Skip to main content

Minimax estimation in systems of observation with Markovian chains by integral criterion


A problem of estimation of states and parameters in stochastic dynamic systems of observation with discrete time containing a Markovian chain is studied. Matrices of transient probabilities and observation plans are random with unknown distribution with a given compact carrier. Observations, on the basis of which the estimation is made, are available at a fixed interval of time [0, T]. As a loss function, we have a conditional mathematical expectation with respect to the available observations of 2-norm of the estimation error of a signal process on [0, T]. The problem is in constructing an estimate minimizing losses correspondent to the worst distribution of the pair “a matrix of transient probabilities—a matrix of observation plan” form a set of allowable distributions. For a correspondent minimax problem is demonstrated the existence of a saddle point and is obtained a form of the wanted minimax estimation. The applicability of the obtained results is illustrated by a numerical example of the estimation of a state of TCP under the conditions of uncertainty of communication channel parameters.

This is a preview of subscription content, access via your institution.


  1. Sinitsyn, I.N., Fil'try Kalmana i Pugacheva (Kalman and Pugachev Filters), Moscow: Logos, 2007.

    Google Scholar 

  2. Wonham, W.N., Some Applications of Stochastic Differential Equations to Optimal Nonlinear Filtering, SIAM J. Control, 1965, no. 2, pp. 347–369.

  3. Elliott, R.J., Aggoun, L., and Moore, J.B., Hidden Markov Models: Estimation and Control, Berlin: Springer, 1995.

    MATH  Google Scholar 

  4. Bar-Shalom, Y., Li, R., Li, X.-R., and Kirubarajan, T., Estimation with Applications to Tracking and Navigation, New York: Wiley, 2001.

    Book  Google Scholar 

  5. Miller, B.M., Avrachenkov, K.E., Stepanyan, K.V., and Miller, G.B., Flow Control as Stochastic Optimal Control Problem with Incomplete Information, Proc. INFOCOM'2005, Miami, 2005, pp. 1328–1337.

  6. Cvitanic, J., Liptser, R.Sh., and Rozovskii, B., Tracking Volatility, Proc. 39th IEEE Conf. Decision Control, Sydnay, 2000, pp. 1189–1193.

  7. Elliott, R.J., Malcolm, W.P., and Tsoi, A., HMM Volatility Estimation, Proc. 41th IEEE Conf. Decision Control, Las Vegas, 2002, pp. 398–404.

  8. de Souza, C.E. and Fragoso, M.D., Robust H 1 Filtering for Uncertain Markovian Jump Linear Systems, Int. J. Robust Nonlinear Control, 2002, vol. 12, no. 5, pp. 435–446.

    MATH  Article  Google Scholar 

  9. Miller, B.M. and Runggaldier, W.J., Kalman Filtering for Linear Systems with Coefficients Driven by a Hidden Markov Jump Process, Syst. Control Lett., 1997, vol. 31, pp. 93–102.

    MathSciNet  MATH  Article  Google Scholar 

  10. Shi, P., Boukas, E.K., and Agarwal, R.K., Kalman Filtering for Continuous-Time Uncertain Systems with Markovian Jumping Parameters, IEEE Trans. Automat. Control., 1999, vol. 44, no. 8, pp. 1592–1597.

    MathSciNet  MATH  Article  Google Scholar 

  11. Cappe, O., Moulines, V., and Ryden, T., Inferences in Hidden Markov Models, Berlin: Springer, 2005.

    Google Scholar 

  12. Merhav, N. and Ziv, J., Estimating with Partial Statistics the Parameters of Ergodic Finite Markov Sources, IEEE Trans. Inform. Theory, 1989, vol. 35, no. 2, pp. 326–333.

    MathSciNet  MATH  Article  Google Scholar 

  13. Dey, S. and Moore, J.B., Risk-Sensitive Filtering and Smoothing for Hidden Markov Models, Syst. Control Lett., 1995, vol. 25, pp. 361–366.

    MathSciNet  MATH  Article  Google Scholar 

  14. Zhang, L., Shi, P., Boukas, E.K., and Wang, C., Robust l 2-l Filtering for Switched Linear Discrete Time-Delay Systems with Polytopic Uncertainties, IET Control Theory Appl., 2007, vol. 3, no. 1, pp. 722–730.

    MathSciNet  Article  Google Scholar 

  15. Xie, L., Ugrinovskii, V.A., and Petersen, I.R., Finite Horizon Robust State Estimation for Uncertain Finite-Alphabet Hidden Markov Models with Conditional Relative Entropy Constraints, SIAM J. Control Optim., 2008, vol. 47, no. 1, pp. 478–508.

    MathSciNet  Article  Google Scholar 

  16. Borisov, A.V., Minimax A Posteriori Estimation in the Hidden Markov Models, Autom. Remote Control, 2007, vol. 68, no. 11, pp. 1917–1930.

    MathSciNet  MATH  Article  Google Scholar 

  17. Borisov, A.V., Minimax A Posteriori Estimation of the Markov Processes with Finite State Spaces, Autom. Remote Control, 2008, vol. 69, no. 2, pp. 233–246.

    MathSciNet  MATH  Article  Google Scholar 

  18. Liptser, R.S. and Shiryaev, A.N., Statistika sluchainykh protsessov (nelineinaya fil'tratsiya i smezhnye voprosy), Moscow: Nauka, 1974. Translated into English under the title Statistics of Random Processes: Theory and Applications, New York: Springer, 2000.

    Google Scholar 

  19. Wall, J.E., Willsky, A.S., and Sandell, N.R., On the Fixed-Interval Smoothing Problem, Stochastic, 1981, no. 1, pp. 1–42.

  20. Gilbert, E.M., Capacity of a Burst-Noise Channel, Bell Syst. Tech., 1960, no. 5, pp. 1253–1265.

  21. Semenikhin, K.V., Minimax Estimation of Random Elements by the Root-Mean-Square Criterion, Izv. Ross. Akad. Nauk, Teor. Sist. Upravlen., 2003, no. 5, pp. 12–25.

  22. Bulinskii, A.V. and Shiryaev, A.N., Teoriya sluchainykh protsessov (Theory of Random Processes), Moscow: Fizmatlit, 2005.

    Google Scholar 

  23. Kirillov, A.A. and Gvishiani, A.D., Teoremy i zadachi funktsional'nogo analiza (Theorem and Problems of Functional Analysis), Moscow: Nauka, 1988.

    MATH  Google Scholar 

Download references

Author information

Authors and Affiliations


Additional information

Original Russian Text © A.V. Borisov, A.V. Bosov, A.I. Stefanovich, 2011, published in Avtomatika i Telemekhanika, 2011, No. 2, pp. 41–55.

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Borisov, V., Bosov, A.V. & Stefanovich, A.I. Minimax estimation in systems of observation with Markovian chains by integral criterion. Autom Remote Control 72, 255–268 (2011).

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI:


  • Saddle Point
  • Remote Control
  • Hide Markov Model
  • Loss Function
  • Observation Plan