Sage Journals: Discover world-class research

Abstract

A piecewise linear function (PLF) plays a key role in the study of approximation of fuzzy system to an unknown function. PLF is not only the extension of unitary piecewise linear function under multi-variant conditions but also an important bridge between fuzzy system and approximation function. In this paper, a specific PLF is first constructed by the mesh subdivision of a polyhedron, and its analytical expression is obtained. Secondly, we verify that a PLF has the approximation performance to a continuous function based on the maximum norm and matrix norm, and the Mamdani fuzzy system can also approximate to arbitrary accuracy with respect to the maximum norm.

Finally, we give a realization process how the constructed PLF approximate to a continuous function in the two-dimensional Euclidean space through a simulation example. The result shows that PLF is a new effective tool for further study on the general fuzzy system or automatic control theory.

Keywords

Divided polyhedron mesh construction PLF Mamdani fuzzy system approximation

Get full access to this article

View all access options for this article.

References

Kosko

, Fuzzy systems as universal approximators, IEEE Transactions Computers 43(11) (1994), 1329–1333.

Feng

D.X.

, Basis of convex analysis, Beijing, Science Press, 1995.

Wang

G.J.

and Duan

C.X.

, Generalized hierarchical hybrid fuzzy system and its universal approximation, Control Theory and Application 29(5) (2012), 673–680.

Wang

G.J.

, Li

X.P.

and Sui

X.L.

, Universal approximation and its realization of generalized Mamdani fuzzy system based on K-integral norms, Acta Automatica Sinica 40(1) (2014), 143–148.

Ying

, Ding

Y.S.

, Li

S.K.

and Shao

S.H.

, Comparison of necessary conditions for typical Takagi-Sugeno and Mamdani fuzzy systems as universal approximators, IEEE Transactions on Systems, Man, Cybernetics, Part A 29(5) (1999), 508–514.

Zhang

and Wang

S.N.

, Piecewise linear approximation with a hypercube partition, J Tsinghua University (Sci & Tech) 48(1) (2008), 153–156.

Buckley

J.J.

, Fuzzy input-output controllers are universal approximators, Fuzzy Sets and Systems 58(2) (1993), 273–278.

Zeng

, Zhang

N.Y.

and Xu

W.L.

, A comparative study on sufficient conditions for Takagi-Sugeno fuzzy system as universal approximators, IEEE Transactions on Fuzzy Systems 8(6) (2000), 773–780.

Wang

L.X.

and Mendel

J.M.

, Fuzzy basic functions, universal approximation, and orthogonal leastsquare learning, IEEE Transactions Neural Networks 3(5) (1992), 807–814.

10.

Salmeri

, Mencattini

and Rovatti

, Function approximation using non-normalized SISO fuzzy systems, International Journal Approximation Reasoning 26(2) (2001), 223–229.

11.

Wang

P.Z.

, Tan

S.H.

, Song

F.M.

and Liang

, Constructive theory for fuzzy systems, Fuzzy Set and Systems 88(2) (1997), 195–203.

12.

Liu

P.Y.

, Mamdani fuzzy system is universal approximator to a class of random process, IEEE Transactions on Fuzzy Systems 10(6) (2002), 756–766.

13.

Liu

P.Y.

and Li

H.X.

, Approximation of generalized fuzzy system to integrable function, Science in China (Series E) 43(6) (2000), 618–628.

14.

Liu

P.Y.

and Li

H.X.

, Analyses for Lp (μ)-norm approximation capability of the generalized Mamdani fuzzy systems, Information Sciences 138(2) (2001), 195–210.

15.

Peng

W.L.

, Structure and approximation of a multi-dimensional piecewise linear function based on the input space of subdivision fuzzy systems, J System Science & Mathematical Science 34(3) (2014), 340–351.

16.

Tao

Y.J.

, Wang

H.Z.

and Wang

G.J.

, Approximation ability and its realization of the generalized Mamdani fuzzy system in the sense of Kp-integral norm, Acta Electronica Sinica 43(11) (2015), 2284–2229.

Mesh construction of PLF and its approximation process in Mamdani fuzzy system

Abstract

Keywords

Get full access to this article

References