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


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.

This is a preview of subscription content, access via your institution.

We’re sorry, something doesn't seem to be working properly.

Please try refreshing the page. If that doesn't work, please contact support so we can address the problem.

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5


  1. 1.

    Wang, L.X., Mendel, J.: Fuzzy basis functions, universal approximation, and orthogonal least-squares learning. IEEE Trans. Neural Netw. 3(5), 807–814 (1992)

    Article  Google Scholar 

  2. 2.

    Buckley, J.J.: Fuzzy input-output controller are universal approximators. Fuzzy Sets Syst. 58(2), 273–278 (1993)

    MathSciNet  Article  Google Scholar 

  3. 3.

    Kosko, B.: Fuzzy systems are universal approximator. IEEE Trans. Comput. 43(11), 1329–1333 (1994)

    Article  Google Scholar 

  4. 4.

    Zeng, X.J., Singh, M.G.: Approximation theory of fuzzy system-MIMO case. IEEE Trans. Fuzzy Syst. 3(2), 219–235 (1995)

    Article  Google Scholar 

  5. 5.

    Wang, L.X.: Universal approximation by hierarchical systems. Fuzzy Set Syst. 93(1), 223–230 (1998)

    MathSciNet  Article  Google Scholar 

  6. 6.

    Ying, H.: General SISO Takagi-Sugeno fuzzy systems with linear rule consequents are universal approximators. IEEE Trans. Fuzzy Syst. 6(4), 582–587 (1998)

    Article  Google Scholar 

  7. 7.

    Ying, H., Ding, Y.S., Li, S.K., et al.: Comparison of necessary conditions for typical Takagi-Sugeno and Mamdani fuzzy system as universal approximator. IEEE Trans. Syst. Man Cybern. 29(5), 508–514 (1999)

    Article  Google Scholar 

  8. 8.

    Zeng, K., Zhang, N.R., Xu, W.L.: A comparative study on sufficient conditions for Takagi-Sugeno fuzzy systems as universal approximators. IEEE Trans. Fuzzy Syst. 8(6), 773–780 (2000)

    Article  Google Scholar 

  9. 9.

    Liu, P.Y., Li, H.X.: Analyses for Lp (μ)-norm approximation capability of the generalized Mamdani fuzzy systems. Inf. Sci. 138(2), 195–210 (2001)

    Article  Google Scholar 

  10. 10.

    Liu, P.Y.: Mamdani fuzzy system is universal approximator to a class of random process. IEEE Trans. Fuzzy Syst. 10(6), 756–766 (2002)

    Article  Google Scholar 

  11. 11.

    Salmeri, M., Mencattini, A., Rovatti, R.: Function approximation using non-normalized SISO fuzzy systems. Int. J. Approx. Reason. 26(2), 223–229 (2001)

    MATH  Google Scholar 

  12. 12.

    Zeng, X.J., Keane, J.A.: Approximation capabilities of hierarchical fuzzy systems. IEEE Trans. Fuzzy Syst. 13(5), 659–672 (2005)

    Article  Google Scholar 

  13. 13.

    Wang, G.J., Yang, Y., Li, X.P.: Rule number and approximation of the hybrid fuzzy system based on binary tree hierarchy. Int. J. Mach. Learn. Cybern. 9(6), 979–991 (2018)

    Article  Google Scholar 

  14. 14.

    Wang, G.J., Li, X.P.: Universal approximation of polygonal fuzzy neural networks in sense of K-integral norms. Science China. Inf. Sci. 54(11), 2307–2323 (2011)

    MathSciNet  MATH  Google Scholar 

  15. 15.

    Wang, D.G., Song, W.Y., Shi, P., et al.: Approximation to a class of non-autonomous systems by dynamic fuzzy inference marginal linearization method. Inf. Sci. 245, 197–217 (2013)

    MathSciNet  Article  Google Scholar 

  16. 16.

    Wang, D.G., Song, W.Y., Li, H.X.: Approximation properties of ELM-Fuzzy systems for smooth functions and their derivatives. Neurocomputing 149, 265–274 (2015)

    Article  Google Scholar 

  17. 17.

    Wang, G.J., Li, X.P., Sui, X.L.: Universal approximation and its realization of generalized Mamdani fuzzy system based on K-integral norms. Acta Autom. Sin. 40(1), 143–148 (2014)

    Google Scholar 

  18. 18.

    Peng, W.L.: Structure and approximation of a multi-dimensional piecewise linear function based on the input space of subdivision fuzzy systems. J. Syst. Sci. Math. Sci. 34(3), 340–351 (2014)

    MathSciNet  MATH  Google Scholar 

  19. 19.

    Tao, Y.J., Wang, H.Z., Wang, G.J.: Approximation of piecewise linear function in the sense of Kp-integral norm induced by K-quasi-arithmetic operations. J. Syst. Sci. Math. Sci. 36(2), 267–277 (2016)

    MathSciNet  MATH  Google Scholar 

  20. 20.

    Wang, H.Z., Tao, Y.J., Wang, G.J.: Approximation analysis of nonhomogeneous linear T-S fuzzy system based on grid piecewise linear function structure. J. Syst. Sci. Math. Sci. 35(11), 1276–1290 (2015)

    MathSciNet  MATH  Google Scholar 

  21. 21.

    Wang, H.Z., Tao, Y.J., Wang, G.J.: Optimizations of peak points and branch radius of nonlinear T-S fuzzy system based on triangular fuzzy numbers. J. Zhejiang Univ. 43(3), 264–270 (2016)

    MathSciNet  MATH  Google Scholar 

  22. 22.

    Wang, G.J., Li, X.P.: Mesh construction of PLF and its approximation process in Mamdani fuzzy system. J. Intell. Fuzzy Syst. 32(6), 4213–4225 (2017)

    Article  Google Scholar 

  23. 23.

    Wang, G.J., Sui, X.L., Li, X.P.: Approximation and its implementation process of the stochastic hybrid fuzzy system. Int. J. Mach. Learn. Cybern. 8(5), 1423–1437 (2017)

    Article  Google Scholar 

  24. 24.

    Wang, G.J., Li, X.P.: Generalized fuzzy valued θ-Choquet integral and their double null asymptotic additivity. Iran. J. Fuzzy Syst. 9(2), 13–24 (2012)

    MathSciNet  Google Scholar 

  25. 25.

    Long, Z.Q., Liang, X.M., Yang, L.R.: Some approximation properties of adaptive fuzzy systems with variable universe of discourse. Inform. Sci. 180, 2991–3005 (2010)

    Article  Google Scholar 

  26. 26.

    Sugeno, M., Murofushi, T.: Pseudo-additive measures and integrals. J. Math. Anal. Appl. 122, 197–222 (1987)

    MathSciNet  Article  Google Scholar 

  27. 27.

    Wang, G.J.: Polygonal Fuzzy Neural Network and Fuzzy System Approximation. Science Press, Beijing (2017)

    Google Scholar 

Download references


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)

Author information



Corresponding author

Correspondence to Zuqiang Long.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Wang, G., Wang, H. & Long, Z. Norm Approximation of Mamdani Fuzzy System to a Class of Integrable Functions. Int. J. Fuzzy Syst. (2021).

Download citation


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