- About this Journal ·
- Abstracting and Indexing ·
- Aims and Scope ·
- Annual Issues ·
- Article Processing Charges ·
- Articles in Press ·
- Author Guidelines ·
- Bibliographic Information ·
- Citations to this Journal ·
- Contact Information ·
- Editorial Board ·
- Editorial Workflow ·
- Free eTOC Alerts ·
- Publication Ethics ·
- Reviewers Acknowledgment ·
- Submit a Manuscript ·
- Subscription Information ·
- Table of Contents

Abstract and Applied Analysis

Volume 2014 (2014), Article ID 673956, 10 pages

http://dx.doi.org/10.1155/2014/673956

## Delay-Dependent Robust Filtering for a Class of Fuzzy Stochastic Systems

^{1}College of Electronic and Information Engineering, Suzhou University of Science and Technology, Suzhou 215009, China^{2}School of Mechanical and Electric Engineering, Soochow University, Suzhou 215006, China

Received 21 January 2014; Accepted 27 March 2014; Published 16 April 2014

Academic Editor: Shuping He

Copyright © 2014 Ze Li and Xinhao Yang. 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 is concerned with the filtering problem for a kind of Takagi-Sugeno (T-S) fuzzy stochastic system with time-varying delay and parameter uncertainties. Parameter uncertainties in the system are assumed to satisfy global Lipschitz conditions. And the attention of this paper is focused on the stochastically mean-square stability of the filtering error system, and the performance level of the output error with the disturbance input. The method designed for the delay-dependent filter is developed based on linear matrix inequalities. Finally, the effectiveness of the proposed method is substantiated with an illustrative example.

#### 1. Introduction

It is well known that many phenomena in engineering have unavoidable uncertain factors that are modeled by the stochastic differential equation. And in recent years, the stochastic system has been widely studied. A great number of investigations on stochastic systems have been reported in the literature. For example, the adaptive back stepping controller has been addressed in [1, 2] for stochastic nonlinear systems in a strict-feedback form. When the time delay appears, [3, 4] have investigated the stability of the time-delay stochastic neutral networks; controllers under different performance levels have been designed for the stochastic system in [5–7] for the delay-dependent controller, output feedback controller, and controller, respectively. And [8–14] have studied the controlling and filtering problem for stochastic jumping systems. However, the results mentioned above are only suitable for the nonlinear systems which have exact known nonlinear dynamics models. As an efficient technique to linearize the nonlinear differential equations, T-S fuzzy model [15] can offer a good way to represent the nonlinear dynamics models.

By using T-S fuzzy model, nonlinear systems turn into linear input-output relations which could be handled easily by appropriate fuzzy sets. This method can be seen in the stirred tank reactor system in [16] and the truck trailer system in [17]. Nowadays, the researches of T-S fuzzy system have grown into a great number. A lot of results have been reported in the literature. For example, the stability and control problem of T-S fuzzy systems have been investigated in [18–22] and the references therein.

On the other hand, state estimation has been found in many practical applications and it has been extensively studied over decades. It aims at estimating the unavailable state variables or their combination for the given system [23, 24]. As a branch of state estimation theory, the filtering problem has become an important research field. The filtering problem for the T-S fuzzy system has been addressed in [25–30]; [31–33] have considered the filtering problem for delayed T-S fuzzy systems with different method. Moreover, robust filters are investigated in [34–36] for stochastic nonlinear systems.

Following above discussion, T-S fuzzy model could be used to divide the nonlinear stochastic systems into several subsystems. And during the past decade, many problems have been tackled. Reference [37] deals with the robust fault detection problem for T-S fuzzy stochastic systems. And [38, 39] consider the stabilization for the fuzzy stochastic systems with delays. References [40–43] have studied the control problem for fuzzy stochastic systems. An adaptive fuzzy controller has been designed for stochastic nonlinear systems in [44]. Reference [45] addresses the passivity of the stochastic T-S fuzzy system. Solutions to fuzzy stochastic differential equations with local martingales have been addressed in [46]. Then recognizing the value of state estimating when state variables are unavailable, it is important to research the filtering problem for T-S fuzzy stochastic systems. However, there are few results available to the best of the authors knowledge, especially the results on filtering problem for the fuzzy stochastic systems.

