Kalman Filter Gain Elimination in Linear Estimation

In linear estimation, the traditional Kalman filter uses the Kalman filter gain in order to produce estimation and prediction of the n-dimensional state vector using the m-dimensional measurement vector. The computation of the Kalman filter gain requires the inversion of an m x m matrix in every iteration. In this paper, a variation of the Kalman filter eliminating the Kalman filter gain is proposed. In the time varying case, the elimination of the Kalman filter gain requires the inversion of an n x n matrix and the inversion of an m x m matrix in every iteration. In the time invariant case, the elimination of the Kalman filter gain requires the inversion of an n x n matrix in every iteration. The proposed Kalman filter gain elimination algorithm may be faster than the conventional Kalman filter, depending on the model dimensions.

• Nicholas D. Assimakis

• Discrete time
• linear estimation
• Kalman filter
• Kalman filter gain.

[1] B. D .O Anderson, J. B. Moore, Optimal Filtering, Dover Publications, New York, 2005.
[2] R. E. Kalman, “A new approach to linear filtering and prediction problems,” J. Bas. Eng., Trans. ASME, Ser. D, vol. 8, no. 1, pp. 34-45, 1960.
[3] J. E. Potter, “New statistical formulas. Instrumentation Laboratory,” MIT, Cambridge, Massachusetts , Space Guidance Memo 40, 1963.
[4] C. Thornton, “Triangular Covariance Factorizations for Kalman Filtering,” Ph.D. dissertation, University of California at Los Angeles, 1976.
[5] N. Assimakis, M. Adam, A. Douladiris, “Information Filter and Kalman Filter Comparison: Selection of the Faster Filter,” International Journal of Information Engineering, vol. 2, no. 1, pp. 1-5, 2012.
[6] C. D’Souza, R. Zanetti, “Information Formulation of the UDU Kalman Filter,” IEEE Transactions on Aerospace and Electronic Systems, vol. 55, no. 1, pp. 493–498, 2019.
[7] N. Assimakis, A. Kechriniotis, S. Voliotis, F. Tassis, M. Kousteri, “Analysis of the time invariant Kalman filter implementation via general Chandrasekhar algorithm,” International Journal of Signal and Imaging Systems Engineering (IJSISE), vol. 1, no. 1, pp. 51-57, 2008.
[8] M. Morf, G. S. Sidhu, T. Kailath, “Some New Algorithms for Recursive Estimation in Constant, Linear, Discrete-time Systems,” IEEE Trans. Automatic Control, vol. 19, issue. 4, pp. 315–323, 1974.
[9] M. Grewal, J. Kain, “Kalman Filter Implementation With Improved Numerical Properties,” IEEE Transactions on Automatic Control, vol. 55, issue. 9, pp. 2058-2068, 2010.
[10] D. Woodbury, J. Junkins, “On the Consider Kalman Filter. In: Proceedings of the AIAA Guidance,” Navigation and Control Conference, 2010.
[11] N. Assimakis, “A new algorithm for the steady state Kalman filter,” Neural, Parallel and Scientific Computations, vol. 14, no. 1, pp. 69-74, 2006.
[12] N. Assimakis, M. Adam, “FIR implementation of the steady state Kalman filter,” International Journal of Signal and Imaging Systems Engineering (IJSISE), vol.1, no 3/4, pp. 279-286, 2008.
[13] M. Adam, N. Assimakis, “Periodic Kalman filter: Steady state from the beginning,” Journal of Mathematical Sciences: Advances and Applications, vol. 1, no. 3, pp. 505-520, 2008.
[14] M. Skliar, W. F. Ramirez, “Implicit Kalman filtering,” Int. J. Control, nol. 66, no. 3, pp. 393-412, 1995.
[15] M. Skliar, W. F. Ramirez, “Square Root Implicit Kalman Filtering,” 13th Triennial World Congress, USA, 1996.
[16] D. H. Dini, D. P. Mandic, “Class of Widely Linear Complex Kalman Filters,” IEEE Trans. On Neural Networks and Learning Systems, vol. 23, no. 5, pp. 775-786, 2012.
[17] G. Chen, J. Wang, L. S. Shieh, “Interval Kalman filtering,” IEEE Trans. Aerospace Electron. Systems, vol. 33, pp. 250-259, 1997.
[18] G. Chen, Q. Xie, L. S. Shieh, “Fuzzy Kalman filtering,” Journal of Information Sciences, vol. 109, pp. 197-209, 1998.
[19] P. Song, “Monte Carlo Kalman filter and smoothing for multivariate discrete state space models,” The Canadian Journal of Statistics, vol. 28, no. 4, pp. 641-652, 2000.
[20] B. Chen, X. Liu, H. Zhao, J. C. Principe, “Maximum Correntropy Kalman filter,” Automatica, vol. 76, pp. 70–77, 2017.
[21] N. Assimakis, M. Adam, “Discrete time Kalman and Lainiotis filters comparison,” Int. Journal of Mathematical Analysis (IJMA), vol. 1, no. 13, pp. 635-659, 2007.