Page 473 - Mathematical Techniques of Fractional Order Systems

Multiswitching Synchronization Chapter | 15 459

15.4.2 Case 2
For Switch 2 defined by Eq. (15.12), the time derivative of the errors is
given by
q
d e 21
5 y 2 2 x 2 1 u 1 ðtÞ
dt q
5 e 22 1 x 3 2 x 2 1 u 1 ðtÞ
q
d e 22 3 3
5 y 1 2 y 2 αy 2 1 fcosωt 1 u 2 ðtÞ 2 β x 1 1 β x 2 1 β x 3 2 β x
2
3
4 1
1
1
dt q
5 e 21 2 αe 22 2 αx 3 1 x 1 1 f 21 ðx; yÞ 1 u 2
where f 21 are nonliear terms in e 11 and e 12 given as
3
f 21 52 y 1 fcosωt 2 β x 1 1 β x 2 1 β x 3 2 β x 3
1 1 2 3 4 1
Theorem 2: If the control function u 1 ðtÞ and u 2 ðtÞ are chosen such that
u 1 ðtÞ 52 x 3 1 x 2 1 k 1 e 21 1 e 22
ð15:21Þ
u 2 ðtÞ 5 αx 3 2 x 1 2 f 21 ðx; yÞ 1 e 21 1 ðk 2 2 αe 22 Þ
then the drive system 15.9 will achieve multiswitching synchronization with
the response system 15.10

Proof: The method of active control is employed to prove theorem 2.
We redefine the controller in order to eliminate nonlinear terms in e 21
and e 22 , so,
u 1 ðtÞ 52 x 3 1 x 2 1 V 21 ðtÞ
u 2 ðtÞ 5 αx 3 2 x 1 2 f 21 ðx; yÞ 1 V 22 ðtÞ
where V 21 ðtÞ and V 22 ðtÞ are virtual control functions to be determined.
Following the same procedure as in Switch 1, we have

V 11 ðtÞ e 11
5 A
V 12 ðtÞ e 12
Matrix B is chosen as

k 1 1
B 5 ð15:22Þ
1 ðk 2 2 αÞ
This yields
V 21 5 k 1 e 21 1 e 22 ð15:23Þ

V 22 5 e 21 1 ðk 2 2 αÞe 22 ð15:24Þ