Optimal control for linear systems with retarded state and observation and quadratic cost

  • Elena M. Fernandez-Berdaguer
  • E. Bruce Lee
Session 7 Distributed Parameter Systems
Part of the Lecture Notes in Control and Information Sciences book series (LNCIS, volume 62)


Stabilization (feedback control) of linear finite dimensional systems has long been based on the use of input-output models and ideas associated with equivalent systems. When such ideas are applied to the linear hereditary systems or other infinite dimensional systems new results appear which are significant in terms of controller synthesis and analysis.

Recently the input-output model in the hereditary situation was generalized by admitting delays in the observation (or reconstruction) of state type data. Also considerations of various categories of equivalent linear hereditary systems under, for example, the action of the feedback group has been undertaken. Each of these studies has led to new insights into the formulation of optimal control questions and questions of stabilizability.

Here we extend previous results on optimal control (quadratic formulation) of the linear hereditary systems. The main contribution is the extension of the optimal control theory to the infinite horizon case (with delayed observation) where questions of stability with the optimal controller become a significant issue in optimal synthesis.

The setting is the standard semigroup formulation in the Hilbert space M2. The main result is a characterization of the optimal stabilizing feedback controller in terms of the solution of a certain system of operator Riccati equations in the delayed cost formulation.


Riccati Equation Admissible Control Quadratic Cost Controller Synthesis Linear Quadratic Optimal Control Problem 
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  1. 1.
    Beltrami & Wholers Distributions and the Boundary Values of Analytic Functions, Academic Press, New York and London, 1966.Google Scholar
  2. 2.
    C. Bernier and A. Manitius, "On semigroups in RnxLp corresponding to differential equations with delays", Canadian Journal of Mathematics, 30 No. 5 (1978), pp. 897–914.Google Scholar
  3. 3.
    A. V. Balakrishnan, Applied Functional Analysis, Springer-Verlag, New York, Heidelberg, Berlin, 1976.Google Scholar
  4. 4.
    R. F. Curtain and A. J. Pritchard, "The finite dimensional Riccati equation for systems defined by evolution operators", SIAM J. Control & Opt., 14(1976), pp. 951–983.Google Scholar
  5. 5.
    R. F. Curtain and A. J. Pritchard, "An abstract theory for unbounded control action for distributed parameter systems", SIAM J. Control & Opt., 15 (1977), pp. 566–611.Google Scholar
  6. 6.
    M. C. Delfour and S.K. Mitter, "Controllability, observability and optimal feedback control of affine hereditary differential systems", SIAM J. Control, 10, (1972), pp. 298–327.Google Scholar
  7. 7.
    M. C. Delfour and S. K. Mitter, "Hereditary differential systems with constant delays, I. general case". J. Differential Equations, 12 (1972), pp. 231–235.Google Scholar
  8. 8.
    M. C. Delfour and S. K. Mitter, "Hereditary differential systems with constant delays, II. A class of finite systems and the adjoint problem", J. Differential Equations, 18 (1975), pp. 18–28.Google Scholar
  9. 9.
    M. C. Delfour, C. McCalla and S. K. Mitter, "Stability and the infinitetime quadratic cost problem for linear hereditary differential systems", SIAM J. Control, 13 (1975), pp. 48–88.Google Scholar
  10. 10.
    M. C. Delfour, "State theory of linear hereditary differential systems", J. Math. Anal. Applic. 60 (1977) pp. 8–35.Google Scholar
  11. 11.
    M. C. Delfour, "The product space approach in the state space theory of linear time-invariant differential systems with delays in state and control variables", Report CRMA1013, Centre de Recherches de Mathematiques Appliquees. February 1981.Google Scholar
  12. 12.
