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Automated derivation of optimal regulators for nonlinear systems by symbolic manipulation of Poisson series

  • M. K. Özgören
  • R. W. Longman
Contributed Papers

Abstract

Concepts of programmability and compact programmability are defined relative to a class of modified Poisson series. Lie-series-based canonical perturbation methods from astrodynamics are applied to the Hamiltonian system boundary-value problem, and more usual methods are applied to the perturbed Hamilton-Jacobi-Bellman partial differential equation, in order to obtain a complete set of equations for the perturbed optimal feedback control law for both infinite-time and finite-time regulator problems. The relative advantages of each approach are evaluated. A major aim of the paper is to determine the largest class of perturbations, within the set of Poisson series, for which the equations can be derived on a computer by symbolic manipulation. The more general the class, the more accurate the perturbation solution can be, for a given order. The solutions developed are complete; all that remains is to program them in order to have computerized derivations of the optimal nonlinear feedback control laws.

Key Words

Optimal control theory perturbation methods Lie series Poisson series 

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References

  1. 1.
    Von Zeipel, H., Recherches sur le Mouvement des petites Planètes, Arkiv für Astronomi, Matematik, Fysik, Vol. 11, No. 1, 1916.Google Scholar
  2. 2.
    Hori, G.,Theory of General Perturbations with Unspecified Canonical Variables, Astronomical Society of Japan, Vol. 18, No. 4, 1966.Google Scholar
  3. 3.
    Deprit, A.,Canonical Transformations Depending on a Small Parameter, Celestial Mechanics, Vol. 1, pp. 12–30, 1969.Google Scholar
  4. 4.
    Kamel, A. A.,Expansion Formulas in Canonical Transformations Depending on a Small Parameter, Celestial Mechanics, Vol. 1, pp. 190–199, 1969.Google Scholar
  5. 5.
    Mersman, W. A.,A New Algorithm for the Lie Transformation, Celestial Mechanics, Vol. 3, pp. 81–89, 1970.Google Scholar
  6. 6.
    Kamel, A. A., andHassan, S. D.,A Perturbation Treatment for Optimal Slightly Nonlinear Systems with Linear Control and Quadratic Criteria, Journal of Optimization Theory and Applications, Vol. 11, No. 4, 1973.Google Scholar
  7. 7.
    Ozgoren, M. K., Longman, R. W., andCooper, C. A.,Application of Lie Transform Based Canonical Perturbation Methods to the Optimal Control of Bilinear Systems, Proceedings of the AAS/AIAA Astrodynamics Specialist Conference, Nassau, Bahamas, 1975.Google Scholar
  8. 8.
    Lukes, D. L.,Optimal Regulation of Nonlinear Dynamic Systems, SIAM Journal on Control, Vol. 7, No. 1, 1969.Google Scholar
  9. 9.
    Willemstein, A. P.,Optimal Regulation of Nonlinear Dynamical Systems on a Finite Interval, SIAM Journal on Control, Vol. 15, No. 6, 1977.Google Scholar
  10. 10.
    Buric, M. R.,Application of Multilinear Algebra in Optimal Regulation of Nonlinear Polynomial Systems, Proceedings of the 6th Annual Allerton Conference on Communication, Control, and Computing, University of Illinois Press, Urbana, Illinois, 1978.Google Scholar

Copyright information

© Plenum Publishing Corporation 1985

Authors and Affiliations

  • M. K. Özgören
    • 1
  • R. W. Longman
    • 2
  1. 1.Department of Mechanical EngineeringMiddle East Technical UniversityAnkaraTurkey
  2. 2.Department of Mechanical EngineeringColumbia UniversityNew York

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