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Model Order Reduction for Differential-Algebraic Equations: A Survey

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Book cover Surveys in Differential-Algebraic Equations IV

Part of the book series: Differential-Algebraic Equations Forum ((DAEF))

Abstract

In this paper, we discuss the model order reduction problem for descriptor systems, that is, systems with dynamics described by differential-algebraic equations. We focus on linear descriptor systems as a broad variety of methods for these exist, while model order reduction for nonlinear descriptor systems has not received sufficient attention up to now. Model order reduction for linear state-space systems has been a topic of research for about 50 years at the time of writing, and by now can be considered as a mature field. The extension to linear descriptor systems usually requires extra treatment of the constraints imposed by the algebraic part of the system. For almost all methods, this causes some technical difficulties, and these have only been thoroughly addressed in the last decade. We will focus on these developments in particular for the popular methods related to balanced truncation and rational interpolation. We will review efforts in extending these approaches to descriptor systems, and also add the extension of the so-called stochastic balanced truncation method to descriptor systems which so far cannot be found in the literature.

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Notes

  1. 1.

    Usually, the term moments is used to denote the coefficients of the Taylor series at s 0 = 0.

  2. 2.

    It should be noted that by abuse of notation, these factors are neither necessarily upper triangular nor square, but we assume them to be of full rank. In particular, for non-minimal systems, these factors will in general be rectangular as then the Gramians will be rank deficient.

  3. 3.

    For partial convergence results, see [52].

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Acknowledgements

The first author acknowledges support by the collaborative project nanoCOPS: “Nanoelectronic COupled Problems Solutions” funded by the European Union in the FP7-ICT-2013-11 Program under Grant Agreement Number 619166.

The second author was supported by the Research Network KoSMos: Model reduction based simulation of coupled PDAE systems funded by the German Federal Ministry of Education and Science (BMBF), grant 05M13WAA, and by the project Model reduction for elastic multibody systems with moving interactions funded by the German Research Foundation (DFG), grant STY 58/1–2.

The authors gratefully acknowledge the careful proofreading by a reviewer and the editors which greatly enhanced the presentation of this survey.

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  1. Computational Methods in Systems and Control Theory, Max Planck Institute for Dynamics of Complex Technical Systems, Sandtorstraße 1, 39106, Magdeburg, Germany

    Peter Benner

  2. Institut für Mathematik, Universität Augsburg, Universitätsstraße 14, 86159, Augsburg, Germany

    Tatjana Stykel

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  1. Peter Benner
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  1. Institut für Mathematik, Technische Universität Ilmenau, Ilmenau, Germany

    Achim Ilchmann

  2. Fachbereich Mathematik, Universität Hamburg, Hamburg, Germany

    Timo Reis

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Benner, P., Stykel, T. (2017). Model Order Reduction for Differential-Algebraic Equations: A Survey. In: Ilchmann, A., Reis, T. (eds) Surveys in Differential-Algebraic Equations IV. Differential-Algebraic Equations Forum. Springer, Cham. https://doi.org/10.1007/978-3-319-46618-7_3

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