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

23-4 Industrial Communication Systems
where α, β, and γ are parameters, h(y 1 ) = by 1 + 0.5(a − b)(|y 1 + 1| − |y 1 − 1|), a and b are constants which
satisfying a < b < 0. The response system under feedback control is given by
⎧x̃̇ 1 = −βx̃ 2 − γx̃
1,
⎪
⎨x̃̇ 2 = ̃y 1 − x̃ 2 + x̃
1,
⎪
⎪̃̇ y1 = − α̃y 1 + αx̃ 2 − αh( ̃
⎩
y1) + U( t).
(23.5)
Clearly,
x
A = ⎡ f X Y
g X Y y x h y
⎣ ⎢ γ 0⎤
⎡
⎥ = − β 2 ⎤
, ( , ) ⎢ ⎥ , ( , ) = − α 1 + α 2 − α ( 1).
0 1⎦
⎣ y1 + x1⎦
In this case, only one state ỹ 1 in the response system (23.5) is controlled by using one state y 1 from the
driving system, namely
U( t) = −k1 ⎡⎣ ̃y 1( t) − y1( t) ⎤ ⎦ .
To illustrate the synchronization of the two Chua’s circuits, choose α = 10, β = 16, γ = 0.0385, a = − 8/7, and
b = −5/7, and the initial conditions for the driving system and the response system are (18. −26. −17) T
and (2. 10. 10) T , respectively. According to the guidelines in [WWSX06] and based on the above parameters,
it is computed that k 1 > 11.9911 in order to ensure synchronization of the two chaotic systems.
Let k 1 = 30 in the simulation. The synchronization errors of two identical Chua’s circuits (23.4) and (23.5)
under feedback control are shown in Figure 23.2.
Clearly, the errors are almost zero for t > 10.s.
e 3 e 2 e 1
100
0
–100
–200
0 2 4 6 8 10 12 14 16 18 20
50
0
–50
0 2 4 6 8 10 12 14 16 18 20
40
20
0
–20
0 2 4 6 8 10
t
12 14 16 18 20
FIGURE 23.2
Synchronization of two Chua’s circuits under feedback control.
© 2011 by Taylor and Francis Group, LLC