Generalization of Gershgorin's theorem. Analysis and design of control laws

TL;DR

This paper extends Gershgorin's theorem to better estimate eigenvalues of matrices with interval and non-stationary elements, enabling improved stability analysis and control law design for complex network systems.

Contribution

It introduces e-circles for more accurate eigenvalue localization and applies these results to analyze and control large, non-stationary network systems without diagonal dominance.

Findings

Enhanced eigenvalue localization with e-circles

Applicable to large-scale network stability analysis

Design of control laws for non-stationary systems

Abstract

The application of the Gershgorin circle theorem and some of its derivatives to estimate the eigenvalues of a matrix is considered. The obtained results are developed to obtain the localization region of the eigenvalues of a matrix with interval-indefinite constant or non-stationary elements. The concept of e-circles is introduced to obtain more accurate estimates of these regions than when using Gershgorin circles. The obtained results are applied to the stability analysis of network systems, where it is shown that the proposed methods allow one to analyze a network with a much larger number of agents than when using the CVX, Yalmip, eig and lyap methods (functions in MatLab). It is further shown that if the obtained results are applied not to the system itself, but to the result obtained using the Lyapunov function method, then one can study systems with matrices without diagonal dominance. This made it possible to consider a modification of the Demidovich condition for systems with non-stationary parameters and design of a control law for non-stationary systems with matrices without diagonal dominance. All obtained results are illustrated by numerical modeling.