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JOURNAL OF COMPUTERS, VOL. 7, NO. 12, DECEMBER 2012
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Tracking of Super Chaotic System with Static Uncertain Functions and Unknown Parameters Yuqiang. Jin Department of Training, Naval Aeronautical and Astronautical University, Yantai, China Email: [email protected]
Junwei Lei Department of Control Engineering, Naval Aeronautical and Astronautical University, Yantai, China Email: [email protected]
Yong Liang Department of Control Engineering, Naval Aeronautical and Astronautical University, Yantai, China Email: [email protected]
Abstract—The tracking of a four dimension super chaotic system with unknown parameters and uncertain static functions is researched in this paper and a robust adaptive tracking law is designed according to the Lyapunov stability theorem. Especially, the multi-unknown functions and parameters are solved by designing of adaptive turning law. It is a meaningful method both in theory and in engineering practice because the multi-input multi output system considered in this paper is complex and close to the secrete communication situation, in which the tracking and synchronization of chaotic system is usually applied. Index Terms— adaptive,chaos,uncertainty,stabilization, robust, unknown parameters.
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[1-7]. Synchronization of chaos systems with unknown parameters was investigated widely by researchers from various fields. The stability of tracking problem of a kind of single input and single output nonlinear systems , which can be transformed into strict feedback form, was researched in [1,2] under the situation that there exist unknown parameters and uncertain nonlinear functions[33-56]. But it needs the assumption that the bounds of unknown parameters and bound functions of uncertain nonlinear functions are known. Manfeng Hu and Tiegang Gao and E.M.Elabbasy[4,5,6] studied the unknown parameters problem with different strategies and Manfeng Hu used the parameter identification method to cope the unknown parameters in synchronization of chaotic system, but all states of the system were used to construct the control law. Fang Tang and Zheng-Ming Ge[7,8]used adaptive method to solve synchronization problem with © 2012 ACADEMY PUBLISHER doi:10.4304/jcp.7.12.2853-2860
uncertainties. Gauthier, J. P[9]used a simple observer method, which is very effective and novel. Khalil and Hao Lei [10,11,12] used output feedback methods to synchronize chaotic systems but the defect is that the method depend on high gain feedback. A kind of deadzone nonlinearity[34,45] was studied in synchronization of chaotic systems and Her-Terng Yau [35] researched the input nonlinearity situation. Synchronization problem with different structures was considered by Jian Huang [36-38]. The reduce order synchronization problem was studied by Ming-Chung Ho[39]. Tsung-Ying Chiang[41] studied the antisynchronization problem for chaotic system with deadzone nonlinearity. And input nonlinearity was considered by Her-Terng Yau[42] . Sliding mode control strategy was adopt in synchronization by Haitao Yu[43] and Chao-Lin Kuo[44]. Also other control methods were studied by researchers to solve the synchronization problem with uncertainties, such as sliding mode control [13],adaptive fuzzy control[17], active method[14-16,19-33,40], [46,51,52,53] , feedback control [50,54]and robust control control[18, 47 48,49]. The synchronization problem of chaotic systems, which can be transformed into single input and single output nonlinear dynamic system, was researched under the conditions that there are both unknown parameters and unknown nonlinear functions. But only the single input and single output situation is concerned in those references[21-28]. In this paper, the multi-input and multioutput problem is considered and a robust adaptive synchronization law is designed for a four dimension super chaotic system based on Lyapunov stability theorem. II. PROBLEM DESCRIPTION Consider a four dimension super chaotic system with static uncertain nonlinear functions and unknown parameters as follows:
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x&1 = a ( x2 − x1 ) + klb x4 cos x2 + u1
z&i = fi ( x1 ,L , x4 ) + Fi ( x)θi
(1)
x&2 = bx1 − k1 x1 x3 + klb (1 + sin( x2 x3 )) x2 + u2 (2)
ui as
Design the control
x&3 = −cx3 + hx12 + klb(2 − cos ( x1 x2 x3 x4 )) x1 + u3
ui = f 2i ( x)[− f i ( x1 ,L , x4 ) − F ( x)θˆ − qˆ *ψ ( x) − f ( z )]
(3)
i
x&4 = − dx1 + klb x3 (3 + sin( x1 x3 )) + u4
(12)
+ Δi ( x1 ,L , x4 ) + bi ui
i
i
i
zi
(13)
i
f 2i ( x) = bi−1 ,
(4)
Without loss of generality, it can be written as
x& = f ( x) + F ( x)θ + Δ( x, t ) + bu
f zi ( zi ) = ki1 zi +ki 2
(5)
where x = [ x1 , L , xn ] , u = [u1 , L , un ] are n demensi T
T
(6)
x&2 = f 2 ( x1 ,L , x4 ) + F ( x)θ + Δ 2 ( x, t ) + b2u2
(7)
then
zi z&i = zi [Δi ( x) − qˆi*ψ i ( x) + θ%i Fi ( x) − f zi ( zi )] ≤ − z f ( z ) + q * z ψ ( x) − qˆ *ψ ( x) z + θ% F ( x) z i zi
(8)
x&4 = f 4 ( x1 ,L , x4 ) + F ( x)θ + Δ 4 ( x, t ) + b4u4
(9)
f ( x) are known functions of the system. F ( x) are known functions, Δ( x, t ) are uncertain dynamic functions , bi is a known parameter vecter.
where
The objective of tracking problem is to design a control
z&i = fi ( x1 ,L , x4 ) + Δi ( x1 ,L , x4 ) + bi ui (10)
i
i
i
i
q&%i = − q&ˆi sign( zi )
i
i
i
i
(15) and
(16)
Design the adaptive turning law as
q&ˆi = ψ i ( x) zi sign( zi ) &
θ&%i = −θˆi = zi Fi ( x)
(17) (18)
Then
1 zi q%i*ψ i ( x) + (q%i*2 )′ 2 * = zi q%i ψ i ( x) −
(19)
q%i*ψ i ( x) zi sign( zi ) sign( zi )
The assumption of the above system is as follows: Assumption 1: There exists unknown positive
= zi q%i*ψ i ( x) − q%i*ψ i ( x) zi = 0
≤ di for( 1 ≤ i ≤ n ) such that qi* ≤ di And
Δ i ( X , t ) ≤ qi*ψ i ( X )
(11) is a known
1 ziθ%i F ( x) + (θ%i 2 )′ 2 & = θ% F ( x) + θ%θ% i
IV. DESIGN OF ROBUST ADAPTIVE CONTROLLER Consider the situation with only one unknown parameter , the I th subsystem can be written as
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i
q%i* = qi* − qˆi sign( zi ) , then
III. ASSUMPTION
ψi(X )
i
i
u = u ( x,θˆ, qˆ ) , θ&ˆ = g ( x,θˆ) , q&ˆ = g ( x, qˆ ) such that the state is stable and x → 0 . d Define zi = xi − xi then
where di is a unknown constant, smooth function.
i
zi z&i = − zi f zi ( zi ) + zi q%i*ψ i ( x) + ziθ%i Fi ( x) is defined as where θ% θ% = θ − θˆ
law
*
i
It holds
x&3 = f3 ( x1 ,L , x4 ) + F ( x)θ + Δ 3 ( x, t ) + b3u3
parameter qi
(14)
3 +ki 3 zi1/ 3 exp( zi2 / 3 )+ki 4 sign( zi ) 2
-on vectors. It can be expended as
x&1 = f1 ( x1 ,L , x4 ) + F ( x)θ + Δ1 ( x, t ) + b1u1
zi zi + ε i1
i i
& = θ%i F ( x) − θ%iθˆi = z θ% F ( x) − θ% z F ( x) = 0 i i
i i
Define a Lyapunov function as
(20)
JOURNAL OF COMPUTERS, VOL. 7, NO. 12, DECEMBER 2012
n 1 1 1 V = ∑ { zi2 + q%i*2 + θ%i 2 } 2 2 i =1 2
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(21)
Then it is easy to define that
400
(22)
200
x4
V& ≤ − zi f zi ( zi ) ≤ 0
So the tracking can be fulfilled. Consider the situation of multi-uncertain functions, the model can be described as follows
-200
-400 150 100
m
z&i = fi ( x1 ,L , x4 ) + ∑ Fij ( x)θij j =1
r
0
40 20
50
0 -20
0 x3
-50
-40 -60
(23)
+ ∑ Δ ij ( x1 ,L , x4 ) + bi ui
x2
Figure 2. Trajectory of uncontrolled chaotic systems(2).
j =1
Design
ui as
V& ≤ − zi f zi ( zi ) ≤ 0
ui = f 2i ( x)[− f i ( x1 ,L, x4 ) m
− ∑ Fij ( x)θˆij −
(24)
Then the tracking can be fulfilled.
j =1
V.
r
∑ qˆ j =1
ψ ij ( x) − f zi ( zi )]
*
ij
− ki 3
(31)
x&2 = bx1 − k1 x1 x3 + klb (1 + sin( x2 x3 )) x2 + u2
(32)
x&3 = −cx3 + hx12 + klb(2 −
(25)
cos( x1 x2 x3 x4 )) x1 + u3
ij
θ%ij = θij − θˆij
as
and
* ij
q&%ij = − q&ˆij sign( zi )
The unknown nonlinear function satisfies below assumption:
klb x4 cos x2 ≤ q1* x4 , (26)
klb (1 + sin( x2 x3 )) x2 ≤ q2* x2
(35)
klb(2 − cos( x1 x2 x3 x4 )) x1 ≤ q3* x1
(36)
klb x3 (3 + sin( x1 x3 )) ≤ q4* x3
(37)
Design the adaptive turning law as
q&ˆij = ψ ij ( x) zi sign( zi )
(27)
&
θ&%ij = −θˆij = zi Fij ( x)
(28)
Define a Lyapunov function as n
1 1 m 1 r V = ∑ { zi2 + ∑ (q%ij*2 )′ + ∑ (θ%ij 2 )′} (29) 2 j =1 2 j =1 i =1 2 So it is easy to prove that
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(33)
x&4 = −dx1 + klb x3 (3 + sin( x1 x3 )) + u4 (34)
q% = q − qˆij sign( zi ) ,Then * i
x&1 = a ( x2 − x1 ) + klb x4 cos x2 + u1
3 1/ 3 zi1 exp( zi21/ 3 ) − ki 4 sign( zi1 ) 2 θ%
Define
zi zi + ε i1
NUMERICAL SIMULATION
Take the below super chaotic system as an example, it can be described as follows:
f 2i ( x) = bi−1 ,
f zi ( zi ) = − ki1 zi − ki 2
(30)
Choose parameters as a = 10 , b = 40 , c = 2.5 ,
d = 10.6 , k = 1 , h = 4 , klb = −0.2 the system is chaotic.
