Page 477 - Mathematical Techniques of Fractional Order Systems

Multiswitching Synchronization Chapter | 15 463

The synchronization error is a linear system with active control inputs
V 51 ðtÞ and V 52 ðtÞ. The feedback control which would stabilize the system so
that e 51 and e 52 converge to zero at time t-N are to be designed. As a
result, we chose

V 51 ðtÞ e 51
5 E
V 52 ðtÞ e 52
where E is a 2 3 2 constant matrix. In order to make the closed-loop system sta-
ble, the matrix E should be chosen such that the eigenvalues λ i of E satisfies

jargðλ i Þj . 0:5πα; i 5 1; 2; ...: ð15:32Þ
There are varieties of choices for choosing matrix E. A good choice of E is

E 5 k 1 1 ð15:33Þ
1 ðk 2 2 αÞ
Matrix E satisfies the condition (15.35) for k 1 ; k 2 . 0. Therefore, multi-
switching synchronization is achieved.

15.4.6 Case 6

For Switch 6 defined by Eq. (15.16), the time derivative of the errors is
given by
q
d e 61 3
5 y 2 2 β x 1 1 β x 2 1 β x 3 2 β x 1 u 1 ðtÞ
2
3
1
dt q 4 1
5 e 62 2 f 61 ðx; yÞ 1 u 1 ðtÞ
q
d e 62
3
5 y 1 2 y 2 αy 2 1 fcosωt 1 u 2 ðtÞ 2 x 3
1
q
dt
5 e 61 2 αe 62 1 f 62 ðx; yÞ 1 u 2
where f 62 are nonlinear terms in e 11 and e 12 given as
f 61 5 x 2 1 β x 1 1 β x 2 1 β x 3 2 β x 3
3
1
2
4 1
3
f 62 52 αx 3 2 y 1 fcosωt
1
Theorem 6: If the control function u 1 ðtÞ and u 2 ðtÞ are chosen such that
u 1 ðtÞ 5 k 1 e 11 1 e 12
ð15:34Þ
u 2 ðtÞ 52 f 12 ðx; yÞ 1 e 11 1 ðk 2 2 αÞe 12
then the drive system 15.9 will achieve multiswitching synchronization with
the response system 15.10