Mesh construction of PLF and its approximation process in Mamdani fuzzy system
Guijun Wang, Xiaoping Li · Journal of Intelligent & Fuzzy Systems · 2017
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.