# American Institute of Mathematical Sciences

June 2019, 9(2): 173-186. doi: 10.3934/naco.2019013

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

 1 Department of Mathematics, Hamdard University Bangladesh, Gazaria, Munshiganj, Bangladesh 2 Department of Mathematics and Physics, North South University, Bashundhara, Dhaka 1229, Bangladesh

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

Received  July 2017 Revised  July 2018 Published  January 2019

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.

Citation: M. Sumon Hossain, M. Monir Uddin. Iterative methods for solving large sparse Lyapunov equations and application to model reduction of index 1 differential-algebraic-equations. Numerical Algebra, Control & Optimization, 2019, 9 (2) : 173-186. doi: 10.3934/naco.2019013
##### References:

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##### References:
Convergence histories of both Gramians by RKSM for mod-2
Comparison between full system and reduced-order system in frequency domain
Comparison between original system and reduced model in time domain
Largest Hankel singular values of original system and 86 dimensional reduced-order system
Number of differential & algebraic variables and largest eigenvalue of $(-A, E)$ for different models.
 Model differential algebraic eigs$(-A, E)$ inputs/outputs Mod-1 606 6 529 10727 4/4 Mod-2 1 142 8 593 10727 4/4 Mod-3 3 078 18 050 10669 4/4
 Model differential algebraic eigs$(-A, E)$ inputs/outputs Mod-1 606 6 529 10727 4/4 Mod-2 1 142 8 593 10727 4/4 Mod-3 3 078 18 050 10669 4/4
Comparisons between full systems and their reduced models
 Model Dimension Error full reduced absolute relative 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 $21\, 128$ $77$ $5.6 \times 10^{-1}$ $4.3 \times 10^{-2}$
 Model Dimension Error full reduced absolute relative 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 $21\, 128$ $77$ $5.6 \times 10^{-1}$ $4.3 \times 10^{-2}$
Balanced truncation tolerances and dimensions of reduced-order model.
 Model tolerance dimension of ROM $10^{-4}$ 118 $10^{-3}$ 104 Mod-2 $10^{-2}$ 86 $10^{-1}$ 70
 Model tolerance dimension of ROM $10^{-4}$ 118 $10^{-3}$ 104 Mod-2 $10^{-2}$ 86 $10^{-1}$ 70
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