Event-triggered cooperative target tracking in wireless sensor networks

Lu Kelin,Zhou Rui,*,Li Ho

aSchool of Automation Science and Electrical Engineering,Beihang University,Beijing 100083,China

bAVIC Chengdu Aircraft Designamp;Research Institute,Chengdu 610091,China

Event-triggered cooperative target tracking in wireless sensor networks

Lu Kelina,Zhou Ruia,*,Li Haob

aSchool of Automation Science and Electrical Engineering,Beihang University,Beijing 100083,China

bAVIC Chengdu Aircraft Designamp;Research Institute,Chengdu 610091,China

Since the issues of low communication bandwidth supply and limited battery capacity are very crucial for wireless sensor networks,this paper focuses on the problem of eventtriggered cooperative target tracking based on set-membership information filtering.We study some fundamental properties of the set-membership information filter with multiple sensor measurements.First,a sufficient condition is derived for the set-membership information filter,under which the boundedness of the outer ellipsoidal approximation set of the estimation means is guaranteed.Second,the equivalence property between the parallel and sequential versions of the setmembership information filter is presented.Finally,the results are applied to a 1D eventtriggered target tracking scenario in which the negative information is exploited in the sense that the measurements that do not satisfy the triggering conditions are modelled as set-membership measurements.The tracking performance of the proposed method is validated with extensive Monte Carlo simulations.

1.Introduction

In recent years,cooperative target tracking in wireless sensor networks (WSNs) has a wide range of applications in the field of intelligence,surveillance and reconnaissance (ISR).1–3Theapplication of distributed WSNs provides a competent method for battlefield information collection and a robust scheme for moving target tracking4,5in a complex and interference-rich environment.However,the limited network resources in terms of energy and communication bandwidth set a constraint on the ability of WSNs.6–8Since the sensors are dispersed via airdrop or cannon fire in a lot of practical scenarios,9their batteries are difficult to be recharged or replaced.Thus,an energysaving tracking strategy is demanded to conserve network energy and extend network life.

For wireless ad hoc networks,the energy consumption depends heavily on the wireless communication.10Since WSNs face stringent energy limits,forcing all sensors to communicate with the fusion center (FC) at full rate is apparently not affordable.Therefore,an energy-saving strategy in our tracking architecture is to reduce the communication frequency while guarantee an acceptable tracking accuracy.11,12Eventtriggered estimation is a promising alternative to fulfill such imperative requirement.13In contrast to sending the measurements periodically,the local sensors employ various event sampling strategies to determine whether to send up-to-date measurements to theremote FC,such as Matched Sampling,14Integral Sampling,15Send-on-Delta(SOD)16and Variancebased Triggering.17As a result,the event-triggered strategy is to estimate the state based on the intermittent observations18that satisfy the triggering conditions,but this cannot be applied straightforwardly in target tracking applications because the improper design of triggering thresholds might result in the loss of track19,20as the FC receives no observation at a number of consecutive time steps.

In cases that the sensors determine not to send their current measurements,some''negative informationquot;21is implicitly available to the remote FC without additional communication.For example,when the SOD method is employed,no measurement transmission from one specific sensor implies that the value of the current measurement does not deviate too much from the value of the last transmitted measurement,22which suggests that although the exact value of the current measurement is unknown to the FC,it lies in a set formulated by the SOD triggering condition.In order to better exploit this''negative informationquot;,as well as to integrate it into the Bayesian state estimation framework,several set-membership state estimators23–25have been proposed,where the uncertainty of the negative information is modelled as a set of Gaussian densities.In Ref.26,a generalization of the standard Kalman filter is developed to solve the problem of set-membership measurement for the single sensor.The work in Ref.27reveals that the information form of set-membership filtering consists of advantageous properties especially when multiple set-membership measurements are received.Despite the great deal of effort that has been dedicated to it,several problems remain open,which are of significant importance in studying the setmembership filter and its application to the problem of eventtriggered cooperative target tracking in WSNs.The first issue is that the boundedness of the set of the estimation means has not been investigated systematically when the set-membership measurement is considered.27,28Another issue is to explore the equivalence between the parallel and recursive implementations of the set-membership information filter,and particularly since the Minkowski sum of ellipsoids29might not be an ellipsoid,some approximation of the exact estimation result is inevitable.

