Page 130 - Mathematical Techniques of Fractional Order Systems
P. 130
118 Mathematical Techniques of Fractional Order Systems
Proof: Let w 1 αðÞ 5 1; αA 0; 1 be a unitary weight function and
½
α
w 2 αðÞ 5 a ; αA 0; 1 be an exponential weight function. At first, we derive
½
the solution of (4.17) associated with the weight function w αðÞ 5 w 2 αðÞ and
then prove this corollary by using the mutual relationship which holds
between the two weight functions in Laplace domain. Considering
w αðÞ 5 w 2 αðÞ, taking the Laplace transform of (4.17) gives
x 0ðÞ
W 2 2lnsð ÞXsðÞ 2 W 2 2lnsÞ 5 AX sðÞ 1 BU sðÞ ð4:76Þ
ð
s
Solving (4.76) for XsðÞ results
ð
21 W 2 2lnsÞ 21
XsðÞ 5 W 2 2lnsðð ÞI2AÞ x 0ðÞ 1 W 2 2lnsðð ÞI2AÞ BU sðÞ ð4:77Þ
s
Note that the relation w 2 αðÞ 5 e αlna w 1 αðÞ holds between the two weight
functions. From the frequency shift property of the Laplace transform we
obtain
W 2 sðÞ 5 W 1 s 2 lnað Þ
ð4:78Þ
W 2 2lnsð Þ 5 W 1 2ln asðÞÞ
ð
Thereby, rewriting (4.77) in terms of W 1 gives
ð
21 W 1 2lnasÞ
XsðÞ 5 W 1 2lnasðð ÞI2AÞ x 0ðÞ 1
s ð4:79Þ
21
ð W 1 2lnasð ÞI2AÞ BU sðÞ
Replacing W 1 2lnasð Þ 5 as 2 1 in (4.79) and writing the resultant
lnas
expression in terms of Φ 1 asðÞ and Φ 2 asðÞ which are derived from (4.47)
yield
XsðÞ 5 aΦ 1 asðÞx 0ðÞ 1 aΦ 2 asðÞBsU sðÞ ð4:80Þ
By using the time scale property of the Laplace transform we obtain
0
xtðÞ 5 φ t=a x 0ðÞ 1 φ t=a TBu tðÞ ð4:81Þ
2
1
Note that the original weight function involves a scaling term, i.e.,
w αðÞ 5 cw 2 αðÞ. Substituting p k tðÞ obtained from Lemma 7 in (4.53) and
(4.54) gives φ tðÞ and φ tðÞ, respectively. Therefore, replacing φ t=a and
1
1
2
φ t=a with their expressions in (4.81) and using Lemma 1 afterwards, veri-
2
fies the solution (4.75).
Consider the scalar differential equation
c wðαÞ
0 D t xðtÞ 52 λxtðÞ; λ . 0 ð4:82Þ
which is in fact a special case of the more general system (4.17) and there-
fore its solution is readily available from Corollary 1 in case of an
