Automation and Remote Control

, Volume 79, Issue 10, pp 1767–1779 | Cite as

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

  • I. B. YadykinEmail author
  • A. B. Iskakov
Control Problems for the Development of Large-Scale 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 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.


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


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  1. 1.
    Simonchini, V., Computational Methods for Linear Matrix Equations, SIAM Rev., 2014, vol. 58, no. 3, pp. 377–441.MathSciNetCrossRefGoogle Scholar
  2. 2.
    Polyak, B.T. and Shcherbakov, P.S., Robastnaya ustoichivost’ i upravlenie (Robust Stability and Control), Moscow: Nauka, 2002.Google Scholar
  3. 3.
    Lancaster, P., Explicit Solution of Linear Matrix Equations, SIAM Rev., 1970, vol. 12, no. 4, pp. 544–566.MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Sylvester, J., Sur l’équation en matrices px=xq, Comptes Rendus de l’Acad. Sci., 1884, pp. 67–71.Google Scholar
  5. 5.
    Lyapunov, A., Problème général de la stabilité du mouvement, Commun. Soc. Math. Kharkov, 1893. Reprinted in Ann. of Math. Studies, 1949, vol.17.Google Scholar
  6. 6.
    Talbot, A., The Evaluation of Integrals of Products of Linear Systems Responses, Quart. J. Mech. Appl. Math., 1959, vol. 12, no. 4, pp. 488–503.MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Daletskii, Yu.L. and Krein, M.G., Ustoichivost’ reshenii differentsial’nykh uravnenii v banakhovom prostranstve (Stability of Solutions of Differential Equations in a Banach Space), Moscow: Nauka, 1970.Google Scholar
  8. 8.
    Godunov, S.K., Lektsii po sovremennym aspektam lineinoi algebry (Lectures on Modern Aspects of Linear Algebra), Novosibirsk: Nauchnaya Kniga, 2002.Google Scholar
  9. 9.
    Demidenko, G.V., Matrichnye uravneniya (Matrix Equations), Novosibirsk: Novosib. Gos. Univ., 2009.Google Scholar
  10. 10.
    Yadykin, I.B., On Properties of Gramians of Continuous Control Systems, Autom. Remote Control, 2010, vol. 71, no. 6, pp. 1011–1021.MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Yadykin, I.B., Iskakov, A.B., and Akhmetzyanov, A.V., Stability Analysis of Large-Scale Dynamical Systems by Sub-Gramian Approach, Int. J. Robust Nonlin. Control, 2014, vol. 24, no. 8–9, pp. 1361–1379.MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Yadykin, I.B., On Spectral Decompositions of Solutions to Discrete Lyapunov Equations, Dokl. Math., 2016, vol. 93, no. 3, pp. 344–347.MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Yadykin, I.B., Kataev, D.E., Iskakov, A.B., and Shipilov, V.K., Characterization of Power Systems Near Their Stability Boundary Using the Sub-Gramian Method, Control Eng. Practice, 2016, vol. 53, pp. 173–183.CrossRefGoogle Scholar
  14. 14.
    Yadykin, I.B. and Iskakov, A.B., Energy Approach to Stability Analysis of the Linear Stationary Dynamic Systems, Autom. Remote Control, 2016, vol. 77, no. 12, pp. 2132–2149.MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Benner, P. and Damm, T., Lyapunov Equations, Energy Functionals, and Model Order Reduction of Bilinear and Stochastic Systems, SIAM J. Control Optim., 2011, vol. 49, no. 2, pp. 686–711.MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Barinov, V.A. and Sovalov, S.A., Rezhimy energosistem: metody analiza i upravleniya (Modes of Power Systems: Methods of Analysis and Control), Moscow: Energoatomizdat, 1990.Google Scholar
  17. 17.
    Voropai, N.I., Uproshchenie matematicheskikh modelei dinamiki elektroenergeticheskikh sistem (Simplifying Mathematical Models of Electric Power Systems Dynamics), Novosibirsk: Nauka, 1981.Google Scholar
  18. 18.
    Kundur, P., Power Systems Stability and Control, New York: McGrow-Hill, 1994.Google Scholar
  19. 19.
    Rogers, M.G., Power Systems Oscillations, Norwell: Kluwer, 2000.CrossRefGoogle Scholar
  20. 20.
    Klein, M., Rogers, G.J., and Kundur, P., A Fundamental Study of Inter-Area Oscillations in Power Systems, IEEE Trans. Power Syst., 1991, vol. 6, no. 3, pp. 914–921.CrossRefGoogle Scholar
  21. 21.
    Bialek, J.W., Why Has it Happened Again? Comparison between the UCTE Blackout in 2006 and Blackouts of 2003, Proc. IEEE Lausanne Power Tech Int. Conf., Lausanne, Switzerland, 2007.Google Scholar
  22. 22.
    Neuman, P., Island Operations of Parallel Synchronous Generators—Simulators Case Study for Large Power Plants, IFAC-PapersOnLine, 2016, vol. 49, no. 27, pp. 164–169.CrossRefGoogle Scholar
  23. 23.
    Weber, H. and Al Ali, S., Influence of Huge Renewable Power Production on Inter Area Oscillations in the European ENTSO-E-System, IFAC-PapersOnLine, 2016, vol. 49, no. 27, pp. 12–17.CrossRefGoogle Scholar
  24. 24.
    Ruhle, O., Eigenvalue Analysis—All Information on Power System Oscillation Behavior Rapidly Analyzed, Siemens PTI. Newsletter, 2006, no. 99. Analysis.pdfGoogle Scholar
  25. 25.
    Yadykin, I.B. and Iskakov, A.B., Spectral Decompositions for the Solutions of Sylvester, Lyapunov, and Krein Equations, Dokl. Math., 2017, vol. 95, no. 1, pp. 103–107.CrossRefzbMATHGoogle Scholar
  26. 26.
    Polyak, B.T. and Tremba, A.A., Analytic Solution of a Linear Differential Equation with Identical Roots of the Characteristic Polynomial, Proc. XII Rus. Conf. on Control Problems, VSPU-2014, Moscow: Inst. Probl. Upravlen., 2014, pp. 212–217.Google Scholar
  27. 27.
    Antoulas, A.C., Approximation of Large-Scale Dynamical Systems, Philadelphia: SIAM, 2005.CrossRefzbMATHGoogle Scholar
  28. 28.
    Gradshtein, I.S. and Ryzhik, I.M., Tablitsy integralov, summ, ryadov i proizvedenii (Tables of Integrals, Sums, Series, and Products), Moscow: Fizmatlit, 1963.Google Scholar

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© Pleiades Publishing, Ltd. 2018

Authors and Affiliations

  1. 1.Trapeznikov Institute of Control SciencesRussian Academy of SciencesMoscowRussia
  2. 2.Skolkovo Institute of Science and TechnologyCenter for Research, Innovation, and Education for Energy SystemsMoscowRussia

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