Synchronization of Uncertain Chaotic Systems with Unknown ...
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JOURNAL OF COMPUTERS, VOL. 8, NO. 1, JANUARY 2013
Synchronization of Uncertain Chaotic Systems with Unknown Response Systems Xinyu Wang
Institute of Science and Technology for Opto-electronic Information, Yantai University, China, Email: [email protected]
Hongxing Wang and Junwei Lei
Dept of Electronic Information and Engineering, Naval Aeronautical and Astronautical University Yantai, China Dept of Control Engineering, Naval Aeronautical and Astronautical University Yantai, China Email: {wanghongxing1024, leijunwei}@126.com
Abstract—Considering the situation with the same dimension but different structure in driven system and response system, a new kind of robust adaptive synchronization controller is designed based on the Lyapunov stability theorem . Both the unknown parameters and uncertain functions are considered and an adaptive turning law is constructed to handle the unknown parameters and a robust control law is adopted to deal with the uncertain functions in the system. At last, a four dimension chaotic system is taken as an example to testify the rightness of the proposed method. And numerical simulation result shows that the proposed method can realize the synchronization between driven chaotic system and response system with unknown parameters and uncertain nonlinear functions. Index Terms—chaos, adaptive, uncertainty, synchronization, unknown parameter , robust
I. 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[1-8]. Synchronization of chaos systems with unknown parameters is investigated widely by researchers from various fields. M.T. Yassen[9] used Lyapunov stability theorem and adaptive control method to realize synchronization between chaotic systems with unknown parameters, and the situation considered is a new kind of chaotic system with the same structure in driven system as its of response system. Rizk Yassen [10]and Qiang Jia[11]also researched synchronization problem with the uncertain parameters in chaotic system and designed an adaptive controller, but it is mostly based on the situation with linear parameters uncertainties. Rongwei Guo[12]designed a kind of simple nonlinear adaptive feedback synchronization controller with LaSalle theorem. The synchronization of two chaotic system is realized by this new and novel method, but it did not consider the situation of any uncertainty . Also the
© 2013 ACADEMY PUBLISHER doi:10.4304/jcp.8.1.178-185
chaotic system is acquired to satisfy a global Lipscitz conditions. In fact, input uncertainties usually exist in actual control systems, especially, unknown control direction is an special complex input certainty. but the situations of chaos systems with input uncertainties are neglected in most papers . Among all methods proposed to solve synchronization problem of chaotic systems[6-15], most of these researches are based on the situation that there only exist static uncertainties between driver system and response system. So the unmodelled dynamics of synchronization between chaotic system with different structure are seldom considered, especially for the situation that there exist static uncertainties and unknown parameters and dynamic uncertainties simultaneously. But it is very possible for the actual system that driven systems have different structure with response systems, or parameters may be changed unexpected because of the disturbance of environment , or the system model is inevitably inaccurate because of the dynamic uncertainties. The above situations are more possible to happen when synchronization of chaotic systems are used in the application of secure communication. So it is meaningful to study the synchronization of chaotic systems with both static uncertainties and dynamic uncertainties. Considering the situation that the driven system and response system have different structure, the synchronization problem of chaotic system was researched with totally unknown parameters both in the response system and in the driven system. But the uncertain nonlinear function situation was neglected . Based on the Lyapunov stability theorem and LMI method, Ju H. Park designed a new feedback synchronization controller for a class of chaotic system with the same structure between driven system and response system. In this paper, a general kind of chaotic system with the same dimension but different structure is studied and both the driven system and response system contains unknown parameters and unknown nonlinear functions. A new kind of robust adaptive synchronization controller is designed
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based on the Lyapunov stability theorem and robust and adaptive method. Also, the numerical simulation is done and simulation results testifies the effectiveness of the proposed method.
p3
y&3 = f y 3 ( y1 ,L , y4 ) + ∑ Fy 3 j ( y1 ,L , y4 )θ y1 j j =1
p4
+ ∑ Δ y 3 j ( y, t ) + b3u3 p3
Considering the following driven system and response system, the response system has unknown parameters and the driven system has both unknown parameters and uncertain nonlinear functions as follows. The driven system can be written as x& = f x ( x) + Fx ( x)θ x + Δ x ( x, t )
(1)
The response system can be described as y& = f y ( y ) + Fy ( y )θ y + Δ y ( y, t ) + bu
(9)
j =1
II. PROBLEM DESCRIPTION
(2)
Take a four dimension system as an example, it can be expanded as follows. The driven system can be expanded as
y& 4 = f y 4 ( y1 ,L , y4 ) + ∑ Fy 4 j ( y1 ,L , y4 )θ y 4 j j =1
p4
+ ∑ Δ y 4 j ( y, t ) + b3u4
(10)
j =1
Then θ x and θ y
are the unknown parameters . the
control target is to design a synchronization law
u = u ( x, y, θˆx , qˆ x , θˆy , qˆ y )
,
q&ˆ x = f ( x, y, qˆ x )
θˆy = f ( x, y,θˆy )
,
&
θˆx = f ( x, y,θˆx ) &
, ,
qˆ& y = f ( x, y, qˆ y ) such that the Synchronization of state can be achieved, it means that y → x .
p1
x&1 = f x1 ( x1 ,L , x4 ) + ∑ Fx1 j ( x1 ,L , x4 )θ x1 j j =1
p2
+ ∑ Δ x1 j ( x, t )
III. ASSUMPTIONS (3)
j =1
p1
x&2 = f x 2 ( x1 ,L , x4 ) + ∑ Fx 2 j ( x1 ,L , x4 )θ x 2 j j =1
p2
+ ∑ Δ xij ( x, t )
(4)
It is necessary to give the following typical assumptions to make the analysis of the system more convenient. Assumption 1: The driven system and response system have the same structure and it means that they also have the same dimension and f xi = f yi .
