Optimized Power Allocation in Nonlinear SensorNetworks via Semidefinite Programming

Umar Rashid, Hoang Duong Tuan and Ha Hoang KhaSchool of Elect. Engg. and Telecom, University of New South Wales, Australia

Email: {umar, h.d.tuan, h.k.ha}@unsw.edu.au

Abstract—This paper presents an efficient technique for powerallocation to the sensor nodes in a nonlinear sensor network(NSN). We minimize mean square error of the estimation of arandom scalar parameter subject to a constraint on total amountof power consumed by the sensor nodes. This estimation is carriedout at fusion center (FC) which receives the local observationsfrom the sensors located at different positions. We convert theoptimization problem into a convex one, and then use semidefiniteprogramming to find the global optimal solution. The simulationresults show that our approach outperforms the previous workboth for the channel with white noise and the one with colorednoise. The proposed strategy also gives better results in caseof nonlinear model when compared to the strategy of assigningequal power to sensor nodes.

Index terms. Nonlinear sensor network, distributed estima-tion, semi-definite programming, convex optimization.

I. INTRODUCTION

Nonlinear sensor networks (NSN) hold key to a widerange of future applications and have the potential to playa significant role in modern technological realm. Deploymentof these wireless networks ensures controlling the instrumen-tation in industrial automation, sensing data remotely in a datacollecting environment and providing surveillance in defenserelated applications [4] . These networks were first used inmilitary applications. However, recently their potential has alsobeen exploited in many other areas due to the advancement infabrication and wireless communication technology [1].

A typical NSN consists of geographically distributed nodeswhich make their own local observations in a noisy envi-ronment, and then send these observations to a central unit,called fusion center (FC). In order to transmit their respectiveobservations, these nodes require some finite amount of power.This power requirement becomes pronounced if the channelbetween sensor nodes and the fusion center is noisy [2]. Inorder to mitigate the effects of noise in the channel, thetransmitter has to use more power. But there are always somelimitations on the amount of power which can be used. In fact,power requirement is the most important constraint in nonlin-ear sensor networks. Therefore, some methods or algorithmsmust be found in order to achieve reliable communicationunder certain constraints on the total amount of power beingused by the network [12].

This paper offers an efficient strategy of allocating powerto the sensor nodes in an optimized manner which couldguarantee a reliable communication. We address problemof estimating a random parameter in a distributed fashion,

where all sensor nodes send their data to fusion center whichmakes the final estimate of the parameter of interest. TheFC makes use of Bayesian estimator in which use of priorknowledge about the statistics of random parameter leads toan accurate estimate. We also consider nonlinear model apartfrom the linear one representing the measurement of randomparameter of interest, e.g. range of a target. We use UnscentedKalman Filtering as well as Extended Kalman Filtering [9] forestimation of the parameter in nonlinear measurement system.

Examples of early work in this area of power alloca-tion include the decentralized detection in power constrainednetworks [3], measurement of link inefficiency of sensornetwork in energy-constrained scenario [10], optimizing theperformance of large-scale wireless sensor networks operatingunder energy and reliability constraints [8]. [5] has consideredonly linear sensor networks (LSNs) for allocation of powerto sensor nodes so as to minimize distortion. Moreover, ithas assumed only uncorrelated noise model in the channelbetween sensor node and FC. [7] also discusses exactly thesame problem of minimizing mean-square error (MSE) undersome power constraints both for uncorrelated and correlatednoise model. The method used by the authors of [7], however,is not optimized for any case. They have assumed that statisticsof the output of the sensors are known at the FC, whichis an unnatural assumption. Moreover, their formulations fornonlinear sensing functions are not clear and do not correspondto any practical scenario. Whereas, our technique which isbased on semidefinite programming, not only achieves betterglobal optimal solution than those achieved in [7], but weconsider a practically justified nonlinear model.

The rest of the paper is organized as follows. Section IIdescribes the system model and the formulation of the prob-lem, Section III discusses the formulation of the optimizationproblem into a semidefinite program. Section IV gives thesimulation setup and results. This is followed by Section Vwhich presents the concluding remarks about the work donein the paper.

