Error Bounds for Compositions of Piecewise Affine Approximations

Nonlinear expressions are often approximated by piecewise affine (PWA) functions to simplify analysis or reduce computational costs. To reduce computational complexity, multivariate functions can be represented as compositions of functions with one or two inputs, which can be approximated individually. This paper provides efficient methods to generate PWA approximations of nonlinear functions via functional decomposition. The key contributions focus on intelligent placement of breakpoints for PWA approximations without requiring optimization, and on bounding the error of PWA compositions as a function of the error tolerance for each component of that composition. The proposed methods are used to systematically construct a PWA approximation for a complicated function, either to within a desired error tolerance or to a given level of complexity.