Weak convergence of hybrid filtering problems involving nearly completely decomposable hidden markov chains

SIAM Journal on Control and Optimization

Volumn 41, Issue 6, 2003, Pages 1820-1842

  Yin, G ^a   Dey, S ^b  

^a WAYNE STATE UNIVERSITY   (United States)

^b UNIVERSITY OF MELBOURNE   (Australia)

Author keywords

Filtering; Markov chain; Near complete decomposability; Weak convergence

Indexed keywords

HYBRID FILTERS; TRANSITION PROBABILITY MATRIX;

CONVERGENCE OF NUMERICAL METHODS; DISCRETE TIME CONTROL SYSTEMS; GAUSSIAN NOISE (ELECTRONIC); MARKOV PROCESSES; MATRIX ALGEBRA; PROBABILITY;

WAVE FILTERS;

EID: 0348220627     PISSN: 03630129     EISSN: None     Source Type: Journal    
DOI: 10.1137/S0363012901388464     Document Type: Article

References (34)

1. Y. BAR-SHALOM AND X. R. LI, Estimation and Tracking: Principles, Techniques, and Software, Artech House Publishers, Norwood, MA, 1996.
2. D. P. BERTSEKAS, Dynamic Programming and Stochastic Control, Academic Press, New York, 1976.
3. T. R. BIELECKI AND L. STETTNER, Ergodic control of a singularly perturbed Markov process in discrete time with general state and compact action spaces, Appl. Math. Optim., 38 (1998), pp. 261-281.
4. T. BJÖRK, Finite-dimensional optimal filters for a class of Itô processes with jumping parameters, Stochastics, 4 (1980), pp. 167-183.
5. G. BLANKENSHIP, Singularly perturbed difference equations in optimal control problems, IEEE Trans. Automat. Control, 26 (1981), pp. 911-917.
6. O. L. V. COSTA, Linear minimum mean square error estimation for discrete-time Markov jump linear systems, IEEE Trans. Automat. Control, 39 (1994), pp. 1685-1689.
7. P. J. COURTOIS, Decomposability. Queueing and Computer System Applications, Academic Press, New York, 1977.
8. D. P. DE FARIAS, J. C. GEROMEL, J. B. R. DO VAL, AND O. L. V. COSTA, Output feedback control of Markov jump linear systems in continuous time, IEEE Trans. Automat. Control, 45 (2000), pp. 944-949.
9. S. DEY, Reduced-complexity filtering for partially observed nearly completely decomposable Markov chains, IEEE Trans. Signal Process., 48, (2000), pp. 3334-3344.
10. A. DOUCET, N. J. GORDON, AND V. KRISHNAMURTHY, Particle filtering for state estimation for jump Markov linear systems, IEEE Trans. Signal Process., 49 (2001), pp. 613-624.
11. F. DUFOUR AND P. BERTRAND, The filtering problem for continuous-time linear systems with Markovian switching coefficients, Systems Control Lett., 23 (1994), pp. 453-461.
12. F. DUFOUR AND R. J. ELLIOTT, Adaptive control of linear systems with Markov perturbations, IEEE Trans. Automat. Control, 43 (1997), pp. 351-372.
13. S. N. ETHIER AND T. G. KURTZ, Markov Processes: Characterization and Convergence, John Wiley, New York, 1986.
14. A. IL'IN, R. Z. KHASMINSKII, AND G. YIN, Singularly perturbed switching diffusions: Rapid switchings and fast diffusions, J. Optim. Theory Appl., 102 (1999), pp. 555-591.
15. M. IOSIFESCU, Finite Markov Processes and Their Applications, John Wiley, Chichester, UK, 1980.
16. R. Z. KHASMINSKII, G. YIN, AND Q. ZHANG, Asymptotic expansions of singularly perturbed systems involving rapidly fluctuating Markov chains, SIAM J. Appl. Math., 56 (1996), pp. 277-293.
17. H. J. KUSHNER, Approximation and Weak Convergence Methods for Random Processes, with Applications to Stochastic Systems Theory, MIT Press, Cambridge, MA, 1984.
18. H. J. KUSHNER, Weak Convergence Methods and Singularly Perturbed Stochastic Control and Filtering Problems, Birkhäuser Boston, Boston, 1990.
19. H. J. KUSHNER AND G. YIN, Stochastic Approximation Algorithms and Applications, Springer-Verlag, New York, 1997.
20. R. S. LIPTSER AND A. N. SHIRYAYEV, Statistics of Random Processes I &: II, Springer-Verlag, New York, 2001.
21. B. M. MILLER AND W. J. RUNGGALDIER, Kalman filtering for linear systems with coefficients driven by a hidden Markov jump process, Systems Control Lett., 31 (1997), pp. 93-102.
22. A. A. PERVOZVANSKII AND V. G. GAITSGORI, Theory of Suboptimal Decisions: Decomposition and Aggregation, Kluwer, Dordrecht, The Netherlands, 1988.
23. R. G. PHILLIPS AND P. V. KOKOTOVIC, A singular perturbation approach to modelling and control of Markov chains, IEEE Trans. Automat. Control, 26 (1981), pp. 1087-1094.
24. W. J. RUNGGALDIER AND C. VISENTIN, Combined filtering and parameter estimation: Approximation and robustness, Automatica J. IFAC, 26 (1990), pp. 401-404.
25. S. P. SETHI AND Q. ZHANG, Hierarchical Decision Making in Stochastic Manufacturing Systems, Birkhäuser Boston, Boston, 1994.
26. H. A. SIMON AND A. ANDO, Aggregation of variables in dynamic systems, Econometrica, 29 (1961), pp. 111-138.
27. D. N. C. TSE, R. G. GALLAGER, AND J. N. TSITSIKLIS, Statistical multiplexing of multiple time-scale Markov streams, IEEE J. Selected Areas Comm., 13 (1995), pp. 1028-1038
28. C. YANG, Y. BAR-SHALOM, AND C.-F. LIN, Discrete-time point process filter for mode estimation, IEEE Trans. Automat. Control, 37 (1992), pp. 1812-1816.
29. G. YIN AND Q. ZHANG, Continuous-Time Markov Chains and Applications: A Singular Perturbation Approach, Springer-Verlag, New York, 1998.
30. G. YIN AND Q. ZHANG, Singularly perturbed discrete-time Markov chains, SIAM J. Appl. Math., 61 (2000), pp. 834-854.
31. G. YIN, Q. ZHANG, AND G. BADOWSKI, Asymptotic properties of a singularly perturbed Markov chain with inclusion of transient states, Ann. Appl. Probab., 10 (2000), pp. 549-572.
32. G. YIN, Q. ZHANG, AND G. BADOWSKI, Decomposition and aggregation of large-dimensional Markov chains in discrete time, in Proceedings of the 40th IEEE Conference on Decision and Control, IEEE Control Systems Society, Piscataway, NJ, 2001, pp. 1687-1692.
33. Q. ZHANG, Hybrid filtering for linear systems with non-Gaussian disturbances, IEEE Trans. Automat. Control, 45 (2000), pp. 50-61.
34. Q. ZHANG AND G. YIN, On nearly optimal controls of hybrid LQG problems, IEEE Trans. Automat. Control, 44 (1999), pp. 2271-2282.