Page 92 - Mathematical Techniques of Fractional Order Systems

80 Mathematical Techniques of Fractional Order Systems

and, borrowing the terminology adopted for integer order systems (Dorato
et al., 1994), will be called the system component and input component of
the output, respectively, because Y Σ ðwÞ is characterized by exactly the same
modes as the system (3.1) and Y U ðwÞ by exactly the same modes as the input
(3.2).

If the fractional order system is asymptotically stable (Petra ´ˇ s, 2009), so is
also the system component, and its time-domain counterpart, obtainable by
inverse Laplace transformation of the s-domain expression corresponding to
Y Σ ðwÞ via (3.3), tends asymptotically to zero. In this case, Y Σ ðwÞ can right-
fully be referred to as the transient response to input UðwÞ. Also, if the input
is persistent, then the time-domain counterpart of the input component Y U ðwÞ
also persists and can rightfully be referred to as the steady-state response or,
more in general, the asymptotic response.
Since very efficient and fast algorithms exist today to find the roots of a
polynomial (Akritas et al., 2008; Jenkins and Traub, 1970) and the related
computer programs are readily available, the easiest way to check the stabil-
ity of a fractional order system is probably to determine numerically the pre-
cise location of the roots of AðwÞ and see whether some of them lie in the
instability sector (3.10). Nevertheless, the problem of finding the root distri-
bution with respect to suitable contours (in particular, the perimeter of circu-
lar sectors with bounded radius, because upper bounds on the “size” of
polynomial roots can be determined easily (Hirst and Macey, 1997)) is cer-
tainly of interest for other purposes, such as root clustering or D-stability
analysis (see Yedavalli, 2014; Gutman and Jury, 1981 and bibliographies
therein), transient characterization, and stability margin evaluation. This
problem is discussed in the following section.

Remark 1: If, contrary to Assumption 1,AðwÞ and CðwÞ are not co-prime,
they may be factored as:

AðwÞ 5 AðwÞI A ðwÞ; CðwÞ 5 CðwÞI C ðwÞ; ð3:16Þ
where I A ðwÞ is the factor of AðwÞ containing all and only the roots of AðwÞ
that are roots of CðwÞ too (with their multiplicities), and I C ðwÞ is the factor
of CðwÞ containing all and only the roots of CðwÞ that are roots of AðwÞ too
(with their multiplicities). Clearly, if all of the common roots are simple
I A ðwÞ 5 I C ðwÞ. Let IðwÞ9I A ðwÞI C ðwÞ. Since the three pairs ½AðwÞ; Cðwފ,
½AðwÞ; Iðwފ and ½IðwÞ; Cðwފ are co-prime, YðwÞ can uniquely be expressed
(Ferrante et al., 2000)as
X ðwÞ X ðwÞ X I ðwÞ
YðwÞ 5 GðwÞUðwÞ 5 A 1 C 1 ; ð3:17Þ
AðwÞ CðwÞ IðwÞ