globally optimized power allocation in multiple sensor fusion for linear and nonlinear networks

IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 60, NO. 2, FEBRUARY 2012 903

Globally Optimized Power Allocation in MultipleSensor Fusion for Linear and Nonlinear Networks

Umar Rashid, Student Member, IEEE, Hoang Duong Tuan, Member, IEEE, Pierre Apkarian, Member, IEEE, andHa Hoang Kha, Member, IEEE

Abstract—The present paper is concerned with a sensor net-work, where each sensor is modeled by either a linear or nonlinearsensing system. These sensors team up in observing either staticor dynamic random targets and transmit their observationsthrough noisy communication channels to a fusion center (FC) forlocating/tracking the targets. Physically, the network is limited byenergy resource. According to the available sum power budget, wedevelop a novel technique for power allocation to the sensor nodesthat enables the FC produce the best linear estimate in terms ofthe mean square error (MSE). Regardless of whether the sensormeasurements are linear or nonlinear, the targets are scalar orvectors, static or dynamic, the corresponding optimization prob-lems are shown to be semidefinite programs (SDPs) of tractableoptimization and thus are globally and efficiently solved by anyexisting SDP solver. In other words, new tractably computationalalgorithms of distributed Bayes filtering are derived with fullmultisensor diversity achieved. Intensive simulation shows thatthese algorithms clearly outperform previously known algorithms.

Index Terms—Bayes filtering, data fusion, linear and non-linear sensor network, linear fractional transformation (LFT),power allocation, semidefinite programming (SDP), unscentedtransformations.

I. INTRODUCTION

S ENSOR networks (SNs) hold the key to a wide range offuture applications and have the potential to play a signif-

icant role in the realm of modern technology. Deployment ofthese networks ensures controlling instrumentation in industrialautomation, sensing data remotely in a data collecting environ-ment and providing surveillance in defense related applications[7]. These networks were initially used in military applicationsbut their potential has recently been exposed in other areas ofscience and engineering such as process monitoring in indus-trial plants, navigational and guidance systems, radar tracking,sonar ranging [24], [1], [22], [17].

An SN is said to be linear (LSN) when each of its sensors ismodeled by a linear input-output system. On the other hand, themost popular sensors are range and/or bearing sensing (see e.g.,

Manuscript received February 17, 2011; revised June 22, 2011, September 21,2011, and September 28, 2011; accepted October 03, 2011. Date of publicationOctober 31, 2011; date of current version January 13, 2012. The associate editorcoordinating the review of this manuscript and approving it for publication wasDr. Hing Cheung So.

U. Rashid, H. D. Tuan, and H. H. Kha are with the Faculty of Engineeringand Information Technology, University of technology, Sydney, NSW 2007,Australia (e-mail: [email protected]; [email protected];[email protected]).

P. Apkarian is with the ONERA-CERT, Toulouse, France (e-mail: [email protected]).

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

Digital Object Identifier 10.1109/TSP.2011.2174230

[4], [5], [12]), which are nonlinear input-output systems and ac-cordingly an SN of such sensors is called nonlinear (NSN). Typ-ically, sensors are geographically distributed and operating in anamplify-and-forward (AF) mode [10], [11]. Through orthogonalnoisy wireless communication channels, they send their ownlocal measurements of a target to a central system, called thefusion center (FC). The FC fuses these local measurements toproduce a global estimate of the target. Obviously, the sensorsconsume power during transmission of their observations to theFC, which must be economical because of low battery power ofthe sensors. Power efficiency is highly crucial for the networklifetime. An optimized power allocation to minimize estimatedistortion of scalar parameters has been considered in [8] and[29] for LSNs. Instead of the optimal linear minimum squareerror estimator (LMMSE), the FC filtering in [8], [29] is thebest linear unbiased estimator (BLUE) for accommodation oftractable convex optimization. Again for LSNs with FC filteringby LMMSE estimator, convex optimization for power allocationhas been given in [2] but it is still not attractive enough for com-putational implementation. An NSN has been touched in [9] butunfortunately its nonlinear sensing modeling does not seem tocorrespond to any practically used scenario. Its assumption thatsensor output statistics are known at the FC does not look quiteconventional as well. It is fair enough to say that decentralizedestimation by NSNs for a static target has practically not beenconsidered and tracking a dynamic object by NSNs has beencompletely open for research.

