Skip to main content
SpringerLink
Log in

Robust Stabilization for ℓ p Gap Perturbations

  • Conference paper
Robust Control Theory

Part of the book series: The IMA Volumes in Mathematics and its Applications ((IMA,volume 66))

  • 370 Accesses

Abstract

This paper studies robust stabilization of linear feedback systems. The special features of this study are: (1) the input and output signal spaces of systems are assumed to be any ℓ p spaces; (2) system perturbations are measured by the gap function.

The authors would like to thank Tryphon Georgiou, Yoshito Ohta and Bo Bernhardsson for helpful discussions.

Department of Electrical & Electronic Engineering, University of Science & Technology, Clearwater Bay, Kowloon, Hong Kong. This author was supported by the Institute for Mathematics and its Applications, University of Minnesota, with funds provided by the National Science Foundation of USA.

Dept. of Elect. & Comp. Eng., University of Waterloo, Waterloo, Ontario, Canada N2L 3G1. Supported by the Natural Sciences and Engineering Research Council of Canada.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 84.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 109.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. M. A. Dahleh, BIBO stability robustness in the presence of coprirne factor perturbation, IEEE Trans. Automat. Contr. 37 (1992), pp. 352–355.

    Article  MathSciNet  Google Scholar 

  2. M. A. Dahleh and J. B. Pearson, ℓ 1 optimal feedback controller for MIlvIO discrete time systems, IEEE Trans. Automat. Contr. AC-32 (1987), pp. 314–322.

    Google Scholar 

  3. M. A. Dahleh and J. B. Pearson, Optimal rejection of persistent disturbances, robust stability and mixed sensitivity minimization, IEEE Trans. Automat. Contr. 33 (1988), pp. 722–731.

    Article  MathSciNet  MATH  Google Scholar 

  4. W. N. Dale and M. C. Smith, Stabilizability and existence of system representations for discrete-time time-varing systems, Proc. IEEE Conf. Decision and Control (1990), pp. 2737–2742.

    Google Scholar 

  5. C. A. Desoer, R. W. Liu, J. Murray, and R. Saeks, Feedback system design: the fractional representation approach to analysis and synthesis, IEEE Trans. Automat. Contr. AC-25 (1980), pp. 399–412.

    Article  MathSciNet  Google Scholar 

  6. A. K. El-Sakkary, The gap metric: robustness of stabilization of feedback systems, IEEE Trans. Automat. Contr. AC-30 (1985), pp. 240–247.

    Article  MathSciNet  Google Scholar 

  7. A. K. El-Sakkary, Estimating robustness on the Riemann sphere, Int. J. Control 49 (1989), pp. 561–567.

    MathSciNet  MATH  Google Scholar 

  8. A. Feintuch, The gap metric for time-varying systems, Systems & Control Letters 16 (1991), pp. 277–279.

    Article  MathSciNet  MATH  Google Scholar 

  9. C. Foias, T. T. Georgiou, and M. C. Smith, Geometric techniques for robust stabilization of linear time-varying systems, Proc. IEEE Conf. Decision and Control (1990), pp. 2868–2873.

    Google Scholar 

  10. F. Forgo, Nonconvex Programming, Akademiai Kiado, Budapest, 1988.

    MATH  Google Scholar 

  11. T. T. Georgiou, On the computation of the gap metric, Systems & Control Letters 11 (1988), pp. 253–257.

    Article  MathSciNet  MATH  Google Scholar 

  12. T. T. Georgiou and M. C. Smith, Optimal robustness in the gap metric, IEEE Trans. Automat. Contr. 35 (1990), pp. 673–685.

    Article  MathSciNet  MATH  Google Scholar 

  13. T. T. Georgiou and M. C. Smith, Graphs, causality and stabilizability: linear, shift-invariant systems on E 2 [0, ∞), Proc. IEEE Conf. Decision and Control (1992), pp. 1024–1029.

    Google Scholar 

  14. T. T. Georgiou and M. C. Smith, Robust stabilization in the gap metric: controller design for distributed systems, IEEE Trans. Automat. Contr. 37 (1992), pp. 1133–1143.

    Article  MathSciNet  Google Scholar 

  15. K. Glover and D. C. Mcfarlane, Robust stabilization of normalized coprime factor plant description with H ∞-bounded uncertainty, IEEE Trans. Automat. Contr. AC-34 (1989), pp. 821–830.

    Article  MathSciNet  Google Scholar 

  16. P. K. Kamthan and M. Gupta, Sequence Spaces and Series, Marcel Dekker, Inc., New York, 1981.

    MATH  Google Scholar 

  17. T. Kato, Perturbation theory for nullity, deficiency and other quantities of linear operators, J. Analyse Math. 6 (1958), pp. 261–322.

