Advances in Computational Mathematics

, Volume 44, Issue 6, pp 1821–1844 | Cite as

Balanced truncation model order reduction in limited time intervals for large systems

  • Patrick KürschnerEmail author
Open Access


In this article we investigate model order reduction of large-scale systems using time-limited balanced truncation, which restricts the well known balanced truncation framework to prescribed finite time intervals. The main emphasis is on the efficient numerical realization of this model reduction approach in case of large system dimensions. We discuss numerical methods to deal with the resulting matrix exponential functions and Lyapunov equations which are solved for low-rank approximations. Our main tool for this purpose are rational Krylov subspace methods. We also discuss the eigenvalue decay and numerical rank of the solutions of the Lyapunov equations. These results, and also numerical experiments, will show that depending on the final time horizon, the numerical rank of the Lyapunov solutions in time-limited balanced truncation can be smaller compared to standard balanced truncation. In numerical experiments we test the approaches for computing low-rank factors of the involved Lyapunov solutions and illustrate that time-limited balanced truncation can generate reduced order models having a higher accuracy in the considered time region.


Lyapunov equation Rational Krylov subspaces Model order reduction Balanced truncation Matrix exponential 

Mathematics Subject Classification 2010

15A16 15A18 15A24 65F60 93A15 93C 



Open access funding provided by Max Planck Society. I thank the referees for their helpful comments. Moreover, I am grateful for the constructive discussions with Maria Cruz Varona, Serkan Gugercin, and Stefan Guettel.


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Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (, which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Authors and Affiliations

  1. 1.Computational Methods in Systems and Control TheoryMax Planck Institute for Dynamics of Complex Technical SystemsMagdeburgGermany

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