Journal of Applied Mathematics
Volume 2008 (2008), Article ID 528934, 13 pages
Research Article

Extending the Root-Locus Method to Fractional-Order Systems

Farshad Merrikh-Bayat1 and Mahdi Afshar2

1Department of Electrical Engineering, Zanjan University, Zanjan, Iran
2Department of Mathematics, Zanjan Azad University, Zanjan, Iran

Received 16 September 2007; Revised 11 March 2008; Accepted 14 May 2008

Academic Editor: Alberto Tesi

Copyright © 2008 Farshad Merrikh-Bayat and Mahdi Afshar. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


The well-known root-locus method is developed for special subset of linear time-invariant systems known as fractional-order systems. Transfer functions of these systems are rational functions with polynomials of rational powers of the Laplace variable s. Such systems are defined on a Riemann surface because of their multivalued nature. A set of rules for plotting the root loci on the first Riemann sheet is presented. The important features of the classical root-locus method such as asymptotes, roots condition on the real axis, and breakaway points are extended to fractional case. It is also shown that the proposed method can assess the closed-loop stability of fractional-order systems in the presence of a varying gain in the loop. Three illustrative examples are presented to confirm the effectiveness of the proposed algorithm.