Skip to main content
SpringerLink
Log in

Robust Adaptive Control

  • Chapter
Adaptive and Learning Systems

Abstract

Recent developments in robust adaptive control in the presence of bounded as well as state-dependent disturbances are discussed in the paper. The various approaches that have been suggested in the literature using modifications of a standard adaptive law, as well as arguments involving persistent excitation of the reference input, are presented. While the paper is partly tutorial, it also contains several new results due to the authors. These include a new adaptive law, the proof of global boundedness of all signals in the presence of bounded disturbances using persistent excitation, and the extension of these results to the adaptive control problem with state-dependent disturbances. Both local results and global results reported in the literature are discussed, but a greater emphasis is placed on those that are global in character. The implications of these results for future research on the reduced order model problem, are discussed towards the end of the paper.

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 39.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 54.99
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. K.S. Narendra, Y.H. Lin and L.S. Valavani, “Stable Adaptive Controller Design–Part II:Proof of Stability,” IEEE Trans. on Autom. Control, AC-25, pp. 440–448, June 1980.

    Google Scholar 

  2. A.S. Morse, “Global Stability of Parameter Adaptive Control Systems,” IEEE Trans. on Autom. Control, AC-25, pp. 433–439, June 1980.

    Google Scholar 

  3. K.S. Narendra and Y.H. Lin, “Stable Discrete Adaptive Control,” IEEE Trans. on Autom. Control, AC-25, pp. 456–461, June 1980.

    Google Scholar 

  4. G.C. Goodwin, P.J. Raoradge and P.E. Caines, “Discrete time Multivariable Adaptive Control,” IEEE Transactions on Automatic Control, vol. 25, pp. 449–456, June 1980.

    Article  MATH  Google Scholar 

  5. I.G. Malkin, “Theory of Stability of Motion,” Tech. Report Tr. 3352, U.S. Atomic Energy Commission, English ed. 1958.

    Google Scholar 

  6. A.N. Krylov and N.N. Bogoliuboff, Introduction to Nonlinear Mechanics, (English Translation), Princeton University Press, Princeton, New Jersey, 1943.

    Google Scholar 

  7. K.S. Narendra, A.M. Annaswamy and R.P. Singh, “A General Approach to the Stability Analysis of Adaptive Systems,” International Journal of Control, vol. 41, No. 1, 1985.

    Google Scholar 

  8. K.S. Narendra and A.M. Annaswamy, “Persistent Excitation and Stability Properties of Adaptive Systems,” Proceedings of the Fourth Yale Workshop on Applications of Adaptive Systems Theory, New Haven, May 1985.

    Google Scholar 

  9. R.E. Kalman and J.E. Bertran, “Control Systems Analysis and Design via the `Second Method’ of Lyapunov,” Journal of Basic Engineering, pp. 371–392, June 1960.

    Google Scholar 

  10. J.P. LaSalle, “Some Extensions of Liapunov’s Second Method,” IRE Transactions on Circuit Theory, pp. 520–527, Dec. 1960.

    Google Scholar 

  11. J.P. LaSalle, “Asymptotic Stability Criteria,” Proceedings of Symposium on Appl. Math., Hydrodynamic Instability, Providence, R.I., vol. 13, pp. 299–307, 1962.

    Article  MathSciNet  Google Scholar 

  12. A.P. Morgan and K.S. Narendra, “On the Uniform Asymptotic Stability of Certain Linear Nonautonomous Differential Equations” SIAM Journal of Control and Optimization, Vol. 15, pp. 5–24, Jan. 1977.

    Article  MathSciNet  MATH  Google Scholar 

  13. A.P. Morgan and K.S. Narendra, “On the Stability of Nonautonomous Differential Equations x = [A -I- B(t)lx with Skew-Symmetric Matrix B(t)” SIAM Journal of Control and Optimization, Vol. 15, pp. 163–176, Jan. 1977.

