Elsevier

Automatica

Volume 47, Issue 7, July 2011, Pages 1482-1488
Automatica

Brief paper
A delay decomposition approach to L2L filter design for stochastic systems with time-varying delay

https://doi.org/10.1016/j.automatica.2011.02.021Get rights and content

Abstract

This paper investigates the problem of L2L filter design for a class of stochastic systems with time-varying delay. The addressed problem is the design of a full order linear filter such that the error system is asymptotically mean-square stable and a prescribed L2L performance is satisfied. In order to develop a less conservative filter design, a new Lyapunov-Krasovskii functional (LKF) is constructed by decomposing the delay interval into multiple equidistant subintervals, and a new integral inequality is established in the stochastic setting. Then, based on the LKF and integral inequality, the delay-dependent conditions for the existence of L2L filters are obtained in terms of linear matrix inequalities (LMIs). The resulting filters can ensure that the error system is asymptotically mean-square stable and the peak value of the estimation error is bounded by a prescribed level for all possible bounded energy disturbances. Finally, two examples are given to illustrate the effectiveness of the proposed method.

Introduction

The problem of state and signal estimation is central to a wide range of applications in signal processing and control. Over the past decades, considerable attention has been given to methods that are based on the minimization of the variance of the estimation error, i.e., the celebrated Kalman filtering approach (Anderson & Moore, 1979). One of the underlying assumptions of these methods is that the exogenous disturbances impinging on the system under consideration are stochastic in nature, but have known statistical properties. In many cases, however, the statistical nature of the external disturbances is not easily known. To solve this difficulty, some alternative filtering approaches have been developed, such as H filtering (Emara-Shabaik et al., 2010, Green and Limebeer, 1995, Liu and Wang, 2009, Simon, 2006), L2L filtering (Grigoriadis and Watson, 1997, Palhares and Peres, 2000), and L1 filtering (Nagpal et al., 1994, Tseng, 2006).

In recent years, the stochastic filtering and control problems with system models expressed by Itô-type stochastic differential equations have received considerable attention (see, e.g. Gershon et al., 2001, Hinrichsen and Pritchard, 1998, Xu and Chen, 2002a, Zhang et al., 2005, and the references therein). Such models are encountered in many areas of application, e.g., population models, nuclear fission and heat transfer, immunology, etc. (Mohler & Kolodziej, 1980). Meanwhile, time delay is often encountered in various engineering systems. In many cases, time delay is a source of instability and performance deterioration. The presence of time delay greatly complicates the stochastic filtering and control designs, and makes them more difficult (Deng et al., 2010, Gu, 2001, Gu et al., 2003, Hale and Lunel, 1993, Han, 2005, Liu et al., 2010, Wu et al., 2004). Therefore, studying the filtering and control problems of stochastic systems with time delay is of theoretical and practical importance, and has attracted a rapid growing interest in the past few decades (Gao et al., 2006, Liu et al., 2007, Liu et al., 2008, Mao, 1996, Mao et al., 1998, Xia et al., 2007, Xu and Chen, 2002b, Xu and Chen, 2003). In particular, the delay-independent L2L filtering results for uncertain stochastic systems with time-varying delay were presented in Gao et al. (2006). The delay-dependent L2L filtering results for stochastic systems with a constant time delay were given in Xia et al. (2007). Recently, the delay-range-dependent L2L filtering design was also developed in Zhou, Chen, Li, and Lin (2009) for stochastic systems with time-varying interval delay. Despite these efforts, there is room for further improvement. Yet, how to further reduce the conservatism and computational load remains an important and challenging problem.

