American Institute of Mathematical Sciences

Advanced
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

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 & 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, (1979).   Google Scholar

[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.  doi: 10.1109/TAC.2005.846599.  Google Scholar

[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.  doi: 10.2478/v10006-009-0004-8.  Google Scholar

[4]

Y. Chocheyras, Near field three dimensional time delay and doppler target motion analysis,, in, (1989), 2649.  doi: 10.1109/ICASSP.1989.267012.  Google Scholar

[5]

T. Kailath, A. H. Sayed, and B. Hassibi, "Linear Estimation",, Prentice-Hall, (1999).   Google Scholar

[6]

R. E. Kalman, A new approach to linear filtering and prediction problems,, Trans. ASME, 82 (1960), 35.   Google Scholar

[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.   Google Scholar

[8]

H. Kwakernaak, Optimal filtering in linear systems with time delays,, IEEE Trans. on Automatic Control, 12 (1967), 169.  doi: 10.1109/TAC.1967.1098541.  Google Scholar

[9]

X. Lu, H. S. Zhang, W. Wang and K. L. Teo, Kalman filtering for multiple time-delay systems,, Automatica, 41 (2005), 1455.  doi: 10.1016/j.automatica.2005.03.018.  Google Scholar

[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, (2008).   Google Scholar

[11]

G. A. Medrano-Cerda, Filtering for linear system involving time delays in the noise process,, IEEE Trans. on Automatic Control, 28 (1983), 801.  doi: 10.1109/TAC.1983.1103318.  Google Scholar

[12]

C. L. Su and C. N. Lu, Interconnected network state estimation using randomly delayed measurements,, IEEE Trans. on Power Systems, 16 (2001), 870.  doi: 10.1109/59.962439.  Google Scholar

[13]

A. Subramanian and A. H. Sayed, Multiobjective filter design for uncertain stochastic time-delay systems,, IEEE Trans. on Automatic Control, 49 (2004), 149.  doi: 10.1109/TAC.2003.821422.  Google Scholar

[14]

S. L. Sun, Linear minimum variance estimators for systems with bounded random measurement delays and packet dropouts,, Signal Processing, 89 (2009), 1457.  doi: 10.1016/j.sigpro.2009.02.002.  Google Scholar

[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.  doi: 10.1109/TCSII.2004.829572.  Google Scholar

[16]

E. Yaz and A. Ray, Linear unbiased state estimation under randomly varying bounded sensor delay,, Applied Mathematics Letters, 11 (1998), 27.  doi: 10.1016/S0893-9659(98)00051-2.  Google Scholar

[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.  doi: 10.1109/TAC.2006.874983.  Google Scholar

[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, 6 (2008), 378.   Google Scholar

show all references

References:
[1]

B. D. O. Anderson and J. B. Moore, "Optimal Filtering,", Prentice-Hall, (1979).   Google Scholar

[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.  doi: 10.1109/TAC.2005.846599.  Google Scholar

[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.  doi: 10.2478/v10006-009-0004-8.  Google Scholar

[4]

Y. Chocheyras, Near field three dimensional time delay and doppler target motion analysis,, in, (1989), 2649.  doi: 10.1109/ICASSP.1989.267012.  Google Scholar

[5]

T. Kailath, A. H. Sayed, and B. Hassibi, "Linear Estimation",, Prentice-Hall, (1999).   Google Scholar

[6]

R. E. Kalman, A new approach to linear filtering and prediction problems,, Trans. ASME, 82 (1960), 35.   Google Scholar

[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.   Google Scholar

[8]

H. Kwakernaak, Optimal filtering in linear systems with time delays,, IEEE Trans. on Automatic Control, 12 (1967), 169.  doi: 10.1109/TAC.1967.1098541.  Google Scholar

[9]

X. Lu, H. S. Zhang, W. Wang and K. L. Teo, Kalman filtering for multiple time-delay systems,, Automatica, 41 (2005), 1455.  doi: 10.1016/j.automatica.2005.03.018.  Google Scholar

[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, (2008).   Google Scholar

[11]

G. A. Medrano-Cerda, Filtering for linear system involving time delays in the noise process,, IEEE Trans. on Automatic Control, 28 (1983), 801.  doi: 10.1109/TAC.1983.1103318.  Google Scholar

[12]

C. L. Su and C. N. Lu, Interconnected network state estimation using randomly delayed measurements,, IEEE Trans. on Power Systems, 16 (2001), 870.  doi: 10.1109/59.962439.  Google Scholar

[13]

A. Subramanian and A. H. Sayed, Multiobjective filter design for uncertain stochastic time-delay systems,, IEEE Trans. on Automatic Control, 49 (2004), 149.  doi: 10.1109/TAC.2003.821422.  Google Scholar

[14]

S. L. Sun, Linear minimum variance estimators for systems with bounded random measurement delays and packet dropouts,, Signal Processing, 89 (2009), 1457.  doi: 10.1016/j.sigpro.2009.02.002.  Google Scholar

[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.  doi: 10.1109/TCSII.2004.829572.  Google Scholar

[16]

E. Yaz and A. Ray, Linear unbiased state estimation under randomly varying bounded sensor delay,, Applied Mathematics Letters, 11 (1998), 27.  doi: 10.1016/S0893-9659(98)00051-2.  Google Scholar

[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.  doi: 10.1109/TAC.2006.874983.  Google Scholar

[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, 6 (2008), 378.   Google Scholar

[1]

Z. G. Feng, Kok Lay Teo, N. U. Ahmed, Yulin Zhao, W. Y. Yan. Optimal fusion of sensor data for Kalman filtering. Discrete & Continuous Dynamical Systems - A, 2006, 14 (3) : 483-503. doi: 10.3934/dcds.2006.14.483
[2]

