## 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
In conclusion, combined with (50), (52) and (54), the nominal fault-tolerant controller of quick-loop (47) can be further developed as \[\dot{\tilde{\Xi}} =Q(-F^{T}(\tilde{\omega},\
u,\dot{\
u})\tilde{\omega}-k_{\tilde{ \Xi}}\tilde{\Xi}),\ \dot{\theta}_{1}=\Psi_{1}({\Phi^{\times}}^{T}\tilde{\omega}-k_{\dot{ \theta}_{1}}\dot{\theta}_{1}),\] \[\dot{\theta}_{2} =\Psi_{2}({\Theta^{\times}}^{T}\tilde{\omega}-k_{\dot{\theta}_{ 2}}\dot{\theta}_{2}),\ \dot{\kappa}_{d}=P(\tilde{\omega}^{T}\tanh(\tilde{\omega}/\varsigma_{1})-k_{ \kappa_{d}}\dot{\kappa}_{d})\] \[u_{\tau} =-K\tilde{\omega}+F(\tilde{\omega},\
u,\dot{\
u})\tilde{\Xi}-{ \Phi^{\times}}\dot{\theta}_{1}-\kappa_{d}\tanh(\tilde{\omega}/\varsigma_{1})- \Theta^{\times}\dot{\theta}_{2}\] (55)  
Eventually, based on the Lyapunov stability theorem, a theoretical analysis is performed to summarize the main results of this section and guarantee the stability of the closed-loop control system. Theorem 3.2: _Considering the quick-loop dynamics of a hypersonic vehicle (7) suffered from the variation of inertial matrix (8), eccentric moment (9), system uncertainty (11), actuator failures (12) and external disturbances, \(d_{f}\), adaptive feedback controls (55) without resorting to the exact knowledge of inertial matrix \(J\) by means of the adaptive laws (50), (52) and (54) developed to ensure that the quick-loop system of HSV can track expected signals, converging to a tiny compact set of the origin, and all the closed signals are bounded._  
Proof: First, let us choose a positive definite Lyapunov function candidate as  
\[V_{f}=\frac{1}{2}\tilde{\omega}^{T}J\tilde{\omega}+\frac{1}{2}\tilde{\Xi}^{T }Q^{-1}\tilde{\Xi}+\frac{1}{2}\theta_{1}^{T}\Psi_{1}^{-1}\tilde{\theta}_{1}+ \frac{1}{2}\theta_{2}^{T}\Psi_{2}^{-1}\tilde{\theta}_{2}+\frac{1}{2}\tilde{ \kappa}_{d}^{T}P^{-1}\tilde{\kappa}_{d} \tag{56}\]  
where the estimate errors of \(\tilde{\Xi},\tilde{\theta_{1}},\ \tilde{\theta_{2}}\) and \(\kappa_{d}\) are defined as \(\tilde{\Xi}=\Xi-\hat{\Xi},\tilde{\theta_{1}}=\theta_{1}-\tilde{\theta_{1}},\ \tilde{\theta_{2}}=\theta_{2}-\tilde{\theta_{2}},\tilde{\kappa_{d}}= \kappa_{d}-\kappa_{d}\). Then, invoking Assumption 2 and (55), differentiating (56) gives  
\[\dot{V}_{f} =\tilde{\omega}^{T}J(-J^{-1}(\tilde{\omega}+\
u)^{\times}J(\tilde{ \omega}+\
u)-\hat{\
u}+J^{-1}u_{\tau}+J^{-1}\Phi^{\times}\tilde{\rho}+J^{-1}D_{ f}+J^{-1}\Theta^{\times}\tilde{\rho})\] \[\quad+\tilde{\Xi}^{T}Q^{-1}\dot{\tilde{\Xi}}+\tilde{\theta_{1}}^{ T}\Psi_{1}^{-1}\dot{\theta_{1}}+\tilde{\theta_{2}}^{T}\Psi_{2}^{-1}\dot{ \theta_{2}}+\kappa_{d}^{T}P^{-1}\kappa_{d}\] \[=\tilde{\omega}^{T}J(-J^{-1}F(\tilde{\omega},\
u,\hat{\
u})\tilde {\Xi}+J^{-1}\Phi^{\times}\tilde{\rho}+J^{-1}\Theta^{\times}\tilde{\rho}+J^{-1 }D_{f}+J^{-1}(-K\tilde{\omega}-\hat{\kappa_{d}}\tanh(\tilde{\omega}/\zeta_{1})\] \[D_{f}=d_{f}-\lambda u_{\tau}\]0  
\[D_{f}=d_{f}-\lambda u_{\tau}\]1  
To continue, submitting the adaptive laws (50), (52) and (54) into (57) can further give  
\[D_{f}=d_{f}-\lambda u_{\tau}\]2 \[D_{f}=d_{f}-\lambda u_{\tau}\]3 \[D_{f}=d_{f}-\lambda u_{\tau}\]4  
Subsequently, by taking the following inequalities  
\[D_{f}=d_{f}-\lambda u_{\tau}\]5 \[D_{f}=d_{f}-\lambda u_{\tau}\]6  
into consideration, (59) can be further deduced as  
\[D_{f}=d_{f}-\lambda u_{\tau}\]7 \[D_{f}=d_{f}-\lambda u_{\tau}\]8  
Furthermore, two indirect designed parameters are adopted as  
\[D_{f}=d_{f}-\lambda u_{\tau}\]9 \[\dot{k}_{d}=P(\tilde{\omega}^{T}\tanh(\tilde{\omega}/\varsigma_{1})-k_{\kappa _{d}}\dot{\kappa}_{d})\]0  
Finally, (60) is reshaped as  
\[\dot{k}_{d}=P(\tilde{\omega}^{T}\tanh(\tilde{\omega}/\varsigma_{1})-k_{\kappa _{d}}\dot{\kappa}_{d})\]1  
By multiplying \(e^{Ht}\) into both sides of (63) and integrating at the interval \([t_{0},t)\), we have  
\[\dot{k}_{d}=P(\tilde{\omega}^{T}\tanh(\tilde{\omega}/\varsigma_{1})-k_{\kappa _{d}}\dot{\kappa}_{d})\]2  
where \(t_{0}\) is the initial actuation time and \(t>t_{0}\).