Discrete Dynamics in Nature and Society

Volume 2010, Article ID 392915, 21 pages

http://dx.doi.org/10.1155/2010/392915

## Improved Results on Fuzzy Filter Design for T-S Fuzzy Systems

^{1}State Key Laboratory of Advanced Design and Manufacture for Vehicle Body, Institute of Space Technology, Hunan University, Changsha 410082, China^{2}School of Computer and Communication, Hunan University, Changsha 410082, China^{3}Faculty of Engineering and Information Technology, University of Technology Sydney, NSW 2007 Sydney, Australia

Received 12 July 2010; Accepted 30 July 2010

Academic Editor: Leonid Berezansky

Copyright © 2010 Jiyao An 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

The fuzzy filter design problem for T-S fuzzy systems with interval time-varying delay is investigated. The delay is considered as the time-varying delay being either differentiable uniformly bounded with delay derivative in bounded interval or fast varying (with no restrictions on the delay derivative). A novel Lyapunov-Krasovskii functional is employed and a tighter upper bound of its derivative is obtained. The resulting criterion thus has advantages over the existing ones since we estimate the upper bound of the derivative of Lyapunov-Krasovskii functional without ignoring some useful terms. A fuzzy filter is designed to ensure that the filter error system is asymptotically stable and has a prescribed performance level. An improved delay-derivative-dependent condition for the existence of such a filter is derived in the form of linear matrix inequalities (LMIs). Finally, numerical examples are given to show the effectiveness of the proposed method.

#### 1. Introduction

During the last decades, the filtering problem has attracted many researchers to study through various methodologies, see, for example, [1–20] and the references therein, in which these methods mostlyconsist of two main approaches, namely, the Kalman filtering approach [1–3] and the filtering approach [4–17]. In contrast with the Kalman filtering, the filtering approach does not require the exact knowledge of the statistics of the external noise signals and it is insensitive to the uncertainties both in the exogenous signastatistics and in dynamic models. This advantage renders the filtering approach very appropriate to some practical applications. Recently, the filter design contains two cases of filtering technique, that is, filtering technique [18–20] and the filtering technique [4–17].

On the other hand, Takagi-Sugeno (T-S) fuzzy model can provide an effective way to represent a complex nonlinear system into a weighted sum of some simple linear subsystems [8, 21, 22], which has been an increasing interest in the study of T-S fuzzy systems. In recent years, T-S fuzzy model approach has been extended to filter or controller design [4–6, 9, 10, 12, 15–21, 23–35]. For instance, the stability analysis and stabilization synthesis problems of T-S fuzzy systems were studied in [21, 29, 30, 33–35], while fuzzy controllers were designed in [23–28]. One set of fuzzy filters for a class of T-S fuzzy systems was designed in [32]. However, the above-mentioned works use common Lyapunov-Krasovskii functional, and the results under a common Lyapunov method are quite conservative. To reduce the conservatism, a fuzzy weighting-dependent Lyapunov method has been proposed in [6], which is effective in reducing conservatism of previous results on fuzzy systems. More recently, Lin et al. [4] and Su et al. [5] have concerned with filtering of nonlinear continuous-time state-space models with time-varying delays via T-S fuzzy model approach. However, some negative semidefinite terms are ignored and the lower bound of time delay is restricted to be zero, see, for example, [4–6] and the references therein. Qiu et al. [36] investigated the problem of delay-dependent robust stability and filtering design for a class of uncertain continuous-time nonlinear systems with time-varying state delay represented by T-S fuzzy models. However, there is room for further investigation to reduce the conservativeness of the filter design. This motivates the current research.

In this paper, we discuss the fuzzy filter design problem for T-S fuzzy systems with interval time-varying delay. Our aim is to design a suitable fuzzy filter, which ensures both the fuzzy stability and a prescribed performance level of the filter error system. By constructing a Lyapunov-Krasovskii functional, estimating the time derivative of the Lyapunov-Krasovskii functional less conservatively, and adopting convex optimization approach, an improved delay-derivative-dependent condition for the solvability of fuzzy filter design problem is proposed in terms of linear matrix inequalities (LMIs). Two examples are used to compare with the previous literatures and demonstrate the effectiveness of the proposed method.

