# Engineering Applications of Adaptive Kalman Filtering Based on Singular Value Decomposition (SVD)
## 5 Numerical Experiments: Applications in Navigation  
The computer round-off for floating-point arithmetic is always demarcated by a single parameter \(\epsilon_{roundoff}\), called the unit round-off error, which is demarcated as \(1+\epsilon_{roundoff\equiv 1}\) in machine precision. In MATLAB, the parameter \"eps\" satisfies the definition above [2]. Round-off errors are a side effect of computer arithmetic using fixed or floating-point data words with a fixed number of bits [2]. The next example demonstrates that round-off can still be a problem when implementing KF in MATLAB environments and how a problem that is well-conditioned can be made ill-conditioned by the filter implementation [2]. The problem is taken from [PERSON] and [PERSON], and it is significant, as it shows the filter's observational behavior. The example does not make use of the Algorithm 1 adaptive computation from Equations (29) and (30), and the resulting algorithm is called K-SVD. **Example 1**. : _Let \(\mathrm{I_{n}}\) denote the \(n\times n\) identity matrix. Considering the filtering problem with the measurement sensitivity matrix:_  
\[H=\left[ \begin{array}{ccc}1&1&1\\ 1&1&1+\delta\end{array} \right],\]  
_and covariance matrices._  
\(P_{0}=I_{3}\) and \(R=\delta^{2}I_{2}\); to simulate round-off, it is assumed that:  
\(\delta^{2}<e_{roundoff}\) but \(\delta>e_{roundoff}\). In this case, although \(H\) has rank = 2 in machine precision, the product \(H_{k}P_{0}^{-}H_{k}^{T}\) with round-off will equal:  
\[\left[ \begin{array}{cc}3&3+\delta\\ 3+\delta&3+2\delta\end{array} \right],\]  
which is singular, and the results remain unchanged when \(R\) is added to \(H_{k}P_{0}^{-}H_{k}^{T}\). All the KF tests were carried out with the same precision (64-bit floating-point) in MATLAB running on a standard personal computer with an Intel(R) Core(TM) processor, i5-4200U CPU @ 1.60 GHz 2.30 GHz, 8.0 GB RAM, where the unit round-off error is \(2^{-53}\sim 1.11.10^{-16}\). The MATLAB eps function is twice the unit round-off error [2, 22]. Hereafter, the set of numerical experiments is repeated for various values of the ill-conditioned parameter \(\delta\); as such, delta tends toward the machine precision limit \(\delta\rightarrowe_{roundoff}\). This example does not prove the robustness of the solution of the UD, square root Kalman (SQRK), and K-SVD methods. Full implementation would require temporal updates; however, observational updates have created difficulties with traditional implementation [2]. Figure 1 illustrates how the conventional KF and a few other choice implementation methods work on various ill-conditioned experiments as the conditioned parameter \(\delta\to 0\). For this particular example, the square root-based (SQRK-based Carlson), Cholesky-based (UD-based Bierman), and K-SVD-based methods appear to degrade more gracefully than the others. The SQRK, UD decomposition, and K-SVD solutions also retain nine digits (\(\approx 30\) bits) of accuracy at \(\delta\approx\sqrt{\epsilon}\), while the other methods have no bits of accuracy in the calculated solution [1]. The other methods worth mentioning in this research are conventional KF, Potter square root filter, and Joseph stabilization. The problem analyzed corresponds to an ill-conditioned intermediate result where \(R^{*}=HPH^{T}+R\) for inversion in the Kalman gain formula in Equation (8). In this case, although \(H\) has rank = 2 in machine precision, the product \(HP_{0}\ H^{T}+R\) is not invertible, and the observational filter updates fail. The results are provided graphically in Figure 1, showing that the K-SVD-based method performs as well as the UD-based and SQRK-based Kalman filter methods. Examples 2 and 3 as outlined were run one time--i.e., the research team performed a simulation.