Research ArticleAn Improved Unscented Kalman Filter for Discrete NonlinearSystems with Random Parameters

Yue Wang1 Zhijian Qiu1 and Xiaomei Qu2

1School of Economic Mathematics Southwestern University of Finance and Economics Chengdu Sichuan 611130 China2College of Computer Science and Technology Southwest University for Nationalities Chengdu Sichuan 610041 China

Correspondence should be addressed to Yue Wang wangyue 001163com

Received 29 December 2016 Accepted 6 February 2017 Published 26 February 2017

Academic Editor Delfim F M Torres

Copyright copy 2017 Yue Wang et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

This paper investigates the nonlinear unscented Kalman filtering (UKF) problem for discrete nonlinear dynamic systems withrandom parameters We develop an improved unscented transformation by incorporating the random parameters into the statevector to enlarge the number of sigma points The theoretical analysis reveals that the approximated mean and covariance viathe improved unscented transformation match the true values correctly up to the third order of Taylor series expansion Basedon the improved unscented transformation an improved UKF method is proposed to expand the application of the UKF fornonlinear systemswith randomparameters An application to themobile source localizationwith time difference of arrival (TDOA)measurements and sensor position uncertainties is provided where the simulation results illustrate that the improved UKFmethodleads to a superior performance in comparison with the normal UKF method

1 Introduction

Since Kalman proposed his famous recursive method tosolve discrete dynamic filtering problems [1] the Kalmanfilter has been widely used in many areas ranging fromengineering to economics [2 3] However the linearity ofthe dynamic system as one of the basic requirements of theKalman filter is hard to satisfy in actual implementationOn the other hand discrete dynamic systems with randomparameters arise in many applications such as missile trackestimation satellite navigation maneuvering target trackingand economic forecast [4ndash8]

For the linear discrete dynamic system where the statetransition and measurement matrices are random param-eters De Koning [4] provides a linear minimum variancerecursive estimator without rigorous derivation By convert-ing the system to a linear dynamic system with deterministicparameter matrices and state-dependent noises Luo et al[6] propose the optimal linear filtering with the form of amodified Kalman filtering

Unlike the filtering problem in linear dynamic systemswith randomparameters which has been extensively studied

the research on the filtering for nonlinear dynamic systemswith random parameters has seldom been reported in theliteratureThemain challenge lies in how to transform a prob-ability density function (pdf) through a general nonlinearfunction which contains random parameters

In order tomake the Kalman filter applicable to nonlineardynamic systems the extended Kalman filter (EKF) [9]based on the first-order approximation in the Taylor seriesexpansion of a nonlinear function is proposed Althoughthe EKF method maintains the computationally efficientupdate form of the Kalman filter its estimation accuracy maybe unsatisfactory due to neglecting the higher-order termsof nonlinear system function [10] Also the EKF requiresthe calculation of the Jacobian matrix which is difficult toimplement in practical applications [11]

Since the seminal work of Julier et al in 1995 [12] theunscented Kalman filter (UKF) which is an extension of theKalman filter reducing the linearization errors of the EKFhas been an object of great interest in nonlinear filtering[13ndash17] The UKF is a derivative-free filter which combinesthe concept of unscented transformation with the linearupdate structure of the Kalman filtering [12] In addition the

HindawiDiscrete Dynamics in Nature and SocietyVolume 2017 Article ID 7905690 10 pageshttpsdoiorg10115520177905690

2 Discrete Dynamics in Nature and Society

UKFmethod is based on statistical approximations of systemequations without requiring the calculation of the Jacobianmatrix [13]

The fundamental of UKF is the unscented transformationwhich uses a set of sigma points to propagate the mean andcovariancematrix [13]The sigma points are deterministicallycalculated from the mean and square-root decompositionof the covariance matrix of the a priori random variableBy applying the nonlinear system function to each sigmapoint to obtain transformed vectors the ensemble mean andcovariance of the transformed vectors give an estimate ofthe true mean and covariance In theory if the nonlinearsystem function is completely specified the unscented trans-formation can capture the posterior mean and covarianceof a random variable with symmetric pdf accurately up tothe third order of Taylor series expansion [18] leading to asuperior performance in comparison with the EKF method

However when the nonlinear system function containsrandom parameters the step of transforming the sigmapoints to transformed vectors is unrealizable One methodto deal with this problem is to neglect the randomness ofthese parameters by using the nominal parameters in thetransforming procedure In intuition this will lead to unex-pected error in the corresponding unscented transformationespecially when the true parameters are far from the nominalparameters

The key motivation of this paper is to overcome the dif-ficulty of unscented transformation with random parametersand expand the application of the UKF for the nonlinear stateestimation Hence we focus on the design of an improvedunscented transformation by regarding the random param-eters as part of the state to augment the state vector Inthe corresponding improved unscented transformation thenumber of sigma points is enlarged due to the increaseddimension of the state vector In addition each improvedsigma point contains samples of both the state and therandom parameters so there are no unknown parametersinvolved in the step of transforming the sigma points

Our main contributions in this paper include the fol-lowing (i) An improved unscented transformation and thecorresponding improved UKF method are proposed to dealwith the nonlinear dynamic systemwith random parameters(ii) the performance analysis is provided which shows thatthe approximated mean and covariance via the improvedunscented transformation match the true values correctlyup to the third order of Taylor series expansion when thenonlinear system function contains random parameters

An application of the proposed improvedUKFmethod tothemobile source localization problem is provided where weconsider source localization by a network of passive sensorsusing noisy time difference of arrival (TDOA)measurementsExtensive studies for the static source localization by TDOAmeasurements can be found in [19 20] However in practicethe emitter sourcemay not be static which could bemountedon-board of dynamic mobile platforms [21] In additionaccurate sensor location information may not be available[20] For instance in modern localization applicationssensors could be deployed randomly in a field and theirnominal locations may not be accurate The mobile source

localization problem with sensor location uncertainties canbe described as a discrete nonlinear dynamic system withrandom parameters

The Monte Carlo simulations are carried out to comparethe localization performance of the improved UKF methodand the normal UKF method The results indicate that theroot of mean squared error (RMSE) of mobile localization isevidently decreased by the improved UKFmethod especiallywhen the sensor position uncertainty level is high whichcorroborates that the proposed improved UKF method canimprove the estimation accuracy of the nonlinear dynamicsystems with random parameters

The remainder of the paper is organized as followsSection 2 briefly describes the nonlinear system modeland the unscented Kalman filter Section 3 presents theimproved unscented Kalman filteringmethod for the nonlin-ear dynamic systemwith random parameters An applicationto the mobile source localization problem and the MonteCarlo simulation results are provided in Section 4 andSection 5 gives some concluding remarks

Throughout this paper the transpose and inverse ofmatrix X are denoted by X1015840 and Xminus1 respectively the ldquominusrdquoand ldquo+rdquo superscripts denote that the estimate is a priori anda posteriori estimate respectively sdot means the Euclideannorm and 119864(sdot) is the mathematical expectation

2 Nonlinear and Unscented Kalman Filter

The unscented Kalman filter provides a suboptimal solutionfor the stochastic filtering problem of a nonlinear discrete-time dynamic system in the form

x119896 = 119891 (x119896minus1) + w119896y119896 = ℎ (x119896) + k119896 (1)

where 119896 is the discrete-time instant x119896 isin R119899119909 is thestate vector y119896 isin R119899119910 is the measurement output and119891(sdot) and ℎ(sdot) are the process and measurement functionsrespectively The vectors w119896 and k119896 are two zero-mean whiteGaussian noise processes with covariance matrixes Q119908 andQV respectivelyThe noise termsw119896 and k119896 are assumed to beuncorrelated

The goal of the stochastic filtering is to estimate the statex119896 as new measurements y119896 are acquired When the processand measurement functions are linear to the state vectorthe celebrated Kalman filter provides the optimal solutionwith respect to the minimum variance (MV) criterion [22]However in the case of nonlinear systems such optimalsolution tends to be computationally intractable [18 23]Therefore suboptimal approaches are developed such asextended Kalman filter (EKF) [9] and unscented Kalmanfilter (UKF) [13]

The EKF works on the principle that a linearized trans-formation of means and covariances is approximately equalto the true nonlinear transformation but this approximationcould be unsatisfactory in practice In the following we willbriefly describe the unscented transformation which is thekey technique of the UKF

Discrete Dynamics in Nature and Society 3

21 Unscented Transformation Suppose that we have a vectorx isin R119899119909 with known mean x and covariance P and its pdf issymmetric around its mean vector For a nonlinear functiony = ℎ(x) we want to approximate the mean and covarianceof y The first step of the unscented transformation is tofind a set of deterministic vectors called sigma points whoseensemble weighted mean and covariance are equal to x andP respectively which can be realized as follows

x(119895) = x + x(119895) 119895 = 1 2119899119909 (2)

where

x(119895) = (radic119899119909P)1015840119895 119895 = 1 119899119909

x(119899119909+119895) = minus(radic119899119909P)1015840119895 119895 = 1 119899119909 (3)

radic119899119909P is the matrix square root of 119899119909P such that(radic119899119909P)1015840radic119899119909P = 119899119909P and (radic119899119909P)119895 is the 119895th row of radic119899119909PThe weighting coefficients are given as

W(119895) = 12119899119909 119895 = 1 2119899119909 (4)

Then we transform each sigma point in (2) using thenonlinear function ℎ(sdot) and approximate the mean of y bytaking the weighted sum of the transformed sigma pointsSpecifically let y denote the truemean of yThe approximatedmean of y is denoted as y and computed as follows

y = 2119899119909sum119895=1

W(119895)y(119895) y(119895) = ℎ (x(119895)) 119895 = 1 2119899119909 (5)

Let Py denote the true covariance of y The approximatedcovariance of y is denoted as Py and computed as follows

Py = 2119899119909sum119895=1

W(119895) (y(119895) minus y) (y(119895) minus y)1015840 (6)

Theunscented transformations (5) and (6) aremore accu-rate than the linearizationmethod for propagatingmeans andcovariances of nonlinear functions which is summarized inthe following proposition

Proposition 1 (see [18]) For random vector x with a sym-metric pdf around its mean if the nonlinear function y =ℎ(x) is completely specified then the approximated mean andcovariance of y via the unscented transformations (5)-(6)match the true mean and covariance of y correctly up to thethird order

22 Unscented Kalman Filter The unscented transformationcan be generalized to give the unscented Kalman filter whichkeeps the structure of the Kalman filter that includes oneprediction (or a time update) step and one correction (or ameasurement update) step

For the discrete nonlinear system (1) the unscentedKalman filter is initialized as follows

x+0 = 119864 (x0) P+0 = 119864 [(x0 minus x+0 ) (x0 minus x+0 )1015840] (7)

Suppose that at time step 119896 x+119896minus1 andP+119896minus1 are givenTheUKFalgorithm can be summarized in Algorithm 2

Algorithm 2 (the unscented Kalman filter)

(1) Choose the sigma points x(119895)119896minus1

as specified in (2) withappropriate changes that is

x(119895)119896minus1= x+119896minus1 + x(119895)

119896minus1 119895 = 1 2119899119909

x(119895)119896minus1= (radic119899119909P+119896minus1)1015840119895 119895 = 1 119899119909

x(119899119909+119895)119896minus1

= minus (radic119899119909P+119896minus1)1015840119895 119895 = 1 119899119909(8)

(2) Use the completely specified motion function 119891(sdot) totransform the sigma points into x(119895)

119896vectors

x(119895)119896= 119891 (x(119895)

119896minus1) (9)

(3) Combine the x(119895)119896

vectors to obtain the predictionestimate at time 119896

xminus119896 = 12119899119909 2119899119909sum119895=1x(119895)119896 (10)

and the corresponding error covariance

Pminus119896 = 12119899119909 2119899119909sum119895=1 (x(119895)119896 minus xminus119896 ) (x(119895)119896 minus xminus119896 )1015840 +Q119908 (11)

(4) Use the completely specified measurement functionℎ(sdot) to transform the sigma points into y(119895)119896

vectors

y(119895)119896= ℎ (x(119895)

119896minus1) (12)

(5) Combine the y(119895)119896

vectors to obtain the predictedmeasurement at time 119896

y119896 = 12119899119909 2119899119909sum119895=1y(119895)119896 (13)

and the corresponding covariance

P119910119896= 12119899119909 2119899119909sum119895=1 (y(119895)119896 minus y119896) (y(119895)119896 minus y119896)1015840 +QV (14)

4 Discrete Dynamics in Nature and Society

(6) Estimate the cross covariance between xminus119896 and y119896

P119909119910119896= 12119899119909 2119899119909sum119895=1 (x(119895)119896 minus xminus119896 ) (y(119895)119896 minus y119896)1015840 (15)

(7) The measurement update of the state estimate can beperformed following the normal Kalman filter

K119896 = P119909119910119896(P119910119896)minus1

x+119896 = xminus119896 + K119896 (y119896 minus y119896) P+119896 = Pminus119896 minus K119896P

119910

119896K1015840119896 (16)

The algorithm above assumes that the motion andmeasurement functions are completely known In practicalapplications the motion and measurement functions maycontain parameters that are not specified andmay be randomwith known distributions In such cases the previous UKFalgorithm can not be directly applied

3 Improved Unscented Kalman Filter

In this section we propose an improved UKFmethod to dealwith the filtering problem in nonlinear dynamic systems withrandomparameters In the following wewill firstly introducethe improved unscented transformation which propagatesthe mean and covariance of a random vector through anonlinear function with random parameters

31 Improved Unscented Transformation Suppose that wehave a vector x isin R119899119909 with known mean x and covarianceP and its pdf is symmetric around its mean vector For anonlinear function y = ℎ(x 120579) the random parameter 120579 isinR119899120579 has known mean 120579 and covariance P120579 and also has asymmetric pdf In addition the parameter 120579 is uncorrelatedwith x

In the normal unscented transformation a set of sigmapoints x(119895) (2) is generated and each sigma point is trans-formed by the known nonlinear function ℎ(sdot) to approximatethe mean and covariance of y However when the functionℎ(sdot) is nonlinear with respect to random parameter 120579 the stepof transforming the sigma points is unrealizable

In order to deal with this problem we augment therandom parameter 120579 onto the state vector as follows

x = [x1015840 1205791015840]1015840 isin R119899119909+119899120579 (17)

The mean and covariance of the augment vector x are119864 (x) = x = [x1015840 1205791015840]1015840 Cov (x) = P = (P 0

0 P120579) (18)

Based on the augmentmodel the number of sigma pointsis enlarged to 2(119899119909 + 119899120579) The new sigma points are generatedas follows

x(119895) = x + x(119895)119896 119895 = 1 2 (119899119909 + 119899120579) (19)

where x(119895) = (radic2 (119899119909 + 119899120579) P)1015840119895 119895 = 1 119899119909 + 119899120579x(119899119909+119899120579+119895) = minus(radic2 (119899119909 + 119899120579) P)1015840119895 119895 = 1 119899119909 + 119899120579

(20)

The corresponding weighting coefficients are given as

W(119895) = 12 (119899119909 + 119899120579) 119895 = 1 2 (119899119909 + 119899120579) (21)

Because we regard both the vector x and the randomparameter 120579 as the unknown parameters in the augmentmodel each sigma point can be divided into two parts thatis

x(119895) = [x(119895) 120579(119895)] (22)

where x(119895) = x(119895)(1 119899119909) represents the sigma point of thevector x and 120579(119895) = x(119895)(119899119909 + 1 119899119909 + 119899120579) represents the sigmapoint of the parameter 120579 As a result the transformed sigmapoints can be computed as follows

y(119895) = ℎ (x(119895) 120579(119895)) 119895 = 1 2 (119899119909 + 119899120579) (23)

Let y denote the true mean of y The approximated mean ofthe improved unscented transformation is denoted as y andcomputed as follows

y = 2(119899119909+119899120579)sum119895=1

W(119895)y(119895) (24)

Let Py denote the true covariance of y The approximatedcovariance of the improved unscented transformation isdenoted as Py and computed as follows

Py = 2(119899119909+119899120579)sum119895=1

W(119895) (y(119895) minus y) (y(119895) minus y)1015840 (25)

The improved unscented transformations (24) and (25)incorporate the information of the random parameter 120579 sothey are intuitively more accurate than the normal unscentedtransformations (5) and (6) which ignore the randomnessof 120579 The theoretical analysis of the improved unscentedtransformations is given in the following theorems

Theorem3 For random vector xwith a symmetric pdf aroundits mean if the nonlinear function y = ℎ(x 120579) containsunknown random parameter 120579 the approximated mean of yvia the improved unscented transformation (24) matches thetrue mean of y correctly up to the third order

Proof We firstly expand y = ℎ(x 120579) in a Taylor series aroundx = [x1015840 1205791015840]1015840 as follows

y = ℎ (x 120579)= ℎ (x 120579) + 119863xℎ + 121198632xℎ + 131198633xℎ + sdot sdot sdot (26)

Discrete Dynamics in Nature and Society 5

where x = x minus x and the operation119863119896xℎ is defined as

119863119896xℎ = (119899119909+119899120579sum119895=1

x119895 120597ℎ120597x119895 1003816100381610038161003816100381610038161003816100381610038161003816x=x)119896 (27)

The mean of y can therefore be expanded as

y = 119864(ℎ (x 120579) + 119863xℎ + 121198632xℎ + 131198633xℎ + sdot sdot sdot)= ℎ (x 120579) + 119864(119863xℎ + 121198632xℎ + 131198633xℎ + sdot sdot sdot) (28)

We can see that

119864 (119863xℎ) = 119864(119899119909+119899120579sum119895=1

x119895 120597ℎ120597x119895 1003816100381610038161003816100381610038161003816100381610038161003816x=x)= 119899119909+119899120579sum119895=1

119864 (x119895) 120597ℎ120597x119895 1003816100381610038161003816100381610038161003816100381610038161003816x=x = 0(29)

due to 119864(x) = 0 Because both x and 120579 have a symmetric pdfaround the mean vector we can verify that

119864 (1198633xℎ) = 119864[[(119899119909+119899120579sum119895=1 x119895 120597ℎ120597x119895 1003816100381610038161003816100381610038161003816100381610038161003816x=x)3]] = 0 (30)

Similarly all of the odd terms in (28) will be equal to zerowhich leads to the simplification

y = ℎ (x 120579) + 119864( 121198632xℎ + 141198634xℎ + sdot sdot sdot) (31)

Now we compute the value of y by expanding each y(119895) in(24) in a Taylor series around x as follows

y = 12 (119899119909 + 119899120579)sdot 2(119899119909+119899120579)sum119895=1

(ℎ (x 120579) + 119863x(119895)ℎ + 121198632x(119895)ℎ + sdot sdot sdot) (32)

Because x(119895) = minusx(119899119909+119899120579+119895) (119895 = 1 119899119909 + 119899120579) for any integer119896 gt 0 we have2(119899119909+119899120579)sum119895=1

1198632119896+1x(119895) ℎ = 2(119899119909+119899120579)sum

119895=1

[[(119899119909+119899120579sum119894=1 x(119895)119894 120597ℎ120597x119895 1003816100381610038161003816100381610038161003816100381610038161003816x=x)2119896+1]]= 2(119899119909+119899120579)sum119895=1

[119899119909+119899120579sum119894=1

(x(119895)119894 )2119896+1 1205972119896+1ℎ120597x2119896+1119895 1003816100381610038161003816100381610038161003816100381610038161003816x=x]= 119899119909+119899120579sum119894=1

[[2(119899119909+119899120579)sum119895=1

(x(119895)119894 )2119896+1 1205972119896+1ℎ120597x2119896+1119895 1003816100381610038161003816100381610038161003816100381610038161003816x=x]] = 0(33)

Therefore all of the odd terms in (32) evaluate to zero and wehave

y= ℎ (x 120579)+ 12 (119899119909 + 119899120579) 2(119899119909+119899120579)sum

119895=1

( 121198632x(119895)ℎ + 141198634x(119895)ℎ + sdot sdot sdot) (34)

After some tedious calculation we can verify that

12 (119899119909 + 119899120579) 2(119899119909+119899120579)sum119895=1

121198632x(119895)ℎ = 12 119899119909+119899120579sum119894119895=1

P1198941198951205972ℎ120597x119894120597x119895 1003816100381610038161003816100381610038161003816100381610038161003816x=x= 12119864 (1198632xℎ) (35)

It can be seen that y (the approximated mean of y) matchesthe true mean of y correctly up to the third order

Theorem4 For random vector xwith a symmetric pdf aroundits mean if the nonlinear function y = ℎ(x 120579) containsunknown random parameter 120579 the approximated covarianceof y via the improved unscented transformation (25) matchesthe true covariance of y correctly up to the third order

Proof The true covariance of y is given as

Py = 119864 [(y minus y) (y minus y)1015840] (36)

By substituting (26) and (31) into (36) and using the same typeof reasoning in the proof ofTheorem 3 we have that all of theodd-powered terms evaluate to zero This leads to

Py = 119864 [119863xℎ (119863xℎ)1015840] + 119864[[119863xℎ (1198633xℎ)10158403

+ 1198632xℎ (1198632xℎ)101584022 + 1198633xℎ (119863xℎ)10158403 ]] + 119864[1198632xℎ2 ]sdot 119864 [1198632xℎ2 ]1015840 + sdot sdot sdot(37)

The first term on the right side of the equation above can bewritten as

119864 [119863xℎ (119863xℎ)1015840] = HPH1015840 (38)

whereH is the partial derivative matrix at x = x

6 Discrete Dynamics in Nature and Society

On the other hand after some tedious calculation theapproximate covariance Py in (25) can be written as

Py = 12 (119899119909 + 119899120579) 2(119899119909+119899120579)sum119895=1

(119863x(119895)ℎ) (119863x(119895)ℎ)1015840 +HOT

= 12 (119899119909 + 119899120579)sdot 2(119899119909+119899120579)sum119895=1

119899119909+119899120579sum119894119896=1

(x(119895)119894 120597ℎ (x)120597x119894 )(x(119895)119896 120597ℎ (x)120597x119896 )1015840+HOT(39)

whereHOTmeans higher-order terms (ie terms to the forthpower and higher) Now recall that x(119895)119894 = minusx(119899119909+119899120579+119895)119894 andx(119895)119896 = minusx(119899119909+119899120579+119895)119896 for 119895 = 1 119899119909 + 119899120579 therefore thecovariance approximation becomes

Py

= 1119899119909 + 119899120579 119899119909+119899120579sum119895=1 119899119909+119899120579sum119894119896=1 (x(119895)119894 120597ℎ (x)120597x119894 )(x(119895)119896 120597ℎ (x)120597x119896 )1015840+HOT = HPH1015840 +HOT(40)

Comparing (40) with the true covariance of y from (37) wesee that Py in (25) approximates the true covariance of y upto the third order

32 Improved Unscented Kalman Filter The improvedunscented transformation developed in the previoussubsection can be generalized to give the improvedunscented Kalman filter for the discrete-time nonlinearsystem which contains random parameters that is

x119896 = 119891 (x119896minus1 120579) + w119896y119896 = ℎ (x119896 120579) + k119896 (41)

where the motion function 119891(sdot) and the measurement func-tion ℎ(sdot) are nonlinear with respect to not only the state vectorx119896 but also the random parameter 120579 The other notations arethe same as that of model (1) Assume that the parameter 120579 isinR119899120579 has knownmean 120579 and covarianceP120579 and is uncorrelatedwith the state vector x119896 as well as the noise vectors w119896 and k119896

