Advertisement

Journal of Optimization Theory and Applications

, Volume 44, Issue 2, pp 333–362 | Cite as

Optimal feedback control of bilinear systems

  • E. P. Ryan
Contributed Papers

Abstract

Feedback synthesis of optimal constrained controls for single-input bilinear systems is considered. Quadratic cost functionals (with and without quadratic control penalization) are modified by the inclusion of additional nonnegative state penalizing functions in the respective cost integrands. The latter functions are chosen so as to regularize the problems, in the sense that feedback solutions of particularly simple form are obtained. Finite and infinite time horizon problem formulations are treated, and associated aspects of feedback stabilization of bilinear systems are discussed.

Key Words

Bilinear control systems optimal feedback control bangbang control saturating control stabilizing control 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Mohler, R. R., andRink, R. E.,Control with a Multiplicative Mode, Transactions of the ASME, Journal of Basic Engineering, Vol. 91D, pp. 201–206, 1969.Google Scholar
  2. 2.
    Mohler, R. R.,Bilinear Control Processes, Academic Press, New York, New York, 1973.Google Scholar
  3. 3.
    Bruni, C., DiPillo, G., andKoch, G.,Bilinear Systems: An Appealing Class of Nearly Linear Systems, IEEE Transactions on Automatic Control, Vol. AC-19, pp. 334–348, 1974.Google Scholar
  4. 4.
    Frankena, J. F., andSivan, R.,A Nonlinear Optimal Control Law for Linear Systems, International Journal of Control, Vol. 30, pp. 159–178, 1979.Google Scholar
  5. 5.
    Ryan, E. P.,Optimal Feedback Control of Saturating Systems, International Journal of Control, Vol. 35, pp. 521–534, 1982.Google Scholar
  6. 6.
    Denn, M. M.,The Optimality of an Easily Implementable Feedback Control System: An Inverse Problem in Optimal Control Theory, AIChE Journal, Vol. 13, pp. 926–931, 1967.Google Scholar
  7. 7.
    Denn, M. M.,Optimization by Variational Methods, McGraw-Hill, New York, New York, 1969.Google Scholar
  8. 8.
    Clarke, F. H.,Optimization and Nonsmooth Analysis, John Wiley and Sons, New York, New York, 1983.Google Scholar
  9. 9.
    Brockett, R. W.,Finite-Dimensional Linear Systems, John Wiley and Sons, New York, New York, 1970.Google Scholar
  10. 10.
    Fillippov, A. F.,Application of the Theory of Differential Equations with Discontinuous Right-Hand Sides to Nonlinear Problems in Automatic Control, Proceedings of First IFAC Congress, Moscow, USSR, 1960; Butterworths, London, London, pp. 923–927, 1961.Google Scholar
  11. 11.
    Brunovský, P.,The Closed-Loop Time-Optimal Control, I: Optimality, SIAM Journal on Control and Optimization, Vol. 12, pp. 624–634, 1974.Google Scholar
  12. 12.
    Hájek, O.,Discontinuous Differential Equations, I, Journal of Differential Equations, Vol. 32, pp. 149–170, 1979.Google Scholar
  13. 13.
    Hájek, O.,Discontinuous Differential Equations, II, Journal of Differential Equations, Vol. 32, pp. 171–185, 1979.Google Scholar
  14. 14.
    LaSalle, J., andLefschetz, S.,Stability by Lyapunov's Direct Method with Applications, Academic Press, New York, New York, 1961.Google Scholar
  15. 15.
    LaSalle, J.,Stability Theory for Ordinary Differential Equations, Journal of Differential Equations, Vol. 4, pp. 57–65, 1968.Google Scholar
  16. 16.
    Slemrod, M.,Stabilization of Bilinear Control Systems with Applications to Nonconservative Problems in Elasticity, SIAM Journal on Control and Optimization, Vol. 16, pp. 131–141, 1978.Google Scholar
  17. 17.
    Ryan, E. P., andBuckingham, N. J.,On Asymptotically Stabilizing Control of Bilinear Systems, IEEE Transactions on Automatic Control, Vol. AC-28, pp. 863–864, 1984.Google Scholar
  18. 18.
    Quinn, J. P.,Stabilization of Bilinear Systems by Quadratic Feedback Controls, Journal of Mathematical Analysis and Applications, Vol. 75, pp. 66–80, 1980.Google Scholar
  19. 19.
    Longchamp, R.,Stable Feedback Control of Bilinear Systems, IEEE Transactions on Automatic Control, Vol. AC-25, pp. 302–306, 1980.Google Scholar
  20. 20.
    Gutman, P. O.,Stabilizing Controls for Bilinear Systems, IEEE Transactions on Automatic Control, Vol. AC-26, pp. 917–922, 1981.Google Scholar
  21. 21.
    Brockett, R. W.,Lie Algebras and Lie Groups in Control Theory, Geometric Methods in Systems Theory, Edited by D. Q. Mayne and R. W. Brockett, Reidel, Dordrecht, Holland, pp. 43–82, 1973.Google Scholar

Copyright information

© Plenum Publishing Corporation 1984

Authors and Affiliations

  • E. P. Ryan
    • 1
  1. 1.School of MathematicsUniversity of BathBathEngland

Personalised recommendations