In this paper,we study the boundedness of the set of the estimation means with the information form,where a sufficient condition is proved,under which the outer ellipsoidal approximation set is asymptotically bounded.We also present the equivalence between the parallel and sequential setmembership information filters.Finally,the set-membership information filter is applied to the problem of event-triggered cooperative target tracking,and the performance of the proposed tracking strategy is further validated with extensive Monte Carlo simulations.

2.Problem formulation

For a random vector x∈Rn,we use E(x)and Cov(x)to denote its mean and covariance respectively.For a matrix A∈RnXn,we define tr(A)as its trace.Given S∈RnXngt;0,i.e.,S is positive definite,an ellipsoidal set X=ε(c,S)is represented as

For two ellipsoidal sets X and Y,let X⊕Y denotes their Minkowski sum,namelyand we have

We consider a linear time-invariant dynamic system that evolves in discrete time and is perturbed by Gaussian white noise as follows:

where vi~N(0,Ri)denotes the measurement noise,Rithe covariance of the measurement noise for theith sensor;N is the number of the sensors;Hiis the measurement model matrix.We also assume that(F,H)is detectable,30whereand R=diag(R1,R2,...,RN).In addition to the stochastic measurement noise vi,we further consider that the obtained measurement consists of an unknown but bounded error ei(k),namely

where the uncertainty of ei(k)is confined to an ellipsoidal set as

With respect to the fact that the uniqueness of the measurementcannot be maintained due to the uncertainty of ei(k),the set-membership31measurementwill be used as a replacement:

To cope with the set-membership uncertainty of Zi,one feasible manner is to model the uncertainty by a set of Gaussian densities,which gives rise to the set-membership Kalman filter32and set-membership information filter.27We know that the standard information filter embodies an algebraic reformulation of the Kalman filter,which provides an easier update phase for the distributed estimation architecture by estimating the information about the state rather than the state itself.33More exactly,the information state

where P is estimation covariance matrix and^x the estimation mean;and the information matrix

are the quantities to be calculated at the prediction and update steps.In the presence of additional set-membership uncertainties,an ellipsoidal setof estimation means has to be processed in its information form,which is obtained by an affine transformation34:and thus an ellipsoidal set including all possible information states is produced.

For conventional information filter,the measurement of the estimation accuracy of the state is fully represented as the information matrix Y,while for set-membership information filter,simultaneous consideration of stochastic and setmembership estimation uncertainty should be introduced,26where the confidence in stochastic uncertainty is still quantified as Y,and the confidence in set-membership uncertainty is quantified as the trace of SY.

The information matrix Y converges to one unique solution to a Riccati equation35as time iterates,provided that(F,Q)is stabilizable and(F,H)is detectable.However,the asymptotic behavior of the confidence in the set-membership uncertainty,i.e.,tr(SY),has not been investigated theoretically yet.Besides,since the Minkowski sum of ellipsoidal sets is difficult to calculate analytically,24the exact set of estimation means can only be approximated with an outer ellipsoidal set^Y.In this regard,the boundedness property of tr(S^Y)is of importance as well since the exact set is contained in the outer approximation set.

Based on the presented notations,we are now ready to introduce the problems to be solved in this work:

(1)Investigate the asymptotic boundedness of the trace of the set

(2)Explore the equivalence between the parallel and sequential set-membership information filters.

(3)Apply the set-membership information filter to the problem of event-triggered cooperative target tracking in WSNs.

3.Asymptotic boundedness of outer ellipsoidal set

In this section,we analyze the boundedness of the outer ellipsoidal approximation set^Y.Here we define that an ellipsoidal set X=ε(c,S)is bounded if tr(S)is bounded.The setmembership information filter is conceptualized as two distinct phases:''Predictionquot;and''Updatequot;.The prediction phase produces a set of priori estimates of the state with the following linear transformation:

where

In the update phase,the priori set with the information form is updated with the set-membership measurements by the operation of Minkowski sum27as

where

As for the calculation of information matrix,it follows that of the standard information filter:

where

From Eqs.(14)and(15),it can be inferred that the set of estimation means shares the same covariance,since the estimation error covariance is independent of Ziin the setmembership information filtering framework.

Due to the fact that the Minkowski sum of ellipsoids in Eq.(12)does not yield an ellipsoid,an outer ellipsoidal approximation is calculated according to the following result34:

with

for each set of pigt;0(i=1,2,...,N).Consequently Y(k|k)in Eq.(12)can be externally approximated as

Now we are in the position to introduce the result on the asymptotic boundedness property of the outer ellipsoidal approximation set^Y.

