Tracking of Super Chaotic System with Static Uncertain Functions and Unknown Parameters

(1)

Tracking of Super Chaotic System with Static

Uncertain Functions and Unknown Parameters

Yuqiang. Jin

Department of Training, Naval Aeronautical and Astronautical University, Yantai, China

Junwei Lei

Department of Control Engineering, Naval Aeronautical and Astronautical University, Yantai, China

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. I

NTRODUCTION

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

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

dead-zone 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

anti-synchronization problem for chaotic system with

dead-zone 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

method

[14-16,19-33,40]

, fuzzy control

[17]

, active

control

[46,51,52,53]

, feedback control

[50,54]

and robust

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 input and

multi-output problem is considered and a robust adaptive

synchronization law is designed for a four dimension

super chaotic system based on Lyapunov stability

theorem.

II. P

ROBLEM

D

ESCRIPTION

(2)

1

(

2 1

)

lb 4

cos

2 1

x

&

=

a x

−

x

+

k x

x

+

u

(1)

2 1 1 1 3 lb

(1 sin(

2 3

))

2 2

x

&

=

bx

−

k x x

+

k

+

x x

x

+

u

(2)

2

3 3 1

1 2 3 4 1 3

2 cos

(

))

lb

x

cx

hx

k

x x x x

x

u

= −

+

−

+

&

(

(3)

4 1 lb 3

(3 sin(

1 3

))

4

x

&

= −

dx

+

k x

+

x x

+

u

(4)

Without loss of generality, it can be written as

( )

( , )

x

&

=

f x

+

F x

θ

+ Δ

x t

+

bu

(5)

where

x

=

[ ,

x

₁

_L

,

x

_n

]

T

,

u

=

[ ,

u

₁

_L

,

u

_n

]

T

are

n

demen

si

-on

vectors. It can be expended as

1 1

( ,

1

,

4

)

( )

1

( , )

1 1

x

&

=

f x

_L

x

+

F x

θ

+ Δ

x t

+

b u

(6)

2 2

( ,

1

,

4

)

( )

2

( , )

2 2

x

&

=

f x

_L

x

+

F x

θ

+ Δ

x t

+

b u

(7)

3 3

( ,

1

,

4

)

( )

3

( , )

3 3

x

&

=

f x

L

x

+

F x

θ

+ Δ

x t

+

b u

(8)

4 4

( ,

1

,

4

)

( )

4

( , )

4 4

x

&

=

f x

L

x

+

F x

θ

+ Δ

x t

+

b u

(9)

where

f x

( )

are known functions of the system.

( )

F x

are known functions,

Δ

( , )

x t

are uncertain

dynamic functions ,

b

i

is a known parameter vecter.

The objective of tracking problem is to design a control

law

u

=

u x

( , , )

θ

ˆ ˆ

q

,

θ

&

ˆ

=

_{g x}

_{( , )}

θ

ˆ

,

q

ˆ

&

=

g x q

( , )

ˆ

such

that the state is stable and

x

→

0 .

Define

z

_i

= −

x

_i

x

_id

then

1 4 1 4

( ,

,

)

( ,

,

)

i i i i i

z

_&

=

f x

_L

x

+ Δ

x

_L

x

+

b u

(10)

III. A

SSUMPTION

The assumption of the above system is as follows:

Assumption 1: There exists unknown positive

parameter

q

_i*

≤

d

_i

for(

1 ≤ ≤

i

n

) such that

q

*_i

≤

d

_i

*

( , )

( )

i

X t

q

i

ψ

i

X

Δ

≤

(11)

where

d

i

is a unknown constant,

ψ

i

( )

X

is a known

smooth function.

