Approximability of triangular fuzzified nonlinear T–S fuzzy systems to p-integrable functions based on piecewise linear functions

The piecewise linear function (PLF) is an extension of subsection linear function with one variable on Euclidean space, and it plays an important bridging role in characterizing the relationship between the fuzzy systems and the approximated function. In this article, the determinant linear expression of PLF is introduced by subdividing the input space of p-integral functions and solving linear equations systems, and analyzes and represents the vertex coordinates of these small polyhedrons obtained after subdivision through geometric methods. A triangular fuzzified nonlinear Takagi Sugeno (TFNL T–S) fuzzy system is established by introducing the fuzzy rules of the nonlinear output and the determinant coefficients of PLF, and it was proved through the p-integral norm and matrix norm that TFNL T–S fuzzy system has approximation performance for p-integrable functions when all parameters of a linear part in the consequents of the rules take non-zero constants. Finally, the numerical simulations were conducted on the approximation performance of TFNL T–S fuzzy systems through simulation examples. The results show that the proposed TFNL T–S fuzzy system can indeed approximate a given p-integrable function with arbitrary accuracy.