A study of QR decomposition and Kalman filter implementations

Abstract: With the rapid development of new technologies during the last decades, the demand of complex algorithms to work in real-time applications has increased considerably. To achieve the real time expectations and to assure the stability and accuracy of the systems, the application of numerical methods and matrix decompositions have been studied as a trade-off between complexity, stability and accuracy. In the first part of this thesis, a survey of state-of-the-art QR Decomposition methods applied to matrix inversion is done. Stability and accuracy of these methods are analyzed analytically and the complexity is studied in terms of operations and level of parallelism. Besides, a new method called Modified Gaussian Elimination (MGE) is proposed. This method is shown to have better accuracy and less complexity than the previous methods while keeping good stability in real time applications. In the second part of this thesis, different techniques of extended Kalman Filter implementations are discussed. The EKF is known to be numerically unstable and various methods have been proposed in the literature to improve the performance of the filter. These methods include square-root and unscented versions of the filter that make use of numerical methods such as QR, LDL and Cholesky Decomposition. At the end of the analysis, the audience/reader will get some idea about best implementation of the filter given some specifications.