Comparison of Sub-Gramian Analysis with Eigenvalue Analysis for Stability Estimation of Large Dynamical Systems

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...

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Bibliographic Details
Published in:Automation and remote control Vol. 79; no. 10; pp. 1767 - 1779
Main Authors: Yadykin, I. B., Iskakov, A. B.
Format: Journal Article
Language:English
Published: Moscow Pleiades Publishing 01.10.2018
Springer Nature B.V
Subjects:
CAE) and Design
Calculus of Variations and Optimal Control; Optimization
Computer-Aided Engineering (CAD
Control
Control Problems for the Development of Large-Scale Systems
Dynamic stability
Dynamical systems
Eigenvalues
Mathematical analysis
Mathematics
Mathematics and Statistics
Mechanical Engineering
Mechatronics
Oscillators
Resonant interactions
Robotics
Stability analysis
Systems Theory
spectral expansions
resonant interactions
stability boundary estimation
small signal stability analysis
sub-Gramians
Lyapunov equations
large-scale systems
ISSN:0005-1179, 1608-3032
Online Access:Get full text
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Summary: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.
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ISSN:0005-1179
1608-3032
DOI:10.1134/S000511791810003X