And
choose the initial state as x1 (0) = 1 , x2 (0) = −1 , x3 (0) = −2 , x4 (0) = 2 , the simulation result of free movement of chaotic system is as following figures1 and figure2.
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The above figure 4 shows the stabilization process of chaotic state x2 without unknown parameters. 150
0.5
100 0 x3
50
0 x3
-0.5
-50 50
-1 30 20
0
10 0 -10
-50
x2
-20
-1.5 x1
Figure 1．Trajectory of uncontrolled chaotic systems(1).
-2
0
2
4
6 t
8
10
12
Figure5. Trajectory of state x3.
The above figure 5 shows the stabilization process of chaotic state x3 without unknown parameters 2
1.5
1 x4
The above figure 1 shows the behavior of uncontrolled chaotic system states x1,x2 and x3 . The above figure 2 shows the chaotic behavior of uncontrolled states x2, x3 and x4. Consider a simple situation to do the simulation first, assume there is not unknown parameters, then it means that all the parameters are known for the controller designer. The objective of the controller is to design a controller law such that all the states of the chaotic system can converged to zero for any initial states of chaotic system. The stabilization process can see the below figures.
0.5
1
0
0.8 -0.5
0
2
4
6 t
0.6
8
10
12
x1
Figure6. Trajectory of state x4. 0.4
0.2
0
-0.2
0
2
4
6 t
8
10
12
Figure3. Trajectory of state x1.
The above figure 3 shows the stabilization process of chaotic state x1.
The above figure 6 shows the stabilization process of chaotic state x4 without unknown parameters Consider the situation with unknown parameters , the simulation is more complex and The stabilization of chaotic system can see figure7, figure 8 figure 9 and figure 10 . 4
0.2
3.5
0.1
3 2.5
0
x1
2 -0.1 x2
1.5 -0.2
1
-0.3
0.5 0
-0.4 -0.5 -0.5
0
2
4
6 t
8
10
12
0
2
4
6 t
8
Figure 7.. Trajectory of state x1. Figure4. Trajectory of state x2.
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The above figure 7 shows the stabilization process of chaotic state x1 with unknown parameters
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the method proposed in this paper, the simulation result is as below figure 11, figure 12, figure 13 and figure 14. The below figure 11 shows the tracking process of
0.2 4
0.1 3.5
0 3
-0.1
x1
x2
2.5
-0.2
2
-0.3 1.5
-0.4 1
-0.5
0
2
4
6 t
8
10
12
0.5 0
0.2
0.4
0.6
0.8
1
1.2
1.4
t
Figure8. Trajectory of state x2.
Figure 11. Tracking of state x1
The above figure 8 shows the stabilization process of chaotic state x2 with unknown parameters
chaotic state x1 with unknown parameters. 1.2
1
1
0
0.8 0.6
-1
0.4 x2
-2 x3
0.2 -3 0 -4 -0.2 -5
-0.4
-6 -7
-0.6
0
0.2
0.4
0.6
0.8
1
1.2
1.4
t 0
2
4
6 t
8
10
12
Figure 12. Tracking of state x2
Figure 9. Trajectory of state x3
The above figure 9 shows the stabilization process of chaotic state x3 with unknown parameters
The above figure 12 shows the tracking process of chaotic state x2 with unknown parameters.
10
1 0
8 -1 6
x3
x4
-2 4
-3 -4
2 -5 0 -6 -2
0
2
4
6 t
8
10
12
Figure 10. Trajectory of state x4.
The above figure 10 shows the stabilization process of chaotic state x4 with unknown parameters Without loss of generality, set the desired value as 1. Consider that a, b, c, d are unknown constants, using
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-7
0
0.2
0.4
0.6
0.8
1
1.2
1.4
t
Figure 13. Tracking of state x3.
The above figure 13 shows the tracking process of chaotic state x4 with unknown parameters.
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12
10
x4
8
6
4
2
0
0
0.2
0.4
0.6
0.8
1
1.2
1.4
t
Figure 14. Tracking of state x4.
