## 4 FTGC Design-Based LOS Angular Rate and the Target Motion Estimation Information
### LOS Angular Rate Reconfiguration and Target Acceleration Estimation  
As mentioned previously, the strapdown seeker can only provide the body-LOS angles, \(q_{Bz}\) and \(q_{By}\). According to (11), it has  
\[\left\{ \begin{array}{l}q_{z}=\arcsin(L_{gb}(2,1)\cos q_{By}\cos q_{Bz}+L_{gb }(2,2)\sin q_{Bz}-L_{gb}(2,3)\sin q_{By}\cos q_{Bz})\\ q_{y}=-\arctan\Big{(}\frac{L_{gb}(3,1)\cos q_{By}\cos q_{Bz}+L_{gb}(3)\sin q_{ Bz}-L_{gb}(3,3)\sin q_{By}\cos q_{Bz}}{L_{gb}(1)\cos q_{Bz}\cos q_{Bz}+L_{gb}(1,2)\sin q_{ Bz}-L_{gb}(1,3)\sin q_{By}\cos q_{Bz}}\Big{)}\end{array} \right. \tag{20}\]  
Figure 5: Control structure diagram of the flight vehicle with a strapdown seeker. That is to say, the body-LOS angles will uniquely determine the LOS angles. Although direct differentiation of \(q_{z}\) and \(q_{y}\) can obtain the LOS angular rates, it may magnify the measurement noise. Therefore, the ESO technique is applied. Letting \(\mathbf{x}_{0}= \begin{bmatrix}q_{z}&q_{y}\end{bmatrix} ^{T}\), \(\mathbf{a}_{\text{Ms}}= \begin{bmatrix}a_{\text{sy}}&a_{\text{sz}}\end{bmatrix} ^{T}\) and \(\mathbf{a}_{\text{Ts}}= \begin{bmatrix}a_{\text{Tsy}}&a_{\text{Tsz}}\end{bmatrix} ^{T}\), (6) will furthermore yield  
\[\left\{ \begin{array}{l}\dot{\mathbf{x}}_{0}=\mathbf{x}_{1}\\ \dot{\mathbf{x}}_{1}=f_{o}(\mathbf{x}_{0},\mathbf{x}_{1})+g_{o}\mathbf{a}_{\text{Ms}}-g_{o}\bm {a}_{\text{Ts}}\\ \mathbf{y}_{0}=\mathbf{x}_{0}\end{array} \right. \tag{21}\]  
in which \(\mathbf{f}_{o}(\mathbf{x}_{0},\mathbf{x}_{1})\) and \(\mathbf{g}_{o}\) are given as  
\[\mathbf{f}_{o}= \begin{bmatrix}-\frac{2\hat{R}}{R}\dot{q}_{z}-\dot{q}_{y}^{2}\cos q _{z}\sin q_{z}\\ -\frac{2\hat{R}}{R}\dot{q}_{y}+2\dot{q}_{y}\dot{q}_{z}\tan q_{z}\end{bmatrix}  \cdot\mathbf{g}_{o}= \begin{bmatrix}-\frac{1}{R}&0\\ 0&\frac{1}{R\cos q_{z}}\end{bmatrix}  \tag{22}\]  
**Remark 1**. : _The body-aligned accelerometer can measure the apparent acceleration in the body reference frame while the gravity acceleration in the inertia frame can also be calculated. With the measured body-LOS angles and calculated LOS angles, \(\mathbf{a}_{\text{Ms}}\) can be obtained easily._  
In what follows, a linear ESO is designed as [34]  
\[\left\{ \begin{array}{l}\dot{\mathbf{e}}_{o}=\mathbf{e}_{1}-\mathbf{\beta}_{0}\mathbf{e}_{o} \\ \dot{\mathbf{e}}_{1}=-g_{o}\mathbf{e}_{2}-\mathbf{\beta}_{1}\mathbf{e}_{o}\\ \dot{\mathbf{e}}_{2}=-\mathbf{\beta}_{2}\mathbf{e}_{o}-\dot{\mathbf{a}}_{\text{Ts}}\end{array} \right. \tag{23}\]  
where \(\mathbf{\beta}_{0}\), \(\mathbf{\beta}_{1}\), and \(\mathbf{\beta}_{2}\) are the diagonal observer gains to be designed. The observing errors are designed as \(\mathbf{e}_{1}=\mathbf{z}_{1}-\mathbf{x}_{1}\) and \(\mathbf{e}_{2}=\mathbf{z}_{2}-\mathbf{a}_{\text{Ts}}\). Considering (21)-(23), this gives  
\[\left\{ \begin{array}{l}\dot{\mathbf{e}}_{o}=\mathbf{e}_{1}-\mathbf{\beta}_{0}\mathbf{e}_{o} \\ \dot{\mathbf{e}}_{1}=-g_{o}\mathbf{e}_{2}-\mathbf{\beta}_{1}\mathbf{e}_{o}\\ \dot{\mathbf{e}}_{2}=-\mathbf{\beta}_{2}\mathbf{e}_{o}-\dot{\mathbf{a}}_{\text{Ts}}\end{array} \right. \tag{24}\]  
Note that (24) can be rewritten as  
\[ \begin{bmatrix}\dot{\mathbf{e}}_{o}\\ \dot{\mathbf{e}}_{1}\\ \dot{\mathbf{e}}_{2}\end{bmatrix} = \begin{bmatrix}-\mathbf{\beta}_{0}&\mathbf{E}_{2}&0\\ -\mathbf{\beta}_{1}&0&-g_{o}\\ -\mathbf{\beta}_{2}&0&0\end{bmatrix}\begin{bmatrix}\mathbf{e}_{o}\\ \mathbf{e}_{1}\\ \mathbf{e}_{2}\end{bmatrix} + \begin{bmatrix}0\\ 0\\ -\dot{\mathbf{a}}_{\text{Ts}}\end{bmatrix}  \tag{25}\]  
Apparently, if the selection of the observer gains, \(\mathbf{\beta}_{0}\), \(\mathbf{\beta}_{1}\), and \(\mathbf{\beta}_{2}\), makes the matrix  
\[\mathbf{A}_{o}= \begin{bmatrix}-\mathbf{\beta}_{0}&\mathbf{E}_{2}&0\\ -\mathbf{\beta}_{1}&0&-\mathbf{g}_{o}\\ -\mathbf{\beta}_{2}&0&0\end{bmatrix} .\]  
be Hurtwig, the observer errors will be Uniform Ultimate Boundedness (UUB), as long as \(\left|\dot{\mathbf{a}}_{\text{Ts}}\right|<\infty\).