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Revision on the Frequency Domain Conditions for Strict Positive Realness  

Moghaddam Mojtaba Hakimi (Department of Electrical Engineering, Ferdowsi University of Mashhad)
Khaloozadeh Hamid (Faculty of Electrical Engineering, K.N.Toosi University of Technology)
Publication Information
International Journal of Control, Automation, and Systems / v.5, no.1, 2007 , pp. 1-7 More about this Journal
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
Frequency domain analysis; maximum modulus principle; strict positive realness; taylor expansion of rational transfer function;
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1 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   DOI   ScienceOn
2 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   DOI   ScienceOn
3 R. V. Churchill and J. W. Brown, Complex Variables and Applications, 5th edition, McGraw-Hill, 1990
4 K. Narendra and J. Taylor, Frequency Domain Criteria for Absolute Stability, Academic, New York, 1973
5 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
6 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   DOI   ScienceOn
7 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   DOI   ScienceOn
8 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
9 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   DOI
10 W.-K. Chen, Passive and Active Filters, John Wiley & Sons, 1986
11 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   DOI   ScienceOn
12 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   DOI   ScienceOn
13 H. K. Khalil, Nonlinear Systems, 2nd edition, Prentice-Hall, 1996
14 P. A. Ioannou and J. Sun, Robust Adaptive Control, Prentice-Hall, Upper Saddle River, New Jersey, 1995
15 R. E. Kalman, 'When is a linear control system optimal,' Trans. ASME, Series D, vol. 86, no. 1, pp. 1-10, 1964   DOI
16 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   DOI
17 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   DOI   ScienceOn
18 P. Kokotovic and M. Arcak, 'Constructive nonlinear control: A historical perspective,' Automatica, vol. 37, pp. 637-662, 2001   DOI
19 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   DOI   ScienceOn
20 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   DOI   ScienceOn
21 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
22 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   DOI   ScienceOn
23 K. J. Astrom and B. Wittenmark, Adaptive Control, 2nd edition, Prentice Hall, 1996
24 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   DOI   ScienceOn
25 B. D. O. Anderson, 'A simplified viewpoint of hyperstability,' IEEE Trans. on Automatic Control, vol. 13, no. 3, pp. 292-294, June 1968   DOI
26 J. J. E. Slotine and W. Li, Applied Nonlinear Control, Prentice Hall, 1991
27 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   DOI   ScienceOn
28 L.-X. Wang, A Course in Fuzzy Systems and Control, Prentice-Hall International, Inc., 1997
29 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   DOI   ScienceOn
30 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   DOI   ScienceOn
31 G. Tao, Adaptive Control Design and Analysis, John Wiley & Sons, 2003