Theorem 1.Assume that(F,Q)is stabilizable and(F,H)is detectable.Letthe n the setis asymptotically bounded for all measurement setswith bounded sizes tr(Sei(k)).

Appendix A shows the proof of Theorem 1.In the following,we present an example to show the boundedness of the outer ellipsoidal set.

4.Equivalence between parallel and sequential set-membership information filters

In this section,the purpose is to examine the equivalence between two versions of set-membership information filter for a multi-sensor system.As shown in Figs.2 and 3,the major difference between the m lies in the manner in which the y process the set-membership measurements at the update step.The first filter which has been introduced in Eq.(12)processes all the set-membership measurements in parallel,while the second filter updates the set-membership measurements in a recursive manner,which offers the benefit that the matrix S^Y(k|k)in Eq.(A6)does not need to be computed at once,but can be computed sensor by sensor.

To aid the analysis,we first suppose that the parallel rule is as follows:

which indicates that the state estimate at time step k has been updated by the parallel measurements from sensors 1,2,...,n(1≤n≤N),namely

Fig.1 Center of set of estimation errors and bounds for the first and second states in set-membership information filter.

Fig.2 Parallel set-membership information filter.

Fig.3 Sequential set-membership information filter.

where ppand piare given in Eq.(A16).In the following,we present the result of the equivalence analysis.

Theorem 2.Let

where

According to Eqs.(16)and(17),we have

Then the following equations hold:

and

Appendix B shows the proof of Theorem 2.

5.Application to event-triggered cooperative target tracking in wireless sensor networks

In this section,we present how the results introduced above can be applied to the problem of event-triggered cooperative target tracking in WSNs.We consider a scenario with one target,ten sensors and a remote FC.A 1D constant velocity model for the target is given as

where x is the target position;dT=1 s is the step size of the discretization;and w is the Gaussian white noise with Qc=diag(1,1).We assume that the sensors are able to obtain the 1D position measurements of the target with a sampling interval of 1 s and the variances of the measurement noise are the same as R1=R2=...=RN=R=1 m2.Furthermore,at each time step,the sensor may or may not send its current measurement to the FC according to a decision variable di(k)that is determined by the SOD16triggering condition with the following form:

where Serepresents the triggering threshold.In this manner,while di(k)=1,the ith sensor forwards its current measurement zi(k)to the FC,otherwise the FC formulates a setmembership pseudo measurement23for the ith sensor as

As a result,the information transmission from the sensors to the FC becomes event-based,namely the communication is triggered only if the discrepancy between the current measurement and the one last sent exceeds a tolerable predefined threshold,and thus the communication cost reduces significantly.

In the following,we use the classical information filter with intermittent observations (IF-IO) as a benchmark to evaluate the performance of the proposed cooperative tracking strategy based on set-membership information filter (IF-SM).We first compare the communication cost between IF-IO and the proposed IF-SM in terms of the communication rate.After that,we compare the estimation accuracy in terms of the mean squared error (MSE).

Table 1 Comparison of communication rates between IF-SM and IF-IO.

Table 1 shows the comparison of communication rate between the two estimators,where CSM,iand CIO,irepresent the communication rates for IF-SM and IF-IO respectively.In the following,we examine the mean difference between the communication rates of IF-SM and IF-IO by Student's t-test.

Let Δi=CSM,i-CIO,ifor i=1,2,...,n where n=11 in this case.We further assume that Δi~N(a,σ)where a and σ are both unknown,and set up the hypothesis as follows:

Fig.4 Comparison of MSE in position between IF-SM and IFIO with Se=1,2,3 m2.

Fig.5 Comparison of MSE in position between IF-SM and IFIO with Se=4,5 m2.

In this regard,the standard deviation S for the sample Δi(i=1,2,...,n)is calculated as

In the following,we compare the estimation accuracy in terms of the MSE.For the set-membership estimator,it is hard to distinguish which point of the set corresponds to the minimum estimation error at each step,since the actual state is always unknown.As a result,the center of the set of the estimation means is empirically regarded as an alternative pointvalued estimation.23,25,32The results are obtained from a 1000-run Monte Carlo simulation.

