In many applications from industry and technology computer simulations are
performed using models which can be formulated by systems of differential
equations. Often the equations underlie additional algebraic constraints.
In this context we speak of descriptor systems. Very important characteristic
values of such systems are the $L_\infty$-norms of the corresponding transfer
functions. The main goal of this thesis is to extend a numerical method for
the computation of the $L_\infty$-norm for standard state space systems to
descriptor systems. For this purpose we develop a numerical method to check
whether the transfer function of a given descriptor system is proper or
improper and additionally use this method to reduce the order of the system
to decrease the costs of the $L_\infty$-norm computation. When computing the
$L_\infty$-norm it is necessary to compute the eigenvalues of certain
skew-Hamiltonian/Hamiltonian matrix pencils composed by the system
matrices. We show how we extend these matrix pencils to
skew-Hamiltonian/Hamiltonian matrix pencils of larger dimension to get more
reliable and accurate results. We also consider discrete-time systems, apply
the extension strategy to the arising symplectic matrix pencils and transform
these to more convenient structures in order to apply structure-exploiting
eigenvalue solvers to them. We also investige a new structure-preserving
method for the computation of the eigenvalues of skew-Hamiltonian/Hamiltonian
matrix pencils and use this to increase the accuracy of the computed
eigenvalues even more. In particular we ensure the reliability of the
$L_\infty$-norm algorithm by this new eigenvalue solver. Finally we describe the
implementation of the algorithms in Fortran and test them using two
real-world examples.
| Identifer | oai:union.ndltd.org:DRESDEN/oai:qucosa.de:bsz:ch1-201001050 |
| Date | 15 July 2010 |
| Creators | Voigt, Matthias |
| Contributors | TU Chemnitz, Fakultät für Mathematik, Prof. Dr. rer. nat. habil. Peter Benner, Dr. Ing. Math. Vasile Sima, Prof. Dr. rer. nat. habil. Peter Benner, Dr. Ing. Math. Vasile Sima |
| Publisher | Universitätsbibliothek Chemnitz |
| Source Sets | Hochschulschriftenserver (HSSS) der SLUB Dresden |
| Language | English |
| Detected Language | English |
| Type | doc-type:masterThesis |
| Format | application/pdf, text/plain, application/zip |
| Rights | Dokument ist für Print on Demand freigegeben |
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