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The normal numbers of the fuzzy systems and their classes

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Abstract

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.

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Authors and Affiliations

  1. Faculty of Electronic Information and Electrical Engineering, Dalian University of Technology, Dalian, 116024, China

    HongXing Li & XueHai Yuan

  2. School of Mathematical Sciences, Beijing Normal University, Beijing, 100875, China

    JiaYin Wang

  3. Department of Automation, North China University of Technology, Beijing, 100144, China

    HongXing Li & YuCheng Li

Authors
  1. HongXing Li
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  2. XueHai Yuan
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  3. JiaYin Wang
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  4. YuCheng Li
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Correspondence to HongXing Li.

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Li, H., Yuan, X., Wang, J. et al. The normal numbers of the fuzzy systems and their classes. Sci. China Inf. Sci. 53, 2215–2229 (2010). https://doi.org/10.1007/s11432-010-4083-9

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  • DOI: https://doi.org/10.1007/s11432-010-4083-9

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