Extending Jacobian matrix in proving stability for nonlinear systems with one equilibrium point such as compressor

TL;DR

This paper introduces an extended Jacobian matrix method to analyze and ensure global stability of nonlinear systems with a single equilibrium point, including industrial models like compressors.

Contribution

It proposes a novel extension of the Jacobian matrix for global stability analysis, integrating eigenvalue characteristics, and validates the approach with examples and benchmark systems.

Findings

The method effectively determines global stability for certain nonlinear systems.

Eigenvalue analysis can predict global instability in these systems.

The approach is applicable to industrial systems like compressors.

Abstract

Global stability of the systems has always been vital of importance; however, this concept has not yet been sufficiently developed for the nonlinear systems. This paper extends the Jacobian matrix so that this method be able to seek the criteria to ensure global stability for a special class of nonlinear systems. In this regard, we propose a new analysis method that utilizes the Jacobian matrix concept, integrating with the characteristics of the negative eigenvalues to analyze the global stability of the nonlinear systems with only one equilibrium point. Also, the positive eigenvalue to analyze the global instability of the nonlinear systems with only one equilibrium point. Some theorems such as Hartman-Grobman and Popov criteria can prove this claim. To this end, several examples and a benchmark systems have been intended to evaluate the efficiency of the proposed method. Results indicate the high potential of the proposed approach in order to develop the global stability analysis. The nonlinear compressor model, categorized in this extensive class, is also investigated as a well-known industrial system besides other several examples. The outcomes demonstrate that extended Jacobian stability analysis can ensure global stability for this class of nonlinear systems under some spatial conditions, discussed in this paper.