Robust finite-horizon filtering with randomly occurred nonlinearities and quantization effects☆
Introduction
The filtering or state estimation problem has long been one of the fundamental problems in the control and signal processing areas which has attracted constant research attention. In the past decade, a number of linear/nonlinear filtering techniques have been developed with respect to various filtering performance criteria, such as the specification, the minimum variance requirement and the so-called admissible variance constraint. For example, extended Kalman filters have been designed in Reif and Unbehauen (1999) for nonlinear deterministic systems and in Reif, Günther, Yaz, and Unbehauen (1999) for nonlinear stochastic systems. Robust filtering problems have been extensively studied in Hung and Yang (2003), Wang, Ho, and Liu (2003), Xie, Soh, and de Souza (1994) and Yang, Wang, Feng, and Liu (2009) for systems with norm-bounded uncertainties and in James and Petersen (1998), Savkin and Petersen, 1995, Savkin and Petersen, 1998 and Seo, Yu, Park, and Lee (2006) for uncertain systems with integral quadratic constraints. Filters with error variance constraints have been exploited in Wang et al. (2003), Xie et al. (1994) and Yang et al. (2009) for systems which are subject to noise with known statistics. Recently, filtering problems have received particular research attention by means of the linear matrix inequality (LMI) approach, see e.g. Gao and Chen (2007), Gong and Su (2008), Shi, Mahmoud, Nguang, and Ismail (2006), Wang, Liu, and Liu (2008), Wang, Ho, Liu, and Liu (2009), Wu, Lam, Paszke, Galkowski, and Rogers (2008) and Wu, Shi, Gao, and Wang (2008). It is worth pointing out that, in most of the literature mentioned above, the infinite-horizon filtering problem has been considered for time-invariant systems.
With respect to time-varying systems, finite-horizon filtering problems have received much research attention due primarily to their insight into applications. Among others, the recursive Riccati difference equation approach has been widely employed to design filters. For example, the bounded real lemma (BRL), based on the Riccati difference equation, was derived in Xie, de Souza, and Wang (1993), which is suitable for offline computation since the boundary conditions are given for the end of the known interval. In Rawson, Hsu, and Rho (1997), by employing the method of Hilbert adjoins, another version of the BRL was obtained, based on which the Riccati difference equation can be solved forward in time, and a reduced order filter was designed for the linear discrete-time system. In Hung and Yang (2003), a robust filter with error variance constraints was designed for a discrete time-varying uncertain system by forwardly solving a recursive Riccati difference equation. For Itô-type stochastic systems, the control problem was studied in Zhang, Huang, and Zhang (2007) for discrete time-varying stochastic systems, where a BRL was obtained in terms of a constrained backward difference equation in the stochastic framework. It is worth mentioning that, in Gershon and Shaked (2005) and Shaked and Suplin (2001), a differential/difference linear matrix inequality (DLMI) approach was proposed to obtain a BRL that allows for time-varying matrices in the state-space description and can therefore be applied to various problems involving time-varying systems.
With the rapid development of network technologies, more and more control systems are executed over communication networks, which have many advantages such as low cost, reduced weight and power requirements, simple installation and maintenance, and high reliability. However, since the network cable is of limited capacity, many challenging issues inevitably emerge, for example, the transmission delay (Basin and Alvarez, 2008, Basin et al., 2007, Gao et al., 2007, Lu et al., 2007, Sun et al., 2008, Yang et al., 2009), data missing (packet dropouts) (Sun et al., 2008, Wang et al., 2003, Wang et al., 2009, Wang et al., 2006, Xiong and Lam, 2007), signal quantization (Gao and Chen, 2007, Tian et al., 2008), scheduling confusion, etc. Nevertheless, one interesting problem that has been largely overlooked is the so-called randomly occurred nonlinearities (RONs). As is well known, a wide class of practical systems are influenced by additive nonlinear disturbances that are caused by environmental circumstances. Such nonlinear disturbances themselves may be subject to random abrupt changes, for example, random failures and repairs of the components, changes in the interconnections of subsystems, sudden environment changes, modification of the operating point of a linearized model of a nonlinear system, etc. In other words, the nonlinear disturbances may occur in a probabilistic way and are randomly changeable in terms of their types and/or intensity. Unfortunately, to the best of the authors’ knowledge, the finite-horizon filtering problem for discrete time-varying Itô-type stochastic systems with RONs has not been fully investigated, not to mention the case where the systems also involve polytopic uncertainties and quantization effects. It is, therefore, the purpose of this paper to reduce this gap by solving a set of recursive linear matrix inequalities motivated by the DLMI approach developed in Gershon and Shaked (2005) and Shaked and Suplin (2001).
