Journal of the Korean Mathematical Society (대한수학회지)

Korean Mathematical Society (대한수학회)
권호 표지
DOI QR코드

DOI QR Code

ADAPTIVE PARTIAL STABILIZATION, LIMIT DYNAMICS AND BIFURCATION ANALYSIS

  • Received : 2011.03.15
  • Published : 2012.11.01
Download PDF
⟨ Previous Next ⟩

Abstract

A class of autonomous control systems with fixed unknown parameters is considered to be stabilized with respect to only a part of the variables. A certain type of such systems can be recursively adaptively partially stabilized. The bifurcation analysis reveals the nature of the closed loop system.

Keywords

References

  1. A. S. Andreyev, Investigation of partial asymptotic stability and instability based on the limiting equations, J. Appl. Math. Mech. 51 (1987), no. 2, 196-201. https://doi.org/10.1016/0021-8928(87)90064-5
  2. K. J. Astrom and B. Wittenmark, Adaptive Control, Adison-Wesley, 1995.
  3. F. Dumortier and R. Roussarie, Geometric singular perturbation theory beyond normal hyperbolicity, Multiple-time-scale dynamical systems (Minneapolis, MN), 29-63, IMA Vol. Math. Appl., 122, Springer, New York, 2001.
  4. B. Fiedler and S. Liebscher, Takens-Bogdanov bifurcation without parameters and oscillatory shock profiles, Global analysis of dynamical systems, 211-259, Inst. Phys., Bristol, 2001.
  5. J. Guckenheimer and P. Holmes, Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields, Springer-Verlag, New York, 2nd edition, 1990.
  6. P. A. Ioannou and J. Sun, Robust Adaptive Control, Prentice-Hall, New Jersey, 1996.
  7. M. Krstic, Invariant manifolds and asymptotic properties of adaptive nonlinear stabilizers, IEEE Trans. Automat. Control 41 (1996), no. 6, 817-829. https://doi.org/10.1109/9.506234
  8. M. Krstic, I. Kanellakopoulos, and P. V. Kokotovic, Adaptive nonlinear control without overparametrization, Systems Control Lett. 19 (1992), no. 3, 177-185. https://doi.org/10.1016/0167-6911(92)90111-5
  9. M. Krstic, I. Kanellakopoulos, and P. V. Kokotovic, Nonlinear and Adaptive Control Design, John Wiley & Sons, 1995.
  10. J. A. Leach, S. Triantafillidis, D. H. Owens, and S. B. Townley, The dynamics of universal adaptive stabilization: computational and analytical studies, Control Theory Adv. Tech. 10 (1995), no. 4, 1689-1716.
  11. I. G. Malkin, Theory of Stability of Motion, Izdat. Nauka, Moscow, 2nd edition, 1996.
  12. A. S. Oziraner, On asymptotic stability and instability relative to a part of variables, J. Appl. Math. Mech. 37 (1973), 659-665.
  13. C. Risito, Sulla stabilita asintotica parziale, Ann. Mat. Pura Appl. (4) 84 (1970), 279- 292. https://doi.org/10.1007/BF02413656
  14. G. R. Rokni Lamooki and S. B. Townley, Adaptive partial stabilization for non- deterministic strict feedback form, Proc. 11th IEEE Int. Conf. on Meth. and Mod. in Aut. and Robo. MMAR, Poland, (2005), 249-254.
  15. G. R. Rokni Lamooki, S. B. Townley, and H. M. Osinga, Bifurcations and limit dynamics in adaptive control systems, Internat. J. Bifur. Chaos Appl. Sci. Engrg. 15 (2005), no. 5, 1641-1664. https://doi.org/10.1142/S021812740501296X
  16. V. V. Rumyantsev, On the stability of motion with respect to the part of the variables, Vestnik Moscow. Univ. Ser. Mat. Mech Fiz. Astron. Khim. 4 (1957), 9-16.
  17. V. V. Rumyantsev and A. S. Oziraner, The Stability and Stabilization of Motion with Respect to Some of the Variables, Nauka, 1987.
  18. S. B. Townley, An example of a globally stabilizing adaptive controller with a generically destabilizing parameter estimate, IEEE Trans. Automat. Control 44 (1999), no. 11, 2238-2241. https://doi.org/10.1109/9.802953
  19. V. I. Vorotnikov, Partial Stability and Control, Birkhauser, 1998.
  20. V. I. Vorotnikov, Partial stability and control: The state-of-the-art and development prospects, Automation and Remote Control 66 (2005), no. 4, 511-561. https://doi.org/10.1007/s10513-005-0099-9
  21. L. Yang, S. A. Neild, D. J. Wagg, and D. W. Virden, Model reference adaptive control of a nonsmooth dynamical system, Nonlinear Dynamics 46 (2006), no. 3, 323-335. https://doi.org/10.1007/s11071-006-9048-6