wilamowski-b-m-irwin-j-d-industrial-communication-systems-2011

Secure Communication Using Chaos Synchronization 23-5
23.2.2 adaptive Control for Chaos Synchronization via Partial States
Sometimes, the determination of suitable controller gain K in (23.3) is complicated. So, an adaptive version
of the controller can be designed to continuously adjust the gain parameters based on the system responses
for synchronization. Based on the results in [WWSX06], the following adaptive controller can be designed:
2
U( t) = − K( t) E2( t), with K( t) = diag( kj( t)) and k̇ j( t) = θ jen+
j( t)
for j = 1, …,
m (23.6)
where θ j is a positive parameter chosen by users.
Again, the two Chua’s circuits (23.4) and (23.5) are considered under the adaptive control (23.6). Let
θ 1 = 10, and the initial condition of the adaptive control (23.6) is k 1 (0) = 0, the synchronization errors of
two identical Chua’s circuits (23.4) and (23.5) and the changing parameter of the adaptive control (23.6)
are shown in Figures 23.3 and 23.4, respectively.
It is observed that the two systems are synchronized.
100
0
e 3 e 2 e 1
–100
–200
0 1 2 3 4 5 6 7 8 9 10
50
0
–50
0 1 2 3 4 5 6 7 8 9 10
40
20
0
–20
0 1 2 3 4
5 6 7 8 9 10
t
FIGURE 23.3
Synchronization of two Chua’s circuits under adaptive control.
150
100
k 1
50
0 0 1 2 3 4 5 6
t
7 8 9 10
FIGURE 23.4
Changing parameter of the adaptive control.
© 2011 by Taylor and Francis Group, LLC