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

Article type: Research Article

Authors: Wang, Guijun^* | Li, Xiaoping

Affiliations: School of Mathematical Sciences, Tianjin Normal University, Tianjin, China

Correspondence: [*] Corresponding author. Guijun Wang, School of Mathematical Sciences, Tianjin Normal University, Tianjin 300387, China. Tel./Fax: +86 022 23009237; E-mail: tjwgj@126.com.

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

DOI: 10.3233/JIFS-16586

Journal: Journal of Intelligent & Fuzzy Systems, vol. 32, no. 6, pp. 4213-4225, 2017