## 4 FTGC Design-Based LOS Angular Rate and the Target Motion Estimation Information
### FTGC Design  
In the previous subsection, the LOS angular rates, \(\dot{q}_{z}\) and \(\dot{q}_{y}\), can be estimated by the ESO, and the attitude vector \(\mathbf{x}_{2}\) and angular velocity \(\mathbf{x}_{3}\) can be measured. Thus, the full states of (13) are available. Furthermore, the ESO can also provide the estimated target acceleration components, \(a_{Tsy}\) and \(a_{Tsz}\). In what follows, the FTGC technique is developed. Recall that (13) is a cascade dynamic system with a matched or unmatched disturbance. The control procedure includes three steps, motivated by the backstepping techniques. * Step 1, Virtual attitude command design With the reconfigured target accelerations \(z_{2}\), and estimated LOS angular rates \(z_{1}\), promoting \(\mathbf{x}_{2}^{\#}\) as the virtual control input, design \(\mathbf{x}_{2c}^{\#}\) as \[\mathbf{x}_{2c}^{\#}=g_{1}^{-1}\bigg{(}-\overset{\kappa}{f}_{1}(\mathbf{x}_{1})-\mathbf{K }_{1}\mathbf{z}_{1}-\hat{d}_{1}\frac{\mathbf{x}_{1}}{\|\mathbf{x}_{1}\|}\bigg{)}\] (28) where \(\overset{\kappa}{f}_{1}(\mathbf{x}_{1})\) is the estimated vector with the ESO's outputs, \(z_{1}\) and \(z_{2}\), replacing \(\mathbf{x}_{1}\) and \(\mathbf{a}_{Tsz}\), substituting in (14). \(\mathbf{K}_{1}=diag([K_{11},K_{12}])\) is the positive gain matrix. \(\hat{d}_{1}\) represents the adaptive parameters, updated by \[\dot{\hat{d}}_{1}=\lambda_{1}\|\mathbf{x}_{1}\|\] (29) where \(\lambda_{1}\) is the positive gain. To the avoid explosion of differential, a low-pass filter is designed, motivated by the DSC technique [1], as \[\tau_{2}\mathbf{x}_{2d}^{\#}=-\mathbf{x}_{2d}^{\#}+\mathbf{x}_{2c}^{\#}-\tau_{2}g_{1}^{T} \mathbf{x}_{1}\] (30) where \(\tau_{2}\) denotes the filter time constant and \(\mathbf{x}_{2d}^{\#}\) is the state of the filter. Letting \(\mathbf{S}_{2}^{\#}=\mathbf{x}_{2}^{\#}-\mathbf{x}_{2d}^{\#}\) and \(\mathbf{S}_{2f}^{\#}=\mathbf{x}_{2c}^{\#}-\mathbf{x}_{2d}^{\#}\), and substituting (28) into (13) will yield \[\dot{\mathbf{x}}_{1}=-\mathbf{K}_{1}\mathbf{x}_{1}-\mathbf{K}_{1}\mathbf{e}_{1}+\overset{\kappa}{ f}_{1}(\mathbf{x}_{1})+g_{1}\mathbf{S}_{2}^{\#}-\hat{d}_{1}\frac{\mathbf{x}_{1}}{\|\mathbf{x}_{1 }\|}+\mathbf{d}_{1}\] (31) in which \[\overset{\kappa}{f}_{1}(\mathbf{x}_{1})=\overset{\kappa}{f}_{1}(\mathbf{x}_{1})- \overset{\kappa}{f}_{1}(\mathbf{x}_{1})\]. The upper bound of \[\mathbf{e}_{1}\] has been analyzed in Section 4.1, which will further yield the boundness of \[\overset{\kappa}{f}_{1}(\mathbf{x}_{1})\]. Therefore, as stated earlier, it can be implied that there must exist an unknown positive constant \[\overset{\kappa}{\delta}_{1}\] satisfying \(\|-\mathbf{K}_{1}e_{1}+\tilde{f}_{1}(\mathbf{x}_{1})+\mathbf{d}_{1}\|\leq\overline{ \delta}_{1}\). Additionally, to analyze the stability of the subsystem, a Lyapunov function candidate is selected as  
\[V_{1}=\frac{\mathbf{x}_{1}^{T}\mathbf{x}_{1}}{2}+\frac{\left(\hat{d}_{1}- \overline{\delta}_{1}\right)^{2}}{2\lambda_{1}} \tag{32}\]  
whose time derivatives can be given by  
\[ \begin{array}{ll}\dot{V}_{1}&=-\mathbf{K}_{1}\mathbf{x}_{1}^{T}\mathbf{x}_{1 }-\mathbf{x}_{1}^{T}\mathbf{K}_{1}\mathbf{e}_{1}+\mathbf{x}_{1}^{T}\dot{f}_{1 }(\mathbf{x}_{1})+\mathbf{x}_{1}^{T}\mathbf{\delta}_{1}\mathbf{\delta}_{2}^{ \sharp}-\mathbf{x}_{1}^{T}\mathbf{\delta}_{1}\mathbf{\delta}_{2}^{\sharp}- \hat{d}_{1}\|\mathbf{x}_{1}\|+\mathbf{x}_{1}^{T}\hat{d}_{1}+(\hat{d}_{1}- \overline{\delta}_{1})\|\mathbf{x}_{1}\|}\\ &\leq-\mathbf{K}_{1}\mathbf{x}_{1}^{T}\mathbf{x}_{1}+\mathbf{x}_{1}^{T} \mathbf{\delta}_{1}\mathbf{\delta}_{2}^{\sharp}-\mathbf{x}_{1}^{T}\mathbf{ \delta}_{1}\mathbf{\delta}_{2}^{\sharp}\end{array}  \tag{33}\]  
As a result, if the boundary layer error \(\mathbf{\xi}_{2}^{\sharp}-\mathbf{\xi}_{2f}^{\sharp}\) converges to a zero vector, i.e., \(\mathbf{\xi}_{2}^{\sharp}-\mathbf{\xi}_{2f}^{\sharp}=0\), it has \(\dot{V}_{1}\leq-\mathbf{K}_{1}\mathbf{x}_{1}^{T}\mathbf{x}_{1}\). According to the Lyapunov theorem and invariant set lemma, the asymptotical stability of this subsystem can be concluded.