Advertisement

The Gap Metric and Internal Stability

  • Avraham Feintuch
Chapter
  • 434 Downloads
Part of the Applied Mathematical Sciences book series (AMS, volume 130)

Abstract

In Section 6.1 we saw that the internal stability of a feedback system {L, C} can be formulated in geometric terms relating the graph of L and C. In this chapter we introduce and study a precise geometric tool, the gap metric, which will allow us to study internal stability from a geometric point of view. This will prepare the grounds for a study of robust stabilization from a point of view seemingly different from that given in Chapter 8.

Keywords

Orthogonal Projection Bounded Operator Feedback System Closed Operator Invertible Operator 
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, Notes, and Remarks

  1. 1.
    Krasnosel’skii, M. A., Vainikko, G. M., Zabreiko, P. P., Rutitskii, Ya. B., Stetsenko, V. Ya.Approximate Solutions of Operator EquationsGroningen, Wolters-Noordhoff, 1972.CrossRefGoogle Scholar
  2. 2.
    Kato, T.Perturbation Theory for Linear OperatorsNew York, Springer-Verlag, 1966.zbMATHGoogle Scholar
  3. 3.
    Foias, C., Georgiou, T. T., Smith, M., Robust stability of feedback systems: A geometric approach using the gap metricSIAM J. Cont. and Optim.31, 6 (1993), 1518–1537.MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Nikol’skii, N.Treatise on the Shift OperatorNew York, Springer-Verlag, 1986.CrossRefzbMATHGoogle Scholar
  5. 5.
    Georgiou, T., On the computation of the gap metricSystem Control Lett. 11 (1988), 253–257.MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Georgiou, T., Smith, M., Optimal robustness in the gap metricIEEE Trans. Aut. Cont.35, (1990), 673–686.MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    El-Sakkary, A., The gap metric: Robustness of stabilization of feedback systemsIEEE Trans. Aut. Cont.30 (1985), 240–247.MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Ober, R., Sefton, J., Stability of linear systems and graphs of linear systemsSystem Control Lett. 17 (1991), 265–280.MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Zhu, S. Q., Hautus, L. J., Praagman, C., Sufficient conditions for robust BIBO stabilization: Given by the gap metricSystem Control Lett.11 (1988), 53–59.MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Zhu, S. Q., Graph topology and gap topology for unstable systemsIEEE Trans. Aut. Cont.34 (1989), 848–855.CrossRefzbMATHGoogle Scholar
  11. 11.
    Feintuch, A., The time-varying gap and co-prime factor perturbationsMCSS8 (1995), 352–374.MathSciNetzbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media New York 1998

Authors and Affiliations

  • Avraham Feintuch
    • 1
  1. 1.Department of Mathematics and Computer ScienceBen-Gurion University of the NegevBeer-ShevaIsrael

Personalised recommendations