State estimation for discrete linear systems with observation time-delayed noise

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January 2011, 7(1): 79-85. doi: 10.3934/jimo.2011.7.79

State estimation for discrete linear systems with observation time-delayed noise

Peng Cui ^1, , Hongguo Zhao ^2, and Jun-e Feng ^3,

1.	School of Control Science and Engineering, Shandong University, Jinan 250000, China
2.	Department of Information Science and Technology, Taishan University, Taian 271021, China
3.	School of Mathematics, Shandong University, Jinan 250000, China

Received July 2010 Revised September 2010 Published January 2011

Abstract
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State estimation problem is discussed for discrete-time systems with delays in measurement noise sequence, which is usually seen in network control and geophysical prospecting systems. An optimal recursive filter is derived via state augmentation. Dimensions of the optimal filter just are the sum of dimensions of state and observation vector. Therefore, they are not related to the size of delay. Besides, a sub-optimal recursive filter with same dimension as the original state is designed. The sub-optimal filter realizes instant optimization at current time. One example shows the effectiveness of the proposed approach.

Keywords: Kalman filtering., optimal estimation, sub-optimal estimation, Time-delayed noise.

Mathematics Subject Classification: Primary: 93E11; Secondary: 62M2.

Citation: Peng Cui, Hongguo Zhao, Jun-e Feng. State estimation for discrete linear systems with observation time-delayed noise. Journal of Industrial and Management Optimization, 2011, 7 (1) : 79-85. doi: 10.3934/jimo.2011.7.79

References:

[1]	B. D. O. Anderson and J. B. Moore, "Optimal Filtering," Prentice-Hall, Englewood Cliffs, New Jersey, 1979.
[2]	M. Basin, J. Rodriguez-Gonzalez and R. Martinez Zúniga, Optimal filtering for linear state delay systems, IEEE Trans. on Automatic Control, 50 (2005), 684-690. doi: 10.1109/TAC.2005.846599.
[3]	A. Calzolari, P. Florchinger and G. Nappo, Nonlinear filtering for markov systems with delayed observations, Int. J. Appl. Math. Comput. Sci., 19 (2009), 49-57. doi: 10.2478/v10006-009-0004-8.
[4]	Y. Chocheyras, Near field three dimensional time delay and doppler target motion analysis, in "International Conference on Acoustics, Speech, and Signal Processing," (1989), 2649-2652. doi: 10.1109/ICASSP.1989.267012.
[5]	T. Kailath, A. H. Sayed, and B. Hassibi, "Linear Estimation", Prentice-Hall, Englewood Cliffs, New Jersey, 1999.
[6]	R. E. Kalman, A new approach to linear filtering and prediction problems, Trans. ASME, Series D, Journal of Basic Engineering, 82 (1960), 35-45.
[7]	R. E. Kalman and R. S. Bucy, New results in linear filtering and prediction theory, Transactions of the ASME-Journal of Basic Engineering, 83 (1961), 95-108.
[8]	H. Kwakernaak, Optimal filtering in linear systems with time delays, IEEE Trans. on Automatic Control, 12 (1967), 169-173. doi: 10.1109/TAC.1967.1098541.
[9]	X. Lu, H. S. Zhang, W. Wang and K. L. Teo, Kalman filtering for multiple time-delay systems, Automatica, 41 (2005), 1455-1461. doi: 10.1016/j.automatica.2005.03.018.
[10]	D. MacMillan, J. Bohm, M. Gipson, R. Haas, A. Niell, T. Nilsson, A. Pany, B. Petrachenko and J. Wresnik, Simulation analysis of the geodetic performance of the future IVS VLBI2010 system, in "American Geophysical Union Fall Meeting," San Francisco, (2008), G33A-0667.
[11]	G. A. Medrano-Cerda, Filtering for linear system involving time delays in the noise process, IEEE Trans. on Automatic Control, 28 (1983), 801-803. doi: 10.1109/TAC.1983.1103318.
[12]	C. L. Su and C. N. Lu, Interconnected network state estimation using randomly delayed measurements, IEEE Trans. on Power Systems, 16 (2001), 870-878. doi: 10.1109/59.962439.
[13]	A. Subramanian and A. H. Sayed, Multiobjective filter design for uncertain stochastic time-delay systems, IEEE Trans. on Automatic Control, 49 (2004), 149-154. doi: 10.1109/TAC.2003.821422.
[14]	S. L. Sun, Linear minimum variance estimators for systems with bounded random measurement delays and packet dropouts, Signal Processing, 89 (2009), 1457-1466. doi: 10.1016/j.sigpro.2009.02.002.
[15]	Z. Wang, D. W. C. Ho and X. Liu, Robust filtering underrandomly varying sensor delay with variance constraints, IEEE Trans. on Circuits and Systtems II: Express Briefs, 51 (2004), 320-326. doi: 10.1109/TCSII.2004.829572.
[16]	E. Yaz and A. Ray, Linear unbiased state estimation under randomly varying bounded sensor delay, Applied Mathematics Letters, 11 (1998), 27-32. doi: 10.1016/S0893-9659(98)00051-2.
[17]	H. S. Zhang, X. Lu, and D. Z. Cheng, Optimal estimation for continuous-time systems with delayed measurements, IEEE Trans. on Automatic Control, 51 (2006), 823-827. doi: 10.1109/TAC.2006.874983.
[18]	H. G. Zhao, H. S. Zhang and C. H. Zhang, Optimal filtering for linear discrete-time systems with single delayed measurement, Int. J. of Control, Automation, and Systems, 6 (2008), 378-385.