Matthieu Canaud, Lyudmila Mihaylova, Jacques Sau, Nour-Eddin El Faouzi. Probability hypothesis density filtering for real-time traffic state estimation and prediction. Networks & Heterogeneous Media, 2013, 8 (3) : 825-842. doi: 10.3934/nhm.2013.8.825
[3]

Claude Stolz. On estimation of internal state by an optimal control approach for elastoplastic material. Discrete & Continuous Dynamical Systems - S, 2016, 9 (2) : 599-611. doi: 10.3934/dcdss.2016014
[4]

Dominique Chapelle, Philippe Moireau, Patrick Le Tallec. Robust filtering for joint state-parameter estimation in distributed mechanical systems. Discrete & Continuous Dynamical Systems - A, 2009, 23 (1&2) : 65-84. doi: 10.3934/dcds.2009.23.65
[5]

Hui Huang, Eldad Haber, Lior Horesh. Optimal estimation of $\ell_1$-regularization prior from a regularized empirical Bayesian risk standpoint. Inverse Problems & Imaging, 2012, 6 (3) : 447-464. doi: 10.3934/ipi.2012.6.447
[6]

H. T. Banks, D. Rubio, N. Saintier, M. I. Troparevsky. Optimal design for parameter estimation in EEG problems in a 3D multilayered domain. Mathematical Biosciences & Engineering, 2015, 12 (4) : 739-760. doi: 10.3934/mbe.2015.12.739
[7]

Sie Long Kek, Kok Lay Teo, Mohd Ismail Abd Aziz. Filtering solution of nonlinear stochastic optimal control problem in discrete-time with model-reality differences. Numerical Algebra, Control & Optimization, 2012, 2 (1) : 207-222. doi: 10.3934/naco.2012.2.207
[8]

Robert J. Elliott, Tak Kuen Siu. Stochastic volatility with regime switching and uncertain noise: Filtering with sub-linear expectations. Discrete & Continuous Dynamical Systems - B, 2017, 22 (1) : 59-81. doi: 10.3934/dcdsb.2017003
[9]

Ming Mei. Stability of traveling wavefronts for time-delayed reaction-diffusion equations. Conference Publications, 2009, 2009 (Special) : 526-535. doi: 10.3934/proc.2009.2009.526
[10]

Lizhong Qiang, Bin-Guo Wang. An almost periodic malaria transmission model with time-delayed input of vector. Discrete & Continuous Dynamical Systems - B, 2017, 22 (4) : 1525-1546. doi: 10.3934/dcdsb.2017073
[11]

Edward Hooton, Pavel Kravetc, Dmitrii Rachinskii. Restrictions to the use of time-delayed feedback control in symmetric settings. Discrete & Continuous Dynamical Systems - B, 2018, 23 (2) : 543-556. doi: 10.3934/dcdsb.2017207
[12]

Martin Gugat, Markus Dick. Time-delayed boundary feedback stabilization of the isothermal Euler equations with friction. Mathematical Control & Related Fields, 2011, 1 (4) : 469-491. doi: 10.3934/mcrf.2011.1.469
[13]

Weijiu Liu. Asymptotic behavior of solutions of time-delayed Burgers' equation. Discrete & Continuous Dynamical Systems - B, 2002, 2 (1) : 47-56. doi: 10.3934/dcdsb.2002.2.47
[14]

Jérôme Fehrenbach, Jacek Narski, Jiale Hua, Samuel Lemercier, Asja Jelić, Cécile Appert-Rolland, Stéphane Donikian, Julien Pettré, Pierre Degond. Time-delayed follow-the-leader model for pedestrians walking in line. Networks & Heterogeneous Media, 2015, 10 (3) : 579-608. doi: 10.3934/nhm.2015.10.579
[15]

Isabelle Schneider, Matthias Bosewitz. Eliminating restrictions of time-delayed feedback control using equivariance. Discrete & Continuous Dynamical Systems - A, 2016, 36 (1) : 451-467. doi: 10.3934/dcds.2016.36.451
[16]

Yanbin Tang, Ming Wang. A remark on exponential stability of time-delayed Burgers equation. Discrete & Continuous Dynamical Systems - B, 2009, 12 (1) : 219-225. doi: 10.3934/dcdsb.2009.12.219
[17]

Yijun Lou, Xiao-Qiang Zhao. Threshold dynamics in a time-delayed periodic SIS epidemic model. Discrete & Continuous Dynamical Systems - B, 2009, 12 (1) : 169-186. doi: 10.3934/dcdsb.2009.12.169
[18]

Francisco de la Hoz, Anna Doubova, Fernando Vadillo. Persistence-time estimation for some stochastic SIS epidemic models. Discrete & Continuous Dynamical Systems - B, 2015, 20 (9) : 2933-2947. doi: 10.3934/dcdsb.2015.20.2933
[19]

Chongyang Liu, Meijia Han, Zhaohua Gong, Kok Lay Teo. Robust parameter estimation for constrained time-delay systems with inexact measurements. Journal of Industrial & Management Optimization, 2017, 13 (5) : 0-0. doi: 10.3934/jimo.2019113
[20]

Ling Yun Wang, Wei Hua Gui, Kok Lay Teo, Ryan Loxton, Chun Hua Yang. Time delayed optimal control problems with multiple characteristic time points: Computation and industrial applications. Journal of Industrial & Management Optimization, 2009, 5 (4) : 705-718. doi: 10.3934/jimo.2009.5.705

2018 Impact Factor: 1.025

Tools

Metrics

  • PDF downloads (6)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2019 American Institute of Mathematical Sciences