    Article  MathSciNet  MATH  Google Scholar 

  14. J.S.-C Yuan and W.M. Wonham, “Probing Signals for Model Reference Identification,” IEEE Transactions on Automatic Control, AC-22, pp. 530–538, August 1977.

    Google Scholar 

  15. B.B. Peterson and K.S. Narendra, “Bounded Error Adaptive Control,” IEEE Trans. on Autom. Control, AC-27, pp. 1161–1168, Dec. 1982.

    Google Scholar 

  16. G. Kreisselmeier and K.S. Narendra, “Stable Model Reference Adaptive Control in the Presence of Bounded Disturbances,” IEEE Trans. on Autom. Control, AC-27, pp. 1169–1175, Dec. 1982.

    Google Scholar 

  17. J.P. LaSalle and S. Lefschetz, Stability by Liapunov’s Direct Method, Academic Press, New York, 1961.

    Google Scholar 

  18. W. Hahn, Theory and Application of Liapunov’s Direct Method, Prentice-Hall International Series in Applied Mathematics, Englewood Cliffs, New Jersey, 1963.

    Google Scholar 

  19. N.N. Bogoliuboff and Yu. A. Mitropolskii, Asymptotic Methods in the Theory of Nonlinear Oscillators, Gordon Breach, New York, 1961.

    Google Scholar 

  20. J.K. Hale, Ordinary Differential Equations, Kreiger, Molaban, Florida, 1980.

    MATH  Google Scholar 

  21. V.I. Arnold, Geometric Methods in the Theory of Differential Equations, Springer-Verlag, New York, 1982.

    Google Scholar 

  22. L.-C. Fu, M. Bodson and S.S. Sastry, “New Stability Theorems for Averaging and Their Applications to the Convergence Analysis of Adaptive Identification and Control Schemes,” Tech. Report UCB/ERL M85/21, UC at Berkeley, March 1985.

    Google Scholar 

  23. D. Hill and P. Moylan, “Connection Between Finite-Gain and Asymptotic Stability,” IEEE Transactions on Automatic Control, vol. AC-25, pp. 931–936, Oct. 1980.

    Google Scholar 

  24. C.A. Desoer, R. Liu and L.V. Auth, Jr., “Linearity and Nonlinearity and Asymptotic Stability in the Large,” IEEE Transactions on Circuit Theory, pp. 117–118, March 1965.

    Google Scholar 

  25. C.A. Desoer and M. Vidyasagar, Feedback Systems: Input-Output Properties, Academic Press, 1985.

    Google Scholar 

  26. R.L. Kosut and C.R. Johnson, Jr., “An Input-Output View of Robustness in Adaptive Control,” Special Issue on Adaptive Control, Automatica, September 1984.

    Google Scholar 

  27. R.L. Kosut and B.D.O. Anderson, “Robust Adaptive Control: Conditions for Local Stability,” Proceedings of the 23rd IEEE CDC, Las Vegas, 1984.

    Google Scholar 

  28. K.S. Narendra and L.S. Valavani, “Stable Adaptive Controller Design-Direct Control,” IEEE Trans. on Autom. Control, AC-23, pp. 570–583, Aug. 1978.

    Google Scholar 

  29. K.S. Narendra and P. Kudva, “Stable Adaptive Schemes for System Identification and Control–Parts IP’ IEEE Trans. on Systems, Man and Cybernetics, SMC-4, pp. 541–560, Nov. 1974.

    Google Scholar 

  30. K.S. Narendra and Y.H. Lin, “Design of Stable Model Reference Adaptive Controllers,” Applications of Adaptive Control, pp. 69–130, Academic Press, New York, 1980.

    Google Scholar 

  31. S. Dasgupta, B.D.O. Anderson and A.C. Tsoi, “Input Conditions for Continuous Time Adaptive Systems Problems,” Tech. Report, Australian National University, 1983.

