Region of Attraction Estimation Using Union Theorem in Sum-of-Squares Optimization

TL;DR

This paper introduces a novel Union theorem-based Sum-of-Squares method for more accurate and less conservative estimation of the Region of Attraction in nonlinear polynomial systems, enhancing control design.

Contribution

It presents a new Union theorem allowing multiple shape functions to improve Lyapunov function representation and Region of Attraction estimation, overcoming existing conservativeness.

Findings

Significantly larger Region of Attraction estimates achieved.

Method effective for systems with non-symmetric or unbounded regions.

Demonstrated improvements through simulations on benchmark examples.

Abstract

Appropriate estimation of Region of Attraction for a nonlinear dynamical system plays a key role in system analysis and control design. Sum-of-Squares optimization is a powerful tool enabling Region of Attraction estimation for polynomial dynamical systems. Employment of a positive definite function called shape function within the Sum-of-Squares procedure helps to find a richer representation of the Lyapunov function and a larger corresponding Region of Attraction estimation. However, existing Sum-of-Squares optimization techniques demonstrate very conservative results. The main novelty of this paper is the Union theorem which enables the use of multiple shape functions to create a polynomial Lyapunov function encompassing all the areas generated by the shape functions. The main contribution of this paper is a novel computationally-efficient numerical method for Region of Attraction estimation, which remarkably improves estimation performance and overcomes limitations of existing methods, while maintaining the resultant Lyapunov function polynomial, thus facilitating control system design and construction of control Lyapunov function with enhanced Region of Attraction using conventional Sum-of-Squares tools. A mathematical proof of the Union theorem along with its application to the numerical algorithm of Region of Attraction estimation is provided. The method yields significantly enlarged Region of Attraction estimations even for systems with non-symmetric or unbounded Region of Attraction, which is demonstrated via simulations of several benchmark examples.