Skip to main content
SpringerLink
Log in

Nash games and mixed H 2/H ∞ control

  • Conference paper
  • First Online:
Robust Control

Part of the book series: Lecture Notes in Control and Information Sciences ((LNCIS,volume 183))

Abstract

The established theory of non-zero sum games is used to solve a mixed H 2/H ∞ control problem. Our idea is to use the two pay-off functions associated with a two player Nash game to represent the H 2 and H ∞ criteria separately. We treat the state feedback problem, and we find necessary and sufficient conditions for the existence of a solution. A full stability analysis is available in the infinite horizon case [13], and the resulting controller is a constant state feedback law which is characterised by the solution to a pair of cross-coupled Riccati differential equations.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. B. D. O. Anderson and J. B. Moore, “Optimal Control-Linear Quadratic Methods,” Prentice-Hall International, 1989

    Google Scholar 

  2. D. Arov and M. G. Krein, “On the evaluation of entropy functionals and their minima in generalized extension problems,” Acta Scienta Math., Vol.45, pp 33–50, 1983, (in Russian)

    MATH  MathSciNet  Google Scholar 

  3. T. Basar, “On the uniqueness of the Nash solution in linear-quadratic differential games,” Int. Journal of Game Theory, Vol. 5, pp 65–90, 1977

    Article  MathSciNet  Google Scholar 

  4. T. Basar and G. J. Olsder, “Dynamic noncooperative game theory,” Academic Press, New York, 1982

    MATH  Google Scholar 

  5. D. S. Bernstein and W. M. Haddad, “LQG control with an H ∞ performance bound: A Riccati equation approach,” IEEE Trans. Automat. Control, Vol. AC-34, No. 3, pp 293–305, 1989

    Article  MathSciNet  Google Scholar 

  6. J. C. Doyle, K. Zhou and B. Bodenheimer, “Optimal control with mixed H 2 and H ∞ performance objectives,” Proc. 1989 American Control Conf., Pittsburgh, Pennsylvania, pp 2065–2070, 1989

    Google Scholar 

  7. H. Dym, “J-contractive matrix functions, reproducing kernel Hilbert spaces and interpolation,” Vol. 71 of Regional Conference Series in Mathematics, Am. Math. Soc., 1989

    Google Scholar 

  8. H. Dym and I. Gohberg, “A maximum entropy principle for contractive interpolants,” J. Funct. Analysis, Vol.65, pp 83–125, 1986

    Article  MATH  MathSciNet  Google Scholar 

  9. K. Glover and D. Mustafa, “Derivation of the maximum entropy H ∞-controller and a state-space formula for its entropy,” Int. J. Control, Vol.50, pp 899–916, 1989

    Article  MATH  MathSciNet  Google Scholar 

  10. I. M. Jaimoukha and D. J. N. Limebeer, “State-space algorithm for the solution of the 2-block super-optimal distance problem,” IEEE Conference on Decision and Control, Brighton, United Kingdom, 1991

    Google Scholar 

  11. H. Kwakernaak, “A polynomial approach to minimax frequency domain optimisation of multivariable feedback systems,” Int. J. Control, Vol.44, No.1, pp 117–156, 1986

    Article  MATH  MathSciNet  Google Scholar 

  12. H. Kwakernaak and R. Sivan, “Linear optimal control systems,” J. Wiley & son, 1972

    Google Scholar 

  13. D. J. N. Limebeer, B. D. O. Anderson and B. Hendel, “A Nash game approach to mixed H 2/H ∞ control,” submitted for publication

    Google Scholar 

  14. D. J. N. Limebeer, G. D. Halikias and K. Glover, “State-space algorithm for the computation of super-optimal matrix interpolating functions,” Int. J. Control, Vol.50, No.6, pp 2431–2466, 1989

    Article  MATH  MathSciNet  Google Scholar 

  15. D. J. N. Limebeer and Y. S. Hung, “An analysis of the pole-zero cancellations in H ∞-optimal control problems of the first kind,” SIAM J. Control Optimiz., Vol.25, pp 1457–1493, 1987

    Article  MATH  MathSciNet  Google Scholar 

  16. D. Mustafa, “Relations between maximum entropy H ∞ control and combined H ∞/LQG control,” Systems and Control Letters, Vol. 12, pp 193–203, 1989

    Article  MATH  MathSciNet  Google Scholar 

  17. D. Mustafa and K. Glover, “Minimum entropy H ∞ control,” Heidelberg, Springer-Verlag, 1990

    MATH  Google Scholar 

  18. D. Mustafa, K. Glover and D. J. N. Limebeer, “Solutions to the H ∞ general distance problem which minimize an entropy integral,” Automatica, Vol. 27, pp. 193–199, 1991

    Article  MATH  MathSciNet  Google Scholar 

  19. M. A. Rotea and P. P. Khargonekar, “H 2-optimal control with an H ∞-constraint—the state feedback case,” Internal Report, Univ. of Minnesota

    Google Scholar 

  20. A. W. Starr and Y. C. Ho, “Nonzero-sum differential games,” J. Optimization Theory and Applications, Vol.3, No. 3, pp 184–206, 1967

    Article  MathSciNet  Google Scholar 

  21. M. C. Tsai, D. W. Gu and I. Postlethwaite, “A state space approach to super-optimal H ∞ control problems,” IEEE Trans. Auto. Control, AC-33, No.9, 1988

    Google Scholar 

  22. N.J. Young, “The Nevanlinna-Pick problem for matrix-valued functions,” J. Operator Theory, Vol.15, pp239–265, 1986

    MATH  MathSciNet  Google Scholar 

  23. K. Zhou, J. C. Doyle, K. Glover and B. Bodenheimer, “Mixed H 2 and H 8 control,” Proc. 1990 American Control Conf., San Diego, pp 2502–2507, 1990

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Electrical Engineering, Imperial College, Exhibition Rd., London

    D. J. N. Limebeer & B. Hendel

  2. Department of Systems Engineering, Australian National University, Canberra, Australia

    B. D. O. Anderson

Authors
  1. D. J. N. Limebeer
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. B. D. O. Anderson
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. B. Hendel
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

L. D. Davisson A. G. J. MacFarlane H. Kwakernaak J. L. Massey Ya Z. Tsypkin A. J. Viterbi Shigeyuki Hosoe

Rights and permissions

Reprints and permissions

Copyright information

© 1992 Springer-Verlag

About this paper

Cite this paper

Limebeer, D.J.N., Anderson, B.D.O., Hendel, B. (1992). Nash games and mixed H 2/H ∞ control. In: Davisson, L.D., et al. Robust Control. Lecture Notes in Control and Information Sciences, vol 183. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0114668

Download citation

  • DOI: https://doi.org/10.1007/BFb0114668

  • Published:

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-55961-0

  • Online ISBN: 978-3-540-47320-6

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics

Search

Navigation