More recently, inspired by the discretized Lyapunov functional method proposed by Gu (2001), the delay decomposition approach has been developed for stability analyses of linear retarded and neutral systems (Han, 2009), linear systems with time-varying delays (Zhang & Han, 2009) and delayed T–S fuzzy systems (Zhao, Gao, Lames, & Du, 2009), respectively. It has been shown that this method can lead to less conservative results. Motivated by this fact, the delay decomposition approach will be employed to deal with the L2L filtering design for a class of stochastic systems with time-varying delay in this work. First, a new LKF is constructed via delay decomposition, and a new integral inequality is established in the stochastic setting. Then, using the LKF and integral inequality, the delay-dependent conditions for the existence of L2L filters are obtained in terms of linear matrix inequalities (LMIs), which can be efficiently solved using the existing LMI optimization techniques (Boyd et al., 1994, Gahinet et al., 1995). The resulting filters can ensure that the error system is asymptotically mean-square stable and the peak value of the estimation error is bounded by a prescribed level for all possible bounded energy disturbances. Finally, two examples are given to show the effectiveness of the proposed method.

Notations: n and m×n denote the n-dimensional Euclidean space and the set of real m×n matrix, respectively. For a real symmetric matrix X, X>0(X0) means that X is positive definite (positive semi-definite). The superscript “T” denotes the transpose of a matrix or a vector. The symbol “” in a matrix stands for the transposed elements in the symmetric positions. λmin() means the minimal eigenvalue of a matrix. E{} denotes the expectation operator. L2[0,) is the space of square-integrable vector functions over [0,). || refers to the Euclidean norm, and stands for the usual L2[0,) norm.

Section snippets

Problem formulation

Consider a class of stochastic time-delay systems described by Itô-type stochastic retarded functional differential equations dx(t)=[Ax(t)+Adx(td(t))+Bυ(t)]dt+[Mx(t)+Mdx(td(t))]dw(t)dy(t)=[Cx(t)+Cdx(td(t))+Dυ(t)]dt+[Nx(t)+Ndx(td(t))]dw(t)z(t)=Hx(t)x(t)=φ(t),t[τ,0] where x(t)n is the state, υ(t)m is the disturbance input which belongs to L2[0,), y(t)p is the measured output, z(t)q is the signal to be estimated, and w(t) is a one-dimensional Brownian motion satisfying E{dw(t)}=0

L2L filter design

In this section, we begin with the L2L performance analysis for the error systems in (7), (8), (9) using the delay decomposition method. Then, an L2L filter design is developed for the systems (1), (2), (3), (4).

For simplicity, let φ̃(t)=Āx̃(t)+ĀdKx̃(td(t))+B̄υ(t)g̃(t)=M̄x̃(t)+M̄dKx̃(td(t)).

Then the Eq. (7) can be rewritten as dx̃(t)=φ̃(t)dt+g̃(t)dw(t).

In order to obtain a less conservative L2L filter design, inspired by the works of Han (2009), Zhang and Han (2009) and Zhao et al.

Numerical examples

In this section, two examples are given to illustrate the effectiveness and benefits of the proposed approach.

Example 1

Consider the systems (1), (2), (3), (4) with the following matrices, which have been used in Gao et al. (2006): A=[1.50.51.22.0],Ad=[0.80.30.20.4],B=[0.50.81.00.2],M=[0.80.20.60.5],C=[0.10.20.30.5],D=[0.10.20.20.3],H=[0.50.30.61.0],Md=Cd=N=Nd=[0000].

First, let us consider the time-invariant delay case, i.e., μ=0. For this case, when τ=0.5 and γ=0.8, the delay-independent result

Conclusions

The problem of L2L filtering for stochastic systems with time-varying delay has been investigated in this paper. The improved delay-dependent LMI conditions for the existence of L2L filters have been obtained using the delay decomposition based LKF and the new stochastic integral inequality. Finally, numerical examples have been presented to illustrate the effectiveness and improvement of the proposed filtering method.

Acknowledgements

The authors gratefully acknowledge the helpful comments and suggestions of the Associate Editor and anonymous reviewers, which have improved the presentation.

Huai-Ning Wu was born in Anhui, China, on November 15, 1972. He received the B.E. degree in automation from the Shandong Institute of Building Materials Industry, Jinan, China and the Ph.D. degree in control theory and control engineering from Xi’an Jiaotong University, Xi’an, China, in 1992 and 1997, respectively.

