# Passivity-Based Synchronization of Unified Chaotic System

Authors: K. Kemih; M. Benslama; H. Baudrand
Journal: Journal of Control Science and Engineering
 (2008-02-10)

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## Abstract

This letter further improves and extends the work of Kemih et al. In detail, feedback passivity synchronization with only one controller for a unified chaotic system is discussed here. It is noticed that the unified system contains the noted Lorenz, Lu, and Chen systems. Numerical simulations are given to show the effectiveness of these methods.


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## Body

## 1. Introduction

Chaotic systems are characterized by being extremely
sensitive to initial conditions, deterministically random, and hence ultimately
unpredictable. Chaotic systems have numerous potential applications in
mechanics, laser and chemical technologies, communications, biology and
medicine, economics, ecology, and so forth [1, 2].During the last two decades, synchronization in
chaotic dynamic systems has received a great deal of interest among scientists
from various research fields, since
Pecora and Carroll [3] introduced a method to synchronize two identical chaotic
systems with different initial conditions. The idea of synchronization is to
use the output of the master system to control the slave system so that the
output of the response system follows the output of the master system
asymptotically. A wide variety of approaches have been proposed for the
synchronization of various chaotic systems which include PC method, OGY method,
active control approach, impulsive
control method, generalized
predictive control, adaptive control method, time-delay feedback
approach, and backstepping design technique [5–16]. Many people have begun to give their attention to passive network theory
[17–20]. The character of passive system is one of the network theory concepts,
which show characteristics of dissipative network. Passive network theory can
be used to analyze the dynamical character of system, such as stabilization and
dynamic characteristics.In this paper, our aim is to synchronize two unified chaotic systems. To achieve this
goal, passive control theory is used to study passive synchronization of two
unified chaotic systems. We first show that the passive synchronization problem
is a passive control problem. In the stable
control region, the error dynamical system is transformed to a passive system,
and then using the state feedback, the error dynamical system can be globally
asymptotically stabilized at zero. Finally, some simulation
results are provided for illustration and verification.


## 2. Propreties of Passive System

Some preliminaries of passivity theory used in
this paper will be shortly reviewed for the consistency of the presentation.
Passivity is applied to nonlinear systems which are modelled by ordinary
differential equations with input vectoru(t) and output vector y(t) [17]:
(1)x˙(t)=f(x(t),u(t)),y(t)=h(x(t)).
The system (1)
is dissipative with the supply rate W(u(t),y(t)),
if it is not able to generate power by itself, that is, the energy stored in the
system is less than or equal to the supplied power:
(2)V(x(t))≥0,V(x(T))−V(x(0))≤∫0TW(u(t),y(t))dt.
Furthermore, the storage function V(x(t)) must satisfy the requirements for a Lyapunov function.
If there exists a positive semidefinite Lyapunov function, such that
(3)∫uT(τ)y(τ)dτ≥∫[∂V(x(τ))∂x(τ)f(x(τ),u(τ))+euT+δyT(τ)y(τ)+ρϕ(x(τ))]dτ
then the system (1)
is passive. A passive system implies that any increase in storage energy is due
solely to an external power supply.Then the equilibrium point of the system:(4)x˙(t)=f(t,x(t),0)
is asymptotically stable in either of the two cases:(i)
ρ>0,(ii)
e+δ>0 and the system is zeros-state observation.The system (1) can be represented as the normal form:
(5)x˙=f(z)+g(z,y)y,y˙=l(z,y)+k(z,y)u.
The nonlinear system (5) may be
rendered bay a state feedback of the form [17]
(6)u=γ12+β(x)υ.