IV. D

ESIGN

O

F

R

OBUST

A

DAPTIVE

C

ONTROLLER

Consider the situation with only one unknown

parameter , the I th subsystem can be written as

1 4

( ,

,

)

( )

( ,

,

)

i i i i

i i i

z

f x

x

F x

x

b u

θ

=

+

+ Δ

+

&

L

(12)

Design the control

u

i

_as

2 1 4

*

( )[

( ,

,

)

ˆ

_ˆ

( )

( )]

i i i

i i i i zi i

u

f

x

f x

x

F x

θ

q

ψ

x

f

z

=

−

L

(13)

1

2i

( )

i

f

x

=

b

−

,

1 2

1

1/ 3 2 / 3

3 4

( )

+

3 +

exp(

)+

( )

2

i

zi i i i i

i i

i i i i i

z

f

z

k z k

z

k

z

k sign z

ε

=

+

(14)

then

* * *

ˆ

[ ( )

( )

( )]

ˆ

( )

i i i i i i i i zi i

i zi i i i i i i i i i

z z

z

x

q

x

F x

f z

z f z

q z

x

q

x z

F x z

ψ

θ

ψ

θ

= Δ

−

+

−

≤ −

+

−

+

%

&

%

It holds

*

( )

i i i zi i i i i i i i

z z

&

= −

z f

z

+

z q

%

ψ

x

+

z

θ

%

F x

(15)

where

θ

%

_i

is defined as

θ θ θ

%

_i

= −

_i

ˆ

_i

and

* *

ˆ

( )

i i i i

q

%

=

q

−

q sign z

, then

ˆ

( )

i i i

q

&%

= −

q sign z

&

(16)

Design the adaptive turning law as

ˆ

_i _i

( )

_i

( )

_i

q

&

=

ψ

x z sign z

(17)

ˆ

_{( )}

i i

z F x

i i

θ

&%

= − =

θ

&

(18)

Then

* *2 * * * *

1 ( )

(

)

2 ( )

( )

0

i i i i

i i i

i i i i i

i i i i i i

z q

x

q

z q

x

q

x z sign z sign z

z q

x

q

x z

ψ

′

+

=

−

=

−

=

%

(19)

And

2

1 ( )

(

)

2 ( )

ˆ

( )

0

i i i

i i i

i i i i

z F x

F x

z F x

θ

θθ

θ

θθ

θ

′

+

=

+

=

−

=

−

=

%

&

%

% %

&

%

(20)

(3)

-60 -40 -20 0 20 40 -50 0 50 100 150 -400 -200 0 200 400 x2 x3 x4

Figure 2. Trajectory of uncontrolled chaotic systems(2).

2 *2 2

1

1 {

}

2

n

i i i

i

V

z

q

θ

=

∑

+

_%

+

%

(21)

Then it is easy to define that

( )

0

i zi i

V

&

≤ −

z f

z

≤

(22)

So the tracking can be fulfilled.

Consider the situation of multi-uncertain functions, the

model can be described as follows

1 4 1 1 4 1

( ,

,

)

( )

( ,

,

)

m

i i ij ij

j

r

ij i i

j

z

f x

x

F x

x

b u

θ

= =

=

+

Δ

+

∑

&

L

(23)

Design

u

i

as

2 1 4

1 * 1

( )[

( ,

,

)

ˆ

( )

ˆ

( )

( )]

i i i

m

ij ij j

r

ij ij zi i

j

u

f

x

f x

x

F x

q

x

f

z

θ

ψ

= =

=

−

∑

L

(24)

1

2i

( )

i

f

x

=

b

−

,

1 2

1 1/ 3 2 / 3

3 1 1 4 1

( )

3 exp(

)

(

)

2

i zi i i i i

i i

i i i i i

z

f

z

k z

k

z

k

z

k sign z

ε

= −

−

+

−

(25)

Define

θ

%

_ij

as

θ

%

_ij

=

θ θ

_ij

−

ˆ

_ij

and

* *

ˆ

( )

i ij ij i

q

%

=

q

−

q sign z

,Then

ˆ

( )

ij ij i

q

&%

= −

q sign z

&

(26)

Design the adaptive turning law as

ˆ

_ij _ij

( )

_i

( )

_i

q

&

=

ψ

x z sign z

(27)

ˆ

_{( )}

ij ij

z F x

i ij

θ

&%

= − =

θ

&

(28)

Define a Lyapunov function as

2 *2 2

1 1 1

1

1 {

(

)

(

) }

2

n m r

i ij ij

i j j

V

z

q

θ

= = =

′

=

∑

+

∑

_%

+

∑

%

(29)

So it is easy to prove that

( )

0

i zi i

V

&

≤ −

z f

z

≤

(30)

Then the tracking can be fulfilled.

