Skip to main content
SpringerLink
Log in

Normal Form, Invariants, and Bifurcations of Nonlinear Control Systems in the Particle Deflection Plane

  • Conference paper
  • First Online:
Dynamics, Bifurcations, and Control

Part of the book series: Lecture Notes in Control and Information Sciences ((LNCIS,volume 273))

  • 469 Accesses

Abstract

This paper addresses the problem of bifurcation analysis for nonlinear control systems. It is a review of some results published in recent years on bifurcations of control systems based on the normal form approach. We address three problems for systems with a single uncontrollable mode, namely the problem of normal forms, bifurcations of control systems, and bifurcation control by state feedback.

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 129.00
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 169.99
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. E. H. Abed and J.-H. Fu, “Local feedback stabilization and bifurcation control, I–II. Stationary bifurcation”, Systems & Control Letters, 8, 1987, 467–473.

    Article  MATH  MathSciNet  Google Scholar 

  2. J.P. Barbot, S. Monaco and D. Normand-Cyrot, “Quadratic forms and approximative feedback linearization in discrete time, International Journal of Control, 67, pp. 567–586, 1997.

    Article  MATH  MathSciNet  Google Scholar 

  3. S. Behtash and S. Sastry, “Stabilization of nonlinear systems with uncontrol-lable linearization”, IEEE Trans. Automat. Contr., vol. AC-33, pp. 585–591.

    Google Scholar 

  4. J. Carr, Applications of Center Manifold Theory, Springer-Verlag, New York, 1981.

    Google Scholar 

  5. G. Chen, “Control and synchronization of chaotic systems (a bibliography)”, ECE Dept., Univ. of Houston, TX-available from ftp: “http://ftp.egr.uh.edu/pub/TeX/chaos.tex”.

  6. G. Chen and J. L. Moiola, “An overview of bifurcation, chaos and nonlinear dynamics in control systems”, J. of Franklin Instit. 331B, 1994, 819–858.

    Article  MATH  MathSciNet  Google Scholar 

  7. F. Colonius, and W. Kliemann (1995). Controllability and stabilization of one-dimensional systems near bifurcation points. Systems & Control Letters, 24, 87–95.

    Article  MATH  MathSciNet  Google Scholar 

  8. K. M. Eveker, C. N. Nett, “Control of compression system surge and rotating stall: a laboratory-based ‘hands-on’ introduction”, Proceedings of the American Control Conference, Baltimore, MD, 1994, 1307–1311.

    Google Scholar 

  9. O. E. Fitch, The control of bifurcations with engineering applications, Ph.D. Dissertation, Naval Postgraduate School, Monterey, CA 93943, 1997.

    Google Scholar 

  10. R. Genesio, A. Tesi, H. O. Wang and E. H. Abed, Control of period doubling bifurcations using harmonic balance, Proc. IEEE Conf. Decision and Control, San Antonio, Texas, 1993, 492–497.

    Google Scholar 

  11. P. Glendinning, Stability, Instability and Chaos: an introduction to the theory of nonlinear differential equations, Cambridge University Press, 1994.

    Google Scholar 

  12. G. Gu, X. Chen, A. Sparks and S. Banda, “Bifurcation stabilization with local output feedback,” SIAM J. Contr. Optimiz., Vol.37, 934–956, 1999.

    Article  MATH  MathSciNet  Google Scholar 

  13. J. Hale and H. Koçak, Dynamics and Bifurcations, Springer-Verlag, New York, 1991.

    MATH  Google Scholar 

  14. B. Hamzi, J.P. Barbot, S. Monaco, and D. Normand-Cyrot, “Normal forms versus Naimark-Sacker bifurcation control, NOLCOS’01, to appear.

    Google Scholar 

  15. W. Kang and A. J. Krener, “Extended quadratic controller normal form and dynamic feedback linearization of nonlinear systems”, SIAM J. Control and Optimization, vol. 30, No. 6, pp 1319–1337, 1992.

    Article  MATH  MathSciNet  Google Scholar 

  16. W. Kang, Extended controller normal form, invariants and dynamic feedback linearization of nonlinear control systems, Ph.D. Dissertation, University of California at Davis, 1991.

    Google Scholar 

  17. W. Kang, Bifurcation and normal form of nonlinear control systems — part I, SIAM J. Control and Optimization, 36, (1998), 193–212.

    Article  MATH  MathSciNet  Google Scholar 

  18. W. Kang, Bifurcation and normal form of nonlinear control systems — part II, SIAM J. Control and Optimization, 36, (1998), 213–232.

