# Engineering Applications of Adaptive Kalman Filtering Based on Singular Value Decomposition (SVD)
## 5 Numerical Experiments: Applications in Navigation
Figure 6:0 Figure 6:1  
To estimate the aircraft position using the above mathematical expressions, it is necessary to clear x and y as a function of trigonometric functions, and the equations are solved using linearization by expansion:  
Figure 6:2  
Figure 6:3  
Writing Equations (50) and (51) in matrix form, we have:  
\[\left( \begin{array}{c}\partial z1\\ \partial z2\end{array} \right)=\left[ \begin{array}{cc}\frac{\partial f1(x,y) }{\partial x}&\frac{\partial f1(x,y)}{\partial y}\\ \frac{\partial f2(x,y)}{\partial x}&\frac{\partial f2(x,y)}{\partial y}\\ \end{array} \right]\left( \begin{array}{c}\partial x\\ \partial y\end{array} \right), \tag{52}\]  
\[\left( \begin{array}{c}\partial z1\\ \partial z2\end{array} \right)=\left[ \begin{array}{cc}\frac{\partial f1(x,y) }{\partial x}&\frac{\partial f1(x,y)}{\partial y}\\ \frac{\partial f2(x,y)}{\partial x}&\frac{\partial f2(x,y)}{\partial y}\\ \end{array} \right]\left( \begin{array}{c}\partial x\\ \partial y\end{array} \right). \tag{53}\]  
Thus, the evaluation of the partial derivatives leads to:  
\[\frac{\partial f1(x,y)}{\partial x}=\frac{x-x_{1}}{z1}=\sin(\theta 1), \tag{54}\]  
\[\frac{\partial f1(x,y)}{\partial y}=\frac{y-y_{1}}{z1}=\cos(\theta 1), \tag{55}\]  
\[\frac{\partial f2(x,y)}{\partial x}=\frac{x-x_{2}}{z2}=-\sin(\theta 2), \tag{56}\]  
\[\frac{\partial f2(x,y)}{\partial y}=\frac{y-y_{2}}{z2}=\cos(\theta 2). \tag{57}\]  
Substituting in (53), we have:  
\[\left( \begin{array}{c}\partial z1\\ \partial z2\end{array} \right)=\left[ \begin{array}{cc}\sin(\theta 1)&\cos( \theta 1)\\ -\sin(\theta 2)&\cos(\theta 2)\end{array} \right]\left( \begin{array}{c} \partial x\\ \partial y\end{array} \right). \tag{58}\]  
Then,  
\[\left( \begin{array}{c}\Delta z1\\ \Delta z2\end{array} \right)=\left[ \begin{array}{cc}\sin(\theta 1)&\cos( \theta 1)\\ -\sin(\theta 2)&\cos(\theta 2)\end{array} \right]\left( \begin{array}{c} \Delta x\\ \Delta y\end{array} \right). \tag{59}\]  
Finally, we notice that the equation above can be generalized further because the cosine and sine terms represent the line of sight into the two DME channels, along with direction cosine between the y- and x-axes [6]. Please note the connection between the observables, z2, and the estimated quantity, x, y, is not linear. Therefore,  
\[H=\left[ \begin{array}{cc}\sin(\theta 1)&\cos(\theta 1)\\ -\sin(\theta 2)&\cos(\theta 2)\end{array} \right]. \tag{60}\]  
Matrix H is sometimes called the linearized matrix. Applying the inverse matrix method, we have:  
\[\left( \begin{array}{c}\Delta x\\ \Delta y\end{array} \right)=H^{-1}\cdot\left( \begin{array}{c}\Delta z1\\ \Delta z2\end{array} \right). \tag{61}\]Figure 21 depicts the results after running Equations (47)-(61) in MATLAB code:  
As seen in Figure 21, the error does not go to infinity when the nominal trajectory goes through the origin, because the assumed aircraft dynamics (i.e., random walk) gives the filter some memory, and all the weight is not placed on the current measurement. From the figure, it can be seen that the aircraft reaches the origin in about 1063 s. What it is illustrated here is the classical error dilution of precision (DOP) when the line of sight is 180\({}^{\circ}\) apart relative to the aircraft. The detected peak is the greatest in the y-axis, and the error occurs due to the mathematical effect of navigating the DME geometry on the position measurement. After that, the position is calculated using conventional KF and K-SVD, and the general formulation is:  
\[\underline{\mathrm{x}}_{k+1}=\mathrm{A.}\underline{\mathrm{x}}_{k}+\mathrm{B.}\underline{\mathrm{u}}_{k}+\underline{\mathrm{w}}_{k}, \tag{62}\]  
\[\underline{\mathrm{z}}_{k}=\mathrm{C_{k}.}\underline{\mathrm{x}}_{k}+ \underline{\mathrm{v}}_{k}. \tag{63}\]  
The covariance of \(\underline{\mathrm{w}}\)k is Qk and that of \(\underline{\mathrm{v}}\)k is Rk, then \(\mathrm{W}\ \sim\ \mathrm{N}(0,\mathrm{Q})\) and \(\mathrm{V}\ \sim\mathrm{N}(0,\mathrm{R})\). The state equations are then:  
\[\left(