Page 110 - Mathematical Techniques of Fractional Order Systems

98 Mathematical Techniques of Fractional Order Systems

1 1
YðsÞ 52 1 ;
b
1 1 s 0:8 s 0:8
so that it tends quickly to the chosen reference input.
By the change of variable w 5 s , G p ðsÞ is transformed into
0:1 b
1
G p ðwÞ 5 ð3:66Þ
1 1 10w 8
and TðsÞ into
b
1
TðwÞ 5 ð3:67Þ
1 1 w 8
so that, according to (3.62), the controller transfer function in w-domain
turns out to be
1 1 10w 8
G c ðwÞ 5 ; ð3:68Þ
w 8
whence
1 1 10s 0:8
G c ðsÞ 5 : ð3:69Þ
b
s 0:8
As is expected, this controller contains an internal model of the input
transform (Francis and Wonham, 1976).

3.8 CONCLUSIONS
It has been shown that the forced response of a fractional order system to an
input belonging to a very numerous class can uniquely be decomposed into a
system component and an input component, as is the case also for integer
order systems. The first is characterized by the same modes as the system
and the second by the same modes as the input. Therefore, if the system is
asymptotically stable and the input is persistent, the input component corre-
sponds to the steady-state or asymptotic response to the selected input
whereas the system component corresponds to the transient response.
To ascertain whether the fractional order system is BIBO stable without
computing numerically the system poles, resort can be made to the
Routh Hurwitz criteria for complex polynomials which allow us to deter-
mine the root distribution with respect to any straight line of the complex
plane. On the basis of these criteria, simple stability and instability condi-
tions have been provided that partly extend previous results of the same kind
presented in the literature.
The response decomposition can be used in various contexts, ranging
from system approximation to system analysis and synthesis. In particular, it
has been applied to find a simplified model that retains the asymptotic
behavior of the original system in the response to characteristic inputs, a