This paper is concerned with the robust filtering problem for discrete time-varying stochastic systems with polytopic uncertainties, RONs and quantization effects. In order to take into account the phenomena of nonlinear disturbances appearing in a random way, we make the first attempt to introduce RONs that are modeled by a Bernoulli distributed white sequence with a known conditional probability. Sufficient conditions are derived for the estimation error of the system under consideration to satisfy the performance constraint. A robust filter is then designed by solving a set of recursive LMIs. The proposed robust filtering technique is a recursive algorithm that is suitable for on-line computation by employing more information at and before the current time to estimate the current state. Finally, a numerical simulation example is used to demonstrate the effectiveness of the filtering technology presented in this paper.
Notation The notation used here is fairly standard except where otherwise stated. denotes the dimensional Euclidean space. refers to the norm of a matrix defined by . The notation (respectively, ), where and are real symmetric matrices, means that is positive semi-definite (respectively, positive definite). represents the transpose of the matrix . denotes the identity matrix of compatible dimension. stands for a block-diagonal matrix. Moreover, we may fix a probability space , where , the probability measure, has total mass 1. stands for the expectation of the random variable with respect to the given probability measure . The set of all nonnegative integers is denoted by and the set of all nonnegative real numbers is represented by . The asterisk in a matrix is used to denote a term that is induced by symmetry. Matrices, if they are not explicitly specified, are assumed to have compatible dimensions.
Section snippets
Problem formulation and preliminaries
Consider the following class of nonlinear discrete time-varying polytopic uncertain stochastic systems defined on : where is the state vector, is the measured output vector, is the state combination to be estimated, and is a one-dimensional, zero-mean Gaussian white noise sequence on a probability space with . Let
Performance analysis of filter
In this section, we will give a BRL for the augmented system (10) to satisfy the performance constraint (9) for all nonlinearities subject to (12).
To derive the BRL for the augmented system (10), we introduce the following lemma. Lemma 1 For a given scalar , the augmented system (10) has the -gain not greater than , i.e., the following criterion is satisfiedfor all and for all , if and only if there exist a family of positive
Design of robust filters
In this section, the robust filter is designed for nonlinear discrete time-varying stochastic systems subject to RONs as well as quantization effects in terms of time-varying LMIs. Lemma 3 Let and be real matrices of appropriate dimensions with satisfy . Then, for any scalar , we have .
The following theorem provides a recursive LMI approach to the addressed design problem of robust filters for the discrete time-varying stochastic system with
An illustrative example
In this section, a numerical example is presented to demonstrate the effectiveness of the method proposed in this paper.