The above figure 14 shows the tracking process of chaotic state x4 with unknown parameters. So the tracking of chaotic system can be realized ideally and good performance is achieved by the above method. VI. CONCLUSIONS The tracking of a four dimension super chaotic system with unknown parameters and uncertain static functions is researched in this paper and a robust adaptive tracking law is designed according to the Lyapunov stability theorem. Especially, the multi-unknown functions and parameters are solved by designing of adaptive turning law. ACKNOWLEDGMENT The authors wish to thank their friend Heidi in Angels (a town of Canada) for her help , and thank Amado for his many helpful suggestions. REFERENCES [1] Junwei Lei, Xinyu Wang, Yinhua Lei, Physics Letters A,Available online 7 February 2009 [2] Junwei Lei, Xinyu Wang, Yinhua Lei, A Nussbaum gain adaptive synchronization of a new hyperchaotic system with input uncertainties and unknown parameters, Communications in Nonlinear Science and Numerical Simulation, Available online 24 December 2008 [3] Xinyu Wang, JunweiLei,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 [4] Manfeng Hu, Zhenyuan Xu,Rong Zhang, and Aihua Hu, “Parameters identification and adaptive full state hybrid projective synchronization of chaotic (hyper-chaotic) systems,” Physics Letters A, 361(2007) , pp. 231-237 [5] Tiegang Gao, Zengqiang Chen, Zhuzhi Yuan, and Dongchuan Yu, “Adaptive synchronization of a new hyperchaotic system with uncertain parameters,” Chaos, Solitons and Fractals 33(2007) ,pp.922-928 [6] E.M.Elabbasy, H.N.Agiza, M.M. El-Dessoky, Adaptive synchronization of a hyperchaotic system with uncertain parameter, Chaos, Solitons and Fractals 30(2006) 11331142.
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Yuqiang Jin was born in Hengshui, Hebei province of China in 1977. He received the B. Eng degree in Electronic Automation and the Master Degree in Control Theory and Control Engineering from Naval Aeronautical Astronautical University, Yantai of China in 2000 and 2003 respectively. After that he continued his study there and received the Doctor degree in Guidance , Control and Navigation in 2006 . He worked in NAAU as an assistant teacher in 2003 and became a lecture in 2005. In 2008, he was promoted to be a vice professor. His present interests are chaotic system, aircraft control and neural networks.
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.
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Yong Liang was born in Yantai, Shandong province of China on Nov, 1976. He received the B. Eng degree in Aeronautical Electronic Device and the Master Degree in Control Theory and Control Engineering from Naval Aeronautical Astronautical University, Yantai of China in 1998 and 2001 respectively. After that he continued his study and received the Doctor degree in Guidance , Control and Navigation in 2010 from The Second Artillery Engineering College in Xian of China . He worked in NAAU as an teacher in 2008 and became a assistant professor in 2009. His present interests are guidance of warship, spacecraft control, variable structure control and adaptive control.
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JOURNAL OF COMPUTERS, VOL. 7, NO. 12, DECEMBER 2012
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Tracking of Super Chaotic System with Static Uncertain Functions and Unknown Parameters Yuqiang. Jin Department of Training, Naval Aeronautical and Astronautical University, Yantai, China Email: [email protected]
Junwei Lei Department of Control Engineering, Naval Aeronautical and Astronautical University, Yantai, China Email: [email protected]
Yong Liang Department of Control Engineering, Naval Aeronautical and Astronautical University, Yantai, China Email: [email protected]
Abstract—The tracking of a four dimension super chaotic system with unknown parameters and uncertain static functions is researched in this paper and a robust adaptive tracking law is designed according to the Lyapunov stability theorem. Especially, the multi-unknown functions and parameters are solved by designing of adaptive turning law. It is a meaningful method both in theory and in engineering practice because the multi-input multi output system considered in this paper is complex and close to the secrete communication situation, in which the tracking and synchronization of chaotic system is usually applied. Index Terms— adaptive,chaos,uncertainty,stabilization, robust, unknown parameters.
