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

Secure Communication Using Chaos Synchronization 23-3
less control information to achieve chaos synchronization. An intuitive idea is to realize the chaos
synchronization only by using part of the states information.
Suppose the driving system is given as
⎧Ẋ
⎪
= − AX + f ( X, Y) + h1( t),
⎨
Ẏ
⎩⎪ = g( X, Y) + h2( t),
(23.1)
where X = [x 1 . .... x n ] T ∈ R n and Y = [x n+1 . .... x n+m ] T ∈ R m are the state vectors of the nonlinear
system, A ∈ R n×n . f : R n × R m → R n , g : R n × R m → R m are nonlinear functions of X and Y, h 1 : R + → R n and
h 2 : R + → R m are linear or nonlinear functions of t.
The response system under control using partial system states is given as follows:
⎧ ̃̇ ̃ ̃ ̃
⎪
X = − AX + f ( X, Y) + h1( t),
⎨
⎪Ỹ̇
= g X̃ Ỹ
⎩ ( , ) + h2( t) + U( t),
(23.2)
where X˜ = [x˜1. .... x˜n] T ∈ R n , Ỹ = [x˜n+1 . .... x˜n+m ] T ∈ R m , and U(t) is the control action which is only
added to states Ỹ of the n + mth order system (23.2).
The synchronization errors are defined as
E = X̃
( t) − X( t), E = Ỹ
( t) − Y( t)
1 2
and they can be denoted as vectors
T
E1 = ⎡ ⎣ e1 ⋯ en⎤ ⎦ , E2 = ⎡ ⎣ en+ 1 ⋯ en+
m⎤ ⎦ .
T
In the linear feedback control schema, the controller is designed as
U( t) = − KE ( t) with K = diag( k ) for j = ,…, m,
(23.3)
2 j 1
where k j is a parameter determined according to the conditions given in [WWSX06].
Obviously, such a feedback controller saves the bandwidth of the communication channel as only m
states of the driving system are required to be transmitted, instead of all the n + m states.
Note that the system (23.1) is quite general as f(X, Y), g(X, Y), h 1 (t), and h 2 (t) are general nonlinear
functions. In fact, it includes the majority of typical nonlinear chaotic systems studied so far and thus
can be easily implemented. Now, the Chua’s circuit given below is taken as an example for illustration:
⎧ẋ
1 = −βx2 − γx1,
⎪
⎨ẋ
2 = y1 − x2 + x1,
⎪
⎩⎪
ẏ
1 = − αy1 + αx2 − αh( y1),
(23.4)
© 2011 by Taylor and Francis Group, LLC