Page 472 - Mathematical Techniques of Fractional Order Systems

458 Mathematical Techniques of Fractional Order Systems

Proof: The method of active control is employed to prove theorem 1.Thisis
predicated on its simplicity, efficiency, and fast convergence (Ojo et al.,
2013a; Ogunjo et al., 2017)
The time derivative of the errors can also be written as
q
0 1
d e 11
dt
B q C 0 1 0 u 1 ðtÞ
B C 5 e 11 1 1
C
q
1 f 12 ðx; yÞ u 2 ðtÞ
B 2α
@ d e 12 A e 12
dt q
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 5 V 11 ðtÞ
u 2 ðtÞ 5 e 11 2 αe 12 2 f 12 ðx; yÞ 1 V 12 ðtÞ
which leads to
q
d e 11
5 e 12 1 V 11 ðtÞ
dt q
q ð15:18Þ
d e 12
5 e 1 1 2 αe 12 1 V 12 ðtÞ
dt q
The synchronization error (Eq. 15.18) is a linear system with active con-
trol inputs V 11 ðtÞ and V 12 ðtÞ. The feedback control which would stabilize the
system so that e 11 and e 12 converge to zero at time t-N are to be designed.
As a result, we chose

V 11 ðtÞ e 11
5 A
V 12 ðtÞ e 12
where A 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 A
satisfies
jargðλ i Þj . 0:5πα; i 5 1; 2; ...: ð15:19Þ

There are varieties of choices for choosing matrix A. A good choice
of A is

1
k 1
A 5 ð15:20Þ
1 ðk 2 2 αÞ
Matrix A satisfies the condition 15.19 for k 1 ; k 2 . 0. Therefore, multi-
switching synchronization is achieved.