# Model Reduction of Second-Order Systems.

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## Abstract

Model Reduction of Second-Order Systems. Younes Chahlaoui, Kyle A Gallivan, Antoine Vandendorpe and Paul Van Dooren 2006 MIMS EPrint: 2008.11 Manchester Institute for Mathematical Sciences School of Mathematics The University of Manchester Reports available from: http://www.manchester.ac.uk/mims/eprints And by contacting: The MIMS Secretary School of Mathematics The University of Manchester Manchester, M13 9PL, UK ISSN 1749-9097 6 Model Reduction of Second-Order Systems Y. Chahlaoui1, K. A. Gallivan1, A. Vandendorpe2, P. Van Dooren2 1 School of Computational Science, Florida State University, Tallahassee, U.S.A. [email protected], [email protected] 2 CESAME, Universite´ catholique de Louvain, Louvain-la-Neuve, Belgium, [email protected], [email protected] 6.1 Introduction In this chapter, the problem of constructing a reduced order system while preserving the second-order structure of the original system is discussed. Af- ter a brief introduction on second-order systems and a review of first order model reduction techniques, two classes of second-order structure preserving model reduction techniques – Krylov subspace-based and SVD-based – are presented. For the Krylov techniques, conditions on the projectors that guar- antee the reduced second-order system tangentially interpolates the original system at given frequencies are derived and an algorithm is described. For SVD-based techniques, a Second-Order Balanced Truncation method is de- rived from second-order Gramians. Second-order systems arise naturally in many areas of engineering (see, for example, [Pre97, WJJ87]) and have the following form:{ Mq¨(t) +Dq˙(t) + Sq(t) = F in u(t), y(t) = F out q(t). (6.1) We assume that u(t) ∈ Rm, y(t) ∈ Rp, q(t) ∈ RN , F in ∈ RN×m, F out ∈ R p×N , and M,D,S ∈ RN×N with M invertible. For mechanical systems the matrices M , D and S represent, respectively, the mass (or inertia), damping and stiffness matrices, u(t) corresponds to

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