Elsevier

Automatica

Volume 46, Issue 10, October 2010, Pages 1682-1688
Automatica

Brief paper
Distributed H-consensus filtering in sensor networks with multiple missing measurements: The finite-horizon case

https://doi.org/10.1016/j.automatica.2010.06.025Get rights and content

Abstract

This paper is concerned with a new distributed H-consensus filtering problem over a finite-horizon for sensor networks with multiple missing measurements. The so-called H-consensus performance requirement is defined to quantify bounded consensus regarding the filtering errors (agreements) over a finite-horizon. A set of random variables are utilized to model the probabilistic information missing phenomena occurring in the channels from the system to the sensors. A sufficient condition is first established in terms of a set of difference linear matrix inequalities (DLMIs) under which the expected H-consensus performance constraint is guaranteed. Given the measurements and estimates of the system state and its neighbors, the filter parameters are then explicitly parameterized by means of the solutions to a certain set of DLMIs that can be computed recursively. Subsequently, two kinds of robust distributed H-consensus filters are designed for the system with norm-bounded uncertainties and polytopic uncertainties. Finally, two numerical simulation examples are used to demonstrate the effectiveness of the proposed distributed filters design scheme.

Introduction

The past few decades have witnessed constant research interest on various aspects of sensor networks due primarily to the fact that sensor networks have been extensively applied in many fields such as information collection, environmental monitoring, industrial automation and intelligent buildings. In particular, the distributed filtering or estimation for sensor networks has been an ongoing research issue that has attracted increasing attention from researchers in the area. Compared to the single sensor, filter i in a sensor network estimates the system state based not only on sensor i’s measurement, but also on its neighboring sensors’ measurements according to the topology of the given sensor network. Such a problem is usually referred to as the distributed filtering or estimation problem. The main difficulty in designing distributed filters lies in how to deal with the complicated coupling between one sensor and its neighboring sensors.

So far, considerable research efforts have been made with respect to distributed filtering and some novel distributed filters are proposed, see e.g. Cattivelli, Lopes, and Sayed (2008), Speranzon, Fischione, Johansson, and Sangiovanni-Vincentelli (2008) and Yu, Chen, Wang, and Yang (2009). Also, the consensus problems of multi-agent networks have stirred a great deal of research interest, and a rich body of research results has been reported in the literature, see e.g. Bliman and Ferrari-Trecate (2008), Lin, Jia, and Li (2008), Li and Zhang (2009), Olfati-Saber and Murray (2004), Sun, Wang, and Xie (2008), Shi and Hong (2009), Xiao and Boyd (2004) and Xiao and Wang (2008). Recently, the consensus problem has also been studied for designing distributed Kalman filters (DKFs) (Olfati-Saber, 2007, Olfati-Saber and Shamma, 2005, Stankovic et al., 2009). For example, a distributed filter has been introduced in Olfati-Saber and Shamma (2005) that allows the nodes of a sensor network to track the average of n sensor measurements using an average consensus based distributed filter called a consensus filter. The DKF algorithm presented in Olfati-Saber and Shamma (2005) has been modified in Olfati-Saber (2007), where another two novel DKF algorithms have been proposed and the communication complexity as well as packet-loss issues have been discussed. As is well known, a variety of robust and/or H filtering approaches have been proposed in the literature to improve the robustness of the filters against parameter uncertainties and exogenous disturbances. In this sense, it seems natural to include the robust and/or H performance requirements for the distributed consensus filtering problems, and this constitutes one of the two motivations for our current investigation.

Virtually all practical engineering systems are time-varying. A finite-horizon filter could provide better transient performance for the filtering process especially when the noise inputs are non-stationary. Therefore, it is of vital importance to consider the filtering problems for a time-varying system over a finite horizon. Some efforts have been made on this issue. For example, in Yang, Wang, and Hung (2002), a robust finite-horizon Kalman filter has been designed for uncertain systems with multiplicative noises by means of two discrete Riccati difference equations. Using the same approach, the robust finite-horizon filtering problem has been investigated for an uncertain system with randomly varying sensor delay in Yang, Wang, Feng, and Liu (2009). In addition to the recursive Riccati equation approach, the difference linear matrix inequality (DLMI) method serves as another effective tool for handling finite-horizon control and filtering problems for time-varying systems. The DLMI approach had been originally proposed in Gershon, Pila, and Shaked (2001) and Gershon, Shaked, and Yaesh (2005), which has proven to be computationally appealing due mainly to the numerical efficiency of LMI algorithms. Up to now, the robust and/or H distributed consensus filtering problem has not been adequately addressed for time-varying systems over a finite horizon, which gives rise to the second motivation of our research.

