IDEAS home Printed from https://ideas.repec.org/a/gam/jmathe/v12y2023i1p36-d1305615.html
   My bibliography  Save this article

Spectral Decomposition of Gramians of Continuous Linear Systems in the Form of Hadamard Products

Author

Listed:
  • Igor Yadykin

    (V.A. Trapeznikov Institute of Control Sciences, 65 Profsoyuznaya, 117997 Moscow, Russia)

Abstract

New possibilities of Gramian computation, by means of canonical transformations into diagonal, controllable, and observable canonical forms, are shown. Using such a technique, the Gramian matrices can be represented as products of the Hadamard matrices of multipliers and the matrices of the transformed right-hand sides of Lyapunov equations. It is shown that these multiplier matrices are invariant under various canonical transformations of linear continuous systems. The modal Lyapunov equations for continuous SISO LTI systems in diagonal form are obtained, and their new solutions based on Hadamard decomposition are proposed. New algorithms for the element-by-element computation of Gramian matrices for stable, continuous MIMO LTI systems are developed. New algorithms for the computation of controllability Gramians in the form of Xiao matrices are developed for continuous SISO LTI systems, given by the equations of state in the controllable and observable canonical forms. The application of transformations to the canonical forms of controllability and observability allowed us to simplify the formulas of the spectral decompositions of the Gramians. In this paper, new spectral expansions in the form of Hadamard products for solutions to the algebraic and differential Sylvester equations of MIMO LTI systems are obtained, including spectral expansions of the finite and infinite cross - Gramians of continuous MIMO LTI systems. Recommendations on the use of the obtained results are given.

Suggested Citation

  • Igor Yadykin, 2023. "Spectral Decomposition of Gramians of Continuous Linear Systems in the Form of Hadamard Products," Mathematics, MDPI, vol. 12(1), pages 1-20, December.
    • Handle: RePEc:gam:jmathe:v:12:y:2023:i:1:p:36-:d:1305615
    as

    Download full text from publisher

    File URL: https://www.mdpi.com/2227-7390/12/1/36/pdf
    Download Restriction: no
    File URL: https://www.mdpi.com/2227-7390/12/1/36/
    Download Restriction: no
    ---><---

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:gam:jmathe:v:12:y:2023:i:1:p:36-:d:1305615. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    We have no bibliographic references for this item. You can help adding them by using this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: MDPI Indexing Manager (email available below). General contact details of provider: https://www.mdpi.com .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.