Pitfalls When Solving Eigenproblems - With Applications in Control Engineering

Vasile Sima, Peter Benner

2015

Abstract

There is a continuous research effort worldwide to improve the reliability, efficiency, and accuracy of numerical computations in various domains. One of the most promising research avenues is to exploit the structural properties of the mathematical problems to be solved. This paper investigates some numerical algorithms for the solution of common and structured eigenproblems, which have many applications in automatic control (e.g., linear-quadratic optimization and H¥-optimization), but also in various areas of applied mathematics, physics, and computational chemistry. Of much interest is to find the eigenvalues and certain deflating subspaces, mainly those associated to the stable eigenvalues. Several simple examples are used to highlight the pitfalls which may appear in such numerical computations, using state-of-the-art solvers. Balancing the matrices and the use of condition numbers for eigenvalues are shown to be essential options in investigating the behavior of the solvers and problem sensitivity.

References

  1. Anderson, E., Bai, Z., Bischof, C., Blackford, S., Demmel, J., Dongarra, J., Du Croz, J., Greenbaum, A., Hammarling, S., McKenney, A., and Sorensen, D. (1999). LAPACK Users' Guide. SIAM, Philadelphia, third edition.
  2. Benner, P. (1999). Computational methods for linearquadratic optimization. Supplemento ai Rendiconti del Circolo Matematico di Palermo, II, (58):21-56.
  3. Benner, P., Byers, R., Losse, P., Mehrmann, V., and Xu, H. (2007). Numerical solution of real skewHamiltonian/Hamiltonian eigenproblems. Technical report, Technische Universität Chemnitz, Chemnitz.
  4. Benner, P., Byers, R., Mehrmann, V., and Xu, H. (2002). Numerical computation of deflating subspaces of skew Hamiltonian/Hamiltonian pencils. SIAM J. Matrix Anal. Appl., 24(1):165-190.
  5. Benner, P. and Kressner, D. (2006). Fortran 77 subroutines for computing the eigenvalues of Hamiltonian matrices II. ACM Trans. Math. Softw., 32(2):352-373.
  6. Benner, P., Kressner, D., Sima, V., and Varga, A. (2010). Die SLICOT-Toolboxen für Matlab. atAutomatisierungstechnik, 58(1):15-25.
  7. Benner, P., Mehrmann, V., Sima, V., Van Huffel, S., and Varga, A. (1999). SLICOT - A subroutine library in systems and control theory. In Applied and Computational Control, Signals, and Circuits, 1, ch. 10, pp. 499-539. Birkhäuser, Boston.
  8. Benner, P., Sima, V., and Voigt, M. (2012a). L¥-norm computation for continuous-time descriptor systems using structured matrix pencils. IEEE Trans. Automat. Contr., AC-57(1):233-238.
  9. Benner, P., Sima, V., and Voigt, M. (2012b). Robust and efficient algorithms for L¥-norm computations for descriptor systems. In 7th IFAC Symposium on Robust Control Design, pp. 189-194.
  10. Benner, P., Sima, V., and Voigt, M. (2013a). FORTRAN 77 subroutines for the solution of skewHamiltonian/Hamiltonian eigenproblems. Part I: Algorithms and applications. www.slicot.org.
  11. Benner, P., Sima, V., and Voigt, M. (2013b). FORTRAN 77 subroutines for the solution of skewHamiltonian/Hamiltonian eigenproblems. Part II: Implementation and numerical results. www.slicot. org.
  12. Bojanczyk, A. W., Golub, G., and Van Dooren, P. (1992). The periodic Schur decomposition: algorithms and applications. In SPIE Conference Advanced Signal Processing Algorithms, Architectures, and Implementations III, 1770, pp. 31-42.
  13. Golub, G. H. and Van Loan, C. F. (1996). Matrix Computations. The Johns Hopkins University Press, Baltimore, Maryland, third edition.
  14. Granat, R., Ka°gström, B., and Kressner, D. (2007). Computing periodic deflating subspaces associated with a specified set of eigenvalues. BIT Numerical Mathematics, 47(4):763-791.
  15. Laub, A. J. (1979). A Schur method for solving algebraic Riccati equations. IEEE Trans. Automat. Contr., AC24(6):913-921.
  16. MathWorks (2014). MATLAB R : The Language of Technical Computing. R2014b. The MathWorks, Inc.
  17. Mehrmann, V. (1991). The Autonomous Linear Quadratic Control Problem. Theory and Numerical Solution. Springer-Verlag, Berlin.
  18. Sima, V. (1996). Algorithms for Linear-Quadratic Optimization. Marcel Dekker, Inc., New York.
  19. Sima, V. (2010). Structure-preserving computation of stable deflating subspaces. In 10th IFAC Workshop “Adaptation and Learning in Control and Signal Processing”.
  20. Sima, V. (2011a). Computational experience with structurepreserving Hamiltonian solvers in optimal control. In 8th International Conference on Informatics in Control, Automation and Robotics, vol. 1, pp. 91-96. SCITEPRESS.
  21. Sima, V. (2011b). Computational experience with structurepreserving Hamiltonian solvers in complex spaces. In 5th International Scientific Conference on Physics and Control.
  22. Sima, V., Benner, P., and Kressner, D. (2012). New SLICOT routines based on structured eigensolvers. In 2012 IEEE International Conference on Control Applications, pp. 640-645. Omnipress.
  23. Van Huffel, S., Sima, V., Varga, A., Hammarling, S., and Delebecque, F. (2004). High-performance numerical software for control. IEEE Control Syst. Mag., 24(1):60-76.
Download


Paper Citation


in Harvard Style

Sima V. and Benner P. (2015). Pitfalls When Solving Eigenproblems - With Applications in Control Engineering . In Proceedings of the 12th International Conference on Informatics in Control, Automation and Robotics - Volume 1: ICINCO, ISBN 978-989-758-122-9, pages 171-178. DOI: 10.5220/0005533301710178


in Bibtex Style

@conference{icinco15,
author={Vasile Sima and Peter Benner},
title={Pitfalls When Solving Eigenproblems - With Applications in Control Engineering},
booktitle={Proceedings of the 12th International Conference on Informatics in Control, Automation and Robotics - Volume 1: ICINCO,},
year={2015},
pages={171-178},
publisher={SciTePress},
organization={INSTICC},
doi={10.5220/0005533301710178},
isbn={978-989-758-122-9},
}


in EndNote Style

TY - CONF
JO - Proceedings of the 12th International Conference on Informatics in Control, Automation and Robotics - Volume 1: ICINCO,
TI - Pitfalls When Solving Eigenproblems - With Applications in Control Engineering
SN - 978-989-758-122-9
AU - Sima V.
AU - Benner P.
PY - 2015
SP - 171
EP - 178
DO - 10.5220/0005533301710178