Assumption 2: The structure of some part of the driven
system is known, it means Fxij , f xi is known.
j =1
Assumption 3: The structure of some part of the p1
response system is known, it means
x&3 = f x 3 ( x1 ,L , x4 ) + ∑ Fx 3 j ( x1 ,L , x4 )θ x 3 j j =1
p2
+ ∑ Δ x 2 j ( x, t )
(5)
j =1
known. Assumption 4: The nonlinear functions of driven system satisfies the follow conditions such as there exists *
a unknown positive parameters qxij ≤ d xij such that for
1 ≤ i ≤ n , 1 ≤ j ≤ p2 , it holds
p1
x&4 = f x 4 ( x1 ,L , x4 ) + ∑ Fx 4 j ( x1 ,L , x4 )θ x 4 j j =1
p2
+ ∑ Δ x 4 j ( x, t )
(6)
And the response system can be expanded as y&1 = f y1 ( y1 ,L , y4 ) + ∑ Fy1 j ( y1 ,L , y4 )θ y1 j j =1
+ ∑ Δ y1 j ( y, t ) + b1u1
(7)
*
a unknown positive parameters q yij ≤ d yij such that for
1 ≤ i ≤ n , 1 ≤ j ≤ p4 , it holds
j =1
Δ yij ( X , t ) ≤ q*yijψ yij ( X )
p3
y& 2 = f y 2 ( y1 ,L , y4 ) + ∑ Fy 2 j ( y1 ,L , y4 )θ y 2 j j =1
+ ∑ Δ y 2 j ( y, t ) + b2u2 j =1
© 2013 ACADEMY PUBLISHER
(11)
smooth and non-negative function. Assumption 5: The nonlinear functions of response system satisfies the follow conditions such as there exists
p3
p4
* Δ xij ( X , t ) ≤ qxij ψ xij ( X )
Where d xij is a known constant and ψ xij ( X ) is known
j =1
p4
Fyij , f yi , bi is
(8)
Where
(12)
d yij is a known constant and ψ yij ( X ) is
known smooth and non-negative function.
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IV. DESIGN OF ADAPTIVE ROBUST SYNCHRONIZATION CONTROLLER
p2
zi ∑ {qˆ xijψ ij ( x) − Δ xij ( x, t )} j =1
Define a new error variable as zi = yi − xi for the driven and response system, the error system can be written as
p2
≤ ∑ {zi qˆ xijψ xij ( x) + qxij * zi ψ xij ( x)}
z&i = f yi ( y1 ,L , y4 ) − f xi ( x1 ,L , x4 ) p3
p4
j =1
j =1
p1
p2
j =1
j =1
p2
= ∑ { zi ψ xij ( x)[ sign( zi )qˆ xij + qxij * ]}
+ ∑ Fyij ( y1 ,L , y4 )θ yij + ∑ Δ yij ( y, t )
j =1
(13)
p4
zi ∑ {Δ yij ( y, t ) − qˆ yijψ yij ( x)}
−∑ Fxij ( x1 ,L , x4 )θ xij − ∑ Δ xij ( x, t ) + bi ui The control
j =1 p4
≤ ∑ {− zi qˆ yijψ yij ( y ) + q yij * zi ψ yij ( y )} (19)
ui is designed as
j =1 p4
= ∑ zi ψ yij ( y ){− sign( zi )qˆ yij + q yij *}
ui = f 2i ( x)[− f yi ( y1 ,L , y4 ) + f xi ( x1 ,L , x4 ) p1
j =1
Fxij ( x1 ,L , x4 )θˆxij
+∑ j =1 p2
p3
j =1
j =1
Define
+ ∑ qˆ xijψ xij ( x) − ∑ Fyij ( y1 ,L , y4 )θˆyij
(14)
p2
− ∑ qˆ yijψ yij ( y ) − f zi ( zi )]
q% xij = qxij * + sign( zi )qˆ xij
(20)
q% yij = q yij * − sign( zi )qˆ yij
(21)
Then
j =1
p2
where
zi ∑ {qˆ xijψ xij ( x) − Δ xij ( x, t )}
f 2i ( x) = bi−1
j =1
f zi ( zi ) = ki1 zi + ki 2 + ki 3 Then
(22)
p2
zi zi + ε i1
≤ ∑ { zi ψ xij ( x)q% xij }
3 1/ 3 zi exp( zi2 / 3 ) + ki 4 sign( zi ) 2
j =1
(15)
p4
zi ∑ {qˆ yijψ yij ( y ) − Δ yij ( y, t )} j =1
p1
zi z&i = zi [− f zi ( zi ) − ∑ Fxij ( x1 ,L , x4 )θ%xij
≤ ∑ { zi ψ yij ( y )q% yij } j =1
p2
+ ∑ {qˆ xijψ ij ( x) − Δ xij ( x, t )}
And consider that
j =1
&
(16)
p3
+ ∑ Fyij ( y1 ,L , y4 )θ%yij p4
&
+ ∑ {Δ yij ( y, t ) − qˆ yijψ yij ( x)}] j =1
Consider that
&
θ&%xij = θ&xij − θˆxij = −θˆxij
(24)
Design the adaptive turning law as
j =1
θ%xij = θ xij − θˆxij , θ%yij = θ yij − θˆyij
(23)
p4
j =1
where
(18)
j =1
θˆxij = − zi Fxij ( x1 ,L , x4 )
(25)
q&% xij = sign( zi )q&ˆ xij
(26)
q&% yij = − sign( zi )q&ˆ yij
(27)
Then it holds (17)
Design the adaptive turning law as
q&ˆ xij = − ziψ xij ( x) qˆ& yij = ziψ yij ( y ) © 2013 ACADEMY PUBLISHER
(28)
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Choose a Laypunov function as n n p1 1 V = ∑ zi2 + ∑∑ (θ%xij ) 2 i =1 i =1 j =1 2
0
(29)
1 + ∑∑ (q%ij ) 2 i =1 j =1 2
-100
x4
p2
n
-200
-300
It is easy to prove that
-400 150
n
V& ≤ ∑ − zi f zi ( zi ) ≤ 0
40
100
20
(30)
0
50 -20
i =1
0
x3
So the synchronization problem of chaotic system with multi-unknown parameters and uncertain functions are solved. V. SIMULATION RESULTS Take the below super chaotic system as an example, it can be described as follows:
x&1 = a ( x2 − x1 ) + klb x4 cos x2 + u1
-40
x2
Figure 2. Trajectory of uncontrolled chaotic systems(2).
