Advertisement

A fast algorithm to solve Kalman’s partial realisation problem for single input, multi-output systems

  • P. R. Graves-Morris
  • J. M. Wilkins
Conference paper
Part of the Lecture Notes in Mathematics book series (LNM, volume 1287)

Abstract

A brief review is given of the solution of the scalar partial realisation problem using Padé approximants. The use of simultaneous Padé approximants in the solution of the single input, multi-output partial realisation problem is then discussed. We show how analogues of Frobenius identities are derived for simultaneous Padé approximants of two series, and we give twelve such identities. We show how some of these identities are combined to construct analogues of Baker’s and Kronecker’s algorithms. These analogues are fast algorithms for simultaneous Padé approximation of two series, and so also for a solution of the single input, two output partial realisation problem.

Keywords

Single Input Degree Reduction Reliable Algorithm Markov Parameter Denominator Polynomial 
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.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Aitken, A.C., Determinants and Matrices, Oliver and Boyd (1959).Google Scholar
  2. 2.
    Baker, G.A. Jr., Essentials of Padé Approximants, Academic Press (New York, 1975).zbMATHGoogle Scholar
  3. 3.
    Baker, G.A. Jr. and Graves-Morris, P.R., Padé Approximants Part I in Encyclopedia of Mathematics, 13 (Addison-Wesley, 1981).Google Scholar
  4. 4.
    Brent, R.P., Gustavson, F.G. and Yun, D.Y.Y., Fast Solution of Toeplitz Systems of Equations and Computation of Padé Approximants, J. Algorithms 1, (1980), 259–295.CrossRefzbMATHMathSciNetGoogle Scholar
  5. 5.
    de Bruin, M.G., Generalised C-Fractions and a Multi-dimensional Padé Table, Thesis, Amsterdam University (1974).Google Scholar
  6. 6.
    de Bruin, M.G., Generalised Padé Tables and Some Algorithms Therein, in Proceedings of First French-Polish Meeting on Padé Approximation and Convergence Acceleration Techniques ed. J. Gilewicz CNRS (Marseille, 1982), 1–10.Google Scholar
  7. 7.
    de Bruin, M.G., Zeros of Polynomials Generated by 4-Term Recurrence Relations, in Rational Approximation and Interpolation eds. P.R. Graves-Morris, E.B. Saff and R.S. Varga, Springer (Heidelberg, 1984) 331–345.CrossRefGoogle Scholar
  8. 8.
    Bultheel, A., Division Algorithms for Continued Fractions and the Pade Table, J. Comp. Appld. Math. 6, (1980), 259–266.CrossRefzbMATHMathSciNetGoogle Scholar
  9. 9.
    Bultheel, A. and van Barel, M., Padé Techniques for Model Reduction in Linear System Theory: a Survey, J. Comp. Appld. Math. 14, (1986), 401–438.CrossRefGoogle Scholar
  10. 10.
    Gragg, W.B. and Lindquist, A., On the Partial Realisation Problem, Lin. Alg. and its Applcns. 50, (1983), 277–319.CrossRefzbMATHMathSciNetGoogle Scholar
  11. 11.
    Graves-Morris, P.R., The Numerical Calculation of Padé Approximants, in Padé Approximation and its Applications, ed. L. Wuytack, Springer (Heidelberg, 1980), 231–245.Google Scholar
  12. 12.
    Graves-Morris, P.R., Toeplitz Equations and Kronecker’s Algorithm, C.S.S.P. 1, (1982), 289–304.zbMATHMathSciNetGoogle Scholar
  13. 13.
    Graves-Morris, P.R., Vector-valued Rational Interpolants II, IMA J. Numer. Analy. 4, (1984), 209–224.CrossRefzbMATHMathSciNetGoogle Scholar
  14. 14.
    Kailath, T., Linear Systems, Prentice-Hall (1980).Google Scholar
  15. 15.
    Kalman, R.E., On Minimal Partial Realizations of a Linear Input/Output Map, in Aspects of Network and Systems Theory, eds. R.E. Kalman and N. de Claris, Holt, Reinhart and Winston (1971), 385–407.Google Scholar
  16. 16.
    McEliece, R.J. and Shearer, J.B., A Property of Euclid’s Algorithm and an Application to Padé Approximation, SIAM J. Appl. Math. 34, (1978) 611–615.CrossRefzbMATHMathSciNetGoogle Scholar
  17. 17.
    MacWilliams, F.J. and Sloane, N.J.A., The theory of Error Correcting Codes, North-Holland (1983).Google Scholar
  18. 18.
    Massey, J.L., Shift-Register Synthesis and BCH Decoding, IEEE Trans. Info. Theory IT-15, (1969), 122–127.CrossRefzbMATHMathSciNetGoogle Scholar
  19. 19.
    Padé, H., Sur la Généralisation des Fractions Continues Algébriques, J. de Math. 10 (1894), 291–329.Google Scholar
  20. 20.
    Warner, D., Hermite Interpolation with Rational Functions, Thesis, University of California, (1974).Google Scholar
  21. 21.
    Wilkins, J.M., Thesis, University of Kent at Canterbury, in preparation. Postcript. Since original submission of this paper, we have become aware that J. van Iseghem has independently derived identity (B) in the same way. By combining it with the vector qd algorithm, she has derived an iterative construction of a different staircase sequence of denominator polynoials in the d-dimensional case [22].Google Scholar
  22. 22.
    van Iseghem, J., Thèse, Université de Lille, (1987); ICAM proceedings, (1987, in press).Google Scholar

Copyright information

© Springer-Verlag 1987

Authors and Affiliations

  • P. R. Graves-Morris
    • 1
  • J. M. Wilkins
    • 2
  1. 1.School of Mathematical SciencesUniversity of BradfordBradfordEngland
  2. 2.D.A.M.T.P.Cambridge UniversityCambridgeEngland

Personalised recommendations