Page 90 - Mathematical Techniques of Fractional Order Systems

78 Mathematical Techniques of Fractional Order Systems

Correspondingly, the Laplace transform YðsÞ of the forced response of
b
the system with transfer function (3.1) to the input with transform (3.2) is
converted into the following rational function of w:
BðwÞDðwÞ
YðwÞ 5 GðwÞUðwÞ 5 : ð3:9Þ
AðwÞCðwÞ
With some abuse of terminology, GðwÞ, UðwÞ, and YðwÞ will simply be
referred to as system, input, and output functions, respectively, because they
are directly related via (3.3) to GðsÞ, UðsÞ, and YðsÞ 5 GðsÞUðsÞ.
b
b
b
b
b
It has been long recognized that the denominator of the rational order
function (3.1) is a multivalued function of s which becomes a single-valued
function on a Riemann surface consisting of q sheets with branch cuts along
the negative real semi-axis. The first, or principal, sheet contains the physical
poles of (3.1) (Radwan et al., 2009) corresponding to the so-called structural,
or relevant, roots of its denominator (Petra ´ˇ s, 2009). The stability of the ratio-
nal order system depends on their location with respect to the imaginary
axis. The right half of the first sheet, corresponding to the unstable region,
maps into the (minor) sector of the w plane defined by
π π
jφ
S9 w 5 ρe :ρAR 1 ; φA 2 ; : ð3:10Þ
2q 2q
As is known, the time-domain expressions of the fractional order system
responses are easily obtained from the partial fraction expansions of the
w-domain expressions (Semary et al., 2016; Valerio et al., 2013). It has right-
fully been observed in this regard that the Mittag Leffler functions play for
fractional order systems a role analogous to that played by the exponential
modes characterizing the time-domain response of integer order systems
(Rivero et al., 2013; Trzaska, 2008).
The next section shows how the expression (3.9) of the forced-response
of a fractional order system can be separated into a component consisting of
the same modes as GðwÞ and a component consisting of the same modes as
UðwÞ.

3.3 DECOMPOSITION OF THE FORCED RESPONSE

The rational output (3.9) can be expanded into elementary partial fractions
corresponding to its poles which are poles of either GðwÞ or UðwÞ.If no
pole-zero cancellation occurs, all (and only) the poles of GðwÞ and UðwÞ are
poles of YðwÞ. However, if p is a pole of both GðwÞ and UðwÞ, the multiplic-
ity of pole p in YðwÞ is the sum of the multiplicities of the same pole in
GðwÞ and UðwÞ, so that at least one elementary fraction appears in the expan-
sion of YðwÞ that is not present in GðwÞ or UðwÞ. This typically happens in