Robust Quantized Generalized H 2 Filtering for Uncertain Discrete-Time Fuzzy Systems

This paper deals with the problem of robust generalized H 2 filter design for uncertain discrete-time fuzzy systems with output quantization. Firstly, the outputs of the system are quantized by a memoryless logarithmic quantizer before being transmitted to a filter. Then, attention is focused on the design of a generalized H 2 filter to mitigate quantization effects, such that the filtering error systems ensure the robust stability with a prescribed generalized H 2 noise attenuation level. Via applying Finsler lemma to introduce some slack variables and using the fuzzy Lyapunov function, sufficient conditions for the existence of a robust generalized


Introduction
Quantization in feedback control systems has received much attention in recent years [1][2][3][4][5][6][7][8].This is mainly due to the wide application of digital computers in control systems.For example, it usually arises in a distributed or network based control system, where information should be quantized before being transmitted through a communication channel under limited bandwidth.However, quantization of a stabilizing controller may lead to limiting cycles and chaotic behavior as described in [1].And there are substantially two reasons to account for these changes in the system's behavior.One is saturation and the other is deterioration of performance near the system equilibrium.Therefore, considerable efforts have been devoted to develop tools for better analysis and design of quantized feedback systems [2][3][4][5][6].In order to find out quantized characteristic, [6] investigated the quantization model from statistical perspective and found that quantization is inherently a nonlinear feature.For stabilization of discretetime SISO linear systems, [2] proposed that the coarsest quantizer that quadratically stabilizes such a linear system is logarithmic, which can be computed by solving a special linear quadratic regulator problem.Unfortunately, the results are difficult to extend to the multiple-input case.By noting that the quantization error can be treated as uncertainty or nonlinearity and can be bounded by a sector bound, [9] proposed a sector bound approach to deal with the quantized feedback control problem.Recently, by recognizing that only a quadratic Lyapunov function is used in the sector bound approach, [5] designed a quantization-dependent Lyapunov function approach which can lead to less conservative results.
On the other hand, as it is well known, Takagi-Sugeno (T-S) model has proved its effectiveness in the study of nonlinear systems.Indeed, it gives a simpler formulation from mathematical point of view to represent the behavior of nonlinear systems [10].On the whole, they are composed of linear models blended together with nonlinear functions.Then, for the stability and stabilization of such T-S fuzzy model, some tools inspired from the study of linear systems are proposed.In particular, there have been a lot of results to study the stability problem, such as in ( [11][12][13][14] and the references therein).Nevertheless, the use of the quadratic Lyapunov function leads to conservative results and reaches quickly its limits.To overcome the drawback, different Lyapunov functions have been proposed ( [15][16][17][18] and the references therein).

