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H∞Control of a Class of Infinite-Dimensional Linear Systems with nonlinear Outputs

  • Mingqing Xiao
  • Tamer Başar
Conference paper
Part of the Annals of the International Society of Dynamic Games book series (AISDG, volume 5)

Abstract

For a class of infinite-dimensional linear systems with nonlinear outputs, and using a differential game-theoretic approach, we obtain a set of necessary and sufficient conditions for the existence of a state-feedback controller under which a given H bound (on disturbance attenuation) is achieved. Characterization of such a controller is given, and the result is applied to a disturbance attenuation problem with control constraints.

Keywords

Mild Solution Differential Game Maximal Monotone Infinitesimal Generator Control Constraint 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. [1]
    Adams, R. A. Sobolev Space. Academic Press, New York 1975.Google Scholar
  2. [2]
    Aubin, J. P. An Introduction to Nonlinear Analysis. Springer-Verlag, New York, 1993.MATHGoogle Scholar
  3. [3]
    Ball, J. A. and J. W. Helton. Viscosity Solutions of Hamilton—Jacobi Equations Arising in Nonlinear H Control. Preprint, 1994.Google Scholar
  4. [4]
    Barbu, V. The H -Problem with Control Constraints. SIAM J.Control and Optimization, 32, 952–964, 1994.MathSciNetMATHCrossRefGoogle Scholar
  5. [5]
    Barbu, V. H Boundary Control with State-Feedback: The Hyperbolic Case. SIAM J.Control and Optimization, 33, 684–701, 1995.MathSciNetMATHCrossRefGoogle Scholar
  6. [6]
    Barbu, V. and Th. Precupanu. Convexity and Optimization in Banach Space. Reidel, Dordrecht, 1986.Google Scholar
  7. [7]
    Barbu, V. and G. Da Prato. Hamilton—Jacobi Equations in Hilbert Spaces. Reidel, Dordrecht, 1983.MATHGoogle Scholar
  8. [8]
    Başar, T. and P. Bernhard. H -Optimal Control and Related Minimax Design Problems. Birkhäuser, Boston, 2nd ed, 1995.Google Scholar
  9. [9]
    Bensoussan, A. and P. Bernhard. On the Standard Problem of H -Optimal Control for Infinite-Dimensional Systems. Proceedings of the Conference on Control and Identification of Partial Differential Equation, South Hadlley, MA, 117–140, 1992.Google Scholar
  10. [10]
    Bensoussan, A., G. D. Prato, M. C. Delfour, and S. K. Mitter. Representation and Control of Infinite-Dimensional Systems, Volume II. Birkhäuser, Boston, 1992.MATHGoogle Scholar
  11. [11]
    Crandall, M. C., H. Ishii, and P. L. Lions. User’s Guide to Viscosity Solutions of Second Order Partial Differential Equations. Bulletin American Mathematical Society, 27, 1–67, 1992.MathSciNetMATHCrossRefGoogle Scholar
  12. [12]
    Crandall, M. C. and P. L. Lions. Viscosity Solutions of Hamilton-Jacobi Equations. Transactions of the American Mathematical Society, 277, 1–42,1983.MathSciNetMATHCrossRefGoogle Scholar
  13. [13]
    Curtain, R. F. and H. J. Zwart. An Introduction to Infinite-Dimensional Linear Systems Theory. Springer-Verlag, New York, 1995.MATHCrossRefGoogle Scholar
  14. [14]
    Doyle, J., K. Glover, P. Khargonekar, and B. Francis. State-Space Solutions to Standard H2 and H -Control Problem. IEEE Transactions on Automatic Control, 34, 831–847, 1989.MathSciNetMATHCrossRefGoogle Scholar
  15. [15]
    Evans, L. C. and P. E. Souganidis. Differential Games and Representation Formulas for Solutions of Hamilton—Jacobi Equations. Indiana University Mathematics Journal, 33, 773–797,1984.MathSciNetMATHCrossRefGoogle Scholar
  16. [16]
    Fleming, W. H. and H. M. Soner. Controlled Markov Processes and Viscosity Solutions. Springer-Verlag, New York, 1993.MATHGoogle Scholar
  17. [17]
    Francis, B. A. A Course in H -Control Theory. Springer-Verlag, New York, 1987.CrossRefGoogle Scholar
