Applied Mathematics, 2013, 4, 7-12
Published Online November 2013 (http://www.scirp.org/journal/am)
http://dx.doi.org/10.4236/am.2013.411A2002
Open Access AM
Chaos Synchronization of Uncertain Lorenz System via
Single State Variable Feedback
Fengxiang Chen1,2*, Tong Zhang1,2
1School of Automotive Studies, Tongji University, Shanghai, China
2Clean Energy Automotive Engineering Center, Tongji University, Shanghai, China
Email: *Fxchen_qjy@hotmail.com
Received May 6, 2013; revised June 6, 2013; accepted June 13, 2013
Copyright © 2013 Fengxiang Chen, Tong Zhang. This is an open access article distributed under the Creative Commons Attribution
License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
ABSTRACT
This paper treats the problem of chaos synchronization for uncertain Lorenz system via single state variable information
of the master system. By the Lyapunov stability theo ry and adaptive technique, the derived contro ller is featured as fol-
lows: 1) only single state variable information of the master system is needed; 2) chaos synchronization can also be
achieved even if the perturbation occurs in some parameters of the master chaotic system. Finally, the effectiveness of
the proposed co ntrollers is also illustrated by the simulations as well as rigorous mathematical proofs.
Keywords: Uncertain Lorenz System; Single State Variable; Chaos Synchronization
1. Introduction
Chaos control and synchronization have been intensively
investigated during last decade [1-3] and still have at-
tracted increasing attention in recent years. Chaos syn-
chronization has many potential applications in secure
communication [1], laser physics, chemical reactor proc-
ess [2], biomedicine and so on. Up to now, numerous
methods have been proposed to cope with the chaos
syn chron izat ion, su ch as backstepping design method [3 ],
adaptive design method [4], impulsive control method
[5], sliding mode control method [6,7], and other control
methods [8-10]. But most of the proposed methods
abovementioned need more single state variable informa-
tion of the master system. However, for instance, the
more state variables transmitted to the slave system
means the more bandwidth and energy consumption in
secure communication system as well as security reduc-
tion. Additionally, controller based on single state vari-
able is simple, efficient and easy to be implemented in
practical applications [11]. For example, in a real engi-
neering case, some state variables may be difficult or
even cannot be detected.
Recently, scholars begin to have attention on the prob-
lem of chaos synchronization via single state variable
controller (hereafter refereed to as “SSVC”) with the
motivation of the above facts. Jiang Zhang [12] gives a
schematic method to design the synchronization control-
ler for a class of chaos system based on backstepping
design, and several elegant results derived. However, the
controllers conceived by several high-degree complex
polynomials. M. T. Yassen [11] provided linear SSVC to
Lu chaotic system, but the gain of the controller is diffi-
cult to be determined due to the fact that it contains the
information of the upper bound of the system trajectory.
In order to overcome the deficiency, he modified the
SSVC based on adaptive technique. Junan Lu [13] gave
out an adaptive SSVC for an unified chaotic system (uni-
fication of Lorenz, Chen and Lu chaotic system). Feng-
xiang Chen [14] proposes a linear SSVC for Lu system
via theory of cascade-connection system. However, lit-
eratures [11-14] did not take the parameters uncertainty
into account, and the synchronization failed when some
uncertainty occurs. On the other hand, for an electrical or
electronic system, parameter uncertainty is inevitably
suffered due to the variation of temperature, humidity,
voltage, or interference of electric and magnetic fields.
Thus, in this pap er, we will provide robust SSVC for un-
certain Lorenz chaotic system based on Lyapunov stabil-
ity theory, and its effectness is validated by both rigorous
theoretical analysis and simulations.
The rest of this paper is organized as follows. In the
next section, the problem statement on the scheme of
master-slave chaos synchronization for uncertain Lorenz
system via single state variable information is presented.
*Corresponding author.
F. X. CHEN, T. ZHANG
8
In Section 3, three controllers are provided to the Lorenz
system without/with parameters perturbation. In Section
4, numerical simulations are provided to illustrate the
effectiveness of the proposed controllers. Finally, some
conclusion remarks are included in Section 5.
2. Problem Formulation
The Lorenz system is a system of ordinary differential
equations (the Lorenz equations defined by (1)) first
studied by Edward Lorenz [15]. It is notable for having
chaotic solutions for certain parameter values and initial
conditions. In particular, the Lorenz attractor is a set of
chaotic solution s of the Lorenz system. Consider the Lo-
renz system:

121
2121
312 3
xxx
3
x
rxxx x
xxxbx



(1)
where ,,rb
are system parameters, 123
,,
x
xx are
state variables, the system generates the chaotic behavior
(see Figure 1) when 10,28,8 3rb
. Hereafter in
this paper, we refer to the system (1) as master system
and assume that 1
x
is the only available state variable.
The related slave system with control inputs are writ-
ten as

10211
2012132
312033
ˆˆˆ
ˆˆˆˆˆ
ˆˆˆˆ
xxxu
x
rx xxx u
xxxbxu



(2)
where 000
,,rb
are system nominal parameters and
12
ˆˆ
,,
3
ˆ
x
x
uD
x4
:RR
are state variables, i.e.
i, and they will be work out
later.

