Mathematical Problems in Engineering
Volume 2008 (2008), Article ID 419046, 13 pages
On the Essential Instabilities Caused by Fractional-Order Transfer Functions
Department of Electrical Engineering, Sharif University of Technology, Tehran, Iran
Received 4 May 2008; Accepted 10 September 2008
Academic Editor: Jerzy Warminski
Copyright © 2008 Farshad Merrikh-Bayat and Masoud Karimi-Ghartemani. 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 exact stability condition for certain class of
fractional-order (multivalued) transfer functions is presented.
Unlike the conventional case that the stability is directly studied
by investigating the poles of the transfer function, in the systems
under consideration, the branch points must also come into
account as another kind of singularities. It is shown that a
multivalued transfer function can behave unstably because of
the numerator term while it has no unstable poles. So, in this
case, not only the characteristic equation but the numerator
term is of significant importance. In this manner, a family of
unstable fractional-order transfer functions is introduced which
exhibit essential instabilities, that is, those which cannot be
removed by feedback. Two illustrative examples are presented;
the transfer function of which has no unstable poles but the
instability occurred because of the unstable branch points of
the numerator term. The effect of unstable branch points is
studied and simulations are presented.