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Iterative methods for solving large sparse Lyapunov equations and application to model reduction of index 1 differential-algebraic-equations

  • Corresponding author: M. Monir Uddin (E-mail: monir.uddin@northsouth.edu)

    Corresponding author: M. Monir Uddin (E-mail: monir.uddin@northsouth.edu)
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  • To implement the balancing based model reduction of large-scale dynamical systems we need to compute the low-rank (controllability and observability) Gramian factors by solving Lyapunov equations. In recent time, Rational Krylov Subspace Method (RKSM) is considered as one of the efficient methods for solving the Lyapunov equations of large-scale sparse dynamical systems. The method is well established for solving the Lyapunov equations of the standard or generalized state space systems. In this paper, we develop algorithms for solving the Lyapunov equations for large-sparse structured descriptor system of index-1. The resulting algorithm is applied for the balancing based model reduction of large sparse power system model. Numerical results are presented to show the efficiency and capability of the proposed algorithm.

    Mathematics Subject Classification: Primary: 58F15, 58F17; Secondary: 53C35.


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  • Figure 1.  Convergence histories of both Gramians by RKSM for mod-2

    Figure 2.  Comparison between full system and reduced-order system in frequency domain

    Figure 3.  Comparison between original system and reduced model in time domain

    Figure 4.  Largest Hankel singular values of original system and 86 dimensional reduced-order system

    Table 1.  Number of differential & algebraic variables and largest eigenvalue of $ (-A, E) $ for different models.

    Modeldifferentialalgebraiceigs$ (-A, E) $inputs/outputs
    Mod-16066 529107274/4
    Mod-21 1428 593107274/4
    Mod-33 07818 050106694/4
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    Table 2.  Comparisons between full systems and their reduced models

    Mod-1$ 7\, 135 $ $ 87 $ $ 3.1\times 10^{-3} $ $ 1.5 \times 10^{-4} $
    Mod-2$ 9\, 735 $ $ 86 $ $ 5.3 \times 10^{-2} $$ 4.7 \times 10^{-4} $
    Mod-3$ 21\, 128 $ $ 77 $$ 5.6 \times 10^{-1} $$ 4.3 \times 10^{-2} $
     | Show Table
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    Table 3.  Balanced truncation tolerances and dimensions of reduced-order model.

    Modeltolerancedimension of ROM
    $ 10^{-4} $118
    $ 10^{-3} $104
    Mod-2 $ 10^{-2} $86
    $ 10^{-1} $70
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