  18. [18]
    Ichikawa, A. Differential games and H -problems. Presented at the MTNS, Kobe, Japan, 1991.Google Scholar
  19. [19]
    Keulen, B., M. Peters, and R. Curtain. H -Control with State-Feedback: The Infinite-Dimensional Case. Journal of Mathematical Systems, Estimation, and Control, 3, 1–39,1993MathSciNetMATHGoogle Scholar
  20. [20]
    Khargonekar, P. P., I. R. Petersen, and M. A. Rotea. H-Optimal Control with State Feedback. IEEE Transactions on Automatic Control, 33, 786–788, 1988.MathSciNetMATHCrossRefGoogle Scholar
  21. [21]
    Lions, J. L. Optimal Control of Systems Governed by Partial Differential Equations. Springer-Verlag, Berlin, 1971.MATHCrossRefGoogle Scholar
  22. [22]
    Lions, J. L. Generalized Solutions of Hamilton—Jacobi Equations. Pitman, Boston, 1982.MATHGoogle Scholar
  23. [23]
    McMillan, C. and R. Triggiani. Min-Max Game Theory and Algebraic Riccati Equations for Boundary Control Problems with Continuous Input-Solution Map. Part II: the General Case. Applied Mathematics Optimization, 29, 1–65, 1994.MathSciNetMATHCrossRefGoogle Scholar
  24. [24]
    McMillan, C. and R. Triggiani. Min-Max Game Theory and Algebraic Riccati Equations for Boundary Control Problems with Analytic Semigroups: The Stable Case. Preprint, 1995.Google Scholar
  25. [25]
    McMillan, C. and R. Triggiani. Algebratic Riccati Equations Arising in Game Theory and in H -Control Problems for a Class of Abstract Systems. Differential Equations with Applications to Mathematical Physics, pp. 239–247,1993.Google Scholar
  26. [26]
    Pazy, A. Semigroup of Linear Operators and Applications to Partial Differential Equations. Springer-Verlag, New York, 1983.CrossRefGoogle Scholar
  27. [27]
    Petersen, I. R. Disturbance attenuation and H -optimization: A Design Method Based on the Algebraic Riccati Equation. IEEE Transactions on Automatic Control, 32,427–129,1987.MATHCrossRefGoogle Scholar
  28. [28]
    Pritchard, A. J. and S. Townley. Robustness Optimization for Abstract, Uncertain Control Systems: Unbounded Inputs and Perturbations. Proceedings of IFAC Symposium on Distributed Parameter Systems (El Jai, Amouroux, eds.), pp. 117–121, 1990.Google Scholar
  29. [29]
    Scherer, G. H -Control by State-Feedback for Plants with Zeros on the Imaginary Axis. Preprint, 1992.Google Scholar
  30. [30]
    Soravia, P. H Control of Nonlinear Systems: Differential Games and Viscosity Solutions. SIAM Journal on Control and Optimization, 34, 1071–1097, 1996.MathSciNetMATHCrossRefGoogle Scholar
  31. [31]
    Stoorvogel, A. A. The H -Control Problem: A State-Space Approach. Prentice Hall, New York, 1992.Google Scholar
  32. [32]
    Tadmor, G. Worst-Case Design in the Time Domain. The Maximum Principle and the Standard H -Problem. MCSS, 3, 301–324, 1990.Google Scholar
  33. [33]
    Zames, G. Feedback and Optimal Sensitivity: Model Reference Transformation, Multiplicative Seminorms and Approximate Inverses. IEEE Transactions on Automatic Control AC-26, 301–320, 1981.MathSciNetMATHCrossRefGoogle Scholar
  34. [34]
    Xiao, M. and T. Başar. Solutions to Generalized Riccati Evolution Equations and H -Optimal Control Problems on Hilbert Spaces. Preprint, 1997.Google Scholar
  35. [35]
    Zhao, Y. The Global Attractor of Infinite-dimensional Dynamical Systems Governed by a Class of Nonlinear Parabolic Variational Inequalities and Associated Control Problems. Applicable Analysis, 54,163–180,1994.MathSciNetMATHCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2000

Authors and Affiliations

  • Mingqing Xiao
    • 1
  • Tamer Başar
    • 1
  1. 1.Coordinated Science LaboratoryUniversity of Illinois at Urbana-ChampaignUrbanaUSA

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