## 3 Preliminaries
### Fault-Tolerant Integrated Guidance and Control Problem Formulation  
The actuator fault may cause significant results, and even the failure of the task. The actuator fault in this paper is described as [32]  
\[\mathbf{\delta}_{F}=\mathbf{\rho}(t)\mathbf{\delta} \tag{12}\]  
where \(\mathbf{\delta}= \begin{bmatrix}\delta_{x}&\delta_{y}&\delta_{z}\end{bmatrix} ^{T}\) is the fin deflection command and \(\mathbf{\delta}_{F}\) is the actual actuator output. \(\mathbf{\rho}(t)=diag(\rho_{x}(t),\rho_{y}(t),\rho_{z}(t))\) is the fin healthy factor matrix indicating the efficiency of the fins. Apparently, \(0\leq\rho_{i}(t)\leq 1,(i=x,y,z)\) always holds. \(\rho_{i}(t)=0\) denotes the complete failure, i.e., this fin cannot work completely. \(\rho_{i}(t)=1\) indicates that no fault occurs in this fin. Let \(\mathbf{x}_{1}= \begin{bmatrix}\dot{q}_{z}&\dot{q}_{y}\end{bmatrix} ^{T},\mathbf{x}_{2}= \begin{bmatrix}\gamma&\alpha&\beta\end{bmatrix} ^{T},\mathbf{x}_{3}=\mathbf{\omega}= \begin{bmatrix}\omega_{x}&\omega_{y}&\omega_{z}\end{bmatrix} ^{T}\). Then, according to Sections 3.1 and 3.2, the integrated guidance and control model of the flight vehicle can be summarized as  
\[\left\{ \begin{array}{l}\dot{\mathbf{x}}_{1}=\mathbf{f}_{1}(\mathbf{x}_{1})+\mathbf{g}_{1} \mathbf{x}_{2}^{\mathbf{\beta}}+\mathbf{d}_{1}\\ \dot{\mathbf{x}}_{2}=\mathbf{f}_{2}(\mathbf{x}_{2})+\mathbf{g}_{2}\mathbf{x}_{3}+\mathbf{d}_{2}\\ \dot{\mathbf{x}}_{3}=\mathbf{f}_{3}(\mathbf{x}_{2},\mathbf{x}_{3})+g_{3}\rho(t)\mathbf{\delta}+\bm {d}_{3}\end{array} \right. \tag{13}\]  
where \(\mathbf{x}_{2}^{\mathbf{\beta}}= \begin{bmatrix}\alpha&\beta\end{bmatrix} ^{T}\). \[\mathbf{f}_{1}(\mathbf{x}_{1})= \begin{bmatrix}-\frac{2\dot{R}}{R}\dot{q}_{z}-\dot{q}_{ y}^{2}\cos q_{z}\sin q_{z}+\frac{g\cos q_{z}}{R}\\ -\frac{2\dot{R}}{R}\dot{q}_{y}+2\dot{q}\dot{q}_{z}\tan q_{z}\end{bmatrix} + \begin{bmatrix}\frac{q_{Bz}}{R}\\ \frac{-q_{Bz}}{R\cos q_{z}}\end{bmatrix}  \tag{14}\]  
\[\mathbf{g}_{1}=\frac{qS}{m} \begin{bmatrix}-\frac{1}{R}&0\\ 0&\frac{1}{R\cos q_{z}}\end{bmatrix}\begin{bmatrix}\mathbf{C}_{y}^{\mathbf{x}}L_{sk}( 2,2)&\mathbf{C}_{E}^{\beta}L_{sk}(2,3)\\ \mathbf{C}_{y}^{\mathbf{x}}L_{sk}(3,2)&\mathbf{C}_{z}^{\beta}L_{sk}(3,3)\end{bmatrix}  \tag{15}\]  
\[\mathbf{f}_{2}(\mathbf{x}_{1})= \begin{bmatrix}\frac{mg\cos\theta\cos\gamma_{x}-qS_{x}^ {\mathbf{\beta}}\mathbf{x}}{mV\cos\beta}\\ \frac{mg\cos\theta\sin\gamma_{x}+qS_{x}^{\mathbf{\beta}}\mathbf{\beta}}{mV}\end{bmatrix}  \tag{16}\]  
\[\mathbf{g}_{2}= \begin{bmatrix}\cos\alpha\sec\beta&-\sin\alpha\sec\beta&0\\ -\cos\alpha\tan\beta&\sin\alpha\tan\beta&1\\ \sin\alpha&\cos\alpha&0\end{bmatrix}  \tag{17}\]  
\[\mathbf{f}_{3}(\mathbf{x}_{2},\mathbf{x}_{3})=-\mathbf{I}^{-1}\mathbf{\tilde{\omega}}\mathbf{I}\mathbf{ \omega}+\mathbf{I}^{-1}qSL \begin{bmatrix}m_{x}^{\mathbf{\alpha}}&m_{x}^{\beta}\\ 0&m_{y}^{\mathbf{\beta}}\\ m_{x}^{\mathbf{\alpha}}&0\end{bmatrix}\begin{bmatrix}\mathbf{\alpha}\\ \mathbf{\beta}\end{bmatrix}  \tag{18}\]\[\mathbf{g}_{3}=\mathbf{I}^{-1}qSL \begin{bmatrix}m_{x}^{\delta_{x}}&0&0\\ 0&m_{y}^{\delta_{y}}&0\\ 0&0&m_{z}^{\delta_{z}}\end{bmatrix}  \tag{19}\]  
Before demonstrating the IGC design, the following assumptions are proposed. **Assumption 1**. : _Ref. [24] the complete failure of the fit is not taken into consideration. It is assumed that \(0<\rho_{i}(t)\leq 1\), \((i=x,y,z)\). Moreover, \(\rho_{i}(t)\) is assumed to be unknown. Furthermore, there is an unknown constant \(\rho_{\min}\) such that \(\lambda_{\min}(\mathbf{g}_{3}\mathbf{\rho}\mathbf{g}_{3}^{-1})\geq\rho_{\min}\)._  
**Assumption 2**. : _Ref. [23] the external disturbance \(d_{i}\) (\(i=1\), 2, 3) in (13) is assumed to be unknown but bounded. That is to say, there exists an unknown positive constant \(d_{\max}\) satisfying that \(\|\mathbf{d}_{i}\|\leq d_{\max}\)._  
**Assumption 3**. : _Ref.