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This paper deals with the control problem of the chaotic system subject to disturbance. The sliding mode surface is designed by singular system approach, and sufficient condition for convergence is given. Then, the adaptive sliding mode controller is designed to make the state arrive at the sliding mode surface in finite time. Finally, Lorenz system is considered as an example to show the effectiveness of the proposed method.

In the past decades, many studies have been devoted to the properties of nonlinear systems with applications [

For different chaos systems, different methods have been employed. In order to realize chaos control or synchronization, some well-known methods have been utilized, such as back-stepping method [

It should be noted that the recent innovation [

The rest of the paper is organized as follows. Section

In this section, we present the formulation of the problem and preliminaries which are necessary for our further investigation.

Let us consider the following chaotic system with disturbance:

When the disturbance

The disturbance

In many physical systems, the parameter

Since this paper employs the singular system to deal with the control of chaotic system, in what follows, we introduce some basics about the singular system.

Let us consider the following singular system:

The system (

regular if

impulse-free if

stable if all the roots of

admissible if it is regular, impulse-free, and stable.

The system (

In this section, we use the singular system approach to design the sliding mode surface for the system (

Let

The system (

The main task of this section is to design

It should be noted that (

Let

In order to make the computation more tractable, here, we set

It is known that (

In this section, the adaptive controller is designed to drive the state of the system (

Let

Consider the Lyapunov function as

It is known that the equilibrium

We show that (

We consider the Lorenz system (

We now complete the simulation by Simulink in Matlab. The initial state of the system (

The attractors of Lorenz system.

Response of the states of the closed-loop system (

The estimation of unknown parameter

We consider the control problem for the chaotic system, where the upper bound of the disturbance is unknown. We give sliding mode surface to guarantee the convergence of the sliding mode dynamic by singular system method and then design adaptive controller to make the closed-loop system reach the sliding mode in finite time. We also present a numerical example to show the validation of the proposed method. In the future work, we will extend the results of the paper to the hyper-chaotic system.

The authors declare that there is no conflict of interests regarding the publication of this paper.

This work is supported by Key Open Lab of Control Engineering of Henan Province (Grant no. KG 2011-13), Education Department of Henan Natural Science Research Key Project of China (Grant no.13A470342), and Science and Technology Research Project of China National Coal Association (Grant no. MTKJ2012-369).