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A unified filter for simultaneous input and state estimation of linear discrete-time stochastic systems. (English) Zbl 1329.93148
Summary: In this paper, we present a unified optimal and exponentially stable filter for linear discrete-time stochastic systems that simultaneously estimates the states and unknown inputs in an unbiased minimum-variance sense, without making any assumptions on the direct feedthrough matrix. We also provide the connection between the stability of the estimator and a system property known as strong detectability, and discuss the global optimality of the proposed filter. Finally, an illustrative example is given to demonstrate the performance of the unified unbiased minimum-variance filter.

MSC:
93E11 Filtering in stochastic control theory
93E10 Estimation and detection in stochastic control theory
93C05 Linear systems in control theory
93E03 Stochastic systems in control theory (general)
93C55 Discrete-time control/observation systems
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References:
[1] Anderson, B. D.O.; Moore, J. B., Detectability and stabilizability of time-varying discrete-time linear systems, SIAM Journal on Control and Optimization, 19, 1, 20-32, (1981) · Zbl 0468.93051
[2] Bernstein, D. S., Matrix mathematics: theory, facts, and formulas, (2009), Princeton University Press, Princeton reference · Zbl 1183.15001
[3] Cheng, Y.; Ye, H.; Wang, Y.; Zhou, D., Unbiased minimum-variance state estimation for linear systems with unknown input, Automatica, 45, 2, 485-491, (2009) · Zbl 1158.93415
[4] Darouach, M.; Zasadzinski, M., Unbiased minimum variance estimation for systems with unknown exogenous inputs, Automatica, 33, 4, 717-719, (1997) · Zbl 0874.93086
[5] Darouach, M.; Zasadzinski, M.; Boutayeb, M., Extension of minimum variance estimation for systems with unknown inputs, Automatica, 39, 5, 867-876, (2003) · Zbl 1036.93058
[6] De Nicolao, G.; Sparacino, G.; Cobelli, C., Nonparametric input estimation in physiological systems: problems, methods, and case studies, Automatica, 33, 5, 851-870, (1997) · Zbl 0874.93008
[7] Draper, N. R.; Smith, H., Applied regression analysis (wiley series in probability and statistics), (1998), Wiley-Interscience
[8] Fang, H.; Shi, Y.; Yi, J., A new algorithm for simultaneous input and state estimation, (American control conference, (2008)), 2421-2426
[9] Fang, H.; Shi, Y.; Yi, J., On stable simultaneous input and state estimation for discrete-time linear systems, International Journal of Adaptive Control and Signal Processing, 25, 8, 671-686, (2011) · Zbl 1227.93112
[10] Gillijns, S.; De Moor, B., Unbiased minimum-variance input and state estimation for linear discrete-time systems, Automatica, 43, 1, 111-116, (2007) · Zbl 1140.93480
[11] Gillijns, S.; De Moor, B., Unbiased minimum-variance input and state estimation for linear discrete-time systems with direct feedthrough, Automatica, 43, 5, 934-937, (2007) · Zbl 1117.93366
[12] Hautus, M. L.J., Strong detectability and observers, Linear Algebra and its Applications, 50, 353-368, (1983) · Zbl 0528.93018
[13] Hou, M.; Patton, R. J., Optimal filtering for systems with unknown inputs, IEEE Transactions on Automatic Control, 43, 3, 445-449, (1998) · Zbl 0904.93037
[14] Hsieh, C., Robust two-stage Kalman filters for systems with unknown inputs, IEEE Transactions on Automatic Control, 45, 12, 2374-2378, (2000) · Zbl 0990.93130
[15] Hsieh, C., Extension of unbiased minimum-variance input and state estimation for systems with unknown inputs, Automatica, 45, 9, 2149-2153, (2009) · Zbl 1175.93213
[16] Hsieh, C., On the global optimality of unbiased minimum-variance state estimation for systems with unknown inputs, Automatica, 46, 4, 708-715, (2010) · Zbl 1193.93164
[17] Kalman, R. E., A new approach to linear filtering and prediction problems, Transactions of the ASME. Series D, Journal of Basic Engineering, 82, 35-45, (1960)
[18] Kerwin, W. S.; Prince, J. L., On the optimality of recursive unbiased state estimation with unknown inputs, Automatica, 36, 9, 1381-1383, (2000) · Zbl 0964.93076
[19] Kitanidis, P. K., Unbiased minimum-variance linear state estimation, Automatica, 23, 6, 775-778, (1987) · Zbl 0627.93065
[20] Li, X.; Gao, H., Robust finite frequency filtering for uncertain 2-D systems: the FM model case, Automatica, 49, 8, 2446-2452, (2013) · Zbl 1364.93808
[21] Palanthandalam-Madapusi, H. J.; Bernstein, D. S., Unbiased minimum-variance filtering for input reconstruction, (American control conference, (2007)), 5712-5717
[22] Patton, R.; Clark, R.; Frank, P. M., Fault diagnosis in dynamic systems: theory and applications, (Prentice-Hall international series in systems and control engineering, (1989), Prentice Hall)
[23] Peters, M. A.; Iglesias, P. A., A spectral test for observability and reachability of time-varying systems, SIAM Journal on Control and Optimization, 37, 5, 1330-1345, (1999) · Zbl 0933.93017
[24] Sayed, A. H., Fundamentals of adaptive filtering, (2003), Wiley
[25] Silverman, L. M., Discrete Riccati equations: alternative algorithms, asymptotic properties, and system theory interpretations, (Control and dynamic systems, Vol. 12, (1976), Academic Press) · Zbl 0362.49014
[26] Sundaram, S.; Hadjicostis, C. N., Distributed function calculation via linear iterative strategies in the presence of malicious agents, IEEE Transactions on Automatic Control, 56, 7, 1495-1508, (2011) · Zbl 1368.93140
[27] Wang, Z.; Dong, H.; Shen, B.; Gao, H., Finite-horizon \(H_\infty\) filtering with missing measurements and quantization effects, IEEE Transactions on Automatic Control, 58, 7, 1707-1718, (2013) · Zbl 1369.93660
[28] Wang, Z.; Liu, Y.; Liu, X., \(H_\infty\) filtering for uncertain stochastic time-delay systems with sector-bounded nonlinearities, Automatica, 44, 5, 1268-1277, (2008) · Zbl 1283.93284
[29] Yong, S. Z., Zhu, M., & Frazzoli, E. (2013a). A unified filter for simultaneous input and state estimation of linear discrete-time stochastic systems. arXiv. (Extended version). Available from: http://arxiv.org/abs/1309.6627.
[30] Yong, S. Z., Zhu, M., & Frazzoli, E. (2013b). Simultaneous input and state estimation for linear discrete-time stochastic systems with direct feedthrough. In Conference on decision and control (pp. 7034-7039), December.
[31] Yong, S. Z., Zhu, M., & Frazzoli, E. (2014). Generalized innovation and inference algorithms for hidden mode switched linear stochastic systems with unknown inputs. In: Conference on decision and control (pp. 3388-3394).
[32] Zhang, W.; Huang, Y.; Xie, L., Infinite horizon stochastic \(H_2\)/\(H_\infty\) control for discrete-time systems with state and disturbance dependent noise, Automatica, 44, 2306-2316, (2008) · Zbl 1153.93030
[33] Zhang, W.; Zhang, H.; Chen, B.-S., Generalized Lyapunov equation approach to state-dependent stochastic stabilization/detectability criterion, IEEE Transactions on Automatic Control, 53, 7, 1630-1642, (2008) · Zbl 1367.93549
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