Page 375 - Mathematical Techniques of Fractional Order Systems

366 Mathematical Techniques of Fractional Order Systems

T
T
where e 1 5 ½e 11 ; e 12 ; e 13 Š and e 2 5 ½e 21 ; e 22 ; e 23 Š . In view of preposi-
tion 1 the control functions U 1 5 C 1 ½A 2 X 2 1 F 2 ðX 2 Þ 2 A 1 X 1 1 F 1 ðX 1 ފ 2
A 3 C 1 ðX 2 1 X 1 Þ 2 F 3 ðX 3 Þ 1 K 1 e 1 and U 2 5 C 2 ½B 2 Y 2 1 G 2 ðY 2 Þ 1 B 1 Y 1 1
G 1 ðY 1 ފ 2 B 3 C 2 ðY 2 1 Y 1 Þ 2 G 3 ðY 3 Þ 1 K 2 e 2 will be
2 3
2 3
a 21 ðx 22 2x 21 Þ1x 24 1a 11 ðx 12 2x 11 Þ1x 14 1a 31 ðx 21 1x 11 Þ2a 31 ðx 22 1x 12 Þ2a 31 e 12
u 11
5;
6 a 22 x 21 2x 21 x 23 1a 23 x 22 2x 11 x 13 1a 13 x 12 2a 33 ðx 22 1x 12 Þ1x 31 x 33 2ða 33 11Þe 12 7
4 5
U 1 5 u 12 54
x 21 x 22 2a 24 x 23 1x 11 x 12 2a 12 x 13 1x 22 x 23 1a 25 x 24 1x 11 x 13 1a 14 x 14
u 13
1a 32 ðx 23 1x 13 Þ1a 32 ðx 24 1x 14 Þ2x 31 x 32
2 3
b 21 ðy 22 2y 21 Þ1y 24 1b 11 y 11 2y 12 1b 31 ðy 21 1y 11 Þ2b 31 ðy 22 1y 12 Þ2y 32 y 33 2b 31 e 22
2 3
u 21 2
b 22 y 21 2y 22 2y 21 y 23 1y 11 2y 12 y 1b 33 ðy 21 1y 11 Þ1ðy 22 1y 12 Þ1y 31 y 33 2b 33 e 21 7 :
6 7
13
5
4
U 2 5 u 22 5 6
y 21 y 22 2b 23 y 23 2b 12 y 12 2b 13 y 13 2b 14 y 14 2y 22 y 23 1b 24 y 24 1y 13 1b 15 y 14
4 5
u 23
1b 32 ðy 23 1y 13 Þ1b 32 ðy 24 1y 14 Þ2y 31 y 32
The error systems are given in the form
α
d e 1
dt α 5 ðA 3 1 K 1 Þe 1
α ð12:23Þ
d e 2 5 ðB 3 1 K 2 Þe 2 :
dt α
The preposition 1 confirms that if we take gain matrices as
2 3 2 3
0 2a 31 0 0 2b 31 0
K 1 5 0 2a 33 2 1 0 , K 2 5 4 2b 33 0 0 , then the systems
5
5
4
0 0 0 0 0 0
(12.17) (12.22) will be dual combination synchronized.
Fig. 12.10A G represent the simulation results towards synchronization
of the considered chaotic systems considering the earlier values of the para-
meters of the systems. The initial conditions of master systems Lu, 4D
Integral order, Chen, Lorenz hyperchaotic system are taken as
ðx 11 ð0Þ; x 12 ð0Þ; x 13 ð0Þ; x 14 ð0ÞÞ 5 ð2 10; 2 14; 12; 10Þ, ðy 11 ð0Þ; y 12 ð0Þ; y 13 ð0Þ;
y 14 ð0ÞÞ 5 ð1:2; 0:6; 0:8; 0:5Þ, ðx 21 ð0Þ; x 22 ð0Þ; x 23 ð0Þ; x 24 ð0ÞÞ 5 ð2 1; 2 3; 2;
5Þ, ðy 21 ð0Þ; y 22 ð0Þ; y 23 ð0Þ; y 24 ð0ÞÞ 5 ð1:5; 3; 2 1; 3Þ and slave systems Lu,
Qi chaotic systems are taken as ðx 31 ð0Þ; x 32 ð0Þ; x 33 ð0ÞÞ 5 ð0:2; 0:5; 0:3Þ
and ðy 31 ð0Þ; y 32 ð0Þ; y 33 ð0ÞÞ 5 ð2 1; 2 1; 2 2Þ, respectively. Hence the ini-
tial condition of the error system will be ðe 11 ð0Þ; e 12 ð0Þ; e 13 ð0Þ;
e 21 ð0Þ; e 22 ð0Þ; e 23 ð0ÞÞ 5 ð11:20; 17:50; 2 28:70; 2 3:70; 2 4:60; 2 5:3Þ.
It is clear from the figures that the error vectors asymptotically converge to
zero as time becomes large which justifies the achievement of dual combina-
tion synchronization among the considered fractional order systems at order
of the derivative α 5 0:98.