Page 361 - Mathematical Techniques of Fractional Order Systems
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352 Mathematical Techniques of Fractional Order Systems
α n
D x 5 Ax with initial condition xð0Þ 5 x 0 , where xAR is a state vector,
then the system is asymptotically stable if and only if argðλ i ðAÞÞ . απ=2,
i 5 1; 2; ::::; n, where argðλ i ðAÞÞ denotes the argument of the eigenvalues
λ i of A.
Proposition 1: If the control functions are chosen as
U 1 5 C 1 ½A 2 X 2 1 F 2 ðX 2 Þ 2 A 1 X 1 1 F 1 ðX 1 Þ 2 A 3 C 1 ðX 2 1 X 1 Þ 2 F 3 ðX 3 Þ 1 K 1 e 1
U 2 5 C 2 ½B 2 Y 2 1 G 2 ðY 2 Þ 1 B 1 Y 1 1 G 1 ðY 1 Þ 2 B 3 C 2 ðY 2 1 Y 1 Þ 2 G 3 ðY 3 Þ 1 K 2 e 2 ;
ð12:8Þ
where K 1 AR n 3 n , K 2 AR n 3 n are the gain matrices, then the dual combina-
tion synchronization will be achieved among considered systems (12.1),
(12.2), (12.3), (12.4), (12.5), and (12.6) if and only if all the eigen
values λ i of A 3 1 K 1 and B 3 1 K 2 satisfy argðλ i Þ . απ=2, where
i 5 1; 2;::::; n:
Proof: : After substituting the values of control function given in Eq. (12.8)
into Eq. (12.7), the error system (12.7) is reduced in the following form
α
D e 1 5 ðA 3 1 K 1 Þe 1
α ð12:9Þ
D e 2 5 ðB 3 1 K 2 Þe 2
In view of theorem 1, it can be concluded that the system (12.9) is
asymptotically stable if and only if all the eigen values λ i of A 3 1 K 1 and
B 3 1 K 2 satisfy argðλ i Þ . απ=2, where i 5 1; 2; ::::; n.
If lim t-N :e: 5 0and thus the considered systems are dual combination
synchronized and hence this completes the proof.
12.3 APPLICATION OF THE SCHEME
In this section the effectiveness of the proposed scheme for dual combination
synchronization among different dimensional fractional order chaotic sys-
tems are demonstrated numerically. Here two kinds of cases are taken to
discuss dual combination synchronization of chaotic systems: dual combina-
tion synchronization with order n . m and dual combination synchronization
with order m . n.
12.3.1 Dual Combination Synchronization With Order n . m
Let us consider the fractional order Lu system (Petras, 2011) and fractional
order Qi system (Xiangjun and Yang, 2010) as the first two master
systems as