be described as
y&1 = a y ( y2 − y1 ) + klb (1 + sin( y2 y3 )) y2 + u1
(38)
y& 2 = by y1 − ky1 y3 + klb y4 cos y2 + u2 (39)
(31)
x&2 = bx1 − k1 x1 x3 + klb (1 + sin( x2 x3 )) x2 + u2
(32)
2 1
x&3 = −cx3 + hx + klb(2 − cos( x1 x2 x3 x4 )) x1 + u3 (33) x&4 = − dx1 + klb x3 (3 + sin( x1 x3 )) + u4 (34) The unknown nonlinear function satisfies below assumption: * 1
klb x4 cos x2 ≤ q x4 , klb (1 + sin( x2 x3 )) x2 ≤ q2* x2
(35)
klb(2 − cos( x1 x2 x3 x4 )) x1 ≤ q3* x1
(36)
y&3 = −c y y3 + hy12 + klb y3 (3 + sin( y1 y3 )) + u3
(40)
y& 4 = − d y y1 + klb(2 − cos( y1 y2 y3 y4 )) y1 + u4
(41)
The
unknown
parameters
are
as ( a y , b y , c y , d y ) = (9, 39, 2.4, − 10.5) states
of
driven
system
are
chosen , initial
chosen
as
( x1 , x2 , x3 , x4 ) = (1, −1, 2, −2) , initial states of
response system are as ( y1 , y2 , y3 , y4 ) = (−3,3, −5,5) .
chosen
0
klb x3 (3 + sin( x1 x3 )) ≤ q4* x3
(37)
Parameters of response system are unknown, and it can
x4
-100
-200
-300
-400 150 40
100
20 0
50 -20 x3
150
0
-40
x2
Figure 3. Trajectory of uncontrolled chaotic systems(3).
x3
100
50
0 40 20
20 10
0
0
-20 x2
-10 -40
-20
x1
Figure 1. Trajectory of uncontrolled chaotic systems(1).
© 2013 ACADEMY PUBLISHER
The simulation result is as follows. The behavior of uncontrolled chaotic system can see fig 1 to fig4. Since it is a four dimension system, so four three-dimension figures are used to show the dynamic behavior.
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140
0
120 -100
x4
100 -200 80 x3&y3
-300
-400 150
60 40
40
100
20
20
0
50 -20 0
x3
-40
0
x2
Figure 4. Trajectory of uncontrolled chaotic systems(4).
-20
0
2
4
6 t
8
10
12
Figure 7. Trajectory of state x3 and y3.
The fellow fig 5 to fig 8 shows the comparison between the trajectory of the driven system and it of response system.
The above fig 7 shows the difference between state x3 and state y3.
20 400 15 300 10 200 100
0 x4&y4
x1&y1
5
-5 -10
0 -100
-15
-200
-20 -300 -25
0
2
4
6 t
8
10
12
-400
0
2
4
6 t
Figure 5. Trajectory of state x1 and y1.
8
10
12
Figure 8. Trajectory of state x4 and y4.
The above fig 5 shows the difference between state x1 and state y1. 50
The above fig 8 shows the difference between state x4 and state y4. As fig 5 to 8 shows, the driven chaotic system can not synchronize with response system without control.
40
x2&y2
30 20
25
10
20
0
15
-10
10 x1&y1
-20 -30
5 0
-40
0
2
4
6 t
8
10
12
Figure 6. Trajectory of state x2 and y2.
-5 -10 -15
The above fig 6 shows the difference between state x2 and state y2.
© 2013 ACADEMY PUBLISHER
0
2
4
6 t
8
Figure 9. Tracing curve of state x1.
10
12
JOURNAL OF COMPUTERS, VOL. 8, NO. 1, JANUARY 2013
The above fig 9 shows the synchronization behavior between state x1 and y1 with different initial states. 40 30
183
The above fig 12 shows the synchronization behavior between state x4 and y4 with different initial states. And according to the above figures (Fig9, 10,11,12) , the driven system can synchronize with response system by the synchronization controller proposed in this paper.
20 0.5 10
-0.5
-10
-1
-20
-1.5 e1
x2&y2
0 0
-2
-30 -40
-2.5 0
2
4
6 t
8
10
12 -3
Figure 10. Tracing curve of state x2.
-3.5 -4
0
2
4
6 t
8
10
12
Figure 13. Error curve of e1.
The above fig 10 shows the synchronization behavior between state x2 and y2 with different initial states. The above fig 13 shows that the synchronization error between x1 and y1 can converged to zero or near zero. 0.5 0 140
-0.5 -1
100
-1.5 e1
120
-2
x3&y3
80
-2.5 60 -3 40 -3.5 20 -4 0 -20
0
2
4
6 t
8
10
12
0
2
4
6 t
8
10
12
Figure 14. Error curve of e2.
Figure 11. Tracing curve of state x3.
The above fig 11 shows the synchronization behavior between state x3 and y3 with different initial states. 50 0 -50
x4&y4
-100 -150 -200 -250 -300 -350
0
2
4
6 t
8
Figure 12. Tracing curve of state x4.
© 2013 ACADEMY PUBLISHER
10
12
The above fig 14 shows that the synchronization error between x2 and y2 can converged to zero or near zero.
