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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Vydané v:	Automation and remote control Ročník 79; číslo 10; s. 1767 - 1779
Hlavní autori:	Yadykin, I. B., Iskakov, A. B.
Médium:	Journal Article
Jazyk:	English
Vydavateľské údaje:	Moscow Pleiades Publishing 01.10.2018 Springer Nature B.V
Predmet:	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
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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.
AbstractList	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.
Author	Yadykin, I. B. Iskakov, A. B.
Author_xml	– sequence: 1 givenname: I. B. surname: Yadykin fullname: Yadykin, I. B. email: jad@ipu.ru organization: Trapeznikov Institute of Control Sciences, Russian Academy of Sciences, Skolkovo Institute of Science and Technology, Center for Research, Innovation, and Education for Energy Systems – sequence: 2 givenname: A. B. surname: Iskakov fullname: Iskakov, A. B. organization: Trapeznikov Institute of Control Sciences, Russian Academy of Sciences, Skolkovo Institute of Science and Technology, Center for Research, Innovation, and Education for Energy Systems
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SubjectTerms	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
Title	Comparison of Sub-Gramian Analysis with Eigenvalue Analysis for Stability Estimation of Large Dynamical Systems
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