Page 368 - Mathematical Techniques of Fractional Order Systems

Dual Combination Synchronization Scheme Chapter | 12 359

T T
state vectors, U 1 5 ½u 11 ; u 12 ; u 13 ; u 14 Š and U 2 5 ½u 21 ; u 22 ; u 23 ; u 24 Š
are the controller to be designed later.
Now choosing the matrix as
2 3 2 3
100 100
7 ;
6 010 7 7 ; 6 010 7
C 1 5 6 C 2 5 6
001 5 001 5
4 4
001 001
the error functions e 1 5 X 3 2 C 1 ðX 2 1 X 1 Þ, e 2 5 Y 3 2 C 2 ðY 2 1 Y 1 Þ can be
obtained as
e 11 5 x 31 2 ðx 21 1 x 11 Þ;
e 12 5 x 32 2 ðx 22 1 x 12 Þ;
e 13 5 x 33 2 ðx 23 1 x 13 Þ;
e 14 5 x 34 2 ðx 23 1 x 13 Þ;
e 21 5 y 31 2 ðy 21 1 y 11 Þ;
e 22 5 y 32 2 ðy 22 1 y 12 Þ;
e 23 5 y 33 2 ðy 23 1 y 13 Þ;
e 24 5 y 34 2 ðy 23 1 y 13 Þ;
T T
where e 1 5 ½e 11 ; e 12 ; e 13 ; e 14 Š and e 2 5 ½e 21 ; e 22 ; e 23 ; e 24 Š . In view of
preposition 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
and U 2 5 C 2 ½B 2 Y 2 1 G 2 ðY 2 Þ 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
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
u 11 2a 21 x 21 1x 22 110x 22 x 23 1a 11 ðx 12 2x 11 Þ1a 31 ðx 21 1x 11 Þ2a 31 ðx 22 1x 12 Þ2x 23 2x 13 2a 31 e 12 2e 14
6 7 6 7
2x 21 20:4x 22 15x 21 x 23 2x 11 x 13 1a 13 x 12 2a 33 ðx 22 1x 12 Þ1x 31 x 33 2ða 33 11Þe 12
6 u 127 6 7
7 ;
U 1 5 6 7 5 6
6 7 6 7
a 22 x 23 25x 21 x 22 1x 11 x 12 2a 12 x 13 1a 32 ðx 23 1x 13 Þ2x 31 x 32
4 u 135 4 5
u 14 a 22 x 23 25x 21 x 22 1x 11 x 12 2a 12 x 13 2a 34 ðx 23 1x 13 Þ2x 31 x 33 2ða 34 11Þe 14
2 3 2 3
u 21 2y 21 2b 21 y 22 2y 23 y 22 1b 11 ðy 12 2y 11 Þ1y 12 y 13 2b 31 ðy 21 1y 11 Þ1y 22 1y 12 2ðb 31 11Þe 21 1e 22
6 7 6 2 7
2y 22 2b 22 y 21 2y 21 y 23 1b 13 y 11 2y 12 2y 11 y 13 2y 21 2y 11 1y 32 y 33 2e 21 2e 22
6 u 227 6 7
7 :
U 2 5 6 7 5 6
6 7 6 7
4 u 235 4 b 23 y 23 1y 21 y 22 112b 12 y 13 1y 11 y 12 1b 32 ðy 22 1y 12 Þ1ðb 33 1b 34 Þðy 23 1y 13 Þ1b 32 e 22 1b 34 e 24 5
u 24 b 23 y 23 1y 21 y 22 112b 12 y 13 1y 11 y 12 2ð11b 35 Þðy 23 1y 13 Þ2e 23 2ðb 35 11Þe 24
The error systems are given in the form
α
d e 1
dt α 5 ðA 3 1 K 1 Þe 1
α
d e 2 5 ðB 3 1 K 2 Þe 2 : ð12:16Þ
dt α
The preposition 1 confirms that if gain matrices are taken as K 1 5
2 3 2 3
0 2a 31 0 21 2b 31 2 1 1 0 0
6 0 2a 33 2 1 0 0 7 6 21 21 0 0 7
6 7 , K 2 5 6 7 ,
4 0 0 0 0 5 4 0 b 32 0 b 34 5
0 0 0 2a 34 2 1 0 0 21 2b 35 2 1
the systems (12.10) (12.15) realize the dual combination synchronization.
In order to simulate the systems, the earlier values of parameters of
the systems are considered and the initial conditions of master systems Lu, Qi,
Newton Leipnik, Volta’s chaotic System are taken as ðx 11 ð0Þ; x 12 ð0Þ;