Balancing Skew-Hamiltonian/Hamiltonian Pencils - With Applications in Control Engineering

Vasile Sima

Abstract

Badly-scaled matrix pencils could reduce the reliability and accuracy of computed results for many numerical problems, including computation of eigenvalues and deflating subspaces, which are needed in many key procedures for optimal and H∞ control, model reduction, spectral factorization, and so on. Standard balancing techniques can improve the results in many cases, but there are situations when the solution of the scaled problem is much worse than that for the unscaled problem. This paper presents a new structure-preserving balancing technique for skew-Hamiltonian/Hamiltonian matrix pencils, and illustrates its good performance in solving eigenvalue problems and algebraic Riccati equations for large sets of examples from well-known benchmark collections with difficult examples.

References

  1. Abels, J. and Benner, P. (1999). CAREX-A collection of benchmark examples for continuous-time algebraic Riccati equations (Version 2.0). SLICOT Working Note 1999-14.
  2. 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: Third Edition. SIAM, Philadelphia.
  3. Benner, P. (2001). Symplectic balancing of Hamiltonian matrices. SIAM J. Sci. Comput., 22(5):1885-1904.
  4. 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.
  5. 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.
  6. Benner, P., Kressner, D., and Mehrmann, V. (2005). SkewHamiltonian and Hamiltonian eigenvalue problems: Theory, algorithms and applications. In Proceedings of the Conference on Applied Mathematics and Scientific Computing, 3-39. Springer-Verlag, Dordrecht.
  7. Benner, P., Sima, V., and Voigt, M. (2012a). L8-norm computation for continuous-time descriptor systems using structured matrix pencils. IEEE Trans. Automat. Contr., AC-57(1):233-238.
  8. Benner, P., Sima, V., and Voigt, M. (2012b). Robust and efficient algorithms for L8-norm computations for descriptor systems. In 7th IFAC Symposium on Robust Control Design (ROCOND'12), 189-194. IFAC.
  9. Benner, P., Sima, V., and Voigt, M. (2013a). FORTRAN 77 subroutines for the solution of skewHamiltonian/Hamiltonian eigenproblems. Part I: Algorithms and applications. SLICOT Working Note 2013-1.
  10. Benner, P., Sima, V., and Voigt, M. (2013b). FORTRAN 77 subroutines for the solution of skewHamiltonian/Hamiltonian eigenproblems. Part II: Implementation and numerical results. SLICOT Working Note 2013-2.
  11. Bruinsma, N. A. and Steinbuch, M. (1990). A fast algorithm to compute the H8-norm of a transfer function. Systems Control Lett., 14(4):287-293.
  12. Jiang, P. and Voigt, M. (2013). MB04BV - A FORTRAN 77 subroutine to compute the eigenvectors associated to the purely imaginary eigenvalues of skewHamiltonian/Hamiltonian matrix pencils. SLICOT Working Note 2013-3.
  13. Kressner, D. (2005). Numerical Methods for General and Structured Eigenvalue Problems. Springer-Verlag, Berlin.
  14. Laub, A. J. (1979). A Schur method for solving algebraic Riccati equations. IEEE Trans. Automat. Contr., AC24(6):913-921.
  15. Leibfritz, F. and Lipinski, W. (2003). Description of the benchmark examples in COMPleib. Technical report, Department of Mathematics, University of Trier, D54286 Trier, Germany.
  16. MATLAB (2016). MATLAB R Primer. R2016a. The MathWorks, Inc., 3 Apple Hill Drive, Natick, MA.
  17. Mehrmann, V. (1991). The Autonomous Linear Quadratic Control Problem. Theory and Numerical Solution. Springer-Verlag, Berlin.
  18. Paige, C. and Van Loan, C. F. (1981). A Schur decomposition for Hamiltonian matrices. Lin. Alg. Appl., 41:11- 32.
  19. Sima, V. (1996). Algorithms for Linear-Quadratic Optimization. Marcel Dekker, Inc., New York.
  20. Sima, V. and Benner, P. (2015a). Pitfalls when solving eigenproblems with applications in control engineering. In Proceedings of the 12th International Conference on Informatics in Control, Automation and Robotics (ICINCO-2015), 21-23 July, 2015, Colmar, France, volume 1, 171-178. SciTePress.
  21. Sima, V. and Benner, P. (2015b). Solving SLICOT benchmarks for continuous-time algebraic Riccati equations by Hamiltonian solvers. In Proceedings of the 2015 19th International Conference on System Theory, Control and Computing (ICSTCC 2015), October 14-16, 2015, Cheile Gradistei, Romania, 1-6. IEEE.
  22. Van Loan, C. F. (1984). A symplectic method for approximating all the eigenvalues of a Hamiltonian matrix. Lin. Alg. Appl., 61:233-251.
  23. Ward, R. C. (1981). Balancing the generalized eigenvalue problem. SIAM J. Sci. Stat. Comput., 2:141-152.
Download


Paper Citation


in Harvard Style

Sima V. (2016). Balancing Skew-Hamiltonian/Hamiltonian Pencils - With Applications in Control Engineering . In Proceedings of the 13th International Conference on Informatics in Control, Automation and Robotics - Volume 1: ICINCO, ISBN 978-989-758-198-4, pages 177-184. DOI: 10.5220/0005981201770184


in Bibtex Style

@conference{icinco16,
author={Vasile Sima},
title={Balancing Skew-Hamiltonian/Hamiltonian Pencils - With Applications in Control Engineering},
booktitle={Proceedings of the 13th International Conference on Informatics in Control, Automation and Robotics - Volume 1: ICINCO,},
year={2016},
pages={177-184},
publisher={SciTePress},
organization={INSTICC},
doi={10.5220/0005981201770184},
isbn={978-989-758-198-4},
}


in EndNote Style

TY - CONF
JO - Proceedings of the 13th International Conference on Informatics in Control, Automation and Robotics - Volume 1: ICINCO,
TI - Balancing Skew-Hamiltonian/Hamiltonian Pencils - With Applications in Control Engineering
SN - 978-989-758-198-4
AU - Sima V.
PY - 2016
SP - 177
EP - 184
DO - 10.5220/0005981201770184