Most of the notations used in the paper are fairly standard.Bold faced upper case letters, e.g.A, denote matrices, whereasbold faced lower case letters denote vectors. R denotes set ofall real numbers. Notation used for a semi-definite matrix isA � 0. diag(a1, a2, . . . , an) indicates an n×n diagonal matrixwith diagonal entries equal to (a1, a2, . . . , an). E[x] means theexpectation of a random variable x. trace() denotes the trace ofa matrix. Rx =E[xxT ] and Rxy =E[xyT ] which represents

978-1-4244-3574-6/10/$25.00 ©2010 IEEE

autocorrelation of vector x and, crosscorrelation between xand y respectively.

II. PROBLEM FORMULATION AND SDP BASED SOLUTION

Consider the problem of estimating a random scalar param-eter, s ∈ R in an NSN. The network consisting of M numberof sensor nodes, which, located at different locations, makesindependent observations of the random parameter of interestin a noisy environment. This noisy measurement yi ∈ R ofeach sensor can be modeled by

yi = fi(s) + ni, i = 1, . . . ,M (1)

where fi() are general functions which represent the relation-ship between the measured signal corrupted by noise and therandom scalar parameter s. ni is ith component of Gaussiannoise vector, n = [n1, n2, . . . , nM ]T , with zero mean and avariance of σ2

ni. This noise vector can be either uncorrelated

(white) or spatially correlated (colored).These observations are transmitted according to amplify

and forward (AF) scheme towards fusion center throughindependent Additive White Gaussian (AWGN) Channels. Atthe fusion center, all these local observations from the sensornodes are combined to produce a final estimate of s. Themessage received at FC from the ith sensor is given as

zi = hiαiyi + wi, i = 1, . . . ,M (2)

where αi is the amplification factor used at sensor node i, hi isthe ith channel gain, and wi is zero-mean Additive Gaussiannoise in channel i with covariance matrix given as Rw =E[wwT ] = diag(σ2

w1, σ2w2, . . . , σ

2wM ). This channel noise is

supposed to be independent of the sensor measurements.In aggregate form, (2) can be written as

z = Ay + w (3)

where w = [w1w2 . . . wM ]T is the noise vector,z = [z1z2 . . . zM ]T is received vector and A =diag(h1α1, h2α2, . . . , hMαM ).

Using the received vector y, the linear mean square estimateof s can be written as

s = s + RszR−1z (z − z)

= s + RszAT (ARyAT + Rw)−1(z − z), (4)

where s is the mean value of s. Consequently, the expressionfor MSE is given by

E[(s − s)2]=σ2s−RsyAα

T(AαRyATα +Rw)−1AαRT

sy (5)

where A =diag(α1, α2, . . . , αM ), Rw =diag

(σ2

w1h21

,σ2

w2h22

, . . . ,σ2

wM

h2M

).

Our objective is to find the diagonal matrix Aα whoseentries will determine the amount of amplification at each ofthe corresponding node in order to adjust the transmissionpowers. The average power of ith sensor is given by

Pi = α2i σ

2si i = 1, . . . ,M (6)

where σ2si is the variance of the random parameter to be

estimated. In an energy constrained environment, it is desirableto minimize MSE subject to a constraint on total amount ofpower. This total power is given by

Ptotal =M∑i=1

Pi = trace(AαRyAT

α

). (7)

Hence the optimization problem for power allocation is givenby

maxAα

trace(RsyAα

T(AαRyATα + Rw)−1AαRT

sy

)s.t.(7). (8)

The optimization problem given in (8) can be convertedinto a semidefinite programming problem. First of all, we willtransform the objective function into a convex function of α2

i s.The utility of convex function lies in the fact that it is feasibleto find a global optimal solution of an optimization problemwith a convex objective function. Using matrix inversionlemma, (AαRyAT

α + Rw)−1 in (8) can be written as

(R−1y +AT

αR−1w Aα)−1=Ry−RyAT

α(Rw+AαRyATα)

