# Engineering Applications of Adaptive Kalman Filtering Based on Singular Value Decomposition (SVD)
## 1 Introduction
In this research, we design an adaptive Kalman filter implementation based on SVD, provide its derivation, and discuss the stability issues numerically. The filter is derived by substituting the SVD of the covariance matrix into the conventional discrete Kalman filter equations after its initial propagation and adaptive estimation of the covariance measurement matrix \(R_{k}\). One application of SVD in navigation is to apply it in calculating position errors to decompose the geometric matrix G, where error analysis with SVD is used to construct the pseudo inverse of G, making the solution numerically stable. While the linear least squares (LLS) is somewhat susceptible to round-up errors, where the normal equation is sometimes close to singular, SVD fixes the round-up problem and produces a solution that is the best approximation in the least-squares sense [11]. Different applications in the area of ionospheric tomography, as the problem to be solved, encounter the same ill-conditioned issue for the design matrix, which is composed of contributions of unknown parameters containing the ionospheric electron density (IED) in the total electron content (STEC) measurements, which is often not square, ill-posed, and ill-conditioned. Many methods have been employed to circumvent the problem of ill-posedness in the GNSS tomographic equation [12]. Those algorithms are iterative and include the Kalman filter, singular value decomposition, and algebraic reconstruction [7; 9]. In the field of industrial electronic applications where numerical stability is necessary, an example is sensorless motor control, where the advantages of estimating the speed/position of the rotor include reduced hardware complexity, lower cost, reduced machine drive size, the elimination of the sensor cable, noise immunity, and increased reliability [13]; for example, a motor without a position sensor is suitable for harsh environments (such as submerged pumps). Other applications include diagnostics, the fault-tolerant control of AC drives, distributed generation and storage systems, robotics, vision and sensor fusion technologies, signal processing and instrumentation applications, and the real-time implementation of the Kalman filter for industrial control systems [13]. We briefly discuss the numerical stability of the Kalman filter related to industrial control systems and sensor fusion for cardiorespiratory signal extraction, but focus on numerical examples in aerospace applications like GNSS navigation. The safe and efficient operation of control systems requires computer-based failure detection algorithms [13], where diagnosis may be used to provide fault-tolerant control. Several failures can occur in electrical systems, and redundant design is one of the solutions to assure the continuity and, therefore, the reliability of operations. Kalman filters can be used in fault-tolerant controls, for example, in aircraft control, home and municipal appliances (gas turbines, cooling systems, and heat pumps), and electric vehicles. A challenge in the real-time implementation of Kalman filters is the high computational complexity, and among the main difficulties is numerical stability when computing the covariance matrices as well the computational load and execution time. There are three possible solutions to circumvent these issues, and one solution is to modify the algorithm, which is the scope of this paper. In addition, an efficient digital architecture to implement the estimator [13], and parallel computing can be used. The observability problem physically means that one or more state variables (or linear combinations) are hidden from the view of the observer (i.e., the measurements) [6]. This is a challenge in the sensorless control of AC drives, which can be accentuated when the stator frequency is close to zero and the rotor voltage carries small values.