# Engineering Applications of Adaptive Kalman Filtering Based on Singular Value Decomposition (SVD)
## 5 Numerical Experiments: Applications in Navigation
\begin{array}{c}\mathrm{x}_{k+1}\\ \mathrm{y}_{k+1}\\ \mathrm{v}_{\mathrm{x}\,k+1}\\ \mathrm{v}_{\mathrm{y}\,k+1}\end{array} \right)=\left[ \begin{array}{cccc}1&0&T& 0\\ 0&1&0&T\\ 0&0&1&0\\ 0&0&0&1\end{array} \right]\left( \begin{array}{c}\mathrm{x}_{k}\\ \mathrm{y}_{k}\\ \mathrm{v}_{\mathrm{x}\,k}\\ \mathrm{v}_{\mathrm{y}\,k}\end{array} \right)+\left( \begin{array}{c}\mathrm{ w}\mathrm{1}_{k}\\ \mathrm{w}\mathrm{2}_{k}\\ \mathrm{d}\mathrm{w}\mathrm{1}_{k}\\ \mathrm{d}\mathrm{w}\mathrm{1}_{k}\end{array} \right) \tag{64}\]  
\[\left( \begin{array}{c}\mathrm{x}_{k+1}\\ \mathrm{y}_{k+1}\\ \mathrm{v}_{\mathrm{x}\,k+1}\\ \mathrm{v}_{\mathrm{y}\,k+1}\end{array} \right)=\left[ \begin{array}{cccc}1&0&T &0\\ 0&1&0&T\\ 0&0&1&0\\ 0&0&0&1\end{array} \right]\left( \begin{array}{c}\mathrm{x}_{k}\\ \mathrm{y}_{k}\\ \mathrm{v}_{\mathrm{x}\,k}\\ \mathrm{v}_{\mathrm{y}\,k}\end{array} \right)+\left( \begin{array}{c}\frac{ \mathrm{7}^{2}}{2}&0\\ 0&\frac{\mathrm{7}^{2}}{2}\\ T&0\\ 0&T\end{array} \right) \tag{65}\]  
where \(\mathrm{x}_{k+1}\) and \(\mathrm{y}_{k+1}\) denote the incremental position in the horizontal and vertical plane, respectively. \[\overline{\mathrm{z}}_{k}\ =\ \left[ \begin{array}{cccc}1&0&0&0\\ 0&1&0&0\end{array} \right]\left( \begin{array}{c}\mathrm{x}_{k}\\ \mathrm{y}_{k}\\ \mathrm{v}_{\mathrm{x}\,k}\\ \mathrm{v}_{\mathrm{y}\,k}\end{array} \right)+\left( \begin{array}{c}\mathrm{ v}_{\mathrm{x}\,k}\\ \mathrm{v}_{\mathrm{y}\,k}\\ \mathrm{v}_{\mathrm{y}\,k}\end{array} \right). \tag{66}\]  
Let Q be the error in the process, and the matrix maintains the values of \(P_{k}\) from Equation (6):  
\[Q=\left[ \begin{array}{cccc}\frac{\mathrm{7}^{2}}{2}&0\\ 0&\frac{\mathrm{7}^{2}}{2}\\ T&0\\ 0&T\end{array} \right]\left[ \begin{array}{cccc}\frac{\mathrm{7}^{2}}{2}&0\\ 0&\frac{\mathrm{7}^{2}}{2}\\ T&0\\ 0&T\end{array} \right]\left[ \begin{array}{c}\mathrm{w}_{k}^{a}\end{array} \right]^{2}, \tag{67}\]  
\[\mathrm{Q}=\left[ \begin{array}{cccc}\frac{\mathrm{7}^{4}}{4}&0&\frac{ \mathrm{7}^{3}}{2}&0\\ 0&\frac{\mathrm{7}^{4}}{4}&0&\frac{\mathrm{7}^{3}}{2}\\ \frac{\mathrm{7}^{3}}{2}&0&\mathrm{7}^{2}&0\\ 0&\frac{\mathrm{7}^{3}}{2}&0&\mathrm{7}^{2}\end{array} \right][\sigma_{amax}]^{2}. \tag{68}\]  
\(\mathrm{v}_{k}\) denotes the measurement error:  
\[\mathrm{v}_{k}\ =\ \left[ \begin{array}{cc}\mathrm{\alpha x}^{2}&0\\ 0&\mathrm{\alpha y}^{2}\end{array} \right]. \tag{69}\]The results from calculating the position using conventional KF and Algorithm 1 without the adaptive computation (K-SVD) from Equations (29) and (30) are shown in Figure 22.  
From Figure 22, is it easy to see that at about 333 m from the origin, the DME presents singularity in the calculation due to DOP, and at that critical point the KF and K-SVD perform fine; however, the KF does not perform as well as K-SVD at the cusp of the circular return. Additionally, the results show how far the RMS is above the KF compared with the K-SVD in this DME example (see Figure 23). The results from Figure 23 indicate that the RMS is lower for the K-SVD filter compared with the conventional KF. When solving the observation equation to obtain the calculated position, it is likely that the error of the observables is manifested by producing erratic oscillations in the position. This makes the computed trajectory uneven, making the motion of the airplane very unstable. By using a prediction system that minimizes that noise, it is possible to produce estimates of unknown or known variables that tend to be more precise than those based on the measurement alone (see Equation (61)). It is demonstrated that advantages can be obtained when combining the position of the airplane based on observables and predictions. Given that the DME was left in place to circumvent the reliability  
Figure 23: RMS for K-SVD vs. the conventional Kalman filter. Figure 22: Position with the conventional Kalman filter and K-SVD vs. the estimated position. problem of GPS and to increase reliability, the best filter to deal with the singularities at the origin is K-SVD.