Advertisement

Robust Stability Analysis of Various Classes of Delay Systems

  • Catherine Bonnet
  • Jonathan R. Partington
Conference paper
Part of the Lecture Notes in Computational Science and Engineering book series (LNCSE, volume 38)

Abstract

This chapter is a review of some work of the authors on the robust stabilization of retarded and neutral delay systems, including the case of fractional delay systems. BIBO-stability and nuclearity conditions are derived and the question of parametrization of all BIBO-stabilizing controllers is addressed.

Keywords

Satisfying Condition Robust Stabilization Delay System Hankel Operator Springer Lecture Note 
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.
    R. Bellman and K. L. Cooke. Differential-difference equations. Academic Press New York, 1963.zbMATHGoogle Scholar
  2. 2.
    D. Brethe and J.-J. Loiseau. Stabilization of linear time-delay systems. JESA-RAIROA-PII, 6, 1025–1047, 1997.Google Scholar
  3. 3.
    C. Bonnet and J. R. Partington. Robust stabilization in the BIBO gap topology. Int. J. Robust Nonlin. Control, 7, 429–447, 1997.Google Scholar
  4. 4.
    C. Bonnet and J. R. Partington. Bézout factors and L 1-optimal controllers for delay systems using a two-parameter compensator scheme. IEEE Trans. Automat. Control, 44, 1512–1521, 1999.MathSciNetzbMATHCrossRefGoogle Scholar
  5. 5.
    C. Bonnet and J. R. Partington. Stabilization of fractional exponential systems including delays. Kybemetica, 37(3), 345–353, 2001.MathSciNetzbMATHGoogle Scholar
  6. 6.
    C. Bonnet and J. R. Partington. Analysis of fractional delay systems of retarded and neutral type. Automatica, 38(7), 1133–1138, 2002.MathSciNetzbMATHCrossRefGoogle Scholar
  7. 7.
    C. Bonnet and J. R. Partington. H∞ and BIBO stabilization of delay systems of neutral type. Systems Control Lett., to appear.Google Scholar
  8. 8.
    C. Bonnet, J. R. Partington and M. Sorine. Robust control and tracking of a delay system with discontinuous nonlinearity in the feedback. Int. J. Control 72(15), 1354–1364, dy1999.Google Scholar
  9. 9.
    C. Bonnet, J. R. Partington and M. Sorine. Robust stabilization of a delay system with saturating actuator or sensor. Int. J. Robust Nonlin. Control 10(7), 579–590, 2000.MathSciNetzbMATHCrossRefGoogle Scholar
  10. 10.
    R. F. Curtain and H. Zwart. An introduction to injinite-dimensional linear systems theory. Texts in Applied Mathematics, 21. Springer-VerlagNew York, 1995Google Scholar
  11. 11.
    H. Dym and T. T. Georgiou and M. C. Smith, Explicit formulas for optimally robust controllers for delay systems. IEEE Trans. Automat. Control, 40, 656–669, 1995.MathSciNetzbMATHCrossRefGoogle Scholar
  12. 12.
    C. Foias, H. Ozbay and A. Tannenbaum. Robust control in infinite dimensional systems-Frequency domain methods. Springer Lecture Notes in Control and Inform. Sci., no. 209, 1996.Google Scholar
  13. 13.
    T. T. Georgiou and M. C. Smith. Robust stabilization in the gap metric: controller design for distributed plants. IEEE Trans. Automat. Control, 37, 1133–1143, 1992.MathSciNetzbMATHCrossRefGoogle Scholar
  14. 14.
    T. T. Georgiou and M. C. Smith. Metric uncertainty and nonlinear feedback stabilization. In Feedback control, nonlinear systems, and complexity. Springer Lecture Notes in Control and Inform. Sci., no. 202, 88–98, 1995.Google Scholar
  15. 15.
    T. T. Georgiou and M. C. Smith. Robustness analysis of nonlinear feedback systems: an input-output approoch. IEEE Trans. Automm. Control, 42(9), 1200–1221, 1997.MathSciNetzbMATHCrossRefGoogle Scholar
  16. 16.
    K. Glover, R. F. Curtain, and J. R. Partington. Realization and approximation of linear infinite dimensional systems with error bounds. SIAM J. Control Optimiz., 26, 863–898, 1988.MathSciNetzbMATHCrossRefGoogle Scholar