## 3. Passive Synchronization of Unified Chaotic System

We
consider unified chaotic system [21]:
(7)x˙1=(25α+10)(x2−x1),x˙2=−x1x3+(29α−1)x2+(28−35α)x1,x˙3=x1x2−8+α3x3,
where α∈[0,1], which contains the canonical Lorenz system
[21, 23] and Chen system [24] as two extremes and Lü system [25] as a special
case. Obviously, when α=0, α=0.8,
and α=1,
it is the Lorenz chaotic attractor, Lü chaotic attractor and Chen chaotic
attractor, respectively. What is interesting is that as the parameter α changes continuously from 0 to 1, the
resulting system remains continuously to be chaotic.We will study the passive synchronization of two
chaotic systems. Let system (7) be the drive system, and the response system is
modelled by the following equation:
(8)y˙1=(25α+10)(y2−y1),y˙2=−y1y3+(29α−1)y2+(28−35α)y1,y˙3=y1y2−8+α3y3.
Let e=(e1,e2,e3)T=(y1−x1,y2−x2,y3−x3)T be the synchronization error, and then the controlled
model of error synchronization system is given by
(9)e˙1=−(25α+10)e1+(25α+10)e2,e˙2=(28−35α−x3)e1+(29α−1)e2−x1e3−e1e3+u,e˙3=x2e1+x1e2−8+α3e3+e1e2.
The error
synchronization system (9) is
already in the normal form of (5), where z1=e1, z2=e3, y=e2, z=[z1,z2]T :
(10)f(z)=[−(25α+10)z1,x2z1−8+α3z2],g(z,y)=[(25α+10),x1+z1]T,l(z,y)=(28−35α−x3)z1+(29α−1)y−x1z2−z1z2,k(z,y)=1.
Our object is to design a
smooth control (6) for the error synchronization system to make the closed-loop system passive. Choose a
storage function candidate:
(11)V(z,y)=W(z)+12y2,
where W(z) is Lyapunov function, with W(0) = 0,
(12)W(z)=12(z12+z22).
The zero dynamics of the system (5) describes those internal dynamics which
are consistent with external constraint y = 0,
that is,
(13)z˙=f(z).
Considering (13) and because α>0 is positive constant,
(14)ddtW(z)=−(25α+10)z12+x2z1z2−8+α3z22,ddtW(z)=−(25α+10)(z1−x2z22(25α+10))2−(8+α3−x224(25α+10))z22.
In
addition, we have
(15)8+α3−x224(25α+10)=(100α2+840α+320)−3x1212(25α+10)>0.
Then
(16)ddtW(z)≤0.
The zero dynamics of error synchronization system is Lyapunov stable. The derivative of V(z, y) along the
trajectory of the error synchronization system (9) is
(17)ddtV(z,y)=∂∂zW(z)z˙+yy˙=∂∂zW(z)f(z)+∂∂zW(z)g(z,y)y+l(z,y)y+k(z,y)y.
The error synchronization system is minimum phase:
(18)ddtW(z)f(z)≤0.
Equation (17) becomes
(19)ddtV(z,y)≤∂∂zW(z)g(z,y)y+(l(z,y)+k(z,y)u)y.
If we select the feedback control (6) of the following form and consider (10):
(20)u=k−1(z,y)[−lT(z,y)−∂W∂zg(z,y)−γ12y+υ]=−(38−10α−x3)z1−(29α−1+γ12)y+υ,
where γ11 is positive constant, and υ is an external signal which is connected with
the reference input, the above inequality can be rewritten as
(21)ddtV(z,y)≤−(γ12+29α−1)y2+υy.
Then by integrating both sides of (21), we have
(22)V(z,y)−V(z0,y0)≤∫0t−(γ12+29α−1)y(τ)2dτ+∫0tυ(τ)y(τ)dτ,V(z,y)≥0, and ρ=V(z0,y0),(23)V(z,y)−V(z0,y0)≤∫0t−(γ12+29α−1)y(τ)2dτ+V(z,y)≥∫0t−(γ12+29α−1)y(τ)2dτ.
It satisfies the passive
definition (3). The error synchronization system (9) is rendered to be output strict passive (OSP) under
the feedback control. If external signal υ = 0, then we want to steer state of error
synchronization system to the
origin.


## 4. Simulation Results

Unified
chaotic system has attractor forα=0.8.
The typical unified chaotic
attractor is shown in Figure 1.Figure 1
The typical unified chaotic attractors.The
goal is to force the two systems to synchronize under control, while they have
different initial conditions. The drive system starts from [8−9 −7] and the response
system from [−6 5 6]. We select γ12=11 and use the controller as in (20): u=−(30−x3)e1−33.2e2.In Figures2 and 3, we note that the system is quickly and perfectly synchronized, also
the bigger γ12 gives the best performance.Synchronization of two unified chaotic systems.
(a)(b)(c)Figure 3
Phase-space trajectory.


## 5. Conclusion

In this paper, we have
shown that the passivity-based synchronization can be used to synchronize
unified chaotic system. The controller is very simple and has linear feedback form for the synchronization
error system. It is noticed that the unified system contains the noted Lorenz, Lu, and Chen
systems. Numerical simulations given show the effectiveness of
these methods.



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*Source: 567807-2008-03-11.xml*