1123
ˆˆˆ
,,, ,
ii
uuxxxx
,3,i1,2
tion Our target is to find out the func1,2,3
i,ui
such that the trajectory of th e slave system (2) is going to
asymptotically approach the master system (1) and fi-
nally implement chaos synchronization in the sense that
-20 -10 010 20
-50
0
50
0
10
20
30
40
50
x
1
x
2
x
3
Figure 1. Lorenz chaos phenomenon at 10,
28,r
83b.
lim0, 1,2,3
i
tei

where .
ˆ
ii
exx
i
3. Controller Design
According to the parameter uncertainties, we can classify
the synchronization system into 8 cases listed in Table 1.
If we introduce the set ,,
BC defined as following:

00 0000
,,,, ,,,
A
rbr brr bb
 


00 0000
,,,, ,,,Brbrb rrbb
 

 
22
00 000
,,,, ,0Crbrbrrbb


Obviously, it is an equivalent partition for the element
from case 1 to case 8. i.e., ,
case 1Acase 2B
,
3i
. Consequently, in this section, we are only
going to investigate the chaos synchronization under
three different classifications defined by the set
8caseCi
,,
A
BC.
C1:
00 0
,,,, ,rbr bA

Theorem 1: The two Lorenz chaotic systems (1) and
(2) can be synchronized under the control law as follows:
1
201011313
31212
0
ˆˆˆ ˆ
ˆˆˆ
u
urxrxxxxx
uxxxx


(3)
Proof: Subtracting Equation (1) from Equation (2),
then the error dynamic system is obtained as
1021
2213
31203
ee
eexe
exebe
e
 

(4)
Choosing the Lyapunov function as
222
123
0
122
2
e
Ve


e
2
(5)
and taking the derivative along the trajectory of the sys-
tem (4), it yields
22
1 12212312303
22 2
121203
22 2 2
220
Veeeexeexeebe
eeeebe
 
 
(6)
The proof is completed.
C2:
00 0
,,,, ,rbr bB

For this case, the system (1) can be equivalent to the
perturbation system as
Table 1. Synchronization system classification based on
parameter uncertainties.
Case 1Case 2Case 3Case 4Case 5 Case 6 Case 7Case 8
0
0
0
rr
bb
0
0
0
rr
bb
0
0
0
rr
bb
0
0
0
rr
bb
0
0
0
rr
bb
0
0
0
rr
bb
0
0
0
rr
bb
0
0
0
rr
bb
Open Access AM
F. X. CHEN, T. ZHANG
Open Access AM
9
1

1021
201213
31203
xxx
x
rx xxx
xxxbx



1
(7) where . 0h
Proof: Subtracting Equation (1) from Equation (2),
then the error dynamic system is obtained as
10211 11
2213
31203
eee e
eexe
exebe


 

where


102
x
x

 . e
(9)
Since the trajectory of the master system (1) is
bounded due to the property of chaos system, 1
must
be bounded, i.e. 11
M
1
,..Rst M
 . Choosing the Lyapunov function as
Theorem 2: The two Lorenz chaotic systems (1) and
(2) can be synchronized under the control law as follows:
22
22 *
123
00
11
22
22
e
Vee
h


 

 (10)

11111
201011313
31212
11
ˆˆ
ˆˆˆ ˆ
ˆˆˆ
ˆ
uxxxx
urxrxxxxx
uxxxx
hx x
 



(8) where *
is a constant and will be determined later.
Taking the derivative along the trajectory of the system
(4), it yields

*
1
22 2
11
112 2123 123 0300 0
*1
11 1
222
1203 00 0
*
11 1
222
1203 00
22 2 2
132
22
132
22
e
e
Veeeexeexeebe h
e
ee
eebe
ee
eebe

 

 

 
 
 
(11)
If we se lect *1
 , then . 0V
Theorem 3: The two Lorenz chaotic systems (1) and
(2) can be synchronized under the control law as follows:
Comment 1: Since the controller input component
11
ee (see Equation (8)) is not continuous at 10e
, it
leads to chattering in the viewpo int of engineering appli-
cation. In order to overcome this defect, a continuous
function

1
2arctanke is adopted to substitute the
discontinuity function 11
ee based on the conception
of variable structure controller design theory, and thus
the chattering will be eliminated.
10 11
201011313
31212
ˆ
ˆˆˆ ˆ
ˆˆˆ
uMxx
urxrxxxxx
uxxxx



