Page 342 - Mathematical Techniques of Fractional Order Systems

332 Mathematical Techniques of Fractional Order Systems

We define the active control functions ψ ðtÞ as follows :
i
8
ψ 5 V 1 1 ða 2 1 cÞy 1 2 a 2 y 2 1 dx 1 2 x 1 z 1 2 w 1
> 1
>
<
ψ 5 V 2 2 a 3 x 2 1 x 2 z 2 1 ð1 2 bÞz 1 2 w 2 1 x 1 y 1 ð11:21Þ
2
2
ψ 5 V 3 2 x 1 a 4 x 2 1 a 4 w 1 1 x 1 1 k
> 3 2
>
:
ψ 5 V 4 1 a 5 y 2 1 a 1 y 1 2 a 1 w 2
4
The terms V i ðtÞ are linear functions of the error term e i ðtÞ. With the
choice of ψ ðtÞ given by (11.21), the error system (11.20) becomes
i
q
8
d e 1
> 52 a 2 e 1 1 V 1 ðtÞ
>
dt
> q
>
>
>
>
q
>
d e 2
>
>
> 52 e 2 1 V 2 ðtÞ
>
dt
> q
<
q ð11:22Þ
d e 3
>
> 52 a 4 e 3 1 V 3 ðtÞ
> q
> dt
>
>
>
> q
d e 4
>
>
>
> 52 a 1 e 4 1 V 4 ðtÞ
dt
> q
:
The control terms V i ðtÞ are chosen so that the system (11.22) becomes
stable. There is not a unique choice for such functions.
We choose
2 3 2 3
V 1 e 1
V 2 e 2
6 7 6 7
5 A
6 7 6 7 ð11:23Þ
V 3 e 3
4 5 4 5
V 4 e 4
where A is a 4 3 4 real matrix, chosen so that all eigenvalues λ i of the
system (11.23) satisfy the condition
απ
jargðλ i Þj . ð11:24Þ
2
if we choose
2 3
a 2 2 61 0 0 0
0 21 0 0 7
A 5 6 7 ð11:25Þ
6
4 0 0 a 4 2 5 0 5
0 0 0 a 1 2 26
Then the eigenvalues of the linear system (11.23) are 1, 1, 1, and
1. Hence the condition (11.24) is satisfied for and we get the required syn-
chronization. Numerical simulations for switch 1 are performed to demonstrate
the theoretical results. The parameter values for which the system (11.14) and
(11.15) exhibits chaotic behavior are taken as a 5 36; b 5 3; c 5 28;
d 52 16; k 5 0:5; a 1 5 25; a 2 5 60; a 3 5 40; a 4 5 4; a 5 5 5. The fractional