Abstract
This paper studies the structure preserving (second-order to second-order) model order reduction of second-order systems applying the projection onto the dominant eigenspace of the Gramians of the systems. The projectors which create the reduced order model are generated cheaply from the low-rank Gramian factors. The low-rank Gramian factors are computed efficiently by solving the corresponding Lyapunov equations of the system using the rational Krylov subspace method. The efficiency of the theoretical results are then illustrated by numerical experiments.
Highlights
In this paper we consider model order reduction of second-order linear time-invariant (LTI) continuous-time system of the formMz (t) + Dż(t) + Kz(t) = Hu(t), y(t) = Lz(t), (1)where M, D, K ∈ Rn×n are large and sparse matrices, H ∈ Rn×p is the input matrix and L ∈ Rm×n is the output matrix
Mathematical models nowadays are often generated by the finite element method (FEM) and most of the systems are composed of large distinct devices which lead the systems to get larger and increases the complexity of the systems
There are some remarkable techniques for structure preserving model order reduction (SPMOR) of second-order systems; such as Balanced truncation (BT) [5, 6], second order Arnoldi method [7], moment matching approximation based on Krylov subspace [8]
Summary
In this paper we consider model order reduction of second-order linear time-invariant (LTI) continuous-time system of the form. It is necessary to approximate the system (1) by a lower dimensional system without losing the essential dynamics of the original system, which is efficient for practical implementation In control literature, this procedure of approximation is known as model order reduction (MOR) [3, 4]. There are some remarkable techniques for SPMOR of second-order systems; such as Balanced truncation (BT) [5, 6], second order Arnoldi method [7], moment matching approximation based on Krylov subspace [8]. The Gramian factors are computed by solving the continuous-time algebraic Lyapunov equations obtained from the second-order system To solve such Lyapunov equations efficiently, we modify the rational Krylov subspace method (RKSM) which was originated in [10] for solving the Lyapunov equations of standard systems. For a vectors or matrices ‖. ‖2, ‖. ‖∞and ‖. ‖F denote 2-norm, infinity-norm and Frobenius norm respectively
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