Adaptive Backstepping Sliding-mode Synchronization of Uncertain

Adaptive Backstepping Sliding-mode Synchronization of Uncertain

Multi-scroll Chaotic System

Yu-ye Wang, Hong-zhen Xu, Li-li Guo

Information and Communication Engineering College

Harbin Engineering University

Harbin, Heilongjiang, China

E-mail: wyuye2002@https://www.sodocs.net/doc/8d11960553.html,

Abstract

A synchronization method of adaptive backstepping sliding-mode control is presented based on the multi-scroll chaotic systems with hysteresis nonlinear function and unknown parameters. The strategy is designed by a step-by-step procedure interlacing. At each step, a coordinate transformation， the design of

a virtual control via a Lyapunov technique and the definition of a tuning function are derived, then, the globally control of the system is obtained. As a result, synchronization of the chaotic systems is achieved by controller. The synchronization error system arrives at the sliding-mode in finite time. Then, the controller can drive the error system to reach the origin. The identification of the unknown parameters is obtained at the same time. Because of using the adaptive arithmetic and sliding-mode control, the controller is robustness of parameter uncertainties and disturbances. The analyses and simulation results proved the effectiveness and feasibility of the method. 1. Introduction

The chaotic system has many potential application fields especially in secure communication. Chaos _________________________________________ synchronization is the key to the application of secure communication. Since the pioneering work by Pecoraet al. in 1991 [1], chaos synchronization has been researched extensively. Many effective methods have been already proposed such as drive-response synchronization, feedback synchronization, impulse synchronization, adaptive synchronization and so on.

Most of the studies about synchronization of chaotic system are focus on Lorenz, Chua, R?ssler and Chen before. Recently, people are looking for new chaotic systems like Lü system, unified chaotic system, Liu system, Qi system and chaotic system based on hysteresis nonlinear function and so on [2]. The chaotic system based on hysteresis nonlinear can be made by second-order system and hysteresis function.

In recent years, adaptive backstepping method of chaos synchronization has attracted many researchers’ attention [3-6]. Adaptive backstepping method is a powerful and systematic technique that recursively interlaces the choice of a Lyapunov function with the design of control. It can be applied to any uncertain chaotic system which is strict-feedback form or can be transformed into this type [3,4]. The primary advantage is only one controller is required to guarantee the stabilization of the chaotic system regardless of the order of the system .

In this paper, based on the multi-scroll chaotic systems with hysteresis nonlinear and unknown parameters, a synchronization method of adaptive backstepping control is presented. It use only one controller and has robustness. The method can make the control of the system and the estimate of the parameters achieved at the same time.

This work was supported by the Scientific Research

Foundation of Harbin Engineering University

(HEUFT06023), Postdoctoral Science Foundation of

Heilongjiang Province of China (LBH-Z06056) and

Research Fund of Harbin of China (2007RFQXG025).

2008 International Conference on Computer and Electrical Engineering

2. A new chaotic system

A chaotic system based on hysteresis function was proposed in literature [2] which is described by

111111111

()()()x y y

ax by af x f x hys kx =??

=?++??=? (1) Where x 1, y 1 are the states; a , b are parameters; k is proportional coefficient, f (x 1) is a feedback control variable which is a hysteresis nonlinear function with transfer characteristic shown in Figure 1. The hysteresis function is described by

11

0 ()0[()] 1 ()1[(_)]

0()1x t hys x t x t hys x t x t ≤??

=≥??<

Where 1[(_)]hys x t

is represented the value of 1[()]hys x t at a previous of time t.

hys 1[x (t )]1

1

x (t )

Figure 1. Transfer characteristic of 1[()]hys x t

The system (1) has a 2-scroll chaotic attractor as shown in Figure 2. when a =1，b =0.125，k =1.

x 1

y 1

Figure 2. Phase plot of 2-scroll chaotic attractor

Double hysteresis function is given as Figure 3. by regarding hysteresis function as extending to the third quadrant to realize the 3-scroll chaotic system.

hys[x (t )]101

-1

-1

x (t )

Figure 3. Transfer characteristic of double-hysteresis

function

Define

11

1 ()1[()]0 ()0

[(_)]1()0x t hys x t x t hys x t x t ???≤???=≥???<

11[()][()][()]hys x t hys x t hys x t ?=+

The 3-scroll chaotic system as shown in Figure 4. is gained by double-hysteresis function replacing hysteresis function in system (1) [7]. It can be described as system (4).

11111111()()()

x y y

ax by af x f x hys kx =??

=?++??=?

(4) Where a =1，b =0.125，k =1.

x 1

y 1

Figure 4. Phase plot of 3-scroll chaotic attractor

3. Adaptive backstepping sliding-mode synchronization of the chaotic system

According to the 3-scroll chaotic system (4), the response system with the control input in the second state is proposed as follows.

()()()x y y

ax by af x u f x hys kx =??

=?+++??=? (5) Consider the parameters of drive system are unknown which need to be estimated, and the drive system is written as follows:

11111111()()()

x y y

ax by af x f x hys kx =??