Following the augment method of the improvedunscented transformation we give the augment model of thediscrete-time nonlinear system (41) as follows

x119896 = 119891 (x119896minus1) + w119896y119896 = ℎ (x119896) + k119896 (42)

where

x119896 = [x1015840119896 1205791015840]1015840 w119896 = [w1015840119896 01015840]1015840 (43)

With the initial condition (7) the improved unscentedKalman filter can be initialized as followsx+0 = [(x+0 )1015840 1205791015840]1015840

P+0 = (P+0 0

0 P120579) (44)

Suppose that at time step 119896 x+119896minus1 and P+119896minus1 are given For thesake of brevity we denote 119899 = 119899119909 + 119899120579 The improved UKFalgorithm can be summarized in Algorithm 5

Algorithm 5 (the improved unscented Kalman filter)

(1) Choose 2 119899 sigma points x(119895)119896minus1 as specified in (19) withappropriate changes that isx(119895)119896minus1 = x+119896minus1 + x(119895)119896minus1 119895 = 1 2 119899x(119895)119896minus1 = (radic 119899P+

119896minus1)1015840119895 119895 = 1 119899x( 119899+119895)119896minus1 = minusx(119895)119896minus1 119895 = 1 119899

(45)

(2) Transform the sigma points x(119895)119896minus1 into x(119895)119896 as followsx(119895)119896 = 119891 (x(119895)119896minus1 120579(119895)119896minus1) (46)

where x(119895)119896minus1= x(119895)119896minus1(1 119899119909) and 120579(119895)119896minus1 = x(119895)119896minus1(119899119909 + 1 119899)

(3) Combine the x(119895)119896 vectors to obtain the predictionestimate at time

xminus

119896 = 12 119899 2 119899sum119895=1

x(119895)119896 (47)

and the corresponding error covariance

Pminus119896 = 12 119899 2 119899sum119895=1

(x(119895)119896 minus xminus119896)(x(119895)119896 minus xminus119896)1015840 + Q119908 (48)

where Q119908 = ( Q119908 00 0 )

(4) Transform the sigma points x(119895)119896minus1 into y(119895)119896 vectorsy(119895)119896 = ℎ (x(119895)119896minus1 120579(119895)119896minus1) (49)

(5) Combine the y(119895)119896 vectors to obtain the predictedmeasurement at time 119896y119896 = 12 119899 2 119899sum

119895=1

y(119895)119896 (50)

Discrete Dynamics in Nature and Society 7

and the corresponding covariance

P119910119896= 12 119899 2 119899sum119895=1

(y(119895)119896 minus y119896)(y(119895)119896 minus y119896)1015840 +QV (51)

(6) Estimate the cross covariance between xminus119896 and y119896P119909119910119896= 12 119899 2 119899sum119895=1

(x(119895)119896 minus xminus119896)(y(119895)119896 minus y119896)1015840 (52)

(7) The measurement update of the state estimate can beperformed as follows

K119896 = P119909119910119896(P119910119896)minus1 x+119896 = xminus119896 + K119896 (y119896 minus y119896)

P+119896 = Pminus119896 minus K119896P119910

119896K1015840119896 (53)

The proposed improved UKF method provides a sub-optimal solution for the stochastic filtering problem of anonlinear discrete-time dynamic system which contains ran-dom parameters The main difference between the improvedUKF algorithm and the normal UKF algorithm lies in theconstruction of the sigma pointsThe number of sigma pointsof the improved UKF algorithm is larger than that of thenormal UKF algorithm so generally the proposed improvedUKF method requires higher computational complexityHowever the advantage of the improved UKF method isthe increased accuracy of the state estimation which can beverified by the numerical results in the following section

4 An Application to MobileSource Localization

Consider a wireless sensor network with 119899 passive sensornodes distributed on a 2D plane to localize one movingsource Let the state vector of the mobile source be x119896 =[u1015840119896 k1015840119896]1015840 isin R4 which includes the position and velocitycomponents

Without loss of generality let the first sensor be thereference The TDOA measurement model [20] betweensensor pair 119894 (119894 = 2 119899) and 1 is

1199051198941 (u119896) = 10038171003817100381710038171003817u119896 minus s011989410038171003817100381710038171003817119888 minus 10038171003817100381710038171003817u119896 minus s01

10038171003817100381710038171003817119888 + Δ119905119894119896 (54)

where s0119894 is the true position of the 119894th sensor 1199030119894 (u119896) =u119896 minus s0119894 is the true distance from the source to the 119894thsensor and 119888 is the signal propagation speedThe vectorΔt119896 =[Δ1199052119896 Δ119905119899119896]1015840 is zero-meanGaussian noisewith covarianceQ119905 In simplicity (54) is expressed in the vector form

t119896 = [11990521 (u119896) 1199051198991 (u119896)] (55)

In practice the true sensor positions s0 = [s010158401 s010158404 ]1015840are not known and only their nominal values s =

Table 1 Nominal positions (in meters) of sensors

Sensor number 119894 119909119894 1199101198941 100 1002 100 minus1003 minus100 minus1004 minus100 100

[s10158401 s10158402 s10158404]1015840 are available for source localization Moreprecisely s0119894 and s119894 are related by

s0119894 = s119894 + Δs119894 (56)

where Δs119894 is the sensor position noise and the correspondingnoise vector Δs = [Δs10158401 Δs10158402 Δs1015840119899]1015840 is assumed to be zero-mean Gaussian with covariance matrixQ119904

Because the TDOAmeasurement vector (55) is nonlinearwith respect to the state vector x119896 as well as the true sensorlocations s0 the mobile source localization problem can bedescribed as the following discrete-time nonlinear system

t119896 = f (x119896 s0) + Δt119896 (57)

where s0 can be regarded as unknown random parametersThus we can apply the proposed improved UKF method todynamically estimate the state vector

In order to show the performance of the proposedimproved UKF method we apply it to two localization sce-narios and compare it with the normal UKF method whichignores the random uncertainties in the sensor locationsthrough Monte Carlo simulations

41 Scenario 1 This simulation scenario contains 119899 = 4sensors and their nominal positions are given in Table 1 Themotion model of the mobile source is assumed to be a nearlyconstant velocity model as

x119896 =(1 0 001 00 1 0 0010 0 1 00 0 0 1 ) x119896minus1

+(001 00 00101 00 01)w119896minus1(58)

where w119896minus1 is zero-mean Gaussian noise with covariancematrix Q119908 = 001I The initial value of the state x0 isa Gaussian vector with zero mean and covariance P+0 =diag100 100 001 001 The trajectory of the mobile sourcein 100 time instants is shown in Figure 1

The TDOA measurement covariance Q119905 = (05119888)2Twhere T is the (119899 minus 1) times (119899 minus 1)matrix with 1 in the diagonalelements and 05 otherwise The localization accuracy isevaluated by the root of mean squared error which is defined

8 Discrete Dynamics in Nature and Society

minus5

minus48

minus46

minus44

minus42

minus4

minus38

minus36

y-ax

is (m

)

13 135 14 145 15 155 16 165 17 175125x-axis (m)

Figure 1 The trajectory of the mobile source in Scenario 1

The normal UKFThe improved UKF

0

01

02

03

04

05

06

07

08

09

1

The R

MSE

of m

obile

sour

ce lo

caliz

atio

n

10 20 30 40 50 60 70 80 90 1000k

Figure 2 Comparison of the mobile localization accuracy with lowlevel sensor position uncertainties in Scenario 1

as RMSE(u119896) = radicsum119871119897=1 u119896 minus u1198962119871 where u119896 = x119896(1 2) isthe estimate of the source position at ensemble 119897 and 119871 = 1000is the number of ensemble runs

In theMonte Carlo simulations we consider three sensorposition uncertainty levels which are referred to as the lowlevel with 120590119904 = 01 the moderate level with 120590119904 = 1 and thehigh level with 120590119904 = 2

Figure 2 plots the simulation results with time instantvarying from 1 to 100 when the sensor position uncertaintylevel is low It is evident from Figure 2 that RMSE of theproposed improved UKF method (represented by dashedline) is smaller than that of the normal UKF approach(represented by solid line) which ignores the sensor positionuncertainties

Figure 3 plots the RMSEs of the two methods in the caseof moderate sensor position uncertainty level that is 120590119904 =1 We can see that both of them are generally larger than

The normal UKFThe improved UKF

0

01

02

03

04

05

06

07

The R

MSE

of m

obile

sour

ce lo

caliz

atio

n

10 20 30 40 50 60 70 80 90 1000k

Figure 3 Comparison of themobile localization accuracywithmildlevel sensor position uncertainties in Scenario 1

The normal UKFThe improved UKF

02

04

06

08

1

12

14

16

18

2

22

The R

MSE

of m

obile

sour

ce lo

caliz

atio

n

10 20 30 40 50 60 70 80 90 1000k

Figure 4 Comparison of themobile localization accuracywith highlevel sensor position uncertainties in Scenario 1

the corresponding RMSEs in Figure 2 As we expected theperformance of the proposed improvedUKFmethod ismuchbetter than that of the normal UKF approach In additionthe improvement is larger than that of the low level case inFigure 2

Finally we take the high sensor position uncertainty levelthat is120590119904 = 2The numerical simulation results are illustratedin Figure 4 which indicates that the RMSE of the normalUKF approach is nearly twice asmuch as that of the proposedimproved UKF method Comparing the results in Figures 2ndash4 we can see that the influence of the random sensor positionuncertainties on the localization performance is quite evidentin Scenario 1 The proposed improved UKF method can

Discrete Dynamics in Nature and Society 9

The normal UKFThe improved UKF

0

005

01

015

02

025

03

035

04

045

The R

MSE

of m

obile

sour

ce lo

caliz

atio

n

10 20 30 40 50 60 70 80 90 1000k

Figure 5 Comparison of the mobile localization accuracy with lowlevel sensor position uncertainties in Scenario 2

efficiently decrease the RMSE especially when the sensorposition uncertainty level is high

42 Scenario 2 In this simulation scenario we considerthe sensor geometry is random and the number of sensors119899 = 4 The true position of each sensor is uniformlydistributed in a square of [minus100 100]m times [minus100 100]mat each ensemble in Monte Carlo simulations The TDOAmeasurement covariance Q119905 = (01119888)2T The motion modeland the initial value of the state are generally the same as thoseof Scenario 1

In the first case the sensor position uncertainty levelis considerably small that is 120590119904 = 01 Figure 5 plots theRMSE of the normal UKF approach and that of the proposedimproved UKF method with time instant varying from 1 to100 which are calculated from 119871 = 1000Monte Carlo runsIt can be seen that the RMSE of the proposed improved UKFmethod is smaller than that of the normal UKF approach

Thenwe consider amoderate sensor position uncertaintylevel that is 120590119904 = 1 Figure 6 plots the RMSEs of the twomethods in this case where we can see that the performanceof the proposed improvedUKFmethod is still better than thatof the normal UKF approach with a larger improvement as inFigure 5

For the high sensor position uncertainty level that is120590119904 = 2 the simulation results are illustrated in Figure 7which shows that the RMSE of the normal UKF approach isconsiderably larger than that of the proposed improved UKFmethod

The Monte Carlo simulation results in Figures 5ndash7 revealthat the RMSEs of the proposed improved UKF are consis-tently smaller than those of the normal UKF approach whichcorroborates that the proposed improved UKF method cangenerally improve the estimation accuracy of the nonlineardynamic system when it contains random parameters

The normal UKFThe improved UKF

0

02

04

06

08

1

12

14

16

18

2

The R

MSE

of m

obile

sour

ce lo

caliz

atio

n

10 20 30 40 50 60 70 80 90 1000k

Figure 6 Comparison of themobile localization accuracywithmildlevel sensor position uncertainties in Scenario 2

The normal UKFThe improved UKF

0

05

1

15

2

25

3

35

The R

MSE

of m

obile

sour

ce lo

caliz

atio

n

10 20 30 40 50 60 70 80 90 1000k

Figure 7 Comparison of themobile localization accuracy with highlevel sensor position uncertainties in Scenario 2

5 Conclusions

In this paper an improved unscented Kalman filteringmethod is explored for discrete nonlinear dynamic systemswith randomparameters By augmenting the randomparam-eters into the state vector we enlarge the number of sigmapoints in the improved unscented transformation Theo-retical analysis indicates that the approximated mean andcovariance via the improved unscented transformationmatchthe true values correctly up to the third order of Taylor seriesexpansion An application to the mobile source localizationusing TDOA measurements is provided where the sensorpositions suffer from randomuncertaintiesTheMonte Carlo

10 Discrete Dynamics in Nature and Society

simulation results show that the proposed improved UKFmethod can considerably improve the estimation accuracy

Competing Interests

The authors declare that there are no competing interestsregarding the publication of this paper

Acknowledgments

This work was supported by the Fundamental ResearchFunds for the Central Universities Southwestern Universityof Finance and Economics under Grant no JBK1507059and Southwest University for Nationalities under Grant no2016NGJPY05

References

[1] R E Kalman ldquoA new approach to linear filtering and predictionproblemsrdquo Journal of Basic Engineering vol 82 no 1 pp 35ndash451960

[2] G Bolzon R Fedele and G Maier ldquoParameter identificationof a cohesive crack model by Kalman filterrdquo Computer Methodsin Applied Mechanics and Engineering vol 191 no 25-26 pp2847ndash2871 2002

[3] M Ball and A Wood ldquoTrend growth in post-1850 Britisheconomic history the Kalman filter and historical judgmentrdquoJournal of the Royal Statistical Society vol 45 no 2 Article ID143152 pp 143ndash152 1996

[4] W L De Koning ldquoOptimal estimation of linear discrete-timesystems with stochastic parametersrdquo Automatica vol 20 pp113ndash115 1984

[5] B Sinopoli L Schenato M Franceschetti K Poolla M IJordan and S S Sastry ldquoKalman filtering with intermittentobservationsrdquo IEEE Transactions on Automatic Control vol 49no 9 pp 1453ndash1464 2004

[6] Y Luo Y Zhu D Luo J Zhou E Song and DWang ldquoGloballyoptimal multisensor distributed random parameter matriceskalman filtering fusion with applicationsrdquo Sensors vol 8 no 12pp 8086ndash8103 2008

[7] S Chen Y Li G Qi and A Sheng ldquoAdaptive Kalman esti-mation in target tracking mixed with random one-step delaysstochastic-bias measurements and missing measurementsrdquoDiscrete Dynamics in Nature and Society vol 2013 Article ID716915 14 pages 2013

[8] S Chen Z Zhou and S Li ldquoAn efficient estimate and forecastof the implied volatility surface a nonlinear Kalman filterapproachrdquo Economic Modelling vol 58 pp 655ndash664 2016

[9] S Bolognani L Tubiana and M Zigliotto ldquoExtended Kalmanfilter tuning in sensorless PMSM drivesrdquo IEEE Transactions onIndustry Applications vol 39 no 6 pp 1741ndash1747 2003

[10] J J LaViola Jr ldquoA comparison of unscented and extendedkalman filtering for estimating quaternion motionrdquo in Proceed-ings of the AmericanControl Conference pp 2435ndash2440DenverColo USA June 2003

[11] X Qu and L Xie ldquoRecursive source localisation by time dif-ference of arrival sensor networks with sensor position uncer-taintyrdquo IET Control Theory and Applications vol 8 no 18 pp2305ndash2315 2014

[12] S J Julier J K Uhlmann and H F Durrant-Whyte ldquoNewapproach for filtering nonlinear systemsrdquo in Proceedings of theAmerican Control Conference pp 1628ndash1632 June 1995

[13] S J Julier and J K Uhlmann ldquoUnscented filtering and nonlin-ear estimationrdquo Proceedings of the IEEE vol 92 no 3 pp 401ndash422 2004

[14] Q Pan F Yang and L Ye ldquoSurvey of a kind of non linear filtersUKFrdquo Control and Decision vol 20 no 5 pp 481ndash489 2005

[15] C Eberle and C Ament ldquoThe Unscented Kalman Filter esti-mates the plasma insulin from glucose measurementrdquo BioSys-tems vol 103 no 1 pp 67ndash72 2011

[16] S J Julier ldquoSome relations between extended and unscentedKalman filtersrdquo IEEE Transactions on Signal Processing vol 60no 2 pp 545ndash555 2012

[17] HM T Menegaz J Y Ishihara G A Borges and A N VargasldquoA systematization of the unscented Kalman filter theoryrdquo IEEETransactions on Automatic Control vol 60 no 10 pp 2583ndash2598 2015

[18] D Simon Optimal State Estimation Kalman H Infinity andNonlinear Approaches Wiley-Interscience 2006

[19] KW Lui F K Chan andH C So ldquoSemidefinite programmingapproach for range-difference based source localizationrdquo IEEETransactions on Signal Processing vol 57 no 4 pp 1630ndash16332009

[20] X Qu and L Xie ldquoAn efficient convex constrained weightedleast squares source localization algorithm based on TDOAmeasurementsrdquo Signal Processing vol 119 pp 142ndash152 2016

[21] G Yanni and W Qi ldquoThe nonsequential fusion method forlocalization from unscented kalman filter by multistation arraybuoysrdquo Discrete Dynamics in Nature and Society vol 2016Article ID 7670609 8 pages 2016

[22] A H Jazwinski Stochastic Processes and Filtering Theory Aca-demic Press Cambridge Mass USA 1970

[23] F Daum ldquoNonlinear filters beyond the kalman filterrdquo IEEEAerospace and Electronic Systems Magazine vol 20 no 8 pp57ndash69 2005

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Function Spaces

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Discrete Dynamics in Nature and Society 3

21 Unscented Transformation Suppose that we have a vectorx isin R119899119909 with known mean x and covariance P and its pdf issymmetric around its mean vector For a nonlinear functiony = ℎ(x) we want to approximate the mean and covarianceof y The first step of the unscented transformation is tofind a set of deterministic vectors called sigma points whoseensemble weighted mean and covariance are equal to x andP respectively which can be realized as follows

x(119895) = x + x(119895) 119895 = 1 2119899119909 (2)

where

x(119895) = (radic119899119909P)1015840119895 119895 = 1 119899119909

x(119899119909+119895) = minus(radic119899119909P)1015840119895 119895 = 1 119899119909 (3)

radic119899119909P is the matrix square root of 119899119909P such that(radic119899119909P)1015840radic119899119909P = 119899119909P and (radic119899119909P)119895 is the 119895th row of radic119899119909PThe weighting coefficients are given as

W(119895) = 12119899119909 119895 = 1 2119899119909 (4)

Then we transform each sigma point in (2) using thenonlinear function ℎ(sdot) and approximate the mean of y bytaking the weighted sum of the transformed sigma pointsSpecifically let y denote the truemean of yThe approximatedmean of y is denoted as y and computed as follows

y = 2119899119909sum119895=1

W(119895)y(119895) y(119895) = ℎ (x(119895)) 119895 = 1 2119899119909 (5)

Let Py denote the true covariance of y The approximatedcovariance of y is denoted as Py and computed as follows

Py = 2119899119909sum119895=1

W(119895) (y(119895) minus y) (y(119895) minus y)1015840 (6)

Theunscented transformations (5) and (6) aremore accu-rate than the linearizationmethod for propagatingmeans andcovariances of nonlinear functions which is summarized inthe following proposition

Proposition 1 (see [18]) For random vector x with a sym-metric pdf around its mean if the nonlinear function y =ℎ(x) is completely specified then the approximated mean andcovariance of y via the unscented transformations (5)-(6)match the true mean and covariance of y correctly up to thethird order

22 Unscented Kalman Filter The unscented transformationcan be generalized to give the unscented Kalman filter whichkeeps the structure of the Kalman filter that includes oneprediction (or a time update) step and one correction (or ameasurement update) step

For the discrete nonlinear system (1) the unscentedKalman filter is initialized as follows

x+0 = 119864 (x0) P+0 = 119864 [(x0 minus x+0 ) (x0 minus x+0 )1015840] (7)

Suppose that at time step 119896 x+119896minus1 andP+119896minus1 are givenTheUKFalgorithm can be summarized in Algorithm 2

Algorithm 2 (the unscented Kalman filter)

(1) Choose the sigma points x(119895)119896minus1

as specified in (2) withappropriate changes that is

x(119895)119896minus1= x+119896minus1 + x(119895)

119896minus1 119895 = 1 2119899119909

x(119895)119896minus1= (radic119899119909P+119896minus1)1015840119895 119895 = 1 119899119909

x(119899119909+119895)119896minus1

= minus (radic119899119909P+119896minus1)1015840119895 119895 = 1 119899119909(8)

(2) Use the completely specified motion function 119891(sdot) totransform the sigma points into x(119895)

119896vectors

x(119895)119896= 119891 (x(119895)

119896minus1) (9)

(3) Combine the x(119895)119896

vectors to obtain the predictionestimate at time 119896

xminus119896 = 12119899119909 2119899119909sum119895=1x(119895)119896 (10)

and the corresponding error covariance

Pminus119896 = 12119899119909 2119899119909sum119895=1 (x(119895)119896 minus xminus119896 ) (x(119895)119896 minus xminus119896 )1015840 +Q119908 (11)

(4) Use the completely specified measurement functionℎ(sdot) to transform the sigma points into y(119895)119896

vectors

y(119895)119896= ℎ (x(119895)

119896minus1) (12)

(5) Combine the y(119895)119896

vectors to obtain the predictedmeasurement at time 119896

y119896 = 12119899119909 2119899119909sum119895=1y(119895)119896 (13)

and the corresponding covariance

P119910119896= 12119899119909 2119899119909sum119895=1 (y(119895)119896 minus y119896) (y(119895)119896 minus y119896)1015840 +QV (14)

4 Discrete Dynamics in Nature and Society

(6) Estimate the cross covariance between xminus119896 and y119896

P119909119910119896= 12119899119909 2119899119909sum119895=1 (x(119895)119896 minus xminus119896 ) (y(119895)119896 minus y119896)1015840 (15)

(7) The measurement update of the state estimate can beperformed following the normal Kalman filter

K119896 = P119909119910119896(P119910119896)minus1

x+119896 = xminus119896 + K119896 (y119896 minus y119896) P+119896 = Pminus119896 minus K119896P

119910

119896K1015840119896 (16)

The algorithm above assumes that the motion andmeasurement functions are completely known In practicalapplications the motion and measurement functions maycontain parameters that are not specified andmay be randomwith known distributions In such cases the previous UKFalgorithm can not be directly applied

3 Improved Unscented Kalman Filter

In this section we propose an improved UKFmethod to dealwith the filtering problem in nonlinear dynamic systems withrandomparameters In the following wewill firstly introducethe improved unscented transformation which propagatesthe mean and covariance of a random vector through anonlinear function with random parameters

