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\begin{document}

\title{Approximation factor of the piecewise linear functions in Mamdani fuzzy
systems and its realization process}



\author[1]{jie Tao}%
\author[1]{chunfeng suo}%
\author[2]{Guijun Wang}%
\affil[1]{Tonghua Normal University}%
\affil[2]{Tianjin Normal University}%


\vspace{-1em}



  
  \date{May 07, 2024}


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\begin{abstract}
A piecewise linear function is not only an extension of a segmented
linear function of one variable in the case of multivariate variables,
but also an important bridge to study the approximation of a continuous
function by Mamdani and Takagi-Sugeno fuzzy systems. In this paper, the
concepts of a piecewise linear function and subdivision number are
introduced in hyperplane, the analytic expression of the piecewise
linear function is given by matrix determinant, and a new definition of
the approximation factor is first proposed by m-mesh subdivision.
Secondly, by the method of generating small polyhedron from
three-dimensional cube, the change rule of vertex coordinates of
n-dimensional subdivision polyhedron is studied, the vertex coordinates
of small polyhedron are obtained by rotating component coordinates of
their respective coordinate axes. Furthermore, the calculation methods
of algebraic cofactor and matrix norm for the corresponding determinant
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 subdivision number, but the approximation
precision has something to do with the subdivision number. In addition,
the realization process of a specific binary piecewise linear function
approaching a continuous function according to infinite norm in two
dimensions space is given by an example.%
\end{abstract}%



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