Page 445 - Mathematical Techniques of Fractional Order Systems

430 Mathematical Techniques of Fractional Order Systems

in this experiment. The Chen Lee system was adopted to generate the
attractor used in AVA stenosis detection. The master and slave systems are
defined as:
a 0
0 1
0 1 2x 3 0 1
Dx 1 b x 1
x 3 0 C
B
5 ;
Dx 2 x 2
@ A B 1 C@ A ð14:58aÞ
@ 0 c A
x 2
2
Dx 3 x 3
a 0
0 1
0 1 2y 3 0 1 0 1
Dy 1 b y 1 u 1
y 3
B 0 C
5 1 :
Dy 2 1 y 2 u 2
@ A B C@ A @ A ð14:58bÞ
@ A
y 2 0 c
Dy 3 y 3 u 3
2
The error between the master and slave system was modeled using frac-
tional derivatives as (Chen et al., 2013a,b):
0 a 1
0 0 0 e 2 e 3 e α 1
Γð2 1 αÞ 1
B C
q B C 11α B Γð1 1 αÞ C
0 1 0 1
D e 1 B b C e 1 B C
q B 0 0 C B 2 11α C
D e 2
@ A @ e 11α A ðe 3 Þ e C;
Γð2 1 αÞ C 2 1 B 2
5 B
q B C 11α B C
D e 2 e
B c C 3 @ aΓð2 1 αÞ A
0 0
@ A
Γð2 1 αÞ 0
ð14:59Þ
where q 5 1 2 α and 0 , q , 1. The discrete version of error system is
denoted by : e 1 ½iŠ, e 2 ½iŠ and e 3 ½iŠ. The dynamic error equations in the discrete
case are defined by:
α
a 11α e 2 ½iŠe 3 ½iŠe ½iŠ
1
Φ 1 i ½Š 5 ðe 1 ½iŠÞ 2 ; ð14:60aÞ
Γð2 1 αÞ Γð1 1 αÞ
b ðe 3 ½iŠ
2
Φ 2 i ½Š 5 1 e 11α i ½Š; ð14:60bÞ
2
Γð2 1 αÞ aΓð2 1 αÞ
c
Φ 3 i ½Š 5 e 11α i ½Š: ð14:60cÞ
3
Γð2 1 αÞ
The performance index was defined as (Chen et al., 2013a,b):
q ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2
2
2
Ψ 5 φ 1 φ 1 φ ; ð14:61Þ
1 2 3
where φ 5 maxðΦ j ½iŠÞ, Φ j ½iŠAℝ n22 , j 5 1; 2; 3, and n is the number of
j
samples. This index was used to evaluate the degree of AVA stenosis and
surgical improvement rate (SIR) after the PTA treatment. The same proce-
dure was applied to AVS on 13 patients who also received PTA treatment