Problems of States Estimation (Filtration) of Extremal Fuzzy Processes

  • Gia Sirbiladze
Part of the IFSR International Series on Systems Science and Engineering book series (IFSR, volume 28)


A fuzzy-integral model of an extremal fuzzy process is considered. The model describes the evolution of one class of weakly structurable fuzzy dynamic systems, i.e., of the so-called extremal fuzzy dynamic systems constructed in Chap. 5. In the present chapter, problems of an optimal filtering of continuous as well as of discrete extremal fuzzy processes are solved by means of “past” evaluating information. Sufficient conditions are established for the existence of an optimal estimating fuzzy process. Using only one “past” evaluating fuzzy state of the considered system, variants of fuzzy observers (representations of an optimal estimating extremal fuzzy process) are constructed in terms of approximation of piecewise-constant and extremal measurable filtration functions for continuous and discrete extremal fuzzy dynamic systems. The results obtained are illustrated by a numerical example.


Time Moment Optimal Estimate Fuzzy Function Compatibility Function Fuzzy State 
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  1. 11.
    Chen, G., Xie, Q., Shieh, L.S.: Fuzzy Kalman filtering. Inform. Sci. 109(1–4), 197–209 (1998)MathSciNetMATHGoogle Scholar
  2. 12.
    Chen, G., Wang, J., Shieh, L.: Interval Kalman filtering. IEEE Trans. Aero. Electron. Syst. 33(1), 250–259 (1997)CrossRefGoogle Scholar
  3. 27.
    Dubois, D., Prade, H.: Possibility Theory. An Approach to Computerized Processing of Uncertainty. Plenum Press, New York (1988)MATHCrossRefGoogle Scholar
  4. 31.
    Elliott, R.J.: Stochastic Calculus and Applications. Applications of Mathematics (New York), vol. 18. Springer, New York (1982)Google Scholar
  5. 44.
    Grabisch, M., Murofushi, T., Sugeno M. (eds.): Fuzzy Measures and Integrals. Theory and Applications. Studies in Fuzziness and Soft Computing, vol. 40. Physica, Heidelberg (2000)Google Scholar
  6. 54.
    Kalman, R.E.: A new approach to linear filtering and prediction problems. J. Basic Eng. 82, 35–45 (1960)CrossRefGoogle Scholar
  7. 55.
    Kalman, R.E., Bucy, R.S.: New results in linear filtering and prediction theory. Trans. ASME Ser. J. Basic Eng. 83, 95–108 (1961)MathSciNetCrossRefGoogle Scholar
  8. 94.
    Nachtegael, M., Van der Weken, D., Van De Ville, D., Kerre, E.E. (eds.): Fuzzy filters for image processing. Studies in Fuzziness and Soft Computing, vol. 122. Springer, Berlin (2003).Google Scholar
  9. 113.
    Simon, D.: Kalman filtering for fuzzy discrete time dynamic systems. Appl. Soft Comput. 3(3), 191–207 (2003)CrossRefGoogle Scholar
  10. 128.
    Sirbiladze, G., Gachechiladze, T.: Restored fuzzy measures in expert decision-making. Inform. Sci. 169(1–2), 71–95 (2005)MathSciNetMATHCrossRefGoogle Scholar
  11. 134.
    Sirbiladze, G., Sikharulidze, A.: Weighted fuzzy averages in fuzzy environment. I. Insufficient expert data and fuzzy averages. International Journal of Uncertainty, Fuzziness and Knowledge-Based Systems 11(2), 139–157 (2003)Google Scholar
  12. 135.
    Sirbiladze, G., Sikharulidze, A.: Weighted fuzzy averages in fuzzy environment. II. Generalized weighted fuzzy expected values in fuzzy environment. International Journal of Uncertainty, Fuzziness and Knowledge-Based Systems 11(2), 159–172 (2003)Google Scholar
  13. 138.
    Sirbiladze, G., Zaporozhets, N.: About two probability representations of fuzzy measures on a finite set. J. Fuzzy Math. 11(3), 549–565 (2003)MathSciNetMATHGoogle Scholar
  14. 141.
    Sugeno, M.: Theory of fuzzy integrals and its applications. PhD thesis, Tokyo Institute of Technology (1979)Google Scholar
  15. 160.
    Weng, Z., Chen, G., Shieh, L.S., Larsson, J.: Evolutionary programming Kalman filter. Inform. Sci. 129(1–4), 197–210 (2000)MathSciNetMATHGoogle Scholar
  16. 184.
    Zimmermann, H.-J.: Fuzzy Sets, Decision Making and Expert Systems. Kluwer, Boston (1987)Google Scholar

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© Springer Science+Business Media, LLC 2013

Authors and Affiliations

  • Gia Sirbiladze
    • 1
  1. 1.Department of Computer SciencesIv. Javakhishvili Tbilisi St. UniversityTbilisiGeorgia

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