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

TL;DR

This work tackles the challenge of approximating multi-dimensional nonlinear systems with PWA models when the subregion structure is unknown. It introduces a cut-based partitioning scheme using hinging hyperplanes to divide the augmented domain and learns local affine models $f_p(x)=J_p x+K_p$ on each subregion, with an objective that minimizes a relative squared error. A bi-level optimization framework iteratively refines the partition (via hyperplanes) and the local parameters, and can enforce continuity through a rank-one condition on neighboring Jacobians. Case studies show the method can achieve desired accuracy with fewer regions than some existing approaches and can maintain continuity, offering a scalable offline tool for nonlinear function approximation in control contexts.

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.

Figures (4)

• Figure 1: A schematic view cutting the domain using a hypersphere.
• Figure 2: A schematic view of two cut arrangements in Example 1.
• Figure 3: Illustration of the cut arrangement and the resulting subregions in Example 2.
• Figure 4: Cut-Based PWA approximation of the nonlinear function $\dot{s} = \sin(s+u^2)$ using local modes in (\ref{['fig:casef']}) shown in the same color as their corresponding subregion in (\ref{['fig:casec']}). The hyperplanes are defined by 8 pairs of points on $\mathcal{D}$ to form the cutting arrangement $\mathcal{H}$ (see Fig. \ref{['fig:def2']}), resulting in partitioning the domain into 16 subregions $\mathcal{R}$ (see Fig. \ref{['fig:def3']}).