Page 475 - Mathematical Techniques of Fractional Order Systems

Multiswitching Synchronization Chapter | 15 461

This function results in control functions given by Eq. (15.25) to give
multiswitching synchronization with switch 3.

15.4.4 Case 4

For Switch 4 defined by Eq. (15.14), the time derivative of the errors is
given by
q
d e 41
5 y 2 2 x 3 1 u 1 ðtÞ
dt q
5 e 42 1 u 1 ðtÞ
q
d e 42 3 3
5 y 1 2 y 2 αy 2 1 fcosωt 1 u 2 ðtÞ 1 β x 1 1 β x 2 1 β x 3 2 β x
3
2
1
1
4 1
dt q
5 e 41 2 αe 42 2 f 41 ðx; yÞ 1 u 2
where f 41 are nonliear terms in e 11 and e 12 given as
3
f 41 5 x 2 2 αx 3 2 y 1 fcosωt 1 β x 1 1 β x 2 1 β x 3 2 β x 3
3
1
2
4 1
1
Theorem 4: If the control function u 1 ðtÞ and u 2 ðtÞ are chosen such that
u 1 ðtÞ 5 k 1 e 41 1 e 42
ð15:29Þ
u 2 ðtÞ 52 f 41 ðx; yÞ 1 e 41 1 ðk 2 2 αÞe 42
then the drive system 15.9 will achieve multiswitching synchronization with
the response system 15.10
Proof: We redefine u 1 ðtÞ and u 2 ðtÞ to eliminate all the nonlinear terms in
e 41 and e 42

u 1 ðtÞ 5 V 41 ðtÞ
u 2 ðtÞ 52 f 41 ðx; yÞ 1 V 42 ðtÞ

V 41 ðtÞ e 41
5 D
V 42 ðtÞ e 42
where D is a 2 3 2 constant matrix. In order to make the closed-loop system
stable, the matrix A should be chosen such that the eigenvalues λ i of D
satisfies
jargðλ i Þj . 0:5πα; i 5 1; 2; ...: