New approach to finite frequency Filter Design for two-dimensional T-S fuzzy systems

This paper considers the problem of filter design for two-dimensional (2D) discrete-time non-linear systems in Takagi-Sugeno (T-S) fuzzy mode. The problem to be solved in the paper is to find a H∞ filter model such that the filtering error system is asymptotically stable. A numerical example is employed to illustrate the validity of the proposed methods.


Introduction
In the last few decades, many researchers have investigated two-dimensional (2-D) systems including continuous, discrete and continuous-discrete settings as these systems have great applications in engineering fields such as process control, multi-dimensional digital filtering, image processing, see for instance [1][2][3][4][5][6].
Among the most of the existed literature on filtering problems, the disturbances are considered in the entire frequency (EF) domain, which will bring over design in the filtering design. While many practical engineering problems are more suitable to be considered in finite frequency (FF) ranges.. So, the standard design approaches for the whole frequency domain may bring conservatism (see, [11][12][13][14] and the references therein).
The main objective of this paper is to design a filter for discrete T-S fuzzy Fornasini-Marchesini Models with disturbance in FF domain. The aim is to find a filter 2D T-S fuzzy FM model over FF ranges for a given 2D T-S FM such that the associated filtering error meets a prescribed H ∞ attenuation level. Then, the sufficient conditions for the existence of the desired filter are derived to ensure the asymptotic stability and FF H ∞ performance. A systematic filter design scheme is proposed, which could reduce the conservatism of the results compared to the one considered in EF domain. Finally, a simulation example demonstrates the usefulness of the proposed method.
Notations Superscript "T " stands for matrix transposition. In symmetric block matrices or long matrix expressions, we use an asterisk " * " to represent a term that is induced by symmetry. Notation P > 0 means that matrix P is positive. I denotes an identity matrix with appropriate dimension. Generally, sym{A} denotes A + A T , diag{..} * e-mail: khaddouj.benmeziane@isga.ma * * e-mail: abderrahim.elamrani@usmba.ac.ma * * * e-mail: ismail.boumhidi@usmba.ac.ma stands for block diagonal matrix. The l 2 norm for a 2D signal u(i, j) is given by in the l 2 space is an energy-bounded signal.

Problem description
In this paper, we consider a class of 2D nonlinear discretetime systems described by the following T-S FMLSS fuzzy model Plant Rule l: IF θ 1 (k) isÑ l 1 , θ 2 (k) isÑ l 2 , ... and θ α (k) isÑ l α , Then, where (Ñ l 1 , ...,Ñ l α ) are the fuzzy sets; l is the number of IF-THEN rules (l = 1, 2, ..., r); θ(k) = [θ 1 (k), θ 2 (k), ..., θ α (k)] are the premise variables; k = {(i, j + 1), (i + 1, j)}; x(i, j) ∈ R n is the state vector; y(i, j) ∈ R n y is the measured output; z i, j ∈ R n z is the signal to be estimated; u(i, j) ∈ R p is the noise input (that belongs to l 2 {[0, ∞), [0, ∞)}); (A 1l , B 1l , A 2l , B 2l , C l , D l , E l ) are known real matrices with appropriate dimensions. The frequency spectrum of the exogenous noise u(i, j) is assumed to belong to a known rectangular region Ω, where The weighting fuzzy functions are given by whereÑ lη (θ η (k)) is the grade of membership of θ η (k) iñ N lη . One can see that the fuzzy basis functions satisfy: Via using inference product, singleton fuzzifer and centeraverage defuzzifer, nonlinear system (1) can be described by: In order to estimate the signal z i, j a fuzzy Roesser-type filter is designed as follows: wherex i, j is the filter state vector;ẑ i, j is an estimation of z i, j ; and (Ȃ 1l ,Ȃ 2l ,B 1l ,B 2l ,C l ) are the parameters of filter matrices that are appropriately dimensioned real matrices to be determined.
The defuzzified output of system (7) is obtained as follows:x i+1, j+1 where Defining the augmented state vector ξ i, j : we can obtain the following error system : ; The problem addressed in this work can be formulated as follows: Given a scalar γ and a rectangular FF domain in (3), the filtering error system (10) is said to have a FF gain γ, if it satisfies the following two conditions: • Error system (10) is asymptotically stable when u(i, j) ≡ 0.
• under the zero boundary condition, the inequality holds for all solutions of (10) with

Preliminaries
We introduce the following technical lemmas that are useful for deriving our results.

FF performance analysis
On the basis of Lemmas 1 and 2, we give the following theorem, which can guarantee the asymptotical stability and the H ∞ performance of error system (10) in the FF domain of input noise.
Theorem 1 Let γ > 0 be a given scalar and a rectangular FF domain (2), a filter of form (8) exists such that the filtering error system in (10) Proof 1 First, we prove that (16) is equivalent to (18). Condition (16) can be rewritten as At this stand, by using Lemma 2, (20) is equivalent to We chose M and G are expressed as the following structures: which, using Schur complement, leads to given (18 which is rewritten in the form T +Û TVÛ < 0 (25) We chose F follows: Using Lemma 1,[25][26] are equivalent to (19).

Fuzzy Filter Design
Theorem 1 fails to provide explicit solutions to filter realization. On the basis of Theorem 2, we are trying to find methods for parameterizing fuzzy filters for the discrete T-S fuzzy system (1). Via specifying the structure of the slack matrices in Theorem 2, we can derive the following result.

Numerical Example
Consider a 2D discrete-time T-S fuzzy system with two rules, given by [13] Plant Rule 1: IF θ 1 (k) isÑ 1 1 , θ 2 (k) isÑ 1 2 , Then, Plant Rule 2: IF θ 1 (k) isÑ 2 1 , θ 2 (k) isÑ 2 2 , Then, The normalized membership function : The aim is to design a fuzzy discrete filter in the form of (8)  Furthermore, under zero boundary conditions, and let the disturbance input be the following form u(e jµ 1 , e jµ 2 ) = 0.4, π 3 ≤ µ 1 , µ 2 ≤ 5π 7 0, otherwise. Figures 1-3 show the trajectories of states filters vectorsx 1 ,x 2 and filtering error system of e i, j , respectively, it is clear that effectively, the 2D system is asymptotically stable and converges towards zero. From Figures  1-3, the asymptotic stability of the error system can be clearly observed, while under the zero boundary conditions and the disturbance input (38), we can work out ||e i, j || 2 ||u i, j || 2 = 0.2205. So, the condition (9) is satisfied, which means that the error filtering system has a prescribed FF H ∞ gain γ = 0.2357.

Conclusion
This paper has concerned with the problem of the filter design for discrete T-S fuzzy systems in FMLSS model with FF disturbances. Assuming the disturbances is dominated in a known FF domain, a fuzzy-filtering method is proposed to reduce the design conservatism by making use of the frequency information of disturbances. A simulation example has been given to illustrate the effectiveness of the proposed method.