Region of Attraction Estimation Using Invariant Sets and Rational Lyapunov Functions

TL;DR

This paper introduces a novel method for estimating the region of attraction of nonlinear systems using invariant sets and rational Lyapunov functions, with an algorithm that iteratively enlarges these estimates for polynomial systems.

Contribution

It proposes a new approach combining invariant sets and rational Lyapunov functions, with an algorithm to systematically enlarge the region of attraction estimates for polynomial systems.

Findings

Provides conditions linking invariant sets and Lyapunov functions.

Develops an algorithm guaranteeing enlargement of the RA estimate.

Connects the method to fundamental RA estimation results.

Abstract

This work addresses the problem of estimating the region of attraction (RA) of equilibrium points of nonlinear dynamical systems. The estimates we provide are given by positively invariant sets which are not necessarily defined by level sets of a Lyapunov function. Moreover, we present conditions for the existence of Lyapunov functions linked to the positively invariant set formulation we propose. Connections to fundamental results on estimates of the RA are presented and support the search of Lyapunov functions of a rational nature. We then restrict our attention to systems governed by polynomial vector fields and provide an algorithm that is guaranteed to enlarge the estimate of the RA at each iteration.