Among various fuzzy modeling themes, the Takagi-Sugeno (T-S) model [1] has been one of the most popular modeling frameworks. T-S fuzzy models can be as universal approximator then any smooth nonlinear control systems cab be approximated by T-S fuzzy models and also any smooth nonlinear state feedback controller can be approximated by the parallel distributed compensation (PDC) controller [2]. The predictive controller and T-S model-based linearization controller are, respectively, studied in [3,4]. However,most of these results are in the time domain instead of frequency domain.

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 [5] the stability of the Mamdani fuzzy control system is explored based on the Popov’s criterion, which is a frequency domain-based sufficient condition, so as to guarantee the stability of nonlinear feedback systems. Popov’s criterion is a frequency response method and it evaluate absolutely stable for a system that the forward path is a linear timeinvariant system, and the feedback part is a memoryless nonlinearity. In this paper, Popov’s criterion is utilized to drive the frequency domain-based sufficient condition, which provide a graphical interpretation for the stability analysis of the T-S fuzzy control systems.

The systems considered in this work have the interesting structure shown in Figure 1. The forward path is a linear time-invariant system, and the feedback part is a memoryless nonlinearity, i.e., a nonlinear static mapping.

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 Figure 1 has a special structure. If the feedback path simply contains a constant gain, i.e., , then the stability of the whole system, a linear feedback system, can be simply determined by examining the eigenvalues of the closed-loop system matrix. However, the stability analysis of the whole system with an arbitrary nonlinear feedback function is much more difficult. for using Popov’s criterion we usually require the nonlinearity to satisfy a so-called sector condition, whose definition is given below [6,7].

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 Figure 1 by equations of (1), (2) and (3) will be absolute stability.

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

Many researchers attempted to seek conditions that guarantee the stability of the nonlinear system in Figure 1. Popov’s criterion imposes additional conditions on the linear subsystem, leading to a sufficient condition for asymptotic stability reminiscent of Nyquist’s criterion(a necessary and sufficient condition) in linear system analysis.

A number of versions have been developed for Popov’s criterion [7]. The following basic version is fairly simple and useful.

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 [7].

Remark 1. If

and

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

Let us consider in Figure 2 which can be described by the following state model:

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 (Figure 2):

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 (Figure 3):

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 Figure 2, we observe that:

equations in (9) are equivalent the following equations:

from, we get:

in (11) is equivalent in Figure 3 and represents the functional mapping achived by the following fuzzy rules:

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 Figure 3, i.e.,. Then belong to the sector.

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 Figure 3, i.e.,. Then belong to the sector where .

Proof. clearly we have:

and also it is obvious that:

Then belongs to the sector where.

As 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 Figure 5. When the phase magnitudes are and, the corresponding log magnitudes are and, respectively. Then and are:

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 Figure 6, this figure reveals that the system is stable because Popov’s plot is below line of. Popov’s plot is obtained for. but for is the same, because functions of and are even functions.

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 Figure 1, Then Popov’s criterion imposes conditions for stability. We conclude T-S fuzzy control system in the Figure 2 is absolute stable.