Abstract and Applied Analysis

Volume 2014 (2014), Article ID 523462, 17 pages

http://dx.doi.org/10.1155/2014/523462

## Robust Nonfragile Filtering for Uncertain T-S Fuzzy Systems with Interval Delay: A New Delay Partitioning Approach

^{1}Key Laboratory of Embedded and Network Computing of Hunan Province, College of Computer Science and Electronic Engineering, Hunan University, Changsha 410082, China^{2}Department of Applied Mathematics, Faculty of Mathematics, University of Waterloo, Waterloo, ON, Canada N2L 3G1

Received 16 April 2014; Revised 27 July 2014; Accepted 6 August 2014; Published 3 September 2014

Academic Editor: Alexander Domoshnitsky

Copyright © 2014 Xianzhong Xia 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.

#### Abstract

This paper investigates the problem of robust nonfragile fuzzy filtering for uncertain Takagi-Sugeno (T-S) fuzzy systems with interval time-varying delays. Attention is focused on the design of a filter such that the filtering error system preserves a prescribed performance, where the filter to be designed is assumed to have gain perturbations. By developing a delay decomposition approach, both lower and upper bound information of the delayed plant states can be taken into full consideration; the proposed delay-fractional-dependent stability condition for the filter error systems is obtained based on the direct Lyapunov method allied with an appropriate and variable Lyapunov-Krasovskii functional choice and with tighter upper bound of some integral terms in the derivation process. Then, a new robust nonfragile fuzzy filter scheme is proposed, and a sufficient condition for the existence of such a filter is established in terms of linear matrix inequalities (LMIs). Finally, some numerical examples are utilized to demonstrate the effectiveness and reduced conservatism of the proposed approach.

#### 1. Introduction

During the past several years, fuzzy systems of the T-S model [1, 2] have attracted great interests from the stability and control community [3]. It is well known that the problem of filtering is both theoretically and practically important in control and signal processing [4, 5]. The main advantage of filtering is that no statistical assumption on the noise signals is needed and, thus, it is more general than classical Kalman filtering [6]. Moreover, the filter is designed by minimizing signal estimation error for the bounded disturbances and noises of the worst cases, which is more robust than classical Kalman filtering [7, 8]. For the fuzzy filtering problem based on T-S fuzzy models, some important results have been obtained; see for example, [9–14], and the references therein.

Among the literatures, An et al. [11] designed some filters for uncertain systems with time-varying distributed delays. In [12, 13], some new delay-dependent filter design schemes have been proposed for continuous-time T-S fuzzy systems. Huang et al. [14] improved some existing results on filter design for T-S fuzzy systems with time delay. And the filter has been shown to be much more robust against unmodeled dynamics [10]. Moreover, Li and Gao [15] proposed a new comparison model by employing a new approximation for delayed state, and then lifting method and simple LK functional method are used to analyze the scaled small gain of this comparison model and developed reduced-order filtering [16] and finite frequency filtering [17] for discrete-time systems and for 2-D systems [18], and then these new method can also be extended to T-S fuzzy systems case.

On the other hand, the nonfragile control and filtering problems have been attractive topics in theory analysis and practical implement. The nonfragile concept is proposed to this new problem: how to design a controller or filter that will be insensitive to some error in gains [19–21]. For the nonfragile filtering problem, some numerically effective design methods have been obtained [21–31]. Yang et al. [21–24] focused on the nonfragile filtering problem for linear systems and fuzzy system, respectively. However, the time delays are not considered [21–24]. Most recently, Chang and Yang [31] proposed the design of nonfragile filter for discrete-time T-S fuzzy systems with multiplicative gain variations and investigated fuzzy modeling and control for a class of inverted pendulum system in [32]; however, they are also not considered as the time delay case. However, time delay, as a source of instability and poor performance, often appears in many dynamic systems, for example, chemical process, biological systems, nuclear reactor, rolling mill systems and communication networks [2, 3], and networked control systems. In particular, a special type of time delay, interval time-varying delays, that is, and , is not restricted to be zero in practical engineering systems as NCS. Xia and Li [30] concerned with the nonfragile filter design problem for uncertain discrete-time T-S fuzzy systems with time delay, whereas the delay is constant case. Li et al. [26] investigated the problem of nonfragile robust filtering for a class of T-S fuzzy time-delay systems, whereas the delay is limited to and . Moreover, when , which does not allow the fast time-varying delay, the restriction will limit the application scope. Therefore, the robust nonfragile fuzzy filtering for uncertain nonlinear systems via T-S fuzzy models with interval time-varying delays has not only important theoretical interest but also practical value. And, to best of our knowledge, few results on robust nonfragile fuzzy filtering for the above fuzzy systems have been reported in the literatures. This motivates the present research.