The present paper develops an efficient strategy for allocatingpower to the sensor nodes in a globally optimized manner. Thenovelty of our approach is accentuated by a combination of com-putationally tractable semidefinite programming [6], [23] (forglobally optimal power allocation) with unscented transforma-tion [16] and linear fractional transformation (LFT) [13], [14],[26], [30], [31](for nonlinear statistic function approximation).The major contributions of the paper are as follows.

• Unlike all previous works [2], [8], [9], [29] mainly of LSNsfor locating a static target, the globally optimal decentral-ized Bayes filtering for both LSNs and NSNs are showncomputationally tractable in our approach. Further exten-sive computation show its full diversity as well.

• Unlike previous works [25], [27], [28] which mainly focuson LSNs for tracking a dynamic target, and which, at theend, do not admit computationally tractable and optimalsolutions of decentralized estimation, the globally optimalpower allocation at each time instant is solved by com-putationally tractable SDP in both LSNs and NSNs. Inother words, the globally optimal distributed Bayes fil-tering is solved by a sequence of tractably computationalSDPs in our approach with multisensor diversity showncomputationally.

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904 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 60, NO. 2, FEBRUARY 2012

• Active sensor selection for the best performance of FCfiltering is another important issue [32]. In general, theproblem of optimally selecting a fixed number of sensornodes among a set is a very difficult combinatoric problemand thus is not tractable at all. However, our globally op-timal power allocation for sensor nodes actually leads to aquite efficient and practical sensor node selection. In fact,our simulation shows that the power allocation is ratherconcentrated at a few active sensor nodes, which meansthat only these sensor nodes contribute to the FC filteringperformance. Other sensor nodes with zero or almost zeropower allocation obviously have no impact in the FC fil-tering performance and thus should be put to sleep to pro-long the network lifetime. In short, the optimal selection oflinear or nonlinear sensor nodes in an SN can be effectivelysolved via our globally optimal power allocation.

The remainder of the paper is organized as follows.Sections II and III address distributed Bayes filtering forstatic and dynamic objects, respectively. Section IV presentssimulation results to validate superior performance of theproposed power allocation strategy. This is followed by theconcluding remarks in Section V.

Most of the notations used in the paper are fairly standard.Bold symbols are used to represent vectors and matrices. By

it means is a positive definite matrix.or is a diagonal matrix with ordered diag-onal entries , which may be scalars or matrices.

for a vector with nonnegative components is component-wise understood. Trace of a square matrix is expressed by

. is the expectation operator. For a random vari-able(RV) , the notation is referred to its expectation ,while is its autocovariance and

is its cross-covariance withanother RV . Accordingly, for a deterministic map , momentsof the RV whenever is a RV are and

.means is Gaussian RV with the moments and .

II. GLOBAL OPTIMIZED DECENTRALIZED BAYES FILTERING

In SN context, localization of a static target is based on theknowledge of its statistics along with the sensor noisy obser-vations. The goal is to produce an estimator that has MMSEunder constraints on transmission power at the sensors. Targetlocalization is immensely important to the modern researcharena such as target localization for active sonar systems [19],video sensor nodes selection for localization in wireless camerasensor networks [20] and source localization based on rangeand bearing information [12].

Consider a target in -dimensional space(i.e.,rough initial information on expressed by and is given),which is observed by spatially distributed sensors. The sen-sors send their noise corrupted observations to a FC over wire-less flat-fading time-orthogonal communication channels [10],[11]. Thus, all these interactions can be compactly modeled bythe following behavioral equations:

(1)

(2)

where with each compo-nent a(linear or nonlinear) deterministic function for ex-pression of -sensor measuring quantity such as range and/orbearing. Accordingly, is the sensor ob-servations and with diagonal is a corruptnoise, which is uncorrelated with the source . These observa-tions are relayed to the FC and so is called therelay matrix defined by

(3)

which includes the channel gains between th sensor nodeand FC and amplifier coefficients to control the transmitpower of th sensor node. with diagonal isthe communication noise. It follows that the power consumedby th sensor node is

Hence, for , the sum power consumedby the entire SN is is normally con-strained by a fixed budget

(4)

By [18, Th. 12.1], the LMMSE estimate for based on FCoutput is

(5)

while the covariance of LMMSE estimator of given is

(6)

This covariance is also the covariance of the estimator error

so

Using the Inverse Matrix Lemma [15]

Therefore

(7)

