Reliable Computing

, Volume 6, Issue 3, pp 231–246 | Cite as

Composite Interval Control Systems: Some Strong Kharitonov-Like Properties

  • Long Wang

Abstract

The interval model plays an important role in parametric robust control. This paper deals with the strict positive realness of a class of composite interval control systems, and establishes a strong Kharitonov-like extreme point criterion. We also discuss the robust strict positive realness of the shifted interval plant-controller family, and improve some previous results in the literature. Furthermore, for the collection of Popov plots of an interval transfer function family, we prove that a large portion of its outer boundary comes from the sixteen Kharitonov transfer functions. Finally, we study the beta boundedness of interval systems. A strong Kharitonov-like vertex result is established, namely, that all members in the interval system family are beta bounded if and only if the sixteen Kharitonov critical vertices are beta bounded.

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References

  1. 1.
    Ackermann, J., Bartlett, A., Kaesbauer, D., Sienel, W., and Steinhauser, R.: Robust Control: Systems with Uncertain Physical Parameters, Springer-Verlag, London and Berlin, 1994.Google Scholar
  2. 2.
    Anderson, B. D. O., Dasgupta, S., Khargonekar, P., Kraus, F. J., and Mansour, M.: Robust Strict Positive Realness: Characterization and Construction, IEEE Trans. on Circuits and Systems 37(7) (1990), pp. 869-876.Google Scholar
  3. 3.
    Barmish, B. R.: New Tools for Robustness Analysis, in: Proc. of IEEE Conf. on Decision and Control, San Francisco, 1988, pp. 1-6.Google Scholar
  4. 4.
    Barmish, B. R. and Shi, Z.: Robust Stability of a Class of Polynomials with Coefficients Depending Multilinearly on Perturbations, IEEE Trans. on Automatic Control 35(9) (1990), pp. 1040-1043.Google Scholar
  5. 5.
    Barmish, B. R., Tempo, R., Hollot, C. V., and Kang, H.: Extreme Point Result for Robust Stability of a Diamond of Polynomials, IEEE Trans. on Automatic Control 37(9) (1992), pp. 1460-1462.Google Scholar
  6. 6.
    Bartlett, A. C., Hollot, C. V., and Huang, L.: Root Location of an Entire Polytope of Polynomials: It Suffices to Check the Edges, Mathematics of Control, Signal and Systems 1(1) (1988), pp. 61-71.Google Scholar
  7. 7.
    Bhattacharyya, S. P., Chapellat, H., and Keel, L. H.: Robust Control: The Parametric Approach, Prentice-Hall, Englewood Cliffs, 1995.Google Scholar
  8. 8.
    Chapellat, H. and Bhattacharyya, S. P.: A Generalization of Kharitonov's Theorem: Robust Stability of Interval Plants, IEEE Trans. on Automatic Control 34(3) (1989), pp. 306-311.Google Scholar
  9. 9.
    Chapellat, H., Dahleh, M., and Bhattacharyya, S. P.: On Robust Nonlinear Stability of Interval Control Systems, IEEE Trans. on Automatic Control 36(1) (1991), pp. 59-67.Google Scholar
  10. 10.
    Dahleh, M.: On Robust Popov Criterion for Interval Lur'e Systems, IEEE Trans. on Automatic Control 38(9) (1993), pp. 1400-1405.Google Scholar
  11. 11.
    Dasgupta, S.: A Kharitonov-Like Theorem for Systems under Nonlinear Passive Feedback, in: Proc. of IEEE Conf. on Decision and Control, Philadelphia, 1987, pp. 2062-2063.Google Scholar
  12. 12.
    Dasgupta, S.: Kharitonov's Theorem Revisited, Systems and Control Letters 11(4) (1988), pp. 381-384.Google Scholar
  13. 13.
    Dasgupta, S. and Bhagwat, A. S.: Conditions for Designing Strictly Positive Real Transfer Functions for Adaptive Output Error Identification, IEEE Trans. on Circuits and Systems 34(7) (1987), pp. 731-737.Google Scholar
  14. 14.
    Francis, B. A.: A Course in H-Infinity Control Theory, Springer-Verlag, Berlin, 1987.Google Scholar
  15. 15.
    Garloff, J. and Wagner, D. G.: Hadamard Products of Stable Polynomials Are Stable, Journal of Mathematical Analysis and Applications 202(8) (1996), pp. 797-809.Google Scholar
  16. 16.
    Garloff, J. and Zettler, M.: Robustness Analysis of Polynomials with Polynomial Parameter Dependency Using Bernstein Expansion, IEEE Trans. on Automatic Control 43(3) (1998), pp. 425-431.Google Scholar
  17. 17.
    Gupta, S.: Robust Stability Analysis Using LMTS: Beyond Small Gain and Passivity, International Journal of Robust and Nonlinear Control 6(10) (1996), pp. 953-968.Google Scholar
  18. 18.
    Hollot, C. V. and Tempo, R.: On the Nyquist Envelope of an Interval Plant Family, IEEE Trans. on Automatic Control 39(2) (1994), pp. 391-396.Google Scholar
  19. 19.
    Kharitonov, V. L.: Asymptotic Stability of an Equilibrium Position of a Family of Systems of Linear Differential Equations, Differential'nye Uravneniya 14(11) (1978), pp. 2086-2088.Google Scholar
  20. 20.
    Narendra, K. S. and Taylor, J. H.: Frequency Domain Criteria for Absolute Stability, Academic Press, New York, 1973.Google Scholar
  21. 21.
    Polyak, B. T. and Tsypkin, Y. Z.: Robust Absolute Stability of Continuous Systems, International Journal of Robust and Nonlinear Control 3(2) (1993), pp. 231-239.Google Scholar
  22. 22.
    Šiljak, D. D.: Parameter Space Methods for Robust Control Design: A Guided Tour, IEEE Trans. on Automatic Control 34(7) (1989), pp. 674-688.Google Scholar
  23. 23.
    Tsypkin, Y. Z. and Polyak, B. T.: Frequency Domain Criteria for l p Robust Stability of Continuous Linear Systems, IEEE Trans. on Automatic Control 36(12) (1991), pp. 1464-1469.Google Scholar
  24. 24.
    Vicino, A. and Tesi, A. Strict Positive Realness: New Results for Interval Plant-Controller Families, IEEE Trans. on Automatic Control 38(6) (1993), pp. 982-987.Google Scholar
  25. 25.
    Vidyasagar, M.: Nonlinear Systems Analysis, Prentice-Hall, Englewood Cliffs, 1978.Google Scholar
  26. 26.
    Wang, J. Z.: Controller Approximation and Model Reduction with Some Performance, PhD Thesis, Department of Mechanics and Engineering Science, Peking University, Beijing, 1998.Google Scholar
  27. 27.
    Wang, L. and Huang, L.: Extreme Point Results for Strict Positive Realness of Transfer Function Families, Systems Science and Mathematical Sciences 7(4) (1994), pp. 166-178.Google Scholar
  28. 28.
    Zadeh, L. A. and Desoer, C. A.: Linear System Theory: A State Space Approach, McGraw-Hill, New York, 1963.Google Scholar

Copyright information

© Kluwer Academic Publishers 2000

Authors and Affiliations

  • Long Wang
    • 1
  1. 1.Center for Systems and Control, Department of Mechanics and Engineering SciencePeking UniversityBeijingP.R.China

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