The rest of this paper is organized as follows: The fuzzy filtering problem is formulated in Section 2; the fuzzy performance analysis is derived in Section 3; and fuzzy filter design is addressed in Section 4. Numerical examples are provided in Section 5, and Section 6 concludes this paper.

#### 2. Problem Formulation

Consider a nonlinear system with interval time-varying delay which could be approximated by a class of T-S fuzzy systems with interval time-varying delays. The T-S fuzzy model with plant rules can be described by:

**Plant rule**: IF is andand is , **THEN**
where , and is the state vector; is the measurements vector; is the disturbance signal vector which belongs to ; is the signal vector to be estimated; is the continuous initial vector function defined on ; The system coefficient matrices are constant real matrices with appropriate dimensions, where and is the number of IF-THEN rules; are the premise variables; are the fuzzy sets. For the sake of convenience, we denote .

The time-varying delay is assumed to be either differentiable with where and are given bounds, or fast-varying (with no restrictions on the delay derivative).

The fuzzy system (2.1) is supposed to have singleton fuzzifier, product inference and centroid defuzzifier. The final output of the fuzzy system is inferred as follows: where for , and is the membership function of in . Here . Here, we assume that , and

Our aim is to design the following fuzzy filter.

**Rule **: IF is andand is , **THEN**
where is the filter state, is the estimation of in fuzzy system (2.1), the constant matrices , , , are the filter matrices to be determined. The final fuzzy filter of fuzzy system (2.1) is thus inferred as follows

Defining the augmented state vector , , from (2.3) and (2.6), we can then obtain the following filtering error system: where

So far, *the fuzzy ** filter design problem* for fuzzy system (2.3) can be stated as follows. Given a scalar , design a suitable fuzzy filter in the form of (2.5) such that the filtering error system (2.7) has a prescribed performance , and the following two purposes are satisfied: (i)the system (2.7) with is asymptotically stable;(ii)the performance is guaranteed for all nonzero and a prescribed under the condition , for all . If this is the case, we say that the fuzzy filter design problem is solved.

#### 3. Fuzzy Performance Analysis

In this section, we propose the sufficient criterion for the filter error system (2.7) satisfying a prescribed performance level for fuzzy system (2.1) or (2.3).

Theorem 3.1. * Given scalars , and , the filter error system (2.7), for all differentiable delay with , is asymptotically stable and has a prescribed performance level if there exist real symmetry matrices , , , , , , and real matrices , with appropriate dimensions such that the two LMIs (3.1) where , are feasible.
**
where
**
where
**
Thus, from (2.8) and the above definition, we have
**
where
**
with
**
Next, based on Theorem 3.1, we calculate the feasibility of the LMIs .**Due to , there exist a nonsingular real matrix and a real matrix such that .Let us define
**
left- and right-multiply , defined in (4.8) by and , respectively, and take , and
**
By replacing in defined in (3.1) with , one yields
**
Note that if LMIs (4.1) and (4.2) hold, from (4.8), we arrive at , then .**On the other hand, from (4.1), notice that , applying Schur complement yields .**So far, we conclude from Theorem 3.1 that the filter, that is,
**
with defined in (4.12), guarantees that the filter error system (2.7) is asymptotically stable and has a prescribed performance level . **And, performing an irreducible linear transformation in (4.14) yields
**Therefore, the desired filter (2.5) with the filter matrices in (4.4) is readily obtained from (4.15). This completes the proof.*

Similar to Corollary 3.2, when is unknown, by substituting into (4.2), the following result is then obtained.