  17. 17.
    K. Glover, J. Lam and J. R. Partington Rational approximation of a class of infinite-dimensional systems I: Singular values of Hankel operators Math. Control Sig. Sys., 3, 325–344, 1990.MathSciNetzbMATHCrossRefGoogle Scholar
  18. 18.
    K. Glover, J. Lam and J. R. Partington Rational approximation of a class of infinite-dimensional systems II: Optimal convergence rates of L∞ approximants Math. Control Sig. Sys., 4, 233–246, 1991.MathSciNetzbMATHCrossRefGoogle Scholar
  19. 19.
    H. Glüsing-Luerssen. A behavioral approach to delay-differential systems. SIAM J. Control Optimiz., 35(2), 480–499, 1997.CrossRefGoogle Scholar
  20. 20.
    G. Gripenberg, S.-O. Londen and O. Staffans. Volterra integral and functional equations. Cambridge University Press Cambridge, 1990.zbMATHCrossRefGoogle Scholar
  21. 21.
    E. Hille and R. S. Phillips. Functional analysis and semi-groups. Amer. Math. Soc, 1957.Google Scholar
  22. 22.
    R. Hotzel. Some stability conditions for fractional delay systems. J. Mathematical Systems Estimation Control, 8(4), 1–19, 1998.MathSciNetGoogle Scholar
  23. 23.
    J.-J. Loiseau and H. Mounier. Stabilisation de l’équation de la chaleur commandée en flux. In Systèmes Différentiels Fractionnaires, Modèles, Méthodes el Applications, ESAIM proceedings, 5, 131–144, 1998.MathSciNetzbMATHGoogle Scholar
  24. 24.
    P. M. Mäkilä and J. R. Partington. Robust stabilization-BlBO stability, distance notions and robustness optimization. Automatica, 23, 681–693, 1993.CrossRefGoogle Scholar
  25. 25.
    P. M. Mäkilä and J. R. Partington. Shift operator induced approximations of delay systems. SIAM J. Control Optimiz., 37(6), 1897–1912, 1999.zbMATHCrossRefGoogle Scholar
  26. 26.
    D. Matignon. Stability properties for generalized fractional differential systems. In: Systèmes Différentiels Fractionnaires, Modèles, Méthodes et Applications. Vol. 5. ESAIM proceedings, 145–158, 1998.MathSciNetzbMATHGoogle Scholar
  27. 27.
    D. C. McFarlane and K. Glover. Robust controller design using normalized coprime factor descriptions. Springer-Verlag, 1989.Google Scholar
  28. 28.
    H. Özbay. Introduction to feedback Control theory. CRC Press, 1999.Google Scholar
  29. 29.
    J. R. Partington. An introduction to Hankel operators. Cambridge UniversilY Press, 1988Google Scholar
  30. 30.
    J. R. Partington. Linear operators and linear systems. Cambridge University Press, to appear.Google Scholar
  31. 31.
    J. R. Partington and K. Glover. Robust stabilization of delay systems by approximation of coprime factors. Systems Control Lett., 14(4), 325–331, 1990.MathSciNetzbMATHGoogle Scholar
  32. 32.
    J. R. Partington and G. K. Sankaran. Algebraic construction of normalized coprime factors for delay systems. Math. Control Sig. Sys. 15(1) 1–12, 2002.MathSciNetzbMATHCrossRefGoogle Scholar
  33. 33.
    A. Quadrat. On a generalization of the Youla-Kucera parametrization. Part I: the fractional ideal approach to SISO systems. Systems Control Lett., 50, 135–148, 2003.MathSciNetzbMATHCrossRefGoogle Scholar
  34. 34.
    M. S. Verma. Coprime fractional representations and stability of non-linear feedback systems. Int. J. Control, 48, 897–918, 1988.zbMATHCrossRefGoogle Scholar
  35. 35.
    M. Vidyasagar. Control System Synthesis. MIT Press, 1985.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2004

Authors and Affiliations

  • Catherine Bonnet
    • 1
  • Jonathan R. Partington
    • 2
  1. 1.INRIA Rocquencourt, Domaine de VoluceuuLe Chesnay cedexFrance
  2. 2.University of Leeds, School of MathematicsLeedsUK

Personalised recommendations