IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS—I: REGULAR PAPERS, VOL. 58, NO. 9, SEPTEMBER 2011 2237

A Stochastic Sampled-Data Approach to Distributed� Filtering in Sensor Networks

Bo Shen, Zidong Wang, and Xiaohui Liu

Abstract—In this paper, the problem of distributed filteringin sensor networks using a stochastic sampled-data approach isinvestigated. A set of general nonlinear equations described bysector-bounded nonlinearities is utilized to model the systemand sensors in networks. Each sensor receives the informationfrom both the system and its neighbors. The signal received byeach sensor is sampled by a sampler separately with stochasticsampling periods before it is employed by the corresponding filter.By converting the sampling periods into bounded time-delays, thedesign problem of the stochastic sampled-data based distributed

filters amounts to solving the filtering problem fora class of stochastic nonlinear systems with multiple boundedtime-delays. Then, by constructing a new Lyapunov functionaland employing both the Gronwall’s inequality and the Jensonintegral inequality, a sufficient condition is derived to guaranteethe performance as well as the exponential mean-square sta-bility of the resulting filtering error dynamics. Subsequently, thedesired sampled-data based distributed filters are designedin terms of the solution to certain matrix inequalities that can besolved effectively by using available software. Finally, a numericalsimulation example is exploited to demonstrate the effectivenessof the proposed sampled-data distributed filtering scheme.

Index Terms—Distributed filtering, filtering, Jensonintegral inequality, sampled-data, sensor networks, stochasticsampling.

I. INTRODUCTION

S ENSOR networks are composed of small nodes withsensing, computation, and wireless communications capa-

bilities. These nodes, also called as sensor nodes, are usuallydistributed spatially and coordinated to perform some specificapplication-oriented global tasks. Research on sensor net-works was initially motivated by military applications such asbattle-filed surveillance and enemy tracking. Nowadays, due tothe rapid developments of technologies in computing and com-

Manuscript received August 28, 2010; revised November 21, 2010; acceptedJanuary 17, 2011. Date of publication March 10, 2011; date of current versionSeptember 14, 2011. This work was supported in part by the Engineeringand Physical Sciences Research Council (EPSRC) of the U.K. under GrantGR/S27658/01, the National Natural Science Foundation of China under Grants61028008 and 60974030, the National 973 Program of China under Grant2009CB320600, and the Alexander von Humboldt Foundation of Germany.This paper was recommended by Associate Editor X. Yu.

B. Shen is with the School of Information Science and Technology, DonghuaUniversity, Shanghai 200051, China (e-mail: shenbodh@gmail.com).

Z. Wang is with the Department of Information Systems and Computing,Brunel University, Uxbridge, UB8 3PH, U.K. He is also with the School ofInformation Science and Technology, Donghua University, Shanghai 200051,China (e-mail: Zidong.Wang@brunel.ac.uk).

X. Liu is with the Department of Information Systems and Computing, BrunelUniversity, Uxbridge, UB8 3PH, U.K. (e-mail: Xiaohui.Liu@brunel.ac.uk).

Color versions of one or more of the figures in this paper are available onlineat http://ieeexplore.ieee.org.

Digital Object Identifier 10.1109/TCSI.2011.2112594

munication, the sensor networks have come to pervade everyaspect of our lives such as infrastructure security, environmentand habitat monitoring, industrial sensing, and traffic control[1], [3], [4]. Recently, distributed filtering problem has becomea research focus for wireless networks. In distributed filteringproblem, each sensor can receive not only the measurementsfrom system but also the information from its neighboringsensors according to the topology of the given sensor network.Hence, how to deal with the complicated coupling between onesensor and its neighbors becomes the main issue in designingdistributed filters. It should be pointed out that some initialefforts have been made on this topic, see, e.g., [10] and thereferences therein.

In real-world sensor networks, the signal received by eachsensor has to be sampled before it is transferred for processingby the filter since filters are, by nature, digital devices. The tra-ditional approach is to use periodic sampling technique to ob-tain a discrete-time system for modeling the real plant. How-ever, such a discrete-time model might not capture the inter-sample behavior of the real system, especially for the case whenthe sampling period is time-varying. On this account, consid-erable research efforts have been made on various aspects ofsampled-data systems, see [2] for control problems and [11],[12] for filtering problems. It is worth mentioning that, in [5], anew approach to dealing with sampled-data control problemshas been proposed by converting the sampling period into atime-varying but bounded delay, and then the sampled-datacontrol problem has been investigated by recurring to thecontrol theory for the time-delay systems. Based on this method,the sampled-data control problem has been thoroughly in-vestigated in [7] where the stochastic sampling has been takeninto account.

