Page 471 - Mathematical Techniques of Fractional Order Systems

Multiswitching Synchronization Chapter | 15 457

There are six possible switching combinations that could exist for this
drive-response system given in Eqs. (15.9) and (15.10). They are:

e 11 5 y 1 2 x 1
Switch 1 ð15:11Þ
e 12 5 y 2 2 x 2
e 21 5 y 1 2 x 1
Switch 2 ð15:12Þ
e 22 5 y 2 2 x 3
e 31 5 y 1 2 x 2
Switch 3 ð15:13Þ
e 32 5 y 2 2 x 1
e 41 5 y 1 2 x 2
Switch 4 ð15:14Þ
e 42 5 y 2 2 x 3
e 51 5 y 1 2 x 3
Switch 5 ð15:15Þ
e 52 5 y 2 2 x 1
e 61 5 y 1 2 x 3
Switch 6 ð15:16Þ
e 62 5 y 2 2 x 2

15.4.1 Case 1
For Switch 1 defined by Eq. (15.11), the time derivative of the errors is
given by
q
d e 11
5 y 2 1 u 11 ðtÞ 2 x 2
dt q
5 e 12 1 u 1 t
q
d e 12
3
5 y 1 2 y 2 αy 2 1 fcosωt 1 u 2 ðtÞ 2 x 3
1
q
dt
3
5 e 11 1 x 1 2 y 2 αðe 12 1 x 2 Þ 1 fcosωt 1 u 12 ðtÞ 2 x 3
1
5 e 12 2 αe 12 1 f 12 ðx; yÞ 1 u 2 ðtÞ
where f 12 ðx; yÞ are the nonlinear terms in e 11 and e 12 given as
3
f 12 ðx; yÞ 5 x 1 2 y 2 αx 2 1 fcosωt 2 x 3
1
Theorem 1: 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
ð15:17Þ
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