# Engineering Applications of Adaptive Kalman Filtering Based on Singular Value Decomposition (SVD)
## 7 Conclusions and Future Scope  
The computation of the novel Kalman-SVD algorithm is based on the computation of the singular value decomposition of an unsymmetrical matrix without explicitly forming its left orthogonal factor that has a high dimension [19]. K-SVD does a truncation that keeps only the top \(n\) right singular vectors by setting \(V^{T}\) (where superscript T denotes the transpose of a matrix and n denotes the number of rows), equal to the n rows of \(V^{T}\) (n \(\times\) n matrix). That is equivalent to removing or zeroing out m number of singular values from the original matrix (m + n) by -n, where (m + n) denotes the rows, therefore reducing the dimension, which is far less than the original dimension, thus filtering out the noise.  
The other conclusion of this research is related to the robustness of the adaptive algorithm. Compared with other KF-based decomposition algorithms, K-SVD adaptive can handle round-off errors, which makes the algorithm useful in critical applications, making it possible to deal with singular systems that are ill-conditioned.  
From the analysis of example 3, we can conclude that the accuracy of position and speed can be improved by choosing the model that performs the best for the worst-case scenario, which in this case is the highly dynamic conditions of maneuvering a vehicle. The tracking capabilities can also be increased by adding to the K-SVD algorithm the adaptive computation of the measurement noise matrix as a time-variant matrix. From the analysis of example 4, we can conclude that K-SVD is suitable to deal with real dynamic conditions for ground navigation systems. From the analysis of the literature, it seems that one of the drawbacks of using K-SVD and KF in general is their high computational load. In that sense, K-SVD can be used in applications where the KF is currently used with the advantages of robustness and accuracy.  
The algorithm developed in this research also has superior accuracy compared with regular decomposition-based KF algorithms by adapting the measurement matrix at each step and at the same time using the simplest form to update the error of the covariance matrix P, which is the most common equation to compute the algorithm. Additionally, the proposed Algorithm 1 does not require matrix inverting, and the system of interest does not need to have a square shape.  
Our effort in this paper is guided by the increasing utilization of Kalman filtering in operations that demand high reliability, where the robustness of the algorithm is a necessary condition to perform critical operations. More work can be done in future K-SVD adaptive applications by including the Joseph stabilized form instead of the conventional form of the Riccati equation to avoid the propagation of antisymmetric problems and fine-tuning the process noise covariance matrix \(Q_{k}\). We believe that the key points and principles derived here will find their way to applications in high-speed signal processing by combining K-SVD, K-SVD adaptive and SVD compressive sensing [25]. For future research, we intend to explore particular technology uses.  
**Author Contributions:** [PERSON] designed the mathematical model of the proposed algorithm and wrote the paper, [PERSON] performed the experiments, and [PERSON] analyzed the data. All authors have read and agreed to the published version of the manuscript.  
**Funding:** This research received no external funding.  
**Conflicts of Interest:** The authors declare no conflict of interest.