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ACKNOWLEDGMENT
7
The authors wish to thank their friend Heidi in Angels (a town of Canada) for her help, and thank Amado for his helpful suggestions. This paper is supported by National Nature Science Foundation of China 61174031 and 61102167.
6 5
e4
4 3 2
REFERENCE
1 0 -1
0
2
4
6 t
8
10
12
Figure 15. Error curve of e3.
The above fig 15 shows that the synchronization error 1 0 -1
e3
-2 -3 -4 -5 -6 -7
0
2
4
6 t
8
10
12
Figure 16. Error curve of e4.
between x3 and y3 can converged to zero or near zero. The above fig 16 shows that the synchronization error between x4 and y4 can converged to zero or near zero. And the above figures(fig 13,14,15,16) shows that the synchronization error can be controlled into a small field near 0 . So the synchronization controller designed in this paper is effective and efficient. V. CONCLUSIONS A robust adaptive controller is designed to solve the synchronization problem of a general kind of chaotic system with both unknown parameters and uncertain functions in driven system and response system. An adaptive turning law is constructed to handle the unknown parameters and a robust control law is adopted to deal with the uncertain functions in the system. Detailed numerical simulation is done to testify the adaptive synchronization strategy and simulation results shows that the synchronization can be realized without needing all accurate information of the driven and response system. So this adaptive synchronization controller is robust and it is an big advantage that is benefit to the engineering application.
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[1] Tsung-Ying Chiang, Jui-Sheng Lin Teh-Lu Liao, Jun-Juh Yan, Antisynch -ronization of uncertain unified chaotic systems with dead-zone nonlinearity , Nonlinear Analysis 68 (2008) 2629–2637, Physics Letters A 350 (2006) 36–43 [2] Bing Wang,Robust adaptive control of uncertain general low triangular nonlinear system and its application in electronic systems,[D],[Ph. D. Thesis].China University of Science and Technology, 2006 [3] Gang Cheng, Research on robust adaptive control of uncertain nonlinear system, [D], [Ph.D.Thesis], Zhejing University ,2006 [4] GE S S, Wang C, Adaptive control of uncertain chus’s circuits[J]. IEEE Trans Circuits System. 2000, 47(9): 13971402 [5] Alexander L, Fradkov, Markov A Yu. Adaptive synchronization of chaotic systems based on speed gradient method and passification[J]. IEEE Trans Circuits System 1997,44(10):905-912 [6] Dong X. Chen L. Adaptive control of the uncertain Duffing oscillator[J], Int J Bifurcation and chaos. 1997,7(7):1651-1658 [7] Tao Yang, Chun-Mei Yang and Lin-Bao Yang, A Detailed Study of Adaptive Contorl of Chaotic Systems with Unknown Parameters[J] . Dynamics and Control. 1998,(8):255-267 [8] Junwei Lei, Xinyu Wang, Yinhua Lei, How many parameters can be identified by adaptive synchronization in chaotic systems? [J] Physics Letters A, Volume 373, Issue 14, 23 March 2009, Pages 1249-1256 [9] M.T. Yassen，Adaptive chaos control and synchronization for uncertain new chaotic dynamical system[J]，Physics Letters A 350 (2006) 36-43 [10] Awad El Gohary, Rizk Yassen，Adaptive control and synchronization of a coupled dynamic system with uncertain parameters [J]，Chaos, Solitons and Fractals 29 (2006) 1085-1094 [11] Qiang Jia，Adaptive control and synchronization of a new hyperchaotic system with unknown parameters [J]， Physics Letters A 362 (2007) 424-429 [12] Rongwei Guo ，A simple adaptive controller for chaos and hyperchaos synchronization，Physics Letters A，360 (2009) 38-53 [13] Junwei Lei, Xinyu Wang, Yinhua Lei, A Nussbaum gain adaptive synchronization of a new hyperchaotic system with input uncertainties and unknown parameters, [J] Communications in Nonlinear Science and Numerical Simulation, Volume 14, Issue 8, August 2009, Pages 34393448 [14] Xinyu Wang,Junwei Lei, Yinhua Lei,Trigonometric RBF Neural Robust Controller Design for a Class of Nonlinear System with Linear Input Unmodeled Dynamics[J]. Applied Mathmatical and Computation,185 (2007) 989– 1002 [15] Xinyu Wang, Hongxing Wang, Hong Lei, Junwei Lei , Stable Robust Control for Chaotic Systems Based on Linear-paremeter-neural-networks, The 2nd International
JOURNAL OF COMPUTERS, VOL. 8, NO. 1, JANUARY 2013
Conference on Natural Computation (ICNC'06) and the 3rd International Conference on Fuzzy Systems and Knowledge Discovery (FSKD'06), Sep 24-28, 2006, Xi-an, China
Xinyu Wang was born in Yantai, Shandong province of China in 1965.She received the B. Eng degree in Physics from Liaoning Normal College in 1985. She received the Master Degree in Secure Communication from Naval Aeronautical Astronautical University, Yantai of China in 2006 . After that she continued her study there and received the Doctor degree in Communication and Information System in 2010 . She worked as a teacher in Yantai University in Shandong province of China in 1985 and was promoted to be a vice professor in 2000 and became a professor in 2009. Now her present interests are chaotic system, communication, physics and information.
© 2013 ACADEMY PUBLISHER
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Junwei Lei was born in Chibi, Hubei province of China on 9th Nov, 1981. He received the B. Eng degree in Missile Control and Testing and the Master Degree in Control Theory and Control Engineering from Naval Aeronautical Astronautical University, Yantai of China in 2003 and 2006 respectively. After that he continued his study there and received the Doctor degree in Guidance , Control and Navigation in 2010 . He worked in NAAU as an assistant teacher in 2009 and became a lecture in 2010. His present interests are neural networks, chaotic system control, variable structure control and adaptive control.
