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This paper is concerned with the time-delayed impulsive control and synchronization of general chaotic system based on T-S fuzzy model. By utilizing impulsive control theory, time-delayed feedback control technique, and T-S fuzzy model, some useful and new conditions are derived to guarantee the stability and synchronization of the addressed chaotic system. Finally, some numerical simulations are given to illustrate the effectiveness of the derived results.

Chaos may well be considered together with relativity and quantum mechanics as one of the three monumental discoveries of the twentieth century [

Although chaos is a very attractive subject for study, it was once believed to be neither predictable nor controllable due to its intrinsic topological complexity. However, recent research has shown that chaos can actually be useful under certain circumstances, such as enhanced mixing of chemical reactants. On the other hand, chaos should be weakened or completely suppressed when it is harmful. Therefore, it is very necessary and important to investigate chaos and chaos control.

Currently, many different techniques and methods based on conventional control theory have been extensively investigated to achieve chaos control [

Fuzzy control as a nontraditional method of control, in recent years, has received much attention as a powerful tool for the nonlinear control. Compared with the traditional control paradigm, the advantages of the fuzzy control paradigm are twofold. Firstly, a precise mathematical model of the controlled system is not required. In addition, a satisfactory nonlinear controller can often be developed empirically without using complicated mathematics [

On the other hand, the main drawback of fuzzy control systems is the lack of a systematic modeling and control design methodology. Particularly, stability analysis of fuzzy system is not easy, and parameter tuning is generally a time-consuming procedure, due to the nonlinear and multiparametric nature of fuzzy control systems [

The T-S fuzzy dynamic model originates from Takagi and Sugeno. In this framework, a nonlinear dynamical system is first approximated by a T-S fuzzy model. In this type of fuzzy model, local dynamics in different state-space regions are represented by linear models. The overall model of the system is achieved by fuzzy blending of these linear models. The control design is carried out based on the fuzzy model. For each local linear model, a feedback control is designed. The resulting overall controller, which is nonlinear in general, is again a fuzzy blending of each individual controller [

In addition, impulsive control method has been also widely used to stabilize and synchronize chaotic systems in recent years [

The delayed feedback control, as one of the important chaos control means, was introduced by Lithuanian physicist K. Pyragas in 1992 to stabilize an unstable periodic orbit of a nonlinear control system [

Although impulsive control, fuzzy control, and delayed feedback control are widely proposed in study of chaos and chaos control, to the best of our knowledge, there are few results on the stabilization and synchronization of chaotic system based on T-S fuzzy model by using the so-called time-delayed impulsive control which combine the superiorities of impulsive control and time-delayed feedback control. In addition, most previous researches have been restricted to linear impulsive control functions (e.g., see [

The paper is organized as follows. In Section

A T-S fuzzy model is described by a set of if-then rules, which characterize local relations of a nonlinear system in the state space. The main feature of a T-S model is to express the local dynamics of each fuzzy rule by a linear state-space system model, and the overall fuzzy system is modeled by fuzzy “blending” of these local linear system models through some suitable membership functions.

Consider the following chaotic system:

Rule

Using the singleton fuzzifier, product fuzzy inference, and weighted average defuzzifier (see [

It is clear that

If the input control signal in system (

Rule

Finally, for convenience and further study, we introduce the following lemmas.

For any two real column vectors

Assume that

In this section, we will design a time-delayed impulsive controller for chaotic system (

Control rule

From (

Evidently, the control input

In the following, we discuss the globally asymptotical stability of the origin of system (

Let

Furthermore, denote

From

The following theorem is provided to guarantee that the origin of system (

Under assumption

Choose a Lyapunov function as follows:

For

Then for

In the following, we consider the following two cases.

It follows from (

It follows from (

From the above discussion, we know that the trivial solution

If

If

Similarly, in this case, inequality (

Let

Under assumption

In the following, we consider the case where the impulsive controller is reduced to linearly impulsive controller; that is,

Let

In addition, let

System (

Choose

For system (

If

If

From Corollary

The synchronization of chaotic system based on T-S fuzzy model has been extensively studied in previous researches. For instance, [

Rule

Using the same methods in Section

Let

We observe that the synchronization of driving system (

For driving-driven systems (

Systems (

Systems (

Theorem

If the impulsive controllers

Evidently, Theorems

In this section, based on the results obtained in the previous sections, some numerical simulations are presented to show the effectiveness and feasibility of our results.

Consider the Rössler system described by the following form:

The chaotic behavior of system (

Now, we can construct an exact T-S fuzzy model of system (

From (

Control Rule 1: if

Control Rule 2: if

From system (

Choose

Time response of system (

The plots of the control inputs.

In the second simulation, we study time-delayed impulsive synchronization of the chaotic system (

The plot of the synchronization errors.

The driving system is given in (

Then from (

Let

The curve of

The curve of

The curve of

Time response of the control inputs.

In the paper, by using impulsive control method and the delayed feedback control mean, the stabilization and synchronization of chaotic system based on T-S fuzzy model are proposed; some useful and new conditions are obtained to ensure the stability and synchronization of the addressed chaotic system. In particular, some sufficient conditions which lead to the globally exponential stability of chaotic system are also obtained. It is noted that the impulsive functions can be nonlinear in this letter, which extend some previous results. Besides, in view of applying time-delayed feedback control technique, the delayed impulsive controllers are proposed in this paper. Finally, some numerical simulations are represented to show the effectiveness and feasibility of the developed methods.

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

This work was supported by the Natural Science Foundation of Xinjiang (Grant no. 2013211B06).