Research Journal of Applied Sciences, Engineering and Technology 4(20): 4007-4011, 2012
ISSN: 2040-7467
© Maxwell Scientific Organization, 2012
Submitted: December 20, 2011
Accepted: April 23, 2012
Published: October 15, 2012
Synchronization of Chaotic Systems by Using Limited Nussbaum Gain Method
Aijun Zhou, Guang Ren, Chengyong Shao and Yong Liang
Dalian Maritime University, Dalian, China
Abstract: In this study, we propose a novel kind of limited Nussbaum gain method to solve the unknown
control direction problem of chaotic systems. It is necessary to consider the unknown control direction situation
when chaos synchronization is used for secure communication. Also it is very useful to design a small gain with
a Nussbaum method if the Nussbaum gain method is used in a real engineering system. The smaller a gain is
designed, the safer and more stable a Nussbaum gain method is used. According to the above principles, a
limited gain Nussbaum method is used to solve the uncertainties of chaotic system, such as unknown
parameters, unknown nonlinear functions and unknown input or control directions. At last, detailed numerical
simulation is done to testify the rightness and effectiveness of the proposed method.
Keywords: Chaos, limited gain, nussbaum gain, synchronization, unknown control direction
INTRODUCTION
Chaos systems have complex dynamical behaviors
that possess some special features such as being
extremely sensitive to tiny variations of initial conditions
and having bounded trajectories with a positive leading
Lyapunov exponent and so on Hu et al. (2007), Gao et al.
(2007), Elabbasy et al. (2006) Tang and Wang (2005) Ge
and Yang (2007), Gauthier et al. (1992), Khalil and
Saberi (1987) and Hao and Wei (2006). Synchronization
of chaos systems with unknown parameters was
investigated widely by researchers from various fields.
There are also many papers researched others kinds of
uncertainties such as input uncertainty, unknown
functions and unmolded dynamics. It is also necessary to
research those uncertain situation above since they are
very useful if the synchronization of chaos is used for
secure communication.
There are also many problems should be considered
if a real communication system is constructed with
chaotic synchronization theory (Hao and Wei, 2007;
Wang-Long and Kuo-Ming, 2007). The unknown control
direction problem is one of those problems should be
researched and it is also a very meaningful research since
this problem is very difficult not only in theory but also in
engineering. In fact, we have to solve it firstly in theory.
In the future, maybe it can be used in engineering. But it
is a great challenge and it is real exist in practice. To solve
this problem, Nussbaum proposed a method which was
named by his name later. It is a effective method for some
simple system at first, so the Nussbaum gain method is
mostly used in some low order systems.
There is another problem should be considered when
the Nussbaum gain method is used to design a real system
and is used in engineering practice. It is that the gain of
the system should be as small as possible that it because
the energy of real system is limit and also if the gain can
be designed to be smaller, the system is comparatively
more stable. So for Nussbaum gain method, the gain
should be limit at least. Then it can be defined as limit
gain problem for Nussbaum gain method.
It is obvious that it is very useful to design a limit
Nussbaum gain method for the synchronization of
uncertain chaotic system for secure communication. But
it is also very difficult to solve the limit gain problem. In
this study, a new kind of novel Nussbaum gain method is
proposed under the above background.
METHODOLOGY
Problem description: Take the universe chaos system for
an example, the driven system can be described as follow:
x& = f x ( x ) + Fx ( x )θ x + ∆ ( x , t )
(1)
The response system can be written as:
y& = f y ( y ) + bu
(2)
If it is a three dimension chaotic system, it can be
expanded as:
x&1 = f x1 ( x1 ,..., x 4 ) +
p1
∑F
j =1
x1 j ( x1 ,..., x 4 )θ x1 j +
j =1
x1 j
( x, t )
(3)
x& 2 = f x 2 ( x1 ,..., x 4 ) +
p2
∑F
j =1
x2 j
( x1 ,..., x 4 )θ x 2 j +
Corresponding Author: Aijun Zhou, Dalian Maritime University, Dalian, China
4007
q1
∑∆
q2
∑∆
j =1
x2 j
( x, t )
(4)
Res. J. Appl. Sci. Eng. Technol., 4(20): 4007-4011, 2012
x& 3 = f x 3 ( x1 ,..., x 4 )
+
p3
∑
j =1
Fx 3 j ( x1 ,..., x 4 )θ x 3 j +
z&i = f yi ( y1 ,..., y4 ) − f xi ( x1 ,..., x4 )
q3
∑
j =1
∆ x 3 j ( x, t )
(5)
−
q1
j =1
j =1
(9)
Design a nonlinear sliding mode as:
And the slave system can be written as:
y& 1 = f y1 ( y1 ,..., y 4 ) + b1u1
p1
∑ Fxij ( x1 ,..., x4 )θ x1 j − ∑ ∆ xij ( x, t ) + bi ui
y& 2 = f y 2 ( y1 ,..., y 4 ) + b2 u2
(7)
y& 3 = f y 3 ( y1 ,..., y 4 ) + b3 u3
(8)
{∫ ∫ z dtdt} ⎤⎥⎦
(∫ z1dt ) ⎞⎟⎠ ∫ z dt
Si = zi ⎡⎢ wi1 + wi 2
⎣
(6)
+ ⎛⎜⎝ wi 3 + wi 4
where, 2x are unknown parameters and )xij(x, t) are
unknown functions and bi are unknown input coefficients.
