A Consistency Criterion for Optimizing Defuzzification in Fuzzy Control

  • Hyei Kyung Lee
  • Eric Paillet
  • Werner Peeters
Part of the Mathematical Modelling: Theory and Applications book series (MMTA, volume 24)

Abstract

Throughout the literature about fuzzy control, various defuzzification methods have been proposed, as well as been classified according to their properties, such as continuity, scale-invariance, core consistency and much more. However, the choice of a suitable defuzzification operator still remains an arbitrary one. We do not claim to have found the “perfect defuzzifier”, but in this article, we would like to add one particular new criterion, that we would like to call the Consistency Criterion, which can be used to measure the suitability of a certain defuzzification process, or at least compare several defuzzification operators, even parametric classes with infinitely many members, that contain some very commonly used ones, such as D.P. Filev and R.R. Yager’s BADD-defuzzification ([3]) as the most commonly used class throughout this text. A surprising result is that the minima of a measure of non-consistency yielded with respect to the parameters occurring in the rule base or other parameters of the problem, are certainly not reached in the most “natural” choice values for the parameters, but for some surprising, transcendent numbers.

Keywords

fuzzy control defuzzification consistency antecedent rule base 

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References

  1. 1.
    D. Dubois, J. Lang and H. Prade. Fuzzy sets in approximate reasoning part 2: Logical ap-proaches. Fuzzy Sets and Systems 40, pp. 203-244, 1991.MathSciNetMATHCrossRefGoogle Scholar
  2. 2.
    D. Dubois and H. Prade. Fuzzy sets in approximate reasoning part 1: Inference with possibility distributions. Fuzzy Sets and Systems 40, pp. 143-202,1991.MathSciNetMATHCrossRefGoogle Scholar
  3. 3.
    D.P. Filev and R.R. Yager. A generalized defuzzification method via BADD distributions. Internat. J. Intelligent Systems 6, pp. 687-697, 1991.MATHCrossRefGoogle Scholar
  4. 4.
    E.E. Kerre. A comparative study of the behaviour of some popular fuzzy implication operators. In: L.A. Zadeh and J. Kacprzyk, eds., Fuzzy Logic For The Mamagement of Uncertainty. Wiley, New York, 1992.Google Scholar
  5. 5.
    R. Lowen. Fuzzy Set Theory: Basic Concepts, Techniques and Bibliography. Kluwer Academic, Dordrecht, 1996.MATHGoogle Scholar
  6. 6.
    E.H. Mamdani and S.Assilian. An experiment in linguistic synthesis with a fuzzy logic con-troller. Int. Journal of Man-Machine Studies 7, pp. 1-13, 1975.MATHCrossRefGoogle Scholar
  7. 7.
    A.M. Norwich and I.B. Turksen. A model for the measurement of membership and the conse-quences of its empirical implementation. Fuzzy Sets and Systems 12, pp. 1-25, 1985.MathSciNetCrossRefGoogle Scholar
  8. 8.
    D. Ruan, E.E. Kerre, G. De Cooman, B. Cappelle and F. Vanmassenhove. Influence of the fuzzy implication operator on the method-of-cases inference rule. Internat. J. Approx. Reasoning, 4, pp. 307-318, 1990.MATHCrossRefGoogle Scholar
  9. 9.
    T.A. Runkler and M. Glesner. A set of axioms for defuzzification strategies — towards a the-ory of rational defuzzification operators. Second IEEE International Conference on Fuzzy Systems, San Francisco, pp. 1161-1166, 1994.Google Scholar
  10. 10.
    T.A. Runkler and M. Glesner. A set of axioms for defuzzification strategies — towards a the-ory of rational defuzzification operators. Second IEEE International Conference on Fuzzy Systems, San Francisco, pp. 1161-1166, 1993.Google Scholar
  11. 11.
    M. Sugeno. An introductory survey of fuzzy control. Inform. Sci 36, pp. 59-83, 1985.MathSciNetMATHCrossRefGoogle Scholar
  12. 12.
    W. Van Leekwijck and E.E. Kerre. Defuzzification: criteria and classification. Fuzzy Sets and Systems 108, pp. 159-178, 1999.MathSciNetMATHCrossRefGoogle Scholar
  13. 13.
    R.R. Yager and D.P. Filev. SLIDE: A simple adaptive defuzzification method. IEEE Trans. Fuzzy Systems 1(1), pp. 69-78, 1993.Google Scholar
  14. 14.
    L.A. Zadeh. Fuzzy sets. Inform. Control 8, pp. 338-353, 1965.MathSciNetMATHGoogle Scholar
  15. 15.
    L.A. Zadeh. Outline of a new approach to the analysis of complex systems and decision processes. IEEE Trans. Syst. Man. Cybernet., 3, pp. 28-44, 1973.Google Scholar
  16. 16.
    H.J. Zimmermann. Fuzzy Set Theory And Its Applications. Kluwer Academic, Boston/Dordrecht/London, 1996.MATHGoogle Scholar

Copyright information

© Springer 2008

Authors and Affiliations

  • Hyei Kyung Lee
    • 1
  • Eric Paillet
    • 1
  • Werner Peeters
    • 1
  1. 1.Dept. of Mathematics and Computer ScienceUniversity of AntwerpBelgium

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