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This paper develops a general analysis and design theory for nonlinear time-varying systems represented by impulsive T-S fuzzy control model, which extends conventional T-S fuzzy model. In the proposed, model impulse is viewed as control input of T-S model, and impulsive distance is the major controller to be designed. Several criteria on general stability, asymptotic stability, and exponential stability are established, and a simple design algorithm is provided with stability of nonlinear time-invariant systems. Finally, the numerical simulation for the predator-prey system with functional response and impulsive effects verify the effectiveness of the proposed methods.

Most plants in engineering, science, and industries have inherent nonlinearity and are difficult to design and control using general nonlinear systems. In order to overcome this kind of difficulties, many researchers have developed various schemes, among which a successful approach is fuzzy control combined with the linguistic knowledge representation. For instance, one can see the control of an unmanned helicopter [

However, it should be admitted that the stability of the TS-type FLC is still an open problem. It is well known that the parallel distributed compensation (PDC) technique in the framework of ordinary T-S model has been the most popular controller design approach and belongs to a continuous input control way. It is important to point out that there exist many systems that cannot commonly endure continuous control inputs, or they have impulsive dynamical behavior due to abrupt jumps at certain instants during the evolving processes such as communication networks, biological population management, prey (pest) management, and chemical control [

This paper introduces stability analysis and design of time-varying nonlinear systems based on impulsive fuzzy model. The main contribution of this paper lies in three aspects. Firstly, we generalize the model in [

The rest of this paper is organized as follows. Section

Firstly, we recall T-S model proposed by Takagi and Sugeno [

IF

In general, the stable problems of (

IF

(1) There are the distinct differences between (

(2) In our opinion, the system (

(3) It should be noted that, in this paper, we only investigate the first case, that is, the systems, in which impulses are said to be controller of nonimpulsive fuzzy plants, will be considered. In this case, we have

By using a singleton fuzzifier, product inference, and a center-average defuzzifier, the following dynamic global model of (

According to the above discussions, it is important to remember that

It should be noted that (

Now, we will give several definitions to be used in the sequel.

Let

For

A function

Now, we will study various stabilities of the impulsive fuzzy system (

Assume that

for each

for all

system (

system (

system (

Let

Similar to (

Similarly, when

It should be noted that Theorem

Suppose an

then

system (

system (

system (

To prove this theorem, we only need to check all the conditions of Theorem

For

(1) The condition (ii) of the theorem is very weak, because it is easy to choose

(2) Meanwhile, it is known that impulsive distance should be as large as possible, because by doing so implementation cost of impulsive control plants may be reduced. From this, it follows that

(3) Compared with PDC control technique of T-S model, our proposed approach here has such merits as simple, and that system plant is easy to be implemented by digital devices.

It should be pointed out that

Suppose that an

the origin of impulsive fuzzy systems (

system (

system (

Similar to the proof of Theorem

For

Then from (

According to Theorem

Notice the following algorithm, and let

Set a threshold

Calculate

Stop if

Repeat from Step

Repeat from Step

Once algorithm succeeds, we may determine the bound of impulsive distance by

In this section, we present a design example to show how to perform the impulsive fuzzy control on the predator-prey systems with functional response and impulsive effects. Especially, the biological systems are very much complex, nonlinear, and uncertain ones, which should be represented by fuzzy logical method with the linguistic description. So, the predator-prey systems with functional response and impulsive effects are investigated by the proposed impulsive T-S design model.

Now, consider a predator-prey system with functional response and impulsive effects as follows:

With little loss of generality, suppose that density-dependent prey growth and functional responses of preys both are nonlinear function, that is,

Hereafter, we can construct a fuzzy design model for representing (

Using defuzzification, product inference, and singleton fuzzifier, the global dynamics of the system (

Choose the parameters of system (

State diagram of predator-prey system with parameters

The impulsive control technique was analyzed in the framework of the fuzzy systems based on T-S model, and the proposed design approach is suitable for very complex and nonlinear system with impulsive effects. First, we approximate a nonlinear plant with a Takagi-Sugeno fuzzy model, in which the local dynamics in different state space regions are represented by linear impulsive models. Then, the overall impulsive fuzzy system is obtained by blending each local linear impulsive system. The design procedure is conceptually simple and straightforward. Meanwhile, the various stability results of the impulsive fuzzy system are derived by Lyapunov method, and the stability design algorithm is given. Finally, numerical example for predator-prey systems with functional response and impulsive effects is given to illustrate application of impulsive fuzzy control scheme, and simulation results show the effectiveness of the proposed method.

This work was supported by the National Natural Science Foundation of China, projects nos. 60604007 and 50975300 and the Natural Science Foundation of Guangxi, China, project no. 0899017.