International Journal of Control, Automation, and Systems

Institute of Control, Robotics and Systems (제어로봇시스템학회)
권호 표지

Revision on the Frequency Domain Conditions for Strict Positive Realness

  • Published : 2007.02.28
Download PDF
⟨ Previous Next ⟩

Abstract

In this paper, the necessary and sufficient conditions for strict positive realness of the rational transfer functions directly from basic definitions in the frequency domain are studied. A new frequency domain approach is used to check if a rational transfer function is a strictly positive real or not. This approach is based on the Taylor expansion and the Maximum Modulus Principle which are the fundamental tools in the complex functions analysis. Four related common statements in the strict positive realness literature which is appeared in the control theory are discussed. The drawback of these common statements is analyzed through some counter examples. Moreover a new necessary condition for strict positive realness is obtained from high frequency behavior of the Nyquist diagram of the transfer function. Finally a more simplified and completed conditions for strict positive realness of single-input single-output linear time-invariant systems are presented based on the complex functions analysis approach.

Keywords

References

  1. K. Narendra and J. Taylor, Frequency Domain Criteria for Absolute Stability, Academic, New York, 1973
  2. W.-K. Chen, Passive and Active Filters, John Wiley & Sons, 1986
  3. B. D. O. Anderson, 'A simplified viewpoint of hyperstability,' IEEE Trans. on Automatic Control, vol. 13, no. 3, pp. 292-294, June 1968 https://doi.org/10.1109/TAC.1968.1098910
  4. K. J. Astrom and B. Wittenmark, Adaptive Control, 2nd edition, Prentice Hall, 1996
  5. J. J. E. Slotine and W. Li, Applied Nonlinear Control, Prentice Hall, 1991
  6. P. Ioannou and G. Tao, 'Frequency domain conditions for strictly positive real functions,' IEEE Trans. on Automatic Control, vol. AC-32, no. 1, pp. 53-54, Jan. 1987
  7. H. K. Khalil, Nonlinear Systems, 2nd edition, Prentice-Hall, 1996
  8. P. A. Ioannou and J. Sun, Robust Adaptive Control, Prentice-Hall, Upper Saddle River, New Jersey, 1995
  9. G. Tao, Adaptive Control Design and Analysis, John Wiley & Sons, 2003
  10. C. H. Huang, P. A. Ioannou, J. Margulies, and M. G. Safonov, 'Design of strictly positive real systems using constant output feedback,' IEEE Trans. on Automatic Control, vol. 44, no. 3, pp. 569-573, March 1999 https://doi.org/10.1109/9.751352
  11. P. Kokotovic and M. Arcak, 'Constructive nonlinear control: A historical perspective,' Automatica, vol. 37, pp. 637-662, 2001 https://doi.org/10.1016/S0005-1098(01)00002-4
  12. R. E. Kalman, 'When is a linear control system optimal,' Trans. ASME, Series D, vol. 86, no. 1, pp. 1-10, 1964 https://doi.org/10.1115/1.3687052
  13. I. Hodaka, N. Sakamoto, and M. Suzuki, 'New results for strict positive realness and feedback stability,' IEEE Trans. on Automatic Control, vol. 45, no. 4, pp. 813-819, April 2000 https://doi.org/10.1109/9.847128
  14. V. Chellaboina and W. M. Haddad, Exponentially Dissipative Nonlinear Dynamical Systems: A Nonlinear Extension of Strict Positive Realness, Hindawi Publishing Corporation, http://dx.doi.org/10.1155/S1024123X03202 015, pp. 25-45, 2003
  15. C. W. Wu, 'Synchronization in arrays of coupled nonlinear systems: Passivity, circle criterion, and observer design,' IEEE Trans. on Circuits and System, vol. 48, no. 10, pp. 1257-1261, October 2001 https://doi.org/10.1109/81.956024
  16. L. Wang and W. Yu, 'On Hurwitz stable polynomials and strictly positive real functions,' IEEE Trans. on Circuits and System, vol. 48, no. 1, pp. 127-128, January 2001 https://doi.org/10.1109/81.903198
  17. D. Henrion, 'Linear matrix inequality for robust strictly positive real design,' IEEE Trans. on Circuits and System, vol. 49, no. 7, pp. 1017-1020, July 2002 https://doi.org/10.1109/TCSI.2002.800838
  18. A. Betser and E. Zeheb, 'Design of robust strictly positive real transfer func-tions,' IEEE Trans. on Circuits and System, vol. 40, no. 9, pp. 573-580, September 1993 https://doi.org/10.1109/81.244906
  19. Y. K. Foo and Y. C. Soh, 'Strict positive realness of a family of polytopic plants,' IEEE Trans. on Automatic Control, vol. 38, no. 2, pp. 287-289, February 1993 https://doi.org/10.1109/9.250474
  20. H. J. Marquez and C. J. Damaren, 'On the design of strictly positive real transfer functions,' IEEE Trans. on Circuits and System, vol. 42, no. 4, pp. 214-218, April 1995 https://doi.org/10.1109/81.382475
  21. H. J. Marquez and P. Agathoklin, 'On the existence of robust strictly positive real rational functions,' IEEE Trans. on Circuits and System, vol. 45, no. 9, pp. 962-967, September 1998 https://doi.org/10.1109/81.721261
  22. L.-X. Wang, A Course in Fuzzy Systems and Control, Prentice-Hall International, Inc., 1997
  23. J. H. Taylor, 'Strictly positive-real functions and the Lefschetz-Kalman Yakubovich (LKY) Lemma,' IEEE Trans. on Circuits and System, vol. 21, no. 2, pp. 310-311, March 1974 https://doi.org/10.1109/TCS.1974.1083816
  24. G. Tao and P. A. Ioannou, 'Strictly positive real matrices and the Lefschetz-Kalman-Yakubovich Lemma,' IEEE Trans. on Automatic Control, vol. 33, no. 12, pp. 1183-1185, December 1988 https://doi.org/10.1109/9.14449
  25. R. Lozano and S. M. Joshi, 'Strictly positive real transfer functions revisited,' IEEE Trans. on Automatic Control, vol. 35, no. 11, pp. 1243-1245, November 1990 https://doi.org/10.1109/9.59811
  26. B. D. O. Anderson, M. Mansour, and F. J. Kraus, 'A new test for strict positive realness,' IEEE Trans. on Circuit and System, vol. 42, no. 4, pp. 226-229, April 1996
  27. Z. Bai and W. Freund, 'Eigenvalue based characterization and test for positive realness of scalar transfer functions,' IEEE Trans. on Automatic Control, vol. 45, no 12, pp. 2396-2402, December 2000 https://doi.org/10.1109/9.895582
  28. W. Gao and Y. Zhou, 'Eigenvalue based algorithms for testing positive realness of SISO systems,' IEEE Trans. on Automatic Control, vol. 48, no. 11, pp. 2051-2054, December 2003 https://doi.org/10.1109/TAC.2003.819302
  29. R. Shorten and C. King, 'Spectral conditions for positive realness of single-input single-output systems,' IEEE Trans. on Automatic Control, vol. 49, no. 10, pp. 1875-1877, October 2004 https://doi.org/10.1109/TAC.2004.835593
  30. R. V. Churchill and J. W. Brown, Complex Variables and Applications, 5th edition, McGraw-Hill, 1990
  31. J. T. Wen, 'Time domain and frequency domain conditions for strict positive realness,' IEEE Trans. on Automatic Control, vol. 33, no. 10, pp. 988-992, 1988 https://doi.org/10.1109/9.7263