Chebyshev Approximation and Higher Order Derivatives of Lyapunov Functions for Estimating the Domain of Attraction

TL;DR

This paper introduces Chebyshev approximation methods for estimating the domain of attraction of nonlinear systems, offering improved accuracy and efficiency over traditional Taylor expansion techniques.

Contribution

It develops new Chebyshev approximation techniques, including handling remainders and higher order derivatives, to better estimate the domain of attraction for non-polynomial systems.

Findings

Chebyshev approximation improves estimation accuracy over Taylor expansion.

The proposed methods are computationally efficient for higher order derivatives.

Numerical examples validate the effectiveness of the Chebyshev-based approaches.

Abstract

Estimating the Domain of Attraction (DA) of non-polynomial systems is a challenging problem. Taylor expansion is widely adopted for transforming a nonlinear analytic function into a polynomial function, but the performance of Taylor expansion is not always satisfactory. This paper provides solvable ways for estimating the DA via Chebyshev approximation. Firstly, for Chebyshev approximation without the remainder, higher order derivatives of Lyapunov functions are used for estimating the DA, and the largest estimate is obtained by solving a generalized eigenvalue problem. Moreover, for Chebyshev approximation with the remainder, an uncertain polynomial system is reformulated, and a condition is proposed for ensuring the convergence to the largest estimate with a selected Lyapunov function. Numerical examples demonstrate that both accuracy and efficiency are improved compared to Taylor approximation.