Iterative Cut-Based PWA Approximation of Multi-Dimensional Nonlinear Systems

TL;DR

This paper presents an iterative, cut-based method for approximating multi-dimensional nonlinear systems with piecewise affine functions, improving accuracy without prior system knowledge, and demonstrating effectiveness through case studies.

Contribution

It introduces a novel iterative approach using hinging hyperplanes for PWA approximation, handling complex multi-dimensional nonlinear systems without prior information.

Findings

Method achieves desired approximation accuracy iteratively.

Performance surpasses existing approaches in case studies.

Allows flexible subregion definitions and continuity constraints.

Abstract

PieceWise Affine (PWA) approximations for nonlinear functions have been extensively used for tractable, computationally efficient control of nonlinear systems. However, reaching a desired approximation accuracy without prior information about the behavior of the nonlinear systems remains a challenge in the function approximation and control literature. As the name suggests, PWA approximation aims at approximating a nonlinear function or system by dividing the domain into multiple subregions where the nonlinear function or dynamics is approximated locally by an affine function also called local mode. Without prior knowledge of the form of the nonlinearity, the required number of modes, the locations of the subregions, and the local approximations need to be optimized simultaneously, which becomes highly complex for large-scale systems with multi-dimensional nonlinear functions. This paper introduces a novel approach for PWA approximation of multi-dimensional nonlinear systems, utilizing a hinging hyperplane formalism for cut-based partitioning of the domain. The complexity of the PWA approximation is iteratively increased until reaching the desired accuracy level. Further, the tractable cut definitions allow for different forms of subregions, as well as the ability to impose continuity constraints on the PWA approximation. The methodology is explained via multiple examples and its performance is compared to two existing approaches through case studies, showcasing its efficacy.