31 Improved Unscented Transformation Suppose that wehave a vector x isin R119899119909 with known mean x and covarianceP and its pdf is symmetric around its mean vector For anonlinear function y = ℎ(x 120579) the random parameter 120579 isinR119899120579 has known mean 120579 and covariance P120579 and also has asymmetric pdf In addition the parameter 120579 is uncorrelatedwith x

In the normal unscented transformation a set of sigmapoints x(119895) (2) is generated and each sigma point is trans-formed by the known nonlinear function ℎ(sdot) to approximatethe mean and covariance of y However when the functionℎ(sdot) is nonlinear with respect to random parameter 120579 the stepof transforming the sigma points is unrealizable

In order to deal with this problem we augment therandom parameter 120579 onto the state vector as follows

x = [x1015840 1205791015840]1015840 isin R119899119909+119899120579 (17)

The mean and covariance of the augment vector x are119864 (x) = x = [x1015840 1205791015840]1015840 Cov (x) = P = (P 0

0 P120579) (18)

Based on the augmentmodel the number of sigma pointsis enlarged to 2(119899119909 + 119899120579) The new sigma points are generatedas follows

x(119895) = x + x(119895)119896 119895 = 1 2 (119899119909 + 119899120579) (19)

where x(119895) = (radic2 (119899119909 + 119899120579) P)1015840119895 119895 = 1 119899119909 + 119899120579x(119899119909+119899120579+119895) = minus(radic2 (119899119909 + 119899120579) P)1015840119895 119895 = 1 119899119909 + 119899120579

(20)

The corresponding weighting coefficients are given as

W(119895) = 12 (119899119909 + 119899120579) 119895 = 1 2 (119899119909 + 119899120579) (21)

Because we regard both the vector x and the randomparameter 120579 as the unknown parameters in the augmentmodel each sigma point can be divided into two parts thatis

x(119895) = [x(119895) 120579(119895)] (22)

where x(119895) = x(119895)(1 119899119909) represents the sigma point of thevector x and 120579(119895) = x(119895)(119899119909 + 1 119899119909 + 119899120579) represents the sigmapoint of the parameter 120579 As a result the transformed sigmapoints can be computed as follows

y(119895) = ℎ (x(119895) 120579(119895)) 119895 = 1 2 (119899119909 + 119899120579) (23)

Let y denote the true mean of y The approximated mean ofthe improved unscented transformation is denoted as y andcomputed as follows

y = 2(119899119909+119899120579)sum119895=1

W(119895)y(119895) (24)

Let Py denote the true covariance of y The approximatedcovariance of the improved unscented transformation isdenoted as Py and computed as follows

Py = 2(119899119909+119899120579)sum119895=1

W(119895) (y(119895) minus y) (y(119895) minus y)1015840 (25)

The improved unscented transformations (24) and (25)incorporate the information of the random parameter 120579 sothey are intuitively more accurate than the normal unscentedtransformations (5) and (6) which ignore the randomnessof 120579 The theoretical analysis of the improved unscentedtransformations is given in the following theorems

Theorem3 For random vector xwith a symmetric pdf aroundits mean if the nonlinear function y = ℎ(x 120579) containsunknown random parameter 120579 the approximated mean of yvia the improved unscented transformation (24) matches thetrue mean of y correctly up to the third order

Proof We firstly expand y = ℎ(x 120579) in a Taylor series aroundx = [x1015840 1205791015840]1015840 as follows

y = ℎ (x 120579)= ℎ (x 120579) + 119863xℎ + 121198632xℎ + 131198633xℎ + sdot sdot sdot (26)

Discrete Dynamics in Nature and Society 5

where x = x minus x and the operation119863119896xℎ is defined as

119863119896xℎ = (119899119909+119899120579sum119895=1

x119895 120597ℎ120597x119895 1003816100381610038161003816100381610038161003816100381610038161003816x=x)119896 (27)

The mean of y can therefore be expanded as

y = 119864(ℎ (x 120579) + 119863xℎ + 121198632xℎ + 131198633xℎ + sdot sdot sdot)= ℎ (x 120579) + 119864(119863xℎ + 121198632xℎ + 131198633xℎ + sdot sdot sdot) (28)

We can see that

119864 (119863xℎ) = 119864(119899119909+119899120579sum119895=1

x119895 120597ℎ120597x119895 1003816100381610038161003816100381610038161003816100381610038161003816x=x)= 119899119909+119899120579sum119895=1

119864 (x119895) 120597ℎ120597x119895 1003816100381610038161003816100381610038161003816100381610038161003816x=x = 0(29)

due to 119864(x) = 0 Because both x and 120579 have a symmetric pdfaround the mean vector we can verify that

119864 (1198633xℎ) = 119864[[(119899119909+119899120579sum119895=1 x119895 120597ℎ120597x119895 1003816100381610038161003816100381610038161003816100381610038161003816x=x)3]] = 0 (30)

Similarly all of the odd terms in (28) will be equal to zerowhich leads to the simplification

y = ℎ (x 120579) + 119864( 121198632xℎ + 141198634xℎ + sdot sdot sdot) (31)

Now we compute the value of y by expanding each y(119895) in(24) in a Taylor series around x as follows

y = 12 (119899119909 + 119899120579)sdot 2(119899119909+119899120579)sum119895=1

(ℎ (x 120579) + 119863x(119895)ℎ + 121198632x(119895)ℎ + sdot sdot sdot) (32)

Because x(119895) = minusx(119899119909+119899120579+119895) (119895 = 1 119899119909 + 119899120579) for any integer119896 gt 0 we have2(119899119909+119899120579)sum119895=1

1198632119896+1x(119895) ℎ = 2(119899119909+119899120579)sum

119895=1

[[(119899119909+119899120579sum119894=1 x(119895)119894 120597ℎ120597x119895 1003816100381610038161003816100381610038161003816100381610038161003816x=x)2119896+1]]= 2(119899119909+119899120579)sum119895=1

[119899119909+119899120579sum119894=1

(x(119895)119894 )2119896+1 1205972119896+1ℎ120597x2119896+1119895 1003816100381610038161003816100381610038161003816100381610038161003816x=x]= 119899119909+119899120579sum119894=1

[[2(119899119909+119899120579)sum119895=1

(x(119895)119894 )2119896+1 1205972119896+1ℎ120597x2119896+1119895 1003816100381610038161003816100381610038161003816100381610038161003816x=x]] = 0(33)

Therefore all of the odd terms in (32) evaluate to zero and wehave

y= ℎ (x 120579)+ 12 (119899119909 + 119899120579) 2(119899119909+119899120579)sum

119895=1

( 121198632x(119895)ℎ + 141198634x(119895)ℎ + sdot sdot sdot) (34)

After some tedious calculation we can verify that

12 (119899119909 + 119899120579) 2(119899119909+119899120579)sum119895=1

121198632x(119895)ℎ = 12 119899119909+119899120579sum119894119895=1

P1198941198951205972ℎ120597x119894120597x119895 1003816100381610038161003816100381610038161003816100381610038161003816x=x= 12119864 (1198632xℎ) (35)

It can be seen that y (the approximated mean of y) matchesthe true mean of y correctly up to the third order

Theorem4 For random vector xwith a symmetric pdf aroundits mean if the nonlinear function y = ℎ(x 120579) containsunknown random parameter 120579 the approximated covarianceof y via the improved unscented transformation (25) matchesthe true covariance of y correctly up to the third order

Proof The true covariance of y is given as

Py = 119864 [(y minus y) (y minus y)1015840] (36)

By substituting (26) and (31) into (36) and using the same typeof reasoning in the proof ofTheorem 3 we have that all of theodd-powered terms evaluate to zero This leads to

Py = 119864 [119863xℎ (119863xℎ)1015840] + 119864[[119863xℎ (1198633xℎ)10158403

+ 1198632xℎ (1198632xℎ)101584022 + 1198633xℎ (119863xℎ)10158403 ]] + 119864[1198632xℎ2 ]sdot 119864 [1198632xℎ2 ]1015840 + sdot sdot sdot(37)

The first term on the right side of the equation above can bewritten as

119864 [119863xℎ (119863xℎ)1015840] = HPH1015840 (38)

whereH is the partial derivative matrix at x = x

6 Discrete Dynamics in Nature and Society

On the other hand after some tedious calculation theapproximate covariance Py in (25) can be written as

Py = 12 (119899119909 + 119899120579) 2(119899119909+119899120579)sum119895=1

(119863x(119895)ℎ) (119863x(119895)ℎ)1015840 +HOT

= 12 (119899119909 + 119899120579)sdot 2(119899119909+119899120579)sum119895=1

119899119909+119899120579sum119894119896=1

(x(119895)119894 120597ℎ (x)120597x119894 )(x(119895)119896 120597ℎ (x)120597x119896 )1015840+HOT(39)

whereHOTmeans higher-order terms (ie terms to the forthpower and higher) Now recall that x(119895)119894 = minusx(119899119909+119899120579+119895)119894 andx(119895)119896 = minusx(119899119909+119899120579+119895)119896 for 119895 = 1 119899119909 + 119899120579 therefore thecovariance approximation becomes

Py

= 1119899119909 + 119899120579 119899119909+119899120579sum119895=1 119899119909+119899120579sum119894119896=1 (x(119895)119894 120597ℎ (x)120597x119894 )(x(119895)119896 120597ℎ (x)120597x119896 )1015840+HOT = HPH1015840 +HOT(40)

Comparing (40) with the true covariance of y from (37) wesee that Py in (25) approximates the true covariance of y upto the third order

32 Improved Unscented Kalman Filter The improvedunscented transformation developed in the previoussubsection can be generalized to give the improvedunscented Kalman filter for the discrete-time nonlinearsystem which contains random parameters that is

x119896 = 119891 (x119896minus1 120579) + w119896y119896 = ℎ (x119896 120579) + k119896 (41)

where the motion function 119891(sdot) and the measurement func-tion ℎ(sdot) are nonlinear with respect to not only the state vectorx119896 but also the random parameter 120579 The other notations arethe same as that of model (1) Assume that the parameter 120579 isinR119899120579 has knownmean 120579 and covarianceP120579 and is uncorrelatedwith the state vector x119896 as well as the noise vectors w119896 and k119896

Following the augment method of the improvedunscented transformation we give the augment model of thediscrete-time nonlinear system (41) as follows

x119896 = 119891 (x119896minus1) + w119896y119896 = ℎ (x119896) + k119896 (42)

where

x119896 = [x1015840119896 1205791015840]1015840 w119896 = [w1015840119896 01015840]1015840 (43)

With the initial condition (7) the improved unscentedKalman filter can be initialized as followsx+0 = [(x+0 )1015840 1205791015840]1015840

P+0 = (P+0 0

0 P120579) (44)

Suppose that at time step 119896 x+119896minus1 and P+119896minus1 are given For thesake of brevity we denote 119899 = 119899119909 + 119899120579 The improved UKFalgorithm can be summarized in Algorithm 5

Algorithm 5 (the improved unscented Kalman filter)

(1) Choose 2 119899 sigma points x(119895)119896minus1 as specified in (19) withappropriate changes that isx(119895)119896minus1 = x+119896minus1 + x(119895)119896minus1 119895 = 1 2 119899x(119895)119896minus1 = (radic 119899P+

119896minus1)1015840119895 119895 = 1 119899x( 119899+119895)119896minus1 = minusx(119895)119896minus1 119895 = 1 119899

(45)

(2) Transform the sigma points x(119895)119896minus1 into x(119895)119896 as followsx(119895)119896 = 119891 (x(119895)119896minus1 120579(119895)119896minus1) (46)

where x(119895)119896minus1= x(119895)119896minus1(1 119899119909) and 120579(119895)119896minus1 = x(119895)119896minus1(119899119909 + 1 119899)

(3) Combine the x(119895)119896 vectors to obtain the predictionestimate at time

xminus

119896 = 12 119899 2 119899sum119895=1

x(119895)119896 (47)

and the corresponding error covariance

Pminus119896 = 12 119899 2 119899sum119895=1

(x(119895)119896 minus xminus119896)(x(119895)119896 minus xminus119896)1015840 + Q119908 (48)

where Q119908 = ( Q119908 00 0 )

(4) Transform the sigma points x(119895)119896minus1 into y(119895)119896 vectorsy(119895)119896 = ℎ (x(119895)119896minus1 120579(119895)119896minus1) (49)

(5) Combine the y(119895)119896 vectors to obtain the predictedmeasurement at time 119896y119896 = 12 119899 2 119899sum

119895=1

y(119895)119896 (50)

Discrete Dynamics in Nature and Society 7

and the corresponding covariance

P119910119896= 12 119899 2 119899sum119895=1

(y(119895)119896 minus y119896)(y(119895)119896 minus y119896)1015840 +QV (51)

(6) Estimate the cross covariance between xminus119896 and y119896P119909119910119896= 12 119899 2 119899sum119895=1

(x(119895)119896 minus xminus119896)(y(119895)119896 minus y119896)1015840 (52)

(7) The measurement update of the state estimate can beperformed as follows

K119896 = P119909119910119896(P119910119896)minus1 x+119896 = xminus119896 + K119896 (y119896 minus y119896)

P+119896 = Pminus119896 minus K119896P119910

119896K1015840119896 (53)

The proposed improved UKF method provides a sub-optimal solution for the stochastic filtering problem of anonlinear discrete-time dynamic system which contains ran-dom parameters The main difference between the improvedUKF algorithm and the normal UKF algorithm lies in theconstruction of the sigma pointsThe number of sigma pointsof the improved UKF algorithm is larger than that of thenormal UKF algorithm so generally the proposed improvedUKF method requires higher computational complexityHowever the advantage of the improved UKF method isthe increased accuracy of the state estimation which can beverified by the numerical results in the following section

4 An Application to MobileSource Localization

Consider a wireless sensor network with 119899 passive sensornodes distributed on a 2D plane to localize one movingsource Let the state vector of the mobile source be x119896 =[u1015840119896 k1015840119896]1015840 isin R4 which includes the position and velocitycomponents

Without loss of generality let the first sensor be thereference The TDOA measurement model [20] betweensensor pair 119894 (119894 = 2 119899) and 1 is

1199051198941 (u119896) = 10038171003817100381710038171003817u119896 minus s011989410038171003817100381710038171003817119888 minus 10038171003817100381710038171003817u119896 minus s01

10038171003817100381710038171003817119888 + Δ119905119894119896 (54)

where s0119894 is the true position of the 119894th sensor 1199030119894 (u119896) =u119896 minus s0119894 is the true distance from the source to the 119894thsensor and 119888 is the signal propagation speedThe vectorΔt119896 =[Δ1199052119896 Δ119905119899119896]1015840 is zero-meanGaussian noisewith covarianceQ119905 In simplicity (54) is expressed in the vector form

t119896 = [11990521 (u119896) 1199051198991 (u119896)] (55)

In practice the true sensor positions s0 = [s010158401 s010158404 ]1015840are not known and only their nominal values s =

Table 1 Nominal positions (in meters) of sensors

Sensor number 119894 119909119894 1199101198941 100 1002 100 minus1003 minus100 minus1004 minus100 100

[s10158401 s10158402 s10158404]1015840 are available for source localization Moreprecisely s0119894 and s119894 are related by

s0119894 = s119894 + Δs119894 (56)

where Δs119894 is the sensor position noise and the correspondingnoise vector Δs = [Δs10158401 Δs10158402 Δs1015840119899]1015840 is assumed to be zero-mean Gaussian with covariance matrixQ119904

Because the TDOAmeasurement vector (55) is nonlinearwith respect to the state vector x119896 as well as the true sensorlocations s0 the mobile source localization problem can bedescribed as the following discrete-time nonlinear system

t119896 = f (x119896 s0) + Δt119896 (57)

where s0 can be regarded as unknown random parametersThus we can apply the proposed improved UKF method todynamically estimate the state vector

In order to show the performance of the proposedimproved UKF method we apply it to two localization sce-narios and compare it with the normal UKF method whichignores the random uncertainties in the sensor locationsthrough Monte Carlo simulations

41 Scenario 1 This simulation scenario contains 119899 = 4sensors and their nominal positions are given in Table 1 Themotion model of the mobile source is assumed to be a nearlyconstant velocity model as

x119896 =(1 0 001 00 1 0 0010 0 1 00 0 0 1 ) x119896minus1

+(001 00 00101 00 01)w119896minus1(58)

where w119896minus1 is zero-mean Gaussian noise with covariancematrix Q119908 = 001I The initial value of the state x0 isa Gaussian vector with zero mean and covariance P+0 =diag100 100 001 001 The trajectory of the mobile sourcein 100 time instants is shown in Figure 1

The TDOA measurement covariance Q119905 = (05119888)2Twhere T is the (119899 minus 1) times (119899 minus 1)matrix with 1 in the diagonalelements and 05 otherwise The localization accuracy isevaluated by the root of mean squared error which is defined

8 Discrete Dynamics in Nature and Society

minus5

minus48

minus46

minus44

minus42

minus4

minus38

minus36

y-ax

is (m

)

13 135 14 145 15 155 16 165 17 175125x-axis (m)

Figure 1 The trajectory of the mobile source in Scenario 1

The normal UKFThe improved UKF

0

01

02

03

04

05

06

07

08

09

1

The R

MSE

of m

obile

sour

ce lo

caliz

atio

n

10 20 30 40 50 60 70 80 90 1000k

Figure 2 Comparison of the mobile localization accuracy with lowlevel sensor position uncertainties in Scenario 1

as RMSE(u119896) = radicsum119871119897=1 u119896 minus u1198962119871 where u119896 = x119896(1 2) isthe estimate of the source position at ensemble 119897 and 119871 = 1000is the number of ensemble runs

In theMonte Carlo simulations we consider three sensorposition uncertainty levels which are referred to as the lowlevel with 120590119904 = 01 the moderate level with 120590119904 = 1 and thehigh level with 120590119904 = 2

Figure 2 plots the simulation results with time instantvarying from 1 to 100 when the sensor position uncertaintylevel is low It is evident from Figure 2 that RMSE of theproposed improved UKF method (represented by dashedline) is smaller than that of the normal UKF approach(represented by solid line) which ignores the sensor positionuncertainties

Figure 3 plots the RMSEs of the two methods in the caseof moderate sensor position uncertainty level that is 120590119904 =1 We can see that both of them are generally larger than

The normal UKFThe improved UKF

0

01

02

03

04

05

06

07

The R

MSE

of m

obile

sour

ce lo

caliz

atio

n

10 20 30 40 50 60 70 80 90 1000k

Figure 3 Comparison of themobile localization accuracywithmildlevel sensor position uncertainties in Scenario 1

The normal UKFThe improved UKF

02

04

06

08

1

12

14

16

18

2

22

The R

MSE

of m

obile

sour

ce lo

caliz

atio

n

10 20 30 40 50 60 70 80 90 1000k

Figure 4 Comparison of themobile localization accuracywith highlevel sensor position uncertainties in Scenario 1

the corresponding RMSEs in Figure 2 As we expected theperformance of the proposed improvedUKFmethod ismuchbetter than that of the normal UKF approach In additionthe improvement is larger than that of the low level case inFigure 2

Finally we take the high sensor position uncertainty levelthat is120590119904 = 2The numerical simulation results are illustratedin Figure 4 which indicates that the RMSE of the normalUKF approach is nearly twice asmuch as that of the proposedimproved UKF method Comparing the results in Figures 2ndash4 we can see that the influence of the random sensor positionuncertainties on the localization performance is quite evidentin Scenario 1 The proposed improved UKF method can

Discrete Dynamics in Nature and Society 9

The normal UKFThe improved UKF

0

005

01

015

02

025

03

035

04

045

The R

MSE

of m

obile

sour

ce lo

caliz

atio

n

10 20 30 40 50 60 70 80 90 1000k

Figure 5 Comparison of the mobile localization accuracy with lowlevel sensor position uncertainties in Scenario 2

efficiently decrease the RMSE especially when the sensorposition uncertainty level is high

42 Scenario 2 In this simulation scenario we considerthe sensor geometry is random and the number of sensors119899 = 4 The true position of each sensor is uniformlydistributed in a square of [minus100 100]m times [minus100 100]mat each ensemble in Monte Carlo simulations The TDOAmeasurement covariance Q119905 = (01119888)2T The motion modeland the initial value of the state are generally the same as thoseof Scenario 1

In the first case the sensor position uncertainty levelis considerably small that is 120590119904 = 01 Figure 5 plots theRMSE of the normal UKF approach and that of the proposedimproved UKF method with time instant varying from 1 to100 which are calculated from 119871 = 1000Monte Carlo runsIt can be seen that the RMSE of the proposed improved UKFmethod is smaller than that of the normal UKF approach

Thenwe consider amoderate sensor position uncertaintylevel that is 120590119904 = 1 Figure 6 plots the RMSEs of the twomethods in this case where we can see that the performanceof the proposed improvedUKFmethod is still better than thatof the normal UKF approach with a larger improvement as inFigure 5

For the high sensor position uncertainty level that is120590119904 = 2 the simulation results are illustrated in Figure 7which shows that the RMSE of the normal UKF approach isconsiderably larger than that of the proposed improved UKFmethod

The Monte Carlo simulation results in Figures 5ndash7 revealthat the RMSEs of the proposed improved UKF are consis-tently smaller than those of the normal UKF approach whichcorroborates that the proposed improved UKF method cangenerally improve the estimation accuracy of the nonlineardynamic system when it contains random parameters

The normal UKFThe improved UKF

0

02

04

06

08

1

12

14

16

18

2

The R

MSE

of m

obile

sour

ce lo

caliz

atio

n

10 20 30 40 50 60 70 80 90 1000k

Figure 6 Comparison of themobile localization accuracywithmildlevel sensor position uncertainties in Scenario 2

The normal UKFThe improved UKF

0

05

1

15

2

25

3

35

The R

MSE

of m

obile

sour

ce lo

caliz

atio

n

10 20 30 40 50 60 70 80 90 1000k

Figure 7 Comparison of themobile localization accuracy with highlevel sensor position uncertainties in Scenario 2

5 Conclusions

In this paper an improved unscented Kalman filteringmethod is explored for discrete nonlinear dynamic systemswith randomparameters By augmenting the randomparam-eters into the state vector we enlarge the number of sigmapoints in the improved unscented transformation Theo-retical analysis indicates that the approximated mean andcovariance via the improved unscented transformationmatchthe true values correctly up to the third order of Taylor seriesexpansion An application to the mobile source localizationusing TDOA measurements is provided where the sensorpositions suffer from randomuncertaintiesTheMonte Carlo

10 Discrete Dynamics in Nature and Society

simulation results show that the proposed improved UKFmethod can considerably improve the estimation accuracy

Competing Interests

The authors declare that there are no competing interestsregarding the publication of this paper

Acknowledgments

This work was supported by the Fundamental ResearchFunds for the Central Universities Southwestern Universityof Finance and Economics under Grant no JBK1507059and Southwest University for Nationalities under Grant no2016NGJPY05

References

[1] R E Kalman ldquoA new approach to linear filtering and predictionproblemsrdquo Journal of Basic Engineering vol 82 no 1 pp 35ndash451960

[2] G Bolzon R Fedele and G Maier ldquoParameter identificationof a cohesive crack model by Kalman filterrdquo Computer Methodsin Applied Mechanics and Engineering vol 191 no 25-26 pp2847ndash2871 2002