As a consequence, this paper will focus on the robust fuzzy delay-dependent filter design for a T-S fuzzy stochastic system with time-varying delay and norm-bounded parameter uncertainties by using the Lyapunov-Krasovskii functional technique and some useful free-weighting matrices. The obtained sufficient conditions are expressed in terms of linear matrix inequality (LMI) approach. The remainder of this paper is organized as follows. The filter design problem is formulated in Section 2. And Section 3 gives our main results. In Section 4, a numerical example is shown to illustrate the effectiveness of the proposed methods. Finally, we conclude the paper in Section 5.

*Notation*. The notation used in this paper is fairly standard. The superscript “” stands for matrix transposition. Throughout this paper, for real symmetric matrices and , the notation (resp., ) means that the matrix is positive semidefinite (resp., positive definite). denotes the -dimensional Euclidean space and denotes the set of all real matrices. stands for an identity matrix of appropriate dimension, while denotes a vector of ones. The notation is used as an ellipsis for terms that are induced by symmetry. stands for a block-diagonal matrix. denotes the Euclidean norm for vectors and denotes the spectral norm for matrices. represents the space of square-integrable vector functions over . stands for the mathematical expectation operator. Matrix dimensions, if not explicitly stated, are assumed to be compatible for algebraic operations.

#### 2. Problem Formulation and Preliminaries

Consider the time-delay T-S fuzzy stochastic system with time-varying parameter uncertainties as the following form: where is the system state; is a given differential initial function on ,; is a scalar zero mean Gaussian white noise process with unit covariance; is the measured output; is a signal to be estimated; is the noise signal which belongs to ; is a continuous differentiable function representing the time-varying delay in , which is assumed to satisfy for all , In the considered fuzzy stochastic system, , , , , , , , , and are known constant matrices with appropriate dimensions. , , , and represent the unknown time-varying parameter uncertainties and are assumed to satisfy where , , , and are known real constant matrices and the unknown time-varying matrix function satisfying And using the fuzzy theory, there always have for all , The fuzzy filters we considered are as follows: in which the fuzzy rules have the same representations as in (1). and . , , and are the filters needed to be determined.

*Remark 1. *It is worth to mention that there are two approaches for the filter design in fuzzy systems. The implementation of the filter could be chosen to depend on or not depend on the fuzzy rules when the fuzzy model is available or not. And it is obvious to see that the former filter related to the fuzzy rules is less conserve and more complex. So we assume that the fuzzy is known here, which means the fuzzy-rule-dependent filter is investigated in this paper as in (6).

Let and .

And the filtering error dynamic system can be written as where

We intend to design sets of fuzzy filters in the form of (6) in this paper, such that for any scalar and a prescribed level of noise attenuation , the filtering error system () could be mean square stable. Moreover, the error system () satisfies performance.

Throughout the paper, we adopt the following definitions and lemmas, which help to complete the proof of the main results.

*Definition 2. *The system () is said to be robust stochastic mean-square stable if there exists for any such that
when , for any uncertain variables. And in addition,
for any initial conditions.

*Definition 3. *The robust stochastic mean-square stable system () is said to satisfy the performance, for the given scalar and any nonzero , and the system () satisfies
and for any uncertain variables, where

Lemma 4. *For the given matrices with and positive scalar , the following inequality holds:
*

#### 3. Robust Stochastic Stabile

First, we define the following variables for convenience:

Theorem 5. *The filtering error system () is robust stochastic mean square stable and (11) is satisfied for any time-varying delay , if there exist matrices , , , , , , such that the following matrix inequalities hold:
**
where
*

*Proof. *Define the following Lyapunov-Krasovskii candidate for system ():

When ,

By using the Newton-Leibnitz formula, the following equations can be got for any matrices , with appropriate dimensions:
where
And is a new vector defined as follows:

By the above formulas (19) and Lemma 4, we can deduce that
where

During the analysis, it can be seen that
And applying the Schur complement to (15), we can derive the following inequality with :
From (22)–(26), we can get that
which ensures that system () with is robustly stochastically stable according to Definition 2 and [47]. By Itô's formula, it is easy to derive

Now we establish the performance of the filtering error system . It is easy to obtain

Then applying the Schur complement formula to (15), we can get
for all , where

Therefore, for all , , which means
Then using the Schur complement to the first formula in (15), we have , which guarantees
Therefore, for any zero mean Gaussian white noise process with unit covariance.