(14)
where


2
112
11 0223
212
3
M
MM


M
,

12 10
Proof: Subtracting Equation (1) from Equation (2),
then the error dynamic syste m is ob tained as
0,1 2,0,32,0,2.b
 

C3:

00 0
,,,, ,rbr bC

In this case, the system (1) can be equivalent to the
perturbation system as
1021 1
22132
312033
eee M
eexe
exebe

 

1
e
(15)

1021 1
201213 2
312033
xxx
xrxxxx
xxxbx



(12)
Choosing the Lyapunov function as
where 222
123
0
122
2
e
Ve


e
(16)




102
201
203
1
x
x
rrx
bbx

 
 
 
(13) and taking the derivative along the trajectory of the sys-
tem (4), it yields

22 2
1 12212312303
1102 233
22 2 2
2222
VeeeexeexeebeM
eee
 
 
(17)
Since the trajectory of the master system (1) is
bounded, i
must be bounded, i.e. ,
i
M
R

.. ii
s
tM . Additionally, if 0
, 0, rr0
bb
then , , and respectively.
10 20 3
0Note that , for any ,
thus
21
2ab ab

2,,RabR

F. X. CHEN, T. ZHANG
10






22 2
121203
2
121221
11 01122223333
222
112 2033
2
11212
11 02233
22
2
12 322
2
VeeeebeM
ee
eebeM
22
e

 



 
 
 
 
(18)
Note that

12 1
0,1 2,0,3 2,0,,b
 

0
and


2
112
11 0223
2M


 
12
3
(19)
then . The proof is completed.
0V
Comment 2: Although the component of input 1
will increase to infinity as 11
u
ˆ
x
x, so the law is only a
conceptional controller and can not be implemented in
real application. But this does not mean that the control-
ler law is meaningless. In real implementation, we can
modify the as
1
u


011 11
12
011 11
ˆˆ
ˆˆ
M
xxxx r
u
M
xxrxxr


2
3
(20)
With the modified controller, the synchronization error
will not approach to zero, but in the vicinity of the
, the details see [16] and simulation 3. On the
other hand, the errors of 23
is independent of
and , but decided by the following subsystem:
0,0,0
1
u,ee 1
e
2213
31203
eexe
exebe
 

(21)
which means that an y effor t on the 1 will not have any
beneficial to attenuate the fluctuation of .
u
23
,ee
4. Numerical Simulation
In this section, three numerical simulations named as
Simulation 1, Simulation 2 and Simulation 3 are carried
out to illustrate the effectiveness of the proposed con-
troller. For sake of simplification, we refer the controller
defined by (3) to as nominal controller, the controller
defined by (8) to as adaptive controller, and the control-
ler defined by (14) to as robust controller in the follow-
ing sim ul a ti on s.
Simulation 1:
In this subsection, we are going to validate the effec-
tiveness of nominal controller designed under the condi-
tion . Thus, three parameters of
Lorenz master system and slave system are chosen iden-
tically as
00 0
,,,,,rbr bA

00 0
10,28,8 3rr bb

 
123
02,00,0xxx
 
ˆˆ ˆ
01.1,00.2,xx

. Initial
states of the master and slave system are taken as
and
0
 
0x
12 3
, respectively, and they
will be kept the same through out the following simula-
tions. Taking the control input as the nominal controller,
the simulation results are shown in Figure 2. As we can
1
0246810
-0. 5
0
0. 5
1
Time(Sec.)
e
1
(t)
0246810
-0. 5
0
0. 5
1
Time(Sec.)
e
2
(t)
0 246810
-1
-0. 5
0
0. 5
Time(Sec)
e
3
(t )
Figure 2. Synchronization errors with the nominal control-
ler for Lorenz system without uncertainty.
see that the synchronization is achieved after about 3
seconds. Which means that the nominal controller is ef-
fective for the synchronization system without uncer-
tainty; however, this is not the case when the parameter
perturbations are occurred and the results are shown in
the subsequently Simulation 2 and Simulation 3.
Simulation 2:
In this subsection, we add the parameter 30% pertur-
bations to
, i.e., 13,
. The adaptive controller de-
fined by (8) is taken as