RASHID et al.: GLOBALLY OPTIMIZED POWER ALLOCATION IN MULTIPLE SENSOR FUSION 905

We are now in a position to formulate the problem of minimiza-tion of MSE, subject to the power budget constraint (4) as

subject to (8)

which by (7) is equivalent to (9), shown at the bottom of thepage. Note that by Schur’s complement [15]

while (because is diag-onal) with . This leads to the followingSDP formulation for (9):

(10)

(11)

Alternatively, the problem of minimization of the total powerconsumption under MSE threshold is also formulated by thefollowing SDP

(12)

(13)

Both SDPs (10)–(11) and (12)–(13) are computationallytractable and can be globally solved by any existing SDPsolver such as YALMIP [21], provided that the sensor outputcovariance and its cross-covariance with the sourcecan be calculated. The remainder of this section is devoted tothe computational issue for these covariance matrices to makeSDPs (10)–(11) and (12)–(13) completely realizable.

A. Decentralized Bayes Filtering for LSN

In LSNs, model (1)–(2) is completely linear, i.e., the input-output system (1) of the sensor measurements is represented by

(14)

where is a matrix representing effects of path loss,fading and shadowing, which is known to the FC. Therefore,their analytical forms are available

(15)

Theorem 1: The optimal decentralized Bayes for locating thetarget by a LSN modeled by (14), (2) under the power con-straint (4) is (5) where and are defined by (15) whileis found from the SDP

(16)

B. Decentralized Bayes Filtering for NSN

Due to nonlinearity of sensing map in (1), analytical ex-pressions of and for computational implementation of(9) and (12) are not expected. However, we now present theirefficient and attractive approximations. The first one is the un-scented transformation-based approximation [16], which worksreasonably well for moderately nonlinear maps whereas, thesecond one is the linear fractional transformation (LFT)-basedapproximation [26], which works reasonably well for higherorder nonlinear or fractional maps . Without loss of generality,assume that admits form

(17)

where is moderately nonlinear in (e.g., a range function)while is highly nonlinear (e.g., a bearing function) andadmits a tractable LFT [13], [14], [26], [30], [31]

(18)

with deterministic matrices and simple non-linear feedback capturing all nonlinearity ofin (1) but nevertheless a nonlinear map in .

As shown in [26], for either higher order nonlinearity or formsinvolving fractional terms like the above , the above equiv-alent LFT model (18), consisting of a linear model and a simplenonlinear structure in the virtual feedback, performs better thanthe conventional global unscented transformations. In the LFTsetting, better approximations for moments are obtained by ap-plying unscented transformations in the feedback looponly. It is well known from robust control (see, e.g., [33]) thatany smooth nonlinear map in (17) admits an LFT (18). Thename of LFT (18) comes from the following actual linear frac-tional form of

subject to (9)


1) Unscented Transformations for Moderately NonlinearMaps: Unlike linearizing the deterministic map in (17)as done in the Extended Kalman Filter(EKF), the unscentedtransformation [16] provides a first order approximation forits distribution moments as follows. Since , it admitsCholesky decomposition

(19)

Accordingly, regression pointsare defined

(20)

Clearly

and thereby transform forapproximations

(21)

2) Linear Fractional Transformations for Highly NonlinearMaps: For the regression points definedby (20) set

(22)

Utilize the following alternative approximations to (21)

(23)

3) SDP Based Decentralized Bayes Filtering for NSN:Having the moment approximations (21) and (22) we can easilyapproximate the moments of in (1) as follows:

(24)

Theorem 2: The optimal decentralized Bayes for locating thetarget by NSN modeled by (1), (17) under the power constraint(4) is (5) with and defined by (24) and found fromthe SDP

(25)

Remark: It is obvious that in (14) is a particular(linear)case of (17) with and in (18). Then (15) is(23) corresponding to . Thus, Theorem 1 is a particularcase of Theorem 2 with .

A more general representation of than (17) is

(26)

where like and are at the same structure as in (17), i.e.,is moderately nonlinear while is represented by LFT (18).

Then, still using approximations (21) and (23) leads to the fol-lowing approximation instead of (24) [see (27) at the bottom ofthe next page].