−1AαRy

Simplifying above expression, we get

ATα(Rw + AαRyAT

α)−1Aα = [Ry + RyATαR−1

w AαRy]−1,(9)

it is clear that maximizing R−1y (R−1

y + ATαR−1

w Aα)−1R−1y

is same as minimizing the right hand side of (5). Moreover,since Aα and R−1

w are diagonal matrices, we can write

AαR−1w Aα = A2

αR−1w

Introducing another variable, Xα, such that

Xα = A2α = diag(α2

1 α22 . . . α2

M), Xα ≥ 0

Now, defining a new auxiliary variable G, the optimizationproblem in (8) becomes (using Schur’s Complement)

minXα,G

trace (G)

s.t.

[G Rsy

RTsy Ry + RyXαR−1

w Ry

]� 0

trace (XαRy) = Ptotal (10)

which has linear equality constraint and a linear matrix non-negativity constraint. As a result, we have formulated theoriginal nonconvex optimization problem into a semidefiniteprogram by a series of intelligent change of variables whichmakes it computationally tractable. Modern softwares e.g.,SDPT3 [13] can be used for solving convex optimizationproblems so as to generate an optimal solution or a certificateshowing infeasibility.

III. LINEAR MODEL AND SIMULATION RESULTS

Here we assume, without loss of generality, that both therandom parameter of interest and the observation vector havezero mean. In the linear model, we replace fi(s) with a scalingfactor as shown in the following equation:

yi = qis + ni, i = 1, . . . ,M (11)

where qi is a scaling factor in this linear relationship. Fur-thermore, it is assumed that the fusion center has the a-prioriknowledge of the statistics of the random parameter and thechannel noise. The vector form of the above equation can bewritten as

y = qs + n (12)

where q is an M -dimensional vector. Expressions for Rsy canbe derived as follows:

Rsy = E[syT ] = E[s2qT + snT ] = σ2sq

T .

As it can be seen, Rsy is an M -dimensional row vector.Similarly Ry can be expressed as

Ry = E[yyT ] = E[s2qqT+sqnT+nsqT+nnT ] = σ2sQ+Rn

where Q is an M × M matrix. Scaling factors qi’s are

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10.04

0.06

0.08

0.1

0.12

0.14

0.16

Total Transmit Power

Mea

n S

quar

e E

rror

Approach of [5]Approach of [6]Proposed Approach

Fig. 1. Mean Square Error versus total transmit power with uncorrelatednoise, β = 0.

assumed to be equal to 1 in linear case. Fading coefficientshi’s of the channel between sensors and the fusion centerare independent and identically distributed random variableswith zero mean and unit variance. Noise of local channelsof sensors is modeled as a first order autoregressive processAR(1), satisfying the following difference equation:

ni = βni−1 + εi i = 2, . . . ,M (13)

where εi is a white noise sequence with zero mean andvariance σ2

ε = 0.5, β is AR coefficient. Moreover, we chooseM=20, σ2

s = 1 and σ2w,i = 0.01 for any i.

A. White Noise

In case of white noise we take β = 0 in the aboveautoregressive model. The resulting curves have been plottedbetween the mean square error (MSE) versus total transmitpower in Fig. 1. These results indicate that the proposedalgorithm performs better than the strategies proposed in [5]and [7].

B. Colored Noise

Figs. 2-4 show the estimation performance of the estimatorin terms of mean square error (MSE) of three different powerallocation schemes as a function of the total transmitted powerby all the sensor nodes for spatially correlated noise. In thiscase, the amount of correlation in noise vector across sensorsis quantified in terms of value of autoregressive parameter,e.g. β = −0.9. We can see that the proposed scheme outper-forms the existing schemes for power allocation in distributedestimation.