Mr Lei studied in Canada Military Force and Language School in Base Bodern of Toronto from Jan to Jun in 2010.
JOURNAL OF COMPUTERS, VOL. 8, NO. 1, JANUARY 2013
Synchronization of Uncertain Chaotic Systems with Unknown Response Systems Xinyu Wang
Institute of Science and Technology for Opto-electronic Information, Yantai University, China, Email: [email protected]
Hongxing Wang and Junwei Lei
Dept of Electronic Information and Engineering, Naval Aeronautical and Astronautical University Yantai, China Dept of Control Engineering, Naval Aeronautical and Astronautical University Yantai, China Email: {wanghongxing1024, leijunwei}@126.com
Abstract—Considering the situation with the same dimension but different structure in driven system and response system, a new kind of robust adaptive synchronization controller is designed based on the Lyapunov stability theorem . Both the unknown parameters and uncertain functions are considered and an adaptive turning law is constructed to handle the unknown parameters and a robust control law is adopted to deal with the uncertain functions in the system. At last, a four dimension chaotic system is taken as an example to testify the rightness of the proposed method. And numerical simulation result shows that the proposed method can realize the synchronization between driven chaotic system and response system with unknown parameters and uncertain nonlinear functions. Index Terms—chaos, adaptive, uncertainty, synchronization, unknown parameter , robust
I. 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[1-8]. Synchronization of chaos systems with unknown parameters is investigated widely by researchers from various fields. M.T. Yassen[9] used Lyapunov stability theorem and adaptive control method to realize synchronization between chaotic systems with unknown parameters, and the situation considered is a new kind of chaotic system with the same structure in driven system as its of response system. Rizk Yassen [10]and Qiang Jia[11]also researched synchronization problem with the uncertain parameters in chaotic system and designed an adaptive controller, but it is mostly based on the situation with linear parameters uncertainties. Rongwei Guo[12]designed a kind of simple nonlinear adaptive feedback synchronization controller with LaSalle theorem. The synchronization of two chaotic system is realized by this new and novel method, but it did not consider the situation of any uncertainty . Also the
© 2013 ACADEMY PUBLISHER doi:10.4304/jcp.8.1.178-185
chaotic system is acquired to satisfy a global Lipscitz conditions. In fact, input uncertainties usually exist in actual control systems, especially, unknown control direction is an special complex input certainty. but the situations of chaos systems with input uncertainties are neglected in most papers . Among all methods proposed to solve synchronization problem of chaotic systems[6-15], most of these researches are based on the situation that there only exist static uncertainties between driver system and response system. So the unmodelled dynamics of synchronization between chaotic system with different structure are seldom considered, especially for the situation that there exist static uncertainties and unknown parameters and dynamic uncertainties simultaneously. But it is very possible for the actual system that driven systems have different structure with response systems, or parameters may be changed unexpected because of the disturbance of environment , or the system model is inevitably inaccurate because of the dynamic uncertainties. The above situations are more possible to happen when synchronization of chaotic systems are used in the application of secure communication. So it is meaningful to study the synchronization of chaotic systems with both static uncertainties and dynamic uncertainties. Considering the situation that the driven system and response system have different structure, the synchronization problem of chaotic system was researched with totally unknown parameters both in the response system and in the driven system. But the uncertain nonlinear function situation was neglected . Based on the Lyapunov stability theorem and LMI method, Ju H. Park designed a new feedback synchronization controller for a class of chaotic system with the same structure between driven system and response system. In this paper, a general kind of chaotic system with the same dimension but different structure is studied and both the driven system and response system contains unknown parameters and unknown nonlinear functions. A new kind of robust adaptive synchronization controller is designed
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179
based on the Lyapunov stability theorem and robust and adaptive method. Also, the numerical simulation is done and simulation results testifies the effectiveness of the proposed method.
p3
y&3 = f y 3 ( y1 ,L , y4 ) + ∑ Fy 3 j ( y1 ,L , y4 )θ y1 j j =1
p4
+ ∑ Δ y 3 j ( y, t ) + b3u3 p3
Considering the following driven system and response system, the response system has unknown parameters and the driven system has both unknown parameters and uncertain nonlinear functions as follows. The driven system can be written as x& = f x ( x) + Fx ( x)θ x + Δ x ( x, t )
(1)
The response system can be described as y& = f y ( y ) + Fy ( y )θ y + Δ y ( y, t ) + bu
(9)
j =1
II. PROBLEM DESCRIPTION
(2)
Take a four dimension system as an example, it can be expanded as follows. The driven system can be expanded as
y& 4 = f y 4 ( y1 ,L , y4 ) + ∑ Fy 4 j ( y1 ,L , y4 )θ y 4 j j =1
p4
+ ∑ Δ y 4 j ( y, t ) + b3u4
(10)
j =1
Then θ x and θ y
are the unknown parameters . the
control target is to design a synchronization law
u = u ( x, y, θˆx , qˆ x , θˆy , qˆ y )
,
q&ˆ x = f ( x, y, qˆ x )
θˆy = f ( x, y,θˆy )
,
&
θˆx = f ( x, y,θˆx ) &
, ,
qˆ& y = f ( x, y, qˆ y ) such that the Synchronization of state can be achieved, it means that y → x .
p1
x&1 = f x1 ( x1 ,L , x4 ) + ∑ Fx1 j ( x1 ,L , x4 )θ x1 j j =1
p2
+ ∑ Δ x1 j ( x, t )
III. ASSUMPTIONS (3)
j =1
p1
x&2 = f x 2 ( x1 ,L , x4 ) + ∑ Fx 2 j ( x1 ,L , x4 )θ x 2 j j =1
p2
+ ∑ Δ xij ( x, t )
(4)
It is necessary to give the following typical assumptions to make the analysis of the system more convenient. Assumption 1: The driven system and response system have the same structure and it means that they also have the same dimension and f xi = f yi .