The control objective is to design a synchronization law
u = u(x, y) such that the synchronization can be fulfilled
then it has y6x.
This study proposes a novel kind of limited
Nussbaum gain method to solve the unknown control
direction problem of chaotic systems. According to the
above simulation results, it is easy to make a conclusion
that the proposed limited Nussbaum gain method is
effective for the unknown control direction situation of
chaotic system synchronizations. Moreover, the limit gain
strategy is very useful if a Nussbaum gain method is
applied in a real engineering system.
Assumptions: To make the below context more easy to
understand, it is necessary to make two assumptions for
the chaotic system.
2
i
(10)
2
i
where, wij>0.
Without considering the input uncertainty, assume
the situation that b-1i = 1, then there exists a idea nonlinear
sliding mode adaptive control as:
uip = bi−1 (uic + uid )
(11)
where,
{[
{∫ ∫ z dtdt}]
+ ⎛⎜⎝ w + w ( ∫ z dt ) ⎟⎞⎠ z +
z ∫ z dt ∫ z dt } / ⎡⎢ w + w ( ∫ ∫ z dtdt ) ⎤⎥
⎣
⎦
uic = − zi 2 wi 2 ∫ zi dt
i
2
i3
i4
i
i
(12)
2
2 wi 4
i
i
i
i1
i2
i
And uid is designed as:
uid =
Assumption 1: The response system has the some
dimension as the driven system.
− k i1 Si − d$i1 sgn( Si ) − d$i 2 Si
2
⎡w + w
⎤
i 2 ∫ ∫ zi dtdt
⎢⎣ i1
⎥⎦
{
}
(13)
define,
Assumption 2: There exists a positive constant dij such
that the nonlinearities of the driven system satisfies:
f yi ( y1 ,..., y 4 ) − f xi ( x1 ,..., x 4 )
−
p1
∑F
j =1
xij
( x1 ,.., x 4 )θ x1 j −
~
d ij = d ij − d$ij
then,
q1
∑∆
j =1
xij
(14)
~&
&
d ij = − d$ij
( x, t )
≤ d i1 + d i 2 Si
(15)
Define the turning law as:
where, Si is sliding mode and it is defined as bellows.
Since the driven system is a chaotic system and
chaotic systems are bounded, so it is easy to be satisfied
by many common chaotic systems.
&
&
d$i1 = Si , d$i 2 = Si
2
(16)
Choose a Lyapunov function as:
Design of nonlinear sliding mode: Define a new variable
as zi = yi-xi, the error system can be written as:
4008
Vi =
1 2 ~2 ~2
( S + d i1 + d i 2 )
2 i
(17)
Res. J. Appl. Sci. Eng. Technol., 4(20): 4007-4011, 2012
According to the assumption, it has:
~
~
S&i Si ≤ d i1 Si + d i 2 Si
2
− k i1 Si
2
Choose a = 20, b = 14,c = 10.6 = 2.8,klb =0, there
exists a chaotic attractors. a, b, c, h are unknown
parameters and klb is coefficients of uncertain nonlinear
functions.
Assume that the structure of the response system is
known and its structure is as follows:
(18)
It is easy to prove that:
V&i ≤ 0
(19)
y& 1 = a y ( y 2 − y1 ) + b1u1
Limit nussbaum gain strategy: Considering the situation
that the control direction is unknown and the Nussbaum
gain control strategy can be designed as:
y& 2 = by y1 − k y y1 y 3 + u2
ui = − N ( k i )uip
y& 3 = − c y y 3 + h y y12 + u3
(20)
Choose parameters as (ay, by, cy, ky, hy) = (10,4, 2.5, 1, 4)
And set the initial state of response system as (y1, y2,
y3) = (1,-1,2)
And assume the unknown control direction as:
And the Nussbaum turning law can be designed as:
l&i = k l zi u d
(21)
While design the limit gain function as:
k i = f si (li )
⎧ 1 0 < t < 2⎫
bi = ⎨
⎬
t>2 ⎭
⎩− 1
(22)
The simulation results can be show as follows:
Figure 1 shows that the state x1 of driven system can
synchronize with the state y1 of response system.
Figure 2 shows that the state x2 of driven system can
Where fs(l) can be chosen as a triangle function.