In this paper, we will investigate the problem of robust nonfragile fuzzy filter designs for uncertain T-S fuzzy systems with interval time-varying delays. Our objective is to design a fuzzy filter with the gain perturbations such that the filtering error system is asymptotically stable with a prescribed performance. Firstly, based on the Lyapunov stability theory and Finsler lemma, a delay-fractional-dependent sufficient condition is derived since a new LK functional is constructed by developing a variable delay-decomposition method and estimating tightly the upper bound of its derivative through some improved inequalities techniques. Then, based on the above conditions, a sufficient condition for the solvability of the aforementioned system is developed in terms of LMIs. Finally, some numerical examples are provided to illustrate the feasibility and advantage of the proposed design method.

#### 2. Problem Formulation

Consider a nonlinear system with interval time-varying delays which could be approximated by a time-delay T-S fuzzy model with plant rules.

*Plant Rule *. IF is and … and is , THEN
where is the state vector; is the measurement vector; is the disturbance signal vector which belongs to ; is the signal vector to be estimated; is the continuous initial vector function defined on ; denote the premise variables; and represent the fuzzy sets, , and is the number of IF-THEN rules. In what follows, we define for brevity, and we denote the coefficient matrices of system (1) as
where denotes the nominal part of , and the uncertainty is considered as time varying but norm bounded; that is, stands for the uncertain part, are constant real matrices, and are unknown time-varying matrices satisfying .

The time-varying delay is assumed to be either differentiable case satisfied with , where are given bounds, or fast-varying case (i.e., , but no constraints on the delay derivatives, is unknown).

Let , in which is the membership function of in . It is assumed that , and then . By fuzzy blending, the final output of the fuzzy system (1) is inferred as follows:

Motivated by the parallel distributed compensation (PDC), in this paper, we consider the following full order nonfragile fuzzy filter.

*Rule *. IF is and … and is , THEN
where and are the state and output of the filter, respectively. The filter matrices are to be determined, and , denotes the time-varying parameters of fuzzy filter, and are unknown time-varying matrices satisfying . For simplicity, we define ; ; ; and . The defuzzified output of fuzzy filter system (4) is inferred as follows:

Defining the augmented state vector , , and , we can obtain the following filtering error system: where

Then the robust fuzzy filter design problem to be addressed in this paper can be expressed as follows.

*Robust Nonfragile Fuzzy ** Filtering Problem.* Given uncertain fuzzy system (3), design a suitable robust nonfragile fuzzy filter in the form of (5) such that the following two requirements are satisfied simultaneously:(R1)the filtering error system (6) with is asymptotically stable;(R2)the performance is guaranteed for all nonzero and a prescribed under zero initial condition.

The following lemmas will be useful in establishing our main results.

Lemma 1 1 (integral inequalities, Gu et al. [3] and Zhang et al. [33]). *Let be a vector-valued function with first-order continuous-derivative entries. Then, for any matrices , , , and some given scalars , the following integral inequality holds.*(1)*When and are constant values,
*(2)*When and are time-varying, ,
*(3)*When are time-varying, , and is any symmetric matrix,
with and .*

*Lemma 2 (Wang et al. [34]). Let , and be real matrices of appropriate dimensions with being a matrix function satisfying . Then, for any scalar , we have . Furthermore, for any scalar such that , we have .*

*3. Main Results*

*In this section, we provide a delay-fractional-dependent sufficient condition for the solvability of robust nonfragile fuzzy filtering problem for the system (1), which is formulated in the previous section.*

*The following proposition will be useful in establishing our main results.*

*Proposition 3. For real matrices and , with compatible dimensions, the following inequalities are equivalent, where is an extra slack nonsingular matrix:*

*
where .*

*Proof. *See the Appendix.

