Approximation factor of the piecewise linear functions in Mamdani fuzzy system and its realization process

Piecewise linear function (PLF) is not only a generalization of univariate segmented linear function in multivariate case, but also an important bridge to study the approximation of continuous function by Mamdani and Takagi-Sugeno fuzzy systems. In this paper, the definitions of the PLF and subdivision are introduced in the hyperplane, the analytic expression of PLF is given by using matrix determinant, and the concept of approximation factor is first proposed by using m-mesh subdivision. Secondly, the vertex coordinates and their changing rules of the n-dimensional small polyhedron are found by dividing a three-dimensional cube, and the algebraic cofactor and matrix norm of corresponding determinants of piecewise linear functions are given. Finally, according to the method of solving algebraic cofactors and matrix norms, it is proved that the approximation factor has nothing to do with the number of subdivisions, but the approximation accuracy has something to do with the number of subdivisions. Furthermore, the process of a specific binary piecewise linear function approaching a continuous function according to infinite norm in two dimensions space is realized by a practical example, and the validity of PLFs to approximate a continuous function is verified by t-hypothesis test in Statistics.