In response to the above discussion, in this paper, we aim to deal with the distributed H-consensus filtering problem for sensor networks with multiple missing measurements. The main contributions can be summarized as follows: (1) the concept of H-consensus is introduced to quantify the consensus degree over a finite horizon; (2) the distributed filtering problem is addressed for a class of time-varying systems in the sensor network represented by a directed graph; and (3) a set of random variables is introduced to model the probabilistic data missing occurring in the process of information transmission from the system to each sensor. By resorting to the DLMI technique, the filter parameters can be designed in a recursive way, subject to the H-consensus performance constraint, via the measurements and estimates from the system state as well as its neighbors. Based on the analysis and synthesis results established, we further discuss the robust distributed H-consensus filtering problem for a system with norm-bounded uncertainties and polytopic uncertainties, respectively, in terms of a set of DLMIs which can be solved by using available software. Finally, two numerical simulation examples are exploited to show the effectiveness of the distributed filtering techniques proposed in this paper.

Notation

The notation used here is fairly standard except where otherwise stated. Rn and Rn×m denote, respectively, the n dimensional Euclidean space and the set of all n×m real matrices. A refers to the norm of a matrix A defined by A=trace(ATA). The notation XY (respectively, X>Y), where X and Y are real symmetric matrices, means that XY is positive semi-definite (respectively, positive definite). MT represents the transpose of the matrix M. I denotes the identity matrix of compatible dimension. diagn{Ai} stands for the block-diagonal matrix diag{A1,A2,,An} and diagn{A} means the block-diagonal matrix diag{A,A,,A} with n blocks. diagni{A} represents the block-diagonal matrix with n blocks, where the ith block is A and all others are zero matrices. vecn{xi} denotes [x1x2xn]. E{x} stands for the expectation of the stochastic variable x. Prob{} means the occurrence probability of the event “”. l2[0N1] is the space of square summable vector-value functions f(k) in an interval [0N1] with the norm f2=(k=0N1f(k)2)12. In symmetric block matrices, “” is used as an ellipsis for terms induced by symmetry. Matrices, if they are not explicitly specified, are assumed to have compatible dimensions.

Section snippets

Problem formulation and preliminaries

Consider a sensor network whose topology is represented by a directed graph G=(V,E,A) of order n with the set of nodes (sensors) V={1,2,,n}, set of edges EV×V, and a weighted adjacency matrix A=[aij] with nonnegative adjacency elements aij. An edge of G is denoted by (i,j). The adjacency elements associated with the edges of the graph are positive, i.e., aij>0(i,j)E. Moreover, we assume aii=1 for all iV, and therefore (i,i) can be regarded as an additional edge. The set of neighbors of

Finite-horizon distributed H-consensus filtering

In this section, we investigate the distributed H-consensus filtering problem for system (1) with n sensors whose topology is determined by the given graph G=(V,E,A). For convenience of later analysis, we denote e(k)=vecnT{eiT(k)},x̄(k)=vecnT{xT(k)},z̃(k)=vecnT{z̃iT(k)},Ā(k)=diagn{A(k)},B̄(k)=vecnT{BT(k)},D̄(k)=vecnT{DiT(k)},M̄(k)=diagn{M(k)},Eni(k)=diagni{Ci(k)},Gβ(k)=diagn{βiCi(k)},αi=βi(1βi). Then, the error dynamics governed by (5) can be rewritten in the following compact form {e(k+1)=(A

Robust distributed H-consensus filtering for uncertain systems

In this section, the problem of robust finite-horizon distributed H-consensus filtering is considered for two classes of uncertain systems, i.e., systems with norm-bounded uncertainties and systems with polytopic uncertainties.

Illustrative examples

Consider the sensor network (with 6 nodes) whose topology is represented by a directed graph G=(V,E,A) with the set of nodes V={1,2,3,4,5,6}, set of edges E={(1,1),(1,3),(1,5),(2,1),(2,2),(2,4),(3,3),(3,6),(4,2),(4,4),(4,6),(5,3),(5,5),(6,1),(6,4),(6,6)} and the adjacency matrix A=[aij]6×6 where adjacency elements aij=1 when (i,j)E; otherwise, aij=0.

The nominal time-varying system considered here is given by {x(k+1)=[00.40.60.7sin(6k)]x(k)+[0.51]v(k)z(k)=[0.10.1]x(k) with the initial value x(0

Bo Shen received his B.S. degree in Mathematics from Northwestern Polytechnical University, Xi’an, China, in 2003. He is currently pursuing a 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,

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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 a 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 the program committees for many international conferences.

Y.S. Hung received his B.Sc. (Eng.) in Electrical Engineering and B.Sc. in Mathematics, both from the University of Hong Kong, and his M.Phil. and Ph.D. degrees from the University of Cambridge. He has worked at the University of Cambridge and the University of Surrey before he joined the University of Hong Kong, where he is now a professor in the Department of Electrical and Electronic Engineering. He has authored and co-authored over 150 publications in books, journals and conferences. His research interests include control systems, robotics, computer vision and bioinformatics.

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, 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 Masayuki Fujita under the direction of Editor Ian R. Petersen.

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