Assumption 2: The structure of some part of the driven
system is known, it means Fxij , f xi is known.
j =1
Assumption 3: The structure of some part of the p1
response system is known, it means
x&3 = f x 3 ( x1 ,L , x4 ) + ∑ Fx 3 j ( x1 ,L , x4 )θ x 3 j j =1
p2
+ ∑ Δ x 2 j ( x, t )
(5)
j =1
known. Assumption 4: The nonlinear functions of driven system satisfies the follow conditions such as there exists *
a unknown positive parameters qxij ≤ d xij such that for
1 ≤ i ≤ n , 1 ≤ j ≤ p2 , it holds
p1
x&4 = f x 4 ( x1 ,L , x4 ) + ∑ Fx 4 j ( x1 ,L , x4 )θ x 4 j j =1
p2
+ ∑ Δ x 4 j ( x, t )
(6)
And the response system can be expanded as y&1 = f y1 ( y1 ,L , y4 ) + ∑ Fy1 j ( y1 ,L , y4 )θ y1 j j =1
+ ∑ Δ y1 j ( y, t ) + b1u1
(7)
*
a unknown positive parameters q yij ≤ d yij such that for
1 ≤ i ≤ n , 1 ≤ j ≤ p4 , it holds
j =1
Δ yij ( X , t ) ≤ q*yijψ yij ( X )
p3
y& 2 = f y 2 ( y1 ,L , y4 ) + ∑ Fy 2 j ( y1 ,L , y4 )θ y 2 j j =1
+ ∑ Δ y 2 j ( y, t ) + b2u2 j =1
© 2013 ACADEMY PUBLISHER
(11)
smooth and non-negative function. Assumption 5: The nonlinear functions of response system satisfies the follow conditions such as there exists
p3
p4
* Δ xij ( X , t ) ≤ qxij ψ xij ( X )
Where d xij is a known constant and ψ xij ( X ) is known
j =1
p4
Fyij , f yi , bi is
(8)
Where
(12)
d yij is a known constant and ψ yij ( X ) is
known smooth and non-negative function.
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IV. DESIGN OF ADAPTIVE ROBUST SYNCHRONIZATION CONTROLLER
p2
zi ∑ {qˆ xijψ ij ( x) − Δ xij ( x, t )} j =1
Define a new error variable as zi = yi − xi for the driven and response system, the error system can be written as
p2
≤ ∑ {zi qˆ xijψ xij ( x) + qxij * zi ψ xij ( x)}
z&i = f yi ( y1 ,L , y4 ) − f xi ( x1 ,L , x4 ) p3
p4
j =1
j =1
p1
p2
j =1
j =1
p2
= ∑ { zi ψ xij ( x)[ sign( zi )qˆ xij + qxij * ]}
+ ∑ Fyij ( y1 ,L , y4 )θ yij + ∑ Δ yij ( y, t )
j =1
(13)
p4
zi ∑ {Δ yij ( y, t ) − qˆ yijψ yij ( x)}
−∑ Fxij ( x1 ,L , x4 )θ xij − ∑ Δ xij ( x, t ) + bi ui The control
j =1 p4
≤ ∑ {− zi qˆ yijψ yij ( y ) + q yij * zi ψ yij ( y )} (19)
ui is designed as
j =1 p4
= ∑ zi ψ yij ( y ){− sign( zi )qˆ yij + q yij *}
ui = f 2i ( x)[− f yi ( y1 ,L , y4 ) + f xi ( x1 ,L , x4 ) p1
j =1
Fxij ( x1 ,L , x4 )θˆxij
+∑ j =1 p2
p3
j =1
j =1
Define
+ ∑ qˆ xijψ xij ( x) − ∑ Fyij ( y1 ,L , y4 )θˆyij
(14)
p2
− ∑ qˆ yijψ yij ( y ) − f zi ( zi )]
q% xij = qxij * + sign( zi )qˆ xij
(20)
q% yij = q yij * − sign( zi )qˆ yij
(21)
Then
j =1
p2
where
zi ∑ {qˆ xijψ xij ( x) − Δ xij ( x, t )}
f 2i ( x) = bi−1
j =1
f zi ( zi ) = ki1 zi + ki 2 + ki 3 Then
(22)
p2
zi zi + ε i1
≤ ∑ { zi ψ xij ( x)q% xij }
3 1/ 3 zi exp( zi2 / 3 ) + ki 4 sign( zi ) 2
j =1
(15)
p4
zi ∑ {qˆ yijψ yij ( y ) − Δ yij ( y, t )} j =1
p1
zi z&i = zi [− f zi ( zi ) − ∑ Fxij ( x1 ,L , x4 )θ%xij
≤ ∑ { zi ψ yij ( y )q% yij } j =1
p2
+ ∑ {qˆ xijψ ij ( x) − Δ xij ( x, t )}
And consider that
j =1
&
(16)
p3
+ ∑ Fyij ( y1 ,L , y4 )θ%yij p4
&
+ ∑ {Δ yij ( y, t ) − qˆ yijψ yij ( x)}] j =1
Consider that
&
θ&%xij = θ&xij − θˆxij = −θˆxij
(24)
Design the adaptive turning law as
j =1
θ%xij = θ xij − θˆxij , θ%yij = θ yij − θˆyij
(23)
p4
j =1
where
(18)
j =1
θˆxij = − zi Fxij ( x1 ,L , x4 )
(25)
q&% xij = sign( zi )q&ˆ xij
(26)
q&% yij = − sign( zi )q&ˆ yij
(27)
Then it holds (17)
Design the adaptive turning law as
q&ˆ xij = − ziψ xij ( x) qˆ& yij = ziψ yij ( y ) © 2013 ACADEMY PUBLISHER
(28)
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181
Choose a Laypunov function as n n p1 1 V = ∑ zi2 + ∑∑ (θ%xij ) 2 i =1 i =1 j =1 2
0
(29)
1 + ∑∑ (q%ij ) 2 i =1 j =1 2
-100
x4
p2
n
-200
-300
It is easy to prove that
-400 150
n
V& ≤ ∑ − zi f zi ( zi ) ≤ 0
40
100
20
(30)
0
50 -20
i =1
0
x3
So the synchronization problem of chaotic system with multi-unknown parameters and uncertain functions are solved. V. SIMULATION RESULTS Take the below super chaotic system as an example, it can be described as follows:
x&1 = a ( x2 − x1 ) + klb x4 cos x2 + u1
-40
x2
Figure 2. Trajectory of uncontrolled chaotic systems(2).
