# Engineering Applications of Adaptive Kalman Filtering Based on Singular Value Decomposition (SVD)
## 1 Introduction  
By defining what robustness means in the context of this research, we can focus on the engineering applications of the proposed method. We refer to robustness as the \"relative insensitivity of the solution to errors of some sort\" [1]; also, we refer to numerical stability to denote robustness against round-off errors [2]. The design of the Kalman filter (KF) usually aims to \"ensure that the covariance matrix of estimation uncertainty (the dependent variable of the matrix Riccati equation) remains symmetric and positive definite\" [1,3]. The main problem to be solved in this research is the numerical robustness of KF. The proposed algorithm is useful in applications where the influence of round-off errors reduces the accuracy of the numerical solution of the associated Riccati equation. The fidelity of the solution may degrade to the point that it corrupts the Kalman gain when the Riccati equation does not stay symmetric and positive definite, and it can corrupt the estimate [1]. [PERSON] and [PERSON] haveshown that the asymmetry of \(P\) is one of the factors contributing to the numerical instability of the Riccati equation [4]. The problem was circumvented by [PERSON] [5] using unit triangular matrix diagonal (UD) factorization, which was one of several alternatives available to those in need of computationally efficient and stable algorithms. This decomposition of the covariance matrix is of the form:  
\[P=UDUU^{T}, \tag{1}\]  
where \(D\) denotes the diagonal matrix and \(U\) is the upper triangular matrix with ones along its main diagonal. One of the individual characteristics of UD factorization and the Cholesky method is that there is no need to solve simultaneous Equations [6]. Additionally, UD factorization does not use a square root numerical operation [6], but more recently square root filtering using standard matrix decomposition has found greater acceptance [7]. In this matrix,  
\[S=P^{\frac{1}{2}}\text{ such that }P=SS^{T}. \tag{2}\]  
S is obtained in a triangular matrix from the well-known Cholesky decomposition [8; 9], a drawback of which is that it cannot be used if the covariance matrix is singular [6]. Both the square root and Bierman methods can be used only if one has single-dimensional measurements with uncorrelated measurement noise. Generally, one does not have this in practice [8; 9]. To overcome the limitation, another method can be used--which is one of the topics of this research--that will facilitate the solution of this type of problem, as the computation can be performed in parallel because it is formulated in the form of matrix and matrix-vector operations without the need for additional transformations [10], where the solution also is able to deal with ill-conditioned matrices. This method is singular value decomposition (SVD). The singular values (which are positive by definition) or principal standard deviation \(\sigma_{i}\) are the square roots of the diagonal elements. For the discrete time linear problem, which is evaluated in this research, the first KF based on SVD was researched by [PERSON], [PERSON], and [PERSON] [8; 9]. In this paper, the algorithm is further improved by making the filter adaptable to changes in sensor quality. The Riccati equation of the Kalman filter is an important tool for designing navigation systems, particularly those that incorporate sensors with diverse error characteristics, such as the Global Navigation Satellite System (GNSS) and the inertial navigation system (INS). The Riccati equation permits us to predict the performance of a given system design as a function of the subsystem and component error characteristics [2]. Designers can then search for the mixture of components that will reach a predefined set of performance endpoints at the lowest framework cost. The design of the Kalman filter is aimed at making sure the estimation uncertainty of the covariance matrix (the dependent variable of the matrix Riccati equation) remains symmetric and positive definite [1].