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.

中文翻译：

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Mamdani模糊系统中分段线性函数的逼近因子及其实现过程1

分段线性函数（PLF）不仅是单变量分段线性函数在多元情况下的推广，也是研究 Mamdani 和 Takagi-Sugeno 模糊系统对连续函数逼近的重要桥梁。本文在超平面中引入了PLF和细分的定义，利用矩阵行列式给出了PLF的解析表达式，并首先利用m-mesh细分提出了近似因子的概念。其次，通过划分一个三维立方体，找到了n维小多面体的顶点坐标及其变化规律，并给出了分段线性函数对应行列式的代数余因子和矩阵范数。最后，根据求解代数辅因子和矩阵范数的方法，证明近似因子与细分数无关，但近似精度与细分数有关。此外，通过实例实现了特定二元分段线性函数在二维空间根据无穷范数逼近连续函数的过程，并通过统计学中的t-假设检验验证了PLFs逼近连续函数的有效性。