It is noted that the technology of sampled-data systems hasrecently been applied in a variety of research areas, see, e.g.,[6], [9]. Undoubtedly, in nowadays digitalized world, it is ofboth theoretical significance and practical importance to use thestochastic sampled-data approach for analyzing and designingdistributed filters in sensor networks. Nevertheless, this appearsto be a challenging task and three fundamental difficulties areidentified as follows. 1) For a sensor network, each filter is de-signed based on the information from a sensor that communi-cates with its neighbors according to network topology. So, thefirst challenge is how to deal with the coupling among the sensornodes in a proper way. 2) Each sensor is equipped with a sam-pler with separate sampling period. The asynchronous samplinggives rise to significant difficulties in guaranteeing the filteringperformance over the whole sensor network, which leads to thesecond challenge. 3) It is usually the case that the signals trans-mitted within a sensor network are randomly sampled. Hence,

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2238 IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS—I: REGULAR PAPERS, VOL. 58, NO. 9, SEPTEMBER 2011

the third challenge would be how to deal with the stochasticsampling by quantifying its impact on the global filtering per-formance in terms of the sampling occurrence probability.

To handle the three identified challenges, in this paper, weaim to solve the problem of distributed filtering in sensornetworks based on the stochastically sampled data. The consid-ered system and each of sensors in networks are described by aset of sector-bounded nonlinear equations. The signal receivedby each sensor includes measurements from the system and itsneighboring nodes, and is sampled separately by a sampler be-fore it is transmitted to the corresponding filter. The samplingperiod of each sampler is time-varying that is allowed to switchbetween two different values in a random way. By using themethod proposed in [7], the addressed sampled-data fil-tering problem is transformed into the filtering problem fora stochastic nonlinear system with multiple bounded time-de-lays. Subsequently, a new Lyapunov functional is constructedand, by utilizing both the Gronwall’s inequality and the Jensonintegral inequality, an easily implementable distributed filter de-sign scheme is proposed to achieve the desired performance re-quirements.

Notation

Most notation used in this paper is fairly stan-dard. stands for the block-diagonal matrix

and represents theblock-diagonal matrix with blocks, where the th blockis and all others are zero matrices. denotes

. is the space of squareintegrable vector-valued functions in intervalwith the norm . In symmetric blockmatrices, the asterisk is used to denote term that is inducedby symmetry.

II. PROBLEM FORMULATION AND PRELIMINARIES

Consider the following nonlinear continuous-time system:

(1)

together with sensors described as follows:

(2)

where is the state vector, is the signal tobe estimated, is the measurement output receivedby sensor from system (1), and is the exoge-nous disturbance input belonging . , and

( ) are known constant matrices with appropriatedimensions. is the initial value.

Assumption 1: The nonlinear functions and( ) are assumed to satisfy ,

and

(3)

(4)

Fig. 1. Local structure of node �.

where , are known constantmatrices.

The sensor nodes under consideration in this paper are dis-tributed in space according to a fixed network topology repre-sented by a directed graph of order with the setof nodes (sensors) , set of edges ,and an adjacency matrix with nonnegative ad-jacency elements . An edge of is denoted by . Theadjacency elements associated with the edges of the graph arepositive, i.e., . Moreover, for all

. The set of neighbors of node plus the node itselfis denoted by .

The local structure of information communication sur-rounding the node is shown in Fig. 1, where the informationis first collected by sensor from its neighboring nodes , ,

, then sampled by sampler before it enters the filter . In thepresent paper, the information on each node , which is avail-able for its neighboring node (i.e., , has the followingform:

(5)

where is the estimate for on the node . There-fore, the information received by sensor can be described as

(6)

For every ( ), the sampled signal is generatedby a zero-order hold function with a sequence of hold times

(7)

where is a discrete-time signal which is the actual input offilter , denotes the sampling instant of node and satisfies

.On each interval ( ), the following

filter structure is adopted:

(8)

where is the estimate for on the node , andis the parameter of filter to be determined. The

initial values of filters are for all .Remark 1: It follows from the definition of the set of that,

in the model (6), the information from both the node itself andits neighboring nodes is employed. In most engineering prac-tice, the information from neighboring nodes is deemed helpful

SHEN et al.: A STOCHASTIC SAMPLED-DATA APPROACH TO DISTRIBUTED FILTERING IN SENSOR NETWORKS 2239

to improve the filtering performance of the whole sensor net-work.

