Page 476 - Mathematical Techniques of Fractional Order Systems

462 Mathematical Techniques of Fractional Order Systems

We chose D as

1
k 1
D 5 ð15:30Þ
1 ðk 2 2 αÞ
Matrix D satisfies the condition 15.19 for k 1 ; k 2 . 0. Therefore, multi-
switching synchronization is achieved.
15.4.5 Case 5

For Switch 5 defined by Eq. (15.15), the time derivative of the errors is
given by
q
d e 51 3
5 y 2 1 β x 1 1 βx 2 1 βx 3 2 βx 1 u 1 ðtÞ
1
1
dt q
3
5 e 51 1 x 1 1 β x 1 1 βx 2 1 βx 3 2 βx 1 u 1 ðtÞ
1
1
5 e 51 1 f 51 ðx; yÞ 1 u 1
where f 51 is defined as:
f 51 5 x 1 1 β x 1 1 β x 2 1 β x 3 2 β x 3
1
3
2
4 1
q
d e 52 3
5 y 1 2 y 2 αy 2 1 fcosωt 1 u 2 ðtÞ 2 x 2
1
dt q
3
5 e 51 2 x 3 2 y 2 αx 1 1 fcosωt 2 x 2 2 αe 52 1 u 2
1
where f 52 are nonlinear terms in e 11 and e 12 given as
3
f 52 5 x 3 2 y 2 αx 1 1 fcosωt 2 x 2
1
q
d e 52
5 e 51 2 αe 52 1 f 52 ðx; yÞ 1 u 2
dt q
Theorem 5: If the control function u 1 ðtÞ and u 2 ðtÞ are chosen such that
u 1 ðtÞ 52 f 51 ðx; yÞ 1 k 1 e 51 1 e 52
ð15:31Þ
52 f 52 ðx; yÞ 1 e 51 1 ðk 2 2 αÞe 52
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 5.
In order to eliminate the nonlinear term in e 1 and e 2 , we define the active
control input u 1 ðtÞ and u 2 ðtÞ as

u 1 ðtÞ 52 f 51 ðx; yÞ 1 V 51 ðtÞ
u 2 ðtÞ 52 f 52 ðx; yÞ 1 V 52 ðtÞ