[3] M Ball and A Wood ldquoTrend growth in post-1850 Britisheconomic history the Kalman filter and historical judgmentrdquoJournal of the Royal Statistical Society vol 45 no 2 Article ID143152 pp 143ndash152 1996

[4] W L De Koning ldquoOptimal estimation of linear discrete-timesystems with stochastic parametersrdquo Automatica vol 20 pp113ndash115 1984

[5] B Sinopoli L Schenato M Franceschetti K Poolla M IJordan and S S Sastry ldquoKalman filtering with intermittentobservationsrdquo IEEE Transactions on Automatic Control vol 49no 9 pp 1453ndash1464 2004

[6] Y Luo Y Zhu D Luo J Zhou E Song and DWang ldquoGloballyoptimal multisensor distributed random parameter matriceskalman filtering fusion with applicationsrdquo Sensors vol 8 no 12pp 8086ndash8103 2008

[7] S Chen Y Li G Qi and A Sheng ldquoAdaptive Kalman esti-mation in target tracking mixed with random one-step delaysstochastic-bias measurements and missing measurementsrdquoDiscrete Dynamics in Nature and Society vol 2013 Article ID716915 14 pages 2013

[8] S Chen Z Zhou and S Li ldquoAn efficient estimate and forecastof the implied volatility surface a nonlinear Kalman filterapproachrdquo Economic Modelling vol 58 pp 655ndash664 2016

[9] S Bolognani L Tubiana and M Zigliotto ldquoExtended Kalmanfilter tuning in sensorless PMSM drivesrdquo IEEE Transactions onIndustry Applications vol 39 no 6 pp 1741ndash1747 2003

[10] J J LaViola Jr ldquoA comparison of unscented and extendedkalman filtering for estimating quaternion motionrdquo in Proceed-ings of the AmericanControl Conference pp 2435ndash2440DenverColo USA June 2003

[11] X Qu and L Xie ldquoRecursive source localisation by time dif-ference of arrival sensor networks with sensor position uncer-taintyrdquo IET Control Theory and Applications vol 8 no 18 pp2305ndash2315 2014

[12] S J Julier J K Uhlmann and H F Durrant-Whyte ldquoNewapproach for filtering nonlinear systemsrdquo in Proceedings of theAmerican Control Conference pp 1628ndash1632 June 1995

[13] S J Julier and J K Uhlmann ldquoUnscented filtering and nonlin-ear estimationrdquo Proceedings of the IEEE vol 92 no 3 pp 401ndash422 2004

[14] Q Pan F Yang and L Ye ldquoSurvey of a kind of non linear filtersUKFrdquo Control and Decision vol 20 no 5 pp 481ndash489 2005

[15] C Eberle and C Ament ldquoThe Unscented Kalman Filter esti-mates the plasma insulin from glucose measurementrdquo BioSys-tems vol 103 no 1 pp 67ndash72 2011

[16] S J Julier ldquoSome relations between extended and unscentedKalman filtersrdquo IEEE Transactions on Signal Processing vol 60no 2 pp 545ndash555 2012

[17] HM T Menegaz J Y Ishihara G A Borges and A N VargasldquoA systematization of the unscented Kalman filter theoryrdquo IEEETransactions on Automatic Control vol 60 no 10 pp 2583ndash2598 2015

[18] D Simon Optimal State Estimation Kalman H Infinity andNonlinear Approaches Wiley-Interscience 2006

[19] KW Lui F K Chan andH C So ldquoSemidefinite programmingapproach for range-difference based source localizationrdquo IEEETransactions on Signal Processing vol 57 no 4 pp 1630ndash16332009

[20] X Qu and L Xie ldquoAn efficient convex constrained weightedleast squares source localization algorithm based on TDOAmeasurementsrdquo Signal Processing vol 119 pp 142ndash152 2016

[21] G Yanni and W Qi ldquoThe nonsequential fusion method forlocalization from unscented kalman filter by multistation arraybuoysrdquo Discrete Dynamics in Nature and Society vol 2016Article ID 7670609 8 pages 2016

[22] A H Jazwinski Stochastic Processes and Filtering Theory Aca-demic Press Cambridge Mass USA 1970

[23] F Daum ldquoNonlinear filters beyond the kalman filterrdquo IEEEAerospace and Electronic Systems Magazine vol 20 no 8 pp57ndash69 2005

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Differential EquationsInternational Journal of

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Stochastic AnalysisInternational Journal of

4 Discrete Dynamics in Nature and Society

(6) Estimate the cross covariance between xminus119896 and y119896

P119909119910119896= 12119899119909 2119899119909sum119895=1 (x(119895)119896 minus xminus119896 ) (y(119895)119896 minus y119896)1015840 (15)

(7) The measurement update of the state estimate can beperformed following the normal Kalman filter

K119896 = P119909119910119896(P119910119896)minus1

x+119896 = xminus119896 + K119896 (y119896 minus y119896) P+119896 = Pminus119896 minus K119896P

119910

119896K1015840119896 (16)

The algorithm above assumes that the motion andmeasurement functions are completely known In practicalapplications the motion and measurement functions maycontain parameters that are not specified andmay be randomwith known distributions In such cases the previous UKFalgorithm can not be directly applied

3 Improved Unscented Kalman Filter

In this section we propose an improved UKFmethod to dealwith the filtering problem in nonlinear dynamic systems withrandomparameters In the following wewill firstly introducethe improved unscented transformation which propagatesthe mean and covariance of a random vector through anonlinear function with random parameters

31 Improved Unscented Transformation Suppose that wehave a vector x isin R119899119909 with known mean x and covarianceP and its pdf is symmetric around its mean vector For anonlinear function y = ℎ(x 120579) the random parameter 120579 isinR119899120579 has known mean 120579 and covariance P120579 and also has asymmetric pdf In addition the parameter 120579 is uncorrelatedwith x

In the normal unscented transformation a set of sigmapoints x(119895) (2) is generated and each sigma point is trans-formed by the known nonlinear function ℎ(sdot) to approximatethe mean and covariance of y However when the functionℎ(sdot) is nonlinear with respect to random parameter 120579 the stepof transforming the sigma points is unrealizable

In order to deal with this problem we augment therandom parameter 120579 onto the state vector as follows

x = [x1015840 1205791015840]1015840 isin R119899119909+119899120579 (17)

The mean and covariance of the augment vector x are119864 (x) = x = [x1015840 1205791015840]1015840 Cov (x) = P = (P 0

0 P120579) (18)

Based on the augmentmodel the number of sigma pointsis enlarged to 2(119899119909 + 119899120579) The new sigma points are generatedas follows

x(119895) = x + x(119895)119896 119895 = 1 2 (119899119909 + 119899120579) (19)

where x(119895) = (radic2 (119899119909 + 119899120579) P)1015840119895 119895 = 1 119899119909 + 119899120579x(119899119909+119899120579+119895) = minus(radic2 (119899119909 + 119899120579) P)1015840119895 119895 = 1 119899119909 + 119899120579

(20)

The corresponding weighting coefficients are given as

W(119895) = 12 (119899119909 + 119899120579) 119895 = 1 2 (119899119909 + 119899120579) (21)

Because we regard both the vector x and the randomparameter 120579 as the unknown parameters in the augmentmodel each sigma point can be divided into two parts thatis

x(119895) = [x(119895) 120579(119895)] (22)

where x(119895) = x(119895)(1 119899119909) represents the sigma point of thevector x and 120579(119895) = x(119895)(119899119909 + 1 119899119909 + 119899120579) represents the sigmapoint of the parameter 120579 As a result the transformed sigmapoints can be computed as follows

y(119895) = ℎ (x(119895) 120579(119895)) 119895 = 1 2 (119899119909 + 119899120579) (23)

Let y denote the true mean of y The approximated mean ofthe improved unscented transformation is denoted as y andcomputed as follows

y = 2(119899119909+119899120579)sum119895=1

W(119895)y(119895) (24)

Let Py denote the true covariance of y The approximatedcovariance of the improved unscented transformation isdenoted as Py and computed as follows

Py = 2(119899119909+119899120579)sum119895=1

W(119895) (y(119895) minus y) (y(119895) minus y)1015840 (25)

The improved unscented transformations (24) and (25)incorporate the information of the random parameter 120579 sothey are intuitively more accurate than the normal unscentedtransformations (5) and (6) which ignore the randomnessof 120579 The theoretical analysis of the improved unscentedtransformations is given in the following theorems

Theorem3 For random vector xwith a symmetric pdf aroundits mean if the nonlinear function y = ℎ(x 120579) containsunknown random parameter 120579 the approximated mean of yvia the improved unscented transformation (24) matches thetrue mean of y correctly up to the third order

Proof We firstly expand y = ℎ(x 120579) in a Taylor series aroundx = [x1015840 1205791015840]1015840 as follows

y = ℎ (x 120579)= ℎ (x 120579) + 119863xℎ + 121198632xℎ + 131198633xℎ + sdot sdot sdot (26)

Discrete Dynamics in Nature and Society 5

where x = x minus x and the operation119863119896xℎ is defined as

119863119896xℎ = (119899119909+119899120579sum119895=1

x119895 120597ℎ120597x119895 1003816100381610038161003816100381610038161003816100381610038161003816x=x)119896 (27)

The mean of y can therefore be expanded as

y = 119864(ℎ (x 120579) + 119863xℎ + 121198632xℎ + 131198633xℎ + sdot sdot sdot)= ℎ (x 120579) + 119864(119863xℎ + 121198632xℎ + 131198633xℎ + sdot sdot sdot) (28)

We can see that

119864 (119863xℎ) = 119864(119899119909+119899120579sum119895=1

x119895 120597ℎ120597x119895 1003816100381610038161003816100381610038161003816100381610038161003816x=x)= 119899119909+119899120579sum119895=1

119864 (x119895) 120597ℎ120597x119895 1003816100381610038161003816100381610038161003816100381610038161003816x=x = 0(29)

due to 119864(x) = 0 Because both x and 120579 have a symmetric pdfaround the mean vector we can verify that

119864 (1198633xℎ) = 119864[[(119899119909+119899120579sum119895=1 x119895 120597ℎ120597x119895 1003816100381610038161003816100381610038161003816100381610038161003816x=x)3]] = 0 (30)

Similarly all of the odd terms in (28) will be equal to zerowhich leads to the simplification

y = ℎ (x 120579) + 119864( 121198632xℎ + 141198634xℎ + sdot sdot sdot) (31)

Now we compute the value of y by expanding each y(119895) in(24) in a Taylor series around x as follows

y = 12 (119899119909 + 119899120579)sdot 2(119899119909+119899120579)sum119895=1

(ℎ (x 120579) + 119863x(119895)ℎ + 121198632x(119895)ℎ + sdot sdot sdot) (32)

Because x(119895) = minusx(119899119909+119899120579+119895) (119895 = 1 119899119909 + 119899120579) for any integer119896 gt 0 we have2(119899119909+119899120579)sum119895=1

1198632119896+1x(119895) ℎ = 2(119899119909+119899120579)sum

119895=1

[[(119899119909+119899120579sum119894=1 x(119895)119894 120597ℎ120597x119895 1003816100381610038161003816100381610038161003816100381610038161003816x=x)2119896+1]]= 2(119899119909+119899120579)sum119895=1

[119899119909+119899120579sum119894=1

(x(119895)119894 )2119896+1 1205972119896+1ℎ120597x2119896+1119895 1003816100381610038161003816100381610038161003816100381610038161003816x=x]= 119899119909+119899120579sum119894=1

[[2(119899119909+119899120579)sum119895=1

(x(119895)119894 )2119896+1 1205972119896+1ℎ120597x2119896+1119895 1003816100381610038161003816100381610038161003816100381610038161003816x=x]] = 0(33)

Therefore all of the odd terms in (32) evaluate to zero and wehave

y= ℎ (x 120579)+ 12 (119899119909 + 119899120579) 2(119899119909+119899120579)sum

119895=1

( 121198632x(119895)ℎ + 141198634x(119895)ℎ + sdot sdot sdot) (34)

After some tedious calculation we can verify that

12 (119899119909 + 119899120579) 2(119899119909+119899120579)sum119895=1

121198632x(119895)ℎ = 12 119899119909+119899120579sum119894119895=1

P1198941198951205972ℎ120597x119894120597x119895 1003816100381610038161003816100381610038161003816100381610038161003816x=x= 12119864 (1198632xℎ) (35)

It can be seen that y (the approximated mean of y) matchesthe true mean of y correctly up to the third order

Theorem4 For random vector xwith a symmetric pdf aroundits mean if the nonlinear function y = ℎ(x 120579) containsunknown random parameter 120579 the approximated covarianceof y via the improved unscented transformation (25) matchesthe true covariance of y correctly up to the third order

Proof The true covariance of y is given as

Py = 119864 [(y minus y) (y minus y)1015840] (36)

By substituting (26) and (31) into (36) and using the same typeof reasoning in the proof ofTheorem 3 we have that all of theodd-powered terms evaluate to zero This leads to

Py = 119864 [119863xℎ (119863xℎ)1015840] + 119864[[119863xℎ (1198633xℎ)10158403

+ 1198632xℎ (1198632xℎ)101584022 + 1198633xℎ (119863xℎ)10158403 ]] + 119864[1198632xℎ2 ]sdot 119864 [1198632xℎ2 ]1015840 + sdot sdot sdot(37)

The first term on the right side of the equation above can bewritten as

119864 [119863xℎ (119863xℎ)1015840] = HPH1015840 (38)

whereH is the partial derivative matrix at x = x

6 Discrete Dynamics in Nature and Society

On the other hand after some tedious calculation theapproximate covariance Py in (25) can be written as

Py = 12 (119899119909 + 119899120579) 2(119899119909+119899120579)sum119895=1

(119863x(119895)ℎ) (119863x(119895)ℎ)1015840 +HOT

= 12 (119899119909 + 119899120579)sdot 2(119899119909+119899120579)sum119895=1

119899119909+119899120579sum119894119896=1

(x(119895)119894 120597ℎ (x)120597x119894 )(x(119895)119896 120597ℎ (x)120597x119896 )1015840+HOT(39)

whereHOTmeans higher-order terms (ie terms to the forthpower and higher) Now recall that x(119895)119894 = minusx(119899119909+119899120579+119895)119894 andx(119895)119896 = minusx(119899119909+119899120579+119895)119896 for 119895 = 1 119899119909 + 119899120579 therefore thecovariance approximation becomes

Py

= 1119899119909 + 119899120579 119899119909+119899120579sum119895=1 119899119909+119899120579sum119894119896=1 (x(119895)119894 120597ℎ (x)120597x119894 )(x(119895)119896 120597ℎ (x)120597x119896 )1015840+HOT = HPH1015840 +HOT(40)

Comparing (40) with the true covariance of y from (37) wesee that Py in (25) approximates the true covariance of y upto the third order

32 Improved Unscented Kalman Filter The improvedunscented transformation developed in the previoussubsection can be generalized to give the improvedunscented Kalman filter for the discrete-time nonlinearsystem which contains random parameters that is

x119896 = 119891 (x119896minus1 120579) + w119896y119896 = ℎ (x119896 120579) + k119896 (41)

where the motion function 119891(sdot) and the measurement func-tion ℎ(sdot) are nonlinear with respect to not only the state vectorx119896 but also the random parameter 120579 The other notations arethe same as that of model (1) Assume that the parameter 120579 isinR119899120579 has knownmean 120579 and covarianceP120579 and is uncorrelatedwith the state vector x119896 as well as the noise vectors w119896 and k119896

Following the augment method of the improvedunscented transformation we give the augment model of thediscrete-time nonlinear system (41) as follows

x119896 = 119891 (x119896minus1) + w119896y119896 = ℎ (x119896) + k119896 (42)

where

x119896 = [x1015840119896 1205791015840]1015840 w119896 = [w1015840119896 01015840]1015840 (43)

With the initial condition (7) the improved unscentedKalman filter can be initialized as followsx+0 = [(x+0 )1015840 1205791015840]1015840

P+0 = (P+0 0

0 P120579) (44)

Suppose that at time step 119896 x+119896minus1 and P+119896minus1 are given For thesake of brevity we denote 119899 = 119899119909 + 119899120579 The improved UKFalgorithm can be summarized in Algorithm 5

Algorithm 5 (the improved unscented Kalman filter)

(1) Choose 2 119899 sigma points x(119895)119896minus1 as specified in (19) withappropriate changes that isx(119895)119896minus1 = x+119896minus1 + x(119895)119896minus1 119895 = 1 2 119899x(119895)119896minus1 = (radic 119899P+

119896minus1)1015840119895 119895 = 1 119899x( 119899+119895)119896minus1 = minusx(119895)119896minus1 119895 = 1 119899

(45)

(2) Transform the sigma points x(119895)119896minus1 into x(119895)119896 as followsx(119895)119896 = 119891 (x(119895)119896minus1 120579(119895)119896minus1) (46)

where x(119895)119896minus1= x(119895)119896minus1(1 119899119909) and 120579(119895)119896minus1 = x(119895)119896minus1(119899119909 + 1 119899)

(3) Combine the x(119895)119896 vectors to obtain the predictionestimate at time

xminus

119896 = 12 119899 2 119899sum119895=1

x(119895)119896 (47)

and the corresponding error covariance

Pminus119896 = 12 119899 2 119899sum119895=1

(x(119895)119896 minus xminus119896)(x(119895)119896 minus xminus119896)1015840 + Q119908 (48)

where Q119908 = ( Q119908 00 0 )

(4) Transform the sigma points x(119895)119896minus1 into y(119895)119896 vectorsy(119895)119896 = ℎ (x(119895)119896minus1 120579(119895)119896minus1) (49)

(5) Combine the y(119895)119896 vectors to obtain the predictedmeasurement at time 119896y119896 = 12 119899 2 119899sum

119895=1

y(119895)119896 (50)

Discrete Dynamics in Nature and Society 7

and the corresponding covariance

P119910119896= 12 119899 2 119899sum119895=1

(y(119895)119896 minus y119896)(y(119895)119896 minus y119896)1015840 +QV (51)

(6) Estimate the cross covariance between xminus119896 and y119896P119909119910119896= 12 119899 2 119899sum119895=1

(x(119895)119896 minus xminus119896)(y(119895)119896 minus y119896)1015840 (52)

(7) The measurement update of the state estimate can beperformed as follows

K119896 = P119909119910119896(P119910119896)minus1 x+119896 = xminus119896 + K119896 (y119896 minus y119896)

P+119896 = Pminus119896 minus K119896P119910

119896K1015840119896 (53)

The proposed improved UKF method provides a sub-optimal solution for the stochastic filtering problem of anonlinear discrete-time dynamic system which contains ran-dom parameters The main difference between the improvedUKF algorithm and the normal UKF algorithm lies in theconstruction of the sigma pointsThe number of sigma pointsof the improved UKF algorithm is larger than that of thenormal UKF algorithm so generally the proposed improvedUKF method requires higher computational complexityHowever the advantage of the improved UKF method isthe increased accuracy of the state estimation which can beverified by the numerical results in the following section

4 An Application to MobileSource Localization

Consider a wireless sensor network with 119899 passive sensornodes distributed on a 2D plane to localize one movingsource Let the state vector of the mobile source be x119896 =[u1015840119896 k1015840119896]1015840 isin R4 which includes the position and velocitycomponents

Without loss of generality let the first sensor be thereference The TDOA measurement model [20] betweensensor pair 119894 (119894 = 2 119899) and 1 is

1199051198941 (u119896) = 10038171003817100381710038171003817u119896 minus s011989410038171003817100381710038171003817119888 minus 10038171003817100381710038171003817u119896 minus s01

10038171003817100381710038171003817119888 + Δ119905119894119896 (54)

where s0119894 is the true position of the 119894th sensor 1199030119894 (u119896) =u119896 minus s0119894 is the true distance from the source to the 119894thsensor and 119888 is the signal propagation speedThe vectorΔt119896 =[Δ1199052119896 Δ119905119899119896]1015840 is zero-meanGaussian noisewith covarianceQ119905 In simplicity (54) is expressed in the vector form

t119896 = [11990521 (u119896) 1199051198991 (u119896)] (55)

In practice the true sensor positions s0 = [s010158401 s010158404 ]1015840are not known and only their nominal values s =

Table 1 Nominal positions (in meters) of sensors

Sensor number 119894 119909119894 1199101198941 100 1002 100 minus1003 minus100 minus1004 minus100 100

[s10158401 s10158402 s10158404]1015840 are available for source localization Moreprecisely s0119894 and s119894 are related by

s0119894 = s119894 + Δs119894 (56)

where Δs119894 is the sensor position noise and the correspondingnoise vector Δs = [Δs10158401 Δs10158402 Δs1015840119899]1015840 is assumed to be zero-mean Gaussian with covariance matrixQ119904

Because the TDOAmeasurement vector (55) is nonlinearwith respect to the state vector x119896 as well as the true sensorlocations s0 the mobile source localization problem can bedescribed as the following discrete-time nonlinear system

t119896 = f (x119896 s0) + Δt119896 (57)

where s0 can be regarded as unknown random parametersThus we can apply the proposed improved UKF method todynamically estimate the state vector

In order to show the performance of the proposedimproved UKF method we apply it to two localization sce-narios and compare it with the normal UKF method whichignores the random uncertainties in the sensor locationsthrough Monte Carlo simulations

41 Scenario 1 This simulation scenario contains 119899 = 4sensors and their nominal positions are given in Table 1 Themotion model of the mobile source is assumed to be a nearlyconstant velocity model as

x119896 =(1 0 001 00 1 0 0010 0 1 00 0 0 1 ) x119896minus1

+(001 00 00101 00 01)w119896minus1(58)

where w119896minus1 is zero-mean Gaussian noise with covariancematrix Q119908 = 001I The initial value of the state x0 isa Gaussian vector with zero mean and covariance P+0 =diag100 100 001 001 The trajectory of the mobile sourcein 100 time instants is shown in Figure 1

The TDOA measurement covariance Q119905 = (05119888)2Twhere T is the (119899 minus 1) times (119899 minus 1)matrix with 1 in the diagonalelements and 05 otherwise The localization accuracy isevaluated by the root of mean squared error which is defined

8 Discrete Dynamics in Nature and Society

minus5

minus48

minus46

minus44

minus42

minus4

minus38

minus36

y-ax

is (m

)

13 135 14 145 15 155 16 165 17 175125x-axis (m)

Figure 1 The trajectory of the mobile source in Scenario 1

The normal UKFThe improved UKF

0

01

02

03

04

05

06

07

08

09

1

The R

MSE

of m

obile

sour

ce lo

caliz

atio

n

10 20 30 40 50 60 70 80 90 1000k

Figure 2 Comparison of the mobile localization accuracy with lowlevel sensor position uncertainties in Scenario 1

as RMSE(u119896) = radicsum119871119897=1 u119896 minus u1198962119871 where u119896 = x119896(1 2) isthe estimate of the source position at ensemble 119897 and 119871 = 1000is the number of ensemble runs

In theMonte Carlo simulations we consider three sensorposition uncertainty levels which are referred to as the lowlevel with 120590119904 = 01 the moderate level with 120590119904 = 1 and thehigh level with 120590119904 = 2

