Recent and New Results in Rational L2-Approximation

  • Laurent Baratchart
Part of the NATO ASI Series book series (volume 34)


The problem under consideration in this paper is the best approximation of a stable linear constant dynamical system by one whose Mac-Millan degree does not exceed a prescribed n. This problem can be formulated in many ways, according to the criterion which is used. In particular, striking results have been obtained concerning the Hankel norm, and a bibliographical account can be found in (KL, 81) and (Gl, 84). The criterion here will be the l 2 norm of the discrete transfer function, or equivalently the L2 \( (\frac{{d\omega }}{{1 + {\omega ^{2}}}}) \) norm of the continuous time transfer function. This problem has already been examined in the scalar case in (Ro, 78) (Ruc, 78) (De, 80) (Du, 73) and in the multivariable case in (Ba, 82) (Bs, 85) (Ba, *). In fact, the present paper can be considered as a sequel to (Ba, *), to which we refer on several occasions. Thereby, it does not really need a new introduction. Let us mention, however, that while studying qualitative generic properties as in (Ba, *), we focuse mainly on uniqueness, and give an account of the case of finite sequences. Though we do not develop here effective applications of the differential theory, we hope our results will help clarifying the situation.


Hardy Space Implicit Function Theorem Finite Sequence Good Approximant Real Algebraic Variety 
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  1. (Ba, 82).
    Baratchart L. “Une structure différentielle pour certaines classes de systèmes, application à l’approximation L 2” Thèse de docteur-Ingénieur E.N.S.M.P Paris.Google Scholar
  2. (Ba, 84).
    Baratchart L. “On the parametrization of linear constant systems” SIAM J. cont. & opt., vol. 23 , n 0 5.Google Scholar
  3. (Ba, *).
    Baratchart L., “Existence and generic properties for L 2-approximants of linear systems” to appear in I.M.A. Journal.Google Scholar
  4. (BS, 85).
    Baratchart L., Steer S.“Rosencher type equations for L 2-approximation of linear constant systems” Proc. 24th C.D.C., Fort Lauderdale, FL.Google Scholar
  5. (CR, 79).
    Coste M., Roy M. F. “Topologies for real algebraic geometry” in “Topos theoretic methods in geometry”, A. Kock ed., Arhus Universiteit, pp. 29–100.Google Scholar
  6. (De, 80).
    Delia Dora J. “Contribution à l’approximation de fonctions de la variable complexe au sens de Hermite-Padé et de Hardy” Thèse d’état Univ. Scient. & Médicale de Grenoble.Google Scholar
  7. (Du, 73).
    Duc-Jacquet M. “Approximation des fonctionelles linéaires sur les espaces Hilbel tiens à noyaux reproduisants” Thèse d’état Univ. Scient. & Médicale de Grenoble.Google Scholar
  8. (Gl, 84).
    Glover K. “All optimal Hankel norm approximations of linear multivariable systems and their L bounds” Int. J. Cont. vol.39 , n 0 6 pp. 1115–1195.Google Scholar
  9. (Hi, 76).
    Hirsch M.W. “Differential Topology” Graduate Texts in Math. Springer-Verlag New-York.Google Scholar
  10. (HK, 74).
    Hazewinkel M., Kaiman R.E. “Moduli and canonical forms for linear systems” Report n 0 7504 Econometric Institute Erasmus Univ. Rotterdam.Google Scholar
  11. (KFA, 69).
    Kaiman R. E., Falb P. L., Arbib M. A. “Topics in mathematical system theory” Mc. Graw-Hill, New York.Google Scholar
  12. (KL, 81).
    Kung S., Lin D. ‘Optimal Hankel norm model reduction: multivariable systems’ IEEE trans. Aut. Cont. vol. 26. n 0 4, pp. 832–854.Google Scholar
  13. (Ro, 78).
    Rosencher E. “Approximation rationelle des filtres à 1 ou 2 indices: une approche Hilbertienne” Thèse de docteur-Ingénieur. Univ. Paris IX-Dauphine Paris.Google Scholar
  14. (Ruc, 78).
    Ruckebush G. “Sur l’approximation rationelle des filtres” Rapport n 0 35 C.M. A. Ecole Polytechnique.Google Scholar
  15. (Ruc, 66).
    Rudin W. “Real and complex analysis” Mc. Graw-Hill New York.Google Scholar
  16. (Se, 54).
    Seidenberg A. “A new decision method for elementary algebra” Ann. of Math. 60, pp. 365–374.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1987

Authors and Affiliations

  • Laurent Baratchart
    • 1
  1. 1.Institut National de Recherche en Informatique et AutomatiqueValbonneFrance

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