*Remark 6. *The system we studied is a time-varying delay system containing the information of both the lower bound and the upper bound of time delay. By such a consideration, delay-dependent result is more reliable and approaches to reality that not all the delays begin with 0 moment.

*Remark 7. *It is worth mentioning that Theorem 5 can be easily extended to investigate the robust filtering design problem for the systems () with parameter uncertainties.

Now we are in a position to present a sufficient condition for the solvability of robust filtering problem.

Theorem 8. *Consider the uncertain T-S fuzzy stochastic time-varying delay system () and a constant scalar . The robust filtering problem is solvable if there exist scalars and matrices , , , , , , , ; , , , , , , , and such that the following LMIs hold:
**
where
*

When the LMIs (34)–(38) are feasible, the time-dependent filter we desired here can be chosen as where and are nonsingular matrices satisfying .

*Proof. *Similar to [33], we know that is nonsingular. Therefore, there always exist nonsingular matrices and such that holds. Then we define the nonsingular matrices and as follows:
Define . Then there is

Now using Lemma 4 and recalling (36), we can deduce that
We can deduce that
which is equivalent to (15). Therefore, it is easy to see that the condition in Theorem 5 and the LMIs in (34)–(37) are equivalent. Finally, it can be concluded that the filtering error system is stochastically stable with performance level .

*Remark 9. *The desired filters can be constructed by solving the LMIs in (34)–(38), which can be implemented by using standard numerical algorithms, and no tuning of parameters will be involved.

*Remark 10. *In the proof of above Theorem, we adopt (25), (26), and Newton-Leibnitz formula to reduce the conservatism. Moreover, the results obtained in Theorems 5 and 8 can be further extended based on fuzzy or piecewise Lyapunov-Krasovskii function.

#### 4. Numerical Example

In this section, a numerical example is provided to show the effectiveness of the results obtained in the previous section.

*Example 1. *Consider the T-S fuzzy stochastic system () with model parameters given as follows:
And the parameter uncertainties are shown as:
The membership functions are

By using the Matlab LMI Control Toolbox, we have the robust filtering problem which is solvable to Theorem 8. It can be calculated that for any , , the robust filtering problem can be solved. A desired fuzzy filter can be constructed as in the form of (6) with

The simulation results of the state response of the plant and the filter are given in Figure 1, where the initial condition is , . Figure 2 shows the simulation results of the signal , and the exogenous disturbance input is given by , , which belongs to .

#### 5. Conclusion

This paper considers the robust filter design problem for the uncertain T-S fuzzy stochastic system with time-varying delay. An LMI approach has been developed to design the fuzzy filter ensuring not only the robust stochastic mean-square stability but also a prescribed performance level of the filtering error system for all admissible uncertainties. A numerical example has been provided to show the effectiveness of the proposed filter design methods.

#### Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

#### Acknowledgment

This work is supported by the National Natural Science Foundation of China under Grants nos. 61203048, 61304047, and 61203047.

