Estimation Properties of Kalman Filter for the System with Unobservable Bias

관측 불가능한 바이어스가 있는 시스템의 칼만필터 추정특성

  • Published : 2001.10.01

Abstract

By showing the existence of the ARE solution and the convergence property of the DRE solution, this paper proves that a Kalman filter for the linear system with the unobservable bias is stable. It is also shown that the Kalman filter has a biased steady state estimation error whose covariance is affected mainly by the unobservable bias. Finally, the results are illustrated through a 2nd order system example including the inertial navigation system.

Keywords

References

  1. B. Friedland, 'Treatment of bias in recursive filtering,' IEEE Trans. Automatic Control, vol. AC-14, pp. 359-367, Aug., 1969 https://doi.org/10.1109/TAC.1969.1099223
  2. M. B. Ignagni, 'An alternate derivation and extension of Friedland's two-stage Kalman filter,' IEEE Trans. Automatic Control, vol. 26, pp. 746-750, 1981 https://doi.org/10.1109/TAC.1981.1102697
  3. A. T. Alouani, P. Xia, T. R. Rice, and W. D. Blair, 'On the optimality of the two stage state estimation in the presence of random bias,' IEEE Trans. Automatic Control, vol. 38, pp. 1279-1282, Aug., 1993 https://doi.org/10.1109/9.233168
  4. C. S. Hseih and F. C. Chen, 'Optimal solution of the two stage Kalman estimator,' in Proc. 34th Conf. Decision and Control, New Orleans, pp. 1532-1537, Dec. 1995 https://doi.org/10.1109/CDC.1995.480355
  5. R. E. Kalman and R. S. Bucy, 'New results in linear filtering and prediction theory,' Trans. ASME, Journal of Basic Engineering, vol. 82, 1960, pp. 95-108, Mar., 1961
  6. W. M. Wonham, 'On a matrix Riccati equation of stochastic control,' SIAM J. Control, vol. 6, pp. 681-698, 1968 https://doi.org/10.1137/0306044
  7. K. Martensson, 'On the matrix Riccati eqation,' Inform. Sci. vol. 3, pp. 17-49, 1971 https://doi.org/10.1016/S0020-0255(71)80020-8
  8. V. Kucera, 'On nonnegative definite solutions to matrix quadratic equations,' Automatica, vol. 8, pp. 413-423, 1972 https://doi.org/10.1016/0005-1098(72)90100-8
  9. S. C. Chan, G. C. Goodwin, and K. S. Sin, 'Convergency properties of the Riccati difference equation in optimal filtering of nonstabilizable systems,' IEEE Trans. Automatic Control, vol. AC-29, pp. 110-118, 1984 https://doi.org/10.1109/TAC.1984.1103465
  10. M. A. Poubelle, I. R. Peterson, M. R. Gevers, and R. R. Bitmead, 'A miscellany of results on an equation of Count J. F. Riccati,' IEEE Trans. Automatic Control, vol. AC-31, pp. 651-654, 1986 https://doi.org/10.1109/TAC.1986.1104355
  11. P. Lancaster, L. Rodman, Algebraic Riccati Equations, Oxford University Press, 1995
  12. S. Bittanti, A. J. Laub, J. C. Willems, The Riccati Equations, Springer-Verlag, Berlin, 1991
  13. D. O. Anderson, 'Stability properties of Kalman-Bucy Filters,' J. Franklin Inst., vol. 291, no. 2, pp. 137-144, Feb., 1971 https://doi.org/10.1016/0016-0032(71)90016-0
  14. G. D. Nicolao and M. R. Gevers, 'Difference and differential Riccati equations : A note on the convergence to the strong solution,' IEEE Trans. Automatic Control, vol. AC-37, pp. 1055-1057, 1992 https://doi.org/10.1109/9.148372
  15. R. Bitmead, M. R. Gevers, I. R. Peterson, and R. J. Kaye, 'Monotonicity and stabilizablity properties of solutions of the Riccati difference equation : Proposition, lemmas, theorems, fallacious conjectures and counter examples,' Syst. Contr. Lett., vol. 5, pp. 309-315, Feb., 1985 https://doi.org/10.1016/0167-6911(85)90027-1
  16. H. Kwakernaak and R. Sivan, Linear Optimal Control systems, Jhon Wiley & Sons, New York, pp. 110-111, 1972
  17. D. O. Anderson and J. B. Moore, Linear Optimal Control, Prentic-Hall, Englewood Cliff, pp. 31-38, 1971
  18. E. Kreindler and A. Jameson, 'Conditions for non-negativeness of partitioned matrices,' IEEE Trans. Automatic Control, pp. 147-148, Feb., 1972