    Google Scholar 

  32. K.S. Narendra and A.M. Annaswamy, “Persistent Excitation and Robust Adaptive Algorithms,” Proceedings of the Third Yale Workshop on Applications of Adaptive Systems Theory, Yale University, June 1983.

    Google Scholar 

  33. S. Boyd and S.S. Sastry, “On Parameter Convergence in Adaptive Control,” Systems and Control Letters“, vol. 3, pp. 311–319, 1983.

    MathSciNet  MATH  Google Scholar 

  34. B. Egardt, Stability of Adaptive Controllers, Springer- Verlag, Berlin, 1979.

    Book  MATH  Google Scholar 

  35. K.S. Narendra and B.B. Peterson, “Bounded Error Adaptive Control - Part I,” Tech. Report No. 8005, Center for Systems Science, Yale University, New Haven, Dec. 1980.

    Google Scholar 

  36. K.S. Narendra and A.M. Annaswamy, “Robust Adaptive Control in the Presence of Bounded Disturbances,” Tech. Report 8406, Center for Systems Science, Yale University, New Haven, Ct., 1984. (Also presented at the 1984 ACC conference, San Diego, 1984.); To appear in the IEEE Transactions on Automatic Control.

    Google Scholar 

  37. P.A. Ioannou and P.V. Kokotovic, “Adaptive Systems with Reduced Models,” Springer-Verlag, New York, 1983.

    Google Scholar 

  38. K.S. Narendra and A.M. Annaswamy, “A New Adaptive Law for Robust Adaptive Control Without Persistent Excitation,” Tech. Report No. 8510, Center for Systems Science, Yale University, New Haven, September 1985.

    Google Scholar 

  39. P.A. Ioannou and K. Tsakalis, “A Robust Direct Adaptive Controller,” Tech. Report 85–07–1, Univ. of Southern California, May 1985.

    Google Scholar 

  40. J.S.W. Wong, “Remarks on Global Asymptotic Stability of Certain Quasi-linear Differential Equations,” Proc. Amer. Math. Soc., vol. 17, No. 4, August 1966.

    Google Scholar 

  41. M. Bodson and S.S. Sastry, “Small Signal I/O Stability of Nonlinear Control Systems - Applications to the Robustness of a MRAC Scheme,” Tech. Report, UC at Berkeley, September 1985.

    Google Scholar 

  42. B.D.O. Anderson and R.M. Johnstone, “Adaptive Systems and Time-varying Plants,” International Journal of Control, vol. 37, No. 2, pp. 367–377, Feb. 1983.

    MathSciNet  MATH  Google Scholar 

  43. K.S. Narendra, I.H. Khalifa and A.M. Annaswamy, “Error Models for Stable Hybrid Adaptive Systems–Part II,” IEEE Transactions on Automatic Control, AC-30, No. 4, pp. 339–347, April 1985.

    Article  MathSciNet  Google Scholar 

  44. L. Praly, “Global Stability of a Direct Adaptive Control Scheme Robust with Respect to a Graph Topology,” this volume, Plenum Press, 1985.

    Google Scholar 

  45. G. Kreisselmeier and B.D.O. Anderson, “Robust Model Reference Adaptive Control,” DFVLR Tech. Report, Oberpfaffenhofen, W. Germany, 1984.

    Google Scholar 

  46. K.S. Narendra and A.M. Annaswamy, “Robust Adaptive Control Using a Reduced Order Model,” Tech. Report, New Haven, Ct. (In preparation.).

    Google Scholar 

  47. R.L. Kosut and B. Friedlander, “Robust Adaptive Control: Conditions for Global Stability,” IEEE Transactions on Automatic Control, vol. AC-30, pp. 610–624, April 1985.

    Google Scholar 

  48. K.J. Aström, “Analysis of Rohrs Counter-example to Adaptive Control,” Proceedings of the 22nd IEEE CDC, San Antonio, Texas, pp. 982–987, Dec. 1983.

