## 4 FTGC Design-Based LOS Angular Rate and the Target Motion Estimation Information
### FTGC Design
* Step 3, Fin deflection command In this step, a control law is designed for fin deflections \(\delta\) with fault tolerance ability to track the virtual angular velocity command \(\mathbf{x}_{3d}\), provided by the low-pass filter (37). With the definition of \(\mathcal{S}_{3}\), and considering the potential actuator fault (12), the dynamics of the tracking error can be given by \[\dot{\mathcal{S}}_{3}=f_{3}(\mathbf{x}_{2},\mathbf{x}_{3})+\mathcal{g}_{3}\rho(t) \delta+\mathbf{d}_  
can be derived. Therefore, the time derivative of \(V_{3}\) is  
\[ \begin{split}\dot{V}_{3}&=\dot{V}_{2}+S_{3f}^{T}\Big{(} \dot{x}_{3c}-\frac{\mathbf{S}_{3f}}{\tau_{3}}+g_{2}^{T}\mathbf{S}_{2}\Big{)}+S_ {3}^{T}\mathbf{S}_{3}+(c_{3}-\rho_{\min}\xi_{3})\phi_{m}\|\mathbf{S}_{3}\|\\ &\leq-\chi_{21}\|x_{1}\|^{2}-\chi_{22}\|S_{2}\|^{2}-\chi_{23}\| \mathbf{S}_{2f}\|^{2}+\chi_{24}+S_{2}^{T}\mathbf{S}_{2}\mathbf{S}_{3}+S_{3f}^{ T}\dot{x}_{3c}-\frac{S_{3f}^{T}\mathbf{S}_{3f}}{\tau_{3}}\\ &+S_{3}^{T}\Big{(}f_{3}(x_{2},x_{3})+g_{3}\rho(t)\delta+\delta_{4 }-\dot{x}_{3d}\Big{)}+(c_{3}-\rho_{\min}\xi_{3})\phi_{m}\|\mathbf{S}_{3}\|\\ &\leq-\chi_{21}\|x_{1}\|^{2}-\chi_{22}\|S_{2}\|^{2}-\chi_{23}\| \mathbf{S}_{2f}\|^{2}+\chi_{24}+S_{3f}^{T}\dot{x}_{3c}-\frac{S_{3f}^{T}\mathbf{ S}_{3f}}{\tau_{3}}\\ &+S_{3}^{T}\mathbf{S}_{3}\rho(t)\delta+c_{3}\phi_{m}\|\mathbf{S}_ {3}\|+(c_{3}-\rho_{\min}\xi_{3})\phi_{m}\|\mathbf{S}_{3}\|\\ \end{split}  \tag{46}\]  
Substituting the control law (42) into (46) will further yield  
\[ \begin{split}\dot{V}_{3}&\leq-\chi_{21}\|x_{1}\|^{2}- \chi_{22}\|\mathbf{S}_{2}\|^{2}-\chi_{23}\|\mathbf{S}_{2f}\|^{2}+\chi_{24}+S_ {3f}^{T}\dot{x}_{3c}-\frac{S_{3f}^{T}\mathbf{S}_{3f}}{\tau_{3}}-\rho_{\min}\| \mathbf{S}_{3}\|^{2}\\ &\leq-\chi_{21}\|x_{1}\|^{2}-\chi_{22}\|\mathbf{S}_{2}\|^{2}- \chi_{23}\|\mathbf{S}_{2f}\|^{2}-\rho_{\min}\|\mathbf{S}_{3}\|^{2}-\left( \frac{1}{5}-\frac{1}{2\rho_{2}}\right)\|\mathbf{S}_{3f}\|^{2}+\frac{\mu_{2}}{ 2}\|\dot{x}_{3c}\|^{2}+\chi_{24}\end{split}  \tag{47}\]  
where \(\mu_{2}\) is an arbitrary constant. If \(\tau_{3}\leq 2\mu_{2}\) is satisfied, the following compact set is defined by  
\[ \begin{split}\mathrm{Q}_{521}&=\left\{x_{1}:\|x_{1} \|\leq\sqrt{(\chi_{24}+\mu_{2}\|\dot{x}_{3c}\|^{2}/2)/\chi_{21}}\right\}\\ \mathrm{Q}_{522}&=\left\{\mathbf{S}_{2}:\|\mathbf{S}_ {2}\|\leq\sqrt{(\chi_{24}+\mu_{2}\|\dot{x}_{3c}\|^{2}/2)/\chi_{22}}\right\}\\ \mathrm{Q}_{523}&=\left\{\mathbf{S}_{2f}:\|\mathbf{S}_ {2}f\|\leq\sqrt{(\chi_{24}+\mu_{2}\|\dot{x}_{3c}\|^{2}/2)/\chi_{23}}\right\} \\ \mathrm{Q}_{524}&=\left\{\mathbf{S}_{3}:\|\mathbf{S}_ {2}\|\leq\sqrt{(\chi_{24}+\mu_{2}\|\dot{x}_{3c}\|^{2}/2)/\rho_{\min}}\right\} \\ \mathrm{Q}_{525}&=\left\{\mathbf{S}_{3f}:\|\mathbf{S}_ {2}f\|\leq\sqrt{(\chi_{24}+\mu_{2}\|\dot{x}_{3c}\|^{2}/2)/(1/\tau_{3}-1/2\mu _{2})}\right\}\end{split} \]  
which can be small enough by selecting proper control parameters. Thus, it can be concluded that the LOS angular rates and tracking errors are also UUB, and then the control target can be achieved. **Remark 2**. : _The low-pass filters in (30) and (37) both have extra items \(\tau_{2}g_{1}^{T}x_{1}\) and \(\tau_{3}g_{2}^{T}\mathbf{S}_{2}\), compared with the low-pass filters described in [35; 36]. These extra items help to eliminate the influence of the boundary layer errors caused by the low-pass filter. For example, as in (39), the item \(S_{2}^{T}\mathbf{S}_{2}\mathbf{S}_{3f}\) may break the stability of the system. To avoid this, the item \(\tau_{3}g_{2}^{T}\mathbf{S}_{2}\) in (37) will counteract the same item in (39)._