Consider the following class of nonlinear discrete time-varying polytopic uncertain stochastic system with the initial value . We choose the nonlinear function as
Conclusions
In this paper, we have studied the robust filtering problem for discrete time-varying stochastic systems with polytopic uncertainties, RONs and quantization effects. The RONs have been modeled by a Bernoulli distributed white sequence with a known conditional probability. Sufficient conditions have been derived for the estimation error of the system under consideration to satisfy the performance constraint. A robust filter has then been designed by solving a set of recursive LMIs. The
Bo Shen received his B.S. degree in Mathematics from Northwestern Polytechnical University, Xi’an, China, in 2003. He is currently pursuing the Ph.D. degree in the School of Information Science and Technology, Donghua University, Shanghai, China. From August 2009 to February 2010, he was a Research Assistant in the Department of Electrical and Electronic Engineering, the University of Hong Kong, Hong Kong. He is now a Visiting Ph.D. Student in the Department of Information Systems and
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Bo Shen received his B.S. degree in Mathematics from Northwestern Polytechnical University, Xi’an, China, in 2003. He is currently pursuing the Ph.D. degree in the School of Information Science and Technology, Donghua University, Shanghai, China. From August 2009 to February 2010, he was a Research Assistant in the Department of Electrical and Electronic Engineering, the University of Hong Kong, Hong Kong. He is now a Visiting Ph.D. Student in the Department of Information Systems and Computing, Brunel University, UK. His research interest is primarily in nonlinear stochastic control and filtering. He is a very active reviewer for many international journals.
Zidong Wang was born in Jiangsu, China, in 1966. He received the B.Sc. degree in Mathematics in 1986 from Suzhou University, Suzhou, China, and the M.Sc. degree in Applied Mathematics in 1990 and the Ph.D. degree in Electrical Engineering in 1994, both from Nanjing University of Science and Technology, Nanjing, China.
He is currently Professor of Dynamical Systems and Computing in the Department of Information Systems and Computing, Brunel University, UK. From 1990 to 2002, he held teaching and research appointments in universities in China, Germany and the UK. Prof. Wang’s research interests include dynamical systems, signal processing, bioinformatics, control theory and applications. He has published more than 100 papers in refereed international journals. He is a holder of the Alexander von Humboldt Research Fellowship of Germany, the JSPS Research Fellowship of Japan, and the William Mong Visiting Research Fellowship of Hong Kong.
Prof. Wang serves as an Associate Editor for 11 international journals, including IEEE Transactions on Automatic Control, IEEE Transactions on Control Systems Technology, IEEE Transactions on Neural Networks, IEEE Transactions on Signal Processing, and IEEE Transactions on Systems, Man, and Cybernetics—Part C. He is a Senior Member of the IEEE, a Fellow of the Royal Statistical Society and a member of program committees for many international conferences.
Huisheng Shu received his B.Sc. degree in Mathematics in 1984 from Anhui Normal University, Wuhu, China, and the M.Sc. degree in Applied Mathematics in 1990 and the Ph.D. degree in control theory in 2005, both from Donghua University, Shanghai, China. He is currently a Professor at Donghua University, Shanghai, China. He has published 20 papers in refereed international journals. His research interests include the mathematical theory of stochastic systems, robust control and robust filtering.
Guoliang Wei received his B.Sc. degree in Mathematics in 1997 from Henan Normal University, Xinxiang, China, and the M.Sc. degree in Applied Mathematics in 2005 and the Ph.D. degree in Control Engineering in 2008, both from Donghua University, Shanghai, China.
Dr. Wei is now an Alexander von Humboldt research fellow in the Institute for Automatic Control and Complex Systems, University of Duisburg–Essen, Germany. From March 2009 to February 2010, he was a postdoctoral research fellow in the Department of Information Systems and Computing, Brunel University, UK, sponsored by the Leverhulme Trust of the UK. From June to August 2007, he was a research assistant at the University of Hong Kong. From March to May 2008, he was a research assistant at the City University of Hong Kong.
Dr. Wei’s research interests include nonlinear systems, stochastic systems and bioinformatics. He has published more than 20 papers in refereed international journals. He is a very active reviewer for many international journals.
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This work was supported in part by the Engineering and Physical Sciences Research Council (EPSRC) of the UK under Grant GR/S27658/01, the Royal Society of the UK, the National 973 Program of China under Grant 2009CB320600, the National Natural Science Foundation of China under Grant 60974030, the Shanghai Natural Science Foundation of China under Grant 10ZR1421200, and the Alexander von Humboldt Foundation of Germany. The material in this paper was not presented at any conference. This paper was recommended for publication in revised form by Associate Editor Brett Ninness under the direction of Editor Torsten Söderström.