Advancing Nonlinear System Stability Analysis with Hessian Matrix Analysis

TL;DR

This paper presents a new method for analyzing the global stability of nonlinear systems by incorporating Hessian matrix eigenvalues and Taylor series boundary errors, offering a more flexible and robust approach than traditional Jacobian-based methods.

Contribution

It introduces a novel stability criterion that includes Hessian eigenvalues and Taylor boundary errors, applicable regardless of system dimension or equilibrium count.

Findings

Validated on two industrial benchmark systems.

Enhanced stability analysis surpasses traditional Jacobian methods.

Applicable to a wide range of nonlinear systems.

Abstract

This paper introduces an innovative method for ensuring global stability in a broad array of nonlinear systems. The novel approach enhances the traditional analysis based on Jacobian matrices by incorporating the Taylor series boundary error of estimation and the eigenvalues of the Hessian matrix, resulting in a fresh criterion for global stability. The main strength of this methodology lies in its unrestricted nature regarding the number of equilibrium points or the system's dimension, giving it a competitive edge over alternative methods for global stability analysis. The efficacy of this method has been validated through its application to two established benchmark systems within the industrial domain. The results suggest that the expanded Jacobian stability analysis can ensure global stability under specific circumstances, which are thoroughly elaborated upon in the manuscript. The proposed approach serves as a robust tool for assessing the global stability of nonlinear systems and holds promise for advancing the realms of nonlinear control and optimization.