Estimation of non-symmetric and unbounded region of attraction using shifted shape function and R-composition

TL;DR

This paper introduces a novel numerical method using shifted shape functions and R-composition to estimate the region of attraction for nonlinear polynomial systems, especially effective for non-symmetric and unbounded regions.

Contribution

It proposes a shifted shape function approach combined with R-composition to improve ROA estimation accuracy and efficiency for complex systems.

Findings

Enhanced ROA estimation for non-symmetric and unbounded regions.

Improved accuracy over existing methods with reasonable computational cost.

Validated effectiveness through benchmark examples.

Abstract

A general numerical method using sum of squares programming is proposed to address the problem of estimating the region of attraction (ROA) of an asymptotically stable equilibrium point of a nonlinear polynomial system. The method is based on Lyapunov theory, and a shape function is defined to enlarge the provable subset of a local Lyapunov function. In contrast with existing methods with a shape function centered at the equilibrium point, the proposed method utilizes a shifted shape function (SSF) with its center shifted iteratively towards the boundary of the newly obtained invariant subset to improve ROA estimation. A set of shifting centers with corresponding SSFs is generated to produce proven subsets of the exact ROA and then a composition method, namely R-composition, is employed to express these independent sets in a compact form by just a single but richer-shaped level set. The proposed method denoted as RcomSSF brings a significant improvement for general ROA estimation problems, especially for non-symmetric or unbounded ROA, while keeping the computational burden at a reasonable level. Its effectiveness and advantages are demonstrated by several benchmark examples from literature.