## 4 FTGC Design-Based LOS Angular Rate and the Target Motion Estimation Information
### FTGC Design
* Step 2, Virtual angular velocity command design The STT maneuver is adopted and the command bank angle is set to be zero. Treating \(\mathbf{x}_{3}\) as the virtual control input of the second dynamical equation of (13), this step aims to design a virtual angular velocity command \(\mathbf{x}_{3d}\) to drive \(\mathbf{x}_{2}\) to converge to \(\mathbf{x}_{2d}= \begin{bmatrix}0&(\mathbf{x}_{2d}^{\sharp})^{T}\end{bmatrix} ^ {T}\). Defining \(\mathbf{\delta}_{2}=\mathbf{x}_{2}-\mathbf{x}_{2d}\) and \(\mathbf{\delta}_{2f}=\mathbf{x}_{2c}-\mathbf{x}_{2d}\), it will come to \[\dot{\mathbf{\xi}}_{2}=\mathbf{f}_{2}(\mathbf{x}_{2})+\mathbf{\delta}_{2}\mathbf{ x}_{3}+\mathbf{d}_{2}-\dot{\mathbf{x}}_{2d}\] (34) where \(\dot{\mathbf{x}}_{2d}\) can be calculated through (30). Following the steps of the backstepping techniques, the virtual control input, \(\mathbf{x}_{3}\), is designed as \[\mathbf{x}_{3c}=\mathbf{g}_{2}^{-1}\bigg{(}-\mathbf{f}_{2}(\mathbf{x}_{2})-\mathbf{ K}_{2}\mathbf{\delta}_{2}-(\mathbf{g}_{1}^{T}\mathbf{x}_{1})^{\star}+\dot{ \mathbf{x}}_{2d}-\hat{d}_{2}\frac{\mathbf{\delta}_{2}}{\|\mathbf{\delta}_{2}\|} \bigg{)}\] (35) where \(\left(\mathbf{g}_{1}^{T}\mathbf{x}_{1}\right)^{\star}= \begin{bmatrix}0& \mathbf{x}_{1}^{T}\mathbf{\delta}_{1}\end{bmatrix} ^{T}\). \(\mathbf{K}_{2}\) is the positive-definite gain matrix and \(\hat{d}_{2}\) is the estimate value of \(d_{2\max}\), provided by the following adaptive law: \[\dot{\hat{d}}_{2}=\lambda_{2}\|\mathbf{\delta}_{2}\|\] (36) where \(\lambda_{2}\) is the positive real number. Similarly, a low-pass filter \[\tau_{3}\dot{\mathbf{x}}_{3d}=-\mathbf{x}_{3d}+\mathbf{x}_{3c}-\tau_{3} \mathbf{g}_{2}^{T}\mathbf{\delta}_{2}\] (37) is adopted. Define \(\mathbf{\delta}_{3}=\mathbf{x}_{3}-\mathbf{x}_{3d}\) and \(\mathbf{\delta}_{3f}=\mathbf{x}_{3c}-\mathbf{x}_{3d}\), and select a Lyapunov function as \[V_{2}=V_{1}+\frac{1}{2}\mathbf{\delta}_{2f}^{T}\mathbf{\delta}_{2f}+\frac{1}{2 }\mathbf{\delta}_{2}^{T}\mathbf{\delta}_{2}+\frac{1}{2\lambda_{2}}\big{(}\hat{d }-d_{2\max}\big{)}^{2}\] (38) The time derivative of \(V_{2}\) can be directly given, considering (30), (33)-(36), by \[\begin{array}{ll}\dot{V}_{2}&=\dot{V}_{1}+\mathbf{\delta}_{2f}^{T}\dot{ \mathbf{\delta}}_{2f}+\mathbf{\delta}_{2}^{T}\mathbf{\delta}_{2}+(\hat{d}_{2}-d _{2\max})\|\mathbf{\delta}_{2}\|\\ &\leq-\mathbf{K}_{1}\mathbf{x}_{1}^{T}\mathbf{x}_{1}-\mathbf{K}_{2}\mathbf{ \delta}_{2}^{T}\mathbf{\delta}_{2}-\mathbf{x}_{1}^{T}\mathbf{\delta}_where [PERSON]'s inequality is applied and \(\mu_{1}\) is an arbitrary constant which only helps to proceed with the analysis. If the boundary layer error, \(\mathcal{S}_{3}\) and \(\mathcal{S}_{3f}\), can be ignored, and as long as \(\mu_{1}\leq\tau_{2}/2\) is satisfied, it will eventually yield  
\[\dot{V}_{2}\leq-\chi_{21}\
orm{\mathbf{x}_{1}}^{2}-\chi_{22}\
orm{\mathbf{S}_{2}}^{2}- \chi_{23}\
orm{\mathbf{S}_{2f}}^{2}+\chi_{24} \tag{40}\]  
where \(\chi_{21}=\lambda_{\min}(\mathbf{K}_{1})\), \(\chi_{22}=\lambda_{\min}(\mathbf{K}_{2})\), \(\chi_{23}=1/\tau_{2}-1/2\mu_{1}\), \(\chi_{24}=\mu_{1}\
orm{\dot{\mathbf{x}}_{2c}}^{2}/2\). Define the following compact set as  
\[ \begin{split}\mathcal{Q}_{S11}&=\left\{\mathbf{x}_{1}: \
orm{\mathbf{x}_{1}}\leq\sqrt{\chi_{24}/\chi_{21}}\right\}\\ \mathcal{Q}_{S12}&=\left\{\mathcal{S}_{2}:\
orm{\mathbf{S}_ {2}}\leq\sqrt{\chi_{24}/\chi_{22}}\right\}\\ \mathcal{Q}_{S13}&=\left\{\mathcal{S}_{2f}:\
orm{ \mathbf{S}_{2f}}\leq\sqrt{\chi_{24}/\chi_{23}}\right\}\end{split} \]  
and it can be analyzed that the system trajectories will eventually converge to the compact subset defined by \(Q_{S1}=Q_{S11}\cap Q_{S12}\cap Q_{S13}\), which contains the origin and can be set to an arbitrarily small size by selecting the appropriate control parameters. Therefore, according to the discussion above, we can conclude that \(\lim_{t\rightarrow\infty}\mathbf{x}_{1}=\lim_{t\rightarrow\infty}\mathcal{S}_{2 }=\lim_{t\rightarrow\infty}\mathcal{S}_{2f}=0\).