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Research ArticleAdaptive Finite-Horizon Group Estimationfor Networked Navigation Systems with RemoteSensing Complementary Observations underMixed LOSNLOS Environments

Chao Gao1 Jianhua Lu1 Guorong Zhao1 and Shuang Pan2

1Department of Control Engineering Naval Aeronautical and Astronautical University Yantai 264001 China2Department of Strategic Missile amp Underwater Weapon Naval Submarine Academy Qingdao 266071 China

Correspondence should be addressed to Chao Gao gaochaoshd163com

Received 4 August 2016 Revised 25 October 2016 Accepted 26 October 2016

Academic Editor R Aguilar-Lopez

Copyright copy 2016 Chao Gao 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

Networked navigation system (NNS) enables a wealth of new applications where real-time estimation is essential In this paperan adaptive horizon estimator has been addressed to solve the navigational state estimation problem of NNS with the features ofremote sensing complementary observations (RSOs) andmixed LOSNLOS environments In our approach it is assumed that RSOsare the essential observations of the local processor but suffer from random transmission delay a jump Markov system has beenmodeled with the switching parameters corresponding to LOSNLOS errors An adaptive finite-horizon group estimator (AFGE)has been proposed where the horizon size can be adjusted in real time according to stochastic parameters and random delaysFirst a delay-aware FIR (DFIR) estimator has been derived with observation reorganization and complementary fusion strategiesSecond an adaptive horizon group (AHG) policy has been proposed to manage the horizon size The AFGE algorithm is thusrealized by combining AHG policy and DFIR estimator It is shown by a numerical example that the proposed AFGE has a morerobust performance than the FIR estimator using constant optimal horizon size

1 Introduction

Networked navigation system (NNS) [1 2] which determinesits navigational state individually from the cooperative infor-mation taken with respect to spatial neighbors has been usedfor a variety of purposes such as cooperative localization[3] multivehicle cooperative guidance [4] and spatially dis-tributed estimation [5] NNS typically takes advantage of stateestimations for accurate localization essentially based on theremote sensing observations (RSOs) which is transmitted bythe cooperative NNSs through a wireless mediumThus dur-ing the design of navigational state estimator several prob-lematic issues in NNS should be considered induced by theobservation features and the transmission environment Inthis research the impact of mixed LOSNLOS environmentand complementary feature of RSOs will be investigated

Remote sensing observations (RSOs) such as imagerelative measurement and sink knowledge always have

complementary feature which means every observed datajust comprises partial of system dynamics thus one use-ful observation of estimator can only be obtained by theintegration of multiple synchronous RSOs Note that theseRSOs are ubiquitous in large-scale systems [6] especially inwireless networked system (WNS) [7] and thus the stateestimation based on the sensors of RSO data is broughtforward In [8] a particle filter based track-before-detect(TkBD) scheme was proposed for multiple sensors wherethe sensors were asynchronous and heterogeneous howeverthe TkBD schemes are computationally expensive and aremainly applicable in scene of low signal-to-noise ratiosEarlier research of RSO focused on optical sensors [9] inwhich the sensor noise is assumed to be Gaussian Morerecent efforts focus on the radar problem where the pixelintensity is either Rayleigh or Rician distributed dependingon the absence or presence of objectives [10] Although theextant literature considers various types of RSO sensors

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2016 Article ID 6489165 14 pageshttpdxdoiorg10115520166489165

2 Mathematical Problems in Engineering

the researches on the complementary property of wirelessnetworks are quite few let alone the packet-based RSO ofNNS that is the navigational knowledge of the sender andthe relative measurements