Corollary 4.2. *Given scalars and , the fuzzy filter design problem, for all differentiable delay with , is solvable if there exist matrices , , , , , , , and real matrices , , col , ; with appropriate dimensions such that the LMIs: (4.1) and (4.2) where , are feasible. Meanwhile, a desired filter in the form of (2.5) is given by the filter matrices in (4.4).**Moreover, if the above LMIs are feasible with then the fuzzy filter design problem, for all fast-varying delay , is solvable in which a desired filter in (2.5) is given by the filter matrices in (4.4).*

*Remark 4.3. *Notice that for any scalar , if , then . The fact played a key role in the existing results in [4, 5], respectively. But there existed some coupled matrix variables in the LMIs in [4, 5].Therefore, to solve filter design problem, [4, 5] must use decoupling technique similar to [42] to convert the conditions in [4, 5] into another form, respectively. These decoupling approaches were shown as [4, 5], respectively. Furthermore, because of a scalar being predescribed, the constraint may lead to considerable conservativeness of these results. Examples below show that for different yields different . From simulation results in Table 2, we can see that if or , the conditions in [4, 5] are unsolvable when , while our result works. Meanwhile, the scalar is not needed in this paper. Examples 5.1 and 5.2 below show that our approach yields less conservative results.

#### 5. Numerical Examples

In this section, three examples are given to show the effectiveness of the proposed method in this paper.

*Example 5.1. *Consider the following fuzzy system borrowed from [4, 5]:
where
For and , choosing and in Table 1 and applying Theorems 4.1, the results are -dependent (see Table 1). Moreover, for unknown and , that is, fast-varying delay case, according to Corollary 4.2, by setting , we get the optimal attenuation level after 38 iterations.

For , unknown and , to compare with the recently developed fuzzy filter, it is worthwhile to point out that a given scalar is needed in [4, 5] while the scalar is any value in our results. Thus, we consider different and to find the minimum index . The results obtained by various methods in the literature and in this paper are listed in Table 2. Moreover, for the case of no additional prescribed scalar, in order to demonstrate the advantages of the proposed approach over the existing results, a detailed comparison between the minimum performance levels obtained by the methods in [4, 36, 37] and in this paper for different cases is summarized in Table 3. From Tables 2 and 3, it can be seen that stability conditions obtained in this paper are less conservative than the existing ones.

As an example, for given , according to Theorem 4.1, solve LMIs in (4.1) and (4.2), and get the minimum performance level after 32 iterations, and then compute the fuzzy filter matrices from (4.4) as follows

In order to further show the merit of our method, let us consider the following numerical example.

*Example 5.2. * Consider the following fuzzy system with interval time-varying delay:
where
To compare with the ones existing in [4, 5], we assumed that is unknown and . According to Corollary 4.2, choose and the simulations are run for two cases. In the first case, we compute the minimum index for the given different and in [4, 5] or any in this paper. In the second case, we compute the maximum values of for the given different and in [4, 5] or any in this paper. The simulation results are shown by Tables 4 and 5, respectively. It can also be clearly seen that our approach has less conservative results than the results in the literatures.

As an example, we assume that , the solutions can be obtained after 20 iterations in which the fuzzy filter in the form of (2.5) is given by the following filter matrices as

Next, we will give another example to illustrate that our methods are reduced more conservative than the existing results.

*Example 5.3. *Consider the linear system (3.24) with the following parameters
To compare with those results in the previous literatures, assume that is unknown. For , unknown or , the result of Corollary 3.3 coincides with the one in [41] (the latter are less conservative than those of [39]). Comparison with various existing methods in the literature for the admissible upper-bound , which guarantee the stability of the system (3.24) is listed in Table 6. It is clear that our results are much less conservative than those in [37–39].

#### 6. Conclusion

This paper deals with the problem of fuzzy filter design for T-S fuzzy systems with interval time-varying delay through T-S fuzzy models. By constructing a novel Lyapunov-Krasovskii functional and estimating the time derivative of the Lyapunov-Krasovskii functional less conservatively, an improved filter design scheme is proposed. Three numerical examples are used to illustrate the design procedure and the merit of the proposed method.

#### Acknowledgment

This work was supported in part by the Cultivation Fund of the Key Scientific and Technical Innovation Project, Ministry of Education of China (no. 708067), the Special Scientific Research Fund for Doctor Subjects of Universities (no. 200805320022), and in part by the Program for Changjiang Scholars and Innovative Research Team in University (no. 531105050037).

