Page 604 - Mathematical Techniques of Fractional Order Systems
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574 Mathematical Techniques of Fractional Order Systems
The combined Lyapunov function for the subsystems (19.18) and (19.19)
will be:
1
V 2 5 V 1 1 z 2 2
2
q 2 q
.D V 2 #2 az 2 z 1 z 2 1 z 2 D z 2
1
2 2
#2 az 2 az 2 z 1 z 2 1 z 2 faðz 3 1 α 2 Þ 1 dz 1 g
1 2
Again, virtual controller α 2 is chosen as,
1
α 2 5 ð1 2 dÞz 1
a
which results into
q 2 2
D V 2 #2 az 2 az 1 az 2 z 3
1 2
Further, stepping back towards the third subsystem in (19.17), one can
write,
q q
D z 3 5 cx 2 2 x 2 x 4 2 x 3 1 u 2 D α 2 ð19:20Þ
The Lyapunov function for z 1 ; z 2 ; z 3 Þ subsystem defined by (19.18),
ð
(19.19) and (19.20), is chosen as:
1
V 3 5 V 2 1 z 2
2 3
and its fractional order time derivative is given by,
q 2 2 q
D V 3 #2 az 2 az 1 az 2 z 3 1 z 3 ðcx 2 2 x 2 x 4 2 x 3 1 u 2 D α 2 Þ
1 2
The final control law can be chosen as,
q
u 52 kz 3 2 cx 2 1 x 2 x 4 1 x 3 2 az 2 1 D α 2 ð19:21Þ
which leads to,
2
2
q
D V 3 #2 az 2 az 2 kz 2
1 2 3
From the above two equations, it’s concluded that the system is stable and
T
the transformed states ½ z 1 z 2 z 3 z 4 converge to zero as t-N.Using
the definition in Theorem 4, the asymptotic stabilization of state vector
T
½ x 1 x 2 x 3 x 4 can be ensured.
19.4.2 Controller Design for Synchronization
Here, a controller is designed using the backstepping approach for synchroni-
zation of fractional order Lorenz Stenflo system in a master slave configu-
ration. First, a master and slave dynamics are defined and then error