    M. C. Delfour, "The linear quadratic optimal control problem with delays in state and control variables: A state space approach", CRMA 1012 Centre de Recherches de Mathematiques Appliquees. March 1981. (Revised as CRMA-118 Sept. 1982.)Google Scholar
  13. 13.
    R. Datko, "Extending a theorem of A.M. Lyapunov to Hilbert spaces", J. Math. Anal. Appl., 32 (1970), pp. 610–616.Google Scholar
  14. 14.
    R. E. Edwards, Functional Analysis, Holt, Rinehart and Winston, Inc. 1965.Google Scholar
  15. 15.
    J. S. Gibson "The Riccati Integral Equations for Optimal Control Problems on Hilbert Spaces", SIAM J. Control and Opt. Vol. 17, (1979)Google Scholar
  16. 16.
    — "Linear Quadratic Optimal Control of Hereditary Differential Systems: Infinite dimensional Riccati Equations and Numerical Approximations", Siam J. Control and Opt., 21 (1983), pp. 95–139.Google Scholar
  17. 17.
    J. Hale, Theory of Functional Differential Equations, Springer-Verlag, New York, 1977.Google Scholar
  18. 18.
    A. Ichikawa, "Evolution equations with delay", Control Theory Centre. Reprot, No. 52 (revised), University of Warwick, January 1977.Google Scholar
  19. 19.
    A. Ichikawa, "Optimal control and filtering of evolution with delay in control and observation", Control Theory Centre. Report, No. 53, June 1977.Google Scholar
  20. 20.
    E. W. Kamen, An operator theory of linear functional differential equations", Journal of Differential Equations, 27 (1978), pp. 274–297.Google Scholar
  21. 21.
    H. Koivo and E. B. Lee, "Controller synthesis for linear systems with retarded state and control variables and quadratic cost", Automatica 8, (1972), pp. 203–208.Google Scholar
  22. 22.
    E. B. Lee, "Generalized quadratic optimal controllers for linear hereditary systems", IEEE Trans. on Automat. Contr., AC-25 (1980), pp. 528–531.Google Scholar
  23. 23.
    E. B. Lee and S. H. Żak, "On spectrum placement for linear time invariant delay systems." IEEE Trans. on Automat. Contr. AC—27 (1982 pp. 446–449.Google Scholar
  24. 24.
    E. B. Lee and L. Markus, Foundations Optimal Control Theory, John Wiley and Sons, Inc. New York 1967.Google Scholar
  25. 25.
    J. L. Lions, Optimal Control of Systems governed by Partial Differential Equations, Springer-Verlag, New York, Heidelberg, Berlin, 1971.Google Scholar
  26. 26.
    A. Manitius and A. W. Olbrot, "Finite spectrum assignment problem for systems with delays," IEEE Trans. Automat. Contr., Vol. AC-24 (1979), pp. 541–553.Google Scholar
  27. 27.
    A. Manitius, "Optimal control of hereditary systems", In "Control Theory and Topics in Functional Analysis". Vol. II, pp. 43–178, International Atomic Energy Agency, Vienna 1976.Google Scholar
  28. 28.
    A. S. Morse, "Ring models for delay differential systems", Automatica, 12 (1976), pp. 529–531.Google Scholar
  29. 29.
    A. W. Olbrot, "Stabilizability, detectability and spectrum assignment for linear systems with general time delays", IEEE Trans. Automat. Contr. Vol. AC-23 (1978( pp. 887–890.Google Scholar
  30. 30.
    A. W. Olbrot, "On non-degeneracy and related problems for linear constant time-lag systems", Ricerche di Automatica, Vol. 3, No. 3 (1972), pp. 203–220.Google Scholar
  31. 31.
    V. N. Popov, "Pointwise degeneracy of linar, time-invariant, delay-differential equations", Journal of Differential equations, 11 (1972), pp. 541–561.Google Scholar
  32. 32.
    W. Rudin, Functional Analysis, McGraw-Hill Book Company, 1973.Google Scholar
  33. 33.
    R. B. Vinter and R. H. Kwong, "The infinite time quadratic control problem for linear systems with state and control delays: an evolution equation approach", SIAM J. Control and Opt. Vol 19 (1981) pp. 139–153.Google Scholar

Copyright information

© Springer-Verlag 1984

Authors and Affiliations

  • Elena M. Fernandez-Berdaguer
    • 1
  • E. Bruce Lee
    • 1
  1. 1.Department of Electrical EngineeringUniversity of MinnesotaMinneapolis

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