Page 135 - Mathematical Techniques of Fractional Order Systems
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Exact Solution of Linear Fractional Distributed Order Systems Chapter | 4 123
1
0.9 a = 0.4
0.8
a = 0.8
0.7
a = 1.2
0.6
a = 1.4
0.5
0.4
0.3
0.2
0.1
0
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8
t
FIGURE 4.3 Solution of (4.82) with x 0ðÞ 5 1, λ 5 8 and the weight functions
α
½
w αðÞ 5 a ; αA 0; 1 for aA 0:4; 0:8; 1:2; 1:4f g.
6
a = 0.4
5
a = 0.8
4
a = 1.2
3
a = 1.4
2
1
0
–1
–2
0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1
t
FIGURE 4.4 The first elements of the solutions of (4.17) with x 0ðÞ 5 0, utðÞ 5 HtðÞ,
α
ðA; BÞ 5 ðA 1 ; B 1 Þ and the weight functions w αðÞ 5 a ; αA 0; 1 in which constants a are given in (4.96).
½
4.5 EXACT RESPONSE OF A DISTRIBUTED ORDER RC
ELECTRICAL CIRCUIT
In this section, we consider an RC electrical circuit with a distributed order
capacitor, and use Corollary 2 to describe its current step response in the
time domain. To this end, consider the voltage-driven RC circuit shown in
Fig. 4.5.