*Then, we divide the delay interval and into four segments: , where . For simplicity, we denote , , , and . For the T-S fuzzy filter error system (6), based on the Lyapunov stability theorem, we will give a sufficient condition for the solvability of the fuzzy filter design problem for the system (1) by using the novel delay decomposition approach.*

*Theorem 4. Given scalars and , the filter error system (6), for all differentiable delay with , is asymptotically stable and has a prescribed performance level if there exist real symmetric matrices, , , , , the nonsingular matrix and matrices , with appropriate dimensions, and positive scalars , such that the inequalities in (12) hold:
wherewith
A suitable filter in the form of (4) can be given by
*

*Proof. *The delay-dependent LK functional can be constructed as follows:
where denotes the function defined on the interval and

with , being real symmetry matrices to be determined.

Taking the derivative of (16) with respect to along the trajectory of the filtering error system (6), we have

For any , it is a fact that or , (). In the case of ; that is, , , suitably using the integral inequalities in Lemma 1, the following inequalities are true:
with and .

It follows from (18)-(19) that
with , where

and is defined as follows:
Since , applying the convex combination method, we conclude that if
then
For , by Schur complement, we have
By using Proposition 3, we have the following inequality, which is equivalent to (27):

Let , and, using Lemma 2, for the case of , we have the following equation:

where

If (12) when hold, then , which implies that the first term of (23) is true. Similar to the above process, if (12) when hold and we also have , then (24) hold.

Meanwhile, if ; that is, , , similar to the above deduction process, we also can obtain the conclusion that if (12) hold, then (24) hold.

So far, when assuming the zero disturbance input, from (18)–(19), we can obtain that
whereBy Schur complement, the inequalities in (25) imply . We can conclude that filtering error system (6) with is asymptotically stable. Now, to establish the performance for system (6), assume zero-initial condition and consider the following index:
Under zero initial condition and (24), we have , which means that . Thus, this completes the proof.

*Remark 5. *It may be noted that, in the above, no approximation of the delay term is involved excepting exploiting a convex combination of the uncertain terms involved. In fact, Lemma 1 plays a key effect on the present results, which is different from the common Jensen’s inequality. Although their similarity can be established following the equivalency results in [41], if is uncertain and required to be approximated with its lower or upper bound then use of (9) or (10) would be beneficial since the free variables and are introduced. Such a feature leads to less conservative results compared to the existing ones as is shown in the next section using numerical examples.

*Remark 6. *In Proposition 3, by introduction of the auxiliary slack matrix variable , matrices *, * and are decoupled. This novel technique is proposed in this paper to transform the nonlinear matrix inequalities (25) into a set of LMIs, which is different from the existing literatures.

*Remark 7. *In the proof of Theorem 4, the interval is divided into two variable subintervals and ; meanwhile, the lower bound of the delay is also divided into two equal subintervals and for the sake of simplification. Therefore, the information of delayed state and can be fully taken into account. And it is clear that the LK functional (16) are more general than the existing ones [26, 35, 37]. Moreover, since the variable delay decomposition approach in this paper is introduced in constructing the LK functional and the upper bound of its derivative is also estimated by suitably utilizing integral inequalities in Lemma 1, the proposed result is much less conservative and is more general than some existing ones. Meanwhile, the stability criteria of proposed approach are also different when the tuning delay-fractional parameter is varying. Examples below show that the proposed method yields less conservatism than the existing ones and also show that the delay decomposition is different; the maximum upper bound of the delay may be different.

Without considering the filter gain uncertainties, that is, and , the following corollary gives a delay-fractional-dependent condition of designing a standard fuzzy filter for the uncertain system (1) as [12–14], which system uncertainties have not been considered.

*Corollary 8. For uncertain system (1), given scalars and , the filter error system (6), for all differentiable delay with , is asymptotically stable and has a prescribed performance level if there exist real symmetric matrices , , , , the nonsingular matrix and matrices , , , with appropriate dimensions, and positive scalars , , such that the inequalities in (33) hold:
*