Page 374 - Mathematical Techniques of Fractional Order Systems

Dual Combination Synchronization Scheme Chapter | 12 365

where u 21 ; u 22 ; u 23 are control functions. The phase portraits of system
(12.22) at α 5 0:98 are depicted through Fig. 12.2.
Now again comparing the systems (12.17) (12.22) with systems (12.1)
(12.6), we get

2 3 2 3 2 3
2a 11 a 11 0 1 0 b 11 21 0 0
0 0 1 0 0
6 7 6 7 6 7
6 a 13 0 7 6 2x 11 x 13 7 6 0 7
7 ;
A 1 5 6 7 ; F 1 ðX 1 Þ5 6 7 ; B 1 5 6
6 0 0 2a 12 7 6 7 6 0 2b 12 7
4 0 5 4 x 11 x 12 5 4 2b 13 b 14 5
0 0 0 a 14 x 11 x 13 0 0 1 b 15
2 3 2 3 2 3
0 2a 21 a 21 0 1 0
6 2 7 6 0 7 6 7
13 7 ;
6 2y 12 y 7 6 a 22 a 23 0 7 6 2x 21 x 23 7
G 1 ðY 1 Þ5 6 7 ; A 2 5 6 7 ; F 2 ðX 2 Þ5 6
0 0 0
6 7 6 7 6 7
4 5 4 2a 24 0 5 4 x 21 x 22 5
0 0 0 0 a 25 x 22 x 23
2 3 2 3
0 1 0
2b 21 b 21
2 3
2a 31 a 31 0
6 21 0 7 6 7
b 22
6 0 7 6 2y 21 y 23 7
6 0 0 5;
7
B 2 5 6 7 ; G 2 ðY 2 Þ5 6 7 ; A 3 54 a 33
6 0 0 2b 213 7 6 7
0 0 2a 32
4 0 5 4 y 21 y 22 5
0 0 0 b 24 2y 22 y 23
2 3 2 3
0 2b 31 b 31 0
6 7 6 21 0 5;
7
b 33
F 3 ðX 3 Þ54 2x 31 x 33 5; B 3 54
0 0
x 31 x 32 2b 32
2 3 2 3 2 3
y 32 y 33 u 11 u 21
6 7 6 7 6 7
G 3 ðY 3 Þ54 2y 31 y 33 5; U 1 5 u 12 5; U 2 5 u 22 5
4
4
y 31 y 32 u 13 u 23
where X 1 5 ½x 11 ; x 12 ; x 13 ; x 14 Š, Y 1 5 ½y 11 ; y 12 ; y 13 ; y 14 Š, X 2 5
½x 21 ; x 22 ; x 23 ; x 24 Š, Y 2 5 ½y 21 ; y 22 ; y 23 ; y 24 Š, X 3 5 ½x 31 ; x 32 ; x 33 Š,
T
Y 3 5 ½y 31 ; y 32 ; y 33 Š are the state vectors, U 1 5 ½u 11 ; u 12 ; u 13 Š and
T
U 2 5 ½u 21 ; u 22 ; u 23 Š are the controller to be designed later.
Now choosing the matrix as
2 3 2 3
1 0 0 0 1 0 0 0
C 1 5 0 1 0 0 5 ; C 2 5 0 1 0 0 5 ;
4
4
0 0 1 1 0 0 1 1
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 2 ðx 24 1 x 14 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 2 ðy 24 1 y 14 Þ;