Defining for , filter (8) can bewritten as

(9)

Letting estimate error and filtering error beand , respectively, we obtain the fol-

lowing system that governs the filtering error dynamics for thenode :

(10)

where and.

Define a function in the whole period of time asfollows: , , .Then, the filtering error system can be rewritten as

(11)

Setting

(12)

we arrive at the following augmented filtering error system forthe whole sensor network

(13)

In this paper, the sampling periods for the input signal of eachfilter are taken as two values and , and they switch be-tween these two values in a random way. Here, without lossof generality, we assume that . Such a phe-nomena is refereed to as the stochastic sampling [7], whichcan be represented by utilizing a set of random variables( ) with the probabilities and

where is a known constant.

Following the similar line in [7], the filtering error system(13) can be rewritten as

(14)where and aredefined by

,,

,,

respectively, and is a random variable obeyingand

with .In the sequel, for the purpose of simplicity, we denote

(15)

To this end, the filtering error system (14) can be furtherrewritten as the following compact form:

(16)

Definition 1: The system (16) with is said to beexponentially mean-square stable if there exist two constants

and such that

2240 IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS—I: REGULAR PAPERS, VOL. 58, NO. 9, SEPTEMBER 2011

where is the initial function of system (16) defined as, .

In this paper, we are interested in looking for the filter param-eters ( ) such that the following two require-ments are simultaneously satisfied:

a) The filtering error system (16) with is exponen-tially mean-square stable.

b) Under the zero-initial condition, the filtering errorsatisfies

(17)

for all nonzero , where is a given disturbanceattenuation level.

III. MAIN RESULTS

Theorem 1: Let the filter parameters ( ) begiven. The filtering error system (16) with is exponen-tially mean-square stable if there exist matrices , ,

, , ( ) and scalars ,, satisfying (18) at the bottom of the page, where

(19)

Proof: Consider the system (16) in the disturbance-freecase, i.e., . Denote

(20)

Construct the following Lyapunov functional:

(21)

where

(22)

with and .

Defining the infinitesimal operator of as follows:

(23)

(18)

SHEN et al.: A STOCHASTIC SAMPLED-DATA APPROACH TO DISTRIBUTED FILTERING IN SENSOR NETWORKS 2241

we obtain

(24)

It follows from the Jenson integral inequality [8] that

(25)

and

(26)

Substituting (25) and (26) into (24) yields

(27)where , , , and are given at the bottom of the page.

Noting (3) and (4), we know that and satisfy

(28)

and

(29)

respectively. The latter inequality further implies

(30)

Now, it follows readily from (27), (28), and (30) that

(31)

where is given in (32) at the bottom of the next page.By using the Schur complement formula, it follows immedi-

ately from (18) that , and conse-quently, we have

(33)

where .

2242 IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS—I: REGULAR PAPERS, VOL. 58, NO. 9, SEPTEMBER 2011

From the definition of Lyapunov functional (21), one has

(34)

The nonlinear function can be rewritten as

where satisfies

. Denoting

, we have . Similarly, we can

obtain scalars ( , 2) such that .Subsequently, one has

(35)

where

. Then, it follows from (34) and (35) that

(36)

where and

By noting

together with (33) and (36), we have

By resorting to Gronwall’s inequality, it immediately followsthat

(37)which, from Definition 1, means that the filtering error system(16) with is exponentially mean-square stable. Theproof is now complete.

Next, let us analyze the performance for the filteringerror system (16).

Theorem 2: Let the filter parameters ( )and the disturbance attenuation level be given. The fil-tering error system (16) with is exponentially stablein the mean square with filtering error satisfying (17) under thezero initial condition if there exist matrices , ,

, , ( ) and scalars ,, satisfying (38) at the bottom of the next page,

where

(39)

and , , , , , , , , and are definedin Theorem 1.

Proof: First, it can be easily seen that (38) implies (18) andhence the exponential mean-square stability of the filtering errorsystem (16) with is guaranteed. It remains to furtherconsider the performance for the filtering error system (16)under the zero initial condition.

