Informatics and Applications
2023, Volume 17, Issue 2, pp 41-49
ROBUSTNESS INVESTIGATION OF THE NUMERICAL APPROXIMATION OF THE WONHAM FILTER
Abstract
The properties of the optimal continuous Markov chain state filtering problem decision given be the linear observations noisy Wiener process, assuming incomplete information about its intensity, are investigated. The uncertainty of the observation system is set by the upper bound of the noise intensity. Numerical implementation of the optimal solution in the statement with complete information provided by the Wonham filter does not guarantee stability. It is shown that the Wonham filter in the statement with uncertainty is robust with respect to the noise intensity if the model parameters do not lead to its divergence. In the general case, the instability of the Euler-Maruyama numerical scheme of the Wonham filter is preserved. Simple heuristic techniques that provide stable approximations of the Wonham filter show the workability for a wider set of parameters. However, in the statement with uncertainty, it is possible to give examples when such heuristic filters show unacceptably low quality.
The best solution is provided by discretized filters, approximations of the Wonham filter implemented for a specific model approximating the initial continuous observation system. A series of numerical experiments has shown that these filters have robustness for all sets of parameters. If there are no divergent trajectories among the modeled trajectories of the Wonham filter in the calculations, then the discretized filters lose a little. If there are divergent trajectories, then the gain of discretized filters is absolute.
[+] References (20)
- Kalman, R. E., and R. S. Bucy. 1961. New results in linear filtering and prediction theory. J. Basic Eng. - T. ASME 83(1):95-108. doi: 10.1115/1.3658902.
- Wonham, W. M. 1965. Some application of stochastic differential equations to optimal nonlinear filtering. SIAMJ. Control 2(3):347-369. doi: 10.1137/030202.
- Liptser, R. S., andA. N. Shiryaev. 2001. Statistics of random processes II. Applications. Berlin: Springer-Verlag. 402 p.
- D'Appolito, J.A., and C. E. Hutchinson. 1972. A minimax approach to the design of low sensitivity state estimators. Automatica 8(5):599-608. doi: 10.1016/0005- 1098(72)90031-3.
- Morris, J. 1976. The Kalman filter: A robust estimator for some classes of linear quadratic problems. IEEE T. Inform. Theory 22(5):526-534. doi: 10.1109/TIT.1976.1055611.
- Vandelinde, V. 1977. Robust properties of solutions to linear-quadratic estimation and control problems. IEEE T. Automat. Contr. 22(1):138-139. doi: 10.1109/ TAC.1977.1101433.
- Poor, V., and D. Looze. 1981. Minimax state estimation for linear stochastic systems with noise uncertainty. IEEE T. Automat. Contr. 26(4):902-906. doi: 10.1109/ TAC.1981.1102756.
- Verdu, S., andH. Poor. 1984. Minimax linear observers and regulators for stochastic systems with uncertain second- order statistics. IEEE T. Automat. Contr. 29(6):499-511. doi: 10.1109/TAC.1984.1103576.
- Verdu, S., and H. Poor. 1984. On minimax robustness: A general approach and applications. IEEE T. Inform. Theory 30(2):328-340. doi: 10.1109/TIT.1984.1056876.
- Pankov, A. R. 1994. Control strategies in a linear stochastic system with non-Gaussian perturbations. Automat. Rem. Contr. 55(6):832-840.
- Miller, G.B., and A. R. Pankov. 2006. Minimax filtering in linear stochastic uncertain discrete-continuous systems. Automat. Rem. Contr. 67(3):413-427.
- Miller, G.B., and A. R. Pankov. 2007. Minimax control of a process in a linear uncertain-stochastic system with incomplete data. Automat. Rem. Contr. 68(11):2042-2055.
- Elliott, R. J., L. Aggoun, and J. B. Moore. 1995. Hidden Markov models: Estimation and control. New York, NY: Springer-Verlag. 396 p.
- Kloden, P. E., and E. Platen. 1992. Numerical solution of stochastic differential equations. Berlin: Springer. 636 p.
- Yin, G., Q. Zhang, and Y. Liu. 2004. Discrete-time approximation of Wonham filters. J. Control Theory Applications 2:1-10. doi: 10.1007/s11768-004-0017-7.
- Borisov, A. V. 2020. Chislennye skhemy fil'tratsii markovskikh skachkoobraznykh protsessov po diskretizovannym nablyudeniyam II: Sluchay additivnykh shumov [Numerical schemes of Markov jump process filtering given discretized observations II: Additive noise case]. Informati- ka i ee Primeneniya - Inform. Appl. 14(1):17-23. doi: 10.14357/19922264200103.
- Borisov, A. V. 2020. L1-optimal filtering of Markov jump processes. II. Numerical analysis of particular realizations schemes. Automat. Rem. Contr. 81(12):2160-2180.
- Borisov, A. V., and A. V. Bosov. 2022. Practical implementation of the stabilization problem solution for a linear system with discontinuous random drift by indirect observations. Automat. Rem. Contr. 83(9):1417-1432. doi: 10.31857/S0005231022090069.
- Davis, M. H. A. 1977. Linear estimation and stochastic control. London: Chapman and Hall. 224 p.
- Borisov, A. V. 2018. Fil'tratsiya sostoyaniy markovskikh skachkoobraznykh protsessov po diskretizovannym nablyudeniyam [Filtering of Markov jump processes by discretized observations]. Informatika i ee Primeneniya - Inform. Appl. 12(3):115-121. doi: 10.14357/ 19922264180316.
[+] About this article
Title
ROBUSTNESS INVESTIGATION OF THE NUMERICAL APPROXIMATION OF THE WONHAM FILTER
Journal
Informatics and Applications
2023, Volume 17, Issue 2, pp 41-49
Cover Date
2023-07-10
DOI
10.14357/19922264230206
Print ISSN
1992-2264
Publisher
Institute of Informatics Problems, Russian Academy of Sciences
Additional Links
Key words
Markov jump process; stochastic filtering; robust estimation; Wonham filter; Euler-Maruyama numerical scheme; discretized filters
Authors
A. V. Bosov 
Author Affiliations
Federal Research Center "Computer Science and Control" of the Russian Academy of Sciences, 44-2 Vavilov Str., Moscow 119333, Russian Federation
|