    Article  MATH  MathSciNet  Google Scholar 

  19. W. Kang, Bifurcation control via state feedback for systems with a single uncontrollable mode, SIAM J. Control and Optimization, 38, (2000), 1428–1452.

    Article  MATH  MathSciNet  Google Scholar 

  20. B. Hamzi, W. Kang, and J-P. Barbot, On the Control of Hopf Bifurcations, Proc. IEEE Conference on Decision and Control, Sydney, Australia, December 12–15, 2000.

    Google Scholar 

  21. W. Kang, Extended controller form and invariants of nonlinear control systems with a single input, J. of Mathematical Systems, Estimation and Control, 6 (1996), 27–51.

    MATH  Google Scholar 

  22. W. Kang, Bifurcation and topology of equilibrium sets, IEEE Conf. on Decision and Control, Kobe, Japan, 1996, 2151–2155.

    Google Scholar 

  23. W. Kang, F. Papoulias, Bifurcation and normal forms of dive plane reversal of submersible vehicles, Proc. International Offshore and Polar Engineering Conference, Honolulu, Hawaii, May 25–30, 1997, 62–69.

    Google Scholar 

  24. A. J. Krener, W. Kang and D. E. Chang, Control Bifurcations, preprint.

    Google Scholar 

  25. A. J. Krener, The Feedbacks which Soften the Primary Bifurcation of MG 3, PRET Working Paper D95-9-11, 1995.

    Google Scholar 

  26. M. Krstic, J. M. Protz, J. D. Paduano, P. V. Kokotovic, Backstepping designs for jet engine stall and surge control, Proc. IEEE Conf. Decision and Control, New Orleans, LA, 1995, 3049–3055.

    Google Scholar 

  27. D.-C. Liaw and E. H. Abed, Stability analysis and control of rotating stall, Proc. IFAC Nonlinear Control Systems Design Symposium, Bordeaux, France, June, 1992.

    Google Scholar 

  28. F. E. McCaughan, Bifurcation analysis of axial flow compressor stability, SIAM J Applied Mathematics, vol. 20, 1990, 1232–1253.

    Article  MathSciNet  Google Scholar 

  29. F. K. Moore and E. M. Greitzer, A theory of post-stall transients in axial compression systems — Part I: Development of equations, ASME J. Engineering for Gas Turbines and Power, vol. 108, 1986, 68–76.

    Article  Google Scholar 

  30. M. A. Pinsky and B. Essary, Normal forms, averaging and resonance control of flexible structures, J. of Dynamic Systems, Measurement and Control, v. 116, N3, 1994, pp. 357–366.

    Article  MATH  Google Scholar 

  31. M. A. Pinsky and I. Shumyatsky, Feedback stabilization of bifurcation phenomena and its application to the control of voltage instabilities and collapse, Proc. NOLCOS’95, Lake Tahoe, California, 1995, pp. 58–63.

    Google Scholar 

  32. I. A. Tall and W. Respondek, “Normal forms, canonical forms, and invariants of single input nonlinear systems under feedback,” Proc. IEEE Conf. on Decision and Control, Sydney, Australia, 2000.

    Google Scholar 

  33. H. O. Wang, E. H. Abed, Bifurcation control of a chaotic system, Automatica, vol. 31, No. 9, 1995, 1213–1226.

    Article  MATH  MathSciNet  Google Scholar 

  34. S. Wiggins, Introduction to Applied Nonlinear Dynamical Systems and Chaos, Springer-Verlag, 1990.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, Naval Postgraduate School, Monterey, CA, 93943, USA

    Wei Kang

Authors
  1. Wei Kang
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Institut für Mathematik, Universität Augsburg, Universitätsstraße, 86150, Augsburg, Germany

    Fritz Colonius

  2. Fachbereich Mathematik, J.W. Goethe-Universität, Postfach 11 19 32, 60054, Frankfurt am Main, Germany

    Lars Grüne

Rights and permissions

Reprints and permissions

Copyright information

© 2002 Springer-Verlag Berlin Heidelberg

About this paper

Cite this paper

Kang, W. (2002). Normal Form, Invariants, and Bifurcations of Nonlinear Control Systems in the Particle Deflection Plane. In: Colonius, F., Grüne, L. (eds) Dynamics, Bifurcations, and Control. Lecture Notes in Control and Information Sciences, vol 273. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-45606-6_5

Download citation

  • DOI: https://doi.org/10.1007/3-540-45606-6_5

  • Published:

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-42890-9

  • Online ISBN: 978-3-540-45606-3

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics

Search

Navigation