×

On complementary root locus of biproper transfer functions. (English) Zbl 1183.93074

Summary: This paper addresses the root locus (locus of positive gain) and the complementary root locus (locus of negative gain) of biproper transfer functions (transfer functions with the same number of poles and zeros). It is shown that the root locus and complementary root locus of a biproper transfer function can be directly obtained from the plot of a suitable strictly proper transfer function (transfer function with more poles than zeros). There exists a lack of sources on the complementary root locus plots. The proposed procedure avoids the problems pointed out by Eydgahi and Ghavamzadeh, is a new method to plot complementary root locus of biproper transfer functions, and offers a better comprehension on this subject. It also extends to biproper open-loop transfer functions, previous results about the exact plot of the complementary root locus using only the well-known root locus rules.

MSC:

93B60 Eigenvalue problems
93B51 Design techniques (robust design, computer-aided design, etc.)
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] W. R. Evans, “Graphical analysis of control systems,” Transactions of the American Institute of Electrical Engineers, vol. 67, no. 1, pp. 547-551, 1948.
[2] W. R. Evans, “Control system synthesis by root locus method,” Transactions of the American Institute of Electrical Engineers, vol. 69, no. 1, pp. 66-69, 1950.
[3] F. Merrikh-Bayat and M. Afshar, “Extending the root-locus method to fractional-order systems,” Journal of Applied Mathematics, vol. 2008, Article ID 528934, 13 pages, 2008. · Zbl 1262.93010
[4] F. Merrikh-Bayat, M. Afshar, and M. Karimi-Ghartemani, “Extension of the root-locus method to a certain class of fractional-order systems,” ISA Transactions, vol. 48, no. 1, pp. 48-53, 2009.
[5] D. L. Spencer, L. Philipp, and B. Philipp, “Root loci design using Dickson’s technique,” IEEE Transactions on Education, vol. 44, no. 2, pp. 176-184, 2001.
[6] M. C. M. Teixeira, “Direct expressions for Ogata’s lead-lag design method using root locus,” IEEE Transactions on Education, vol. 37, no. 1, pp. 63-64, 1994.
[7] M. C. M. Teixeira and E. Assun, “On lag controllers: design and implementation,” IEEE Transactions on Education, vol. 45, no. 3, pp. 285-288, 2002.
[8] M. C. M. Teixeira, E. Assun, and M. R. Covacic, “Proportional controllers: direct method for stability analysis and MATLAB implementation,” IEEE Transactions on Education, vol. 50, no. 1, pp. 74-78, 2007.
[9] J. R. C. Piqueira and L. H. A. Monteiro, “All-pole phase-locked loops: calculating lock-in range by using Evan’s root-locus,” International Journal of Control, vol. 79, no. 7, pp. 822-829, 2006. · Zbl 1330.93111
[10] J. C. Basilio and S. R. Matos, “Design of PI and PID controllers with transient performance specification,” IEEE Transactions on Education, vol. 45, no. 4, pp. 364-370, 2002.
[11] V. A. Oliveira, L. V. Cossi, M. C. M. Teixeira, and A. M. F. Silva, “Synthesis of PID controllers for a class of time delay systems,” Automatica, vol. 45, no. 7, pp. 1778-1782, 2009. · Zbl 1184.93048
[12] V. A. Oliveira, M. C. M. Teixeira, and L. Cossi, “Stabilizing a class of time delay systems using the Hermite-Biehler theorem,” Linear Algebra and Its Applications, vol. 369, pp. 203-216, 2003. · Zbl 1109.93363
[13] K. S. Narendra, “Inverse root locus, reversed root locus or complementary root locus?” IRE Transactions on Automatic Control, vol. 26, no. 3, pp. 359-360, 1961.
[14] A. M. Eydgahi and M. Ghavamzadeh, “Complementary root locus revisited,” IEEE Transactions on Education, vol. 44, no. 2, pp. 137-143, 2001.
[15] C. T. Chen, Analog and Digital Control System Design, Saunders College, Orlando, Fla, USA, 1993.
[16] R. C. Dorf and R. H. Bishop, Modern Control Systems, Addison-Wesley, Reading, Mass, USA, 1998. · Zbl 0907.93001
[17] J. J. D’Azzo and C. H. Houpis, Linear Control System Analysis and Design: Conventional and Modern, McGraw-Hill, New York, NY, USA, 1995. · Zbl 0366.93001
[18] G. F. Franklin, J. D. Powell, and A. Emami-Naeini, Feedback Control of Dynamic Systems, Addison-Wesley, Reading, Mass, USA, 1994. · Zbl 0615.93001
[19] B. C. Kuo, Automatic Control Systems, Prentice-Hall, Englewood Cliffs, NJ, USA, 1995.
[20] K. Ogata, Modern Control Engineering, Prentice-Hall, Englewood Cliffs, NJ, USA, 1997. · Zbl 0756.93060
[21] J. Rowland, Linear Control Systems: Modeling, Analysis, and Design, John Wiley & Sons, New York, NY, USA, 1986. · Zbl 1037.93003
[22] M. C. M. Teixeira, E. Assun, and E. R. M. D. Machado, “A method for plotting the complementary root locus using the root-locus (positive gain) rules,” IEEE Transactions on Education, vol. 47, no. 3, pp. 405-409, 2004.
[23] L. H. A. Monteiro and J. J. Da Cruz, “Simple answers to usual questions about unusual forms of the Evans’ root locus plot,” Controle y Automa, vol. 19, no. 4, pp. 444-449, 2008.
[24] M. C. M. Teixeira, H. F. Marchesi, and E. Assun, “Signal-flow graphs: direct method of reduction and MATLAB implementation,” IEEE Transactions on Education, vol. 44, no. 2, pp. 185-190, 2001.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.