Page 89 - Mathematical Techniques of Fractional Order Systems

Fractional Order System Chapter | 3 77

3.2 NOTATION AND PRELIMINARIES
The transfer function of a continuous-time LTI strictly-proper rational order
system can be written as

m m21 1
b m s 1 b m21 s q 1 ... 1 b 1 s 1 b 0
q
q
GðsÞ 5 n n21 1 ; ð3:1Þ
b
a n s 1 a n21 s q 1 ... 1 a 1 s 1 a 0
q
q
where q, m, n are positive integers, m , n, q $ 1 is the least common denom-
inator (lcd) of the (commensurate) fractional exponents of the Laplace vari-
able s. The numerator and denominator coefficients of (3.1) are assumed to
be real.
Consider now the class of inputs whose rational order Laplace transform
can be written as
k k21 1
d k s 1 d k21 s q 1 ... 1 d 1 s 1 d 0
q
q
UðsÞ 5 ‘ ‘21 1 ; ð3:2Þ
b
c ‘ s 1 c ‘21 s q 1 ... 1 c 1 s 1 c 0
q
q
where k and ‘ are positive integers, k , ‘, and the numerator and denomina-
tor coefficients are real. It follows that the lcd of the fractional exponents of
both (3.1) and (3.2) is q. This assumption is not much restrictive because it
is always possible to express the fractional powers of s in (3.1) and (3.2) in
terms of the same lcd, even if this lcd may be larger than the lcd of either
function. The class of inputs (3.2) is fairly numerous and includes all inputs
whose Laplace transform has only integer powers of s, such as the singular-
ity and harmonic inputs.
By the change of variable
1
w 5 s ; ð3:3Þ
q
functions (3.1) and (3.2) are transformed, respectively, into the following
strictly-proper rational functions of w:

BðwÞ DðwÞ
GðwÞ 5 ; UðwÞ 5 ; ð3:4Þ
AðwÞ CðwÞ
where

m
BðwÞ 5 b m w 1 b m21 w m21 1 ... 1 b 1 w 1 b 0 ; ð3:5Þ
n
AðwÞ 5 a n w 1 a n21 w n21 1 .. . 1 a 1 w 1 a 0 ; ð3:6Þ
k
DðwÞ 5 d k w 1 d k21 w k21 1 ... 1 d 1 w 1 d 0 ; ð3:7Þ
‘
CðwÞ 5 c ‘ w 1 c ‘21 w ‘21 1 ... 1 c 1 w 1 c 0 : ð3:8Þ