Page 120 - Mathematical Techniques of Fractional Order Systems
P. 120
108 Mathematical Techniques of Fractional Order Systems
gives
c wðαÞ
L t-s 0 D xðtÞ 5 Wð2 lnsÞðXðsÞ 2 xð0Þ=sÞ ð4:13Þ
t
where WsðÞ is the Laplace transform of ^ wtðÞ. Thus, the impulse response of
distributed order derivative operator (4.9) is given by
dtðÞ 5 L 21 W 2lnsÞ ; t . 0; ð4:14Þ
ð
s-t
in which L 21 : fg denotes the inverse Laplace transform operator. Now, let us
s-t
define the distributed order integrator as
J wðαÞ xtðÞ 5 xtðÞ itðÞ ð4:15Þ
0 t
in which
itðÞ 5 L 21 1=W 2lnsÞ ð4:16Þ
ð
s-t
and denotes the convolution operator. As it can be seen, the impulse
response of distributed order integrator given by (4.16) is the convolution
inverse of that of the corresponding distributed order differentiator (4.14).
By definition, a linear time-invariant system of differential equations of dis-
tributed order is represented by (Jiao et al., 2012, p. 11)
c wðαÞ
0 D t xðtÞ 5 Ax tðÞ 1 Bu tðÞ; ð4:17Þ
n
in which xtðÞAR is called the pseudo state and utðÞAR denotes the input
function of the system. Also, AAR n 3 n is a constant matrix called the
dynamic matrix and BAR n 3 1 is the input vector. The aim of this chapter is
to present the analytical solution of this system in the case that the associated
α
weight function is in exponential form, i.e., w αðÞ 5 ca ; αA 0; 1 in which
½
aAR . 0 and cAR 2 0fg.
Lemma 1: Let c be a real constant. Solution of (4.17) with the weight func-
tion w αðÞ 5 cw 1 αðÞ, dynamic matrix A 5 A 1 and input function utðÞ 5 u 1 tðÞ is
equal to the solution of (4.17) with the weight function w αðÞ 5 w 1 αðÞ,
dynamic matrix A 5 A 1 =c, and input function utðÞ 5 u 1 tðÞ=c.
Proof: This Lemma is immediately proved by inserting the definition of
c wðαÞ
operator D t from (4.9) in (4.17).
0
We conclude this section by recalling a Laplace pair which will be used
later to derive the main results in this chapter.
Lemma 2: (Erdelyi et al., 1954, p. 144): The following Laplace pair holds
true
t k21
e t 1
L t-s 5 k ð4:18Þ
ð k 2 1Þ! ð s21Þ