    Article  MathSciNet  MATH  Google Scholar 

  18. T. Kato, Perturbation Theory for Linear Operator, Springer-Verlag, New York, 1966.

    Google Scholar 

  19. J. L. Kelley and T. P. Srinivasan, Measure and Integral, Volume 1, Springer-Verlag, New York, 1988.

    Book  MATH  Google Scholar 

  20. M. G. Kreìn and M. A. Krasnosel’skiì, Fundamental theorems concerning the extension of Hermitian operators and some of their applications to the theory of orthogonal polynomials and the moment problem, (in Russian) Uspekhi Mat. Nauk 2 (1947), pp. 60–106.

    Google Scholar 

  21. M. G. Kreìn, M. A. Krasnosel’skiì, and D. P. Mil’man, Concerning the deficiency numbers of linear operators in Banach space and some geometric questions, (in Russian) Sbornik Trudov Inst. Mat. Akad. Nauk Ukr. SSR 11 1948.

    Google Scholar 

  22. H. Ozbay, On L 1 optimal control, IEEE Trans. Automat. Contr. 34 (1989), pp. 884–885.

    Article  Google Scholar 

  23. K. Poolla and P. Khargonekar, Stabilizability and stable-proper factorizations for linear time-varing systems, SIAM J. Control & Optimization 25 (1987), pp. 723–736.

    Article  MathSciNet  MATH  Google Scholar 

  24. L. Qiu and E. J. Davison, Feedback stability under simultaneous gap metric uncertainties in plant and controller, Systems & Control Letters 18 (1992), pp. 9–22.

    Article  MathSciNet  MATH  Google Scholar 

  25. L. Qiu and E. J. Davison, Pointwise gap metrics on transfer matrices, IEEE Trans. Automat. Contr. 37 (1992), pp. 741–758.

    Article  MathSciNet  MATH  Google Scholar 

  26. H. L. Royden, Real Analysis, MacMillan Publishing Co., Inc., New York, 1968.

    Google Scholar 

  27. W. Rudin, Functional Analysis, McGraw-Hill Book Company, New York, 1973.

    MATH  Google Scholar 

  28. R. Saeks and J. Murray, Fractional representation, algebraic geometry, and the simultaneous stabilizability problem, IEEE Trans. Automat. Contr. AC-27 (1982), pp. 895–903.

    Article  MathSciNet  Google Scholar 

  29. J. M. Schumacher, A pointwise criterion for controller robustness, Systems & Control Letters 18 (1992), pp. 1–8.

    Article  MathSciNet  MATH  Google Scholar 

  30. M. Vidyasagar, Control System Synthesis: A Factorization Approach, The MIT Press, Cambridge, Massachusetts, 1985.

    MATH  Google Scholar 

  31. M. Vidyasagar, H. Schneider, and B. A. Francis, Algebraic and topological aspects of feedback stabilization, IEEE Trans. Automat. Contr. AC-27 (1982), pp. 880–894.

    Article  MathSciNet  Google Scholar 

  32. G. Vinnicombe, Structured uncertainty and the graph topology, IEEE Trans. Automat. Contr. 38 (1993), pp. 1371–1383.

    Article  MathSciNet  MATH  Google Scholar 

  33. G. Zames and A. K. El-Sakkary, Unstable systems and feedback: the gap metric, Proc. 16th Allerton Conf. (1980), pp. 380–385.

    Google Scholar 

  34. S. Q. Zhu, Graph topology and gap topology for unstable systems, IEEE Trans. Automat. Contr. AC-34 (1989), pp. 848–855.

    Article  Google Scholar 

  35. S. Q. Zhu, Robustness of complementarity and feedback stabilization, (preprint) 1992.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Electrical & Electronic Engineering, University of Science & Technology, Clearwater Bay, Kowloon, Hong Kong, China

    Li Qiu

  2. Dept. of Elect. & Comp. Eng., University of Waterloo, Waterloo, Ontario, Canada, N2L 3G1

    Daniel E. Miller

Authors
  1. Li Qiu
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Daniel E. Miller
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Department of Electrical and Computer Engineering, University of Toronto, M5S 1A4, Toronto, Ontario, Canada

    Bruce A. Francis

  2. Department of Electrical Engineering and Computer Science, University of Michigan, 48109, Ann Arbor, MI, USA

    Pramod P. Khargonekar

Rights and permissions

Reprints and permissions

Copyright information

© 1995 Springer-Verlag New York, Inc.

About this paper

Cite this paper

Qiu, L., Miller, D.E. (1995). Robust Stabilization for ℓ p Gap Perturbations. In: Francis, B.A., Khargonekar, P.P. (eds) Robust Control Theory. The IMA Volumes in Mathematics and its Applications, vol 66. Springer, New York, NY. https://doi.org/10.1007/978-1-4613-8451-9_4

Download citation

  • DOI: https://doi.org/10.1007/978-1-4613-8451-9_4

  • Publisher Name: Springer, New York, NY

  • Print ISBN: 978-1-4613-8453-3

  • Online ISBN: 978-1-4613-8451-9

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics

Search

Navigation