Finite frequency H∞ control for wind turbine systems in T-S form

In this work, we study H∞ control wind turbine fuzzy model for finite frequency (FF) interval. Less conservative results are obtained by using Finsler’s lemma technique, generalized Kalman Yakubovich Popov (gKYP), linear matrix inequality (LMI) approach and added several separate parameters, these conditions are given in terms of LMI which can be efficiently solved numerically for the problem that such fuzzy systems are admissible with H∞ disturbance attenuation level. The FF H∞ performance approach allows the state feedback command in a specific interval, the simulation example is given to validate our results.


INTRODUCTION
In recent years, Takagi-Sugeno (TS) fuzzy models [1] described by a set of IF-THEN rules could approximate any smooth nonlinear function to any specified accuracy within any compact set. In other words, it formulates the complex nonlinear systems into a framework that interpolates some affine local models by a set of fuzzy membership functions. Based on this framework, a systematic analysis and design procedure for complex nonlinear systems can be possibly developed in view of the powerful control theories and techniques in linear systems. Thus, it is expected that the TS fuzzy systems can be used to represent a large class of nonlinear systems and many important results on the TS fuzzy systems have been reported in the literature see [2][3][4][5][6][7][8][9][10][11][12].
Furthermore, the interest in the above mentioned literature is that all performances are given in the full frequency interval. However, when the external disturbance belong to a certain frequency range which is known beforehand, it is not favorable to control the system in the full frequency domain, because this may introduce some conservatism and poor system performance. Recently, the control synthesis in a FF interval has been addressed, and there have appeared many results in this domain of fuzzy systems [13][14][15][16][17][18].
In this work, we present a new method for finding solution to problem H ∞ state feedback wind turbine fuzzy model finite frequency specifications of TS model. Less conservative results are obtained by using the gKYP technique, Finslers lemma a to introduce, several separate parameters, and LMI approach, the sufficient conditions are given in terms of LMI which can be efficiently solved numerically for the problem that such fuzzy systems are admissible with H ∞ disturbance attenuation level in a specific interval. Numerical example is given to illustrate the effectiveness the presented results.

. Notations and lemma
In this part, We tell you a few symbols and Finslers lemme which will be hired in this article. Superscript " * " means matrix transposition. Notation Q > 0 means that the matrix Q > 0 is positive definite, symbol I represents the identity matrix where suitable dimension. sym(N ) denotes N + N * , diag{..} means for block diagonal matrix. [19] Let ψ ∈ R n , Z ∈ R n×n , M ∈ R m×n (rank (M) = k < n), M ⊥ ∈ R n×(n−k) be a classification matrix satisfactorily complete column MM ⊥ = 0 such that the following conditions :
By substituting (6) in (5) we obtain the following augmented model: where Let γ > 0, augmented fuzzy systems in (7)is said may be in H ∞ performance, the following index holds: From Parsevals theorems in [20,21] we have the following index holds: withW (τ ) ,Z(τ ) the Fourier transform of w(p) and Z(p).
The problem proposed in this work reads chosen as: The goal is to design a controller in (6) of model (5) such that : • System (7) is asymptotically stable.
• FF index holds: where is defined in Table 1; Table 1. Different frequency ranges is shortened to (10) (full frequency range (EFR)).

3.
FINITE FREQUENCY H ∞ CONTROLLER ANALYSIS Let γ > 0. For the system (7) is asymptotically stable satisfied FF index in (11), if there exists Hermitian parameters 0 < Q = Q T ∈ H n , P = P T ∈ H n in such a way that • Low-frequency range (LFR) : |τ | ≤τ l Finite frequency H ∞ control for wind turbine systems in T-S form (Salma Aboulem) • High-frequency range (HFR) : |τ | ≥τ h If only if all the parameters of the theorem 3. are non-party of membership functions, then the systems are a linears, and theorem 3. is shrunken to lemme in [22] which has proven to be an efficient being to treat the FF method for linear time-invariant models. Let γ > 0, system (7) is asymptotically stable, if there exists First,Ā(µ(p)) is stable, si S = S T > 0 in such a way that By applying the lemma 2.1. from (18) and (19), we obtain the inequality : who is nothing (16). Moreover, we consider the middle-frequency case. Applying lemma 3., the equation (12) are given by: By Schur complement, the following inequality with Applying the terms (2) and some easy manipulation we obtain exactly the inequalities (12), (13) and (14).

FINITE FREQUENCY H ∞ CONTROLLER DESIGN
Let γ > 0, system (7) is asymptotically stable, if there exists parameters 0 < Q = Q T ∈ H n , 0 < S = S T ∈ H n , P ∈ H n , Y (h), G such that the LMI (23) (24) feasible : -LFM : |τ | ≤τ lΨ -HFR : |τ | ≥τ hΨ The matrices gains are obtained by we get that (16) is equal to (23). Somewhere else, pre/post-multiplying (17) by invertible parameters Ξ = diag{G −1 , G −1 , I, I} and its transpose from the left and right we get that (17) is equal to (24). Then, theorem 4. is resolved the FF H ∞ performance for fuzzy continuous systems. Let γ > 0, system (7) is asymptotically stable, if there exists parameters 0 < Q = Q T ∈ H n , 0 < W = W T ∈ H n , P ∈ H n , G ∈ H n such that: The matrices gains are obtained by The proposed formulas following are: h i h jΨij so we gave theorem 4.. : We propose that the linear parameter equations (29) to non-real defined variables. by virtue of [23], the LMIs in non-real parameters can be transformd to an LMIs for greatmeasure in real parameters. While the equations Ω 1 + jΩ 2 < 0 is equivalent to Ω 1 Ω 2 −Ω 2 Ω 1 < 0, which involved the LMIs in (29) can be taken into account.

EXAMPLE
To demonstrate the effectiveness of FF proposed methods in this work. we provide a problem in the generator of the wind turbine. The variables in the wind turbine are assumed varying in the operating range: φ 1 ≤ φ ≤ φ 2 and ∇ 1 ≤ ∇ ≤ ∇ 2 , Consequently the nonlinear system (1) can be represented by the following four IF-THEN rules [24] with the numerical values given in Table 2  Therefore, the wind turbine system is given by the following approximated fuzzy model T-S : Rule 1: IF ∇ isÑ 1 (p)) and φ isM 1 (p)) THEṄ Rule 2: IF ∇ isÑ 1 (p)) and φ isM 2 (p)) THEṄ Rule 3: IF ∇ isÑ 2 (p)) and φ isM 1 (p)) THEṄ Rule 4: IF ∇ isÑ 2 (p)) and φ isM 1 (p)) THEṄ Numerical value: When the membership parameters are given by: Finite frequency H ∞ control for wind turbine systems in T-S form (Salma Aboulem) To illustrate the advantage of our method, we show in Table 3 the state feedback H ∞ performance, which shows the conservativeness of our method in this work. We suppose that ( 2 ≤ ω ≤ 6 ), let the disturbance be w(p) = (2 + p 1.

CONCLUSION
In this work , an effective finite frequency approach fuzzy systems has been studied and applied for the state feedback problem in disturbed wind turbine. founded on gKYP lemma and lyapunov function for stability with the states feedback control , a sufficient stability conditions proposed to deal with problem of control in specific domain. Based on this, new conditions have been given to guarantee the standard H ∞ performance has been revealed which has been illustrated by numerical examples.