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

M Sumon Hossain,M Monir Uddin

doi:10.3934/naco.2019013

Abstract

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.

Highlights

Lyapunov or Lyapunov-like equations play important roles in various disciplines of science and engineering such as system and control theory, optimization, linear algebra, differential equations, boundary value problem, signal processing, power system, structural dynamics and so on
We briefly review some basic concepts and results including balanced truncation (BT) and Rational Krylov Subspace Method (RKSM) for the solutions of Lyapunov equations that will be used throughout this paper
We have introduced a method to compute the low-rank Gramian factors by solving the Lyapunov equations that arise from the large-scale sparse index-1 descriptor systems

Summary

Introduction

Lyapunov or Lyapunov-like equations play important roles in various disciplines of science and engineering such as system and control theory, optimization, linear algebra, differential equations, boundary value problem, signal processing, power system, structural dynamics and so on (see, e.g., [29, 15, 27, 20]). In control theory besides the stability analysis and stabilization of systems, Lyapunov equations are used in computing balancing transformation, Gramian based model reduction, H2 optimal control and Riccati based optimal control. The LR-ADI (low-rank-alternating direction implicit) method and the Krylov subspace method are frequently used for the solution of Lyapunov equations of large-scale sparse systems. Freitas et al in [12] discusses a balancing based method for the model reduction of index-1 descriptor system (4) They apply the LR-ADI to solve the Lyapunov equations. The computed low-rank Gramian factors are applied to the balancing based model reduction. The reduced order model can be obtained by truncating the states which are associated with the set of the least HSVs. The system in the balanced state space form is obtained by applying transformation matrix. Output: Matrices E, A, B, Cand Dof stable reduced-order model

Compute the reduced-order system

Solve the small Lyapunov equation

11 Truncate small eigenvalues Π2 and construct the full Gramian factor

Form reduced-order matrices as follows:

Conclusions