Norm Approximation of Mamdani Fuzzy System to a Class of Integrable Functions

Abstract

The core of fuzzy system is to bypass the establishment of a definite mathematical model to carry out logical reasoning and intelligent calculation for fuzzy information; its main method is to process data information and language information based on a set of IF-THEN rules. Although it does not depend on accurate mathematical model, it has the ability of logical reasoning, numerical calculation and non-linear function approximation. In this paper, the analytic expression of the piecewise linear function (PLF) is first introduced by the determinant of coefficient matrix of linear equations in hyperplane, and an new Kp-norm is proposed by the K-quasi-subtraction operator. Secondly, it is proved that PLFs can approximate to a \( \hat{\mu }_{p} \)-integrable function in Kp-norm by real analysis method, and the Mamdani fuzzy system can also approximate to a PLF, and then an analytic representation of the upper bound of a subdivision number is given. Finally, using the three-point inequality of Kp-norm it is verified that Mamdani fuzzy system can approximate a \( \hat{\mu }_{p} \)-integrable function to arbitrary accuracy, and the approximation of the Mamdani fuzzy system to a class of integrable function is confirmed by t-hypothesis test method in statistics.

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Funding

This work has been supported by the National Natural Science Foundation of China (Grant No. 61374009). Research Project of Education Department of Hunan Province, China (Grant No. 17A031)

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Correspondence to Zuqiang Long.

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Wang, G., Wang, H. & Long, Z. Norm Approximation of Mamdani Fuzzy System to a Class of Integrable Functions. Int. J. Fuzzy Syst. (2021). https://doi.org/10.1007/s40815-020-01008-3

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Keywords

  • Piecewise linear function (PLF)
  • Subdivision number
  • \( \hat{\mu }_{p} \)-integrable function
  • K p-norm
  • Mamdani fuzzy system
  • Approximation