#### References

- D. S. Bernstein and W. M. Haddad, “Steady-state Kalman filtering with an ${H}_{\infty}$ error bound,”
*Systems & Control Letters*, vol. 12, no. 1, pp. 9–16, 1989. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - D. Simon, “Kalman filtering for fuzzy discrete time dynamic systems,”
*Applied Soft Computing Journal*, vol. 3, no. 3, pp. 197–207, 2003. View at Publisher · View at Google Scholar · View at Scopus - Fuwen Yang, Z. Wang, and Y. S. Hung, “Robust Kalman filtering for discrete time-varying uncertain systems with multiplicative noises,”
*IEEE Transactions on Automatic Control*, vol. 47, no. 7, pp. 1179–1183, 2002. View at Publisher · View at Google Scholar · View at MathSciNet - C. Lin, Q.-G. Wang, T. H. Lee, and B. Chen, “${H}_{\infty}$ filter design for nonlinear systems with time-delay through T-S fuzzy model approach,”
*IEEE Transactions on Fuzzy Systems*, vol. 16, no. 3, pp. 739–746, 2008. View at Publisher · View at Google Scholar · View at Scopus - Yakun Su, B. Chen, C. Lin, and H. Zhang, “A new fuzzy ${H}_{\infty}$ filter design for nonlinear continuous-time dynamic systems with time-varying delays,”
*Fuzzy Sets and Systems*, vol. 160, no. 24, pp. 3539–3549, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - C. Lin, Q.-G. Wang, T. H. Lee, and Y. He, “Fuzzy weighting-dependent approach to ${H}_{\infty}$ filter design for time-delay fuzzy systems,”
*IEEE Transactions on Signal Processing*, vol. 55, no. 6, pp. 2746–2751, 2007. View at Publisher · View at Google Scholar · View at MathSciNet - E. Fridman and U. Shaked, “An improved delay-dependent ${H}_{\infty}$ filtering of linear neutral systems,”
*IEEE Transactions on Signal Processing*, vol. 52, no. 3, pp. 668–673, 2004. View at Publisher · View at Google Scholar · View at MathSciNet - G. Feng, “A survey on analysis and design of model-based fuzzy control systems,”
*IEEE Transactions on Fuzzy Systems*, vol. 14, no. 5, pp. 676–697, 2006. View at Publisher · View at Google Scholar · View at Scopus - Y.-C. Lin and J.-C. Lo, “Robust mixed ${H}_{2}/{H}_{\infty}$ filtering for time-delay fuzzy systems,”
*IEEE Transactions on Signal Processing*, vol. 54, no. 8, pp. 2897–2909, 2006. View at Publisher · View at Google Scholar · View at Scopus - G. Feng, “Robust ${H}_{\infty}$ filtering of fuzzy dynamic systems,”
*IEEE Transactions on Aerospace and Electronic Systems*, vol. 41, no. 2, pp. 658–670, 2005. View at Publisher · View at Google Scholar · View at Scopus - S. K. Nguang and P. Shi, “Delay-dependent ${\mathscr{H}}_{\infty}$ filtering for uncertain time delay nonlinear systems: an LMI approach,”
*IET Control Theory & Applications*, vol. 1, no. 1, pp. 133–140, 2007. View at Publisher · View at Google Scholar · View at MathSciNet - Y.-C. Lin and J.-C. Lo, “Robust mixed ${H}_{2}/{H}_{\infty}$ filtering for discrete-time delay fuzzy systems,”
*International Journal of Systems Science*, vol. 36, no. 15, pp. 993–1006, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - C.-S. Tseng and B.-S. Chen, “${H}_{\infty}$ fuzzy estimation for a class of nonlinear discrete-time dynamic systems,”
*IEEE Transactions on Signal Processing*, vol. 49, no. 11, pp. 2605–2619, 2001. View at Publisher · View at Google Scholar · View at Scopus - H. Zhang, S. Lun, and D. Liu, “Fuzzy ${H}_{\infty}$ filter design for a class of nonlinear discrete-time systems with multiple time delays,”
