Page 122 - Mathematical Techniques of Fractional Order Systems

110 Mathematical Techniques of Fractional Order Systems

By rewriting (4.24) in the form

ð
W 2lnsÞ XsðÞ 2 x 0ðÞ=s 5 AX sðÞ 1 BU sðÞ ð4:25Þ
and taking the inverse Laplace transform afterwards reveals the fact that
xtðÞ 5 x 1 tðÞ 1 x 2 tðÞ satisfies (4.17). Eq. (4.23) indicates that the solution of
distributed order systems in the Laplace domain may be obtained by simply
substituting the Laplace variable s in the solution of their corresponding inte-
α
ger order systems with W 2lnsÞ 5 Ð 1 w αðÞs dα. In the following Lemma
ð
0
the impulse response of distributed order integrator with a unitary weight
function is presented. This result will be used to derive the analytical solu-
tion of (4.17) later.
Lemma 3: Impulse response of distributed order integrator with the unitary
weight function w αðÞ 5 1 αA 0; 1ŠÞ is given by
½
ð
t
itðÞ 5 e E 1 tðÞ; t . 0 ð4:26Þ
in which E 1 tðÞ is the exponential integral function as defined in (4.1).

Proof: Noting the Laplace pair L t-s E 1 tðÞ 5 ln s 1 1Þ which is presented
ð

s
in Erdelyi et al. (1954, p. 178), and the fact that the Laplace transform of itðÞ
s dα 5
is given by IsðÞ 5 1= Ð 1 α lns in the case of a unitary weight function,
0 s 2 1
the proof is easily obtained via the frequency shift property of the Laplace
transform.
Lemma 4: Let lW 21 :ðÞ denote the lower branch of Lambert W function
defined by (4.2) and define

r 0 52 λlW 21 2e 2π21=λ =λ ð4:27Þ
1
for some λ . 21 .If s jj . r 0 then j lnsj , .
ð
lW 21 2e 2π21 Þ j s 2 1j λ
2π2 1 λ
Proof: Define g λðÞ 52 e λ . It can be easily verified that this function
has the minimum of 2e 2π21 occurring at λ 5 1. Therefore, the inequality
e 2π2 1 λ
2π21
2e #2 , 0 ð4:28Þ
λ
holds for λ . 0. This, would result in

2N , lW 21 2e 2π21=λ =λ # lW 21 2e 2π21 ð4:29Þ

as lW 21 :ðÞ is strictly decreasing. Therefore, considering the condition
λ . 21 2π21 from (4.27) and (4.28) one could write
lW 21 2eð Þ
1 , r 0 ,1 N ð4:30Þ