be described as
y&1 = a y ( y2 − y1 ) + klb (1 + sin( y2 y3 )) y2 + u1
(38)
y& 2 = by y1 − ky1 y3 + klb y4 cos y2 + u2 (39)
(31)
x&2 = bx1 − k1 x1 x3 + klb (1 + sin( x2 x3 )) x2 + u2
(32)
2 1
x&3 = −cx3 + hx + klb(2 − cos( x1 x2 x3 x4 )) x1 + u3 (33) x&4 = − dx1 + klb x3 (3 + sin( x1 x3 )) + u4 (34) The unknown nonlinear function satisfies below assumption: * 1
klb x4 cos x2 ≤ q x4 , klb (1 + sin( x2 x3 )) x2 ≤ q2* x2
(35)
klb(2 − cos( x1 x2 x3 x4 )) x1 ≤ q3* x1
(36)
y&3 = −c y y3 + hy12 + klb y3 (3 + sin( y1 y3 )) + u3
(40)
y& 4 = − d y y1 + klb(2 − cos( y1 y2 y3 y4 )) y1 + u4
(41)
The
unknown
parameters
are
as ( a y , b y , c y , d y ) = (9, 39, 2.4, − 10.5) states
of
driven
system
are
chosen , initial
chosen
as
( x1 , x2 , x3 , x4 ) = (1, −1, 2, −2) , initial states of
response system are as ( y1 , y2 , y3 , y4 ) = (−3,3, −5,5) .
chosen
0
klb x3 (3 + sin( x1 x3 )) ≤ q4* x3
(37)
Parameters of response system are unknown, and it can
x4
-100
-200
-300
-400 150 40
100
20 0
50 -20 x3
150
0
-40
x2
Figure 3. Trajectory of uncontrolled chaotic systems(3).
x3
100
50
0 40 20
20 10
0
0
-20 x2
-10 -40
-20
x1
Figure 1. Trajectory of uncontrolled chaotic systems(1).
© 2013 ACADEMY PUBLISHER
The simulation result is as follows. The behavior of uncontrolled chaotic system can see fig 1 to fig4. Since it is a four dimension system, so four three-dimension figures are used to show the dynamic behavior.
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140
0
120 -100
x4
100 -200 80 x3&y3
-300
-400 150
60 40
40
100
20
20
0
50 -20 0
x3
-40
0
x2
Figure 4. Trajectory of uncontrolled chaotic systems(4).
-20
0
2
4
6 t
8
10
12
Figure 7. Trajectory of state x3 and y3.
The fellow fig 5 to fig 8 shows the comparison between the trajectory of the driven system and it of response system.
The above fig 7 shows the difference between state x3 and state y3.
20 400 15 300 10 200 100
0 x4&y4
x1&y1
5
-5 -10
0 -100
-15
-200
-20 -300 -25
0
2
4
6 t
8
10
12
-400
0
2
4
6 t
Figure 5. Trajectory of state x1 and y1.
8
10
12
Figure 8. Trajectory of state x4 and y4.
The above fig 5 shows the difference between state x1 and state y1. 50
The above fig 8 shows the difference between state x4 and state y4. As fig 5 to 8 shows, the driven chaotic system can not synchronize with response system without control.
40
x2&y2
30 20
25
10
20
0
15
-10
10 x1&y1
-20 -30
5 0
-40
0
2
4
6 t
8
10
12
Figure 6. Trajectory of state x2 and y2.
-5 -10 -15
The above fig 6 shows the difference between state x2 and state y2.
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0
2
4
6 t
8
Figure 9. Tracing curve of state x1.
10
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JOURNAL OF COMPUTERS, VOL. 8, NO. 1, JANUARY 2013
The above fig 9 shows the synchronization behavior between state x1 and y1 with different initial states. 40 30
183
The above fig 12 shows the synchronization behavior between state x4 and y4 with different initial states. And according to the above figures (Fig9, 10,11,12) , the driven system can synchronize with response system by the synchronization controller proposed in this paper.
20 0.5 10
-0.5
-10
-1
-20
-1.5 e1
x2&y2
0 0
-2
-30 -40
-2.5 0
2
4
6 t
8
10
12 -3
Figure 10. Tracing curve of state x2.
-3.5 -4
0
2
4
6 t
8
10
12
Figure 13. Error curve of e1.
The above fig 10 shows the synchronization behavior between state x2 and y2 with different initial states. The above fig 13 shows that the synchronization error between x1 and y1 can converged to zero or near zero. 0.5 0 140
-0.5 -1
100
-1.5 e1
120
-2
x3&y3
80
-2.5 60 -3 40 -3.5 20 -4 0 -20
0
2
4
6 t
8
10
12
0
2
4
6 t
8
10
12
Figure 14. Error curve of e2.
Figure 11. Tracing curve of state x3.
The above fig 11 shows the synchronization behavior between state x3 and y3 with different initial states. 50 0 -50
x4&y4
-100 -150 -200 -250 -300 -350
0
2
4
6 t
8
Figure 12. Tracing curve of state x4.
© 2013 ACADEMY PUBLISHER
10
12
The above fig 14 shows that the synchronization error between x2 and y2 can converged to zero or near zero.