Figure 2 plots the simulation results with time instantvarying from 1 to 100 when the sensor position uncertaintylevel is low It is evident from Figure 2 that RMSE of theproposed improved UKF method (represented by dashedline) is smaller than that of the normal UKF approach(represented by solid line) which ignores the sensor positionuncertainties

Figure 3 plots the RMSEs of the two methods in the caseof moderate sensor position uncertainty level that is 120590119904 =1 We can see that both of them are generally larger than

The normal UKFThe improved UKF

0

01

02

03

04

05

06

07

The R

MSE

of m

obile

sour

ce lo

caliz

atio

n

10 20 30 40 50 60 70 80 90 1000k

Figure 3 Comparison of themobile localization accuracywithmildlevel sensor position uncertainties in Scenario 1

The normal UKFThe improved UKF

02

04

06

08

1

12

14

16

18

2

22

The R

MSE

of m

obile

sour

ce lo

caliz

atio

n

10 20 30 40 50 60 70 80 90 1000k

Figure 4 Comparison of themobile localization accuracywith highlevel sensor position uncertainties in Scenario 1

the corresponding RMSEs in Figure 2 As we expected theperformance of the proposed improvedUKFmethod ismuchbetter than that of the normal UKF approach In additionthe improvement is larger than that of the low level case inFigure 2

Finally we take the high sensor position uncertainty levelthat is120590119904 = 2The numerical simulation results are illustratedin Figure 4 which indicates that the RMSE of the normalUKF approach is nearly twice asmuch as that of the proposedimproved UKF method Comparing the results in Figures 2ndash4 we can see that the influence of the random sensor positionuncertainties on the localization performance is quite evidentin Scenario 1 The proposed improved UKF method can

Discrete Dynamics in Nature and Society 9

The normal UKFThe improved UKF

0

005

01

015

02

025

03

035

04

045

The R

MSE

of m

obile

sour

ce lo

caliz

atio

n

10 20 30 40 50 60 70 80 90 1000k

Figure 5 Comparison of the mobile localization accuracy with lowlevel sensor position uncertainties in Scenario 2

efficiently decrease the RMSE especially when the sensorposition uncertainty level is high

42 Scenario 2 In this simulation scenario we considerthe sensor geometry is random and the number of sensors119899 = 4 The true position of each sensor is uniformlydistributed in a square of [minus100 100]m times [minus100 100]mat each ensemble in Monte Carlo simulations The TDOAmeasurement covariance Q119905 = (01119888)2T The motion modeland the initial value of the state are generally the same as thoseof Scenario 1

In the first case the sensor position uncertainty levelis considerably small that is 120590119904 = 01 Figure 5 plots theRMSE of the normal UKF approach and that of the proposedimproved UKF method with time instant varying from 1 to100 which are calculated from 119871 = 1000Monte Carlo runsIt can be seen that the RMSE of the proposed improved UKFmethod is smaller than that of the normal UKF approach

Thenwe consider amoderate sensor position uncertaintylevel that is 120590119904 = 1 Figure 6 plots the RMSEs of the twomethods in this case where we can see that the performanceof the proposed improvedUKFmethod is still better than thatof the normal UKF approach with a larger improvement as inFigure 5

For the high sensor position uncertainty level that is120590119904 = 2 the simulation results are illustrated in Figure 7which shows that the RMSE of the normal UKF approach isconsiderably larger than that of the proposed improved UKFmethod

The Monte Carlo simulation results in Figures 5ndash7 revealthat the RMSEs of the proposed improved UKF are consis-tently smaller than those of the normal UKF approach whichcorroborates that the proposed improved UKF method cangenerally improve the estimation accuracy of the nonlineardynamic system when it contains random parameters

The normal UKFThe improved UKF

0

02

04

06

08

1

12

14

16

18

2

The R

MSE

of m

obile

sour

ce lo

caliz

atio

n

10 20 30 40 50 60 70 80 90 1000k

Figure 6 Comparison of themobile localization accuracywithmildlevel sensor position uncertainties in Scenario 2

The normal UKFThe improved UKF

0

05

1

15

2

25

3

35

The R

MSE

of m

obile

sour

ce lo

caliz

atio

n

10 20 30 40 50 60 70 80 90 1000k

Figure 7 Comparison of themobile localization accuracy with highlevel sensor position uncertainties in Scenario 2

5 Conclusions

In this paper an improved unscented Kalman filteringmethod is explored for discrete nonlinear dynamic systemswith randomparameters By augmenting the randomparam-eters into the state vector we enlarge the number of sigmapoints in the improved unscented transformation Theo-retical analysis indicates that the approximated mean andcovariance via the improved unscented transformationmatchthe true values correctly up to the third order of Taylor seriesexpansion An application to the mobile source localizationusing TDOA measurements is provided where the sensorpositions suffer from randomuncertaintiesTheMonte Carlo

10 Discrete Dynamics in Nature and Society

simulation results show that the proposed improved UKFmethod can considerably improve the estimation accuracy

Competing Interests

The authors declare that there are no competing interestsregarding the publication of this paper

Acknowledgments

This work was supported by the Fundamental ResearchFunds for the Central Universities Southwestern Universityof Finance and Economics under Grant no JBK1507059and Southwest University for Nationalities under Grant no2016NGJPY05

References

[1] R E Kalman ldquoA new approach to linear filtering and predictionproblemsrdquo Journal of Basic Engineering vol 82 no 1 pp 35ndash451960

[2] G Bolzon R Fedele and G Maier ldquoParameter identificationof a cohesive crack model by Kalman filterrdquo Computer Methodsin Applied Mechanics and Engineering vol 191 no 25-26 pp2847ndash2871 2002

[3] M Ball and A Wood ldquoTrend growth in post-1850 Britisheconomic history the Kalman filter and historical judgmentrdquoJournal of the Royal Statistical Society vol 45 no 2 Article ID143152 pp 143ndash152 1996

[4] W L De Koning ldquoOptimal estimation of linear discrete-timesystems with stochastic parametersrdquo Automatica vol 20 pp113ndash115 1984

[5] B Sinopoli L Schenato M Franceschetti K Poolla M IJordan and S S Sastry ldquoKalman filtering with intermittentobservationsrdquo IEEE Transactions on Automatic Control vol 49no 9 pp 1453ndash1464 2004

[6] Y Luo Y Zhu D Luo J Zhou E Song and DWang ldquoGloballyoptimal multisensor distributed random parameter matriceskalman filtering fusion with applicationsrdquo Sensors vol 8 no 12pp 8086ndash8103 2008

[7] S Chen Y Li G Qi and A Sheng ldquoAdaptive Kalman esti-mation in target tracking mixed with random one-step delaysstochastic-bias measurements and missing measurementsrdquoDiscrete Dynamics in Nature and Society vol 2013 Article ID716915 14 pages 2013

[8] S Chen Z Zhou and S Li ldquoAn efficient estimate and forecastof the implied volatility surface a nonlinear Kalman filterapproachrdquo Economic Modelling vol 58 pp 655ndash664 2016

[9] S Bolognani L Tubiana and M Zigliotto ldquoExtended Kalmanfilter tuning in sensorless PMSM drivesrdquo IEEE Transactions onIndustry Applications vol 39 no 6 pp 1741ndash1747 2003

[10] J J LaViola Jr ldquoA comparison of unscented and extendedkalman filtering for estimating quaternion motionrdquo in Proceed-ings of the AmericanControl Conference pp 2435ndash2440DenverColo USA June 2003

[11] X Qu and L Xie ldquoRecursive source localisation by time dif-ference of arrival sensor networks with sensor position uncer-taintyrdquo IET Control Theory and Applications vol 8 no 18 pp2305ndash2315 2014

[12] S J Julier J K Uhlmann and H F Durrant-Whyte ldquoNewapproach for filtering nonlinear systemsrdquo in Proceedings of theAmerican Control Conference pp 1628ndash1632 June 1995

[13] S J Julier and J K Uhlmann ldquoUnscented filtering and nonlin-ear estimationrdquo Proceedings of the IEEE vol 92 no 3 pp 401ndash422 2004

[14] Q Pan F Yang and L Ye ldquoSurvey of a kind of non linear filtersUKFrdquo Control and Decision vol 20 no 5 pp 481ndash489 2005

[15] C Eberle and C Ament ldquoThe Unscented Kalman Filter esti-mates the plasma insulin from glucose measurementrdquo BioSys-tems vol 103 no 1 pp 67ndash72 2011

[16] S J Julier ldquoSome relations between extended and unscentedKalman filtersrdquo IEEE Transactions on Signal Processing vol 60no 2 pp 545ndash555 2012

[17] HM T Menegaz J Y Ishihara G A Borges and A N VargasldquoA systematization of the unscented Kalman filter theoryrdquo IEEETransactions on Automatic Control vol 60 no 10 pp 2583ndash2598 2015

[18] D Simon Optimal State Estimation Kalman H Infinity andNonlinear Approaches Wiley-Interscience 2006

[19] KW Lui F K Chan andH C So ldquoSemidefinite programmingapproach for range-difference based source localizationrdquo IEEETransactions on Signal Processing vol 57 no 4 pp 1630ndash16332009

[20] X Qu and L Xie ldquoAn efficient convex constrained weightedleast squares source localization algorithm based on TDOAmeasurementsrdquo Signal Processing vol 119 pp 142ndash152 2016

[21] G Yanni and W Qi ldquoThe nonsequential fusion method forlocalization from unscented kalman filter by multistation arraybuoysrdquo Discrete Dynamics in Nature and Society vol 2016Article ID 7670609 8 pages 2016

[22] A H Jazwinski Stochastic Processes and Filtering Theory Aca-demic Press Cambridge Mass USA 1970

[23] F Daum ldquoNonlinear filters beyond the kalman filterrdquo IEEEAerospace and Electronic Systems Magazine vol 20 no 8 pp57ndash69 2005

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MathematicsJournal of

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Mathematical Problems in Engineering

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Differential EquationsInternational Journal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

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Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Discrete Dynamics in Nature and Society 5

where x = x minus x and the operation119863119896xℎ is defined as

119863119896xℎ = (119899119909+119899120579sum119895=1

x119895 120597ℎ120597x119895 1003816100381610038161003816100381610038161003816100381610038161003816x=x)119896 (27)

The mean of y can therefore be expanded as

y = 119864(ℎ (x 120579) + 119863xℎ + 121198632xℎ + 131198633xℎ + sdot sdot sdot)= ℎ (x 120579) + 119864(119863xℎ + 121198632xℎ + 131198633xℎ + sdot sdot sdot) (28)

We can see that

119864 (119863xℎ) = 119864(119899119909+119899120579sum119895=1

x119895 120597ℎ120597x119895 1003816100381610038161003816100381610038161003816100381610038161003816x=x)= 119899119909+119899120579sum119895=1

119864 (x119895) 120597ℎ120597x119895 1003816100381610038161003816100381610038161003816100381610038161003816x=x = 0(29)

due to 119864(x) = 0 Because both x and 120579 have a symmetric pdfaround the mean vector we can verify that

119864 (1198633xℎ) = 119864[[(119899119909+119899120579sum119895=1 x119895 120597ℎ120597x119895 1003816100381610038161003816100381610038161003816100381610038161003816x=x)3]] = 0 (30)

Similarly all of the odd terms in (28) will be equal to zerowhich leads to the simplification

y = ℎ (x 120579) + 119864( 121198632xℎ + 141198634xℎ + sdot sdot sdot) (31)

Now we compute the value of y by expanding each y(119895) in(24) in a Taylor series around x as follows

y = 12 (119899119909 + 119899120579)sdot 2(119899119909+119899120579)sum119895=1

(ℎ (x 120579) + 119863x(119895)ℎ + 121198632x(119895)ℎ + sdot sdot sdot) (32)

Because x(119895) = minusx(119899119909+119899120579+119895) (119895 = 1 119899119909 + 119899120579) for any integer119896 gt 0 we have2(119899119909+119899120579)sum119895=1

1198632119896+1x(119895) ℎ = 2(119899119909+119899120579)sum

119895=1

[[(119899119909+119899120579sum119894=1 x(119895)119894 120597ℎ120597x119895 1003816100381610038161003816100381610038161003816100381610038161003816x=x)2119896+1]]= 2(119899119909+119899120579)sum119895=1

[119899119909+119899120579sum119894=1

(x(119895)119894 )2119896+1 1205972119896+1ℎ120597x2119896+1119895 1003816100381610038161003816100381610038161003816100381610038161003816x=x]= 119899119909+119899120579sum119894=1

[[2(119899119909+119899120579)sum119895=1

(x(119895)119894 )2119896+1 1205972119896+1ℎ120597x2119896+1119895 1003816100381610038161003816100381610038161003816100381610038161003816x=x]] = 0(33)

Therefore all of the odd terms in (32) evaluate to zero and wehave

y= ℎ (x 120579)+ 12 (119899119909 + 119899120579) 2(119899119909+119899120579)sum

119895=1

( 121198632x(119895)ℎ + 141198634x(119895)ℎ + sdot sdot sdot) (34)

After some tedious calculation we can verify that

12 (119899119909 + 119899120579) 2(119899119909+119899120579)sum119895=1

121198632x(119895)ℎ = 12 119899119909+119899120579sum119894119895=1

P1198941198951205972ℎ120597x119894120597x119895 1003816100381610038161003816100381610038161003816100381610038161003816x=x= 12119864 (1198632xℎ) (35)

It can be seen that y (the approximated mean of y) matchesthe true mean of y correctly up to the third order

Theorem4 For random vector xwith a symmetric pdf aroundits mean if the nonlinear function y = ℎ(x 120579) containsunknown random parameter 120579 the approximated covarianceof y via the improved unscented transformation (25) matchesthe true covariance of y correctly up to the third order

Proof The true covariance of y is given as

Py = 119864 [(y minus y) (y minus y)1015840] (36)

By substituting (26) and (31) into (36) and using the same typeof reasoning in the proof ofTheorem 3 we have that all of theodd-powered terms evaluate to zero This leads to

Py = 119864 [119863xℎ (119863xℎ)1015840] + 119864[[119863xℎ (1198633xℎ)10158403

+ 1198632xℎ (1198632xℎ)101584022 + 1198633xℎ (119863xℎ)10158403 ]] + 119864[1198632xℎ2 ]sdot 119864 [1198632xℎ2 ]1015840 + sdot sdot sdot(37)

The first term on the right side of the equation above can bewritten as

119864 [119863xℎ (119863xℎ)1015840] = HPH1015840 (38)

whereH is the partial derivative matrix at x = x

6 Discrete Dynamics in Nature and Society

On the other hand after some tedious calculation theapproximate covariance Py in (25) can be written as

Py = 12 (119899119909 + 119899120579) 2(119899119909+119899120579)sum119895=1

(119863x(119895)ℎ) (119863x(119895)ℎ)1015840 +HOT

= 12 (119899119909 + 119899120579)sdot 2(119899119909+119899120579)sum119895=1

119899119909+119899120579sum119894119896=1

(x(119895)119894 120597ℎ (x)120597x119894 )(x(119895)119896 120597ℎ (x)120597x119896 )1015840+HOT(39)

whereHOTmeans higher-order terms (ie terms to the forthpower and higher) Now recall that x(119895)119894 = minusx(119899119909+119899120579+119895)119894 andx(119895)119896 = minusx(119899119909+119899120579+119895)119896 for 119895 = 1 119899119909 + 119899120579 therefore thecovariance approximation becomes

Py

= 1119899119909 + 119899120579 119899119909+119899120579sum119895=1 119899119909+119899120579sum119894119896=1 (x(119895)119894 120597ℎ (x)120597x119894 )(x(119895)119896 120597ℎ (x)120597x119896 )1015840+HOT = HPH1015840 +HOT(40)

Comparing (40) with the true covariance of y from (37) wesee that Py in (25) approximates the true covariance of y upto the third order

32 Improved Unscented Kalman Filter The improvedunscented transformation developed in the previoussubsection can be generalized to give the improvedunscented Kalman filter for the discrete-time nonlinearsystem which contains random parameters that is

x119896 = 119891 (x119896minus1 120579) + w119896y119896 = ℎ (x119896 120579) + k119896 (41)

where the motion function 119891(sdot) and the measurement func-tion ℎ(sdot) are nonlinear with respect to not only the state vectorx119896 but also the random parameter 120579 The other notations arethe same as that of model (1) Assume that the parameter 120579 isinR119899120579 has knownmean 120579 and covarianceP120579 and is uncorrelatedwith the state vector x119896 as well as the noise vectors w119896 and k119896

Following the augment method of the improvedunscented transformation we give the augment model of thediscrete-time nonlinear system (41) as follows

x119896 = 119891 (x119896minus1) + w119896y119896 = ℎ (x119896) + k119896 (42)

where

x119896 = [x1015840119896 1205791015840]1015840 w119896 = [w1015840119896 01015840]1015840 (43)

With the initial condition (7) the improved unscentedKalman filter can be initialized as followsx+0 = [(x+0 )1015840 1205791015840]1015840

P+0 = (P+0 0

0 P120579) (44)

Suppose that at time step 119896 x+119896minus1 and P+119896minus1 are given For thesake of brevity we denote 119899 = 119899119909 + 119899120579 The improved UKFalgorithm can be summarized in Algorithm 5

Algorithm 5 (the improved unscented Kalman filter)

(1) Choose 2 119899 sigma points x(119895)119896minus1 as specified in (19) withappropriate changes that isx(119895)119896minus1 = x+119896minus1 + x(119895)119896minus1 119895 = 1 2 119899x(119895)119896minus1 = (radic 119899P+

119896minus1)1015840119895 119895 = 1 119899x( 119899+119895)119896minus1 = minusx(119895)119896minus1 119895 = 1 119899

(45)

(2) Transform the sigma points x(119895)119896minus1 into x(119895)119896 as followsx(119895)119896 = 119891 (x(119895)119896minus1 120579(119895)119896minus1) (46)

where x(119895)119896minus1= x(119895)119896minus1(1 119899119909) and 120579(119895)119896minus1 = x(119895)119896minus1(119899119909 + 1 119899)

(3) Combine the x(119895)119896 vectors to obtain the predictionestimate at time

xminus

119896 = 12 119899 2 119899sum119895=1

x(119895)119896 (47)

and the corresponding error covariance

Pminus119896 = 12 119899 2 119899sum119895=1

(x(119895)119896 minus xminus119896)(x(119895)119896 minus xminus119896)1015840 + Q119908 (48)

where Q119908 = ( Q119908 00 0 )

(4) Transform the sigma points x(119895)119896minus1 into y(119895)119896 vectorsy(119895)119896 = ℎ (x(119895)119896minus1 120579(119895)119896minus1) (49)

(5) Combine the y(119895)119896 vectors to obtain the predictedmeasurement at time 119896y119896 = 12 119899 2 119899sum

119895=1

y(119895)119896 (50)

Discrete Dynamics in Nature and Society 7

and the corresponding covariance

P119910119896= 12 119899 2 119899sum119895=1

(y(119895)119896 minus y119896)(y(119895)119896 minus y119896)1015840 +QV (51)

(6) Estimate the cross covariance between xminus119896 and y119896P119909119910119896= 12 119899 2 119899sum119895=1

(x(119895)119896 minus xminus119896)(y(119895)119896 minus y119896)1015840 (52)

(7) The measurement update of the state estimate can beperformed as follows

K119896 = P119909119910119896(P119910119896)minus1 x+119896 = xminus119896 + K119896 (y119896 minus y119896)

P+119896 = Pminus119896 minus K119896P119910

119896K1015840119896 (53)

The proposed improved UKF method provides a sub-optimal solution for the stochastic filtering problem of anonlinear discrete-time dynamic system which contains ran-dom parameters The main difference between the improvedUKF algorithm and the normal UKF algorithm lies in theconstruction of the sigma pointsThe number of sigma pointsof the improved UKF algorithm is larger than that of thenormal UKF algorithm so generally the proposed improvedUKF method requires higher computational complexityHowever the advantage of the improved UKF method isthe increased accuracy of the state estimation which can beverified by the numerical results in the following section

4 An Application to MobileSource Localization

Consider a wireless sensor network with 119899 passive sensornodes distributed on a 2D plane to localize one movingsource Let the state vector of the mobile source be x119896 =[u1015840119896 k1015840119896]1015840 isin R4 which includes the position and velocitycomponents

Without loss of generality let the first sensor be thereference The TDOA measurement model [20] betweensensor pair 119894 (119894 = 2 119899) and 1 is

1199051198941 (u119896) = 10038171003817100381710038171003817u119896 minus s011989410038171003817100381710038171003817119888 minus 10038171003817100381710038171003817u119896 minus s01

10038171003817100381710038171003817119888 + Δ119905119894119896 (54)

where s0119894 is the true position of the 119894th sensor 1199030119894 (u119896) =u119896 minus s0119894 is the true distance from the source to the 119894thsensor and 119888 is the signal propagation speedThe vectorΔt119896 =[Δ1199052119896 Δ119905119899119896]1015840 is zero-meanGaussian noisewith covarianceQ119905 In simplicity (54) is expressed in the vector form

t119896 = [11990521 (u119896) 1199051198991 (u119896)] (55)

In practice the true sensor positions s0 = [s010158401 s010158404 ]1015840are not known and only their nominal values s =

Table 1 Nominal positions (in meters) of sensors

Sensor number 119894 119909119894 1199101198941 100 1002 100 minus1003 minus100 minus1004 minus100 100

[s10158401 s10158402 s10158404]1015840 are available for source localization Moreprecisely s0119894 and s119894 are related by

s0119894 = s119894 + Δs119894 (56)

where Δs119894 is the sensor position noise and the correspondingnoise vector Δs = [Δs10158401 Δs10158402 Δs1015840119899]1015840 is assumed to be zero-mean Gaussian with covariance matrixQ119904

Because the TDOAmeasurement vector (55) is nonlinearwith respect to the state vector x119896 as well as the true sensorlocations s0 the mobile source localization problem can bedescribed as the following discrete-time nonlinear system

t119896 = f (x119896 s0) + Δt119896 (57)

where s0 can be regarded as unknown random parametersThus we can apply the proposed improved UKF method todynamically estimate the state vector

In order to show the performance of the proposedimproved UKF method we apply it to two localization sce-narios and compare it with the normal UKF method whichignores the random uncertainties in the sensor locationsthrough Monte Carlo simulations

41 Scenario 1 This simulation scenario contains 119899 = 4sensors and their nominal positions are given in Table 1 Themotion model of the mobile source is assumed to be a nearlyconstant velocity model as

x119896 =(1 0 001 00 1 0 0010 0 1 00 0 0 1 ) x119896minus1

+(001 00 00101 00 01)w119896minus1(58)

where w119896minus1 is zero-mean Gaussian noise with covariancematrix Q119908 = 001I The initial value of the state x0 isa Gaussian vector with zero mean and covariance P+0 =diag100 100 001 001 The trajectory of the mobile sourcein 100 time instants is shown in Figure 1

The TDOA measurement covariance Q119905 = (05119888)2Twhere T is the (119899 minus 1) times (119899 minus 1)matrix with 1 in the diagonalelements and 05 otherwise The localization accuracy isevaluated by the root of mean squared error which is defined

