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Adaptive control of systems containing uncertain functions and unknown functions with uncertain bounds

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Abstract

We consider dynamical systems containing uncertain elements due to imperfect knowledge about the model and the input. Since these uncertainties may result in unstable behavior, we seek controllers which guarantee that all possible responses of the system are uniformly bounded and approach a desired response. Toward that end, we present a class of adaptive controllers.

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References

  1. 1.

    Kushner, H. J.,On the Status of Optimal Control and Stability for Stochastic Systems, IEEE International Convention Rec., Vol. 14, pp. 143–151, 1966.

  2. 2.

    Astrom, K. J.,Introduction to Stochastic Control Theory, Academic Press, New York, New York, 1970.

  3. 3.

    Lur'e, A. I.,Some Nonlinear Problems in the Theory of Automatic Control (in Russian), Gostekhizdat, Moscow, 1951.

  4. 4.

    Letov, A. M.,Stability of Nonlinear Regulating Systems (in Russian), Izdatel'stvo Tekhnichesko-Teoreticheskoi Literatury, Moscow, 1955.

  5. 5.

    Kalman, R. E., andBertram, J. E.,Control System Analysis and Design via the Second Method of Lyapunov, I: Continuous-Time Systems, Journal of Basic Engineering, Vol. 82, pp. 371–393, 1960.

  6. 6.

    Rang, E. R.,Adaptive Controllers Derived by Stability Considerations, Minneapolis-Honeywell Regulator Company, Memorandum No. MR-7905, 1962.

  7. 7.

    Butchart, R. L., andShakcloth, B.,Synthesis of Model Reference Adaptive Control Systems by Liapunov's Second Method, Proceedings of the 2nd IFAC Symposium on Theory of Self-Adaptive Control Systems, Teddington, England, Plenum Press, pp. 145–152, 1966.

  8. 8.

    Parks, P. C.,Liapunov Redesign of Model Reference Adaptive Control Systems, IEEE Transactions on Automatic Control, Vol. 11, pp. 362–367, 1966.

  9. 9.

    La Salle, J. P., andLefschetz, S.,Stability by Liapunov's Direct Method with Applications, Academic Press, New York, New York, 1961.

  10. 10.

    Hahn, W.,Stability of Motion, Springer-Verlag, Berlin, Germany, 1967.

  11. 11.

    Cesari, L.,Asymptotic Behavior and Stability Problems in Ordinary Differential Equations, Springer-Verlag, New York, New York, 1971.

  12. 12.

    Corless, M., andLeitmann, G.,Adaptive Control for Uncertain Dynamical Systemsd, Mathematical Theory of Dynamical Systems and Microphysics, Edited by A. Blaquière and G. Leitmann, Academic Press, New York, New York, 1984.

  13. 13.

    Molander, P.,Stabilisation of Uncertain Systems, Lund Institute of Technology, Report No. LUTFD2/(TFRT-1020)/1-111, 1979.

  14. 14.

    Thorp, J. S., andBarmish, B. R.,On Guaranteed Stability of Uncertain Systems via Linear Control, Journal of Optimization Theory and Applications, Vol. 35, pp. 559–579, 1981.

  15. 15.

    Barmish, B. R., Corless, M., andLeitmann, G.,A New Class of Stabilizing Controllers for Uncertain Dynamical Systems, SIAM Journal of Control and Optimization, Vol. 21, pp. 246–255, 1983.

  16. 16.

    Johnson, C. D.,Optimal Control of the Linear Regulator with Constant Disturbances, IEEE Transactions on Automatic Control, Vol. AC-13, pp. 416–421, 1968.

  17. 17.

    Sobral, M., Jr., andStefanek, R. G.,Optimal Control of the Linear Regulator Subject to Disturbances, IEEE Transactions on Automatic Control, Vol. AC-15, pp. 498–500, 1970.

  18. 18.

    Bélanger, P. R.,Observation and Control of Linear Systems with Constant Disturbances, IEEE Transactions on Automatic Control, Vol. AC-15, pp. 695–696, 1970.

  19. 19.

    Davison, E. J., andSmith, H. W.,Pole Assignment in Linear Time-Invariant Systems with Constant Disturbances, Automatica, Vol. 7, pp. 489–498, 1971.

  20. 20.

    Davison, E. J., andGoldenberg, A.,Robust Control of a General Servomechanism Problem: The Servo Compensator, Automatica, Vol. 11, pp. 461–471, 1975.

  21. 21.

    Desoer, C. A., andWang, Y. T.,On the Minimum Order of a Robust Servocompensator, IEEE Transactions on Automatic Control, Vol. AC-23, pp. 70–73, 1978.

  22. 22.

