{d}}\) are modified as follows  
\[\dot{k}_{d}=P(\tilde{\omega}^{T}\tanh(\tilde{\omega}/\varsigma_{1})-k_{\kappa _{d}}\dot{\kappa}_{d})\]6 \[\dot{k}_{d}=P(\tilde{\omega}^{T}\tanh(\tilde{\omega}/\varsigma_{1})-k_{\kappa _{d}}\dot{\kappa}_{d})\]7  
where  
\[\mathrm{Proj}(\underline{\hat{\Xi}},\underline{\hat{\Xi}})=\left\{ \begin{array}{ccc}0&if&\hat{\Xi}=\underline{\Xi},\underline{\hat{\Xi}}>0 \\ 0&if&\hat{\Xi}=\underline{\Xi},\underline{\hat{\Xi}}<0\\ Q(-F^{T}(\tilde{\omega},\
u,\hat{\upsilon})\tilde{\omega}-k_{\underline{\hat{ \Xi}}})&\end{array} \right.\] \[\mathrm{Proj}(\hat{\theta_{i}},\hat{\theta_{i}})=\left\{ \begin{array}{ccc}0&if&\hat{\theta_{i}}=\overline{\hat{\theta_{i}}},\hat{ \theta_{i}}>0\\ 0&if&\hat{\theta_{i}}=\underline{\theta_{i}},\hat{\theta_{i}}<0\\ \Psi_{i}({\Phi^{\times}}^{T}\tilde{\omega}-k_{\hat{\theta_{i}}}\hat{\theta_{ i}})&\end{array} \right.\] \[\mathrm{Proj}(\hat{\kappa_{d}},\hat{\kappa_{d}})=\left\{ \begin{array}{ccc}0&if&\hat{\kappa_{d}}=\overline{\kappa_{d}},\hat{ \kappa_{d}}>0\\ 0&if&\hat{\kappa_{d}}=\underline{\kappa_{d}},\hat{\kappa_{d}}<0\\ P(\tilde{\omega}^{T}\tanh(\tilde{\omega}/\varsigma_{1})-k_{\kappa_{d}}\hat{ \kappa}_{d})&\end{array} \right. \tag{67}\]  
Finally, since by assumption \(\omega_{c}\) and \(\dot{\omega_{c}}\) are bounded, it can be obtained that \(F(\tilde{\omega},\omega_{c},\dot{\omega_{c}})\) is bounded as well. Since the boundedness of \(\underline{\hat{\Xi}},\hat{\theta_{1}},\hat{\theta_{2}}\), \(\hat{\kappa_{d}}\) is already guaranteed by the use of the parametric projection operator, the stability of the closed-loop system is ensured and all the signals of the HSV fast-loop are bounded.