# IMM Filtering Algorithms for a Highly Maneuvering Fighter Aircraft: An Overview
## Appendix A  
This appendix presents the detailed derivation of the time evolution function with respect to the state vector. Given the nonlinear dynamic models for the NCT motions (31) and (32),  
\[\mathbf{x}_{k}=\mathbf{A}^{(j)}(\mathbf{x}_{k-1})\mathbf{x}_{k-1}+\mathbf{w}_ {k-1}^{(j)},\quad j=2,3, \tag{11}\]  
For simplicity, we drop the superscript \(j\) in (11). Then, from (33), we have  
\[\mathbf{A}(\mathbf{x}_{k-1})=\begin{bmatrix}1&0&\frac{\sin(\omega T)}{\omega} &-\frac{1-\cos(\omega T)}{\omega}\\ 0&1&\frac{1-\cos(\omega T)}{\omega}&\frac{\sin(\omega T)}{\omega}\\ 0&0&\cos(\omega T)&-\sin(\omega T)\\ 0&0&\sin(\omega T)&\cos(\omega T)\end{bmatrix}, \tag{12}\]  
\[\omega=\frac{\kappa a_{\text{m}}}{\sqrt{\dot{x}_{k-1}^{2}+\dot{y}_{k-1}^{2}}}. \tag{13}\]  
Define  
\[\eta_{1}:=\sin\omega T, \tag{14}\]  
\[\eta_{2}:=\cos\omega T, \tag{15}\]  
\[\eta_{3}:=\frac{\sin\omega T}{\omega}=\frac{\eta_{1}}{\omega} \tag{16}\]  
\[\eta_{4}:=\frac{1-\cos\omega T}{\omega}=\frac{1-\eta_{2}}{\omega}. \tag{17}\]  
Then, from (32), we obtain  
\[\mathbf{f}(\mathbf{x}_{k-1})=\mathbf{A}(\mathbf{x}_{k-1})\mathbf{x}_{k-1}, \tag{18}\]  
where  
\[\mathbf{f}(\mathbf{x}_{k-1})=\mathbf{A}(\mathbf{x}_{k-1})\mathbf{x}_{k-1}= \begin{bmatrix}\mathbf{a}_{1}^{\prime}\mathbf{x}_{k-1}\\ \mathbf{a}_{2}^{\prime}\mathbf{x}_{k-1}\\ \mathbf{a}_{3}^{\prime}\mathbf{x}_{k-1}\\ \mathbf{a}_{4}^{\prime}\mathbf{x}_{k-1}\end{bmatrix}=\begin{bmatrix}f_{1}( \mathbf{x}_{k-1})\\ f_{2}(\mathbf{x}_{k-1})\\ f_{3}(\mathbf{x}_{k-1})\\ f_{4}(\mathbf{x}_{k-1})\end{bmatrix}, \tag{19}\]  
with  
\[\mathbf{a}_{1}=\begin{bmatrix}1&0&\eta_{3}&-\eta_{4}\end{bmatrix}^{\prime}, \tag{20}\]  
\[\mathbf{a}_{2}=\begin{bmatrix}0&1&\eta_{4}&\eta_{3}\end{bmatrix}^{\prime}, \tag{21}\]  
\[\mathbf{a}_{3}=\begin{bmatrix}0&0&\eta_{2}&-\eta_{1}\end{bmatrix}^{\prime}, \tag{22}\]  
\[\mathbf{a}_{4}=\begin{bmatrix}0&0&\eta_{1}&\eta_{2}\end{bmatrix}^{\prime}. \tag{23}\]  
By dropping the subscript \(k-1\) from \(\mathbf{x}_{k-1}\), we obtain  
\[f_{i}(\mathbf{x})=\mathbf{a}_{i}^{\prime}\mathbf{x},\ \ i=1,2,...,4. \tag{24}\]Let  
\[\mathbf{x}=[x\quad y\quad\dot{x}\quad\dot{y}]^{\prime}. \tag{101}\]  
By using (100)-(101), we obtain  
\[f_{1}(\mathbf{x})=x+\eta_{3}\dot{x}-\eta_{4}\dot{y}, \tag{102}\]  
\[f_{2}(\mathbf{x})=y+\eta_{4}\dot{x}+\eta_{3}\dot{y}, \tag{103}\]  
\[f_{3}(\mathbf{x})=\eta_{2}\dot{x}-\eta_{1}\dot{y}, \tag{104}\]  
\[f_{4}(\mathbf{x})=\eta_{1}\dot{x}+\eta_{2}\dot{y}, \tag{105}\]  
By differentiating \(f_{1}(\mathbf{x}),f_{2}(\mathbf{x}),f_{3}(\mathbf{x})\) and \(f_{4}(\mathbf{x})\) with respect to \(\mathbf{x}\), we obtain  
\[\frac{\partial f_{1}(\mathbf{x})}{\partial\mathbf{x}}=\Big{[}1\quad 0\quad \Big{(}\eta_{3}+\dot{x}\frac{\partial\eta_{3}}{\partial\dot{x}}-\dot{y}\frac{ \partial\eta_{4}}{\partial\dot{x}}\Big{)}\quad\Big{(}\dot{x}\frac{\partial \eta_{3}}{\partial\dot{y}}-\eta_{4}-\dot{y}\frac{\partial\eta_{4}}{\partial \dot{y}}\Big{  
This completes the derivative of the nonlinear time evolution function \(\mathbf{f}(\mathbf{x})\) with respect to \(\mathbf{x}\). Hence  
\[\overline{\mathbf{F}}(\mathbf{x})=\Big{[}\begin{array}{cc}\frac{\partial f_{1} (\mathbf{x})}{\partial\mathbf{x}}&\frac{\partial f_{2}(\mathbf{x})}{\partial \mathbf{x}}&\frac{\partial f_{3}(\mathbf{x})}{\partial\mathbf{x}}&\frac{ \partial f_{4}(\mathbf{x})}{\partial\mathbf{x}}\end{array}\Big{]}. \tag{103}\]