\(H_\infty \) estimation for discrete-time piecewise homogeneous Markov jump linear systems. (English) Zbl 1180.93100

Summary: This paper concerns the problem of \(H_\infty\) estimation for a class of Markov Jump Linear Systems (MJLS) with time-varying Transition Probabilities (TPs) in discrete-time domain. The time-varying character of TPs is considered to be finite piecewise homogeneous and the variations in the finite set are considered to be of two types: arbitrary variation and stochastic variation, respectively. The latter means that the variation is subject to a higher-level transition probability matrix. The mode-dependent and variation-dependent \(H_\infty\) filter is designed such that the resulting closed-loop systems are stochastically stable and have a guaranteed \(H_\infty\) filtering error performance index. Using the idea in the recent studies of partially unknown TPs for the traditional MJLS with homogeneous TPs, a generalized framework covering the two kinds of variations is proposed. A numerical example is presented to illustrate the effectiveness and the potential of the developed theoretical results.


93E10 Estimation and detection in stochastic control theory
93C55 Discrete-time control/observation systems
60J75 Jump processes (MSC2010)
Full Text: DOI


[1] Internet traffic report (2008). http://www.internettrafficreport.com; Internet traffic report (2008). http://www.internettrafficreport.com
[2] Boukas, E. K., Stochastic switching systems: Analysis and design (2005), Birkhauser: Birkhauser Basel, Berlin · Zbl 1108.93074
[3] Boukas, E. K.; Liu, Z. K., Robust \(H_\infty\) control of discrete-time Markovian jump linear systems with mode-dependent time-delays, IEEE Transactions on Automatic Control, 46, 12, 1918-1924 (2001) · Zbl 1005.93050
[4] Costa, O. L.V.; Fragoso, M. D.; Marques, R. P., Discrete-time markovian jump linear systems (2005), Springer-Verlag: Springer-Verlag London · Zbl 1081.93001
[5] de Farias, D. P.; Geromel, J. C.; do Val, J. B.R.; Costa, O. L.V., Output feedback control of markov jump linear systems in continuous-time, IEEE Transactions on Automatic Control, 45, 5, 944-949 (2000) · Zbl 0972.93074
[6] Diebold, F. X.; Lee, J. H.; Weinbach, G. C., Regime switching with time-varying transition probabilities, (Hargreaves, C.; Granger, C. W.J.; Mizon, G., Nonstationary time series analysis and cointegration. Nonstationary time series analysis and cointegration, Advanced texts in econometrics (1994), Oxford University Press), 283-302
[7] Iosifescu, M., Finite markov processes and their applications (1980), John Wiley and Sons: John Wiley and Sons Bucharest · Zbl 0436.60001
[8] Kemeny, J. G.; Snell, J. L., Finite markov chains (1960), D. Van Nostrand Company: D. Van Nostrand Company Princeton · Zbl 0112.09802
[9] Krtolica, R.; Ozguner, U.; Chan, H.; Goktas, H.; Winkelman, J.; Liubakka, M., Stability of linear feedback systems with random communication delays, International Journal Control, 59, 4, 925-953 (1994) · Zbl 0812.93073
[10] Liu, H.; Sun, F. C.; He, K. Z.; Sun, Z. Q., Design of reduced-order \(H_\infty\) filter for markovian jumping systems with time delay, IEEE Transactions on Circuits and Systems (II), 51, 11, 607-612 (2004)
[11] Narendra, K. S.; Tripathi, S. S., Identification and optimization of aircraft dynamics, Journal of Aircraft, 10, 4, 193-199 (1973)
[12] Salkin, M. S.; Just, R. E.; Cleveland, O. A., Estimation of nonstationary transition probabilities for agricultural firm size projection, The Annals of Regional Science, 10, 1, 71-82 (1976)
[13] Seiler, P.; Sengupta, R., An \(H_\infty\) approach to networked control, IEEE Transactions on Automatic Control, 50, 3, 356-364 (2005) · Zbl 1365.93147
[14] Wang, Z.; Lam, J.; Liu, X. H., Exponential filtering for uncertain Markovian jump time-delay systems with nonlinear disturbances, IEEE Transactions on Circuits and Systems (II), 51, 262-268 (2004)
[15] Wang, Z.; Lam, J.; Liu, X. H., Robust filtering for discrete-time markovian jump delay systems, IEEE Signal Processing Letters, 11, 8, 659-662 (2004)
[16] Xiong, J. L.; Lam, J.; Gao, H. J.; Daniel, W. C., On robust stabilization of markovian jump systems with uncertain switching probabilities, Automatica, 41, 5, 897-903 (2005) · Zbl 1093.93026
[17] Xu, S.; Chen, T.; Lam, J., Robust \(H_\infty\) filtering for uncertain Markovian jump systems with mode-dependent time-delays, IEEE Transactions on Automatic Control, 48, 5, 900-907 (2003) · Zbl 1364.93816
[18] Zhang, L. Q.; Shi, Y.; Chen, T. W.; Huang, B., A new method for stabilization of networked control systems with random delays, IEEE Transacttions on Automatic Control, 50, 8, 1177-1181 (2005) · Zbl 1365.93421
[19] Zhang, L. X.; Boukas, E. K., Stability and stabilization of markovian jump linear systems with partly unknown transition probability, Automatica, 45, 2, 463-468 (2009) · Zbl 1158.93414
[20] Zhang, L. X.; Boukas, E. K.; Lam, J., Analysis and synthesis of markov jump linear systems with time-varying delays and partially known transition probabilities, IEEE Transactions on Automatic Control, 53, 10, 2458-2464 (2008) · Zbl 1367.93710
[21] Zhang, L. X.; Shi, P.; Wang, C. H.; Gao, H. J., Robust \(H_\infty\) filtering for switched linear discrete-time systems with polytopic uncertainties, International Journal of Adaptive Control & Signal Processing, 20, 6, 291-304 (2006) · Zbl 1127.93324
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