An Optimal Recurrent Logical-Dynamical Filter of a High Order and Its Covariance Approximations


The problem of logical recognition of the operating mode and dynamic estimation of the intra-mode state vector of a discrete-time stochastic Markov system with a random structure is considered. To develop an estimation algorithm implemented at the pace of system time on a computer of limited power, a method for designing a new finite-dimensional optimal structure filter is proposed. Its state vector consists of several latest estimates, and the current estimate is found in the form of an optimal-accuracy dependence on the last measurement and the vector of the previous filter state. The structural functions of the filter are designed in advance, which can be performed by the Monte Carlo method by obtaining their multivariate histograms. Due to the computational complexity of this procedure, algorithms are also proposed for constructing two numerical-analytical approximations of the filter, which take into account only the first two moments of random variables.

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  1. 1

    A. Nemura and E. Klekis, Estimation of Parameters and States of Systems with Jump-like Varying Properties (Mokslas, Vil’nyus, 1988) [in Russian].

    Google Scholar 

  2. 2

    V. A. Bukhalev, Recognition, Estimation and Control in Systems with a Random Hopping Structure (Nauka, Moscow, 1996) [in Russian].

    Google Scholar 

  3. 3

    A. V. Borisov and A. I. Stefanovich, “Optimal state filtering of controllable systems with random structure,” J. Comput. Syst. Sci. Int. 46, 348 (2007).

    MathSciNet  Article  Google Scholar 

  4. 4

    A. Bain and D. Crisan, Fundamentals of Stochastic Filtering (Springer, New York, 2009).

    Google Scholar 

  5. 5

    A. V. Bosov and A. R. Pankov, “Conditionally minimax filtering in a system with switching observation channels,” Automation and Remote Control 56, 835–843 (1995).

  6. 6

    E. A. Rudenko, “Analytical-numerical approximations of the optimal recurrent logical–dynamical low order filter-predictor,” J. Comput. Syst. Sci. Int. 54, 691 (2015).

    MathSciNet  Article  Google Scholar 

  7. 7

    E. A. Rudenko, “Finite-dimensional recurrent algorithms for optimal nonlinear logical–dynamical filtering,” J. Comput. Syst. Sci. Int. 55, 36 (2016).

    MathSciNet  Article  Google Scholar 

  8. 8

    E. A. Rudenko, “Optimal recurrent logical-dynamical finite memory filter,” J. Comput. Syst. Sci. Int. 56, 607 (2017).

    MathSciNet  Article  Google Scholar 

  9. 9

    I. A. Kudryavtseva, E. A. Rudenko, and K. A. Rybakov, “Software for optimal state estimation in stochastic dynamical systems,” Inform. Telekommun. Tekhnol., No. 43, 23–28 (2019).

  10. 10

    A. P. Trifonov and Yu. S. Shinakov, Joint Discrimination of Signals and Assessment of their Parameters against the Background of Interference (Radio Svyaz’, Moscow, 1986) [in Russian].

    Google Scholar 

  11. 11

    V. S. Pugachev and I. N. Sinitsyn, Stochastic Systems Theory (Logos, Moscow, 2004) [in Russian].

    Google Scholar 

  12. 12

    A. N. Shiryaev, Probability (Nauka, Moscow, 1980) [in Russian].

    Google Scholar 

  13. 13

    E. A. Rudenko, “Optimal nonlinear recurrent finite memory filter,” J. Comput. Syst. Sci. Int. 57, 43 (2018).

    MathSciNet  Article  Google Scholar 

  14. 14

    E. A. Rudenko, “Autonomous path estimation for a descent vehicle using recursive gaussian filters,” J. Comput. Syst. Sci. Int. 57, 695 (2018).

    Article  Google Scholar 

Download references


This study was supported by the Russian Foundation for Basic Research, project no. 18-08-00128-a.

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Correspondence to E. A. Rudenko.

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Translated by A. Mazurov

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Rudenko, E.A. An Optimal Recurrent Logical-Dynamical Filter of a High Order and Its Covariance Approximations. J. Comput. Syst. Sci. Int. 59, 854–870 (2020).

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