Science China Information Sciences

, Volume 53, Issue 11, pp 2215–2229 | Cite as

The normal numbers of the fuzzy systems and their classes

  • HongXing LiEmail author
  • XueHai Yuan
  • JiaYin Wang
  • YuCheng Li
Research Papers


ℝ-fuzzy set is defined in this paper, which is regarded as the generalization of the Zadeh fuzzy set. By means of CRI method, some fuzzy systems are constructed by suitably using several kinds of ℝ-fuzzy sets as fuzzy inference antecedents, such as interpolation fuzzy system, Bernstein fuzzy system, Lagrange fuzzy system and Hermite fuzzy system. A notion of the normal number of the fuzzy system is defined here, we have shown that all fuzzy systems are able to be classified as three classes such as the normal fuzzy systems, the regular fuzzy systems and the singular fuzzy systems under the significance of the normal numbers of fuzzy systems. Finally, the generalized Bernstein polynomial is obtained by constructing Bernstein fuzzy system, it is proved that the generalized Bernstein polynomial is uniformly convergent in C[a, b] under a weaker condition, and it is pointed out that there exist generalized Bernstein polynomials to be not convergent in C[a, b] by use of constructing a counterexample.


Open image in new window-fuzzy set fuzzy system the normal number of fuzzy system universal approximation property for fuzzy system 


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Copyright information

© Science China Press and Springer-Verlag Berlin Heidelberg 2010

Authors and Affiliations

  • HongXing Li
    • 1
    • 3
    Email author
  • XueHai Yuan
    • 1
  • JiaYin Wang
    • 2
  • YuCheng Li
    • 3
  1. 1.Faculty of Electronic Information and Electrical EngineeringDalian University of TechnologyDalianChina
  2. 2.School of Mathematical SciencesBeijing Normal UniversityBeijingChina
  3. 3.Department of AutomationNorth China University of TechnologyBeijingChina

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