This article is cited in 1 paper

Control Problems for the Development of Large-Scale Systems

Comparison of sub-Gramian analysis with eigenvalue analysis for stability estimation of large dynamical systems

I. B. Yadykin^ab, A. B. Iskakov<sup>ba

a</sup> Skolkovo Institute of Science and Technology, Center for Research, Innovation, and Education for Energy Systems, Moscow, Russia
^b Trapeznikov Institute of Control Sciences, Russian Academy of Sciences, Moscow, Russia

Abstract: In earlier works, solutions of Lyapunov equations were represented as sums of Hermitian matrices corresponding to individual eigenvalues of the system or their pairwise combinations. Each eigen-term in these expansions are called a sub-Gramian. In this paper, we derive spectral decompositions of the solutions of algebraic Lyapunov equations in a more general formulation using the residues of the resolvent of the dynamics matrix. The qualitative differences and advantages of the sub-Gramian approach are described in comparison with the traditional analysis of eigenvalues when estimating the proximity of a dynamical system to its stability boundary. These differences are illustrated by the example of a system with a multiple root and a system of two resonating oscillators. The proposed approach can be efficiently used to evaluate resonant interactions in large dynamical systems.

Keywords: resonant interactions, large-scale systems, small signal stability analysis, spectral expansions, Lyapunov equations, sub-Gramians, stability boundary estimation.

Presented by the member of Editorial Board: A. I. Mikhal'skii

Received: 09.11.2017

 English version: Automation and Remote Control, 2018, 79:10, 1767–1779

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