Robust Quantized Generalized <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-4.373899pt" id="M1" height="16.2512pt" version="1.1" viewBox="-0.0657574 -11.8773 20.8202 16.2512" width="20.8202pt"><g transform="matrix(.018,0,0,-0.018,0,0)"><path id="g113-73" d="M828 650H570L564 622C646 614 650 609 634 524L604 366H253L281 524C296 608 308 615 382 622L388 650H132L126 622C211 615 214 609 198 524L121 122C106 42 102 35 23 28L17 0H276L282 28C196 35 191 40 206 122L243 324H597L559 123C544 42 537 35 452 28L446 0H710L716 28C632 35 628 43 643 122L719 524C735 610 741 615 822 622L828 650Z"/></g><g transform="matrix(.013,0,0,-0.013,13.86,4.308)"><path id="g50-51" d="M414 144C384 79 371 75 317 75H135L276 221C367 316 408 376 408 465C408 570 327 635 237 635C179 635 131 609 100 575L42 494L67 471C94 510 138 565 205 565C277 565 321 517 321 435C321 348 258 270 195 195C146 137 88 81 33 26V0H411C423 44 433 88 446 135L414 144Z"/></g></svg> Filtering for Uncertain Discrete-Time Fuzzy Systems

Wang, Jidong; Si, Xiaoping; Han, Kezhen

doi:https://doi.org/10.1155/2015/807254

Mathematical Problems in Engineering

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Research Article | Open Access

Volume 2015 | Article ID 807254 | https://doi.org/10.1155/2015/807254

Robust Quantized Generalized Filtering for Uncertain Discrete-Time Fuzzy Systems

Jidong Wang,¹Xiaoping Si,¹and Kezhen Han²

Academic Editor: Stefan Balint

Received13 Mar 2014

Revised31 Jul 2014

Accepted01 Aug 2014

Published02 Mar 2015

Abstract

This paper deals with the problem of robust generalized 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 filter to mitigate quantization effects, such that the filtering error systems ensure the robust stability with a prescribed generalized 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 filter are expressed in terms of linear matrix inequalities (LMIs). Finally, a numerical example is provided to demonstrate the effectiveness of the proposed approach.

1. Introduction

Quantization in feedback control systems has received much attention in recent years [1–8]. 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 [2–6]. 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 discrete-time 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 ([11–14] 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 ([15–18] and the references therein).

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, 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 Riccati-based 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 time-domain input disturbances in and the space of time-domain controlled outputs in , the generalized control problem is considered, where the conventional norm is replaced by an operator norm. Due to the fact that the generalized performance is useful for handling stochastic aspects such as measurement noise and random disturbances, it has been received much attention ([22–26] and the references therein). More recently, based on piecewise Lyapunov functions, [27–29] have done some remarkable works on generalized synthesis of fuzzy systems. Reference [30] provides complete results on the induced and generalized filtering for a class of discrete-time systems with repeated scalar nonlinearities. Taking quantization and packet loss into consideration, a generalized filter has been designed in [31]. For discrete-time fuzzy systems, [32] has proposed a robust generalized controller via basis-dependent Lyapunov functions.

In this paper, we are to tackle the robust quantized generalized filtering problem for a class of nonlinear discrete-time 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 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 discrete-time systems with prescribed generalized 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 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 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 denote its transpose and inverse if it exists, respectively. The matrix inequality means that is square symmetric and is positive (negative) definite. The notation represents the space of square-integrable vector functions over . And denotes for simplicity. Matrices, if their dimensions are not explicitly stated, are assumed to have compatible dimensions.

2. 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:

if is is , then, where is the state variable, is the disturbance input and , is the signal to be estimated, and is the measurement output. are premise variables. , , are fuzzy sets, and is the number of fuzzy rules. The matrices , , , , , and are system matrices of appropriate dimensions with the following parametric uncertainties: where , , , , , , , , , , and for are known constant matrices with appropriate dimensions. And is an uncertain matrix satisfying . It is easy to note that such parameter uncertainty is treated as a particular perturbation of linear fractional form, where the linear fractional representation reduces to norm-bounded one when .

Using a standard singleton fuzzifier, product inference, and centre weighted average defuzzifier, a compact presentation of the overall fuzzy model is given by with where and is the normalized weight for each rule with and .

Then, the filter considered here is given as follows: where and are the state and output of the filter, respectively. The matrices , , , and are filter parameters to be determined. is the input of the filter and is a quantizer which is assumed to be symmetric (i.e., ). 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.

Here, we employ the static time-invariant logarithmic quantizer. According to [4], the set of quantized levels is given as where From (6)-(7), for the designed filter (5), it can be concluded that, for any , holds, where . Therefore, can be depicted as where .

