Finite frequency model reduction for 2-D fuzzy systems in FM model

This paper deals with the problem of H∞ model reduction for two-dimensional (2D) discrete Takagi-Sugeno (T-S) fuzzy systems described by Fornasini-Marchesini local state-space (FM LSS) models, over finite frequency (FF) domain. New design conditions guaranteeing the FF H∞ model reduction are established in terms of Linear Matrix Inequalities (LMIs). To highlight the effectiveness of the proposed H∞ model reduction design, a numerical example is given.


Introduction
During the past decades, much progress has been made for 2D systems in the literature [1,2], many important results based on LMI approach have already been reported. Among these results, stability analysis and stabilization design for 2D systems have been studied in [3], H ∞ filtering problem can be found in [14], model reduction problem in [6].
The H ∞ model reduction aims to find a low-order model for a given model such that the H ∞ norm of the error between these two models is minimized or satisfies the specific performance. Many results on H ∞ model reduction have been reported in the literature [6,[9][10][11].
The point of interest in aforementioned literature is that all performance indices are defined in the entire frequency (EF) domain. However, in major real applications, the design characteristics are usually given in specific frequency domains. So, the standard design approaches for the whole frequency domain may bring conservatism [15,18], [17] [12,13,16].
In this paper, we consider the H ∞ model reduction for 2D T-S fuzzy Fornasini-Marchesini Models. Sufficient conditions for the existence of solutions are parameterized in LMI form. An explicit parameterization of the desired reduced-order models is given. Finally, a numerical example is provided to prove the effectiveness of FF propose 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 * e-mail: rachid.naoual@isga.ma * * e-mail: abderrahim.elamrani@usmba.ac.ma * * * e-mail: ismail.boumhidi@usmba.ac.ma dimension. Generally, sym{A} denotes A + A T , diag{..} 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; 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 ) 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 where µ a 1 , µ b 1 , µ a 2 and µ b 2 are known scalars. A more compact presentation of the Takagi-Sugeno 2D discrete-time fuzzy model systems is given by In this paper, we will approximate T-S fuzzy FM system (1) by the following reduced-order T-S model: wherex i, j ∈ R˘n(n < n) is the state vector;y i, j ∈ R y is the output; and (Ȃ 1l ,Ȃ 2l ,B 1l ,B 2l ,C l ) are the parameters of reduced-order model that are appropriately dimensioned real matrices to be determined.
The reduced-order model can be written in a compact formx i+1, j+1 where Defining the augmented state vector ξ i, j := [x T i, jx T i, j ] T , e i, j = y i, j −y i, j , we can obtain the following approximation error system : Next, some related definitions for 2D T-S fuzzy FM model with FF ranges are given as follows: The problem addressed in this work can be formulated as follows: Given TS fuzzy FM system (3), The objective is to design a suitable TS FM reduced-order model in the form of (6) such that the following two requirements are satisfied: • Error system (8) is asymptotically stable when u(i, j) ≡ 0.
• Letting γ > 0, be a given constant, under the zero−initial condition, equation holds for all solutions of (8) with

Preliminaries
We introduce the following technical lemmas that are useful for deriving our results. (13), we can obtain (14) T

H ∞ Model Reduction 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 (8) in the FF domain of input noise.
Theorem 1 Let γ > 0 be a given scalar and a rectangular FF domain (2), a reduced-order model of form (6) exists such that the error system in (8) Proof 1 First, we prove that (15) is equivalent to (17). Condition (15) can be rewritten as At this stand, by using Lemma 2,(19) is equivalent to We chose M and G are expressed as the following structures: which, using Schur complement, leads to given (17 23) which is rewritten in the form We chose F follows: Using Lemma 1,[24][25] are equivalent to (18).

Finite Frequency H ∞ Model Reduction design
The main objective is to determine the reduced-order matrices such that error system (8) is asymptotically stable and guarantees an H ∞ disturbance attenuation level γ and satisfies the FF in (11).

Numerical Example
Considering a 2D discrete-time T-S fuzzy system with two rules, whose matrices are given as [19] Plant Rule 1: IF Plant Rule 2: IF θ 1 (k) isÑ 2 1 , θ 2 (k) isÑ 2 2 , Then, where The normalized membership function : where To demonstrate the value of our proposed approach, we provide in Table 1 the H ∞ reduced order performance levels, which shows the conservativeness of the FF method.

Frequency
Methods Theorem 2 0.2845 Table 1: Comparison of reduced-order performance obtained in different ranges. Assume that the reduced-order model in (8) has an order FM LSS modeln = 2, for FF domain [ π 8 , π 3 ] × [ π 8 , π 3 ], the reduced-order parameters are given as follows: Furthermore, under zero boundary conditions, by calculation, we have ||e i, j || 2 ||u i, j || 2 = 0.2715, which is below the corresponding prescribed value γ = 0.2845, showing the effectiveness of the model reduction design method.
The trajectories of y i, j ,y i, j and the error system e i, j are shown in Figures 2, 3 and 4. It is clear that effectively, the 2D system is asymptotically stable and converges towards zero. All the simulation results show the effectiveness of the designed reduced-order system.

Conclusion
This paper has investigated the FF H ∞ reduced model design problem for two-dimensional (2D) discrete-time T-S fuzzy systems described by FMLSS model. By applying gKYP lemma for 2D discrete systems, we introduce many slack matrices to provide extra free dimensions in the solution space of the H ∞ optimization.