Fig.4 plots the MSE in position for IF-SM and IF-IO respectively,in which the performance of IF-SM is characterized by the center of theset of estimation means.It is observed that when the triggering threshold is relatively low(Se=1,2 m2),the tracking accuracy,namely the MSEs,of IF-SM and IF-IO is close to each other,since both of the m operate in a high-rate communication environment.However,when the communication rate further decreases(Se≥3 m2),the impact of loss of track becomes more apparent,and in this case,all sensors might be simultaneously triggered off at a number of consecutive time steps.When such a condition happens,the MSE of IF-IO becomes higher than that of IF-SM for the reason that IF-IO is not available to correct the prior state estimation without new measurement.As a result,the discrepancy of MSE between IF-SM and IF-IO becomes larger with the increment of the triggering threshold Se.For example,when Se=3 m2,the MSE of IF-IO is only slightly larger than that of IF-SM(Fig.4(c)).When Se=4,5 m2,the discrepancy of MSE between IF-IO and IF-SM keeps increasing(Fig.5).The results demonstrate that the proposed event-triggered tracking strategy with IF-SM is robust against the change of triggering threshold.

6.Conclusions

This paper considers the problem of event-triggered cooperative target tracking in wireless sensor networks based on setmembership information filtering,where each sensor decides whether to send its current measurement or not according to the Send-on-Delta triggering mechanism.The major contribution of this paper lies in the study of some fundamental properties of the set-membership information filter,where we prove a sufficient condition for the asymptotic boundedness of the outer ellipsoidal approximation set of the estimation means.We also illustrate the equivalence between the parallel and sequential implementations of theset-membership information filter.Finally,we apply the set-membership information filter to a cooperative target tracking scenario.The simulation has proved the performance of the proposed method in the sense that it is robust against the change of triggering threshold.

Acknowledgement

This study was partially supported by the National Natural Science Foundation of China(No.61273349).

Appendix A

Proof of Theorem 1.Let

Substitute Eqs.(13)and(A1)into Eq.(12),and we have

We assume that

where

Then according to Eqs.(16)–(18),we have

where Eq.(A6)holds for ppgt;0 and pigt;0.

Since

wefurther assume that

and thus

Substitute Eq.(A6)into Eq.(A10),and the n we have

As a result,the objective is to investigate the boundedness of tr(SY(k+1|k)).If(F,Q)is stabilizable and(F,H)is detectable,the n P(k|k-1)converges towards the steady covarianceas the solution to a Riccati equation36is as follows:

For steady state,we have

Therefore Eq.(A11)can be rewritten as

Take trace on both sides of Eq.(A14),

and let

and the n Eq.(A15)can be rewritten as

Hence,

which indicates that the set^Y is asymptotically bounded if all measurement sets Ziare with bounded sizes tr(Sei(k)).At the same time,the size of the set Y is also bounded since^Y provides an outer approximation of Y.□

Appendix B

Proof of Theorem 2.For n=1,we have

Then according to Eq.(25),we can derive that

Therefore Eq.(27)holds.

According to Eq.(A16),the parameters in Eq.(26)should be given as

We have

Substitute Eq.(B5)into Eq.(B3),and the n we have

We further substitute Eqs.(B5)and(B6)into Eq.(26),and the n we have

Therefore Eq.(28)holds and the proof is completed.□

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Lu Kelinis a Ph.D.student at School of Automation Science and Electrical Engineering,Beihang University.He received his B.S.degree from Nanjing University of Aeronautics and Astronautics in 2010.His area of research includes target tracking and sensor fusion.

Zhou Ruireceived the Ph.D.degree from Harbin Institute of Technology,Harbin,China.He is currently a professor and Ph.D.supervisor in guidance,navigation and control in Beihang University,Beijing,China.His area of expertise includes autonomous flight control and guidance,constrained trajectory optimization and cooperative control of multiple aerial vehicles.

Li Haoreceived the Ph.D.degree from Northwestern Polytechnical University,Xi'an,China.He is currently with AVIC Chengdu Aircraft Designamp;Research Institute,Chengdu,China.His area of expertise includes cooperative control of multiple aerial vehicles.

4 December 2015;revised 4 May 2016;accepted 13 May 2016

Available online 26 August 2016

Event-triggered;

Information filter;

Set-membership;

Target tracking;

Wireless sensor networks

©2016 Production and hosting by Elsevier Ltd.on behalf of Chinese Society of Aeronautics and Astronautics.This is an open access article under the CC BY-NC-ND license(http://creativecommons.org/licenses/by-nc-nd/4.0/).

*Corresponding author.Tel.:+86 10 82339232.

E-mail addresses:klu@buaa.edu.cn(K.Lu),zhr@buaa.edu.cn(R.Zhou),lynn.lihao@163.com(H.Li).

Peer review under responsibility of Editorial Committee of CJA.