Page 125 - Mathematical Techniques of Fractional Order Systems

Exact Solution of Linear Fractional Distributed Order Systems Chapter | 4 113

Taking the Laplace transform of Eq. (4.41) and substituting (4.47) in it
gives

21 21

1 lns lns lns

XsðÞ 5 I2A x 0ðÞ 1 I2A BsU sðÞ ð4:48Þ
s s21 ss 2 1Þ s21
ð
By a little manipulation of (4.48), we obtain
21 21
1 s 2 1 s21 s21

XsðÞ 5 I2A x 0ðÞ 1 I2A BU sðÞ ð4:49Þ
s lns lns lns
s 2 1
Left multiplying both sides of (4.49) by I 2 A gives
lns
s 2 1 1 s 2 1

I 2 A XsðÞ 5 x 0ðÞ 1 BU sðÞ ð4:50Þ
lns s lns
Thereby

s 2 1
XsðÞ 2 x 0ðÞ=s 5 AX sðÞ 1 BU sðÞ ð4:51Þ
lns
Taking the inverse Laplace transform of (4.51) results in Eq. (4.17) which
proves this Lemma.
It is worth mentioning that the initial impulse resulted from the calcula-
0
tion of the derivative u tðÞ when u 0ðÞ 6¼ 0in (4.45) is taken into account in
the solution.
In order to derive φ tðÞ in a simpler form, define
1
ð t
τ

k
τ
p k tðÞ 5 e E ðÞdτ ð4:52Þ
1
0
By means of (4.52), it is possible to write φ tðÞ as
1
1N
X k
φ tðÞ 5 A p k tðÞ ð4:53Þ
1
k50
Also, from (4.43) it is revealed that
1N
X k21
φ tðÞ 5 A p k tðÞ ð4:54Þ
2
k21
It is observed from (4.53) and (4.54) that obtaining closed form expres-
sions of φ tðÞ and φ tðÞ is contingent on the calculation of function p k tðÞ.In
1 2
the following Lemma, a new Laplace pair is introduced which will be uti-
lized to provide a method for calculation of p k tðÞ in Lemma 7.