This paper analyzes semifinite time stability for a general chaotic system. By cooperating methods terminal sliding mode (TSM) with adaptive feedback control (AFC), a controller based on the two methods is derived to achieve semifinite time stability. The theoretical analysis employs the theories of linear matrix inequalities and Lyapunov functional method. Finally, numerical simulation is given to illustrate the derived theoretical results.

Chaos phenomenon can be found in many physics and engineering systems in practice. However, to improve the system’s performance, it is often desirable to avoid chaos, and various methods are proposed. Due to different emphases, controllers have different merits and drawbacks. For example, TSM establishes terminal sliding mode surface to couple system variables and control them to reach equilibrium points. Its control is effective, but it can only control system states coupled in the sliding mode surface; readers are referred to [

For the system’s structure constructed in this paper, we design controllers from methods TSM and AFC, both of which have unified formats. If TSM is used only, we should design several TSM surfaces. If AFC is used only, its controller is very simple and flexible, but it can only achieve asymptotical stability. Combining their merits and drawbacks, a cooperative controller is proposed in the paper. TSM method finite-timely controls system states, which are coupled in TSM surface, as in [

This design scheme can control main elements of system finite-time stability, firstly, then use AFC method to ensure that other dimensions are asymptotically stable, and finally realize the overall system’s semifinite time stability. Compared with TSM only, this design can greatly reduce the control input and simplify the design process of controller; compared with AFC only, it has obvious advantages in time sequence.

The rest of the paper is organized as follows. In Section

We use the following differential equation to describe general dynamic chaos system:

In this paper, we are committed to solve the stability analysis of chaotic systems which can be organized into the following form:

Consider that

Take dynamic system (

Assume that a continuous, positive-definite function

Assume that there exist positive constants

In this section, controller is designed from TSM and AFC separately with detailed theoretical analysis. TSM portion is used to derive

Take the system's first two parts of the states and add controlling part:

Generally speaking, the process of terminal sliding mode control can be divided into two stages: the first stage is to establish the nonsingular terminal sliding model surface; the second is to design TSM controller, which can make the system variables reach and maintain the TSM surface within finite time.

So in this paper, a nonsingular TSM surface is introduced as follows:

In the system (

Introduce the following Lyapunov function:

From Lemma

When the system states slide on the switch surface

Then

Thus, we can get the following sliding mode dynamics:

For differential equation (

Introduce the following Lyapunov function:

It can be proved easily that

Calculate its derivative along the solution of system (

Then differential equation (

After the finite-time convergence of system (

AFC controller

We introduce the following Lyapunov function:

The derivative of

So if and only if

Add related terms of

After the theoretical analyses are investigated, we discuss the following numerical example to illustrate the derived theoretical results. From [

The Chua oscillator is illustrated in Figures

The time series of uncontrolled Chua oscillator.

Chaotic attractor of uncontrolled Chua oscillator.

With constraint condition (

Time series of Chua oscillator with TSM only.

Time series of Chua oscillator with both TSM and AFC.

According to simulation results, it is easy to find that, the TSM method can effectively control the first two of system states and realize their finite-time stability. Comparing Figures

A controller, cooperating TSM with AFC, is proposed to control a class of chaotic system as described above in this paper. Two methods are complementary in the procedure and finally achieve good effectiveness. Complex TSM method controls main elements of chaotic system to finite-time stability; then, simple AFC method controls dimension elements of chaotic system to asymptotic stability and finally the overall system goes to semifinite time stability. This design scheme can not only guarantee the system’s convergence but also reduce the system’s control-input spending and also further improve their applications in chaos control.

In this paper, effective performance of the simulation results proves the feasibility of this design scheme. The proposed method can be applied in many famous chaotic systems such as Lorenz, liu chaotic system, and Chua’s circuit.

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

The authors are greatly indebted to the anonymous referees for their constructive comments. The work described in this paper was partially supported by the National Natural Science Foundation of China (no. 10971240), the Natural Science Foundation Project of CQ CSTC (nos. cstc2012jjA40052, cstc2013jcyjA0973, and cstc2013jcyjA80013), Applying Basic Research Program of Chongqing Education Committee (nos. KJ120615, KJ120630, KJ130611, and KJ1400505), the Foundation Project of Chongqing Normal University (no. 13XLZ01), and the Program of Chongqing Innovation Team Project in University under Grant no. KJTD201308.