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In this paper, we presented a sufficient condition on the frequency domain for the absolutely stable analysis of the Takagi-Sugeno (T-S)fuzzy control system, based on the Popov’s criterion. we use some numerical examples to illustrate the efficiency of frequency domain-based condition.

Among various fuzzy modeling themes, the Takagi-Sugeno (T-S) model [

On the other hand, the frequency response methods have been well developed and widely used in industrial applications with many advantages. for instance, the effect of noise in a control system can be evaluated in a straightforward way by its frequency response. In addition, Bode and Nyquist plots, which are often used in the frequency response methods, can also provide a graphic insight into the control system under investigation.

Stability is one of the most important nocepts concerning the design of control strategies. In [

The systems considered in this work have the interesting structure shown in

The equations of such systems can be written as:

where, , and. G(p) is transfer function for linear system of. The nonlinear system (1)-(3) has various physical applications. The nonlinear system in

Definition 1. A continuous function is said to belong to the sector, if there exists two nonnegative numbers and such that

Geometricaly, condition (4) implies that the nonlinear function always lies between the two straight lines and. Two properties are implied by Equation (4).

First, it implies that. Secondly, it implies that, such that the graph of lies in the first and third quadrants. Assume that both the nonlinearity is a function belonging to the sector and that the matrix of the linear subsystem in the forward path is stable (Hurwitz matrix). What additional constraints are needed to guarantee the stability of the whole system?

Definition 2. If the piont 0 (origin) is globally asymptotically stable for all nonlinearitys that belong to the sector, Then system in

We will see that Popov’s criterion creat conditions for asymptotic stability.

Popov’s CriterionMany researchers attempted to seek conditions that guarantee the stability of the nonlinear system in

A number of versions have been developed for Popov’s criterion [

Theorem 1. If the system described by (1), (2) and (3) satisfies the conditions:

• The matrix is Hurwitz (i.e., has all its eigenvalus strictly in the left half-plan)and the pair is controllable.

• The nonlinearity belongs to the sector.

• is equivalent:

then the point 0 is globally asymptotically stable.

Proof. see [

Remark 1. If

and

then inequality (5) is equivalent that the polar plot of be below the line.

Let us consider in

where, , and. D is a scalar. We assume that the pair is controllable, i.e., , and that the pair is observable, i.e.,

.

The T-S fuzzy controller consists of the following two rules (

where both and are controller inputs, and, , are the outputs of the two local proportional controllers. Then by using the center-of-gravity method for defuzzification, We can represent the vector controller as:

We use the triangular membership functions and of the following form (

where both and are given by the following equations:

We assume that, which are the proportional gains of the local controllers, are positive and. If both and to be negative then we can recast the nonlinear system by an equivalent system according to Theorem 2. In this case the local proportional gains can be made positive with the plant multiplied by.

Theorem 2. Two Systems in Figures 2 and 4 are equivalent.

Proof. We first observe that from (7) and (8) we have. For example, when, we have:

It is clear that:

from

equations in (9) are equivalent the following equations:

from, we get:

in (11) is equivalent in

Then Two Systems in Figures 2 and 3 are equivalent.

Therefore, if the functional mapping achieved by the T-S fuzzy controller belongs to some sector, then Popov’s criterion can be employed directly.

Theorem 3. Let denote the mapping of the T-S fuzzy control system in

Proof. from relation we have:

We know, and. Consequently, with the assumption that, we have:

If multiply to the last inequality then we have.

Theorem 4. Let denote the mapping of the T-S fuzzy control system in

Proof. clearly we have:

and also it is obvious that:

Then belongs to the sector where.

Numerical ApplicationsAs an example we consider a stable plant is described by:

We obtain proportional gains and, based on the Bode plot of. The Bode plot of is given in the

Phase margins of open loop for gains and are, respectively and. Next, the T-S fuzzy con-

troller rules are:

Hear,. We shall point out that this is chosen only for convenience. In fact, has no effect on the closed-loop stability. Popov’s plot are shown in

In this paper, we presented a condition on the frequency domain for the global stability analysis of the T-S fuzzy control system based on the Popov’s criterion and it’s graphical interpretation. We said T-S fuzzy control system can be like to system of