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[1-7]. Synchronization of chaos systems with unknown parameters was investigated widely by researchers from various fields. The stability of tracking problem of a kind of single input and single output nonlinear systems , which can be transformed into strict feedback form, was researched in [1,2] under the situation that there exist unknown parameters and uncertain nonlinear functions[33-56]. But it needs the assumption that the bounds of unknown parameters and bound functions of uncertain nonlinear functions are known. Manfeng Hu and Tiegang Gao and E.M.Elabbasy[4,5,6] studied the unknown parameters problem with different strategies and Manfeng Hu used the parameter identification method to cope the unknown parameters in synchronization of chaotic system, but all states of the system were used to construct the control law. Fang Tang and Zheng-Ming Ge[7,8]used adaptive method to solve synchronization problem with © 2012 ACADEMY PUBLISHER doi:10.4304/jcp.7.12.2853-2860
uncertainties. Gauthier, J. P[9]used a simple observer method, which is very effective and novel. Khalil and Hao Lei [10,11,12] used output feedback methods to synchronize chaotic systems but the defect is that the method depend on high gain feedback. A kind of deadzone nonlinearity[34,45] was studied in synchronization of chaotic systems and Her-Terng Yau [35] researched the input nonlinearity situation. Synchronization problem with different structures was considered by Jian Huang [36-38]. The reduce order synchronization problem was studied by Ming-Chung Ho[39]. Tsung-Ying Chiang[41] studied the antisynchronization problem for chaotic system with deadzone nonlinearity. And input nonlinearity was considered by Her-Terng Yau[42] . Sliding mode control strategy was adopt in synchronization by Haitao Yu[43] and Chao-Lin Kuo[44]. Also other control methods were studied by researchers to solve the synchronization problem with uncertainties, such as sliding mode control [13],adaptive fuzzy control[17], active method[14-16,19-33,40], [46,51,52,53] , feedback control [50,54]and robust control control[18, 47 48,49]. The synchronization problem of chaotic systems, which can be transformed into single input and single output nonlinear dynamic system, was researched under the conditions that there are both unknown parameters and unknown nonlinear functions. But only the single input and single output situation is concerned in those references[21-28]. In this paper, the multi-input and multioutput problem is considered and a robust adaptive synchronization law is designed for a four dimension super chaotic system based on Lyapunov stability theorem. II. PROBLEM DESCRIPTION Consider a four dimension super chaotic system with static uncertain nonlinear functions and unknown parameters as follows:
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x&1 = a ( x2 − x1 ) + klb x4 cos x2 + u1
z&i = fi ( x1 ,L , x4 ) + Fi ( x)θi
(1)
x&2 = bx1 − k1 x1 x3 + klb (1 + sin( x2 x3 )) x2 + u2 (2)
ui as
Design the control
x&3 = −cx3 + hx12 + klb(2 − cos ( x1 x2 x3 x4 )) x1 + u3
ui = f 2i ( x)[− f i ( x1 ,L , x4 ) − F ( x)θˆ − qˆ *ψ ( x) − f ( z )]
(3)
i
x&4 = − dx1 + klb x3 (3 + sin( x1 x3 )) + u4
(12)
+ Δi ( x1 ,L , x4 ) + bi ui
i
i
i
zi
(13)
i
f 2i ( x) = bi−1 ,
(4)
Without loss of generality, it can be written as
x& = f ( x) + F ( x)θ + Δ( x, t ) + bu
f zi ( zi ) = ki1 zi +ki 2
(5)
where x = [ x1 , L , xn ] , u = [u1 , L , un ] are n demensi T
T
(6)
x&2 = f 2 ( x1 ,L , x4 ) + F ( x)θ + Δ 2 ( x, t ) + b2u2
(7)
then
zi z&i = zi [Δi ( x) − qˆi*ψ i ( x) + θ%i Fi ( x) − f zi ( zi )] ≤ − z f ( z ) + q * z ψ ( x) − qˆ *ψ ( x) z + θ% F ( x) z i zi
(8)
x&4 = f 4 ( x1 ,L , x4 ) + F ( x)θ + Δ 4 ( x, t ) + b4u4
(9)
f ( x) are known functions of the system. F ( x) are known functions, Δ( x, t ) are uncertain dynamic functions , bi is a known parameter vecter.
where
The objective of tracking problem is to design a control
z&i = fi ( x1 ,L , x4 ) + Δi ( x1 ,L , x4 ) + bi ui (10)
i
i
i
i
q&%i = − q&ˆi sign( zi )
i
i
i
i
(15) and
(16)
Design the adaptive turning law as
q&ˆi = ψ i ( x) zi sign( zi ) &
θ&%i = −θˆi = zi Fi ( x)
(17) (18)
Then
1 zi q%i*ψ i ( x) + (q%i*2 )′ 2 * = zi q%i ψ i ( x) −
(19)
q%i*ψ i ( x) zi sign( zi ) sign( zi )
The assumption of the above system is as follows: Assumption 1: There exists unknown positive
= zi q%i*ψ i ( x) − q%i*ψ i ( x) zi = 0
≤ di for( 1 ≤ i ≤ n ) such that qi* ≤ di And
Δ i ( X , t ) ≤ qi*ψ i ( X )
(11) is a known
1 ziθ%i F ( x) + (θ%i 2 )′ 2 & = θ% F ( x) + θ%θ% i
IV. DESIGN OF ROBUST ADAPTIVE CONTROLLER Consider the situation with only one unknown parameter , the I th subsystem can be written as
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i
q%i* = qi* − qˆi sign( zi ) , then
III. ASSUMPTION
ψi(X )
i
i
u = u ( x,θˆ, qˆ ) , θ&ˆ = g ( x,θˆ) , q&ˆ = g ( x, qˆ ) such that the state is stable and x → 0 . d Define zi = xi − xi then
where di is a unknown constant, smooth function.