    Google Scholar 

  49. P. Kokotovic, B. Riedle and L. Praly, “On a Stability Criterion for Continuous Slow Adaptation,” Systems éß Control Letters, vol. 6, pp. 7–14, June 1985.

    Article  MathSciNet  MATH  Google Scholar 

  50. R.L. Kosut, B.D.O. Anderson and I. Mareels, “Stability Theory for Adaptive Systems: Methods of Averaging and Persistent Excitation,” Preprint, 1985.

    Google Scholar 

  51. B.D.O. Anderson, R.R. Bitmead, C.R. Johnson, Jr., and R.L. Kosut, “Stability Theorems for the Relaxation of the Strictly Positive Real Condition in Hyperstable Adaptive Schemes,” Proceedings of the 23rd IEEE CDC, Las Vegas, 1984.

    Google Scholar 

  52. L. Praly, “Robustness of Model Reference Adaptive Control,” Proceedings of the Third Yale Workshop on Applications of Adaptive Systems Theory, Yale University, New Haven, Ct., 1983.

    Google Scholar 

  53. H. Elliott, R. Cristi and M. Das, “Global Stability of Adaptive Pole Placement Algorithms,” IEEE Transactions on Automatic Control, AC-30, pp. 348–356, April 1985.

    Google Scholar 

  54. G.C. Goodwin and E.K. Teoh, “Persistency of Excitation in the Presence of Possibly Unbounded Signals,” IEEE Transactions on Automatic Control, AC-30, pp. 595–597, June 1985.

    Google Scholar 

  55. A.S. Morse, “Recent Problems in Parameter Adaptive Control,” Proc. CNRS colloquium on Development and Utilization of Mathematical Models in Automatic Control, Belle-Isle, France, Sept. 1982.

    Google Scholar 

  56. R.D. Nussbaum, “Some Remarks on a Conjecture in Parameter Adaptive Control,” Systems and Control Letters, vol. 3, pp. 243–246, 1983.

    Article  MathSciNet  MATH  Google Scholar 

  57. D.R. Mudgett and A.S. Morse, “Adaptive Stabilization of Linear Systems with Unknown High-Frequency Gains,” IEEE Transactions on Automatic Control, AC-30, pp. 549–554, June 1985.

    Google Scholar 

  58. T.H. Lee and K.S. Narendra, “Stable Discrete Adaptive Control with Unknown High Frequency Gain,” Tech. Report No. 8408, Center for Systems Science, Yale University, New Haven, Ct., Nov. 1984.( Also submitted to the IEEE Trans. on Autom. Control).

    Google Scholar 

  59. J. Willems and C. Byrnes, “Global Adaptive Stabilization in the Absence of Information on the Sign of the High Frequency Gain,” Proc. INRIA Conf. on Analysis and Optimization of Systems, Springer Lecture Notes in Control and Information Sciences, vol. 62, pp. 49–57, June 1984.

    Article  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Center for Systems Science, Yale University, USA

    Kumpati S. Narendra & Anuradha M. Annaswamy

Authors
  1. Kumpati S. Narendra
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Anuradha M. Annaswamy
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Center for Systems Science, Yale University, New Haven, Connecticut, USA

    Kumpati S. Narendra

Rights and permissions

Reprints and permissions

Copyright information

© 1986 Springer Science+Business Media New York

About this chapter

Cite this chapter

Narendra, K.S., Annaswamy, A.M. (1986). Robust Adaptive Control. In: Narendra, K.S. (eds) Adaptive and Learning Systems. Springer, Boston, MA. https://doi.org/10.1007/978-1-4757-1895-9_1

Download citation

  • DOI: https://doi.org/10.1007/978-1-4757-1895-9_1

  • Publisher Name: Springer, Boston, MA

  • Print ISBN: 978-1-4757-1897-3

  • Online ISBN: 978-1-4757-1895-9

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics

Search

Navigation