Page 124 - Mathematical Techniques of Fractional Order Systems

112 Mathematical Techniques of Fractional Order Systems

s dα results in
in which using the relation s 2 1 5 Ð 1 α
lns 0
1
ð
s dα . λ
α
ð4:40Þ

0
Lemma 4: helps us to start constructing the new representation of the
exact solution of (4.17) with the aid of the Laplace transform as shown in
the next Lemma.
Lemma 5: Consider distributed order system of Eq. (4.17) with a unitary
weight function, i.e., w αðÞ 5 1 αA 0; 1ŠÞ and assume :A: . 21 2π21 ,
ð
½
lW 21 2eð Þ
where ::: is any submultiplicative matrix norm satisfying the inequality
:A 1 A 2 : # :A 1 ::A 2 : for any square matrices A 1 and A 2 . The exact solution
of (4.17) is given by
ð t
xtðÞ 5 φ tðÞx 0ðÞ 1 φ t 2 τÞBu τðÞdτ ð4:41Þ
0
ð
1 2
0
in which
1N
ð t X
k τ

k
τ
φ tðÞ 5 A e E ðÞdτ ð4:42Þ
1
1
0 k50
1N
ð t X
k τ
φ tðÞ 5 A e E 1 ð k11Þ τ ðÞdτ ð4:43Þ
2
0 k50
0
u tðÞ 5 L 21 sU sðÞ ð4:44Þ
s-t
and E 1 tðÞ is the exponential integral function.
Proof: The Laplace transforms of (4.42) and (4.43) are given by
1N k 1N k
1 X lns lns X lns
Φ 1 sðÞ 5 A ; Φ 2 sðÞ 5 A ð4:45Þ
s s21 ss 2 1Þ s21
ð
k50 k50

Let r 0 52 :A:lW 21 2e 2π21=:A: =:A: in which lW 21 :ðÞ denote the
lower branch of the Lambert W function. According to Lemma 4, provided
that the condition s jj . r 0 is satisfied, the inequality j lnsj , 1 holds true.
j s 2 1j :A:
Therefore, the series involved in the both functions in (4.45) is convergent to
1N k 21
X lns lns
A 5 I2A ð4:46Þ
s21 s21
k50
in which I is the identity matrix of appropriate dimensions. Thus, functions
(4.45) could also be written in the following form
21 21
1 lns lns lns

Φ 1 sðÞ 5 I2A ; Φ 2 sðÞ 5 I2A ð4:47Þ
s s21 ss 2 1Þ s21
ð