i
zi z&i = − zi f zi ( zi ) + zi q%i*ψ i ( x) + ziθ%i Fi ( x) is defined as where θ% θ% = θ − θˆ
law
*
i
It holds
x&3 = f3 ( x1 ,L , x4 ) + F ( x)θ + Δ 3 ( x, t ) + b3u3
parameter qi
(14)
3 +ki 3 zi1/ 3 exp( zi2 / 3 )+ki 4 sign( zi ) 2
-on vectors. It can be expended as
x&1 = f1 ( x1 ,L , x4 ) + F ( x)θ + Δ1 ( x, t ) + b1u1
zi zi + ε i1
i i
& = θ%i F ( x) − θ%iθˆi = z θ% F ( x) − θ% z F ( x) = 0 i i
i i
Define a Lyapunov function as
(20)
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n 1 1 1 V = ∑ { zi2 + q%i*2 + θ%i 2 } 2 2 i =1 2
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(21)
Then it is easy to define that
400
(22)
200
x4
V& ≤ − zi f zi ( zi ) ≤ 0
So the tracking can be fulfilled. Consider the situation of multi-uncertain functions, the model can be described as follows
-200
-400 150 100
m
z&i = fi ( x1 ,L , x4 ) + ∑ Fij ( x)θij j =1
r
0
40 20
50
0 -20
0 x3
-50
-40 -60
(23)
+ ∑ Δ ij ( x1 ,L , x4 ) + bi ui
x2
Figure 2. Trajectory of uncontrolled chaotic systems(2).
j =1
Design
ui as
V& ≤ − zi f zi ( zi ) ≤ 0
ui = f 2i ( x)[− f i ( x1 ,L, x4 ) m
− ∑ Fij ( x)θˆij −
(24)
Then the tracking can be fulfilled.
j =1
V.
r
∑ qˆ j =1
ψ ij ( x) − f zi ( zi )]
*
ij
− ki 3
(31)
x&2 = bx1 − k1 x1 x3 + klb (1 + sin( x2 x3 )) x2 + u2
(32)
x&3 = −cx3 + hx12 + klb(2 −
(25)
cos( x1 x2 x3 x4 )) x1 + u3
ij
θ%ij = θij − θˆij
as
and
* ij
q&%ij = − q&ˆij sign( zi )
The unknown nonlinear function satisfies below assumption:
klb x4 cos x2 ≤ q1* x4 , (26)
klb (1 + sin( x2 x3 )) x2 ≤ q2* x2
(35)
klb(2 − cos( x1 x2 x3 x4 )) x1 ≤ q3* x1
(36)
klb x3 (3 + sin( x1 x3 )) ≤ q4* x3
(37)
Design the adaptive turning law as
q&ˆij = ψ ij ( x) zi sign( zi )
(27)
&
θ&%ij = −θˆij = zi Fij ( x)
(28)
Define a Lyapunov function as n
1 1 m 1 r V = ∑ { zi2 + ∑ (q%ij*2 )′ + ∑ (θ%ij 2 )′} (29) 2 j =1 2 j =1 i =1 2 So it is easy to prove that
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(33)
x&4 = −dx1 + klb x3 (3 + sin( x1 x3 )) + u4 (34)
q% = q − qˆij sign( zi ) ,Then * i
x&1 = a ( x2 − x1 ) + klb x4 cos x2 + u1
3 1/ 3 zi1 exp( zi21/ 3 ) − ki 4 sign( zi1 ) 2 θ%
Define
zi zi + ε i1
NUMERICAL SIMULATION
Take the below super chaotic system as an example, it can be described as follows:
f 2i ( x) = bi−1 ,
f zi ( zi ) = − ki1 zi − ki 2
(30)
Choose parameters as a = 10 , b = 40 , c = 2.5 ,
d = 10.6 , k = 1 , h = 4 , klb = −0.2 the system is chaotic.
And
choose the initial state as x1 (0) = 1 , x2 (0) = −1 , x3 (0) = −2 , x4 (0) = 2 , the simulation result of free movement of chaotic system is as following figures1 and figure2.
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The above figure 4 shows the stabilization process of chaotic state x2 without unknown parameters. 150
0.5
100 0 x3
50
0 x3
-0.5
-50 50
-1 30 20
0
10 0 -10
-50
x2
-20
-1.5 x1
Figure 1．Trajectory of uncontrolled chaotic systems(1).
-2
0
2
4
6 t
8
10
12
Figure5. Trajectory of state x3.
The above figure 5 shows the stabilization process of chaotic state x3 without unknown parameters 2
1.5
1 x4
The above figure 1 shows the behavior of uncontrolled chaotic system states x1,x2 and x3 . The above figure 2 shows the chaotic behavior of uncontrolled states x2, x3 and x4. Consider a simple situation to do the simulation first, assume there is not unknown parameters, then it means that all the parameters are known for the controller designer. The objective of the controller is to design a controller law such that all the states of the chaotic system can converged to zero for any initial states of chaotic system. The stabilization process can see the below figures.
0.5
1
0
0.8 -0.5
0
2
4
6 t
0.6
8
10
12
x1
Figure6. Trajectory of state x4. 0.4
0.2
0
-0.2
0
2
4
6 t
8
10
12
Figure3. Trajectory of state x1.
The above figure 3 shows the stabilization process of chaotic state x1.
The above figure 6 shows the stabilization process of chaotic state x4 without unknown parameters Consider the situation with unknown parameters , the simulation is more complex and The stabilization of chaotic system can see figure7, figure 8 figure 9 and figure 10 . 4
0.2
3.5
0.1
3 2.5
0
x1
2 -0.1 x2
1.5 -0.2
1
-0.3
0.5 0
-0.4 -0.5 -0.5
0
2
4
6 t
8
10
12
0
2
4
6 t
8
Figure 7.. Trajectory of state x1. Figure4. Trajectory of state x2.