11
201011313
31212
11
ˆ
arctan 800
ˆˆˆ ˆ
ˆˆˆ
ˆ
100,0 5
ux
urxrxxxxx
uxxxx
xx




1
x

(22)
which has been modified according to the Comment 1.
And the results are shown in Figure 3. We can see that
the synchronization via nominal controller is failed due
to the fluctuation of
1
et. Meanwhile, the synchroniza-
tion via the adaptive controller is achieved after about 4
seconds, and the adaptive variable
goes to a fixed
constant. Obviously, the adaptive controller proposed in
Theorem 2 really eliminates the disturbance suffered fro m
he parameters perturbation, and the synchronization is t
Open Access AM
F. X. CHEN, T. ZHANG
Open Access AM
11
05 10
-4
-2
0
2
4
e
1
(t)
Time(Sec.)
05 10
-0.4
-0.2
0
0. 2
0. 4
e
2
(t)
Time(Sec.)
05 10
-1
-0. 5
0
0. 5
e
3
(t)
Time(Sec.)
05 10
5
10
15
20
25
(t)
Time(Sec.)
A dapti ve Cont roll er
Nomi nal Control l er
A dapti ve Cont roll er
Nomi nal Cont rol l er
A dapti ve Cont roll er
Nomi nal Cont rol l er
Figure 3. Synchronization errors for uncertain Lorenz system via the adaptive controller and nominal controller, respec-
tively.
r
achieved. On the other hand, the perturbation of parame-
0246810
-2
2
te 13
hardly has any impact on the convergence of
3
(23)
is independent of
 
23
,etet due to the subsystem
2213
eexe
exebe
 

31
20
and
proved that the subystem3) is asymptotically stable
1
e.
(2It can be easily to be
s
(i.e. 2233
ˆˆ
,
x
xx as t) with any 1
x
x
. For
example, choosing Lyapunov function as 22
23
Vee
,
then . Weans that the nver-
gence rate of 23
ˆˆ
,
2
20
Veb 
2
3
0ehich mco
x
x is independent of the p
of e bationrtur
and 1
e. : Simulation 3
Ithis susecn btion, we are going to analysis the per-
fost controller designed under the condi-
tio
rmance of robu
n

00 0
,,,, ,rbr bC

. Here we consider the worst
case, i.e. 00 0
,,rrbb
. The parameters of the
maste 13
r system are taken as
(+30% perturbation
of the nom23.8
inal value 10), r
(15% perturbation
of the nominal value 28), 3.2b
(+20% perturbation
of the nominal value 8/3). Rntroller and nominal
controller are here both ado as to compare their
performance on the chaos synchronization and the simu-
lation results are shown in Figure 4. From Figure 4, we
can see that the synchronization via nominal controller
and robust controller are both failed due to the errors do
not converge to zero but undulate in a bounded area. For
robust controller,
obust
pte co
d so
1
et approaches to zero, but this is
not the case for other error component
2
et
and
3
et
,
which has been interpted by Comment 2. For nominal
controller, all error components
re
1t,
e
2
et and
0
Robus t Controll er
Nom in al Control l e r
e
1
(t)
0246810
-5
0
5
e
2
(t)
Robus t Cont rol ler
Nominal Controller
0246810
-10
-5
0
e
3
(t)
Time(Sec.)
Robus t Controll er
Nomi nal Cont rol l er
Figure 4. Synchronization errors for uncertain Lorenz sys-
tem via the robust controller and nominal controller, re-
spectively.
3
et varied in the vicinity of certain value, respectively.
As mentioned in Comment 2, the error component
2
et
and
3
et is independent of and , but decided
y (21). Wh1
e
th 2
1
u
and bich means that bo

et
3
et must
keep the same when different controller (nominal con-
troller and robust controller) acted on the slave system
F. X. CHEN, T. ZHANG
12
and orresponding numerical simulation result are
shown in Figure 4.
5. Conclusion
the c
edback of single state variable from
ing to parameter uncertain
tion is defined to classify the parame-
r
more effectively b.
d I
d National Nature Science
77].
Synchro-
nization and Secure Communication,” Philosophical
Transactions of tho. 1911,
2010, pp. 379-
The paper investigates the synchronization of uncertain
Lorenz system via fe
the master system. Accord
an equivalent partities,
ter uncertain Lorenz system. Then three controllers (no-
minal controller, adaptive controller, and robust control-
ler) are given out to achieve the chaos synchronization
based on Lyapunov stability theory and adaptive tech-
nique. Finally, three numerical simulations are conducted
to validate the effectiveness of the proposed controllers,
and it finds that the adaptive controllers do really elimi-
nate the disturbance caused by parameters perturbation
but it is not the case for nominal controller. For robust
controller, any effort to modify the controller 1
u will
not have beneficial to attenuate the error component

2
et and

3
et which is independent of 1
e and 1
u,
but decided by (21). For the future investigation, we will
use L2 gain theory and passivity theory to design aobust
synchronization controller to attenuate the disturbance
ased on single state variable
6. Acknowledgements
This work is supported by National Special Fund for the
Development of Major Research Equipment annstru-
ments [2012YQ150256], an
Foundation of C hi na [6 1 10 4 0
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