#### References

- H. Deng and M. Krstić, “Output-feedback stochastic nonlinear stabilization,”
*Institute of Electrical and Electronics Engineers. Transactions on Automatic Control*, vol. 44, no. 2, pp. 328–333, 1999. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - Z. Pan and T. Başar, “Backstepping controller design for nonlinear stochastic systems under a risk-sensitive cost criterion,”
*SIAM Journal on Control and Optimization*, vol. 37, no. 3, pp. 957–995, 1999. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - W.-H. Chen and X. Lu, “Mean square exponential stability of uncertain stochastic delayed neural networks,”
*Physics Letters A*, vol. 372, no. 7, pp. 1061–1069, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - R. Yang, Z. Zhang, and P. Shi, “Exponential stability on stochastic neural networks with discrete interval and distributed delays,”
*IEEE Transactions on Neural Networks*, vol. 21, no. 1, pp. 169–175, 2010. View at Publisher · View at Google Scholar · View at Scopus - M. Basin and D. Calderon-Alyarez, “Delay-dependent stability for vector nonlinear stochastic systems with multiple delays,”
*International Journal of Innovative Computing, Information and Control*, vol. 7, no. 4, pp. 1565–1576, 2011. View at Scopus - T. Senthilkumar and P. Balasubramaniam, “Delay-dependent robust ${H}_{\infty}$ control for uncertain stochastic T-S fuzzy systems with time-varying state and input delays,”
*International Journal of Systems Science*, vol. 42, no. 5, pp. 877–887, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - W.-B. Wu, P.-C. Chen, M.-H. Hung, K.-Y. Chang, and W.-J. Chang, “LMI robustly decentralized
*H*_{∞}output feedback controller design for stochastic large-scale uncertain systems with time-delays,”*Journal of Marine Science and Technology*, vol. 17, no. 1, pp. 42–49, 2009. View at Scopus - H. Shen, Z. Wang, X. Huang, and J. Wang, “Fuzzy dissipative control for nonlinear Marko-vian jump systems via retarded feedback,”
*Journal of the Franklin Institute*, 2013. View at Publisher · View at Google Scholar - H. Shen, J. H. Park, L. Zhang, and Z. Wu, “Robust extended dissipative control for sampled-data Markov jump systems,”
*International Journal of Control*, 2013. View at Publisher · View at Google Scholar - H. Shen, X. Song, and Z. Wang, “Robust fault-tolerant control of uncertain fractional-order systems against actuator faults,”
*IET Control Theory & Applications*, vol. 7, no. 9, pp. 1233–1241, 2013. View at Publisher · View at Google Scholar · View at MathSciNet - S. He and F. Liu, “Robust peak-to-peak filtering for Markov jump systems,”
*Signal Processing*, vol. 90, no. 2, pp. 513–522, 2010. View at Publisher · View at Google Scholar · View at Scopus - S. He and F. Liu, “Robust ${L}_{2}$-${L}_{\infty}$ filtering of time-delay jump systems with respect to the finite-time interval,”
*Mathematical Problems in Engineering*, vol. 2011, Article ID 839648, 17 pages, 2011. View at Publisher · View at Google Scholar · View at MathSciNet - S. He and F. Liu, “Finite-time ${H}_{\infty}$ control of nonlinear jump systems with time-delays via dynamic observer-based state feedback,”
*IEEE Transactions on Fuzzy Systems*, vol. 20, no. 4, pp. 605–614, 2012. - S. He, “Resilient ${L}_{2}$-${L}_{\infty}$ filtering of uncertain Markovian jumping systems within the finite-time interval,”
*Abstract and Applied Analysis*, vol. 2013, Article ID 791296, 7 pages, 2013. View at Publisher · View at Google Scholar · View at MathSciNet - 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 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 - K. Tanaka and M. Sano, “Robust stabilization problem of fuzzy control systems and its application to backing up control of a truck-trailer,”