Mathematical Problems in Engineering
For the purpose of analysis and synthesis, estimating the state variables of a dynamic system is important in helping to improve our knowledge about the system concerned.In this meaning,  2 filtering design arises as an efficient strategy whenever the noise input is assumed to have a known power spectral density.The problem has been faced using Riccatibased approaches [19] and by means of LMIs methods [20].In the case where there exists insufficient statistical information about the noise input, the well-known  ∞ filtering design and peak-to-peak filtering method can be employed [21].From another point of view, when the closed-loop system is described in the term of mapping between the space of timedomain input disturbances in  2 and the space of time-domain controlled outputs in  ∞ , the generalized  2 control problem is considered, where the conventional  2 norm is replaced by an operator norm.Due to the fact that the generalized  2 performance is useful for handling stochastic aspects such as measurement noise and random disturbances, it has been received much attention ( [22][23][24][25][26] and the references therein).More recently, based on piecewise Lyapunov functions, [27][28][29] have done some remarkable works on generalized  2 synthesis of fuzzy systems.Reference [30] provides complete results on the induced  2 and generalized  2 filtering for a class of discrete-time systems with repeated scalar nonlinearities.Taking quantization and packet loss into consideration, a generalized  2 filter has been designed in [31].For discretetime fuzzy systems, [32] has proposed a robust generalized  2 controller via basis-dependent Lyapunov functions.
In this paper, we are to tackle the robust quantized generalized  2 filtering problem for a class of nonlinear discretetime systems with norm-bounded uncertainties, which is different from the contents and research techniques in [31].Via applying Finsler lemma in [33] to introduce some slack variables to provide extra free dimensions in the solution space and using the variable definition method in [34] to relax the structural constraints of these introduced elements, the robust quantized generalized  2 filter can be designed by solving a set of linear matrix inequalities and can be easily checked through utilizing available numerical software, such as Matlab and Scilab.The main contribution of this paper is that it deals with the quantized error and model additive uncertainty simultaneously for nonlinear discretetime systems with prescribed generalized  2 performance, which has little related literature on this multiobjective filtering problem to the best of our knowledge.
The paper is organized as follows.The next section provides some useful notations and lemmas.In Section 3, generalized  2 performance is firstly considered, and then, a sufficient condition to mitigate quantization error effects is deduced.According to the previous results, a LMI-based approach is established with the systems uncertainties taken into consideration in the end of Section 3. Section 4 gives the standard robust quantized generalized  2 filter design method.Finally, a numerical example is provided to illustrate the effectiveness of our main results.
Notations.Throughout this paper, the symbol * induces a symmetric structure in LMIs.For a matrix ,   and  −1 denote its transpose and inverse if it exists, respectively.
The matrix inequality  > 0 ( < 0) means that  is square symmetric and  is positive (negative) definite.The notation  2 [0, ∞) represents the space of square-integrable vector functions over [0, ∞).And {} denotes (  + ) for simplicity.Matrices, if their dimensions are not explicitly stated, are assumed to have compatible dimensions.

Problem Statement and Preliminaries
Consider an uncertain nonlinear discrete-time system represented by the following uncertain T-S fuzzy model, where the th rule is described as follows: Using a standard singleton fuzzifier, product inference, and centre weighted average defuzzifier, a compact presentation of the overall fuzzy model is given by  ( + 1) =  ()  () +  ()  () ,  () =  ()  () +  ()  () ,  () =  ()  () +  ()  () , where Then, the filter considered here is given as follows: where   () ∈ R  and   () ∈ R  are the state and output of the filter, respectively.The matrices   ,   ,   , and   are filter parameters to be determined.  () ∈ R  is the input of the filter and (⋅) = [ 1 (⋅)  2 (⋅) ⋅ ⋅ ⋅   (⋅)]  is a quantizer which is assumed to be symmetric (i.e., (−V) = −(V)).Note that filter (5) is not the fuzzy type.The reason for applying the basis-independent filter is to avoid the design difficulty in the presence of quantization error and systems uncertainties.

