On approximate Takagi–Sugeno linear representations of nonlinear functions

The paper discusses some methods of approximating a nonlinear function by a linear Takagi–Sugeno model, namely, by a convex combination of linear functions on a compact set. The known algorithms of approximating are considered as well as some new versions are proposed. They reduce both computational complexity and approximation error, and are applicable to a wider class of functions. The desired approximation is represented as a linear combination of the prescribed “basic” functions, which are assumed here to be the most commonly used in such models. We prove linear dependence of the basic functions and provide a linearly independent subsystem used for the proposed embodiments of the method of least squares. Discussion of the considered methods is presented. Also we calculate approximations for some functions of one and two variables to compare the maximum approximation errors and to see how the errors vary as we increase the number of the basic functions.