*IEEE Transactions on Fuzzy Systems*, vol. 2, no. 2, pp. 119–134, 1994. View at Publisher · View at Google Scholar · View at Scopus - C.-W. Chen, K. Yeh, K. F. R. Liu, and M.-L. Lin, “Applications of fuzzy control to nonlinear time-delay systems using the linear matrix inequality fuzzy Lyapunov method,”
*Journal of Vibration and Control*, vol. 18, no. 10, pp. 1561–1574, 2012. View at Publisher · View at Google Scholar · View at MathSciNet - J. Liu and D. Yue, “Asymptotic and robust stability of T-S fuzzy genetic regulatory networks with time-varying delays,”
*International Journal of Robust and Nonlinear Control*, vol. 22, no. 8, pp. 827–840, 2012. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - L. Wu, J. Su, P. Shi, and J. Qiu, “A new approach to stability analysis and stabilization of discrete-time TCS fuzzy time-varying delay systems,”
*IEEE Transactions on Systems, Man, and Cybernetics, Part C*, vol. 41, no. 1, pp. 273–286, 2011. - Z.-G. Wu, P. Shi, H. Su, and J. Chu, “Reliable
*H*_{∞}control for discrete-time fuzzy systems with infinite-distributed delay,”*IEEE Transactions on Fuzzy Systems*, vol. 20, no. 1, pp. 22–31, 2012. View at Publisher · View at Google Scholar · View at Scopus - B. Zhang, W. X. Zheng, and S. Xu, “Passivity analysis and passive control of fuzzy systems with time-varying delays,”
*Fuzzy Sets and Systems*, vol. 174, pp. 83–98, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - S. He and F. Liu, “Adaptive observer-based fault estimation for stochastic Markovian jumping systems,”
*Abstract and Applied Analysis*, vol. 2012, Article ID 176419, 11 pages, 2012. View at Zentralblatt MATH · View at MathSciNet - Z.-G. Wu, P. Shi, H. Su, and J. Chu, “Asynchronous ${l}_{2}$-${l}_{\infty}$ filtering for discrete-time stochastic Markov jump systems with randomly occurred sensor nonlinearities,”
*Automatica*, vol. 50, no. 1, pp. 180–186, 2014. View at Publisher · View at Google Scholar · View at MathSciNet - H. Chen, “Delay-dependent robust ${H}_{\infty}$ filter design for uncertain neutral stochastic system with time-varying delay,”
*IET Signal Processing*, vol. 7, no. 5, pp. 368–381, 2013. View at Publisher · View at Google Scholar · View at MathSciNet - Z. Li, S. Xu, Y. Zou, and Y. Chu, “Robust ${H}_{\infty}$ filter design of uncertain T-S fuzzy neutral systems with time-varying delays,”
*International Journal of Systems Science*, vol. 42, no. 7, pp. 1231–1238, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - M. Liu, J. You, and X. Ma, “
*H*_{∞}filtering for sampled-data stochastic systems with limited capacity channel,”*Signal Processing*, vol. 91, no. 8, pp. 1826–1837, 2011. View at Publisher · View at Google Scholar · View at Scopus - R. Lu, Y. Xu, and A. Xue, “
*H*_{∞}filtering for singular systems with communication delays,”*Signal Processing*, vol. 90, no. 4, pp. 1240–1248, 2010. View at Publisher · View at Google Scholar · View at Scopus - X. Su, P. Shi, L. Wu, and Y. Song, “A novel approach to filter design for TCS fuzzy discrete-time systems with time-varying delay,”
*IEEE Transactions on Fuzzy Systems*, vol. 20, no. 6, pp. 1114–1129, 2012. - B. Zhang and W. X. Zheng, “
*H*_{∞}filter design for nonlinear networked control systems with uncertain packet-loss probability,”*Signal Processing*, vol. 92, no. 6, pp. 1499–1507, 2012. View at Publisher · View at Google Scholar · View at Scopus - P. Balasubramaniam, V. M. Revathi, and J. H. Park, “${L}_{2}-{L}_{\infty}$ filtering for neutral Markovian switching system with mode-dependent time-varying delays and partially unknown transition probabilities,”
*Applied Mathematics and Computation*, vol. 219, no. 17, pp. 9524–9542, 2013. View at Publisher · View at Google Scholar · View at MathSciNet - Y. Chen, A. Xue, and S. Zhou, “New delay-dependent ${L}_{2}$-${L}_{\infty}$ filter design for stochastic time-delay systems,”
