Skip to main content
SpringerLink
Log in

On spectral decompositions of solutions to discrete Lyapunov equations

  • Control Theory
  • Published:
Doklady Mathematics Aims and scope Submit manuscript

Abstract

A new approach to solving discrete Lyapunov matrix algebraic equations is based on methods for spectral decomposition of their solutions. Assuming that all eigenvalues of the matrices on the left-hand side of the equation lie inside the unit disk, it is shown that the matrix of the solution to the equation can be calculated as a finite sum of matrix bilinear quadratic forms made up by products of Faddeev matrices obtained by decomposing the resolvents of the matrices of the Lyapunov equation. For a linear autonomous stochastic discrete dynamic system, analytical expressions are obtained for the decomposition of the asymptotic variance matrix of system’s states.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. J. Sylvester, C. R. Acad. Sci. 99, 67–71 (1884).

    Google Scholar 

  2. A. M. Lyapunov, General Problem of the Stability of Motion (ONTI, Moscow, 1935; Princeton Univ. Press, Princeton, 1947).

    MATH  Google Scholar 

  3. S. N. Vasil’ev and A. A. Kosov, Autom. Remote Control 72 (6), 1163–1183 (2011).

    Article  MathSciNet  Google Scholar 

  4. G. V. Demidenko, Matrix Equations (Novosib. Gos. Univ., Novosibirsk, 2009) [in Russian].

    MATH  Google Scholar 

  5. Yu. L. Daleckii and M. G. Krein, Stability of Solutions of Differential Equations in Banach Space (Nauka, Moscow, 1970; Am. Math. Soc., Providence, 1974).

    Google Scholar 

  6. S. K. Godunov, Modern Aspects of Linear Algebra (Am. Math. Soc., Providence, 1998; Nauchnaya Kniga, Novosibirsk, 2002).

    MATH  Google Scholar 

  7. R. H. Bartels and G. W. Steward, Commun. ACM 15, 820–826 (1972).

    Article  Google Scholar 

  8. G. H. Golub, S. Nash, and C. Van Loan, IEEE Trans. Autom. Control 24 (6), 909–913 (1979).

    Article  Google Scholar 

  9. Kh. D. Ikramov, Numerical Solution of Matrix Equations (Nauka, Moscow, 1984) [in Russian].

    MATH  Google Scholar 

  10. L. A. Mironovskii and T. N. Solov’eva, Autom. Remote Control 74 (4), 588–603 (2013).

    Article  MathSciNet  Google Scholar 

  11. B. T. Polyak and P. S. Shcherbakov, Robust Stability and Control (Nauka, Moscow, 2002) [in Russian].

    MATH  Google Scholar 

  12. D. K. Faddeev and V. N. Faddeeva, Computational Methods of Linear Algebra (Fizmatgiz, Moscow, 1963; Freeman, San Francisco, 1963).

    MATH  Google Scholar 

  13. M. F. Gardner and J. L. Barnes, Transients in Linear Systems (Wiley, Ney York, 1942; Fizmatgiz, Moscow, 1961).

    MATH  Google Scholar 

  14. A. C. Antoulas, Approximation of Large-Scale Dynamical Systems (SIAM, Philadelphia, 2005).

    Book  MATH  Google Scholar 

  15. H. Kwakernaak and R. Sivan, Linear Optimal Control Systems (Wiley, Ney York, 1972; Mir, Moscow, 1977).

    MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Trapeznikov Institute of Control Sciences, Russian Academy of Sciences, Profsoyuznaya ul. 65, Moscow, 117997, Russia

    I. B. Yadykin

  2. Skolkovo Institute of Science and Technology, ul. Novaya 100, Skolkovo, Moscow oblast, 143025, Russia

    I. B. Yadykin

Authors
  1. I. B. Yadykin
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to I. B. Yadykin.

Additional information

Original Russian Text © I.B. Yadykin, 2016, published in Doklady Akademii Nauk, 2016, Vol. 468, No. 3, pp. 264–267.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Yadykin, I.B. On spectral decompositions of solutions to discrete Lyapunov equations. Dokl. Math. 93, 344–347 (2016). https://doi.org/10.1134/S1064562416030133

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1134/S1064562416030133

Search

Navigation