Spectral Nevanlinna-Pick Interpolation

  • Allen R. Tannenbaum
Part of the Communications and Control Engineering book series (CCE)

Abstract

Let ε be a finite dimensional vector space, let D denote the unit disc, let z 1,…, z n D be mutually distinct, and let F 1,…,F n B(ε), where B(ε), denotes the space of bounded linear operators on ε. For a bounded analytic function F: DB(ε), let ∥Fsp be the spectral radius defined by
$${\left\| F \right\|_{sp}}: = \sup \left\{ {{{\left\| {F\left( z \right)} \right\|}_{sp}}:\,z \in D} \right\},$$
where ∥F(z)∥sp is the spectral radius of the operator F(z) (= the absolute value of the eigenvalue of largest magnitude of F(z)).

Keywords

Doyle 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    H. Bercovici, C. Foias, and A. Tannenbaum, “On the optimal solutions in spectral commutant lifting theory,” Journal of Functional Analysis 101 (1991), pp. 38–49.MathSciNetMATHCrossRefGoogle Scholar
  2. [2]
    H. Bercovici, C. Foias, and A. Tannenbaum, “A spectral commutant lifting theorem,” Trans. AMS 325 (1991), pp. 741–763MathSciNetMATHCrossRefGoogle Scholar
  3. [3]
    H. Bercovici, C. Foias, and A. Tannenbaum, “The structured singular value for linear input/output operators,” SIAM J. Control and Optimization 34 (1996), pp. 1392–1404.MathSciNetMATHCrossRefGoogle Scholar
  4. [4]
    J. C. Doyle, B. Francis, and A. Tannenbaum, Feedback Control Theory, McMillan, New York, 1991.Google Scholar
  5. [5]
    B. Francis, A Course in H Control Theory, Lecture Notes in Control and Information Sciences 88, Springer Verlag, 1987.Google Scholar
  6. [6]
    B. Francis and A. Tannenbaum, “Generalized interpolation theory in control,” Mathematical Intelligencer 10 (1988), pp. 48–53.MathSciNetMATHCrossRefGoogle Scholar
  7. [7]
    J. Garnett, Bounded Analytic Functions, Academic Press, New York, 1981.MATHGoogle Scholar
  8. [8]
    D. Sarason, “Generalized interpolation in H∞,” Transactions of the AMS 127 (1967), pp. 179–203.MathSciNetMATHGoogle Scholar
  9. [9]
    B. Sz.-Nagy and C. Foias, Harmonic Analysis of Operators on Hilbert Space, North Holland, Amsterdam, 1970.Google Scholar
  10. [10]
    A. Tannenbaum, Invariance and System Theory: Algebraic and Geo-metric Aspects, Lecture Notes in Mathematics 845, Springer-Verlag, 1981.Google Scholar
  11. [11]
    A. Tannenbaum, “Spectral Nevanlinna-Pick interpolation theory,” Proc. of 26th IEEE Conference on Decision and Control, Los Angeles, California, December 1987, pp. 1635–1638.Google Scholar
  12. [12]
    A. Tannenbaum, “On the multivariable gain margin problem,” Automatica 22 (1986), pp. 381–384.MathSciNetMATHCrossRefGoogle Scholar
  13. [13]
    G. Zames, “Feedback and optimal sensitivity: model reference transformations, multiplicative seminorms, and approximate inverses,” IEEE Trans. Auto. Control AC-26 (1981), pp. 301–320.Google Scholar

Copyright information

© Springer-Verlag London Limited 1999

Authors and Affiliations

  • Allen R. Tannenbaum
    • 1
  1. 1.Department of Electrical and Computer EngineeringUniversity of MinnesotaMinneapolisUSA

Personalised recommendations