From August 1997 to July 1999, he was a Postdoctoral Researcher in the Department of Electronic Engineering at Beijing Institute of Technology, Beijing, China. In August 1999, he

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    Huai-Ning Wu was born in Anhui, China, on November 15, 1972. He received the B.E. degree in automation from the Shandong Institute of Building Materials Industry, Jinan, China and the Ph.D. degree in control theory and control engineering from Xi’an Jiaotong University, Xi’an, China, in 1992 and 1997, respectively.

    From August 1997 to July 1999, he was a Postdoctoral Researcher in the Department of Electronic Engineering at Beijing Institute of Technology, Beijing, China. In August 1999, he joined the School of Automation Science and Electrical Engineering, Beihang University (formerly Beijing University of Aeronautics and Astronautics). From December 2005 to May 2006, he was a Senior Research Associate in the Department of Manufacturing Engineering and Engineering Management (MEEM), City University of Hong Kong, Hong Kong. From October to December between 2006–2008 and from July to August in 2010, he was a Research Fellow in the Department of MEEM, City University of Hong Kong. He is currently a Professor at Beihang University. His current research interests include robust control and filtering, fault-tolerant control, stochastic systems, distributed parameter systems, and fuzzy/neural modeling and control. He is a member of the Committee of Technical Process Failure Diagnosis and Safety, Chinese Association of Automation.

    Jun-Wei Wang received the B.Sc. degree in Mathematics and Applied Mathematics and the M.Sc. degree in System Theory from Harbin Engineering University, Harbin, China, in 2007 and 2009, respectively. He is now studying for the Ph.D. degree in Control Science and Engineering in Beihang University (Beijing University of Aeronautics and Astronautics), Beijing, China. His research interests include robust control and filtering, stochastic systems, distributed parameter systems, and fuzzy modeling and control.

    Peng Shi received the B.Sc. degree in Mathematics from Harbin Institute of Technology, China in 1982; the M.E. degree in Control Theory from Harbin Engineering University, China in 1985; the Ph.D. degree in Electrical Engineering from the University of Newcastle, Australia in 1994; and the Ph.D. degree in Mathematics from the University of South Australia in 1998. He was awarded the Doctor of Science degree by the University of Glamorgan, UK, in 2006.

    Dr. Shi was a Lecturer at Heilongjiang University, China (1985–1989), in the University of South Australia (1997–1999) and a Senior Scientist in the Defence Science and Technology Organisation, Department of Defence, Australia (1999–2005). He joined in the University of Glamorgan, UK, as a professor in 2004. He has also been a Professor at Victoria University, Australia since 2008. Dr. Shi’s research interests include control system design, fault detection techniques, Markov decision processes, and operational research. He has published a number of papers in these areas. In addition, Dr. Shi is a co-author of the three research monographs, Analysis and Synthesis of Systems with Time-Delays (Berlin, Springer, 2009), Fuzzy Control and Filtering Design for Uncertain Fuzzy Systems (Berlin, Springer, 2006), and Methodologies for Control of Jump Time-Delay Systems (Boston, Kluwer, 2003).

    Dr. Shi currently serves as Editor-in-Chief of Int. J. of Innovative Computing, Information and Control. He is also an Advisory Board Member, Associate Editor and Editorial Board Member for a number of other international journals, including IEEE Transactions on Automatic Control, IEEE Transactions on Systems, Man and Cybernetics-Part B, IEEE Transactions on Fuzzy Systems, and Int. J. of Systems Science. He is the recipient of the Most Cited Paper Award of Signal Processing in 2009. Dr. Shi is a Fellow of Institute of Mathematics and its Applications (UK), and a Senior Member of IEEE.

    This work was supported by the National Natural Science Foundation of China under Grants 61074057, 91016004, 61004023, the Fundamental Research Funds for the Central Universities, China, and the Engineering and Physical Sciences Research Council, UK, EP/F0219195. This paper was not presented at any IFAC meeting. This paper was recommended for publication in revised form by Associate Editor Giuseppe De Nicolao under the direction of Editor Ian R. Petersen.

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