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Filtration of a random process in a statistically uncertain linear stochastic differential system


Minimax filtration of a process in a stochastic linear differential system with uncertain perturbation intensities for dynamics and observation models is studied. The filter is optimized by an integral quality criterion. Minimax filtering equations are derived from the solution of the dual optimization problem. A numerical filter designing method is described and its convergence is proved. Results of numerical experiments are given.

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  1. 1.

    Liptser, R.Sh. and Shiryaev, A.N., Statistika sluchainykh protsessov, Moscow: Nauka, 1974. Translated under the title Statistics of Random Processes, Berlin: Springer-Verlag, 1978.

    Google Scholar 

  2. 2.

    Davis, M.H.A., Linear Estimation and Stochastic Control, London: Chapman, 1977. Translated under the title Lineinoe otsenivanie i stokhasticheskoe upravlenie, Moscow: Nauka, 1984.

    Google Scholar 

  3. 3.

    Pugachev, V.S. and Sinitsyn, I.N., Stokhasticheskie differentsial’nye sistemy. Analiz i fil’tratsiya, Moscow: Nauka, 1990. Translated under the title Stochastic Differential Systems. Analysis and Filtering, Chichester: Wiley, 1987.

    Google Scholar 

  4. 4.

    Sage, A.P. and White, C.C., Optimal Systems Control, Englewood Cliffs: Prentice Hall, 1977. Translated under the title Optimal’noe upravlenie sistemami, Moscow: Radio i Svyaz’, 1982.

    Google Scholar 

  5. 5.

    Bertsekas, D. and Rhodes, I.B., Recursive State Estimation for a Set of Membership Description of Uncertainty, IEEE Trans. Automat. Control, 1971, vol. 16, no. 2, pp. 117–128.

    Google Scholar 

  6. 6.

    Morris, J.M., The Kalman Filter: A Robust Estimator for Some Classes of Linear Quadratic Problems, IEEE Trans. Inf. Theory, 1976, vol. 22, pp. 526–534.

    Google Scholar 

  7. 7.

    Kurzhanskii, A.B., Upravlenie i otsenivanie v usloviyakh neopredelennosti (Control and Estimation under Uncertainty), Moscow: Nauka, 1977.

    Google Scholar 

  8. 8.

    Matasov, A.I., Estimators for Uncertain Dynamic Systems, Dordrecht: Kluwer Academic, 1998.

    Google Scholar 

  9. 9.

    Golybev, G.A., Muravlev, O.V., and Pisarev, V.F., Linear Recursive Filtration of Discrete-Time Dynamic Processes under Incomplete Information on Perturbation Processes, Avtom. Telemekh., 1989, no. 12, pp. 49–59.

  10. 10.

    Kats, I. Ya. and Timofeeva, G.A., A Modified Mismatch Method in Statistically Uncertain Estimation, Avtom. Telemekh., 1994, no. 2, pp. 100–109.

  11. 11.

    Verdu, S. and Poor, H.V., Minimax Linear Observers and Regulators for Stochastic Systems with Uncertain Second-Order Statistics, IEEE Trans. Automat. Control, 1984, vol. 29, no. 6, pp. 499–511.

    Google Scholar 

  12. 12.

    Anan’ev, B.I., Minimax Mean-Square Estimation in Statistically Uncertain Systems, Diff. Uravn., 1984, vol. 20, no. 8, pp. 1291–1297.

    Google Scholar 

  13. 13.

    Bobrik, G.I., Golovan, A.A., and Matasov, A.I., A Kalman Filter for the Guaranteed Approach to Topographic Fixation, Avtom. Telemekh., 1997, no. 10, pp. 34–47.

  14. 14.

    Orlov, Yu. and Basin, M., On Minimax Filtering over Discrete-Continuous Observations, IEEE Trans. Automat. Control, 1995, vol. 40, pp. 1623–1626

    Google Scholar 

  15. 15.

    Kurkin, O.M., Korobochkin, Yu. B., and Shatalov, S.A., Minimaksnaya obrabotka informatsii (Minimax Information Processing), Moscow: Energoatomizdat, 1990.

    Google Scholar 

  16. 16.

    Borisov, A.V. and Pankov, A.R., Minimax Filtering in Dynamic Systems Described by Stochastic Differential Equations with a Measure, Avtom. Telemekh., 1998, no. 6, pp. 139–152.

  17. 17.

    Pankov, A.R. and Miller, G.B., Minimax Linear Recursive Filtering of Stochastically Uncertain Sequencesby an Integral Criterion, Inf. Protsessy, 2001, vol. 1, no. 2, pp. 150–166.

    Google Scholar 

  18. 18.

    Pankov, A.R. and Siemenikhin, K.V., Minimax Estimation of Random Elements with Application to Infinite-Dimensional Statistical Linearization, Trudy II Mezhdunar. konf. “Identifikatsiya i zadachi upravleniya” (Proc. 2 Int. Conf. Identification and Control Problems), Moscow, 2003, pp. 1277–1291.

  19. 19.

    Rotenberg, Ya. N., Avtomaticheskoe upravlenie (Automatic Control), Moscow: Nauka, 1978.

    Google Scholar 

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Additional information

Translated from Avtomatika i Telemekhanika, No. 1, 2005, pp. 59–71.

Original Russian Text Copyright © 2005 by Miller, Pankov.

This work was supported by the Russian Foundation for Basic Research, project no. 02-01-00361.

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Miller, G.B., Pankov, A.R. Filtration of a random process in a statistically uncertain linear stochastic differential system. Autom Remote Control 66, 53–64 (2005).

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  • Filtration
  • Mechanical Engineer
  • Numerical Experiment
  • System Theory
  • Random Process