Defining an augmented state vector and , we can obtain the following filtering error system: where In order to facilitate the later operations, we note Then, we have To this end, the objective of this paper can be summarized as follows.

Given uncertain discrete-time system (1), develop a full-order filter of the form in (5) such that(1)the filtering error system (10)-(11) is stochastically stable with system norm-bounded uncertainties and quantization error effects when ;(2)the filtering error system (10)-(11) has a prescribed generalized disturbance attenuation level ; that is, under the zero initial condition , is satisfied for any nonzero [11].

And, Before ending this section, we introduce the following lemmas, which will be used subsequently.

Lemma 1 (see [35]). Let , , and be real matrices of appropriate dimensions with satisfying . Then, for any scalar , we have

Lemma 2 (Finsler’s lemma, see [33, 36]). Given matrices and , then if and only if there exists matrix , such that

Lemma 3 (see [37, 38]). If the following conditions hold: then the following inequality holds:

3. Generalized Filtering Analysis

The purpose of this section is to lay preliminary results regarding quantized generalized filtering with system matrices uncertainties considered. First, Lemma 4 provides a fundamental analytic condition of generalized filtering. Then, based on the condition, quantization error effects will be dealt with by Lemma 7.

Lemma 4. Consider the system (3) and suppose the filter matrices in (5) are known. Then, the filtering error system (10)-(11) is asymptotically stable with a generalized disturbance attenuation level bound if there exist matrices , , and symmetric matrix such that the following inequalities hold:

Proof. For filtering error system, a fuzzy Lyapunov functional candidate can be constructed as follows: Then, the difference of along solutions of (10)-(11) is computed as Let Then, where and .
For (10), it is obvious that Now, with inequalities (26) and (28) in hand, according to Lemma 2, inequality (26) < 0 for any if there exist , , and such that the following inequality holds. To the end, it is obvious to get inequality (22) < 0 and if inequality (29) < 0.
Then, under the zero initial condition , for any nonzero , one has for any . And, from (30), we can obtain that Now, consider the generalized performance. By using Schur complement equivalence, one has by (21) that Note, for any , It is easy to conclude that inequality (33) < 0 if condition (32) holds, which leads to Finally, together with inequality (31), we can deduce that . This completes the proof.

Remark 5. When , 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 norm bound if there exist matrices , , and , symmetric matrix , and scalars , 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 , inequality (22) can also be rewritten asBy Lemma 1 and some simple matrix transformations, inequality (38) holds if there exist scalars such thatIn the end, applying Schur complement to (39) and performance congruence transformation by , 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 . Then, can also be expressed as in (38), where . 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), some definitions are given to facilitate the later development. Let where Then, taking definitions (40) into inequalities (35)-(36) and following the same line as in the proof of Lemma 7, we can establish the following solution for the robust generalized filtering analytic conditions.

Theorem 10. Suppose that uncertain system (3) and filter (5) matrices are given. Then, the filtering error system (10)-(11) is stable with the generalized disturbance attenuation level bound and the quantization density if there exist matrices , , , symmetric matrix , and constant positive scalars , , , such that the following matrix inequalities hold:where

4. Robust Quantized Generalized Filter Design

In this section, based on Theorem 10, a linearization procedure will be provided to design the generalized filter in the form of (5). Similar to the method in [5], we assume that Therefore, the robust quantized generalized filter design can be given as follows.

Theorem 11. Given quantization density and considering the uncertain discrete-time fuzzy system (3), there will be a filter to mitigate the quantization output error effects and to ensure robust stability of filtering error system (10)-(11) with prescribed generalized performance , if there exist matrices , nonsingular matrix , and positive scalars for such that the following inequalities hold: for , where then, a suitable filter with given generalized performance can be designed by

Proof. From Theorem 10, we can design the robust generalized filter if inequalities (42)-(43) are solvable. To obtain the design conditions of filter (5), one assumes that the matrices variables involved in (43) have the following forms: and let Then, with the aforementioned related matrices considered, substituting the above matrices into inequalities (42)-(43) and utilizing Lemma 3, Theorem 11 can be obtained. The proof is completed.

Remark 12. , 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].

5. 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 . Now the discrete-time fuzzy system considered has two rules: where with membership function assumed in the following: and .

Then, given quantization density , the minimum generalized performance level with fixed parameters , , , , , and by using Matlab LMI control toolbox to solve the convex optimization problem in Theorem 11 is . And the corresponding filter parameters are

To verify the effectiveness of solved filter, the external disturbance is defined as , and the initial conditions are chosen as . By considering , Figure 1 shows the state responses of the plant. Define for ; 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.

6. Conclusions

In this paper, the problem of robust quantized generalized 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 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.

Conflict of Interests

The authors declared that they have no conflict of interests related to this work.

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Copyright © 2015 Jidong Wang et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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