8 Discrete Dynamics in Nature and Society

minus5

minus48

minus46

minus44

minus42

minus4

minus38

minus36

y-ax

is (m

)

13 135 14 145 15 155 16 165 17 175125x-axis (m)

Figure 1 The trajectory of the mobile source in Scenario 1

The normal UKFThe improved UKF

0

01

02

03

04

05

06

07

08

09

1

The R

MSE

of m

obile

sour

ce lo

caliz

atio

n

10 20 30 40 50 60 70 80 90 1000k

Figure 2 Comparison of the mobile localization accuracy with lowlevel sensor position uncertainties in Scenario 1

as RMSE(u119896) = radicsum119871119897=1 u119896 minus u1198962119871 where u119896 = x119896(1 2) isthe estimate of the source position at ensemble 119897 and 119871 = 1000is the number of ensemble runs

In theMonte Carlo simulations we consider three sensorposition uncertainty levels which are referred to as the lowlevel with 120590119904 = 01 the moderate level with 120590119904 = 1 and thehigh level with 120590119904 = 2

Figure 2 plots the simulation results with time instantvarying from 1 to 100 when the sensor position uncertaintylevel is low It is evident from Figure 2 that RMSE of theproposed improved UKF method (represented by dashedline) is smaller than that of the normal UKF approach(represented by solid line) which ignores the sensor positionuncertainties

Figure 3 plots the RMSEs of the two methods in the caseof moderate sensor position uncertainty level that is 120590119904 =1 We can see that both of them are generally larger than

The normal UKFThe improved UKF

0

01

02

03

04

05

06

07

The R

MSE

of m

obile

sour

ce lo

caliz

atio

n

10 20 30 40 50 60 70 80 90 1000k

Figure 3 Comparison of themobile localization accuracywithmildlevel sensor position uncertainties in Scenario 1

The normal UKFThe improved UKF

02

04

06

08

1

12

14

16

18

2

22

The R

MSE

of m

obile

sour

ce lo

caliz

atio

n

10 20 30 40 50 60 70 80 90 1000k

Figure 4 Comparison of themobile localization accuracywith highlevel sensor position uncertainties in Scenario 1

the corresponding RMSEs in Figure 2 As we expected theperformance of the proposed improvedUKFmethod ismuchbetter than that of the normal UKF approach In additionthe improvement is larger than that of the low level case inFigure 2

Finally we take the high sensor position uncertainty levelthat is120590119904 = 2The numerical simulation results are illustratedin Figure 4 which indicates that the RMSE of the normalUKF approach is nearly twice asmuch as that of the proposedimproved UKF method Comparing the results in Figures 2ndash4 we can see that the influence of the random sensor positionuncertainties on the localization performance is quite evidentin Scenario 1 The proposed improved UKF method can

Discrete Dynamics in Nature and Society 9

The normal UKFThe improved UKF

0

005

01

015

02

025

03

035

04

045

The R

MSE

of m

obile

sour

ce lo

caliz

atio

n

10 20 30 40 50 60 70 80 90 1000k

Figure 5 Comparison of the mobile localization accuracy with lowlevel sensor position uncertainties in Scenario 2

efficiently decrease the RMSE especially when the sensorposition uncertainty level is high

42 Scenario 2 In this simulation scenario we considerthe sensor geometry is random and the number of sensors119899 = 4 The true position of each sensor is uniformlydistributed in a square of [minus100 100]m times [minus100 100]mat each ensemble in Monte Carlo simulations The TDOAmeasurement covariance Q119905 = (01119888)2T The motion modeland the initial value of the state are generally the same as thoseof Scenario 1

In the first case the sensor position uncertainty levelis considerably small that is 120590119904 = 01 Figure 5 plots theRMSE of the normal UKF approach and that of the proposedimproved UKF method with time instant varying from 1 to100 which are calculated from 119871 = 1000Monte Carlo runsIt can be seen that the RMSE of the proposed improved UKFmethod is smaller than that of the normal UKF approach

Thenwe consider amoderate sensor position uncertaintylevel that is 120590119904 = 1 Figure 6 plots the RMSEs of the twomethods in this case where we can see that the performanceof the proposed improvedUKFmethod is still better than thatof the normal UKF approach with a larger improvement as inFigure 5

For the high sensor position uncertainty level that is120590119904 = 2 the simulation results are illustrated in Figure 7which shows that the RMSE of the normal UKF approach isconsiderably larger than that of the proposed improved UKFmethod

The Monte Carlo simulation results in Figures 5ndash7 revealthat the RMSEs of the proposed improved UKF are consis-tently smaller than those of the normal UKF approach whichcorroborates that the proposed improved UKF method cangenerally improve the estimation accuracy of the nonlineardynamic system when it contains random parameters

The normal UKFThe improved UKF

0

02

04

06

08

1

12

14

16

18

2

The R

MSE

of m

obile

sour

ce lo

caliz

atio

n

10 20 30 40 50 60 70 80 90 1000k

Figure 6 Comparison of themobile localization accuracywithmildlevel sensor position uncertainties in Scenario 2

The normal UKFThe improved UKF

0

05

1

15

2

25

3

35

The R

MSE

of m

obile

sour

ce lo

caliz

atio

n

10 20 30 40 50 60 70 80 90 1000k

Figure 7 Comparison of themobile localization accuracy with highlevel sensor position uncertainties in Scenario 2

5 Conclusions

In this paper an improved unscented Kalman filteringmethod is explored for discrete nonlinear dynamic systemswith randomparameters By augmenting the randomparam-eters into the state vector we enlarge the number of sigmapoints in the improved unscented transformation Theo-retical analysis indicates that the approximated mean andcovariance via the improved unscented transformationmatchthe true values correctly up to the third order of Taylor seriesexpansion An application to the mobile source localizationusing TDOA measurements is provided where the sensorpositions suffer from randomuncertaintiesTheMonte Carlo

10 Discrete Dynamics in Nature and Society

simulation results show that the proposed improved UKFmethod can considerably improve the estimation accuracy

Competing Interests

The authors declare that there are no competing interestsregarding the publication of this paper

Acknowledgments

This work was supported by the Fundamental ResearchFunds for the Central Universities Southwestern Universityof Finance and Economics under Grant no JBK1507059and Southwest University for Nationalities under Grant no2016NGJPY05

References

[1] R E Kalman ldquoA new approach to linear filtering and predictionproblemsrdquo Journal of Basic Engineering vol 82 no 1 pp 35ndash451960

[2] G Bolzon R Fedele and G Maier ldquoParameter identificationof a cohesive crack model by Kalman filterrdquo Computer Methodsin Applied Mechanics and Engineering vol 191 no 25-26 pp2847ndash2871 2002

[3] M Ball and A Wood ldquoTrend growth in post-1850 Britisheconomic history the Kalman filter and historical judgmentrdquoJournal of the Royal Statistical Society vol 45 no 2 Article ID143152 pp 143ndash152 1996

[4] W L De Koning ldquoOptimal estimation of linear discrete-timesystems with stochastic parametersrdquo Automatica vol 20 pp113ndash115 1984

[5] B Sinopoli L Schenato M Franceschetti K Poolla M IJordan and S S Sastry ldquoKalman filtering with intermittentobservationsrdquo IEEE Transactions on Automatic Control vol 49no 9 pp 1453ndash1464 2004

[6] Y Luo Y Zhu D Luo J Zhou E Song and DWang ldquoGloballyoptimal multisensor distributed random parameter matriceskalman filtering fusion with applicationsrdquo Sensors vol 8 no 12pp 8086ndash8103 2008

[7] S Chen Y Li G Qi and A Sheng ldquoAdaptive Kalman esti-mation in target tracking mixed with random one-step delaysstochastic-bias measurements and missing measurementsrdquoDiscrete Dynamics in Nature and Society vol 2013 Article ID716915 14 pages 2013

[8] S Chen Z Zhou and S Li ldquoAn efficient estimate and forecastof the implied volatility surface a nonlinear Kalman filterapproachrdquo Economic Modelling vol 58 pp 655ndash664 2016

[9] S Bolognani L Tubiana and M Zigliotto ldquoExtended Kalmanfilter tuning in sensorless PMSM drivesrdquo IEEE Transactions onIndustry Applications vol 39 no 6 pp 1741ndash1747 2003

[10] J J LaViola Jr ldquoA comparison of unscented and extendedkalman filtering for estimating quaternion motionrdquo in Proceed-ings of the AmericanControl Conference pp 2435ndash2440DenverColo USA June 2003

[11] X Qu and L Xie ldquoRecursive source localisation by time dif-ference of arrival sensor networks with sensor position uncer-taintyrdquo IET Control Theory and Applications vol 8 no 18 pp2305ndash2315 2014

[12] S J Julier J K Uhlmann and H F Durrant-Whyte ldquoNewapproach for filtering nonlinear systemsrdquo in Proceedings of theAmerican Control Conference pp 1628ndash1632 June 1995

[13] S J Julier and J K Uhlmann ldquoUnscented filtering and nonlin-ear estimationrdquo Proceedings of the IEEE vol 92 no 3 pp 401ndash422 2004

[14] Q Pan F Yang and L Ye ldquoSurvey of a kind of non linear filtersUKFrdquo Control and Decision vol 20 no 5 pp 481ndash489 2005

[15] C Eberle and C Ament ldquoThe Unscented Kalman Filter esti-mates the plasma insulin from glucose measurementrdquo BioSys-tems vol 103 no 1 pp 67ndash72 2011

[16] S J Julier ldquoSome relations between extended and unscentedKalman filtersrdquo IEEE Transactions on Signal Processing vol 60no 2 pp 545ndash555 2012

[17] HM T Menegaz J Y Ishihara G A Borges and A N VargasldquoA systematization of the unscented Kalman filter theoryrdquo IEEETransactions on Automatic Control vol 60 no 10 pp 2583ndash2598 2015

[18] D Simon Optimal State Estimation Kalman H Infinity andNonlinear Approaches Wiley-Interscience 2006

[19] KW Lui F K Chan andH C So ldquoSemidefinite programmingapproach for range-difference based source localizationrdquo IEEETransactions on Signal Processing vol 57 no 4 pp 1630ndash16332009

[20] X Qu and L Xie ldquoAn efficient convex constrained weightedleast squares source localization algorithm based on TDOAmeasurementsrdquo Signal Processing vol 119 pp 142ndash152 2016

[21] G Yanni and W Qi ldquoThe nonsequential fusion method forlocalization from unscented kalman filter by multistation arraybuoysrdquo Discrete Dynamics in Nature and Society vol 2016Article ID 7670609 8 pages 2016

[22] A H Jazwinski Stochastic Processes and Filtering Theory Aca-demic Press Cambridge Mass USA 1970

[23] F Daum ldquoNonlinear filters beyond the kalman filterrdquo IEEEAerospace and Electronic Systems Magazine vol 20 no 8 pp57ndash69 2005

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Differential EquationsInternational Journal of

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Applied MathematicsJournal of

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Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

6 Discrete Dynamics in Nature and Society

On the other hand after some tedious calculation theapproximate covariance Py in (25) can be written as

Py = 12 (119899119909 + 119899120579) 2(119899119909+119899120579)sum119895=1

(119863x(119895)ℎ) (119863x(119895)ℎ)1015840 +HOT

= 12 (119899119909 + 119899120579)sdot 2(119899119909+119899120579)sum119895=1

119899119909+119899120579sum119894119896=1

(x(119895)119894 120597ℎ (x)120597x119894 )(x(119895)119896 120597ℎ (x)120597x119896 )1015840+HOT(39)

whereHOTmeans higher-order terms (ie terms to the forthpower and higher) Now recall that x(119895)119894 = minusx(119899119909+119899120579+119895)119894 andx(119895)119896 = minusx(119899119909+119899120579+119895)119896 for 119895 = 1 119899119909 + 119899120579 therefore thecovariance approximation becomes

Py

= 1119899119909 + 119899120579 119899119909+119899120579sum119895=1 119899119909+119899120579sum119894119896=1 (x(119895)119894 120597ℎ (x)120597x119894 )(x(119895)119896 120597ℎ (x)120597x119896 )1015840+HOT = HPH1015840 +HOT(40)

Comparing (40) with the true covariance of y from (37) wesee that Py in (25) approximates the true covariance of y upto the third order

32 Improved Unscented Kalman Filter The improvedunscented transformation developed in the previoussubsection can be generalized to give the improvedunscented Kalman filter for the discrete-time nonlinearsystem which contains random parameters that is

x119896 = 119891 (x119896minus1 120579) + w119896y119896 = ℎ (x119896 120579) + k119896 (41)

where the motion function 119891(sdot) and the measurement func-tion ℎ(sdot) are nonlinear with respect to not only the state vectorx119896 but also the random parameter 120579 The other notations arethe same as that of model (1) Assume that the parameter 120579 isinR119899120579 has knownmean 120579 and covarianceP120579 and is uncorrelatedwith the state vector x119896 as well as the noise vectors w119896 and k119896

Following the augment method of the improvedunscented transformation we give the augment model of thediscrete-time nonlinear system (41) as follows

x119896 = 119891 (x119896minus1) + w119896y119896 = ℎ (x119896) + k119896 (42)

where

x119896 = [x1015840119896 1205791015840]1015840 w119896 = [w1015840119896 01015840]1015840 (43)

With the initial condition (7) the improved unscentedKalman filter can be initialized as followsx+0 = [(x+0 )1015840 1205791015840]1015840

P+0 = (P+0 0

0 P120579) (44)

Suppose that at time step 119896 x+119896minus1 and P+119896minus1 are given For thesake of brevity we denote 119899 = 119899119909 + 119899120579 The improved UKFalgorithm can be summarized in Algorithm 5

Algorithm 5 (the improved unscented Kalman filter)

(1) Choose 2 119899 sigma points x(119895)119896minus1 as specified in (19) withappropriate changes that isx(119895)119896minus1 = x+119896minus1 + x(119895)119896minus1 119895 = 1 2 119899x(119895)119896minus1 = (radic 119899P+

119896minus1)1015840119895 119895 = 1 119899x( 119899+119895)119896minus1 = minusx(119895)119896minus1 119895 = 1 119899

(45)

(2) Transform the sigma points x(119895)119896minus1 into x(119895)119896 as followsx(119895)119896 = 119891 (x(119895)119896minus1 120579(119895)119896minus1) (46)

where x(119895)119896minus1= x(119895)119896minus1(1 119899119909) and 120579(119895)119896minus1 = x(119895)119896minus1(119899119909 + 1 119899)

(3) Combine the x(119895)119896 vectors to obtain the predictionestimate at time

xminus

119896 = 12 119899 2 119899sum119895=1

x(119895)119896 (47)

and the corresponding error covariance

Pminus119896 = 12 119899 2 119899sum119895=1

(x(119895)119896 minus xminus119896)(x(119895)119896 minus xminus119896)1015840 + Q119908 (48)

where Q119908 = ( Q119908 00 0 )

(4) Transform the sigma points x(119895)119896minus1 into y(119895)119896 vectorsy(119895)119896 = ℎ (x(119895)119896minus1 120579(119895)119896minus1) (49)

(5) Combine the y(119895)119896 vectors to obtain the predictedmeasurement at time 119896y119896 = 12 119899 2 119899sum

119895=1

y(119895)119896 (50)

Discrete Dynamics in Nature and Society 7

and the corresponding covariance

P119910119896= 12 119899 2 119899sum119895=1

(y(119895)119896 minus y119896)(y(119895)119896 minus y119896)1015840 +QV (51)

(6) Estimate the cross covariance between xminus119896 and y119896P119909119910119896= 12 119899 2 119899sum119895=1

(x(119895)119896 minus xminus119896)(y(119895)119896 minus y119896)1015840 (52)

(7) The measurement update of the state estimate can beperformed as follows

K119896 = P119909119910119896(P119910119896)minus1 x+119896 = xminus119896 + K119896 (y119896 minus y119896)

P+119896 = Pminus119896 minus K119896P119910

119896K1015840119896 (53)

The proposed improved UKF method provides a sub-optimal solution for the stochastic filtering problem of anonlinear discrete-time dynamic system which contains ran-dom parameters The main difference between the improvedUKF algorithm and the normal UKF algorithm lies in theconstruction of the sigma pointsThe number of sigma pointsof the improved UKF algorithm is larger than that of thenormal UKF algorithm so generally the proposed improvedUKF method requires higher computational complexityHowever the advantage of the improved UKF method isthe increased accuracy of the state estimation which can beverified by the numerical results in the following section

4 An Application to MobileSource Localization

Consider a wireless sensor network with 119899 passive sensornodes distributed on a 2D plane to localize one movingsource Let the state vector of the mobile source be x119896 =[u1015840119896 k1015840119896]1015840 isin R4 which includes the position and velocitycomponents

Without loss of generality let the first sensor be thereference The TDOA measurement model [20] betweensensor pair 119894 (119894 = 2 119899) and 1 is

1199051198941 (u119896) = 10038171003817100381710038171003817u119896 minus s011989410038171003817100381710038171003817119888 minus 10038171003817100381710038171003817u119896 minus s01

10038171003817100381710038171003817119888 + Δ119905119894119896 (54)

where s0119894 is the true position of the 119894th sensor 1199030119894 (u119896) =u119896 minus s0119894 is the true distance from the source to the 119894thsensor and 119888 is the signal propagation speedThe vectorΔt119896 =[Δ1199052119896 Δ119905119899119896]1015840 is zero-meanGaussian noisewith covarianceQ119905 In simplicity (54) is expressed in the vector form

t119896 = [11990521 (u119896) 1199051198991 (u119896)] (55)

In practice the true sensor positions s0 = [s010158401 s010158404 ]1015840are not known and only their nominal values s =

Table 1 Nominal positions (in meters) of sensors

Sensor number 119894 119909119894 1199101198941 100 1002 100 minus1003 minus100 minus1004 minus100 100

[s10158401 s10158402 s10158404]1015840 are available for source localization Moreprecisely s0119894 and s119894 are related by

s0119894 = s119894 + Δs119894 (56)

where Δs119894 is the sensor position noise and the correspondingnoise vector Δs = [Δs10158401 Δs10158402 Δs1015840119899]1015840 is assumed to be zero-mean Gaussian with covariance matrixQ119904

Because the TDOAmeasurement vector (55) is nonlinearwith respect to the state vector x119896 as well as the true sensorlocations s0 the mobile source localization problem can bedescribed as the following discrete-time nonlinear system

t119896 = f (x119896 s0) + Δt119896 (57)

where s0 can be regarded as unknown random parametersThus we can apply the proposed improved UKF method todynamically estimate the state vector

In order to show the performance of the proposedimproved UKF method we apply it to two localization sce-narios and compare it with the normal UKF method whichignores the random uncertainties in the sensor locationsthrough Monte Carlo simulations

41 Scenario 1 This simulation scenario contains 119899 = 4sensors and their nominal positions are given in Table 1 Themotion model of the mobile source is assumed to be a nearlyconstant velocity model as

x119896 =(1 0 001 00 1 0 0010 0 1 00 0 0 1 ) x119896minus1

+(001 00 00101 00 01)w119896minus1(58)

where w119896minus1 is zero-mean Gaussian noise with covariancematrix Q119908 = 001I The initial value of the state x0 isa Gaussian vector with zero mean and covariance P+0 =diag100 100 001 001 The trajectory of the mobile sourcein 100 time instants is shown in Figure 1

The TDOA measurement covariance Q119905 = (05119888)2Twhere T is the (119899 minus 1) times (119899 minus 1)matrix with 1 in the diagonalelements and 05 otherwise The localization accuracy isevaluated by the root of mean squared error which is defined

8 Discrete Dynamics in Nature and Society

minus5

minus48

minus46

minus44

minus42

minus4

minus38

minus36

y-ax

is (m

)

13 135 14 145 15 155 16 165 17 175125x-axis (m)

Figure 1 The trajectory of the mobile source in Scenario 1

The normal UKFThe improved UKF

0

01

02

03

04

05

06

07

08

09

1

The R

MSE

of m

obile

sour

ce lo

caliz

atio

n

10 20 30 40 50 60 70 80 90 1000k

Figure 2 Comparison of the mobile localization accuracy with lowlevel sensor position uncertainties in Scenario 1

as RMSE(u119896) = radicsum119871119897=1 u119896 minus u1198962119871 where u119896 = x119896(1 2) isthe estimate of the source position at ensemble 119897 and 119871 = 1000is the number of ensemble runs

In theMonte Carlo simulations we consider three sensorposition uncertainty levels which are referred to as the lowlevel with 120590119904 = 01 the moderate level with 120590119904 = 1 and thehigh level with 120590119904 = 2

Figure 2 plots the simulation results with time instantvarying from 1 to 100 when the sensor position uncertaintylevel is low It is evident from Figure 2 that RMSE of theproposed improved UKF method (represented by dashedline) is smaller than that of the normal UKF approach(represented by solid line) which ignores the sensor positionuncertainties

Figure 3 plots the RMSEs of the two methods in the caseof moderate sensor position uncertainty level that is 120590119904 =1 We can see that both of them are generally larger than

The normal UKFThe improved UKF

0

01

02

03

04

05

06

07

The R

MSE

of m

obile

sour

ce lo

caliz

atio

n

10 20 30 40 50 60 70 80 90 1000k

Figure 3 Comparison of themobile localization accuracywithmildlevel sensor position uncertainties in Scenario 1

The normal UKFThe improved UKF

02

04

06

08

1

12

14

16

18

2

22

The R

MSE

of m

obile

sour

ce lo

caliz

atio

n

10 20 30 40 50 60 70 80 90 1000k

Figure 4 Comparison of themobile localization accuracywith highlevel sensor position uncertainties in Scenario 1

the corresponding RMSEs in Figure 2 As we expected theperformance of the proposed improvedUKFmethod ismuchbetter than that of the normal UKF approach In additionthe improvement is larger than that of the low level case inFigure 2

Finally we take the high sensor position uncertainty levelthat is120590119904 = 2The numerical simulation results are illustratedin Figure 4 which indicates that the RMSE of the normalUKF approach is nearly twice asmuch as that of the proposedimproved UKF method Comparing the results in Figures 2ndash4 we can see that the influence of the random sensor positionuncertainties on the localization performance is quite evidentin Scenario 1 The proposed improved UKF method can

Discrete Dynamics in Nature and Society 9

The normal UKFThe improved UKF

0

005

01

015

02

025

03

035

04

045

The R

MSE

of m

obile

sour

ce lo

caliz

atio

n

10 20 30 40 50 60 70 80 90 1000k

Figure 5 Comparison of the mobile localization accuracy with lowlevel sensor position uncertainties in Scenario 2

efficiently decrease the RMSE especially when the sensorposition uncertainty level is high

42 Scenario 2 In this simulation scenario we considerthe sensor geometry is random and the number of sensors119899 = 4 The true position of each sensor is uniformlydistributed in a square of [minus100 100]m times [minus100 100]mat each ensemble in Monte Carlo simulations The TDOAmeasurement covariance Q119905 = (01119888)2T The motion modeland the initial value of the state are generally the same as thoseof Scenario 1