=?++??=? (6) Where a , b are the estimates of a , b .

Subtract (6) from (5), then the error system can be written as follows:

111111()[()()]()()()()

x y y x a y b a e e e ae e x be e y e f x a f x f x u

f x hys kx f x hys kx =??

=?++????

+?+??

=??=? (7) Where 1x e x x =?, 1y e y y =?, a e a a =?,

b e b b =?.

Now the problem of synchronization is transformed into designing a controller u and selecting the appropriate parameters to guarantee the system (7) is steady in the origin.

The adaptive backstepping sliding-mode design procedure includes two steps.

Step I . Let 1x h e =, Its derivative is given by

1

112

()x y h

e e h h β===+ where 11()h β is a stabilizing function for the virtual

control h 2 to be defined later in order to make the

derivative 111V h h = of Lyapunov function 2

1111()2

V h h =

negative definite. Definite 111()h ph β=?, where 0p >

is a control parameter. The virtual control

2111()y y h e h e ph β=?=+

Then 121

h h ph =? . And 2111112V h h ph h h ==?+

become negative (The h 1h 2 will be cancelled in the next step).

Step 2. According to the above derivation we have

211

1111()()[()()]y x a y b a y

h e h ae e x be e y e f x a f x f x u pe β=?=?++??+?++ where a , b is the estimate of a , b.

So the following (h 1,h 2)-subsystem is obtained

1212111121()[()()]()x a y b a h h ph h ae e x be e y e f x a f x f x u p h ph ?=???=?++????+?++???

(8)

Choose Lyapunov function defined by

222212112111(,)()()()222

V h h V h h a a b b =++?+?

Its derivative is

221122221122121121212221

2

212211212

()()[(()())()][()][][(()())(1)()][()][x y a b x y a b V ph h h h h a a a b b b ph h h h ae be a f x f x u p h ph e a x h f x h e b y h ph th h ae be a f x f x u p h t p h e a x h f x h e =?+++?+?=?++?++?++?++?+?=??+?++?++?++++?+

12]

b y h ?

Select the sliding-mode 2s h =, we have the controller and parameters estimate update laws as follows.

1212122

112()()()

(()())(1)()sgn()

x y a f x h x h n a a b y h m b b u ae be a f x f x p h t p h q s ?=?????=???=???????

?+??

(9) where t , q , n and m are positive control parameters. Then

22222122()()V ph th n a a m b b q h =??????? is negative definite.

When 0s ≠,2

V q ≤? . The sliding-mode s arrives at zero in finite time.

Hence, the system 12(,,,)h h a a b b ?? is globally asymptotically stable in the equilibrium (0, 0). Again, according to 1x h e =, 21y h e ph =+, the origin (0, 0) is

still the equilibrium of the error system (,)x y e e . a a

→, b b → because of the globally

asymptotically stable of (,)a a b b ??. And

2111

2(()())(1)()sgn()

y u a f x f x h be p h p t h q s =???????+?

is bounded.

So the synchronization of two uncertain multi-scroll chaotic system is achieved under only one controller and the parameter estimate update laws (9).

4. Numerical simulation

In order to verify the effectiveness of the proposed synchronization approach, the numerical simulations are done using matlab software.

The true values of “unknown” parameters are chosen 1a = and 0.125b =. The design parameters of controller and parameter update laws (9) are chosen 2p t ==, 1q =and 5n m ==. The initial values of two systems are taken as 1(0)0.6x =,

1(0)0.3y = and (0)0.5x =, (0)0.1y =. The initial values of estimate for unknown parameters a , b are assumed (0)3a =,10b =.

Numerical simulation results are shown in Figure 5-8. Figure 5. shows the sliding-mode s arrives at zero in finite time. Figure 6. shows the errors x e and y e reach zero. Figure 7. is the graph of controller u . Figure 8. shows the correctness of the parameters estimate.

t/s

s

Figure 5. T he graph sliding-mode s

t/s

e x , e

y

Figure 6. The graph of the errors x e and y e

t/s

u

Figure 7. The graph of the controller u

t/s

a ,b

Figure 8. The estimates of unknown parameter

a and b

The simulation results with the external disturbance 0.5sin(150)d y t = insert into the second function in the response system (5) are shown in Figure 9. The error system (x e ,y e ) is still asymptotically stable in the origin (0, 0), then the robustness of the synchronization

t/s

e x

, e

y

Figure 9. The graph of the errors x e and y e

with disturbance

5. Conclusion

In this paper, an adaptive backstepping design has been proposed for synchronization of two multi-scroll chaotic systems with unknown parameters. This method can realize chaos synchronization and gain the identification of unknown parameters with only one controller. The advantages of this method are that the synchronization and the identification of unknown parameters are achieved simultaneously and strong robustness of external disturbance is obtained. Numerical simulations are given to verify the effectiveness of the proposed synchronization method.

Acknowledgements

This work was supported by the Scientific Research Foundation of Harbin Engineering University (HEUFT06023), Postdoctoral Science Foundation of Heilongjiang Province of China (LBH-Z06056) and Research Fund of Harbin of China (2007RFQXG025).

References

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