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On Spectral Decomposition of States and Gramians of Bilinear Dynamical Systems

Author

Listed:
  • Alexey Iskakov

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

  • Igor Yadykin

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

Abstract

The article proves that the state of a bilinear control system can be split uniquely into generalized modes corresponding to the eigenvalues of the dynamics matrix. It is also shown that the Gramians of controllability and observability of a bilinear system can be divided into parts (sub-Gramians) that characterize the measure of these generalized modes and their interactions. Furthermore, the properties of sub-Gramians were investigated in relation to modal controllability and observability. We also propose an algorithm for computing the Gramians and sub-Gramians based on the element-wise computation of the solution matrix. Based on the proposed algorithm, a novel criterion for the existence of solutions to the generalized Lyapunov equation is proposed, which allows, in some cases, to expand the domain of guaranteed existence of a solution of bilinear equations. Examples are provided that illustrate the application and practical use of the considered spectral decompositions.

Suggested Citation

  • Alexey Iskakov & Igor Yadykin, 2021. "On Spectral Decomposition of States and Gramians of Bilinear Dynamical Systems," Mathematics, MDPI, vol. 9(24), pages 1-19, December.
    • Handle: RePEc:gam:jmathe:v:9:y:2021:i:24:p:3288-:d:705092
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    Cited by:

    1. Igor Yadykin, 2022. "Spectral Decompositions of Gramians of Continuous Stationary Systems Given by Equations of State in Canonical Forms," Mathematics, MDPI, vol. 10(13), pages 1-19, July.

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