Page 304 - Mathematical Techniques of Fractional Order Systems

Sliding Mode Stabilization and Synchronization Chapter | 10 293

Theorem 2: The adaptive control law synchronizes chaotic and hyperchaotic
systems (identical and nonidentical) with the following control law
(Aghababa, 2015; Komurcugil, 2012):
12α ρ
u i 5 s i 1 f i ðzÞ 2 g i ðyÞ 1 k i D signðe i Þje i j ð10:21Þ
and the following adaptive gain:
_
k i 5 2 signðs i ÞD 12α e i ð10:22Þ
Γ i
where s i is the sliding surface, e i 5 z i 2 y i is the error variable, and Γ i and
k i are the adaptive law and controller gain constants.

Proof: Consider the following sliding surface (Aghababa, 2015):
ρ
s i ðtÞ 5 D α21 e i ðtÞ 1 k i D 2α ðe i ðtÞ 1 signðe i ðtÞÞje i ðtÞj Þ ð10:23Þ
and the following Lyapunov function:
n
n
n
1 X X 1 X
2
VðtÞ 5 :SðtÞ: 1 Γ i k 5 js i j 1 Γ i k 2 ð10:24Þ
1 i i
2 2
i51 i51 i51
so the drive system is defined by:
α
D Z 5 fðZÞ ð10:25Þ
and the response system is given by:
α
D Y 5 gðYÞ 1 U ð10:26Þ
and the definition of (10.25) and (10.26) are similar to (10.9), defining the
error variable as:
e i 5 z i 2 y i ð10:27Þ
Then, the first derivative of (10.24) is given by:
n
n X
_
_
X
VðtÞ 5 signðs i Þ_ s i 1 Γ i k i k i ð10:28Þ
i51 i51
so by implementing the properties of fractional calculus and Section 10.2,
(10.28) becomes:
n
ρ
_
X
VðtÞ 5 signðs i Þ½f i ðzÞ 2 g i ðyÞ 2 u i 2 s i 1 k i D 12α e i 1 k i D 12α signðe i Þje i j Š
i51
n
X
1 Γ i k i k i _
i51
ð10:29Þ
so with the following control and adaptive laws the systems are
synchronized.