Page 605 - Mathematical Techniques of Fractional Order Systems

Control and Synchronization Chapter | 19 575

dynamics are written. Further, controller is designed which enables the states
of the slave system to follow the states of the master system. The master sys-
tem can be defined as,
q
D t x 1 52 x 2 2 ax 1
q
D t x 2 5 ax 3 2 ax 2 1 dx 1
q ð19:22Þ
D t x 3 5 cx 2 2 x 2 x 4 2 x 3
q
D t x 4 5 x 2 x 3 2 bx 4
Similarly, the slave system can be written as,
q
D t y 1 52 y 2 2 ay 1
q
D t y 2 5 ay 3 2 ay 2 1 dy 1
q ð19:23Þ
D t y 3 5 cy 2 2 y 2 y 4 2 y 3 1 u
q
D t y 4 5 y 2 y 3 2 by 4
For the errors defined as, e 1 5 y 1 2 x 1 , e 2 5 y 2 2 x 2 , e 3 5 y 3 2 x 3 , and
e 4 5 y 4 2 x 4 , the error dynamics are written as:

q 1
D t e 1 52 e 2 2 ae 1
q 2
D t e 2 5 ae 3 2 ae 2 1 de 1
ð19:24Þ
D t e 3 5 ce 2 2 e 2 e 4 2 e 3 2 e 2 x 4 2 x 2 e 4 1 u
q 3
q
D t e 4 5 e 2 e 3 2 be 4 1 x 3 e 2 1 x 2 e 3
With the transformation, e 1 5 z 1 , z 2 5 e 2 2 α 1 , where e 2 as the controller
and α 1 as the virtual controller for the first subsystem in (19.24), yields,
q
ð
D z 1 52 z 2 1 α 1 Þ 2 az 1 ð19:25Þ
Lyapunov function for the above dynamics can be written as,
1 2
V 1 5 z 1
2
q
ð
.D V 1 # z 1 ð2 z 2 1 α 1 Þ 2 az 1 Þ
For, α 1 5 0;
q 2
D V 1 #2 az 2 z 1 z 2
1
Next for the second subsystem, and using the transformation,
z 3 5 e 3 2 α 2 ,
q
D z 2 5 aðz 3 1 α 2 Þ 2 ae 2 1 de 1 ð19:26Þ
The new Lyapunov function can be written as,
1
V 2 5 V 1 1 z 2 2
2
q 2
.D V 2 #2 az 2 z 1 z 2 1 z 2 aðz 3 1 α 2 Þ 2 ae 2 1 de 1
1