Generalized 𝐻 2 Filtering Analysis
The purpose of this section is to lay preliminary results regarding quantized generalized  2 filtering with system matrices uncertainties considered.First, Lemma 4 provides a fundamental analytic condition of generalized  2 filtering.Then, based on the condition, quantization error effects will be dealt with by Lemma 7.
Remark 5. When () = 0, it is effortless to prove that the system is stable if (22) holds.And, in the derivation of the above conclusion, three slack variables (), (), and () are introduced, which will reduce some conservatism.Remark 6.It should be emphasized that the existing conditions in the literature for the analysis and design of robust filtering systems are only sufficient [34].Therefore, many efforts have been made in the direction of reducing the conservativeness of the analysis and design methods for improving the systems performance.Therein, the approach of introducing auxiliary variables is widely applied.In particular, the results derived from Lemma 2 perform well to provide extra degree of freedom by adding auxiliary variables [39].However, as stated above, the conservatism can not be still avoided due to sufficiency of obtained conditions.Therefore, in the future work, we will be devoted to solving this topic from other perspectives.
In the following discussion, we will process the quantization output error effects.Based on Lemma 4, a significant conclusion is obtained.Lemma 7. Suppose uncertainty system (3) and corresponding filter (5) with output quantization are given; the filtering error system (10)-( 11) is stable with generalized  2 norm bound  if there exist matrices (), (), and (), symmetric matrix () > 0, and scalars  1 () > 0,  2 () > 0 such that the following matrix inequalities hold: Proof.Assume that the conditions in Lemma 4 are satisfied, and take the quantized error into account; that is, Then, with the definition of Δ Λ(), inequality (22) can also be rewritten as ] By Lemma 1 and some simple matrix transformations, inequality (38) holds if there exist scalars  1 () > 0 such that ] In the end, applying Schur complement to (39) and performance congruence transformation by diag{, , , ,  1 ()}, inequality (36) can be obtained.Following a similar development as in the proof of inequality (36), inequality (35) can be established easily from inequality (21) in Lemma 4.
Remark 8.The quantization error is modeled as a kind of norm-bounded uncertainty in Lemma 7. Notice that quantized error will always exist in a real system; therefore, without loss of generality, we assume that ‖Δ()‖ > 0.Then, Δ() can also be expressed as (Δ()/) in (38), where Δ()/ ≤ .Finally, by Lemma 1, it is easy to get (39).
Remark 9.In order to reduce the complexity in the process of obtaining analytic conditions, Lemma 7 has not considered the uncertainties of system matrices.This problem will be discussed in the following part.
With the aid of aforementioned captions, we will deal with the robustness of system ( 10)- (11) in the back of this section.Before beginning, according to (3), (10), and ( 11 Then, taking definitions (40) into inequalities ( 35)-( 36) and following the same line as in the proof of Lemma 7, we canestablish the following solution for the robust generalized  2 filtering analytic conditions.

Robust Quantized Generalized 𝐻 2 Filter Design
In this section, based on Theorem 10, a linearization procedure will be provided to design the generalized  2 filter in the form of (5).Similar to the method in [5], we assume that Therefore, the robust quantized generalized  2 filter design can be given as follows.
Remark 12.  {1,2,3,4,5} ,  2 are free slack scalar variables, which will be useful to reduce the conservatism of the special forms of (), (), and () by providing extra free dimensions in the solution space for Theorem 11.
Remark 13.In the practical applications, quantization error bound  can be calculated according to the given quantization density .Then, inequalities (46) are strictly LMIs that can be easily and effectively solved via LMI control toolbox [40].

Numerical Example
In this section, we use an example to demonstrate the applicability and effectiveness of the proposed approach.Consider a tunnel diode circuit, whose fuzzy discrete modeling was done in [41] with a sampling time  = 0.02.Now the discrete-time fuzzy system considered has two rules: where with membership function assumed in the following: To verify the effectiveness of solved filter, the external disturbance is defined as () =  −0.05 (4),  = 1, 2, . .., and the initial conditions are chosen as (0) = [0 0]  ,   (0) = [0 0]  .By considering Δ  () = sin(0.1),Figure 1 shows the state responses of the plant.Define Δ() = sin(0.5)for  = 1, 2, . ..; the simulation results of (),   (), and   () are given in Figures 2 and 3, respectively.Finally, from Figure 4, we can see that the designed filter meets the prescribed requirements.

Conclusions
In this paper, the problem of robust quantized generalized  2 filtering for uncertain discrete-time fuzzy systems has been studied.Based on the fuzzy Lyapunov function, a less conservative approach is exploited to derive sufficient conditions for designing a robust filter that guarantees a generalized  2 performance and mitigates the output quantization measurement error simultaneously.Moreover, such filter can be obtained easily by solving a set of linear matrix inequalities.Finally, a simulation example has been given to illustrate the successful application of the proposed method.[3] F. Fagnani and S. Zampieri, "Stability analysis and synthesis for scalar linear systems with a quantized feedback, " IEEE Transactions on Automatic Control, vol.48, no. 9, pp.1569-1584, 2003.