IV. NONLINEAR MODEL AND SIMULATION RESULTS

For the nonlinear case we consider a typical scenario wherewe measure the range of a moving object at a particular instantof time. Movement of the object is restricted to along x-axisonly, whereas the sensors are placed in the form of a uniformlinear array along y-axis. Each sensor is d distance apart fromits adjacent neighbor, with the first sensor being located atthe origin. If s represents the abscissa of the target’s location,then the expression for fi(s) in connection with the rangemeasurement [6] is given by

fi(s) =√

[(i − 1)d]2 + s2 (14)

In order to determine the statistical parameters of the systemsso that we could use Mean Square Error (MSE) estimator, wemake use of Kalman Filtering technique. For the nonlinearcase, however, we assume nonzero-mean random parameter.There exists two ways of applying Kalman Filtering in non-linear systems:(1) Extended Kalman Filtering (EKF) and, (2)Unscented Kalman Filtering (UKF). We discuss both of thesein the following sections.

A. Extended Kalman Filters

In case of EKF, we linearize the nonlinear function acrossthe mean value of the parameter to be estimated. Then we findout its statistical parameters in a normal way. Linearizing the

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10.05

0.1

0.15

0.2

0.25

0.3

Total Transmit Power

Mea

n S

quar

e E

rror

Proposed ApproachApproach of [6]Approach of [5]

Fig. 2. Mean Square Error versus total transmit power with correlated noise,β = −0.9.

above nonlinear function across the mean value of s, we get

fi(s) = fi(s) +∂fi

∂s|s=s (s − s)

=√

[(n − 1)d]2 + s2

− s2√[(n − 1)d]2 + s2

+ss√

[(n − 1)d]2 + s2(15)

If we substitute this linearized function into (1), we can thendetermine the statistical parameters of the observed signal ywhich are given as follows

E[yi] =√

[(n − 1)d]2 + s2 (16)

E[(s − s)(y − y)] =sσ2

s√[(n − 1)d]2 + s2

(17)

Similarly the expression for the autocorrelation matrix ofvector y can be found from which we establish the expressionfor autocovariance:

Ry(n,m) =√

([(n − 1)d]2 + s2)([(m − 1)d]2 + s2)

+s2σ2

s√([(n − 1)d]2 + s2)([(m − 1)d]2 + s2)

+E[wnwm] (18)

and covriance is given as

Cy = Ry − E[y]E[y]T . (19)

B. Unscented Kalman Filter

Here we make use of Unscented transforms in order todetermine the statistical parameters of this nonlinear model.First of all, we find out the sigma or regression points [11] by

s0 = s = 1,

s1 = 1 +

√p + 1

2σs = 1 + 1.22σs,

s2 = 1 −√

p + 12

σs = 1 − 1.22σs,

where p=2 since s is one-dimensional parameter. Then theobservation variable vector x is evaluated at these regressionpoints using (11), i.e.,

yik=

√[(n − 1)d]2 + s2

k (20)

so as to find the approximation of its mean, covariance andcross-covariance with s. Similar expressions can be used forfinding the approximated values of statistical parameters of y.

y =1

p + 1

p∑k=0

yk, (21)

Ry =1

p + 1

p∑k=0

(yk − y)(yk − y))T , (22)

Rsy =1

p + 1

p∑k=0

(sk − s)(yk − y))T (23)

In this case, we take the mean value of s to be equal to 20and that of d is 5. We chose σ2

n = 0.02 and σ2w = 0.5 in our

simulations. The resulting curves are plotted for two strategies.One is based on assigning equal power to all nodes, also calledequal power scheme. The second one is optimized distributionof power resources among the nodes in NSN. The results inFig. 3 clearly shows that there is substantial amount of powersaving in case of proposed technique.

V. CONCLUSION

In this paper, we have addressed the problem of powerallocation in distributed estimation of a random parameterin nonlinear sensor networks. We have formulated the op-timization problem of minimizing mean square error as asemidefinite programming (SDP) because SDP produces betterlower bounds than the existing techniques. Simulation resultsshow that the proposed strategy of power allocation to sensornodes produce improvement in mean square error of theestimator over the previous schemes.

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0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10.1

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0.7

Total Transmit Power

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