# $L_\infty$-Norm Computation for Descriptor Systems

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- Universitätsbibliothek Chemnitz
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## Abstract

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.

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