# Engineering Applications of Adaptive Kalman Filtering Based on Singular Value Decomposition (SVD)
## 5 Numerical Experiments: Applications in Navigation
From the figures above, it can be inferred that with all things being equal (the only difference in the MATLAB source code is the algorithm to compute the Kalman filter), K-SVD adaptive has a better prediction accuracy than the conventional KF and traditional factored KFs. This means that K-SVD adaptive is more robust than the current known factor-based algorithms, at the same time providing a better filtering capability when it performs the same operations. The algorithm captures the information that matters, which is the N-dimensional subspace of \(R^{N}\) spanned by the first columns of V or the right orthogonal factor [8; 9]. By performing a matrix multiplication, the algorithm realizes a low dimensionality reduction that captures much of the variability and corresponds with the selection of the eigenvectors with the largest eigenvalues (since SVD orders them from the largest to smallest). Additionally, by adapting to variations in the covariance measurement error, the algorithm has a higher accuracy. Figures 14-19 represent the results of Table 4. By observing Figure 14, it is easy to see that K-SVD adaptive model 6 performs better for the north position. K-SVD model 5 also performs relatively well compared with the conventional KF. Similar results can be inferred from Figure 15; K-SVD adaptive model 6 is the best performer, followed by K-SVD adaptive models 3 and 5. Conclusive results can also be seen for the downward position RMS error in Figure 5:0 Model 5 performs better, independent of the filter type. Model 6 is also a good performer for downward velocity, and the worst is model 3. Figure 5:1 Vector norm of the absolute errors at the north position of the dynamic state at different speeds for models 3, 5, and 6. Figure 5:2 Plan view of the estimated and true locations at 360 kph using the K-SVD adaptive algorithm. Figure 5:3 Vector norm of the absolute errors at the downward position of the dynamic state at different speeds for models 3, 5, and 6. Figure 5:4 Vector norm of the absolute errors at the north velocity of the dynamic state at different speeds for models 3, 5, and 6. Figure 5:5 Vector norm of the absolute errors at the east position of the dynamic state at different speeds for models 3, 5, and 6. Figure 17 illustrates the results for the north velocity RMS. From this figure, K-SVD adaptive model 6 is the best performer, followed closely by K-SVD model 5. From Figure 18, the best performer is K-SVD adaptive model 5, and finally, regarding the downward speed velocity RMS from Figure 19, K-SVD models 3 and 6 perform relatively well. random variables, \(x_{0}\sim N\left(60,10\text{ km}^{2}\right)\), \(y_{0}\sim N\left(210,10\text{ km}^{2}\right)\), sample time T = 2 s, \(g=10\) m/s, and the maximum deviation in the acceleration for the aircraft is \(\sigma_{a\ max}\)= 0.001 \(\frac{m}{sec^{2}}\). Suppose that two DME stations are found on the x-axis, as shown in Figure 5:6 The positions of the DME stations are identified. The aircraft follows a circular path at the end of the trajectory, at a constant speed with the same velocity as the linear motion, turning for 270\({}^{\circ}\) with a gyro of 20 km. The trajectory is depicted in Figure 21 and assumed to be level, with constant velocity plus small perturbations caused by small random accelerations. This model is only practical for short intervals. For navigation based on estimation, it is necessary to extrapolate the position or initial state of the airplane, which is supposedly known, to a posterior time step using the dynamic equation of the process and knowing the excitation that disturbs the system. To obtain distance z1 and z2, it is necessary to have the following geometric relations for both DMEs; the measurements in terms of x and y are then:  
Figure 5:7  
Figure 5:8 Estimated parameters. Figure 5:9 Distance measuring equipment (DME)-to-DME problem geometry.