V. N

UMERICAL

S

IMULATION

Take the below super chaotic system as an example, it

can be described as follows:

1

(

2 1

)

lb 4

cos

2 1

x

&

=

a x

−

x

+

k x

x

+

u

(31)

x

&

₂

=

bx

₁

−

k x x

_{1 1 3}

+

k

_lb

(1 sin(

+

x x

_{2 3}

))

x

₂

+

u

₂

(32)

2

3 3 1

1 2 3 4 1 3

2 cos(

))

lb

x

cx

hx

k

x x x x

x

u

= −

+

−

+

&

(

(33)

4 1 lb 3

(3 sin(

1 3

))

4

x

&

= −

dx

+

k x

+

x x

+

u

(34)

The unknown nonlinear function satisfies below

assumption:

*

4

cos

2 1 4

lb

k x

x

≤

q x

,

*

2 3 2 2 2

(1 sin(

))

lb

k

+

x x

x

≤

q x

(35)

*

1 2 3 4 1 3 1

2 cos(

))

lb

k

(

−

x x x x

x

≤

q x

(36)

*

3

(3 sin(

1 3

))

4 3

lb

k x

+

x x

≤

q x

(37)

Choose parameters as

a

=

10 ,

b

=

40 ,

c

=

2.5 ,

10.6 d

=

,

k

=

1 ,

h

=

4 ,

k

lb

= −

0.2 the system is

chaotic. And choose the initial state as

1

(0) 1

(4)

-20 -10 0

10 20

30

-50 0 50 -50 0 50 100 150

x1 x2

x3

Figure 1．Trajectory of uncontrolled chaotic systems(1).

0 2 4 6 8 10 12

-0.2 0 0.2 0.4 0.6 0.8 1

t

x1

Figure3. Trajectory of state x1.

0 2 4 6 8 10 12

-0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2

t

x2

0 2 4 6 8 10 12

-2 -1.5 -1 -0.5 0 0.5

t

x3

0 2 4 6 8 10 12

-0.5 0 0.5 1 1.5 2

t

x4

0 2 4 6 8 10 12

-0.5 0 0.5 1 1.5 2 2.5 3 3.5 4

t

x1

Figure 7.. Trajectory of state x1.

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.

The above figure 3 shows the stabilization process of

chaotic state x1.

The above figure 4 shows the stabilization process of

chaotic state x2 without unknown parameters.

The above figure 5 shows the stabilization process of

chaotic state x3 without unknown parameters

The above figure 6 shows the stabilization process of

chaotic state x4 without unknown parameters

(5)

0 2 4 6 8 10 12 -0.5

-0.4 -0.3 -0.2 -0.1 0 0.1 0.2

t

x2

0 2 4 6 8 10 12

-7 -6 -5 -4 -3 -2 -1 0 1

t

x3

Figure 9. Trajectory of state x3

0 2 4 6 8 10 12

-2 0 2 4 6 8 10

t

x4

Figure 10. Trajectory of state x4.

0 0.2 0.4 0.6 0.8 1 1.2 1.4

0.5 1 1.5 2 2.5 3 3.5 4

t

x1

Figure 11. Tracking of state x1

0 0.2 0.4 0.6 0.8 1 1.2 1.4

-0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 1.2

t

x2

Figure 12. Tracking of state x2

0 0.2 0.4 0.6 0.8 1 1.2 1.4

-7 -6 -5 -4 -3 -2 -1 0 1

t

x3

Figure 13. Tracking of state x3.

The above figure 7 shows the stabilization process of

chaotic state x1 with unknown parameters

The above figure 8 shows the stabilization process of

chaotic state x2 with unknown parameters

The above figure 9 shows the stabilization process of

chaotic state x3 with unknown parameters

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

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

chaotic state x1 with unknown parameters.

The above figure 12 shows the tracking process of

chaotic state x2 with unknown parameters.

(6)

0 0.2 0.4 0.6 0.8 1 1.2 1.4 0

2 4 6 8 10 12

t

x4

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. C

ONCLUSIONS

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.

A

CKNOWLEDGMENT

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.

R

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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.

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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 .