III. GLOBAL OPTIMAL DECENTRALIZED BAYES FILTERING FOR

TRACKING DYNAMIC OBJECTS

This section discusses tracking of a dynamic target by SNunder a power constraint. We consider a scenario consisting ofa target moving within a surveillance area. The sensor nodesof a SN are assigned to carry out measurements necessary fortracking the object and send their measurements to the FC tooutput the final estimate of the target’s trajectory. Such a processcan be modeled by the following input-output system:

(28)

(29)

(30)

Here at time instants , (28) is the evolution equation of thetarget state transition, (29) is the measurement equa-tion of all sensors and (30) is the FC equation. ,

and are additive noises.Note the similarity of (29)–(30) and (1)–(2), i.e., the physicalmeaning of variables in (29)–(30) is exactly the same


as in (1)–(2), while like in (2) defined by (3), thematrix is defined by

(31)

where is the channel gain between th sensor node and FC,while is the amplifier coefficient to control the transmitpower of th sensor node at time instant , i.e., at each timeinstant , the sum power of the SN is constrained by a budget

(32)

with . On the other hand,(29)–(30) is also a particular case of (1)–(2) with ,

and .From now on, we adopt the following form of and

:

(33)

where are moderately nonlinear maps whileare highly nonlinear map, which are then transformed to an LFT

(34)

with deterministic matrices andlinear map . We have seen at the end of the previoussection that the linear case of or corresponding to (33)with or in (34), so(33)–(34) is an universal representation for whatever possible(linear or nonlinear) modeling.

Given the initial information

(35)

the problem at the FC level is to track the state based on theinstant information , which is power constrained by (32). Ifone sets , as an initial estimator of ,then the following recursive Bayes filtering to track the target

is most natural. At time instant , suppose is the esti-mator of by using past measurementswith the moments and . The FCprocess two iterations at each time instant .

• Treat in (29)–(30) as

(36)

to produce the LMMSE estimator for under thepower constraint (32).

Like (10), for , solve the followingSDP:

(37)

(38)

for the optimal solution . Then, like (5) and (6), the mo-ments of are approximated by

(39)

(40)

which makes use of the following approximations of and.

— Unscented Transformations. Like (21), make Choleskydecomposition for the definitionof regression pointsby (20)(with ) and then transform

for approximations

(41)

(27)


— Linear fraction transformation. Determine cor-responding regression points by(20)(with ). Set

(42)

Accordingly

(43)

— Finalize (44), shown at the bottom of the page.— Treat and in (28) as and , respectively

(45)

and apply [18, Th. 12.1] to produce the momentsand for the next recursive time

(46)

Like (19), (20), (21), (22), (23), (24) making Cholesky fac-torization to define regres-

sion points by (20) with obviouslyreplaced by , the resultant approximation equation for(46) is

(47)

where

(48)

and (49), shown at the bottom of the page, with

(50)

Theorem 3: The decentralized Bayes filtering for tracking thedynamic target by a NSN modeled by (28)–(30) with and

in form (33), (34), under the sum power constraint (32) and

(44)

(49)


Fig. 1. MSE versus sum transmit power in minimizing MSE for scalarparameter.

initialized condition (35) is the following recursive procedurefor .

• Solve the SDP (37)–(38) with , and ap-proximated by (44)

• Execute (39)–(40);• Execute (47).Remark: It should be noted that Theorem 3 in the particular

case of and while and , i.e.,(28)–(30) is the following linear system:

(51)

is nothing but the decentralized Kalman filter, which is also anew result.

IV. SIMULATION RESULTS

Effectiveness of the proposed strategies is validated via10 000 Monte Carlo channel realizations through simulationresults presented for both static and dynamic targets. Thereare ten sensors and channel gains between thesesensors and the FC are generated by normal distribution. It isalso assumed that

(52)

Following is the graphical demonstration of how the proposedfiltering algorithms discussed in previous sections performunder various scenarios.

TABLE IPOWER ALLOCATION � FOR LOCATING RANDOM SCALAR

Fig. 2. MSE versus logarithm of sum transmit power in minimizing sum powerfor scalar parameter.

