Page 606 - Mathematical Techniques of Fractional Order Systems

576 Mathematical Techniques of Fractional Order Systems

Virtual controller α 2 can be chosen as,

1
α 2 5 ð1 2 dÞz 1 ð19:27Þ
a
which results in,

q 2 2
D V 2 #2 az 2 az 1 az 2 z 3
1 2
Finally, from the third equation,
q q
D z 3 5 ce 2 2 e 2 e 4 2 e 3 2 e 2 x 4 2 x 2 e 4 1 u 2 D α 2
And the final Lyapunov function is:

1 2
V 3 5V 2 1 z 3
2
q 2 2 q
.D V 3 #2az 2az 1az 2 z 3 1z 3 fce 2 2e 2 e 4 2e 3 2e 2 x 4 2x 2 e 4 1u2D α 2 g
1 2
The final controller is:
q
u 52 kz 3 2 ðc 1 aÞe 2 1 e 2 e 4 1 e 3 1 e 2 x 4 1 x 2 e 4 1 D α 2 ð19:28Þ
which gives
q
2
2
D V 3 #2 az 2 az 2 kz 2
1 2 3
The above expression leads to convergence of transformed states
T
z 4 Š to zero as t-N. This further ensures asymptotic conver-
½ z 1 z 2 z 3
T
e 4 Š to zero as proved in the previous section.
gence of errors ½ e 1 e 2 e 3
Convergence of state errors leads to asymptotic synchronization of states of
slave system in (19.23) to the states of master system in (19.22), i.e.,
T T
lim t-N ½ x 1 x 2 x 3 x 4 Š 5 ½ y 1 y 2 y 3 y 4 Š .
19.5 CONTROL AND SYNCHRONIZATION BY USING
ACTIVE BACKSTEPPING TECHNIQUE
In this section, controller design is proposed for the stabilization and syn-
chronization of a fractional order Lorenz Stenflo system by employing an
active backstepping strategy which is a combination of backstepping tech-
nique and active control technique.

19.5.1 Controller Design for Stabilization
Foremost, an appropriate controller is designed to drive the system to
stable state (an unstable orbit or an equilibrium point). Controlled system
can be expressed as,