Choose the Lyapunov function:
Vi =
1
(1 + ki1 )zi2 +
2
pi
1
∑ 2 (θ~ )
j =1
ij
2
+
1
k
2 i2
(∫ z dt )
i
2
(23)
15
10
It is easy to solve the derivative of the Lyapunov
function and get:
5
x1
0
-5
1
V&i ≤ − (bi N ( k i ) + 1)dli
k1
(24)
-10
-15
Use the same method as the traditional Nussbaum
strategy, it is easy to prove that ki is bounded and Si60, so
the synchronization of the system can be fulfilled.
-20
0
2.5
t/s
3.0 3.5
4.0
4.5 5.0
Fig. 1: State x1 and y1
NUMERICAL SIMULATION
20
15
10
5
0
-5
-10
-15
-20
-25
-30
x2
Take the below three dimension chaotic system as an
example to do a numerical simulation and the driven
system can be written as:
x&1 = a ( x 2 − x1 ) + k lb x3 cos x 2
x& 2 = bx1 + cx 2 − x1 x 3 + k lb x 3 cos x 2
x& 3 = x 22 − hx 3 + k lb (1 + sin( x 2 x 3 )) x 2
0.5 1.0 1.5 2.0
0
0.5
1.0
Fig. 2: State x2 and y2
4009
1.5
2.0
t/s
2.5
3.0
3.5
4.0
Res. J. Appl. Sci. Eng. Technol., 4(20): 4007-4011, 2012
10
400
5
300
0
-5
x1
x3
200
100
-10
0
-15
-100
-20
-25
-200
0
0.5
1.0
1.5
2.0
t/s
2.5
3.0
3.5
Fig. 3: State x3 and y3
1.0
1.5
2.0
t/s
2.5
3.0
3.5
4.0
1.5
2.0
t/s
2.5
3.0
3.5
4.0
Fig. 7: State x1 and y1
20
15
10
5
0
-5
-10
-15
-20
-25
-30
12
10
8
6
4
x2
x1
0.5
0
4.0
2
0
-2
-4
-6
0
0.5
1.0
1.5
2.0
t/s
2.5
3.0
3.5
0.5
0
1.0
4.0
Fig. 8: State x2 and y2
Fig. 4: The curve of error z1
500
35
400
30
300
x3
25
x2
20
200
15
100
10
0
5
0
-100
0
-5
0
0.5
1.0
1.5
2.0
t/s
2.5
3.0
3.5
1.0
1.5
2.0
2.5
3.0
3.5
4.0
t/s
Fig. 9: State x3 and y3
Fig. 5: The curve of error z2
300
12
10
8
200
6
x1
400
x3
0.5
4.0
100
4
2
0
0
-2
-100
-4
-6
-200
0
0.5
1.0
1.5
2.0
t/s
2.5
3.0
3.5
0
4.0
0.5
1.0
1.5
2.0
t/s
2.5
3.0
3.5
4.0
Fig. 6: The curve of error z3
Fig. 10: The curve of error z1
synchronize with the state y2 of response system with the
interrupt of input direction changed at 2s.Figure 3 shows
that the state x3 of driven system can synchronize with
the state y3 of response system with the interrupt of input
direction changed at 2s. Figure 4 shows that the
synchronization error z1 can converged to zero with the
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Res. J. Appl. Sci. Eng. Technol., 4(20): 4007-4011, 2012
CONCLUSION
x2
35
30
According to the above simulation results, it is easy
to make a conclusion that the proposed limited Nussbaum
gain method is effective for the unknown control direction
situation of chaotic system synchronizations. What is
worth pointing out is that the limit gain strategy is very
useful if a Nussbaum gain method is applied in a real
engineering system.
25
20
15
10
5
0
-5
-10
0
0.5
1.0
1.5
2.0
t/s
2.5
3.0
3.5
REFERENCES
4.0
Fig. 11: The curve of error z2
500
400
x3
300
200
100
0
-100
-200
0
0.5
1.0
1.5
2.0
t/s
2.5
3.0
3.5
4.0
Fig. 12: The curve of error z3
change of control direction at 2s. Figure 5 shows that the
synchronization error z2 can converged to zero with the
change of control direction at 2s. Figure 6 shows that the
synchronization error z3 can converged to zero with the
change of control direction at 2s.
Choose the coefficients of unknown nonlinear
function as klb = 0.2, the simulation results are show as
follows: Figure 7 shows that the state x1 of driven system
can synchronize with the state y1 of response system.
Figure 8 shows that the state x2 of driven system can
synchronize with the state y2 of response system with the
interrupt of input direction changed at 2s. Figure 9 shows
that the state x3 of driven system can synchronize with the
state y3 of response system with the interrupt of input
direction changed at 2s. Figure 10 shows that the
synchronization error z1 can converged to zero with the
change of control direction at 2s. Figure 11 shows that the
synchronization error z2 can converged to zero with the
change of control direction at 2s. Figure 12 shows that the
synchronization error z3 can converged to zero with the
change of control direction at 2s.It can make a conclusion
that the synchronization can be achieved successfully no
matter there are unknown nonlinear functions or not.
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