A. Decentralized Bayes Filtering for Locating Static Targetsand the Optimized Sensor Selection

1) LSNS: Consider (14), (2) with

(53)

i.e., the sensors are in different channel conditions. The decen-tralized Bayes filter for static target based on SDP (16), is sim-ulated by applying Theorem 1. Depending on the target’s statedimension, there are two possible cases:

• (scalar target). This simple yet interesting case con-sists of sensor nodes transmitting their observations of onedimensional parameter of interest to the FC. Fig. 1 showsthe MSE performances at the FC for four different powerallocation schemes. From the analytical view point, the re-sultant solutions offered by [9] lose their tractability for


Fig. 3. (a) MSE estimation performance for proposed multisensor and single sensor. (b) MSE estimation performance for proposed SDP-based power allocationand equal power allocation schemes.

a relatively general channel environment in which eachsensor experiences a different noise variance. It can be ob-served from Fig. 1 that the proposed LMMSE-based tech-nique (by using Theorem 1) yields better results than thesingle best sensor approach, which assigns all the powerto a single sensor of the highest SNR, the suboptimal so-lution of [9] and the equal power strategy. Our solutiontakes full advantage of using a sensor network over a singlesensor, while all other solution could not; the suboptimalsolution of [9] is actually worse than that by a single sensorand the equal power-based solution is much worse thanthat by a single sensor. Furthermore, Table I with the op-timal power allocations under different sum powers revealsthat our globally optimal decentralized Bayes filters as-signs the power to only a selective number ofsensors (sensors # 4, 2, and 3 for , ,and , respectively). These Bayes filters, thus, ef-fectively solves the optimal sensor selection by SDP in-stead of a hard conventional combinatoric problem of se-lection. For comparison with the result of [8], considerthe problem of minimizing the sum transmit power sub-ject to a threshold of MSE distortion, whose global optimalsolution is computed based on (12) and (13). The corre-sponding simulation is sketched in Fig. 2 from which itis clear that for a given distortion, the proposed solutionrequires a much smaller amount of the sum power thandoes the BLUE-based solution of [8]. This demonstratesthat BLUE is far from the optimal LMMSE, despite thatits theoretical diversity has been shown.

• (random vector). Next, we study estimationperformance of various power allocation schemes for arandom vector . We consider localizing a static objectin in which the vector parameter to be estimatedconsists of three Cartesian coordinates. Observationsperformed by sensor nodes include measurements ofrange, elevation angle and azimuth angle, all of whichare then fused at the FC to estimate the target’s po-sition. A formal representation of such a model is

, where (see theequation at the bottom of the page). Under LSN frame-work, nonlinear maps are linearized at to have thelinear sensor model (14) withwith . From Fig. 3(a), it can be seen that,by exploiting spatial diversity, LSN provides a far betterestimate of the position vector as compared to a singlesensor. In Fig. 3(b), it is demonstrated that our proposedstrategy of optimal power allocation gives lower MSE forthe same transmit power when compared to the scheme ofassigning equal power among nodes.

In another scenario of vector estimation, we consider a model[2], [9] for comparison purpose. According to this model, eachsensor in an LSN measures one scalar component of the -di-mensional parameter. The resulting simulation result is given inFig. 4. Once again, our proposed approach performs by far thebest in terms of the sum power spent for different MSE thresh-olds. Table II describes distribution of power to the sensors. Asexpected, the roles of the sensors are more or less equal so thedecentralized Bayes filters pick almost every sensor for activity.


Fig. 4. MSE versus total transmit power while minimizing MSE for vector-valued parameter.

TABLE IIPOWER ALLOCATION � TO SENSORS FOR LOCATING RANDOM VECTOR

Alternatively, for comparison to the result of [3] we considerthe sum power minimization subject to different MSE thresh-olds. Fig. 5, providing the different performances, clearly showshow ours approach using SDP (12) and (13) outperforms thatby [3], under the same simulation conditions as given in [3]:observation noise variance , channel noise variance

, with 2 sensor nodes and channel gains ,, while

with used to express the correlation between the sensorobservations.

Furthermore, Fig. 6 depicts the plot of MSE versus totalpower by solving corresponding SDP (12) and (13).

Fig. 5. MSE versus total transmit power while minimizing power for vector-valued parameter.

Fig. 6. MSE versus total transmit power while minimizing power for vector-valued parameter.

2) NSNs: Simulation environment for NSN is similar to itscounterpart in the previous subsection, except that the sensorscarry out nonlinear range and bearing measurements here [4],[5], [12].

