Methodology of high-precision non-linear filtering of random processes in fixed structure stochastic dynamic systems (Part 2)

System analysis, control and data processing


Аuthors

Kosachev I. M.1*, Chugai K. N.2**, Rybackov K. A.3***

1. Military Academy of the Republic of Belarus, 220, prospekt Nezavisimosti, Minsk, 220057, Belarus'
2. Scientific Research Institute of the Armed Forces, 4/3, str. Slavinsky, Minsk, 220103, Belarus
3. Moscow Aviation Institute (National Research University), 4, Volokolamskoe shosse, Moscow, А-80, GSP-3, 125993, Russia

*e-mail: kosachev1301@mail.ru
**e-mail: konstantin.ch40@gmail.com
***e-mail: rkoffice@mail.ru

Abstract

The article presents a methodical approach to high-precision nonlinear filtering of multidimensional non-Gaussian random processes in continuous-time stochastic dynamical systems with a fixed structure. The high accuracy of the developed algorithms for the optimal nonlinear filtering problem is stipulated by application of a posteriori higher order central moments of the filtered process. Adaptability of the developed high-precision nonlinear filtering algorithms is ensured by real time computing of a posteriori skewness and excess kurtosis for all phase coordinates of the filtered process, their subsequent comparison with the threshold values corresponding to Gaussian random process. If necessary, a posteriori higher central moments of the process being filtered are used in filtering algorithms.

The methodical approach under consideration to high-precision nonlinear filtering of multidimensional non-Gaussian random processes can be applied for eight basic variants of filtering problems, but engineering algorithms are presented only for the filtering problem with additive noises in mathematical model of the stochastic dynamical system.

The second part of the article describes five stages of the proposed methodical approach (the first three stages were described in the first part of the article):

4. Obtaining formulas linking a posteriori arbitrary order central moments with a posteriori cumulants for the filtered process.

5. Truncation and closure of a system of stochastic differential equations for a posteriori central moments of the filtered process.

6. Synthesis of high-precision optimal non-linear filters.

7. Setting known and computing missing initial conditions for a posteriori central moments of the filtered process.

8. Solving the optimal filtering problem (obtaining optimal estimate of the filtered process).

Keywords:

high-precision filtering, random process, dynamical system, stochastic system, fixed structure

References

  1. Kosachev I.M., Chugai K.N., Rybakov K.A. // Trudy MAI, 2019, no. 105, available at: http://trudymai.ru/eng/published.php?ID=104262

  2. Bain A., Crisan D. Fundamentals of Stochastic Filtering, New York, Springer, 2009, 394 p.

  3. Volkov V.A., Kudryavtseva I.A. Trudy MAI, 2016, no. 89, available at: http://trudymai.ru/eng/published.php?ID=73405

  4. Dashevskii M.L. Avtomatika i telemekhanika, 1967, no. 11, pp. 62 - 81.

  5. Dashevskii M.L. Problemy upravleniya i teoriya informatsii, 1975, no. 4, pp. 317 - 328.

  6. Kosachev I.M. Vestnik Voennoi akademii Respubliki Belarus', 2014, no. 4 (45), pp. 125 - 161.

  7. Pugachev V.S., Sinitsyn I.N. Stochastic Systems: Theory and Applications, Singapore, World Scientific, 2002, 928 p.

  8. Sinitsyn I.N. Fil'try Kalmana i Pugacheva (Kalman and Pugachev Filters), Moscow, Logos, 2007, 776 p.

  9. Drozhzhinov Yu.N., Zav'yalov B.I. Vvedenie v teoriyu obobshchennykh funktsii (Introduction to the Theory of Generalized Functions), Moscow, MIAN, 2006, 164 p.

  10. Mlechin V.V. Teoriya radioelektronnogo preodoleniya. Analiz vozdeistviya pomekh na radiotekhnicheskie sistemy i ustroistva (Theory of Radio-Electronic Overcoming. Analysis of the Impact of Interference on Radio Systems and Devices), Moscow, Radiotekhnika, 2009, 976 p.

  11. Cramér H. Mathematical Methods of Statistics, Princeton, Princeton University Press, 1999, 575 p.

  12. Kazakov I.E., Gladkov D.I. Metody optimizatsii stokhasticheskikh system (Optimization Methods for Stochastic Systems), Moscow, Nauka, 1987, 304 p.

  13. Korn G.A., Korn T.M. Mathematical Handbook for Scientists and Engineers, New York, Dover Publ., 2000, 1152 p.

  14. Tikhonov V.I. Statisticheskaya radiotekhnika (Statistical Radio Engineering), Moscow, Ripol klassik, 2013, 684 p.

  15. Kosachev I.M., Eroshenkov M.G. Analiticheskoe modelirovanie stokhasticheskikh system (Analytical Modeling of Stochastic Systems), Minsk, Nauka i tekhnika, 1993, 264 p.

  16. Bakhvalov N.S., Kornev A.A., Chizhonkov E.V. Chislennye metody. Reshenie zadach i uprazhneniya (Numerical Methods. Problems Solving and Exercises), Moscow, Laboratoriya znanii, 2016, 352 p.

  17. Gatelyuk O.V., Ismailov Sh.K., Manyukova N.V. Chislennye metody (Numerical Methods), Moscow, Yurait, 2019, 140 p.

  18. Rybakov K.A. Statisticheskie metody analiza i fil'tratsii v nepreryvnykh stokhasticheskikh sistemakh (Statistical Methods of Analysis and Filtering for Continuous Stochastic Systems), Moscow, Izd-vo MAI, 2017, 176 p.

  19. Averina T.A. Postroenie algoritmov statisticheskogo modelirovaniya sistem so sluchainoi strukturoi (Construction of Statistical Modeling Algorithms for Systems with Random Structure), Novosibirsk, RITs NGU, 2015, 155 p.

  20. Averina T.A., Karachanskaya E.V., Rybakov K.A. Differentsial'nye uravneniya i protsessy upravleniya, 2018, no. 1, pp. 54 - 76, available at: http://diffjournal.spbu.ru/RU/numbers/¬2018.1/article.1.3.html

  21. Kloeden P.E., Platen E. Numerical Solution of Stochastic Differential Equations, Berlin, Springer, 1995, 668 p.

  22. Kuznetsov D.F. Avtomatika i telemekhanika, 2018, no. 7, pp. 80 - 98.

  23. Kuznetsov D.F. Avtomatika i telemekhanika, 2019, no. 5, pp. 99 - 117.

  24. Rybakov K.A. Giroskopiya i navigatsiya, 2018, vol. 26, no. 4 (103), pp. 82 - 95.

  25. Sychev M.I. Trudy MAI, 2016, no. 90, available at: http://trudymai.ru/eng/published.php?ID=74830

  26. Groshev A.V. Trudy MAI, 2019, no. 104, available at: http://trudymai.ru/eng/published.php?ID=102217



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