Nonlinear H Control Theory: A Literature Survey

  • Joseph A. Ball
  • J. William Helton
Chapter
Part of the Progress in Systems and Control Theory book series (PSCT, volume 4)

Abstract

The central problem of H -control theory roughly is to optimize (by the choice of compensator in a standard feedback configuration) some worst case (i.e. infinity norm) measure of performance while maintaining stability. For the linear, time-invariant, finitedimensional case, rather complete state space solutions are now available, and work has begun on understanding less restrictive settings. A recent new development has been the establishment of a connection with differential games and the perception of the H -problem as formally the same as the earlier well established linear quadratic regulator problem, but with an indefinite performance objective. In this article we review the current state of the art for nonlinear systems. The main focus is on the approach through a global theory of nonlinear J-inner-outer factorization and nonlinear fractional transformations being developed by the authors. It turns out that the critical points arising naturally in this theory can also be interpreted as optimal strategies in a game-theoretic interpretation of the control problem.

Keywords

Manifold Stein Doyle 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [AD]
    V. Anantharam and C. A. Desoer, On the stabilization of nonlinear systems, IEEE Trans. Aut. Control AC-29 (1984), 569–573.Google Scholar
  2. [BC]
    J. A. Ball and N. Cohen, Sensitivity minimization in H norm: parameterization of all solutions, Int. J. Control 46 (1987), 785–816.CrossRefGoogle Scholar
  3. [BFHT1]
    J. A. Ball, C. Foias, J. W. Helton and A. Tannenbaum, On a local nonlinear commutant lifting theorem, Indiana Univ. Math. J. 36 (1987), 693–709.Google Scholar
  4. [BFHT2]
    J. A. Ball, C. Foias, J. W. Helton and A. Tannenbaum, A Poincare-Dulac approach to a nonlinear Beurling-Lax-Halmos theorem, J. Math. Anal. Appl. 139 (1989), 496–514.CrossRefGoogle Scholar
  5. [BGR]
    J. A. Ball, I. Gohberg and L. Rodman, Two-sided Nudelman interpolation problem for rational matrix functions, to appear in Marcel-Dekkar volume dedicated to Mischa Cotlar.Google Scholar
  6. [BH1]
    J. A. Ball and J. W. Helton, A Beurling-Lax theorem for the Lie group U(m, n)which contains most classical interpolation, J. Operator Theory 9 (1983), 107–142.Google Scholar
  7. [BH2]
    J. A. Ball and J. W. Helton, Sensitivity bandwidth optimization for nonlinear feedback systems, in Analysis and Control of Nonlinear Systems (ed. by C. I. Byrnes, C. F. Martin and R. E. Saeks ), North Holland (1988), pp. 123–129.Google Scholar
  8. [BH3]
    J. A. Ball and J. W. Helton, Well-posedness of nonlinear causal feedback systems, Proc. 26th IEEE Conf. on Decision and Control, Los Angeles (1987), pp. 152–154.Google Scholar
  9. [BH4]
    J. A. Ball and J. W. Helton, Shift invariant manifolds and nonlinear analytic function theory, Integral Equations and Operator Theory 11 (1988), 615–725.CrossRefGoogle Scholar
  10. [BH5]
    J. A. Ball and J. W. Helton, Factorization of nonlinear systems: toward a theory for nonlinear H control, Proc. 27th IEEE Conf. on Decision and Control, Austin (1988), 2376–2381.Google Scholar
  11. [BH6]
    J. A. Ball and J. W. Helton, Interpolation problems for null and pole structure of nonlinear systems, Proc. 27th IEEE Conf. on Decision and Control, Austin (1988), 14–19.Google Scholar
  12. [BH7]
    J. A. Ball and J. W. Helton, Interconnection of nonlinear causal systems, IEEE Trans. Aut. Control, to appear.Google Scholar
  13. [BH8]
    J. A. Ball and J. W. Helton, Inner-outer factorization of nonlinear operators, preprint.Google Scholar
  14. [BH9]
    J. A. Ball and J. W. Helton, H -control of nonlinear plants: connections with differential games, in preparation.Google Scholar
  15. [BHS]