*IEEE Transactions on Fuzzy Systems*, vol. 15, no. 3, pp. 453–469, 2007. View at Publisher · View at Google Scholar · View at Scopus - S. Xu and J. Lam, “Exponential ${H}_{\infty}$ filter design for uncertain Takagi-Sugeno fuzzy systems with time delay,”
*Engineering Applications of Artificial Intelligence*, vol. 17, no. 6, pp. 645–659, 2004. View at Publisher · View at Google Scholar · View at Scopus - J. Yoneyama, “${H}_{\infty}$ filtering for fuzzy systems with immeasurable premise variables: an uncertain system approach,”
*Fuzzy Sets and Systems*, vol. 160, no. 12, pp. 1738–1748, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - J. Yang, S. Zhong, G. Li, and W. Luo, “Robust ${H}_{\infty}$ filter design for uncertain fuzzy neutral systems,”
*Information Sciences*, vol. 179, no. 20, pp. 3697–3710, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - H. Gao and C. Wang, “Delay-dependent robust ${H}_{\infty}$ and ${L}_{2}$-${L}_{\infty}$ filtering for a class of uncertain nonlinear time-delay systems,”
*IEEE Transactions on Automatic Control*, vol. 48, no. 9, pp. 1661–1666, 2003. View at Publisher · View at Google Scholar · View at MathSciNet - K. M. Grigoriadis and J. T. Watson Jr., “Reduced-order ${H}_{\infty}$ and ${L}_{2}$-${L}_{\infty}$ filtering via linear matrix inequalities,”
*IEEE Transactions on Aerospace and Electronic Systems*, vol. 33, no. 4, pp. 1326–1338, 1997. View at Google Scholar · View at Scopus - Z. Li and S. Xu, “Fuzzy weighting-dependent approach to robust ${L}_{2}$-${L}_{\infty}$ filter design for delayed fuzzy systems,”
*Signal Processing*, vol. 89, no. 4, pp. 463–471, 2009. View at Publisher · View at Google Scholar · View at Scopus - Y.-Y. Cao and P. M. Frank, “Analysis and synthesis of nonlinear time-delay systems via fuzzy control approach,”
*IEEE Transactions on Fuzzy Systems*, vol. 8, no. 2, pp. 200–211, 2000. View at Publisher · View at Google Scholar · View at Scopus - T. Takagi and M. Sugeno, “Fuzzy identification of systems and its applications to modeling and control,”
*IEEE Transactions on Systems, Man and Cybernetics*, vol. 15, no. 1, pp. 116–132, 1985. View at Google Scholar · View at Scopus - E. Fridman and U. Shaked, “Delay-dependent ${H}_{\infty}$ control of uncertain discrete delay systems,”
*European Journal of Control*, vol. 11, no. 1, pp. 29–39, 2005. View at Publisher · View at Google Scholar · View at MathSciNet - C. Lin, Q.-G. Wang, T. H. Lee, and Y. He, “Design of observer-based ${H}_{\infty}$ control for fuzzy time-delay systems,”
*IEEE Transactions on Fuzzy Systems*, vol. 16, no. 2, pp. 534–543, 2008. View at Publisher · View at Google Scholar · View at Scopus - C. Lin, Q.-G. Wang, T. H. Lee, Y. He, and B. Chen, “Observer-based ${H}_{\infty}$ control for T-S fuzzy systems with time delay: delay-dependent design method,”
*IEEE Transactions on Systems, Man, and Cybernetics, Part B*, vol. 37, no. 4, pp. 1030–1038, 2007. View at Publisher · View at Google Scholar · View at Scopus - C. Lin, Q.-G. Wang, and T. H. Lee, “${H}_{\infty}$ output tracking control for nonlinear systems via T-S fuzzy model approach,”
*IEEE Transactions on Systems, Man, and Cybernetics, Part B*, vol. 36, no. 2, pp. 450–457, 2006. View at Publisher · View at Google Scholar · View at Scopus - X. Song, S. Xu, and H. Shen, “Robust ${H}_{\infty}$ control for uncertain fuzzy systems with distributed delays via output feedback controllers,”
