On Estimating the Probabilistic Region of Attraction for Partially Unknown Nonlinear Systems: An Sum-of-Squares Approach

TL;DR

This paper introduces a probabilistic method using sum-of-squares programming, Gaussian processes, and Chebyshev interpolants to estimate the region of attraction for partially unknown nonlinear systems with quantifiable confidence.

Contribution

It presents a novel tractable approach combining polynomial barrier functions, Gaussian process modeling, and sum-of-squares optimization to estimate attraction regions with probabilistic guarantees.

Findings

Effective estimation of attraction regions demonstrated through numerical examples.

Probabilistic bounds provide confidence levels for the estimates.

Integration of Gaussian processes with sum-of-squares enhances handling of unknown dynamics.

Abstract

Estimating the region of attraction for partially unknown nonlinear systems is a challenging issue. In this paper, we propose a tractable method to generate an estimated region of attraction with probability bounds, by searching an optimal polynomial barrier function. Chebyshev interpolants, Gaussian processes and sum-of-squares programmings are used in this paper. To approximate the unknown non-polynomial dynamics, a polynomial mean function of Gaussian processes model is computed to represent the exact dynamics based on the Chebyshev interpolants. Furthermore, probabilistic conditions are given such that all the estimates are located in certain probability bounds. Numerical examples are provided to demonstrate the effectiveness of the proposed method.