Set

(32)

SHEN et al.: A STOCHASTIC SAMPLED-DATA APPROACH TO DISTRIBUTED FILTERING IN SENSOR NETWORKS 2243

and construct the following Lyapunov functional:

(40)

where is defined in (22) and is defined as

Similar to the proof of Theorem 1, we have

where , , , and are given at the bottom of the page.

To analyze the performance, we introduce the followingperformance index

By considering the zero initial condition, it can be obtainedfrom (28) and (30) that

(41)

where is given the first equation at the bottom of the nextpage.

From (41), it is easily shown from the Schur complement for-mula that (38) implies . Letting , the condition(17) is immediately guaranteed and this completes the proof ofTheorem 2.

(38)

2244 IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS—I: REGULAR PAPERS, VOL. 58, NO. 9, SEPTEMBER 2011

Theorem 3: Let the disturbance attenuation level begiven. The stochastic sampled-data based distributed fil-tering problem is solvable by the filters (8) if there exist matrices

, , , , ,( ) and scalars , , satisfying thelinear matrix inequality (LMI) (42) at the bottom of the page,where

(43)

and , , , , , , , , , , and aredefined in Theorem 1. Furthermore, if the LMI (42) is solvable,the desired filter parameters are given as

(44)

Proof: By noting the relations andand using the inequality , the LMI

(38) is guaranteed by the LMI (42), and the rest of the prooffollows directly from Theorem 2.

Remark 2: In Theorems 1–3, a stochastic sampled-dataapproach has been exploited to analyze and design the dis-tributed filters in sensor networks. We outline our contributionas follows. 1) We have included the topology information ofthe sensor network in the adopted filter structure, and there-fore made use of the space information (i.e., coupling amongthe sensor nodes) in the filter design which is fundamentallydifferent from the traditional central Kalman filter design. 2)To handle the asynchronous sampling issue, we have allowedthe sampling periods of each sampler to randomly switchbetween two different values and then the addressed sam-pled-data filtering problem has been transformed into the

filtering problem for a stochastic nonlinear system withmultiple bounded time-delays. 3) In terms of the samplingoccurrence probability, we have actually quantified the impactof the stochastic sampling on the global filtering performanceand reduced the overall conservatism in filter design.

(42)

SHEN et al.: A STOCHASTIC SAMPLED-DATA APPROACH TO DISTRIBUTED FILTERING IN SENSOR NETWORKS 2245

Fig. 2. Measurements.

Fig. 3. Sampled measurements.

IV. AN ILLUSTRATIVE EXAMPLE

Consider a nonlinear continuous system as follows:

(45)

where is taken as

It is easy to see that satisfies (3) with

The topology of the three sensors is represented by a graphwith the set of nodes , set of edges

and the adja-cency matrix where adjacency elementswhen ; otherwise, . The sensor models aregiven by ,

Fig. 4. Output ���� and its estimates.

Fig. 5. Filtering errors �� ��� (� � �,2,3).

, where ( , 2,3) are chosen as ,

and . Also, ( ,2, 3) meet (4) with matrices

In this example, we set , , ,and . Solve LMI (42) to obtain all parameters of

the desired distributed filters as follows:

The exogenous disturbance input is selected as

,elsewhere.

(46)

Simulation results are presented in Figs. 2–5. The measurementsreceived by sensor are shown in Fig. 2. Fig. 3 plots the sampledmeasurements which are actually employed by the designed fil-ters. The output and its estimates from the filters are de-picted in Fig. 4. The filtering errors are given in Fig. 5.

2246 IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS—I: REGULAR PAPERS, VOL. 58, NO. 9, SEPTEMBER 2011

REFERENCES

[1] C. Alippi and C. Galperti, “An adaptive system for optimal solar en-ergy harvesting in wireless sensor network nodes,” IEEE Trans. Cir-cuits Syst. I, Reg. Papers, vol. 55, no. 6, pp. 1742–1750, Jul. 2008.

[2] T. Chen and B. A. Francis, “Linear time-varying� -optimal control ofsampled-data systems,” Automatica, vol. 27, no. 6, pp. 963–974, Nov.1991.

[3] C. Y. Chong and S. P. Kumar, “Sensor networks: Evolution, opportuni-ties, and challenges,” Proc. IEEE, vol. 91, pp. 1247–1256, Aug. 2003.