In the first case the sensor position uncertainty levelis considerably small that is 120590119904 = 01 Figure 5 plots theRMSE of the normal UKF approach and that of the proposedimproved UKF method with time instant varying from 1 to100 which are calculated from 119871 = 1000Monte Carlo runsIt can be seen that the RMSE of the proposed improved UKFmethod is smaller than that of the normal UKF approach

Thenwe consider amoderate sensor position uncertaintylevel that is 120590119904 = 1 Figure 6 plots the RMSEs of the twomethods in this case where we can see that the performanceof the proposed improvedUKFmethod is still better than thatof the normal UKF approach with a larger improvement as inFigure 5

For the high sensor position uncertainty level that is120590119904 = 2 the simulation results are illustrated in Figure 7which shows that the RMSE of the normal UKF approach isconsiderably larger than that of the proposed improved UKFmethod

The Monte Carlo simulation results in Figures 5ndash7 revealthat the RMSEs of the proposed improved UKF are consis-tently smaller than those of the normal UKF approach whichcorroborates that the proposed improved UKF method cangenerally improve the estimation accuracy of the nonlineardynamic system when it contains random parameters

The normal UKFThe improved UKF

0

02

04

06

08

1

12

14

16

18

2

The R

MSE

of m

obile

sour

ce lo

caliz

atio

n

10 20 30 40 50 60 70 80 90 1000k

Figure 6 Comparison of themobile localization accuracywithmildlevel sensor position uncertainties in Scenario 2

The normal UKFThe improved UKF

0

05

1

15

2

25

3

35

The R

MSE

of m

obile

sour

ce lo

caliz

atio

n

10 20 30 40 50 60 70 80 90 1000k

Figure 7 Comparison of themobile localization accuracy with highlevel sensor position uncertainties in Scenario 2

5 Conclusions

In this paper an improved unscented Kalman filteringmethod is explored for discrete nonlinear dynamic systemswith randomparameters By augmenting the randomparam-eters into the state vector we enlarge the number of sigmapoints in the improved unscented transformation Theo-retical analysis indicates that the approximated mean andcovariance via the improved unscented transformationmatchthe true values correctly up to the third order of Taylor seriesexpansion An application to the mobile source localizationusing TDOA measurements is provided where the sensorpositions suffer from randomuncertaintiesTheMonte Carlo

10 Discrete Dynamics in Nature and Society

simulation results show that the proposed improved UKFmethod can considerably improve the estimation accuracy

Competing Interests

The authors declare that there are no competing interestsregarding the publication of this paper

Acknowledgments

This work was supported by the Fundamental ResearchFunds for the Central Universities Southwestern Universityof Finance and Economics under Grant no JBK1507059and Southwest University for Nationalities under Grant no2016NGJPY05

References

[1] R E Kalman ldquoA new approach to linear filtering and predictionproblemsrdquo Journal of Basic Engineering vol 82 no 1 pp 35ndash451960

[2] G Bolzon R Fedele and G Maier ldquoParameter identificationof a cohesive crack model by Kalman filterrdquo Computer Methodsin Applied Mechanics and Engineering vol 191 no 25-26 pp2847ndash2871 2002

[3] M Ball and A Wood ldquoTrend growth in post-1850 Britisheconomic history the Kalman filter and historical judgmentrdquoJournal of the Royal Statistical Society vol 45 no 2 Article ID143152 pp 143ndash152 1996

[4] W L De Koning ldquoOptimal estimation of linear discrete-timesystems with stochastic parametersrdquo Automatica vol 20 pp113ndash115 1984

[5] B Sinopoli L Schenato M Franceschetti K Poolla M IJordan and S S Sastry ldquoKalman filtering with intermittentobservationsrdquo IEEE Transactions on Automatic Control vol 49no 9 pp 1453ndash1464 2004

[6] Y Luo Y Zhu D Luo J Zhou E Song and DWang ldquoGloballyoptimal multisensor distributed random parameter matriceskalman filtering fusion with applicationsrdquo Sensors vol 8 no 12pp 8086ndash8103 2008

[7] S Chen Y Li G Qi and A Sheng ldquoAdaptive Kalman esti-mation in target tracking mixed with random one-step delaysstochastic-bias measurements and missing measurementsrdquoDiscrete Dynamics in Nature and Society vol 2013 Article ID716915 14 pages 2013

[8] S Chen Z Zhou and S Li ldquoAn efficient estimate and forecastof the implied volatility surface a nonlinear Kalman filterapproachrdquo Economic Modelling vol 58 pp 655ndash664 2016

[9] S Bolognani L Tubiana and M Zigliotto ldquoExtended Kalmanfilter tuning in sensorless PMSM drivesrdquo IEEE Transactions onIndustry Applications vol 39 no 6 pp 1741ndash1747 2003

[10] J J LaViola Jr ldquoA comparison of unscented and extendedkalman filtering for estimating quaternion motionrdquo in Proceed-ings of the AmericanControl Conference pp 2435ndash2440DenverColo USA June 2003

[11] X Qu and L Xie ldquoRecursive source localisation by time dif-ference of arrival sensor networks with sensor position uncer-taintyrdquo IET Control Theory and Applications vol 8 no 18 pp2305ndash2315 2014

[12] S J Julier J K Uhlmann and H F Durrant-Whyte ldquoNewapproach for filtering nonlinear systemsrdquo in Proceedings of theAmerican Control Conference pp 1628ndash1632 June 1995

[13] S J Julier and J K Uhlmann ldquoUnscented filtering and nonlin-ear estimationrdquo Proceedings of the IEEE vol 92 no 3 pp 401ndash422 2004

[14] Q Pan F Yang and L Ye ldquoSurvey of a kind of non linear filtersUKFrdquo Control and Decision vol 20 no 5 pp 481ndash489 2005

[15] C Eberle and C Ament ldquoThe Unscented Kalman Filter esti-mates the plasma insulin from glucose measurementrdquo BioSys-tems vol 103 no 1 pp 67ndash72 2011

[16] S J Julier ldquoSome relations between extended and unscentedKalman filtersrdquo IEEE Transactions on Signal Processing vol 60no 2 pp 545ndash555 2012

[17] HM T Menegaz J Y Ishihara G A Borges and A N VargasldquoA systematization of the unscented Kalman filter theoryrdquo IEEETransactions on Automatic Control vol 60 no 10 pp 2583ndash2598 2015

[18] D Simon Optimal State Estimation Kalman H Infinity andNonlinear Approaches Wiley-Interscience 2006

[19] KW Lui F K Chan andH C So ldquoSemidefinite programmingapproach for range-difference based source localizationrdquo IEEETransactions on Signal Processing vol 57 no 4 pp 1630ndash16332009

[20] X Qu and L Xie ldquoAn efficient convex constrained weightedleast squares source localization algorithm based on TDOAmeasurementsrdquo Signal Processing vol 119 pp 142ndash152 2016

[21] G Yanni and W Qi ldquoThe nonsequential fusion method forlocalization from unscented kalman filter by multistation arraybuoysrdquo Discrete Dynamics in Nature and Society vol 2016Article ID 7670609 8 pages 2016

[22] A H Jazwinski Stochastic Processes and Filtering Theory Aca-demic Press Cambridge Mass USA 1970

[23] F Daum ldquoNonlinear filters beyond the kalman filterrdquo IEEEAerospace and Electronic Systems Magazine vol 20 no 8 pp57ndash69 2005

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Mathematical Problems in Engineering

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Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

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Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Discrete Dynamics in Nature and Society 7

and the corresponding covariance

P119910119896= 12 119899 2 119899sum119895=1

(y(119895)119896 minus y119896)(y(119895)119896 minus y119896)1015840 +QV (51)

(6) Estimate the cross covariance between xminus119896 and y119896P119909119910119896= 12 119899 2 119899sum119895=1

(x(119895)119896 minus xminus119896)(y(119895)119896 minus y119896)1015840 (52)

(7) The measurement update of the state estimate can beperformed as follows

K119896 = P119909119910119896(P119910119896)minus1 x+119896 = xminus119896 + K119896 (y119896 minus y119896)

P+119896 = Pminus119896 minus K119896P119910

119896K1015840119896 (53)

The proposed improved UKF method provides a sub-optimal solution for the stochastic filtering problem of anonlinear discrete-time dynamic system which contains ran-dom parameters The main difference between the improvedUKF algorithm and the normal UKF algorithm lies in theconstruction of the sigma pointsThe number of sigma pointsof the improved UKF algorithm is larger than that of thenormal UKF algorithm so generally the proposed improvedUKF method requires higher computational complexityHowever the advantage of the improved UKF method isthe increased accuracy of the state estimation which can beverified by the numerical results in the following section

4 An Application to MobileSource Localization

Consider a wireless sensor network with 119899 passive sensornodes distributed on a 2D plane to localize one movingsource Let the state vector of the mobile source be x119896 =[u1015840119896 k1015840119896]1015840 isin R4 which includes the position and velocitycomponents

Without loss of generality let the first sensor be thereference The TDOA measurement model [20] betweensensor pair 119894 (119894 = 2 119899) and 1 is

1199051198941 (u119896) = 10038171003817100381710038171003817u119896 minus s011989410038171003817100381710038171003817119888 minus 10038171003817100381710038171003817u119896 minus s01

10038171003817100381710038171003817119888 + Δ119905119894119896 (54)

where s0119894 is the true position of the 119894th sensor 1199030119894 (u119896) =u119896 minus s0119894 is the true distance from the source to the 119894thsensor and 119888 is the signal propagation speedThe vectorΔt119896 =[Δ1199052119896 Δ119905119899119896]1015840 is zero-meanGaussian noisewith covarianceQ119905 In simplicity (54) is expressed in the vector form

t119896 = [11990521 (u119896) 1199051198991 (u119896)] (55)

In practice the true sensor positions s0 = [s010158401 s010158404 ]1015840are not known and only their nominal values s =

Table 1 Nominal positions (in meters) of sensors

Sensor number 119894 119909119894 1199101198941 100 1002 100 minus1003 minus100 minus1004 minus100 100

[s10158401 s10158402 s10158404]1015840 are available for source localization Moreprecisely s0119894 and s119894 are related by

s0119894 = s119894 + Δs119894 (56)

where Δs119894 is the sensor position noise and the correspondingnoise vector Δs = [Δs10158401 Δs10158402 Δs1015840119899]1015840 is assumed to be zero-mean Gaussian with covariance matrixQ119904

Because the TDOAmeasurement vector (55) is nonlinearwith respect to the state vector x119896 as well as the true sensorlocations s0 the mobile source localization problem can bedescribed as the following discrete-time nonlinear system

t119896 = f (x119896 s0) + Δt119896 (57)

where s0 can be regarded as unknown random parametersThus we can apply the proposed improved UKF method todynamically estimate the state vector

In order to show the performance of the proposedimproved UKF method we apply it to two localization sce-narios and compare it with the normal UKF method whichignores the random uncertainties in the sensor locationsthrough Monte Carlo simulations

41 Scenario 1 This simulation scenario contains 119899 = 4sensors and their nominal positions are given in Table 1 Themotion model of the mobile source is assumed to be a nearlyconstant velocity model as

x119896 =(1 0 001 00 1 0 0010 0 1 00 0 0 1 ) x119896minus1

+(001 00 00101 00 01)w119896minus1(58)

where w119896minus1 is zero-mean Gaussian noise with covariancematrix Q119908 = 001I The initial value of the state x0 isa Gaussian vector with zero mean and covariance P+0 =diag100 100 001 001 The trajectory of the mobile sourcein 100 time instants is shown in Figure 1

The TDOA measurement covariance Q119905 = (05119888)2Twhere T is the (119899 minus 1) times (119899 minus 1)matrix with 1 in the diagonalelements and 05 otherwise The localization accuracy isevaluated by the root of mean squared error which is defined

8 Discrete Dynamics in Nature and Society

minus5

minus48

minus46

minus44

minus42

minus4

minus38

minus36

y-ax

is (m

)

13 135 14 145 15 155 16 165 17 175125x-axis (m)

Figure 1 The trajectory of the mobile source in Scenario 1

The normal UKFThe improved UKF

0

01

02

03

04

05

06

07

08

09

1

The R

MSE

of m

obile

sour

ce lo

caliz

atio

n

10 20 30 40 50 60 70 80 90 1000k

Figure 2 Comparison of the mobile localization accuracy with lowlevel sensor position uncertainties in Scenario 1

as RMSE(u119896) = radicsum119871119897=1 u119896 minus u1198962119871 where u119896 = x119896(1 2) isthe estimate of the source position at ensemble 119897 and 119871 = 1000is the number of ensemble runs

In theMonte Carlo simulations we consider three sensorposition uncertainty levels which are referred to as the lowlevel with 120590119904 = 01 the moderate level with 120590119904 = 1 and thehigh level with 120590119904 = 2

Figure 2 plots the simulation results with time instantvarying from 1 to 100 when the sensor position uncertaintylevel is low It is evident from Figure 2 that RMSE of theproposed improved UKF method (represented by dashedline) is smaller than that of the normal UKF approach(represented by solid line) which ignores the sensor positionuncertainties

Figure 3 plots the RMSEs of the two methods in the caseof moderate sensor position uncertainty level that is 120590119904 =1 We can see that both of them are generally larger than

The normal UKFThe improved UKF

0

01

02

03

04

05

06

07

The R

MSE

of m

obile

sour

ce lo

caliz

atio

n

10 20 30 40 50 60 70 80 90 1000k

Figure 3 Comparison of themobile localization accuracywithmildlevel sensor position uncertainties in Scenario 1

The normal UKFThe improved UKF

02

04

06

08

1

12

14

16

18

2

22

The R

MSE

of m

obile

sour

ce lo

caliz

atio

n

10 20 30 40 50 60 70 80 90 1000k

Figure 4 Comparison of themobile localization accuracywith highlevel sensor position uncertainties in Scenario 1

the corresponding RMSEs in Figure 2 As we expected theperformance of the proposed improvedUKFmethod ismuchbetter than that of the normal UKF approach In additionthe improvement is larger than that of the low level case inFigure 2

Finally we take the high sensor position uncertainty levelthat is120590119904 = 2The numerical simulation results are illustratedin Figure 4 which indicates that the RMSE of the normalUKF approach is nearly twice asmuch as that of the proposedimproved UKF method Comparing the results in Figures 2ndash4 we can see that the influence of the random sensor positionuncertainties on the localization performance is quite evidentin Scenario 1 The proposed improved UKF method can

Discrete Dynamics in Nature and Society 9

The normal UKFThe improved UKF

0

005

01

015

02

025

03

035

04

045

The R

MSE

of m

obile

sour

ce lo

caliz

atio

n

10 20 30 40 50 60 70 80 90 1000k

Figure 5 Comparison of the mobile localization accuracy with lowlevel sensor position uncertainties in Scenario 2

efficiently decrease the RMSE especially when the sensorposition uncertainty level is high

42 Scenario 2 In this simulation scenario we considerthe sensor geometry is random and the number of sensors119899 = 4 The true position of each sensor is uniformlydistributed in a square of [minus100 100]m times [minus100 100]mat each ensemble in Monte Carlo simulations The TDOAmeasurement covariance Q119905 = (01119888)2T The motion modeland the initial value of the state are generally the same as thoseof Scenario 1

In the first case the sensor position uncertainty levelis considerably small that is 120590119904 = 01 Figure 5 plots theRMSE of the normal UKF approach and that of the proposedimproved UKF method with time instant varying from 1 to100 which are calculated from 119871 = 1000Monte Carlo runsIt can be seen that the RMSE of the proposed improved UKFmethod is smaller than that of the normal UKF approach

Thenwe consider amoderate sensor position uncertaintylevel that is 120590119904 = 1 Figure 6 plots the RMSEs of the twomethods in this case where we can see that the performanceof the proposed improvedUKFmethod is still better than thatof the normal UKF approach with a larger improvement as inFigure 5

For the high sensor position uncertainty level that is120590119904 = 2 the simulation results are illustrated in Figure 7which shows that the RMSE of the normal UKF approach isconsiderably larger than that of the proposed improved UKFmethod

The Monte Carlo simulation results in Figures 5ndash7 revealthat the RMSEs of the proposed improved UKF are consis-tently smaller than those of the normal UKF approach whichcorroborates that the proposed improved UKF method cangenerally improve the estimation accuracy of the nonlineardynamic system when it contains random parameters

The normal UKFThe improved UKF

0

02

04

06

08

1

12

14

16

18

2

The R

MSE

of m

obile

sour

ce lo

caliz

atio

n

10 20 30 40 50 60 70 80 90 1000k

Figure 6 Comparison of themobile localization accuracywithmildlevel sensor position uncertainties in Scenario 2

The normal UKFThe improved UKF

0

05

1

15

2

25

3

35

The R

MSE

of m

obile

sour

ce lo

caliz

atio

n

10 20 30 40 50 60 70 80 90 1000k

Figure 7 Comparison of themobile localization accuracy with highlevel sensor position uncertainties in Scenario 2

5 Conclusions

In this paper an improved unscented Kalman filteringmethod is explored for discrete nonlinear dynamic systemswith randomparameters By augmenting the randomparam-eters into the state vector we enlarge the number of sigmapoints in the improved unscented transformation Theo-retical analysis indicates that the approximated mean andcovariance via the improved unscented transformationmatchthe true values correctly up to the third order of Taylor seriesexpansion An application to the mobile source localizationusing TDOA measurements is provided where the sensorpositions suffer from randomuncertaintiesTheMonte Carlo

10 Discrete Dynamics in Nature and Society

simulation results show that the proposed improved UKFmethod can considerably improve the estimation accuracy

Competing Interests

The authors declare that there are no competing interestsregarding the publication of this paper

Acknowledgments

This work was supported by the Fundamental ResearchFunds for the Central Universities Southwestern Universityof Finance and Economics under Grant no JBK1507059and Southwest University for Nationalities under Grant no2016NGJPY05

References

[1] R E Kalman ldquoA new approach to linear filtering and predictionproblemsrdquo Journal of Basic Engineering vol 82 no 1 pp 35ndash451960

[2] G Bolzon R Fedele and G Maier ldquoParameter identificationof a cohesive crack model by Kalman filterrdquo Computer Methodsin Applied Mechanics and Engineering vol 191 no 25-26 pp2847ndash2871 2002

[3] M Ball and A Wood ldquoTrend growth in post-1850 Britisheconomic history the Kalman filter and historical judgmentrdquoJournal of the Royal Statistical Society vol 45 no 2 Article ID143152 pp 143ndash152 1996

[4] W L De Koning ldquoOptimal estimation of linear discrete-timesystems with stochastic parametersrdquo Automatica vol 20 pp113ndash115 1984

[5] B Sinopoli L Schenato M Franceschetti K Poolla M IJordan and S S Sastry ldquoKalman filtering with intermittentobservationsrdquo IEEE Transactions on Automatic Control vol 49no 9 pp 1453ndash1464 2004

[6] Y Luo Y Zhu D Luo J Zhou E Song and DWang ldquoGloballyoptimal multisensor distributed random parameter matriceskalman filtering fusion with applicationsrdquo Sensors vol 8 no 12pp 8086ndash8103 2008

[7] S Chen Y Li G Qi and A Sheng ldquoAdaptive Kalman esti-mation in target tracking mixed with random one-step delaysstochastic-bias measurements and missing measurementsrdquoDiscrete Dynamics in Nature and Society vol 2013 Article ID716915 14 pages 2013

[8] S Chen Z Zhou and S Li ldquoAn efficient estimate and forecastof the implied volatility surface a nonlinear Kalman filterapproachrdquo Economic Modelling vol 58 pp 655ndash664 2016

[9] S Bolognani L Tubiana and M Zigliotto ldquoExtended Kalmanfilter tuning in sensorless PMSM drivesrdquo IEEE Transactions onIndustry Applications vol 39 no 6 pp 1741ndash1747 2003

[10] J J LaViola Jr ldquoA comparison of unscented and extendedkalman filtering for estimating quaternion motionrdquo in Proceed-ings of the AmericanControl Conference pp 2435ndash2440DenverColo USA June 2003

[11] X Qu and L Xie ldquoRecursive source localisation by time dif-ference of arrival sensor networks with sensor position uncer-taintyrdquo IET Control Theory and Applications vol 8 no 18 pp2305ndash2315 2014

[12] S J Julier J K Uhlmann and H F Durrant-Whyte ldquoNewapproach for filtering nonlinear systemsrdquo in Proceedings of theAmerican Control Conference pp 1628ndash1632 June 1995

[13] S J Julier and J K Uhlmann ldquoUnscented filtering and nonlin-ear estimationrdquo Proceedings of the IEEE vol 92 no 3 pp 401ndash422 2004

[14] Q Pan F Yang and L Ye ldquoSurvey of a kind of non linear filtersUKFrdquo Control and Decision vol 20 no 5 pp 481ndash489 2005

[15] C Eberle and C Ament ldquoThe Unscented Kalman Filter esti-mates the plasma insulin from glucose measurementrdquo BioSys-tems vol 103 no 1 pp 67ndash72 2011

[16] S J Julier ldquoSome relations between extended and unscentedKalman filtersrdquo IEEE Transactions on Signal Processing vol 60no 2 pp 545ndash555 2012

[17] HM T Menegaz J Y Ishihara G A Borges and A N VargasldquoA systematization of the unscented Kalman filter theoryrdquo IEEETransactions on Automatic Control vol 60 no 10 pp 2583ndash2598 2015

[18] D Simon Optimal State Estimation Kalman H Infinity andNonlinear Approaches Wiley-Interscience 2006

[19] KW Lui F K Chan andH C So ldquoSemidefinite programmingapproach for range-difference based source localizationrdquo IEEETransactions on Signal Processing vol 57 no 4 pp 1630ndash16332009

[20] X Qu and L Xie ldquoAn efficient convex constrained weightedleast squares source localization algorithm based on TDOAmeasurementsrdquo Signal Processing vol 119 pp 142ndash152 2016

[21] G Yanni and W Qi ldquoThe nonsequential fusion method forlocalization from unscented kalman filter by multistation arraybuoysrdquo Discrete Dynamics in Nature and Society vol 2016Article ID 7670609 8 pages 2016

[22] A H Jazwinski Stochastic Processes and Filtering Theory Aca-demic Press Cambridge Mass USA 1970

[23] F Daum ldquoNonlinear filters beyond the kalman filterrdquo IEEEAerospace and Electronic Systems Magazine vol 20 no 8 pp57ndash69 2005

Submit your manuscripts athttpswwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

8 Discrete Dynamics in Nature and Society

minus5

minus48

minus46

minus44

minus42

minus4

minus38

minus36

y-ax

is (m

)

13 135 14 145 15 155 16 165 17 175125x-axis (m)

Figure 1 The trajectory of the mobile source in Scenario 1

The normal UKFThe improved UKF

0

01

02

03

04

05

06

07

08

09

1

The R

MSE

of m

obile

sour

ce lo

caliz

atio

n

10 20 30 40 50 60 70 80 90 1000k

Figure 2 Comparison of the mobile localization accuracy with lowlevel sensor position uncertainties in Scenario 1

as RMSE(u119896) = radicsum119871119897=1 u119896 minus u1198962119871 where u119896 = x119896(1 2) isthe estimate of the source position at ensemble 119897 and 119871 = 1000is the number of ensemble runs

