Page 360 - Mathematical Techniques of Fractional Order Systems

Dual Combination Synchronization Scheme Chapter | 12 351

α
D Y 1 5 B 1 Y 1 1 G 1 ðY 1 Þ; ð12:2Þ
where A 1 AR m 3 m , B 1 AR m 3 m are the linear part of the systems,
T T
X 1 5 ½x 11 ; x 12 ; :::; x 1m Š and Y 1 5 ½y 11 ; y 12 ; :::; y 1m Š are the two state vector
m
m
m
of uncoupled master systems (12.1) and (12.2); F 1 :R -R and G 1 :R -R m
are the two known real vector valued functions.
Next, another two master systems are considered as
α
D X 2 5 A 2 X 2 1 F 2 ðX 2 Þ ð12:3Þ
α
D Y 2 5 B 2 Y 2 1 G 2 ðY 2 Þ; ð12:4Þ
where A 2 AR m 3 m , B 2 AR m 3 m are the linear part of the systems,
T
T
X 2 5 ½x 21 ; x 22 ; :::; x 2m Š and Y 2 5 ½y 21 ; y 22 ; :::; y 2m Š are the two state vector
m
m
m
of uncoupled master systems (12.3) and (12.4); F 2 :R -R and G 2 :R -R m
are the two known real vector valued functions.
Now, the corresponding two slave systems are considered as
α
D X 3 5 A 3 X 3 1 F 3 ðX 3 Þ 1 U 1 ð12:5Þ
α
D Y 3 5 B 3 Y 3 1 G 3 ðY 3 Þ 1 U 2 ; ð12:6Þ
T
T
where X 3 5 ½x 31 ; x 32 ; :::; x 3n Š and Y 3 5 ½y 31 ; y 32 ; :::; y 3n Š are the two state
vector of uncoupled master systems (12.5) and (12.6); A 3 AR n 3 n , B 3 AR n 3 n
n
n
n
n
are the linear part of the systems, F 3 :R -R and G 3 :R -R are the two
known real vector valued functions, U 1 and U 2 are the control functions to
be designed later.
The error function is defined as
e 5 Z 2 CðY 1 XÞ;
T T T T
where e 5 ½e 1 ; e 2 Š , X 5 ½X 1 ; Y 1 Š , Y 5 ½X 2 ; Y 2 Š , Z 5 ½X 3 ; Y 3 Š and

0
C 1
C 5 , then the error function will be
0 C 2
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
The error systems is obtained as
α
D e 1 5A 3 e 1 1A 3 C 1 ðX 2 1X 1 Þ1F 3 ðX 3 Þ2C 1 ½A 2 X 2 1F 2 ðX 2 Þ1A 1 X 1 1F 1 ðX 1 ފ1U 1
α
D e 2 5B 3 e 2 1B 3 C 2 ðY 2 1Y 1 Þ1G 3 ðY 3 Þ2C 2 ½B 2 Y 2 1G 2 ðY 2 Þ1B 1 Y 1 1G 1 ðY 1 ފ1U 2
ð12:7Þ

Theorem 1: (Wang et al., 2010) Consider an autonomous fractional order
linear system as