Finite Representations of Block Hankel Operators and Balanced Realizations

  • K. D. Gregson
  • N. J. Young
Chapter
Part of the Operator Theory: Advances and Applications book series (OT, volume 35)

Abstract

The block Hankel operator Гg corresponding to a rational matrix function g, analytic in D and of McMillan degree d, has rank d. Its non-trivial part, acting from (Ker Гg) to Range Гg, can therefore in principle be represented by a d × d matrix with respect to a pair of orthonormal bases. We show how to obtain such a representation using polynomial methods: that is, we work with the coefficients of the numerator and denominator polynomials and do not require the solution of any polynomial equations. We use this representation to derive an algorithm for the construction of balanced realizations of rational transfer functions.

Keywords

Matrix Polynomial Hankel Operator Minimal Realization Polynomial Matrice Balance Realization 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. 1a.
    V. M. Adamyan, D. Z. Arov and M. G. Krein, Infinite Hankel block matrices and related extension problems, Izv. Akad. Nauk Armyan. SSR Ser. Mat. 6 (1971), 87–112;Google Scholar
  2. 1b.
    V. M. Adamyan, D. Z. Arov and M. G. Krein, Infinite Hankel block matrices and related extension problems.Amer. Math. Soc. Transi. (2) 111 (1978), 133–156.Google Scholar
  3. 2.
    A. C. Allison and N. J. Young, Numerical algorithms for the Nevanlinna-Pick problem, Numer. Math. 42 (1983) 125–145.CrossRefGoogle Scholar
  4. 3.
    J. C Doyle, Advances in multivariable control, ONR/Honeywell workshop, Minneapolis, 1984.Google Scholar
  5. 4.
    B. A. Francis, “A Course in H Control Theory”, Lecture Notes in Control and Information Sciences 88 Springer Verlag, Berlin, 1986.Google Scholar
  6. 5.
    B. A. Francis and J. C. Doyle, Linear control theory with an H optimality criterion, SIAM J. Control and Optimisation, to appear.Google Scholar
  7. 6.
    P. A. Fuhrmann, Realization theory in Hubert space for a class of transfer functions, J. Functional Analysis 18 (338–349) 1975.CrossRefGoogle Scholar
  8. 7.
    K. Glover, All optimal Hankel-norm approximations of linear multi-variable systems and their L-error bounds, Int. J. Control 39 (1984), 1115–1193.CrossRefGoogle Scholar
  9. 8.
    J. W. Helton, Discrete time systems, operator models and scattering theory, J. Functional Analysis 16 (1974) 15–38.CrossRefGoogle Scholar
  10. 9.
    K. Hoffman, “Banach Spaces of Analytic Functions”, Prentice Hall, New Jersey, 1962.Google Scholar
  11. 10.
    T. Kailath, “Linear Systems”, Prentice Hall, N. J., 1980.Google Scholar
  12. 11.
    H. Kwakernaak, A polynomial approach to H-optimization of control systems, in “Modelling, Robustness and Sensitivity Reduction in Control Systems”, ed. R. F. Curtain, Springer Verlag, Heidelberg, 1987, 83–94.Google Scholar
  13. 12.
    S. Kung and D. W. Lin, Optimal Hankel norm model reductions: multivariable systems, I.E.E.E. Trans. Automatic Control 26 (1981), 832–852.CrossRefGoogle Scholar
  14. 13.
    B. C. Moore, Principal component analysis in linear systems: controllability, observability and model reduction, IEEE Trans. Automatic Control 26 (1981) 17–32.CrossRefGoogle Scholar
  15. 14.
    L. Pernebo and L. M. Silverman, Model reduction via balanced state space representation, IEEE Trans. Automatic Control 27 (1982) 382–387.CrossRefGoogle Scholar
  16. 15.
    I. Postlethwaite, D. W. Gu, S. D. O’Young and M. S. Tombs, An application of H-design and some computational improvements, in “Modelling, Robustness and Sensitivity Reduction in Control Systems”, ed. R. F. Curtain, Springer Verlag, Heidelberg, 1987, 305–322.Google Scholar
  17. 16.
    M. Vidyasagar, “Control System Synthesis: a Factorization Approach”, MIT Press, Cambridge, MA, 1985.Google Scholar
  18. 17.
    F. B. Yeh, Numerical solution of matrix interpolation problems, Ph. D. Thesis, Glasgow University, 1983.Google Scholar
  19. 18.
    F. B. Yeh and C. D. Yang, An efficient algorithm for H-optimal sensitivity problems, preprint, Institute of Aeronautics and Astronautics, National Chen Kung University, Tainan, Taiwan.Google Scholar
  20. 19.
    N. J. Young, Interpolation by analytic matrix fuctions, in “Operators and Function Theory”, Proceedings of NATO ASI, edited by S. C. Power, D. Reidel Publishing Co. 1985, 351–383.Google Scholar
  21. 20.
    N. J. Young, The singular value decomposition of an infinite Hankel matrix, Linear Algebra and its Applications 50 (1983) 639–656.CrossRefGoogle Scholar

Copyright information

© Birkhäuser Verlag Basel 1988

Authors and Affiliations

  • K. D. Gregson
    • 1
    • 2
  • N. J. Young
    • 1
    • 2
  1. 1.University of GlasgowGlasgowUK
  2. 2.Mathematics DepartmentUniversity GardensGlasgowUK

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