    J. A. Ball, J. W. Helton and C. H. Sung, Nonlinear solutions of Nevanlinna-Pick interpolation problems, Mich. Math. J. 34 (1987), 375–389.Google Scholar
  16. [BR]
    J. A. Ball and A. C. M. Ran, Global inverse spectral problems for rational matrix functions, Linear Alg. Appl. 86 (1987), 237–282.Google Scholar
  17. [CdeF]
    G. Chen and R. J. P. deFigueiredo, On robust stabilization of nonlinear control systems, Systems & Control Letters 12 (1989), 373–379.CrossRefGoogle Scholar
  18. deFC] R. J. P. deFigueiredo and G. Chen, Optimal disturbance rejection for nonlinear control systems, IEEE Trans. Aut. Control, to appear.Google Scholar
  19. [D]
    J. C. Doyle, Lecture Notes for ONR ( Honeywell Workshop on Advances in Multivariable Control, Minneapolis, MN (1984).Google Scholar
  20. [DK]
    C. A. Desoer and M. G. Kabuli, Right factorization of a class of time varying nonlinear systems, IEEE Trans. Aut. Control 33 (1988), 755–757.CrossRefGoogle Scholar
  21. [FT]]
    C. Foias and A. Tannenbaum, Weighted optimization theory for nonlinear systems, SIAM J. Control and Opt. 27 (1989), 842–860.CrossRefGoogle Scholar
  22. [FT2]
    C. Foias and A. Tannenbaum, Iterative commutant lifting for systems with rational symbol, in The Gohberg Anniversary Collection Vol. II: Topics in Analysis and Operator Theory, OT41 Birkhauser, Basel, (1989), pp. 255–277.Google Scholar
  23. [Fr]
    B. Francis, A Course in H Control Theory, Springer-Verlag, Berlin-New York, 1987.CrossRefGoogle Scholar
  24. [FD]
    B. Francis and J. Doyle, Linear control theory with an Hoe optimality criterion, SIAM J. Control Opt. 25 (1987), 815–844.CrossRefGoogle Scholar
  25. [GGLD]
    M. Green, K. Glover, D. Limebeer and J. Doyle, A J-spectral factorization approach to H control, preprint.Google Scholar
  26. [G]
    K. Glover, All optimal Hankel norm approximations of linear multivariable systems and their Loe error bounds, Int. J. Control 39 (1984), 1115–1193.CrossRefGoogle Scholar
  27. [GD]
    K. Glover and J. C. Doyle, State-space formulas for stabilizing controllers, Systems and Control Letters 11 (1988), 167–172.CrossRefGoogle Scholar
  28. [HK]
    S. Hara and R. Kondo, Characterization and computation of Hoo-optimal controllers in the state space, Proc. 27th IEEE Conf. on Decision and Control, Austin (1988), 20–25.Google Scholar
  29. [H]
    J. Hammer, Fractional representations of nonlinear systems: a simplified approach, Int. J. Control 46 (1987), 455–472.CrossRefGoogle Scholar
  30. [Ki]
    H. Kimura, Conjugation, interpolation and model-matching in H , Int. J. Control (1989), 269–307.Google Scholar
  31. [KP]
    P. R. Khargonekar and K. R. Poola, Uniformly optimal control of linear time-invariant plants: nonlinear time-varying controllers, Systems Control Letters 6 (1986), 303–308.CrossRefGoogle Scholar
  32. [Kr]
    A. J. Krener, Nonlinear controller design via approximate normal forms, in Proc. IMA Conf. on Signal Processing, Minneapolis (1988), to appear.Google Scholar
  33. [LAKG]
    D. J. N. Limebeer, B. D. O. Anderson, P. P. Khargonekar and M. Green, A game theoretic approach to H control for time varying systems, preprint.Google Scholar
  34. [S]
    E. D. Sontag, Smooth stabilization implies coprime factorization, IEEE Trans. Aut. Control, 34 (1989), 435–443.CrossRefGoogle Scholar
  35. [T1]
    G. Tadmor, Worst case design in the time domain: the maximum principle and the standard H problem, preprint.Google Scholar
  36. [T2]
    G. Tadmor, The standard H problem and the maximum principle: the general linear case, preprint.Google Scholar
  37. [V]
    M. S. Verma, Coprime fractional representations and stability of nonlinear feedback systems, Int. J. Control, to appear.Google Scholar

Copyright information

© Birkhäuser Boston 1990

Authors and Affiliations

  • Joseph A. Ball
    • 1
  • J. William Helton
    • 2
  1. 1.Department of MathematicsVirginia TechBlacksburgUSA
  2. 2.Department of MathematicsUniversity of California at San DiegoLa JollaUSA

Personalised recommendations