Page 499 - Mathematical Techniques of Fractional Order Systems

486 Mathematical Techniques of Fractional Order Systems

The state errors of the antisynchronization are defined by
e x 5 x 2 1 x 1 ;
8
>
<
e y 5 y 2 1 y 1 ; ð16:16Þ
>
e z 5 z 2 1 z 1 :
:
It is trivial to see that the state error dynamics of the antisynchronization
are given by:
_ e x 5 _ x 2 1 _ x 1 ;
8
>
<
_ e y 5 _ y 1 _ y ; ð16:17Þ
2
1
>
_ e z 5 _z 2 1 _z 1 :
:
By taking the difference between the unknown system parameters (a, b,
^
c) and the estimation of the unknown parameters (^ a, b, ^ c), the parameter esti-
mation errors are calculated as:
8
e a 5 a 2 ^ a;
>
<
^
e b 5 b 2 b; ð16:18Þ
>
e c 5 c 2 ^ c;
:
Similarly, we get the parameter estimation error dynamics by differentiat-
ing Eq. (16.18). Thus the parameter estimation error dynamics are:
8 _
_ e a 52 ^ a;
<
_
^
_ e b 52 b; ð16:19Þ
_ e c 52 ^ c;
: _
The main aim of this section is to design an adaptive control to antisyn-
chronize the slave no-equilibrium system (16.15) and the master no-
equilibrium system (16.14). Therefore, the adaptive control has been
constructed as follows:
8
u x 5 e z 2 k x e x
>
<
2 2
u y 52 x 1 z 2 x 2 z 2 ^ aðsgnðz 1 Þ 1 sgnðz 2 ÞÞ 2 k y e y ð16:20Þ
1 2
>
: ^ y 1 y 2 2 2 3 3
u z 52 e x 1 bðe 1 e Þ 2 ^ cðy z 1 1 y z 2 Þ 1 z 1 z 2 k z e z
1 2 1 2
It is noted that in (16.20), three positive gain constants are k x , k y , k z .In
addition, we have designed the following parameter update law:
_
8
^ a 5 ðsgnðz 1 Þ 1 sgnðz 2 ÞÞe y
>
>
<
^ _ y 1 y 2
b 52 ðe 1 e Þe z ð16:21Þ
>
_ 2 2
>
:
^ c 5 ðy z 1 1 y z 2 Þe z
1 2