show all references

References:

[1]	B. D. O. Anderson and J. B. Moore, "Optimal Filtering," Prentice-Hall, Englewood Cliffs, New Jersey, 1979.
[2]	M. Basin, J. Rodriguez-Gonzalez and R. Martinez Zúniga, Optimal filtering for linear state delay systems, IEEE Trans. on Automatic Control, 50 (2005), 684-690. doi: 10.1109/TAC.2005.846599.
[3]	A. Calzolari, P. Florchinger and G. Nappo, Nonlinear filtering for markov systems with delayed observations, Int. J. Appl. Math. Comput. Sci., 19 (2009), 49-57. doi: 10.2478/v10006-009-0004-8.
[4]	Y. Chocheyras, Near field three dimensional time delay and doppler target motion analysis, in "International Conference on Acoustics, Speech, and Signal Processing," (1989), 2649-2652. doi: 10.1109/ICASSP.1989.267012.
[5]	T. Kailath, A. H. Sayed, and B. Hassibi, "Linear Estimation", Prentice-Hall, Englewood Cliffs, New Jersey, 1999.
[6]	R. E. Kalman, A new approach to linear filtering and prediction problems, Trans. ASME, Series D, Journal of Basic Engineering, 82 (1960), 35-45.
[7]	R. E. Kalman and R. S. Bucy, New results in linear filtering and prediction theory, Transactions of the ASME-Journal of Basic Engineering, 83 (1961), 95-108.
[8]	H. Kwakernaak, Optimal filtering in linear systems with time delays, IEEE Trans. on Automatic Control, 12 (1967), 169-173. doi: 10.1109/TAC.1967.1098541.
[9]	X. Lu, H. S. Zhang, W. Wang and K. L. Teo, Kalman filtering for multiple time-delay systems, Automatica, 41 (2005), 1455-1461. doi: 10.1016/j.automatica.2005.03.018.
[10]	D. MacMillan, J. Bohm, M. Gipson, R. Haas, A. Niell, T. Nilsson, A. Pany, B. Petrachenko and J. Wresnik, Simulation analysis of the geodetic performance of the future IVS VLBI2010 system, in "American Geophysical Union Fall Meeting," San Francisco, (2008), G33A-0667.
[11]	G. A. Medrano-Cerda, Filtering for linear system involving time delays in the noise process, IEEE Trans. on Automatic Control, 28 (1983), 801-803. doi: 10.1109/TAC.1983.1103318.
[12]	C. L. Su and C. N. Lu, Interconnected network state estimation using randomly delayed measurements, IEEE Trans. on Power Systems, 16 (2001), 870-878. doi: 10.1109/59.962439.
[13]	A. Subramanian and A. H. Sayed, Multiobjective filter design for uncertain stochastic time-delay systems, IEEE Trans. on Automatic Control, 49 (2004), 149-154. doi: 10.1109/TAC.2003.821422.
[14]	S. L. Sun, Linear minimum variance estimators for systems with bounded random measurement delays and packet dropouts, Signal Processing, 89 (2009), 1457-1466. doi: 10.1016/j.sigpro.2009.02.002.
[15]	Z. Wang, D. W. C. Ho and X. Liu, Robust filtering underrandomly varying sensor delay with variance constraints, IEEE Trans. on Circuits and Systtems II: Express Briefs, 51 (2004), 320-326. doi: 10.1109/TCSII.2004.829572.
[16]	E. Yaz and A. Ray, Linear unbiased state estimation under randomly varying bounded sensor delay, Applied Mathematics Letters, 11 (1998), 27-32. doi: 10.1016/S0893-9659(98)00051-2.
[17]	H. S. Zhang, X. Lu, and D. Z. Cheng, Optimal estimation for continuous-time systems with delayed measurements, IEEE Trans. on Automatic Control, 51 (2006), 823-827. doi: 10.1109/TAC.2006.874983.
[18]	H. G. Zhao, H. S. Zhang and C. H. Zhang, Optimal filtering for linear discrete-time systems with single delayed measurement, Int. J. of Control, Automation, and Systems, 6 (2008), 378-385.

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