Controlling uncertain Lü system using backstepping design☆

https://doi.org/10.1016/S0960-0779(02)00205-9Get rights and content

Abstract

In this paper, we discussed how to control Lü system with unknown parameters. Firstly we designed an observer to identify the unknown parameter of Lü system, then we used backstepping design method to control the system, and track any desired trajectory by the same way. At the same time we gave the numerical simulation for the results we had gained.

Introduction

In 1963, Lorenz firstly found a chaotic attractor in a simple three-dimensional autonomous system [1]. So far there are many researchers who studied the chaos theory. During the last decades dynamic chaos theory has been deeply studied and applied to many fields extensively, such as secure communications, optical system, biology and so forth.

In 1999, Chen and Ueta found a similar but non-equivalent chaotic attractor [2], which is known to be the dual of the Lorenz system. Recently Lü et al. [3] reported a new chaotic attractor which bridged the gap between the Lorenz system and chen system [4]. In the following we called it Lü system. In a recent work of Lü et al. [5], [6], the compound structure of Lü chaotic attractor was analysed, and its forming mechanism was explored [5]. The chaos synchronization between two linearly coupled chaotic system was studied in paper [6].

Chaos control is a new field in explorations of chaotic motions and it is crucial in applications of chaos. Until now, many different techniques and methods [7], [8], [9], [10], [11], [12] have been proposed to achieve chaos control, such as OGY method [7], differential geometric method [8] and linear state space feedback [9]. But for some system with unknown parameters, above mentioned methods cannot do with them. So there exists an important problem which is how to realize non-linear control for complex dynamical system with unknown parameters in chaos control field. The problem includes the identification of the unknown parameters and the approach to control chaos. In [13], a new adaptive control method was proposed for adaptive synchronization of two uncertain chaotic systems.

In this paper, firstly we designed an observer to identify the unknown parameter of Lü system. Then through backstepping design method we could control the Lü’s chaotic system. At the same time we used the same method to enable stabilization of chaotic motion to a stable state. Finally, in order to illustrate the effectiveness of the method we gave the computer numerical simulation.

Section snippets

Lü system

Recently, Lü et al. found a new chaotic attractor in a three-dimensional autonomous system which bridged the gap between the canonical Lorenz system and the Chen system. Lü system can be described as followed:ẋ=a(y−x),ẏ=−xz+cy,ż=xy−bz.When the parameters satisfy the following conditions a=36, b=3, and 12.7<c<17.0, 18.0<C<22.0, 23.0<c<28.5, 28.6<c<29.0, 29.2334<c<29.345, system (1) will generate chaotic attractor. For example in Fig. 1 there is a chaotic attractor, where a=36, b=3, c=20.

In

The identification of the unknown parameter

In this section, we will design an observer. Using it we can identify the unknown parameter b of system (1). Because b is a constant number, we haveḃ=0.We can let the unknown parameter b as a state variant, then the system (2) can be changed into the following system:ẋ=a(y−x),ẏ=−xz+cy,ż=xy−bz+u,ḃ=0.We can assume that the scalar variants x, y, z can be obtained as the system outputs. In the following we will design an observer to identify the unknown parameter b. From the third equation of

Controlling Lü system

In the following we will use backstepping design method to control Lü system. At first, we assume that the parameter b of controlled system (2) has been identified, that is, we have got b=b̄. Our purpose is to find a control law u, which can make system (2) stabilize at the origin point. For the virtual control y we design a stabilizing function α1(x) to make the derivative function of V1(x)=x2/2, i.e. V̇1=−ax2+axy be negative definite as y=α1(x). We can let α1(x)=0 and define the error varianty

Tracking Lü system

In this section, we discuss how to find a control law u so that we can track any desired trajectory r(t) with a scalar output y(t). Let ȳ be the deviation between the output y and the desired trajectory r(t), i.e. ȳ=y−r(t).

We define a function U2=(1/2)ȳ2 and calculate its time derivative along the system (2). We haveU̇2=ȳ(−xz+cȳ+cr(t)−ṙ(t)).We can choose an appropriate virtual control z to make (17) be negative definite, such asz=α2=cr(t)−ṙ(t)+(c+1)ȳx,satisfies the above-mentioned

Simulation

We will do some numerical simulations about the results that we have obtained. In the following all the differential equation are solved through using the fourth–fifth order Runge–Kutta method.

In Fig. 2 we can notice that the state variants x, y, z of Lü system is unstable for system (1). But for system (2) Fig. 3 shows that the chaotic system is quickly to its origin point whatever the initial condition is, that is, through using the backstepping design method we can stabilize the Lü system

Conclusion

An approach for controlling Lü chaotic system with unknown parameters using backstepping design method has been presented in this paper. And we have obtained some results as what we expected. Through using backstepping design method, we have known that the control law could drive a chaotic attractor to a stable state, even to any desired trajectory. At the same time, the method was shown to very effective, and the numerical simulation also accorded with the conclusion that we obtained.

Acknowledgements

The authors are grateful to Dr. Lü Jinhu for providing papers and helpful suggestions, and also thank the support of scientific and social practice of graduate student from CAS (Kind: Innovation Research).

References (13)

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Supported by the National Nature Science Foundation of China (Grant No. 10171099).

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