*Signal Processing*, vol. 89, no. 6, pp. 974–980, 2009. View at Publisher · View at Google Scholar · View at Scopus - Z. Li, S. Xu, Y. Zou, and Y. Chu, “Delay-dependent robust ${L}_{2}$-${L}_{\infty}$ filtering of T-S fuzzy systems with time-varying delays,”
*International Journal of Adaptive Control and Signal Processing*, vol. 24, no. 7, pp. 529–539, 2010. View at MathSciNet - H. Gao, J. Lam, and C. Wang, “Robust energy-to-peak filter design for stochastic time-delay systems,”
*Systems & Control Letters*, vol. 55, no. 2, pp. 101–111, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - M. Liu, D. W. C. Ho, and Y. Niu, “Robust filtering design for stochastic system with mode-dependent output quantization,”
*IEEE Transactions on Signal Processing*, vol. 58, no. 12, pp. 6410–6416, 2010. View at Publisher · View at Google Scholar · View at MathSciNet - G. Wang and C. Su, “Delay-distribution-dependent ${H}_{\infty}$ filtering for linear systems with stochastic time-varying delays,”
*Journal of the Franklin Institute*, vol. 350, no. 2, pp. 358–377, 2013. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - L. Wu and D. W. C. Ho, “Fuzzy filter design for Itô stochastic systems with application to sensor fault detection,”
*IEEE Transactions on Fuzzy Systems*, vol. 17, no. 1, pp. 233–242, 2009. View at Publisher · View at Google Scholar · View at Scopus - P. Balasubramaniam and T. Senthilkumar, “Delay-dependent robust stabilization and ${H}_{\infty}$ control for uncertain stochastic T-S fuzzy systems with discrete interval and distributed time-varying delays,”
*International Journal of Automation and Computing*, vol. 10, no. 1, pp. 18–31, 2013. - J. Yang, S. Zhong, W. Luo, and G. Li, “Delay-dependent stabilization for stochastic delayed fuzzy systems with impulsive effects,”
*International Journal of Control, Automation and Systems*, vol. 8, no. 1, pp. 127–134, 2010. View at Publisher · View at Google Scholar · View at Scopus - W. H. Chen, B. S. Chen, and W. Zhang, “Robust control design for nonlinear stochastic partial differential systems with Poisson noise: fuzzy implementation,”
*Fuzzy Sets and Systems*, 2014. View at Publisher · View at Google Scholar - Y. Li, S. Tong, T. Li, and X. Jing, “Adaptive fuzzy control of uncertain stochastic nonlinear systems with unknown dead zone using small-gain approach,”
*Fuzzy Sets and Systems*, vol. 235, pp. 1–24, 2014. View at Publisher · View at Google Scholar · View at MathSciNet - T. Wang, S. Tong, and Y. Li, “Robust adaptive fuzzy output feedback control for stochastic nonlinear systems with unknown control direction,”
*Neurocomputing*, vol. 106, pp. 31–41, 2013. - Z. Xia, J. M. Li, and J. R. Li, “Delay-dependent fuzzy static output feedback control for discrete-time fuzzy stochastic systems with distributed time-varying delays,”
*ISA Transactions*, vol. 51, pp. 702–712, 2012. - Z. Wang, L. Huang, X. Yang, and A. Xin, “Adaptive fuzzy control for stochastic nonlinear systems via sliding mode method,”
*Circuits, Systems, and Signal Processing*, vol. 32, no. 6, pp. 2839–2850, 2013. View at Publisher · View at Google Scholar · View at MathSciNet - Q. Song, Z. Zhao, and J. Yang, “Passivity and passification for stochastic Takagi-Sugeno fuzzy systems with mixed time-varying delays,”
*Neurocomputing*, vol. 122, pp. 330–337, 2013. View at Publisher · View at Google Scholar - W. Fei, H. Liu, and W. Zhang, “On solutions to fuzzy stochastic differential equations with local martingales,”
*Systems & Control Letters*, vol. 65, pp. 96–105, 2014. View at Publisher · View at Google Scholar · View at MathSciNet - X. Mao,
*Stochastic Differential Equations and Applications*, Horwood Publishing, Chichester, UK, 2nd edition, 2008. View at MathSciNet