[4] G. Cimatti, R. Rovatti, and G. Setti, “Chaos-based spreading inDS-UWB sensor networks increases available bit rate,” IEEE Trans.Circuits Syst. I, Reg. Papers, vol. 54, no. 6, pp. 1327–1339, Jun. 2007.

[5] E. Fridman, A. Seuret, and J. P. Richard, “Robust sampled-data stabi-lization of linear systems: An input delay approach,” Automatica, vol.40, no. 8, pp. 1441–1446, Aug. 2004.

[6] H. Gao, W. Sun, and P. Shi, “Robust sampled-data � control forvehicle active suspension systems,” IEEE Trans. Control Syst. Technol.,vol. 18, no. 1, pp. 238–245, Jan. 2010.

[7] H. Gao, J. Wu, and P. Shi, “Robust sampled-data � control withstochastic sampling,” Automatica, vol. 45, no. 7, pp. 1729–1736, Jul.2009.

[8] K. Gu, “An integral inequality in the stability problem of time-delaysystems,” in Proc. 39th IEEE Conf. Decision Control, Sydney, Aus-tralia, Dec. 2000, pp. 2805–2810.

[9] H. Katayama, “Nonlinear sampled-data stabilization of dynamicallypositioned ships,” IEEE Trans. Control Syst. Technol., vol. 18, no. 2,pp. 463–468, Mar. 2010.

[10] B. Shen, Z. Wang, and Y. S. Hung, “Distributed � -consensus fil-tering in sensor networks with multiple missing measurements: Thefinite-horizon case,” Automatica, vol. 46, no. 10, pp. 1682–1688, Oct.2010.

[11] P. Shi, “Filtering on sampled-data systems with parametric uncer-tainty,” IEEE Trans. Autom. Control, vol. 43, no. 7, pp. 1022–1027,Jul. 1998.

[12] Z. Wang, B. Huang, and P. Huo, “Sampled-data filtering with errorcovariance assignment,” IEEE Trans. Signal Process., vol. 49, pp.666–670, Mar. 2001.

Bo Shen received the B.S. degree in mathematicsfrom Northwestern Polytechnical University, Xi’an,China, in 2003. He is currently working toward thePh.D. degree in the School of Information Scienceand Technology, Donghua University, Shanghai,China.

From August 2009 to February 2010, he was aResearch Assistant in the Department of Electricaland Electronic Engineering, the University of HongKong, Hong Kong. He is now a Visiting Ph.D.Student in the Department of Information Systems

and Computing, Brunel University, U.K. His research interests include non-linear control and filtering, stochastic control and filtering, as well as complexnetworks and genetic regulatory networks. He is a very active reviewer formany international journals.

Zidong Wang (SM’03) was born in Jiangsu, China,in 1966. He received the B.Sc. degree in mathematicsin 1986 from Suzhou University, Suzhou, China,and the M.Sc. degree in applied mathematics and thePh.D. degree in electrical and computer engineeringfrom Nanjing University of Science and Technology,Nanjing, China in 1990 and 1994, respectively.

He is currently a Professor of Dynamical Systemsand Computing at Brunel University, U.K. He haspublished more than 120 papers in refereed interna-tional journals. His research interests include dynam-

ical systems, signal processing, bioinformatics, control theory, and applications.Dr. Wang is currently serving as an Associate Editor for 12 international

journals including the IEEE TRANSACTIONS ON AUTOMATIC CONTROL, theIEEE TRANSACTIONS ON NEURAL NETWORKS, the IEEE TRANSACTIONS

ON SIGNAL PROCESSING, the IEEE TRANSACTIONS ON SYSTEMS, MAN,AND CYBERNETICS—PART C: APPLICATIONS AND REVIEWS, and the IEEETRANSACTIONS ON CONTROL SYSTEMS TECHNOLOGY.

Xiaohui Liu received the B.Eng. degree in com-puting from Hohai University, Nanjing, China, in1982 and the Ph.D. degree in computer science fromHeriot-Watt University, Edinburg, U.K., in 1988.

He is currently a Professor of Computing atBrunel University, U.K. He leads the Intelligent DataAnalysis (IDA) Group, performing interdisciplinaryresearch involving artificial intelligence, dynamicsystems, image and signal processing, and statistics,particularly for applications in biology, engineering,and medicine. He serves on editorial boards of four

computing journals, founded the biennial international conference series onIDA in 1995, and has given numerous invited talks in bioinformatics, datamining, and statistics conferences.