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DFG project

Numerical algorithms for generalized eigenvalue problems of even structure with application in robust control of descriptor systems




Project director:
  • Prof. Dr. Peter Benner
    Chemnitz University of Technology,
    Department of Mathematics, 09107 Chemnitz, Germany
    Tel: +49-(0)371-531-38367
    E-mail: benner@mathematik.tu-chemnitz.de
  • Prof. Dr. Volker Mehrmann
    Technische Universität Berlin,
    Institute for Mathematics, 10623 Berlin, Germany
    Tel: +49 (0)30 314-25736
    E-mail: mehrmann(at)math.tu-berlin.de
Researcher:
  • Philip Losse
    Technische Universität Berlin,
    Institute for Mathematics, 10623 Berlin, Germany
    Tel: +49 (0)30-314-79236
    E-mail: losse(at)math.tu-berlin.de
  • Lisa Poppe (until 02/15/2009)
    Technische Universität Berlin,
    Institute for Mathematics, 10623 Berlin, Germany
Graduate research assistant: Duration: 06/01/2006 - 06/30/2010

Project description:
The aim of the project is the development of efficient structure-preserving numerical methods for the solution of generalized eigenvalue problems of even structure, as well as their use in the construction of efficient numerical procedures for the robust control of continuous-time and discrete-time general descriptor systems (under inclusion of time retardations).

The first main point of the project is the investigation of the theoretical basis of the H-theory for singular continuous-time and discrete-time descriptor systems under inclusion of retarded arguments. Especially it has been shown that the solution of this task can be reached by the calculation of generalized Lagrange-subspaces for matrix pencils of even structure. The second main point is the development, investigation and implementation of numerical algorithms for generalized eigenvalue problems of even structure and their error analysis. Especially the existence, uniqueness and stability of generalized Lagrange-subspaces for matrix pencils of even structure, as well as the accompanying perturbation theory and error analysis, will be examined.

Together both project parts give the base of design procedures for optimal H-controllers, both for general linear continuous-time and for discrete-time (retarded) descriptor systems.




Related publications:

@TECHREPORT{BenLMV13,
author = {Peter Benner and Philip Losse and Volker Mehrmann and Matthias Voigt},
title = {Numerical Solution of Eigenvalue Problems for Alternating Matrix Polynomials and Their Application in Control Problems for Descriptor Systems},
institution = {Max Planck Institute Magdeburg},
type = {Preprint},
number = {MPIMD/13-24},
month = {December},
year = {2013},
note = {Available from \url{http://www.mpi-magdeburg.mpg.de/preprints/2013/24/}},
}
Numerical Solution of Eigenvalue Problems for Alternating Matrix Polynomials and Their Application in Control Problems for Descriptor Systems;
Benner, Peter; Losse, Philip; Mehrmann, Volker; Voigt, Matthias;
2013.
Preprint MPIMD/13-24.
@TECHREPORT{BenSV13a,
author = {Peter Benner and Vasile Sima and Matthias Voigt},
title = {{FORTRAN} 77 Subroutines for the Solution of Skew-{H}amiltonian/{H}amiltonian Eigenproblems -- {P}art {I}: Algorithms and Applications},
institution = {Max Planck Institute Magdeburg},
type = {Preprint},
number = {MPIMD/13-11},
month = {July},
year = {2013},
note = {Available from \url{http://www.mpi-magdeburg.mpg.de/preprints/2013/11/}},
}
FORTRAN 77 Subroutines for the Solution of Skew-Hamiltonian/Hamiltonian Eigenproblems - Part I: Algorithms and Applications;
Benner, Peter; Sima, Vasile; Voigt, Matthias;
2013.
Submitted, also available as preprint MPIMD/13-11.
@TECHREPORT{BenSV13b,
author = {Peter Benner and Vasile Sima and Matthias Voigt},
title = {{FORTRAN} 77 Subroutines for the Solution of Skew-{H}amiltonian/{H}amiltonian Eigenproblems -- {P}art {II}: Implementation and Numerical Results},
institution = {Max Planck Institute Magdeburg},
type = {Preprint},
number = {MPIMD/13-12},
month = {July},
year = {2013},
note = {Available from \url{http://www.mpi-magdeburg.mpg.de/preprints/2013/12/}},
}
FORTRAN 77 Subroutines for the Solution of Skew-Hamiltonian/Hamiltonian Eigenproblems - Part II: Implementation and Numerical Results;
Benner, Peter; Sima, Vasile; Voigt, Matthias;
2013.
Submitted, also available as preprint MPIMD/13-12.
@ARTICLE{BenSV12a,
author = {Peter Benner and Vasile Sima and Matthias Voigt},
title = {$\mathcal{L}_\infty$-Norm Computation for Continuous-Time Descriptor Systems Using Structured Matrix Pencils},
journal = {IEEE Transactions on Automatic Control},
volume = {57},
number = {1},
pages = {233--238},
month = {Jan.},
year = {2012},
publisher = {IEEE},
doi = {10.1109/TAC.2011.2161833},
}
L-Norm Computation for Continuous-Time Descriptor Systems Using Structured Matrix Pencils;
Benner, Peter; Sima, Vasile; Voigt, Matthias;
IEEE Transactions on Automatic Control  :  Vol. 57, No. 1, pp. 233 - 238;
IEEE; 2012. ISBN/ISSN: 0018-9286
DOI: 10.1109/TAC.2011.2161833.
@INPROCEEDINGS{BenSV12b,
author = {Peter Benner and Vasile Sima and Matthias Voigt},
title = {Robust and Efficient Algorithms for $\mathcal{L}_\infty$-Norm Computation for Descriptor Systems},
booktitle = {Proceedings of the 7th IFAC Symposium on Robust Control Design},
pages = {195--200},
month = {June},
year = {2012},
publisher = {IFAC},
address = {Aalborg, Denmark},
doi = {10.3182/20120620-3-DK-2025.00114},
}
Robust and Efficient Algorithms for L-Norm Computation for Descriptor Systems;
Benner, Peter; Sima, Vasile; Voigt, Matthias;
Proceedings of the 7th IFAC Symposium on Robust Control Design  :  pp. 195 - 200;
IFAC; 2012.
DOI: 10.3182/20120620-3-DK-2025.00114.
@INPROCEEDINGS{BenV11b,
author = {Peter Benner and Matthias Voigt},
title = {$\mathcal{L}_\infty$-Norm Computation for Discrete-Time Descriptor Systems},
booktitle= {Tagungsband des GMA-Fachausschusses 1.30 \glqq Modellbildung, Identifikation und Simulation in der Automatisierungstechnik\grqq},
pages = {228--240},
editor = {Oliver Sawodny and J\"urgen Adamy},
month = {December},
year = {2011},
publisher = {Technische Universit\"at Darmstadt, Institut f\"ur Automatisierungstechnik und Mechatronik},
isbn = {978-3-9815012-1-6},
}
L-Norm Computation for Discrete-Time Descriptor Systems;
Benner, Peter; Voigt, Matthias;
in Sawodny, Oliver; Adamy, Jürgen: Tagungsband des GMA-Fachausschusses 1.30 „Modellbildung, Identifikation und Simulation in der Automatisierungstechnik“  :  pp. 228 - 240;
Technische Universität Darmstadt, Institut für Automatisierungstechnik und Mechatronik; 2011. ISBN/ISSN: 978-3-9815012-1-6
@MASTERSTHESIS{Voi10,
author = {Matthias Voigt},
title = {$\mathcal{L}_\infty$-Norm Computation for Descriptor Systems},
school = {Chemnitz University of Technology, Faculty of Mathematics, Germany},
month = {July},
year = {2010},
type = {Diploma Thesis},
url = {http://nbn-resolving.de/urn:nbn:de:bsz:ch1-201001050},
}
L-Norm Computation for Descriptor Systems;
Voigt, Matthias;
2010.
Diploma Thesis, Faculty of Mathematics, TU Chemnitz.
@inproceedings{BLMPR08,
title={Gamma-Iteration for Descriptor Systems Using Structured Matrix Pencils},
author={P. Benner and P. Losse and V. Mehrmann and L. Poppe and T. Reis},
year=2008,
booktitle={Proceedings of the International Symposium on Mathematical Theory of Networks and Systems},
adress={Blacksburg VA, USA} }
Gamma-Iteration for Descriptor Systems Using Structured Matrix Pencils
P. Benner, P. Losse, V. Mehrmann, L. Poppe, T. Reis;
Proceedings MTNS08  :  
2008.
@article{LosMPR08,
author = {P. Losse and V. Mehrmann and L. Poppe and T. Reis},
title = {The modified optimal $\mathcal{H}_\infty$ control problem for descriptor systems},
journal = {SIAM Journal on Control and Optimization},
volume = {47 (6)},
pages={2795--2811},
year = {2008}
The modified optimal H control problem for descriptor systems;
P. Losse, V. Mehrmann, L. Poppe; T. Reis;
SIAM Journal on Control and Optimization  :  Vol.47, No.6, pp.2795-2811;
SIAM; 2008. ISBN/ISSN: 0363-0129
@techreport{LosMPR07,
author = {P. Losse and V. Mehrmann and L. Poppe and T. Reis}, title = {Robust Control of Descriptor Systems},
institution= {Institut f{\"u}r Mathematik, TU Berlin},
type = {Preprint}, number = {2007/47}, address = {D-10623 Berlin, FRG},
year = {2007},
note ={{ t url: http://www.math.tu-berlin.de/preprints/}} }
Robust Control of Descriptor Systems;
P. Losse, V. Mehrmann, L. Poppe, T. Reis;
Preprint 2007/47 Fak. für Mathematik TU Berlin; 2007.
Related talks:

P. Losse:
Optimale H-Regelung von Deskriptorsystemen unter Verwendung von Matrixb├╝scheln (in German), Elgersburg Workshop, Elgersburg, Germany, 02/11-02/14/2008.

P. Losse:
Gamma-Iteration for Descriptor Systems Using Structured Matrix Pencils, MTNS 2008, Blacksburg VA, USA, 07/28-08/01/2008.

M. Voigt:
L-Norm Computation for Descriptor Systems, Meeting of the GAMM Activity Group ,,Dynamics and Control Theory'' , Berlin, Germany, 04/16-04/17/2010.

M. Voigt:
L-Norm Computation for Discrete-Time Descriptor Systems, Treffen des GMA Activity Group 1.30: ,,Modeling, Identification and Simulation in Automation Engineering'', Anif/Salzburg, Austria, 09/21-09/23/2011.




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Matthias Voigt, voigtm@mpi-magdeburg.mpg.de