Transactions of the Korean Society of Mechanical Engineers A (대한기계학회논문집A)

The Korean Society of Mechanical Engineers (대한기계학회)
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Robust Kalman Filter Design via Selecting Performance Indices

성능지표 선정을 통한 강인한 칼만필터 설계

  • 정종철 (한양대학교 대학원 정밀기계공학과) ;
  • 허건수 (한양대학교 기계공학부)
  • Published : 2005.01.01
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Abstract

In this paper, a robust stationary Kalman filter is designed by minimizing selected performance indices so that it is less sensitive to uncertainties. The uncertainties include not only stochastic factors such as process noise and measurement noise, but also deterministic factors such as unknown initial estimation error, modeling error and sensing bias. To reduce the effect on the uncertainties, three performance indices that should be minimized are selected based on the quantitative error analysis to both the deterministic and the stochastic uncertainties. The selected indices are the size of the observer gain, the condition number of the observer matrix, and the estimation error variance. The observer gain is obtained by optimally solving the multi-objectives optimization problem that minimizes the indices. The robustness of the proposed filter is demonstrated through the comparison with the standard Kalman filter.

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References

  1. Kalman, R. E. and Bucy, R. S., 1961, 'New Results in Linear Filtering and Prediction Theory,' Trans. of the ASME Series D: J. of Basic Engineering, Vol. 83, No.3, pp. 95-108 https://doi.org/10.1115/1.3658902
  2. Bernstein, D. S. and Haddad, W. M., 1989, 'Steady-State Kalman Filtering with an $H_{\infty}$ Error Bound,' Systems and Control Letters, Vol. 12, No.1, pp. 9-16 https://doi.org/10.1016/0167-6911(89)90089-3
  3. Khargonekar, P. P. and Rotea, M. A., 1992, 'Mixed $H_2$/H_{\infty}$, Filtering,' Proc. of the 31st Conference on Decision and Control, pp. 2299-2304
  4. Chen, X. and Zhou, K., 2002, '$H_{\infty}$ Gaussian Filter on Infinite Time Horizon,' IEEE Trans. on Circuits and Systems-I: Fundamental Theory and Applications, Vol. 49, No.5, pp. 674-679 https://doi.org/10.1109/TCSI.2002.1001957
  5. I. R. Petersen, Optimal guaranteed cost control and filtering for uncertain linear systems, IEEE translation on Automatic Control, AC-37 pp. 1971-1977, 1994 https://doi.org/10.1109/9.317138
  6. Xie, L. and Soh, Y. C., 1994, 'Robust Kalman Filtering for Uncertain Systems,' Systems and Control Letters, Vol. 22, No.2, pp. 123-129 https://doi.org/10.1016/0167-6911(94)90106-6
  7. Fu, M., de Souza, C. E. and Luo, Z. Q., 2001, 'Finite-Horizon Robust Kalman Filter Design,' IEEE Trans. on Automatic Control, Vol. 49, No.9, pp. 2103-2112 https://doi.org/10.1109/78.942638
  8. Geromel, J. C., 1999, 'Optimal Linear Filtering Under Parameter Uncertainty,' IEEE Trans. on Signal Processing, Vol. 47, No. 1, pp. 168- 175 https://doi.org/10.1109/78.738249
  9. Shaked, U., Xie, L. and Soh, Y. C., 2001, 'New Approaches to Robust Minimum Variance Filter Design,' IEEE Trans. on Signal Processing, Vol. 49, No.11, pp. 2620-2629 https://doi.org/10.1109/78.960408
  10. Bertsekas, D. P. and Rhodes, I. B., 1971, 'Recursive State Estimation for a Set-membership Description of Uncertainty,' IEEE Trans. on Automatic Control, Vol. 16, No.2, pp. 117-128 https://doi.org/10.1109/TAC.1971.1099674
  11. Jain, B. N., 1975, 'Guaranteed Error Estimation in Uncertain Systems,' IEEE Trans. on Automatic Control, Vol. 20, No.2, pp. 230-232 https://doi.org/10.1109/CDC.1974.270407
  12. Savkin, A. V. and Petersen, I. R., 1995, 'Recursive State Estimation for Uncertain Systems with an Integral Quadratic Constraint,' IEEE Trans. on Automatic Control, Vol. 40, No.6, pp. 1080-1083 https://doi.org/10.1109/9.388688
  13. Kwon, S., Chung, W. K. and Youm, Y., 2003, 'A Combined Observer for Robust State Estimation and Kalman Filtering,' Proc. of the American Control Conference, pp. 2459-2464 https://doi.org/10.1109/ACC.2003.1243444
  14. Mehra, R. K., 1970, 'On the Identification of Variances and Adaptive Kalman Filtering,' IEEE Trans. on Automatic Control, Vol. 15, No.2, pp. 175-184 https://doi.org/10.1109/TAC.1970.1099422
  15. Sage, A. P. and Melsa, J. L., 1971, Estimation Theory with Application to Communications and Control, McGraw-Hill, Inc.
  16. Lee, B. S., 1997, 'Quantitative Performance Comparison of Observers with Uncertainties,' M. S. Thesis, Hanyang University, Seoul, Korea
  17. Jung, J. and Huh, K., 2002, 'Design of the Well-Conditioned Observer Using the Non-Normality Measure,' Trans. of the KSME, A, Vol. 26, No.6, pp. 1114-1119 https://doi.org/10.3795/KSME-A.2002.26.6.1114
  18. Huh, K. and Stein, J. L., 1994, 'A Quantitative Performance Index for Observer-Based Monitoring Systems,' Trans. of the ASME: J of Dynamic Systems, Measurement, and Control, Vol. 116, pp. 487-497. https://doi.org/10.1115/1.2899243
  19. Golub, G. H. and Van Loan, C. F., 1996, Matrix Computations, 3rd Ed., The Johns Hopkins University Press
  20. Sasa, S., 1998, 'Robustness of a Kalman Filter Against Uncertainties of Noise Covariances,' Proc. of the American Control Conference, pp. 2344-2348 https://doi.org/10.1109/ACC.1998.703050
  21. Huh, K., Jung, J. and Stein, J. L., 2001, 'DiscreteTime Well-Conditioned State Observer Design and Evaluation,' Trans. of the ASME:.J of Dynamic Systems, Measurement, and Control, Vol. 123, No.4, pp.615-622 https://doi.org/10.1115/1.1409553
  22. Boyd, S., El Ghaoui, L., Feron, E. and Balakrishnan, V., 1994, Linear Matrix Inequalities in System and Control Theory, SIAM
  23. El Ghaoui, L., Delebeque, F. and Nikoukhah, R., 1999, LMITOOL Ver. 2.1, available via anonymous ftp to ftp.ensta.fr, directory pub/elghaoui/lmitool-2.1
  24. Oishi, J. and Balakrishnan, V., 2000, 'Linear Controller Design for the NEC Laser Bonder via Linear Matrix Inequality Optimization,' In Advances in Linear Matrix Inequality Methods in Control (Edited by El Ghaoui, L. and Niculescu, S.-L.), SIAM, Philadelphia, pp. 295-307