Model Order Reduction for Differential-Algebraic Equations: A Survey

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


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.


Balanced truncation Differential-algebraic equations Interpolation-based approximation Matrix equations Matrix pencils Model order reduction 

Mathematics Subject Classification (2010)

15A22 15A24 34A09 65D05 65F30 93C05 



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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© Springer International Publishing AG 2017

Authors and Affiliations

  1. 1.Computational Methods in Systems and Control TheoryMax Planck Institute for Dynamics of Complex Technical SystemsMagdeburgGermany
  2. 2.Institut für MathematikUniversität AugsburgAugsburgGermany

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