• (random scalar). Sensors are randomly distributedover a region where they perform range and bearing mea-surement of a target lying on the axis. Thus the target


Fig. 7. MSE versus total transmit power for locating scalar target under non-linear model.

is located by its axis coordinate based on range andbearing sensors

Here are coordinates of -sensor. Thus,the nonlinear sensing map is in the form of (17) with

and , whichadmits an LFT (18) with

Accordingly, the approximation (24) is used in implemen-tation of SDP (25) of Theorem 2. Fig. 7 shows the perfor-mance comparison between the proposed power allocationby Theorem 2, the equal power allocation and the strategyof assigning all power to a single sensor. It is notable thatthe single sensor performance is not even comparable to theproposed and equal power schemes due to lack of spatialdiversity, i.e., a full diversity is achievable for multi non-linear sensors. The proposed scheme performs better thanthe equal power allocation throughout the entire range. Onecan also note that the difference between these two strate-gies become lager for smaller values of , which demon-strates that our scheme ensures best use of the resourcesunder severe power budget constraints.

• (random vector). To determine MSE performancefor vector target in NSN, we consider an object in aplane, which is located by its axis coordinates

Fig. 8. MSE versus total transmit power in nonlinear model of locating a statictarget.

. A typical sensor’s measurements consist ofthe following ranging and bearing:

(54)

where each map in form (17) withadmits an LFT (18) with

Accordingly, (24) is used in implementation of SDP (25)of Theorem 2. Simulation results in Fig. 8 indicate thatthe SDP-based approach outperforms the single sensor ap-proach, and shows better results than the equal power-based scheme.

B. Decentralized Bayes Filtering for Locating DynamicTargets

1) LSN With Nonlinear Dynamics: We begin our discussionof simulation results for dynamic target’s state estimationwith a nonlinear state transition model of the third order.In this example, we consider a typical third-order non-linear autoregressive process described mathematically as

with the noise corruptedobservations . Addressed in previous work [26],this system admits the following state-space formulations withthe state


Fig. 9. (a) True trajectory and estimate of the state �� under single sensor andmultisensor case for a nonlinear dynamic model. (b) MSE versus total transmitpower for a nonlinear dynamic model in an LSN.

with . An equivalent LFT model (34) and (49) forthis third-order nonlinearity is achieved with the following de-terministic parameters:

where is the 2 2 identity matrix. Usingas the initial conditional estimate of the state with covariance

, trajectory of the statefor 50 time steps along with state estimates are shown in Fig. 9where mean square error (MSE) for the first component of thestate variable obtained from Monte Carlo runs is plottedusing Theorem 3. These results suggest that the proposed mul-tisensors approach outperforms the single sensor scheme for themeasured state estimation. Also, the proposed optimal power al-location technique offers less MSE than equal power allocation.

2) NSN: The problem of dynamic target tracking has beenextensively addressed in previous literature (see, e.g., [4] and[5] and the references therein). In this example, we consider avehicle moving along a trajectory as specified in Fig. 10, witha dynamic model based on constant velocity and a coordinatedturn model [4] to account for nonmaneuver and maneuver mo-tion of the target. Corresponding state and the measurementequations are

where is the process noise,denotes the kinetic state of the target

Fig. 10. Trajectory of a maneuvering target and distribution of nodes oversurveillance region.

at time consisting of the target coordinate and itsvelocity , and with the system matrices

Here is the sampling period and is the turn rate ofthe maneuvering target. is initialized as

with initial position and velocity as(250,150) and (15, 15), respectively. The sensor nonlinear mea-surements include range and bearing information of the vehicle,corrupted by noise vector . It is notable thatrange measurement function bears a moderate nonlinear map,hence (41) is used to approximate mean and other moments,whereas for highly nonlinear map of the bearing information,we use (43) to evaluate the required statistics with the sameLFT system matrices as used in the static nonlinear scenario.In other words, implementation of Theorem 3 is sought to trackthe ever evolving state vector under linear state dynamics forthis target tracking problem.

Tracking of the moving vehicle is executed over a 2Dsurveillance region of . A multitudeof sensor nodes , randomly deployed over the regionof interest, takes measurements at a sampling period of 1 s.For process noise, we choose . Similarly, varianceof local as well as global channel noise while measuringrange and bearing are and

, respectively. The target’strajectory, along with sensor nodes distribution have beenshown in Fig. 10. A one-dimensional view of the true and


Fig. 11. Target tracking performance of multisensor and single sensor in termsof estimation error of x-coordinate.

