Page 108 - Mathematical Techniques of Fractional Order Systems

96 Mathematical Techniques of Fractional Order Systems

transfer function of either system at the roots of the denominator of the input
transform. To clarify this, consider two systems whose respective transfer
functions are G 1 ðwÞ 5 B 1 ðwÞ=A 1 ðwÞ and G 2 ðwÞ 5 B 2 ðwÞ=A 2 ðwÞ. Under
suitable co-primeness assumptions, their responses to UðwÞ 5 DðwÞ=CðwÞ can
be decomposed as
B 1 ðwÞ DðwÞ X A 1 ðwÞ X C 1 ðwÞ
Y 1 ðwÞ 5 5 1 ð3:56Þ
A 1 ðwÞ CðwÞ A 1 ðwÞ CðwÞ
and

B 2 ðwÞ DðwÞ X A 2 ðwÞ X C 2 ðwÞ
Y 2 ðwÞ 5 5 1 : ð3:57Þ
A 2 ðwÞ CðwÞ A 2 ðwÞ CðwÞ
ðwÞ 5 X C ðwÞ (equality of the input-dependent compo-
For X C 1 ðwÞ 5 X C 2
nents), from (3.56) and (3.57) we get
ðwÞCðwÞ 1 X C ðwÞA 1 ðwÞ; ð3:58Þ
B 1 ðwÞDðwÞ 5 X A 1
ðwÞCðwÞ 1 X C ðwÞA 2 ðwÞ; ð3:59Þ
B 2 ðwÞDðwÞ 5 X A 2
so that, at the roots of CðwÞ, i.e., for CðwÞ 5 0, we have
B 2 ðwÞ B 1 ðwÞ
G 2 ðwÞ 5 5 5 G 1 ðwÞ; ð3:60Þ
A 2 ðwÞ A 1 ðwÞ
which means that G 2 ðwÞ interpolates G 1 ðwÞ at the poles of UðwÞ. As is well
known, if G 2 ðwÞ and G 1 ðwÞ are realized in a unity-feedback fashion, this
means, in turn, that their forward paths include an internal model of the
(common) input (Francis and Wonham, 1976).
The previous considerations have obvious implications on the so-called
direct or analytic synthesis of control systems (Ferrante et al., 2000) whose
first step consists in choosing an overall, or total, or complementary sensitiv-
ity, system transfer function TðwÞ that satisfies the specifications, the next
step being its realization, possibly by means of a feedback structure with
controller G c ðwÞ and plant G p ðwÞ located in the forward path so that

G c ðwÞG p ðwÞ
TðwÞ 5 ð3:61Þ
1 1 G c ðwÞG p ðwÞ
and

1 TðwÞ
G c ðwÞ 5 : ð3:62Þ
G p ðwÞ 1 2 TðwÞ
In the case of fractional order systems, to profit by the efficient techni-
ques developed for integer order systems, the rational function TðwÞ will be
obtained, via (3.3), from an original fractional order transfer function. To
facilitate the synthesis procedure, it is convenient to choose the lcd of the
fractional powers in this function equal to the lcd of the powers in the