Page 474 - Mathematical Techniques of Fractional Order Systems

460 Mathematical Techniques of Fractional Order Systems

This function results in control functions given by Eq. (15.21) to give
multiswitching synchronization with switch 2.

15.4.3 Case 3
For Switch 3 defined by Eq. (15.13), the time derivative of the errors is
given by
q
d e 31
5 y 2 1 u 1 ðtÞ
dt q
5 e 32 1 x 1 2 x 3 1 u 1 ðtÞ
q
d e 32 3
5 y 1 2 y 2 αy 2 1 fcosωt 1 u 2 2 x 2
1
dt q
5 e 31 2 αe 32 1 f 31 ðx; yÞ 1 u 2
where f 13 are nonliear terms in e 11 and e 12 given as
3
f 13 52 y 2 αx 1 1 fcosωt
1
Theorem 3: If the control function u 1 ðtÞ and u 2 ðtÞ are chosen such that

u 1 ðtÞ 52 x 1 1 x 3 1 k 1 e 31 1 e 32
ð15:25Þ
u 2 ðtÞ 52 f 13 ðx; yÞ 1 e 31 1 ðk 2 2 αÞe 32
then the drive system 15.9 will achieve multiswitching synchronization with
the response system 15.10
Proof: The controller is defined as

u 1 ðtÞ 5 V 31 ðtÞ 2 x 1 1 x 3
u 2 ðtÞ 52 f 13 ðx; yÞ 1 V 32
where V 31 ðtÞ and V 32 ðtÞ are virtual control functions to be determined.
Following the same procedure as in Switch 1, we have

V 31 ðtÞ e 31
5 C
V 32 ðtÞ e 32
Matrix C is chosen as

k 1 1
C 5 ð15:26Þ
1 ðk 2 2 αÞ
This yields
V 31 5 k 1 e 31 1 e 32 ð15:27Þ

V 32 5 e 31 1 ðk 2 2 αÞe 32 ð15:28Þ