Fig. 12. Target tracking performance of multisensor and single sensor in termsof estimation error of y coordinate.

estimated trajectory for single and multiple sensors convertedto Cartesian coordinates is shown in Fig. 13. At s targetsets off from at a constant velocity of 15 m/s andafter 14 s performs a clockwise turn for 6 s at a turn rate of

rad/s. It takes two counterclockwise turns; one after16 s and then after 4 s. Simulation results shown in Figs. 11and 12 suggest that multisensors perform better than a singlesensor even when all available power is allocated entirely to thesingle sensor.

Of particular interest is the observation that for optimized es-timation performance at all time instants, nodes with the bestchannel conditions, determined by the proposed strategy, are se-lected to sense and send their data. This is further elaborated in

Fig. 13. True trajectory and estimates by single and multisensor network.

Fig. 14. Distribution of power among different sensor nodes for the case� �

��.

Fig. 14 where distribution of power among 10 sensor nodes isplotted.

V. CONCLUSION

The problem of power allocation among sensor nodes for lo-cating a static target or for tracking a dynamic target in eitherlinear or nonlinear sensing systems has been addressed. Thesesensors observe the targets and then transmit their noisy obser-vations through noisy wireless channels to the FC where thefinal estimate is carried out. Due to limited energy resources,it is desired to develop an optimized power allocation techniquewhich is able to minimize MSE of the estimate under a givenpower budget. A novel technique based on tractable optimiza-tion (SDP) and approximation (unscented and linear fractionaltransformations) has been proposed. The multisensor diversityhas been fully exploited to arrive at an accurate estimate of the


target’s state. Accompanying simulation results clearly showedthe viability of the theoretical results.

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Umar Rashid (S’10) received the B.Sc. degree in electrical engineering fromthe University of Engineering and Technology, Lahore, Pakistan, in 2007.

He has been a Lecturer/Lab Engineer with the same university since 2008.Currently, he is on study leave to pursue the Ph.D. degree at University of Tech-nology, Sydney, Australia, under the supervision of Prof. Hoang Duong Tuan.His research interests include nonlinear multisensor network, statistical signalprocessing, and target tracking.

Hoang Duong Tuan (M’94) was born in Hanoi, Vietnam. He received thediploma (Hon.) and the Ph.D. degrees, both in applied mathematics, fromOdessa State University, Ukraine, in 1987 and 1991, respectively.

He spent nine academic years in Japan as an Assistant Professor with the De-partment of Electronic-Mechanical Engineering, Nagoya University, from 1994to 1999, and then as an Associate Professor with the Department of Electricaland Computer Engineering, Toyota Technological Institute, Nagoya, from 1999to 2003. He has been with the School of Electrical Engineering and Telecommu-nications, the University of New South Wales, as a Senior Lecturer from 2003 to2006, an Associate Professor from 2007 to 2010, and a Professor from 2011. Heis currently a Professor of Centre for Health Technologies, University of Tech-nology, Sydney. He has been involved in research with the areas of optimization,control, signal processing, wireless communication, and bioinformatics for 20years.

Pierre Apkarian (A’94–M’00) received the Ph.D. degree in control engineeringand Habilitation (HDR) from the École Nationale Supérieure de l’Aéronautiqueet de l’Espace (ENSAE), Toulouse, France, in 1988 and 1997, respectively.

He was a Professor with the Université Paul Sabatier, Toulouse, in both au-tomatic control and applied mathematics, in 1999 and 2001, respectively. Since1988, he has been a Research Scientist at ONERA-Toulouse and a Professor inthe Mathematics Department of the Université Paul Sabatier. His research in-terests include robust and gain-scheduling control theory, LMI, and nonsmoothoptimization techniques with applications in aeronautics.

Dr. Apkarian has served as an Associate Editor for the IEEE TRANSACTIONS

ON AUTOMATIC CONTROL.

Ha Hoang Kha (S’05–M’09) was born in Dong Thap, Vietnam. He received theB.Eng. and M.Eng. degrees from HoChiMinh City University of Technology,in 2000 and 2003, respectively, and the Ph.D. degree from the University ofNew South Wales, Sydney, Australia, in 2009, all in electrical engineering andtelecommunications.

He was a Visiting Research Fellow at the School of Electrical Engineeringand Telecommunications, the University of New South Wales, from April 2009to March 2011. He is currently a Research Postdoctoral Fellow at Centre forHealth Technologies, University of Technology, Sydney. His research inter-ests are in digital signal processing and wireless communications, with a recentemphasis on convex optimization techniques in signal processing for wirelesscommunications.

globally optimized power allocation in multiple sensor fusion for linear and nonlinear networks

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