Estimating Regions of Attraction for Transitional Flows using Quadratic Constraints

TL;DR

This paper introduces new quadratic constraint-based methods for estimating the regions of attraction in nonlinear fluid systems, improving accuracy and reducing conservatism compared to existing approaches.

Contribution

It presents two novel algorithms—an iterative semi-definite programming method and a generalized eigenvalue approach—for analyzing transitional flows with enhanced efficiency and precision.

Findings

Algorithms outperform sum-of-squares methods in accuracy.

Iterative method refines estimates through successive SDP solutions.

Eigenvalue-based method offers lower computational complexity.

Abstract

This letter describes a method for estimating regions of attraction and bounds on permissible perturbation amplitudes in nonlinear fluids systems. The proposed approach exploits quadratic constraints between the inputs and outputs of the nonlinearity on elliptical sets. This approach reduces conservatism and improves estimates for regions of attraction and bounds on permissible perturbation amplitudes over related methods that employ quadratic constraints on spherical sets. We present and investigate two algorithms for performing the analysis: an iterative method that refines the analysis by solving a sequence of semi-definite programs, and another based on solving a generalized eigenvalue problem with lower computational complexity, but at the cost of some precision in the final solution. The proposed algorithms are demonstrated on low-order mechanistic models of transitional flows. We further compare accuracy and computational complexity with analysis based on sum-of-squares optimization and direct-adjoint looping methods.