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.
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Original Russian Text © I.B. Yadykin, A.B. Iskakov, 2018, published in Avtomatika i Telemekhanika, 2018, No. 10, pp. 39–54.
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Yadykin, I.B., Iskakov, A.B. Comparison of Sub-Gramian Analysis with Eigenvalue Analysis for Stability Estimation of Large Dynamical Systems. Autom Remote Control 79, 1767–1779 (2018). https://doi.org/10.1134/S000511791810003X
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DOI: https://doi.org/10.1134/S000511791810003X