1. Introduction

Fuzzy models T-S [1] call for wide observation of various practical industrial applications, mainly as the recognized T-S samples actually approximate nonlinear shapes. The essential characteristic on the sample TS is its general estimation of a nonlinear function. There are large number of results of literature that treat the globally difficulties utilizing the TS fuzzy samples, see [2]–[7].

The our existing sources on model reduction problem and disturbances are based on the whole full frequency (EF) area, which will give several types of model reduction design [8]–[13]. However, most practical industrial applications work in a FF domain. So far, a few applications have been made [14]–[19]. Thus, for this we will present new approaches to solve these problems.

The primary goal of our work is to define a fuzzy model reduction of discrete Model over FF ranges such a way that the augmented model is steady get a reduced H_∞ index in FF areas with disturbance is established as a prerequisite. we have also presented an example of simulation in order to exemplify the efficiency of the suggested method.

Notations :

• ”T” : Matrix transposition
• ” ∗ ” : Matrix symmetry
• M > 0 : matrix M is positive
• sym[D] : D + D^∗
• HeD∗

2. Problem statement

2.1    System formulation

Envisage the nonlinear model presented by :

Rule x: IF ζ₁(µ) is T₁ⁱ,… ζ_n(µ) is Tⁱ_n THEN

with z(µ) is the input; w(µ) is deliberate output; M_x, N_x and J_x are system parametres; ζ₁(µ) ,…, ζ_n(µ) the premise variables. d(µ) is a known disturbance signal located in a following FF areas

We describe the nonlinear system (1) employ singleton fuzzifer, center-average and inference product by the following relation :

with zˆ(µ) is state MR vector; wˆ (µ) is the output MR function, Mˆ _x, Nˆ _x and Jˆ_x are parameters should exist defined.

We get defuzzified for system (6) as following :

Definition 2.1 [20] Consider β > 0, if the following inequality is verified:

From Parseval function [21], we get :

We express the question of this work by : we design an appropriate fuzzy MR system (7) such that a error model is well-posed, stable satises the FF index:

2.2    Preliminaries

Lemma 2.2 [22] Let Φ ∈ R^m, L ∈ R^m×m and S ∈ R^p×m. Then, The following equations are the same:

• Φ^∗LΦ < 0, ∀Φ , 0 : SΦ = 0
• S^⊥TLS^⊥ < 0
• ∃θ ∈ R : L − θS^TS < 0
• ∃D ∈ R^m×p : L + DS + S^TD^T < 0

Lemma 2.3 Error system (10) is stable and FF in (15) is fulfilled, on condition that there are B, 0 < C, satisfying

Proof 2.4 The substantiation of Lemma 2.3 is defined in appendix.

3. FF performance analysis

Theorem 3.1 Error model (10) is stable, FF index (15) is fulfilled, on condition that there are X, B, C, E, Z, T satisfying C > 0,

which is nothing but (18).

4. FF performance Design

Theorem 4.1 Error model (10) is stable, FF index (15) is fulfilled,

Proof 4.2 Parameterise slack matrices E, Z, T in Theorem 3.1 as

Moreover, the Theorem 4.3 is to solve the FFMR problem in the FF        Therefore, us find Theorem 4.1. index (15).

Theorem 4.3 Error model (10) is stable, FF index (15) is fulfilled,          Remark 4.5 If us pick C = 0, we mastery employ theorem 4.3 to

 

Proof 4.4 We propose the following equations :

The comparison result with the technique proposed in Theorem 4.3 illustrate in Table 1, that indicate the little conservation of the method suggest in the paper.

The initial cases are proposed null. Figure 1 illustrate the errors e(k) from various approaches. It illustrate the FF method have the best performance in comparison with the EF approach.

Table 1: H_∞ performance apply from various approaches

Frequency	Methods	βmin	Max error
−π ≤ % ≤ π	[12]	0.9715	0.0915
−π ≤ % ≤ π	Th4.3 (Q=0)	0.5514	0.0423
	Th4.3	0.1305	0.0088

Figure 1: Error response of g_µ.

The ratio W(µ) is presented as :

Figures 2 indicate the the ratio in (37), we could notice whether the error model is stable, knowing that the initial Condition are null, that indicate the little conservation of the method suggest in the paper.

Figure 2: Value of W(µ).

6. Conclusion

This work, we dealt with problematic for the FF model reduction design of nonlinear systems over FF ranges. We have suggested a model reduction process in order to minimize the conservatism design using the frequency information of the disturbances and we have assumed that the disturbances are known in a recognized FF domain. Also, systematic techniques have been suggested for the generation of a model reduction which ensures asymptotic stability and FF H_∞ index, at the basis of a more general linearization procedure.

7. Appendix

Multiply equation (16) of    for right and by its transposition d(µ)

to the left, we get

From Parseval function [21], we get :

seeing that, (40) and the FF of signal input %˜₁ ≤ %˜ ≤ %˜₂, thus (15) is fulfilled.

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