    Grujić, Lj. T., andPorter, B.,Continuous-Time Tracking Systems Incorporating Lur'e Plants with Single Nonlinearities, International Journal of Systems Science, Vol. 11, pp. 177–189, 1980.

  23. 23.

    Porter, B., andGrujić, Lj. T.,Continuous-Time Tracking Systems Incorporating Lur'e Plants with Multiple Nonlinearities, International Journal of Systems Science, Vol. 11, pp. 837–840, 1980.

  24. 24.

    Winsor, C. A., andRoy, R. J.,Design of Model Reference Adaptive Control Systems by Liapunov's Second Method, IEEE Transactions on Automatic Control, Vol. AC-13, p. 204, 1968.

  25. 25.

    Lindorff, D. P., andCarroll, R. L.,Survey of Adaptive Control Using Liapunov Design, International Journal of Control, Vol. 18, pp. 897–914, 1973.

  26. 26.

    Landau, I. D.,A Survey of Model Reference Adaptive Techniques—Theory and Applications, Automatica, Vol. 10, pp. 353–379, 1974.

  27. 27.

    Monopoli, R. V.,Model Reference Adaptive Control with an Augmented Error Signal, IEEE Transactions on Automatic Control, Vol. AC-19, pp. 474–484, 1974.

  28. 28.

    Narendra, K. S., andValavani, L. S.,Stable Adaptive Controller Design—Direct Control, IEEE Transactions on Automatic Control, Vol. AC-23, pp. 570–583, 1978.

  29. 29.

    Narendra, K. S., andValavani, L. S.,A Comparison of Liapunov and Hyperstability Approaches to Adaptive Control of Continuous Systems, IEEE Transactions on Automatic Control, Vol. AC-25, pp. 243–247, 1980.

  30. 30.

    Narendra, K. S., Lin, Y. H., andValavani, L. S.,Stable Adaptive Controller Design, Part II: Proof of Stability, IEEE Transactions on Automatic Control, Vol. AC-25, pp. 440–448, 1980.

  31. 31.

    Morse, A. S.,Global Stability of Parameter-Adaptive Control Systems, IEEE Transactions on Automatic Control, Vol. AC-25, pp. 433–439, 1980.

  32. 32.

    Parks, P. C.,Stability and Convergence of Adaptive Controllers—Continuous Systems, IEEE Proceedings, Vol. 128, Part D, pp. 195–200, 1981.

  33. 33.

    Monopoli, R. V.,Engineering Aspects of Control System Design via the Direct Method of Liapunov, NASA, Report No. CR-654, 1966.

  34. 34.

    Gutman, S., andLeitmann, G.,Stabilizing Control for Linear Systems with Bounded Parameter and Input Uncertainty, Proceedings of the 7th IFIP Conference on Optimization Techniques, Nice, France; Springer-Verlag, Berlin, Germany, 1976.

  35. 35.

    Gutman, S., andLeitmann, G.,Stabilizing Feedback Control for Dynamical Systems with Bounded Uncertainty, Proceedings of the IEEE Conference on Decision and Control, Gainesville, Florida, 1976.

  36. 36.

    Gutman, S.,Uncertain Dynamical Systems—Lyapunov Min-Max Approach, IEEE Transactions on Automatic Control, Vol. 24, pp. 437–443, 1979.

  37. 37.

    Leitmann, G.,Guaranteed Asymptotic Stability for Some Linear Systems with Bounded Uncertainties, Journal of Dynamic Systems, Measurement, and Control, Vol. 101, pp. 212–216, 1979.

  38. 38.

    Leitmann, G.,Deterministic Control of Uncertain Systems, Astronautica Acta, Vol. 7, pp. 1457–1461, 1980.

  39. 39.

    Corless, M., andLeitmann, G.,Continuous State Feedback Guaranteeing Uniform Ultimate Boundedness for Uncertain Dynamic Systems, IEEE Transactions on Automatic Control, Vol. 26, pp. 1134–1144.

  40. 40.

    Leitmann, G.,On the Efficacy of Nonlinear Control in Uncertain Linear Systems, Journal of Dynamic Systems, Measurement, and Control, Vol. 103, pp. 95–102, 1981.

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Dedicated to L. Cesari

This paper is based on research supported by the National Science Foundation, Grant No. ECS-78-13931, and carried out, in part, while G. Leitmann was a recipient of a US Senior Scientist Award of the Alexander von Humboldt Foundation.

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Corless, M., Leitmann, G. Adaptive control of systems containing uncertain functions and unknown functions with uncertain bounds. J Optim Theory Appl 41, 155–168 (1983). https://doi.org/10.1007/BF00934441

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Key Words

  • Uncertain systems
  • deterministic control
  • adaptive control
  • guaranteed stability