Page 500 - Mathematical Techniques of Fractional Order Systems

Dynamics, Synchronization and Fractional Order Form Chapter | 16 487

The 3D chaotic systems with infinite equilibrium (16.14) and (16.15)
with unknown system parameters are globally and exponentially antisynchro-
nized by using the adaptive controller (16.20) and the parameter update law
(16.21), where k x , k y , k z are positive gain constants.
In order to confirm the antisynchronization of the slave system (16.15)
and the master system (16.14) when applying the designed adaptive control
(16.20), the Lyapunov function is selected as follows:
1 2 2 2 2 2 2
Vðe x ; e y ; e z ; e a ; e b ; e c Þ 5 ðe 1 e 1 e 1 e 1 e 1 e Þ: ð16:22Þ
x
b
c
a
y
z
2
From Eq. (16.22), we obtain the differentiation of V:
_
V 5 e x _ e x 1 e y _ e y 1 e z _ e z 1 e a _ e a 1 e b _ e b 1 e c _ e c : ð16:23Þ
It is simple to verify that by combining (16.14), (16.15), and (16.20), syn-
chronization error dynamics are rewritten by:
8
_ e x 52 k x e x
>
<
_ e y 5 ðsgnðz 1 Þ 1 sgnðz 2 ÞÞe a 2 k y e y ð16:24Þ
>
y 1 y 2
: 2 2
_ e z 52 ðe 1 e Þe b 1 ðy z 1 1 y z 2 Þe c 2 k z e z
1 2
Similarly, we get the differentiation of the Lyapunov function by substi-
tuting Eqs. (16.19) and (16.24) into Eq. (16.23):
_
2
2
2
V 52 k x e 2 k y e 2 k z e : ð16:25Þ
x y z
It is simple to verify that the differentiation of V is a negative semidefi-
nite function. Therefore, according to Barbalat’s lemma (Khalil, 2002), we
have e x -0, e y -0, and e z -0 exponentially as t-N. In other words, the
antisynchronization between the slave system with infinite equilibria and the
master system with infinite equilibria is achieved.
An example is presented to illustrate the correction of the proposed anti-
synchronization scheme. In this example, the parameter values of the master
system and the slave system are selected as:
a 5 0:1; b 5 0:1; c 5 1: ð16:26Þ
We assume that the initial states of the master system with infinite equi-
libria are taken as:
x 1 ð0Þ 5 0:1; y 1 ð0Þ 5 0:1; z 1 ð0Þ 5 0:1: ð16:27Þ

We take the following initial states for the slave system with infinite
equilibria:
x 2 ð0Þ 5 0:1; y 2 ð0Þ 5 0:2; z 2 ð0Þ 5 0:3: ð16:28Þ

In this example, the positive gain constants are given by:
k x 5 6; k y 5 6; k z 5 6; ð16:29Þ