*Information Sciences*, vol. 178, no. 22, pp. 4341–4356, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - S. K. Nguang and P. Shi, “${\mathscr{H}}_{\infty}$ output feedback control design for uncertain fuzzy systems with multiple time scales: an LMI approach,”
*European Journal of Control*, vol. 11, no. 2, pp. 157–170, 2005, With discussion. View at Publisher · View at Google Scholar · View at MathSciNet - B. Chen, X. Liu, and S. Tong, “New delay-dependent stabilization conditions of T-S fuzzy systems with constant delay,”
*Fuzzy Sets and Systems*, vol. 158, no. 20, pp. 2209–2224, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - C. Lin, Q.-G. Wang, and T. H. Lee, “Delay-dependent LMI conditions for stability and stabilization of T-S fuzzy systems with bounded time-delay,”
*Fuzzy Sets and Systems*, vol. 157, no. 9, pp. 1229–1247, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - B. Chen and X. Liu, “Delay-dependent robust ${H}_{\infty}$ control for T-S fuzzy systems with time delay,”
*IEEE Transactions on Fuzzy Systems*, vol. 13, no. 4, pp. 544–556, 2005. View at Publisher · View at Google Scholar · View at Scopus - S. Zhou, J. Lam, and A. Xue, “${H}_{\infty}$ filtering of discrete-time fuzzy systems via basis-dependent Lyapunov function approach,”
*Fuzzy Sets and Systems*, vol. 158, no. 2, pp. 180–193, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - L. Wu and Z. Wang, “Fuzzy filtering of nonlinear fuzzy stochastic systems with time-varying delay,”
*Signal Processing*, vol. 89, no. 9, pp. 1739–1753, 2009. View at Publisher · View at Google Scholar · View at Scopus - H. Huang and D. W. C. Ho, “Delay-dependent robust control of uncertain stochastic fuzzy systems with time-varying delay,”
*IET Control Theory & Applications*, vol. 1, no. 4, pp. 1075–1085, 2007. View at Publisher · View at Google Scholar · View at MathSciNet - J. Yoneyama, “Robust stability and stabilizing controller design of fuzzy systems with discrete and distributed delays,”
*Information Sciences*, vol. 178, no. 8, pp. 1935–1947, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - J. Qiu, G. Feng, J. Yang, and Y. Sun, “${H}_{\infty}$ filtering design for continuous-time nonlinear systems with interval time-varying delay via T-S fuzzy models,” in
*Proceedings of the 7th Asian Control Conference (ASCC '09)*, pp. 1006–1011, Hong Kong, August 2009. - Y. He, Q.-G. Wang, C. Lin, and M. Wu, “Delay-range-dependent stability for systems with time-varying delay,”
*Automatica*, vol. 43, no. 2, pp. 371–376, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - X. Jiang and Q.-L. Han, “On ${H}_{\infty}$ control for linear systems with interval time-varying delay,”
*Automatica*, vol. 41, no. 12, pp. 2099–2106, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - H. Shao, “New delay-dependent stability criteria for systems with interval delay,”
*Automatica*, vol. 45, no. 3, pp. 744–749, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - K. Gu, V. Kharitonov, and J. Chen,
*Stability of Time-Delay Systems*, Birkhäuser, Boston, Mass, USA, 2003. - J. Sun, G. P. Liu, J. Chen, and D. Rees, “Improved delay-range-dependent stability criteria for linear systems with time-varying delays,”
*Automatica*, vol. 46, no. 2, pp. 466–470, 2010. View at Publisher · View at Google Scholar · View at Scopus - H. Gao, J. Lam, P. Shi, and C. Wang, “Parameter-dependent filter design with guaranteed
*ℋ*_{∞}performance,”*IEE Proceedings of Control Theory and Application*, vol. 152, pp. 531–537, 2005. View at Publisher · View at Google Scholar · View at Scopus