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ACKNOWLEDGMENT
7
The authors wish to thank their friend Heidi in Angels (a town of Canada) for her help, and thank Amado for his helpful suggestions. This paper is supported by National Nature Science Foundation of China 61174031 and 61102167.
6 5
e4
4 3 2
REFERENCE
1 0 -1
0
2
4
6 t
8
10
12
Figure 15. Error curve of e3.
The above fig 15 shows that the synchronization error 1 0 -1
e3
-2 -3 -4 -5 -6 -7
0
2
4
6 t
8
10
12
Figure 16. Error curve of e4.
between x3 and y3 can converged to zero or near zero. The above fig 16 shows that the synchronization error between x4 and y4 can converged to zero or near zero. And the above figures(fig 13,14,15,16) shows that the synchronization error can be controlled into a small field near 0 . So the synchronization controller designed in this paper is effective and efficient. V. CONCLUSIONS A robust adaptive controller is designed to solve the synchronization problem of a general kind of chaotic system with both unknown parameters and uncertain functions in driven system and response system. An adaptive turning law is constructed to handle the unknown parameters and a robust control law is adopted to deal with the uncertain functions in the system. Detailed numerical simulation is done to testify the adaptive synchronization strategy and simulation results shows that the synchronization can be realized without needing all accurate information of the driven and response system. So this adaptive synchronization controller is robust and it is an big advantage that is benefit to the engineering application.
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[1] Tsung-Ying Chiang, Jui-Sheng Lin Teh-Lu Liao, Jun-Juh Yan, Antisynch -ronization of uncertain unified chaotic systems with dead-zone nonlinearity , Nonlinear Analysis 68 (2008) 2629–2637, Physics Letters A 350 (2006) 36–43 [2] Bing Wang,Robust adaptive control of uncertain general low triangular nonlinear system and its application in electronic systems,[D],[Ph. D. Thesis].China University of Science and Technology, 2006 [3] Gang Cheng, Research on robust adaptive control of uncertain nonlinear system, [D], [Ph.D.Thesis], Zhejing University ,2006 [4] GE S S, Wang C, Adaptive control of uncertain chus’s circuits[J]. IEEE Trans Circuits System. 2000, 47(9): 13971402 [5] Alexander L, Fradkov, Markov A Yu. Adaptive synchronization of chaotic systems based on speed gradient method and passification[J]. IEEE Trans Circuits System 1997,44(10):905-912 [6] Dong X. Chen L. Adaptive control of the uncertain Duffing oscillator[J], Int J Bifurcation and chaos. 1997,7(7):1651-1658 [7] Tao Yang, Chun-Mei Yang and Lin-Bao Yang, A Detailed Study of Adaptive Contorl of Chaotic Systems with Unknown Parameters[J] . Dynamics and Control. 1998,(8):255-267 [8] Junwei Lei, Xinyu Wang, Yinhua Lei, How many parameters can be identified by adaptive synchronization in chaotic systems? [J] Physics Letters A, Volume 373, Issue 14, 23 March 2009, Pages 1249-1256 [9] M.T. Yassen，Adaptive chaos control and synchronization for uncertain new chaotic dynamical system[J]，Physics Letters A 350 (2006) 36-43 [10] Awad El Gohary, Rizk Yassen，Adaptive control and synchronization of a coupled dynamic system with uncertain parameters [J]，Chaos, Solitons and Fractals 29 (2006) 1085-1094 [11] Qiang Jia，Adaptive control and synchronization of a new hyperchaotic system with unknown parameters [J]， Physics Letters A 362 (2007) 424-429 [12] Rongwei Guo ，A simple adaptive controller for chaos and hyperchaos synchronization，Physics Letters A，360 (2009) 38-53 [13] Junwei Lei, Xinyu Wang, Yinhua Lei, A Nussbaum gain adaptive synchronization of a new hyperchaotic system with input uncertainties and unknown parameters, [J] Communications in Nonlinear Science and Numerical Simulation, Volume 14, Issue 8, August 2009, Pages 34393448 [14] Xinyu Wang,Junwei Lei, Yinhua Lei,Trigonometric RBF Neural Robust Controller Design for a Class of Nonlinear System with Linear Input Unmodeled Dynamics[J]. Applied Mathmatical and Computation,185 (2007) 989– 1002 [15] Xinyu Wang, Hongxing Wang, Hong Lei, Junwei Lei , Stable Robust Control for Chaotic Systems Based on Linear-paremeter-neural-networks, The 2nd International
JOURNAL OF COMPUTERS, VOL. 8, NO. 1, JANUARY 2013
Conference on Natural Computation (ICNC'06) and the 3rd International Conference on Fuzzy Systems and Knowledge Discovery (FSKD'06), Sep 24-28, 2006, Xi-an, China
Xinyu Wang was born in Yantai, Shandong province of China in 1965.She received the B. Eng degree in Physics from Liaoning Normal College in 1985. She received the Master Degree in Secure Communication from Naval Aeronautical Astronautical University, Yantai of China in 2006 . After that she continued her study there and received the Doctor degree in Communication and Information System in 2010 . She worked as a teacher in Yantai University in Shandong province of China in 1985 and was promoted to be a vice professor in 2000 and became a professor in 2009. Now her present interests are chaotic system, communication, physics and information.
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Junwei Lei was born in Chibi, Hubei province of China on 9th Nov, 1981. He received the B. Eng degree in Missile Control and Testing and the Master Degree in Control Theory and Control Engineering from Naval Aeronautical Astronautical University, Yantai of China in 2003 and 2006 respectively. After that he continued his study there and received the Doctor degree in Guidance , Control and Navigation in 2010 . He worked in NAAU as an assistant teacher in 2009 and became a lecture in 2010. His present interests are neural networks, chaotic system control, variable structure control and adaptive control.
Mr Lei studied in Canada Military Force and Language School in Base Bodern of Toronto from Jan to Jun in 2010.