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10
12
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The above figure 7 shows the stabilization process of chaotic state x1 with unknown parameters
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the method proposed in this paper, the simulation result is as below figure 11, figure 12, figure 13 and figure 14. The below figure 11 shows the tracking process of
0.2 4
0.1 3.5
0 3
-0.1
x1
x2
2.5
-0.2
2
-0.3 1.5
-0.4 1
-0.5
0
2
4
6 t
8
10
12
0.5 0
0.2
0.4
0.6
0.8
1
1.2
1.4
t
Figure8. Trajectory of state x2.
Figure 11. Tracking of state x1
The above figure 8 shows the stabilization process of chaotic state x2 with unknown parameters
chaotic state x1 with unknown parameters. 1.2
1
1
0
0.8 0.6
-1
0.4 x2
-2 x3
0.2 -3 0 -4 -0.2 -5
-0.4
-6 -7
-0.6
0
0.2
0.4
0.6
0.8
1
1.2
1.4
t 0
2
4
6 t
8
10
12
Figure 12. Tracking of state x2
Figure 9. Trajectory of state x3
The above figure 9 shows the stabilization process of chaotic state x3 with unknown parameters
The above figure 12 shows the tracking process of chaotic state x2 with unknown parameters.
10
1 0
8 -1 6
x3
x4
-2 4
-3 -4
2 -5 0 -6 -2
0
2
4
6 t
8
10
12
Figure 10. Trajectory of state x4.
The above figure 10 shows the stabilization process of chaotic state x4 with unknown parameters Without loss of generality, set the desired value as 1. Consider that a, b, c, d are unknown constants, using
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-7
0
0.2
0.4
0.6
0.8
1
1.2
1.4
t
Figure 13. Tracking of state x3.
The above figure 13 shows the tracking process of chaotic state x4 with unknown parameters.
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12
10
x4
8
6
4
2
0
0
0.2
0.4
0.6
0.8
1
1.2
1.4
t
Figure 14. Tracking of state x4.
The above figure 14 shows the tracking process of chaotic state x4 with unknown parameters. So the tracking of chaotic system can be realized ideally and good performance is achieved by the above method. VI. CONCLUSIONS The tracking of a four dimension super chaotic system with unknown parameters and uncertain static functions is researched in this paper and a robust adaptive tracking law is designed according to the Lyapunov stability theorem. Especially, the multi-unknown functions and parameters are solved by designing of adaptive turning law. ACKNOWLEDGMENT The authors wish to thank their friend Heidi in Angels (a town of Canada) for her help , and thank Amado for his many helpful suggestions. REFERENCES [1] Junwei Lei, Xinyu Wang, Yinhua Lei, Physics Letters A,Available online 7 February 2009 [2] Junwei Lei, Xinyu Wang, Yinhua Lei, A Nussbaum gain adaptive synchronization of a new hyperchaotic system with input uncertainties and unknown parameters, Communications in Nonlinear Science and Numerical Simulation, Available online 24 December 2008 [3] Xinyu Wang, JunweiLei,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 [4] Manfeng Hu, Zhenyuan Xu,Rong Zhang, and Aihua Hu, “Parameters identification and adaptive full state hybrid projective synchronization of chaotic (hyper-chaotic) systems,” Physics Letters A, 361(2007) , pp. 231-237 [5] Tiegang Gao, Zengqiang Chen, Zhuzhi Yuan, and Dongchuan Yu, “Adaptive synchronization of a new hyperchaotic system with uncertain parameters,” Chaos, Solitons and Fractals 33(2007) ,pp.922-928 [6] E.M.Elabbasy, H.N.Agiza, M.M. El-Dessoky, Adaptive synchronization of a hyperchaotic system with uncertain parameter, Chaos, Solitons and Fractals 30(2006) 11331142.
