Page 243 - Mathematical Techniques of Fractional Order Systems

232 Mathematical Techniques of Fractional Order Systems

where
1 ð t q21
q
ðI fÞðtÞ 5 ðt2sÞ fðsÞds:
01
ΓðqÞ 0
Its Laplace transform is given by
n21
X 1
C
q
q
ðkÞ
Lf D fðtÞgðsÞ 5 s FðsÞ 2 f ð0 Þs q212k :
k50
In particular, if 0 , q # 1; then
q
C
1
q
Lf D fðtÞgðsÞ 5 s FðsÞ 2 fð0 Þs q21 ;
and, if 1 , q # 2; then
q
q
C
1
1
0
Lf D fðtÞgðsÞ 5 s FðsÞ 2 fð0 Þs q21 2 f ð0 Þs q22 :
For an n 3 n matrix A the Mittag Leffler function is defined by
N k
X A
E q;p ðAÞ 5 :
Γðkq 1 pÞ
k50
For positive q; p; its Laplace transform is given by
s q2p
p21 q
L t E q;p ð 6 At Þ ðsÞ 5 :
q
ðs I7AÞ
Consider the following linear fractional deterministic control system
C q C p
D xðtÞ 5 A D xðtÞ 1 BuðtÞ 1 fðtÞ; tAJ;
ð8:2Þ
xð0Þ 5 x 0 ; x ð0Þ 5 x ;
0
0
0
n
where p; q; x; u; A and B are defined as above, and f:J-R is a continuous
function. In order to find the solution of the above problem (8.2), take the
Laplace transform on both sides, one can get
q
x ð0Þ 2 As XðsÞ 1 As
s XðsÞ 2 s q21 xð0Þ 2 s q22 0 p p21 xð0Þ 5 BUðsÞ 1 FðsÞ:
Substituting the Laplace transformation of the Mittag Leffler function
and the Laplace convolution operator, one can obtain the solution of the
given problem as
xðtÞ 5 E q2p ðAt q2p Þx 0 2 At q2p E q2p;q2p11 ðAt q2p Þx 0 1 tE q2p;2 ðAt q2p Þx 0
0
ð t
1 ðt2sÞ q21 E q2p;q ðAðt2sÞ q2p Þ½BuðsÞ 1 fðsފds:
0
Similarly, it is easy to obtain the following solution of Eq. (8.1) as a
function x; defined on J with xð0Þ 5 x 0 ; x ð0Þ 5 x such that there exists
0
0
0
vðtÞAFðt; xðtÞÞ almost everywhere on J