Stabilizing control for linear systems with bounded parameter and input uncertainty

  • S. Gutman
  • G. Leitmann
Optimal Control Stochastic
Part of the Lecture Notes in Computer Science book series (LNCS, volume 41)


We consider dynamical systems with norm-bounded uncertainty in (i) the system parameters (model uncertainty) or in (ii) the input (disturbance).

For case (i), the nominal (null uncertainty) system is linear with constant matrices. Such systems with norm-bounded control as well as with a control penalty are treated. However, in the former the treatment is restricted to single input systems in companion form, and in the latter to second order systems. For case (ii), the system is linear with time-varying matrices and norm-bounded control.

Using some results from the theories of differential games and general dynamical systems, we deduce feedback controls which render the origin uniformly asymptotically stable in the large for all admissible parameter uncertainties or input disturbances; these may be both time and state dependent.

The application of the theory is illustrated by examples.


Feedback Control Parameter Uncertainty Differential Game Input Disturbance Input Uncertainty 
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  1. 1.
    Ragade, R. K. and Sarma, I. G., A game-theoretic approach to optimal control in the presence of uncertainty, IEEE Trans. on A.C., Vol. AC-12, No. 4, 1967.Google Scholar
  2. 2.
    Sarma, I. G. and Ragade, R. K., Some considerations in formulating optimal control problems as differential games, Intl. J. Control, Vol 4, pp. 265f., 1966.Google Scholar
  3. 3.
    Bertsekas, D. P. and Rhodes, I. B., Sufficiently informative functions and the minmax feedback control of uncertain dynamic systems, IEEE Trans. on A.C.Google Scholar
  4. 4.
    Chang, S. S. L. and Peng, T. K. C., Adaptive guaranteed cost control of systems with uncertain parameters, IEEE Trans. on A.C., Vol. AC-17, No. 4, 1972.Google Scholar
  5. 5.
    Speyer, J. L. and Shaked, U., Minimax design for a class of linear quadratic problems with parameter uncertainty, IEEE Trans. on A.C., Vol. AC-19, No. 2, 1974.Google Scholar
  6. 6.
    Menga, G. and Dorato, P., Observer-feedback design for linear systems with large parameter uncertainty, IEEE Conference on Decision and Control, pp. 872f., Phoenix, 1974.Google Scholar
  7. 7.
    Davison, E. J., The output control of linear time invariant multivariable systems with unmeasurable arbitrary disturbances, IEEE Trans. on A.C., Vol. AC-17, No. 5, 1972.Google Scholar
  8. 8.
    Blaquière, A., Gérard, F. and Leitmann, G., Quantitative and Qualitative Games, Academic Press, N.Y., 1969.Google Scholar
  9. 9.
    Leitmann, G., Cooperative and Noncooperative Many Player Differential Games, CISM Monograph 190, Springer Verlag, Vienna, 1974.Google Scholar
  10. 10.
    Gutman, S., Differential Games and Asymptotic Behavior of Linear Dynamical Systems in the Presence of Bounded Uncertainty, Ph.D. dissertation, University of California, Berkeley, 1975.Google Scholar
  11. 11.
    Filippov, A. G., Application of the theory of differential equations with discontinuous right-hand sides to non-linear problems in automatic control, First IFAC Congress, pp. 923f., 1960.Google Scholar
  12. 12.
    André, J. and Seibert, P., Über stückweise lineare Differential-gleichungen, die bei Regelungsproblemen auftreten, I and II, Arch. Math., Vol 7, pp. 148f. and 157f., 1956.Google Scholar
  13. 13.
    André, J. and Seibert, P., After end-point motions of general discontinuous control systems and their stability properties, First IFAC Congress, pp. 919f., 1960.Google Scholar
  14. 14.
    Alimov, Y. I., On the application of Lyapunov's direct method to differential equations with ambiguous right sides, Automation and Remote Control, Vol. 22, No. 7, 1961Google Scholar
  15. 15.
    Roxin, E., On generalized dynamical systems defined by a contingent equation, J. Differential Equations, Vol. 1, pp. 188f., 1965.Google Scholar
  16. 16.
    Roxin, E., On asymptotic stability in control systems, Rend. Circ. Mat. di Palermo, Serie II, Tomo XV, 1966.Google Scholar
  17. 17.
    Kwakernaak, H. and Sivan, R., Linear Optimal Control Systems, Wiley-Interscience, N.Y., 1972.Google Scholar
  18. 18.
    Stalford, H. and Leitmann, G., Sufficient conditions for optimality in two-person zero-sum differential games with state and strategy constraints, J. Math. Analysis and Appl., Vol. 33, No. 3, 1971.Google Scholar
  19. 19.
    Stalford, H. and Leitmann, G., Sufficiency conditions for Nash equilibria in N-person differential games, in Topics in Differential Games, ed. A. Blaquiére, pp. 345f., North-Holland, Amsterdam, 1973.Google Scholar
  20. 20.
    Kalman, R. E. and Bertram, J. E., Control system analysis and design via the "second method" of Lyapunov I, J. Basic Engin., ASME Trans., Vol. 82, No. 2, 1960.Google Scholar
  21. 21.
    Gutman, S. and Leitmann, G., On class of linear differential games, J. of Optimization Theory and Application, to appear.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1976

Authors and Affiliations

  • S. Gutman
    • 1
  • G. Leitmann
    • 2
  1. 1.NASAAmes
  2. 2.University of CaliforniaBerkeley

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