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[7] Fang Tang, Ling Wang, An adaptive active control for the modified Chua’s circuit, Physics Letters A ,346(2005) pp.342-346. [8] Zheng-Ming Ge, Cheng-Hsiung Yang, Pragmatical generalized synchronization of chaotic systems with uncertain parameters by adaptive control, Physica D ,231(2007),pp.87-89 [9] Gauthier, J. P, Hammouri, H, & Othman, S. (1992). A simple observer for nonlinear systems, applications to bioreactors. IEEE Transactions on Automatic Control, 37, 875–880. [10] Khalil, H. K., & Saberi, A. (1987). Adaptive stabilization of a class of nonlinear systems using high-gain feedback. IEEE Transactions on Automatic Control, 32, 1031–1035 [11] Hao Lei,Wei Lin,Universal adaptive control of nonlinear systems with unknown growth rate by output feedback,Automatica 42 (2006) 1783-1789 [12] Hao Lei,Wei Lin,Adaptive regulation of uncertain nonlinear systems by output feedback: Auniversal control approach , Systems & Control Letters 56 (2007) 529– 537 [13] Wang-Long Li, Kuo-Ming Chang, Robust synchronization of drive–response chaotic systems via adaptive sliding mode control, Chaos, Solitons and Fractals (2007) 321– 324 [14] Nussbaum, R. D. (1983). Some remarks on the conjecture in parameter adaptive control. Systems and Control Letters, 3(3), 243–246 [15] Ye, X. D., & Jiang, J. P. (1998). Adaptive nonlinear design without a prioriknowledge of control directions. IEEE Transactions on Automatic Control, 43(11), 1617–1621. [16] Alexander L.Fradkov, A Yu.Markov. Adaptive synchronization of chaotic system phased on speed gradient method and passification. IEEE Trans on Circuit Syst.(I),1997,44(10):905-912 [17] Young_Hoon Joo, Leang-San shieh, Guangrong Chen, Hybrid state-space fuzzy model-based controller with dualrate sampling for digital control of chaotic systems, IEEE Trans on Fuzzy Syst.(I),1999,7(4):394-408 [18] Alexander Pogromsky, Henk Nijmeijer. Observer-based robust synchronization of dynamical systems. Int,J. Bifurcation and Chaos,1998,8(11):2243-2254 [19] E.M.Elabbasy,H.N.Agiza,M.M.El-dessoky.Adaptive synchronization of lv System with uncertain parameters. Chaos,solitons and fractals. 2004,(21):657-667 [20] M.T.yassen. Adaptive synchronization of Rosser and lü systems with fully uncertain patameters. Chaos,solitons and fractals.2005,(23):1527-1536 [21] Ju H.Park. Adaptive sysnchronization of Rossler system with uncertain Parameters.Chaos, solitons and fractals.2005,(22):1-6. [22] T.L. Liao,Adaptive synchronization of two Lorenz system.Chaos.solitons and fractals.1998,9(9):1555-1561 [23] Yongguang Yu , Suochun Zhang, Adaptive backstepping synchronization of uncertain chaotic system[J], Chaos, Solitons and Fractals 21 (2004) 643–649 [24] M.T. Yassen,Adaptive chaos control and synchronization for uncertain new chaotic dynamical system[J],Physics Letters A 350 (2006) 36-43 [25] 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 [26] Qiang Jia,Adaptive control and synchronization of a new hyperchaotic system with unknown parameters [J],Physics Letters A 362 (2007) 424-429
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[45] 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 [46] Rong-An Tang , Ya-Li Liu , Ju-Kui Xue , An extended active control for chaos synchronization, Physics Letters A 373 (2009) 1449–1454 [47] Pei Xinzhe, Liu Zhiyuan, Pei Run, Robust trajectory tracking controller design for mobile robots with bounded input, [J] Acta automatica sinica, 2003,29(6):876-881 [48] Gao Ping Jing et al. A simple global synchronization criterion for coupled chaotic systems. Solitons and fractals. 2003,(15):925-935 [49] Jianping Yan, Changpin Li. On synchronization of three chaotic systems. Chaos,Solitons and fracrals.2005,(23):1683-1688 [50] C.Sarasola,F.J.Torrealdea,A.D Anjou. Feedback synchronization of chaotic Systems[J].Int.J.Bifurcation and chaos.2003,13(1):177-191 [51] M.T.Yassen.Chaos synchronization between two different chaotc systems using active control.Chaos,Solitons & Fractals.2005:23(4):131-140 [52] H.N.Agiza,M.T.Yassen. Synchronization of Rossler and Chen chaotic dynamical systems using active control.Physics Letters A.2001,(278):191-197 [53] Ming-chung Ho, Yao-Chen Hung.Synchronization of two different systems by Using generalized active control. Physics Letters A.2002,(301):424-428. [54] Jiang G.P., Tang K.S. A global synchronization criterion for coupled chaotic systems via unidirectional linear error feedback approach. Int. J. Bifurcation Chaos, 2002, 12(10):2239-2253
Yuqiang Jin was born in Hengshui, Hebei province of China in 1977. He received the B. Eng degree in Electronic Automation and the Master Degree in Control Theory and Control Engineering from Naval Aeronautical Astronautical University, Yantai of China in 2000 and 2003 respectively. After that he continued his study there and received the Doctor degree in Guidance , Control and Navigation in 2006 . He worked in NAAU as an assistant teacher in 2003 and became a lecture in 2005. In 2008, he was promoted to be a vice professor. His present interests are chaotic system, aircraft control and neural networks.
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.
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Yong Liang was born in Yantai, Shandong province of China on Nov, 1976. He received the B. Eng degree in Aeronautical Electronic Device and the Master Degree in Control Theory and Control Engineering from Naval Aeronautical Astronautical University, Yantai of China in 1998 and 2001 respectively. After that he continued his study and received the Doctor degree in Guidance , Control and Navigation in 2010 from The Second Artillery Engineering College in Xian of China . He worked in NAAU as an teacher in 2008 and became a assistant professor in 2009. His present interests are guidance of warship, spacecraft control, variable structure control and adaptive control.
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