In theMonte Carlo simulations we consider three sensorposition uncertainty levels which are referred to as the lowlevel with 120590119904 = 01 the moderate level with 120590119904 = 1 and thehigh level with 120590119904 = 2

Figure 2 plots the simulation results with time instantvarying from 1 to 100 when the sensor position uncertaintylevel is low It is evident from Figure 2 that RMSE of theproposed improved UKF method (represented by dashedline) is smaller than that of the normal UKF approach(represented by solid line) which ignores the sensor positionuncertainties

Figure 3 plots the RMSEs of the two methods in the caseof moderate sensor position uncertainty level that is 120590119904 =1 We can see that both of them are generally larger than

The normal UKFThe improved UKF

0

01

02

03

04

05

06

07

The R

MSE

of m

obile

sour

ce lo

caliz

atio

n

10 20 30 40 50 60 70 80 90 1000k

Figure 3 Comparison of themobile localization accuracywithmildlevel sensor position uncertainties in Scenario 1

The normal UKFThe improved UKF

02

04

06

08

1

12

14

16

18

2

22

The R

MSE

of m

obile

sour

ce lo

caliz

atio

n

10 20 30 40 50 60 70 80 90 1000k

Figure 4 Comparison of themobile localization accuracywith highlevel sensor position uncertainties in Scenario 1

the corresponding RMSEs in Figure 2 As we expected theperformance of the proposed improvedUKFmethod ismuchbetter than that of the normal UKF approach In additionthe improvement is larger than that of the low level case inFigure 2

Finally we take the high sensor position uncertainty levelthat is120590119904 = 2The numerical simulation results are illustratedin Figure 4 which indicates that the RMSE of the normalUKF approach is nearly twice asmuch as that of the proposedimproved UKF method Comparing the results in Figures 2ndash4 we can see that the influence of the random sensor positionuncertainties on the localization performance is quite evidentin Scenario 1 The proposed improved UKF method can

Discrete Dynamics in Nature and Society 9

The normal UKFThe improved UKF

0

005

01

015

02

025

03

035

04

045

The R

MSE

of m

obile

sour

ce lo

caliz

atio

n

10 20 30 40 50 60 70 80 90 1000k

Figure 5 Comparison of the mobile localization accuracy with lowlevel sensor position uncertainties in Scenario 2

efficiently decrease the RMSE especially when the sensorposition uncertainty level is high

42 Scenario 2 In this simulation scenario we considerthe sensor geometry is random and the number of sensors119899 = 4 The true position of each sensor is uniformlydistributed in a square of [minus100 100]m times [minus100 100]mat each ensemble in Monte Carlo simulations The TDOAmeasurement covariance Q119905 = (01119888)2T The motion modeland the initial value of the state are generally the same as thoseof Scenario 1

In the first case the sensor position uncertainty levelis considerably small that is 120590119904 = 01 Figure 5 plots theRMSE of the normal UKF approach and that of the proposedimproved UKF method with time instant varying from 1 to100 which are calculated from 119871 = 1000Monte Carlo runsIt can be seen that the RMSE of the proposed improved UKFmethod is smaller than that of the normal UKF approach

Thenwe consider amoderate sensor position uncertaintylevel that is 120590119904 = 1 Figure 6 plots the RMSEs of the twomethods in this case where we can see that the performanceof the proposed improvedUKFmethod is still better than thatof the normal UKF approach with a larger improvement as inFigure 5

For the high sensor position uncertainty level that is120590119904 = 2 the simulation results are illustrated in Figure 7which shows that the RMSE of the normal UKF approach isconsiderably larger than that of the proposed improved UKFmethod

The Monte Carlo simulation results in Figures 5ndash7 revealthat the RMSEs of the proposed improved UKF are consis-tently smaller than those of the normal UKF approach whichcorroborates that the proposed improved UKF method cangenerally improve the estimation accuracy of the nonlineardynamic system when it contains random parameters

The normal UKFThe improved UKF

0

02

04

06

08

1

12

14

16

18

2

The R

MSE

of m

obile

sour

ce lo

caliz

atio

n

10 20 30 40 50 60 70 80 90 1000k

Figure 6 Comparison of themobile localization accuracywithmildlevel sensor position uncertainties in Scenario 2

The normal UKFThe improved UKF

0

05

1

15

2

25

3

35

The R

MSE

of m

obile

sour

ce lo

caliz

atio

n

10 20 30 40 50 60 70 80 90 1000k

Figure 7 Comparison of themobile localization accuracy with highlevel sensor position uncertainties in Scenario 2

5 Conclusions

In this paper an improved unscented Kalman filteringmethod is explored for discrete nonlinear dynamic systemswith randomparameters By augmenting the randomparam-eters into the state vector we enlarge the number of sigmapoints in the improved unscented transformation Theo-retical analysis indicates that the approximated mean andcovariance via the improved unscented transformationmatchthe true values correctly up to the third order of Taylor seriesexpansion An application to the mobile source localizationusing TDOA measurements is provided where the sensorpositions suffer from randomuncertaintiesTheMonte Carlo

10 Discrete Dynamics in Nature and Society

simulation results show that the proposed improved UKFmethod can considerably improve the estimation accuracy

Competing Interests

The authors declare that there are no competing interestsregarding the publication of this paper

Acknowledgments

This work was supported by the Fundamental ResearchFunds for the Central Universities Southwestern Universityof Finance and Economics under Grant no JBK1507059and Southwest University for Nationalities under Grant no2016NGJPY05

References

[1] R E Kalman ldquoA new approach to linear filtering and predictionproblemsrdquo Journal of Basic Engineering vol 82 no 1 pp 35ndash451960

[2] G Bolzon R Fedele and G Maier ldquoParameter identificationof a cohesive crack model by Kalman filterrdquo Computer Methodsin Applied Mechanics and Engineering vol 191 no 25-26 pp2847ndash2871 2002

[3] M Ball and A Wood ldquoTrend growth in post-1850 Britisheconomic history the Kalman filter and historical judgmentrdquoJournal of the Royal Statistical Society vol 45 no 2 Article ID143152 pp 143ndash152 1996

[4] W L De Koning ldquoOptimal estimation of linear discrete-timesystems with stochastic parametersrdquo Automatica vol 20 pp113ndash115 1984

[5] B Sinopoli L Schenato M Franceschetti K Poolla M IJordan and S S Sastry ldquoKalman filtering with intermittentobservationsrdquo IEEE Transactions on Automatic Control vol 49no 9 pp 1453ndash1464 2004

[6] Y Luo Y Zhu D Luo J Zhou E Song and DWang ldquoGloballyoptimal multisensor distributed random parameter matriceskalman filtering fusion with applicationsrdquo Sensors vol 8 no 12pp 8086ndash8103 2008

[7] S Chen Y Li G Qi and A Sheng ldquoAdaptive Kalman esti-mation in target tracking mixed with random one-step delaysstochastic-bias measurements and missing measurementsrdquoDiscrete Dynamics in Nature and Society vol 2013 Article ID716915 14 pages 2013

[8] S Chen Z Zhou and S Li ldquoAn efficient estimate and forecastof the implied volatility surface a nonlinear Kalman filterapproachrdquo Economic Modelling vol 58 pp 655ndash664 2016

[9] S Bolognani L Tubiana and M Zigliotto ldquoExtended Kalmanfilter tuning in sensorless PMSM drivesrdquo IEEE Transactions onIndustry Applications vol 39 no 6 pp 1741ndash1747 2003

[10] J J LaViola Jr ldquoA comparison of unscented and extendedkalman filtering for estimating quaternion motionrdquo in Proceed-ings of the AmericanControl Conference pp 2435ndash2440DenverColo USA June 2003

[11] X Qu and L Xie ldquoRecursive source localisation by time dif-ference of arrival sensor networks with sensor position uncer-taintyrdquo IET Control Theory and Applications vol 8 no 18 pp2305ndash2315 2014

[12] S J Julier J K Uhlmann and H F Durrant-Whyte ldquoNewapproach for filtering nonlinear systemsrdquo in Proceedings of theAmerican Control Conference pp 1628ndash1632 June 1995

[13] S J Julier and J K Uhlmann ldquoUnscented filtering and nonlin-ear estimationrdquo Proceedings of the IEEE vol 92 no 3 pp 401ndash422 2004

[14] Q Pan F Yang and L Ye ldquoSurvey of a kind of non linear filtersUKFrdquo Control and Decision vol 20 no 5 pp 481ndash489 2005

[15] C Eberle and C Ament ldquoThe Unscented Kalman Filter esti-mates the plasma insulin from glucose measurementrdquo BioSys-tems vol 103 no 1 pp 67ndash72 2011

[16] S J Julier ldquoSome relations between extended and unscentedKalman filtersrdquo IEEE Transactions on Signal Processing vol 60no 2 pp 545ndash555 2012

[17] HM T Menegaz J Y Ishihara G A Borges and A N VargasldquoA systematization of the unscented Kalman filter theoryrdquo IEEETransactions on Automatic Control vol 60 no 10 pp 2583ndash2598 2015

[18] D Simon Optimal State Estimation Kalman H Infinity andNonlinear Approaches Wiley-Interscience 2006

[19] KW Lui F K Chan andH C So ldquoSemidefinite programmingapproach for range-difference based source localizationrdquo IEEETransactions on Signal Processing vol 57 no 4 pp 1630ndash16332009

[20] X Qu and L Xie ldquoAn efficient convex constrained weightedleast squares source localization algorithm based on TDOAmeasurementsrdquo Signal Processing vol 119 pp 142ndash152 2016

[21] G Yanni and W Qi ldquoThe nonsequential fusion method forlocalization from unscented kalman filter by multistation arraybuoysrdquo Discrete Dynamics in Nature and Society vol 2016Article ID 7670609 8 pages 2016

[22] A H Jazwinski Stochastic Processes and Filtering Theory Aca-demic Press Cambridge Mass USA 1970

[23] F Daum ldquoNonlinear filters beyond the kalman filterrdquo IEEEAerospace and Electronic Systems Magazine vol 20 no 8 pp57ndash69 2005

Submit your manuscripts athttpswwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Discrete Dynamics in Nature and Society 9

The normal UKFThe improved UKF

0

005

01

015

02

025

03

035

04

045

The R

MSE

of m

obile

sour

ce lo

caliz

atio

n

10 20 30 40 50 60 70 80 90 1000k

Figure 5 Comparison of the mobile localization accuracy with lowlevel sensor position uncertainties in Scenario 2

efficiently decrease the RMSE especially when the sensorposition uncertainty level is high

42 Scenario 2 In this simulation scenario we considerthe sensor geometry is random and the number of sensors119899 = 4 The true position of each sensor is uniformlydistributed in a square of [minus100 100]m times [minus100 100]mat each ensemble in Monte Carlo simulations The TDOAmeasurement covariance Q119905 = (01119888)2T The motion modeland the initial value of the state are generally the same as thoseof Scenario 1

In the first case the sensor position uncertainty levelis considerably small that is 120590119904 = 01 Figure 5 plots theRMSE of the normal UKF approach and that of the proposedimproved UKF method with time instant varying from 1 to100 which are calculated from 119871 = 1000Monte Carlo runsIt can be seen that the RMSE of the proposed improved UKFmethod is smaller than that of the normal UKF approach

Thenwe consider amoderate sensor position uncertaintylevel that is 120590119904 = 1 Figure 6 plots the RMSEs of the twomethods in this case where we can see that the performanceof the proposed improvedUKFmethod is still better than thatof the normal UKF approach with a larger improvement as inFigure 5

For the high sensor position uncertainty level that is120590119904 = 2 the simulation results are illustrated in Figure 7which shows that the RMSE of the normal UKF approach isconsiderably larger than that of the proposed improved UKFmethod

The Monte Carlo simulation results in Figures 5ndash7 revealthat the RMSEs of the proposed improved UKF are consis-tently smaller than those of the normal UKF approach whichcorroborates that the proposed improved UKF method cangenerally improve the estimation accuracy of the nonlineardynamic system when it contains random parameters

The normal UKFThe improved UKF

0

02

04

06

08

1

12

14

16

18

2

The R

MSE

of m

obile

sour

ce lo

caliz

atio

n

10 20 30 40 50 60 70 80 90 1000k

Figure 6 Comparison of themobile localization accuracywithmildlevel sensor position uncertainties in Scenario 2

The normal UKFThe improved UKF

0

05

1

15

2

25

3

35

The R

MSE

of m

obile

sour

ce lo

caliz

atio

n

10 20 30 40 50 60 70 80 90 1000k

Figure 7 Comparison of themobile localization accuracy with highlevel sensor position uncertainties in Scenario 2

5 Conclusions

In this paper an improved unscented Kalman filteringmethod is explored for discrete nonlinear dynamic systemswith randomparameters By augmenting the randomparam-eters into the state vector we enlarge the number of sigmapoints in the improved unscented transformation Theo-retical analysis indicates that the approximated mean andcovariance via the improved unscented transformationmatchthe true values correctly up to the third order of Taylor seriesexpansion An application to the mobile source localizationusing TDOA measurements is provided where the sensorpositions suffer from randomuncertaintiesTheMonte Carlo

10 Discrete Dynamics in Nature and Society

simulation results show that the proposed improved UKFmethod can considerably improve the estimation accuracy

Competing Interests

The authors declare that there are no competing interestsregarding the publication of this paper

Acknowledgments

This work was supported by the Fundamental ResearchFunds for the Central Universities Southwestern Universityof Finance and Economics under Grant no JBK1507059and Southwest University for Nationalities under Grant no2016NGJPY05

References

[1] R E Kalman ldquoA new approach to linear filtering and predictionproblemsrdquo Journal of Basic Engineering vol 82 no 1 pp 35ndash451960

[2] G Bolzon R Fedele and G Maier ldquoParameter identificationof a cohesive crack model by Kalman filterrdquo Computer Methodsin Applied Mechanics and Engineering vol 191 no 25-26 pp2847ndash2871 2002

[3] M Ball and A Wood ldquoTrend growth in post-1850 Britisheconomic history the Kalman filter and historical judgmentrdquoJournal of the Royal Statistical Society vol 45 no 2 Article ID143152 pp 143ndash152 1996

[4] W L De Koning ldquoOptimal estimation of linear discrete-timesystems with stochastic parametersrdquo Automatica vol 20 pp113ndash115 1984

[5] B Sinopoli L Schenato M Franceschetti K Poolla M IJordan and S S Sastry ldquoKalman filtering with intermittentobservationsrdquo IEEE Transactions on Automatic Control vol 49no 9 pp 1453ndash1464 2004

[6] Y Luo Y Zhu D Luo J Zhou E Song and DWang ldquoGloballyoptimal multisensor distributed random parameter matriceskalman filtering fusion with applicationsrdquo Sensors vol 8 no 12pp 8086ndash8103 2008

[7] S Chen Y Li G Qi and A Sheng ldquoAdaptive Kalman esti-mation in target tracking mixed with random one-step delaysstochastic-bias measurements and missing measurementsrdquoDiscrete Dynamics in Nature and Society vol 2013 Article ID716915 14 pages 2013

[8] S Chen Z Zhou and S Li ldquoAn efficient estimate and forecastof the implied volatility surface a nonlinear Kalman filterapproachrdquo Economic Modelling vol 58 pp 655ndash664 2016

[9] S Bolognani L Tubiana and M Zigliotto ldquoExtended Kalmanfilter tuning in sensorless PMSM drivesrdquo IEEE Transactions onIndustry Applications vol 39 no 6 pp 1741ndash1747 2003

[10] J J LaViola Jr ldquoA comparison of unscented and extendedkalman filtering for estimating quaternion motionrdquo in Proceed-ings of the AmericanControl Conference pp 2435ndash2440DenverColo USA June 2003

[11] X Qu and L Xie ldquoRecursive source localisation by time dif-ference of arrival sensor networks with sensor position uncer-taintyrdquo IET Control Theory and Applications vol 8 no 18 pp2305ndash2315 2014

[12] S J Julier J K Uhlmann and H F Durrant-Whyte ldquoNewapproach for filtering nonlinear systemsrdquo in Proceedings of theAmerican Control Conference pp 1628ndash1632 June 1995

[13] S J Julier and J K Uhlmann ldquoUnscented filtering and nonlin-ear estimationrdquo Proceedings of the IEEE vol 92 no 3 pp 401ndash422 2004

[14] Q Pan F Yang and L Ye ldquoSurvey of a kind of non linear filtersUKFrdquo Control and Decision vol 20 no 5 pp 481ndash489 2005

[15] C Eberle and C Ament ldquoThe Unscented Kalman Filter esti-mates the plasma insulin from glucose measurementrdquo BioSys-tems vol 103 no 1 pp 67ndash72 2011

[16] S J Julier ldquoSome relations between extended and unscentedKalman filtersrdquo IEEE Transactions on Signal Processing vol 60no 2 pp 545ndash555 2012

[17] HM T Menegaz J Y Ishihara G A Borges and A N VargasldquoA systematization of the unscented Kalman filter theoryrdquo IEEETransactions on Automatic Control vol 60 no 10 pp 2583ndash2598 2015

[18] D Simon Optimal State Estimation Kalman H Infinity andNonlinear Approaches Wiley-Interscience 2006

[19] KW Lui F K Chan andH C So ldquoSemidefinite programmingapproach for range-difference based source localizationrdquo IEEETransactions on Signal Processing vol 57 no 4 pp 1630ndash16332009

[20] X Qu and L Xie ldquoAn efficient convex constrained weightedleast squares source localization algorithm based on TDOAmeasurementsrdquo Signal Processing vol 119 pp 142ndash152 2016

[21] G Yanni and W Qi ldquoThe nonsequential fusion method forlocalization from unscented kalman filter by multistation arraybuoysrdquo Discrete Dynamics in Nature and Society vol 2016Article ID 7670609 8 pages 2016

[22] A H Jazwinski Stochastic Processes and Filtering Theory Aca-demic Press Cambridge Mass USA 1970

[23] F Daum ldquoNonlinear filters beyond the kalman filterrdquo IEEEAerospace and Electronic Systems Magazine vol 20 no 8 pp57ndash69 2005

Submit your manuscripts athttpswwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

10 Discrete Dynamics in Nature and Society

simulation results show that the proposed improved UKFmethod can considerably improve the estimation accuracy

Competing Interests

The authors declare that there are no competing interestsregarding the publication of this paper

Acknowledgments

This work was supported by the Fundamental ResearchFunds for the Central Universities Southwestern Universityof Finance and Economics under Grant no JBK1507059and Southwest University for Nationalities under Grant no2016NGJPY05

References

[1] R E Kalman ldquoA new approach to linear filtering and predictionproblemsrdquo Journal of Basic Engineering vol 82 no 1 pp 35ndash451960

[2] G Bolzon R Fedele and G Maier ldquoParameter identificationof a cohesive crack model by Kalman filterrdquo Computer Methodsin Applied Mechanics and Engineering vol 191 no 25-26 pp2847ndash2871 2002

[3] M Ball and A Wood ldquoTrend growth in post-1850 Britisheconomic history the Kalman filter and historical judgmentrdquoJournal of the Royal Statistical Society vol 45 no 2 Article ID143152 pp 143ndash152 1996

[4] W L De Koning ldquoOptimal estimation of linear discrete-timesystems with stochastic parametersrdquo Automatica vol 20 pp113ndash115 1984

[5] B Sinopoli L Schenato M Franceschetti K Poolla M IJordan and S S Sastry ldquoKalman filtering with intermittentobservationsrdquo IEEE Transactions on Automatic Control vol 49no 9 pp 1453ndash1464 2004

[6] Y Luo Y Zhu D Luo J Zhou E Song and DWang ldquoGloballyoptimal multisensor distributed random parameter matriceskalman filtering fusion with applicationsrdquo Sensors vol 8 no 12pp 8086ndash8103 2008

[7] S Chen Y Li G Qi and A Sheng ldquoAdaptive Kalman esti-mation in target tracking mixed with random one-step delaysstochastic-bias measurements and missing measurementsrdquoDiscrete Dynamics in Nature and Society vol 2013 Article ID716915 14 pages 2013

[8] S Chen Z Zhou and S Li ldquoAn efficient estimate and forecastof the implied volatility surface a nonlinear Kalman filterapproachrdquo Economic Modelling vol 58 pp 655ndash664 2016

[9] S Bolognani L Tubiana and M Zigliotto ldquoExtended Kalmanfilter tuning in sensorless PMSM drivesrdquo IEEE Transactions onIndustry Applications vol 39 no 6 pp 1741ndash1747 2003

[10] J J LaViola Jr ldquoA comparison of unscented and extendedkalman filtering for estimating quaternion motionrdquo in Proceed-ings of the AmericanControl Conference pp 2435ndash2440DenverColo USA June 2003

[11] X Qu and L Xie ldquoRecursive source localisation by time dif-ference of arrival sensor networks with sensor position uncer-taintyrdquo IET Control Theory and Applications vol 8 no 18 pp2305ndash2315 2014

[12] S J Julier J K Uhlmann and H F Durrant-Whyte ldquoNewapproach for filtering nonlinear systemsrdquo in Proceedings of theAmerican Control Conference pp 1628ndash1632 June 1995

[13] S J Julier and J K Uhlmann ldquoUnscented filtering and nonlin-ear estimationrdquo Proceedings of the IEEE vol 92 no 3 pp 401ndash422 2004

[14] Q Pan F Yang and L Ye ldquoSurvey of a kind of non linear filtersUKFrdquo Control and Decision vol 20 no 5 pp 481ndash489 2005

[15] C Eberle and C Ament ldquoThe Unscented Kalman Filter esti-mates the plasma insulin from glucose measurementrdquo BioSys-tems vol 103 no 1 pp 67ndash72 2011

[16] S J Julier ldquoSome relations between extended and unscentedKalman filtersrdquo IEEE Transactions on Signal Processing vol 60no 2 pp 545ndash555 2012

[17] HM T Menegaz J Y Ishihara G A Borges and A N VargasldquoA systematization of the unscented Kalman filter theoryrdquo IEEETransactions on Automatic Control vol 60 no 10 pp 2583ndash2598 2015

[18] D Simon Optimal State Estimation Kalman H Infinity andNonlinear Approaches Wiley-Interscience 2006

[19] KW Lui F K Chan andH C So ldquoSemidefinite programmingapproach for range-difference based source localizationrdquo IEEETransactions on Signal Processing vol 57 no 4 pp 1630ndash16332009

[20] X Qu and L Xie ldquoAn efficient convex constrained weightedleast squares source localization algorithm based on TDOAmeasurementsrdquo Signal Processing vol 119 pp 142ndash152 2016

[21] G Yanni and W Qi ldquoThe nonsequential fusion method forlocalization from unscented kalman filter by multistation arraybuoysrdquo Discrete Dynamics in Nature and Society vol 2016Article ID 7670609 8 pages 2016

[22] A H Jazwinski Stochastic Processes and Filtering Theory Aca-demic Press Cambridge Mass USA 1970

[23] F Daum ldquoNonlinear filters beyond the kalman filterrdquo IEEEAerospace and Electronic Systems Magazine vol 20 no 8 pp57ndash69 2005

Submit your manuscripts athttpswwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Submit your manuscripts athttpswwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of