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Numerical Linear Algebra Methods for Linear Differential-Algebraic Equations

  • Peter Benner
  • Philip Losse
  • Volker Mehrmann
  • Matthias Voigt
Part of the Differential-Algebraic Equations Forum book series (DAEF)

Abstract

A survey of methods from numerical linear algebra for linear constant coefficient differential-algebraic equations (DAEs) and descriptor control systems is presented. We discuss numerical methods to check the solvability properties of DAEs as well as index reduction and regularization techniques. For descriptor systems we discuss controllability and observability properties and how these can be checked numerically. These methods are based on staircase forms and derivative arrays, obtained by real orthogonal transformations that are discussed in detail. Then we use the reformulated problems in several control applications for differential-algebraic equations ranging from regular and singular linear-quadratic optimal and robust control to dissipativity checking. We discuss these applications and give a systematic overview of the theory and the numerical solution methods. In particular, we show that all these applications can be treated with a common approach that is based on the computation of eigenvalues and deflating subspaces of even matrix pencils. The unified approach allows us to generalize and improve several techniques that are currently in use in systems and control.

Keywords

Canonical form Controllability Descriptor system Differential-algebraic equation Dissipativity Even matrix pencil \(\mathcal{H}_{\infty }\) control Kronecker index \(\mathcal{L}_{\infty }\)-norm Linear-quadratic optimal control Observability 

Notes

Acknowledgements

The author “Philip Losse” was supported by the DFG Research Center Matheon in Berlin. The author “Volker Mehrmann” was supported by the European Research Council through ERC Advanced Grant ModSimConMP. Research of the author “Matthias Voigt” was supported in the framework of Matheon project C-SE1: Reduced order modeling for data assimilation supported by the Einstein Foundation Berlin.

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Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

  • Peter Benner
    • 1
  • Philip Losse
    • 2
  • Volker Mehrmann
    • 3
  • Matthias Voigt
    • 3
  1. 1.Max Planck Institute for Dynamics of Complex Technical SystemsMagdeburgGermany
  2. 2.BerlinGermany
  3. 3.Institut für MathematikTechnische Universität BerlinBerlinGermany

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