Non-line-of-sight (NLOS) propagation might take placein a wireless network whenever there are obstacles hin-dering the radio signal path between the objective and itscooperative neighbors especially in an urban region or theindoor environment Whenever measured by received signalstrength (RSS) or angle of arrival (AOA) NLOS propagationoften induces transmission delay or dropout of the packetsand thus large localization errorsmight occur if not addressed[11] Several strategies have been proposed for mitigating thebad effect of NLOS errors such as statistical identification[12] hypothesis test [13] and robust parameter estimation[14] In [15] the switching behavior of NLOS and the line-of-sight (LOS) has been described by a hidden MarkovModel and an estimator has been designed by combiningKalman filter with the interacting multiple model (IMM)approach These results are further extended to nonlinearAOA measurement by using the MUSIC method in [16]Similar approach has also been used to observe a mobileobjective in mixed LOSNLOS environments by using indi-vidual measurement and LOS detection [17] However littleresearch has been investigated to mitigate the delay anddropout problems caused by NLOS propagation In [18] ajoint AOA and delay estimation method has been proposedfor space-time coherent distributed signals based on searchThe problem of joint AOARSS based state estimation forNNS in mixed LOS and NLOS environment has not beenstudied

In reality almost all networked systems can record theresults over a recent finite horizon for such time-varyingsystems one would be more interested in the optimalestimation based on the finite recent data When it comesto the filter design issue finite-horizon estimation is aneffective approach and gives rise to many available methodsin terms of moving horizon estimation [1 19] finite-horizonHinfin fault estimation [20] and finite-horizon FIR estimation[21] However finite-horizon estimation approach has itsown limitations especially on expensively computationalburden [22] with the reasons in terms of the adoption offinite recent measurements the inversion computation [23]and the singularity problem [24] One of the improvementmethods is to manage the horizon size (window length) andseveral approaches [25] have been proposed Although theexisting approaches are effective with linear time-invariantsystems they can be less effective with wireless networkedsystems with the reason that the constant (ie fixed) horizonsize of the existing methods may be not able to handlethe network-induced constraints owing to mixed LOSNLOSenvironmentThus it is desirable to adapt the horizon size infinite-horizon estimation and this motivates us to investigatethis issue

Summarizing the above analysis it can be concludedthat there is a great need to examine how the mixedLOSNLOS environments affect the performance of thedistributed estimation over a sensor network with RSOdata To address the effects of NLOS errors and random

delays a jump Markov system has been modeled with theswitching parameters corresponding to LOSNLOS errorsAn adaptive finite-horizon group estimator (AFGE) has beenproposed where the horizon size can be adjusted in real timeaccording to stochastic parameters and random delays Anumerical example is proposed to verify the effectiveness ofthe proposed AFGE scheme The main contributions of thisresearch can be highlighted as follows (i) The observationmodel of NNS is a special one that has complementaryproperty and is affected by switching LOSNLOS errors andrandom transmission delay (ii) An observation reorganiza-tion scheme and a complementary fusion strategy have beenintroduced to DFIR estimation procedure to mitigate theimpacts of delay and complementary property respectively(iii) The proposed AFGE scheme contains an AHG policywhich is proposed to manage the horizon size

The remainder of this paper is organized as follows InSection 2 the graph-based network background is intro-duced while the NNS model and some preliminaries arebriefly outlined Section 3 describes the derivation of a newDFIR estimator Section 4 proposes the AHG method andcompletes the AFGE In Section 5 an application of clusteredUAVs is given for the designedAFGE Finally the conclusionsare drawn in Section 6

Notation 119906119866 stands for the Euclidean norm of a genericvector 119906 1199062119866 = 119906119879119866119906 where119866 is a symmetric positive semi-definite matrix Given a generic vector 119906119905 119906119905119905minus119873 = 119906119905minus119873119905 ≜col(119906119905minus119873 119906119905minus119873+1 119906119905) a generic matrix 119860119905

119905minus119873 = 119860 119905minus119873119905 ≜col(119860 119905minus119873 119860 119905minus119873+1 119860 119905) 119905 = 119873119873 + 1 L1199101 119910119899denotes the linear space spanned by 1199101 1199101198992 Preliminaries and Problem FormulationIn this section we introduce some preliminaries related to thedescription of communication network and then present theproblem setup

21 Graph-BasedNetworkDescription Suppose that119873Σ vehi-cles form into an Ad hoc network where every vehicle acts asa networked node and carries out the uniform communica-tion protocol that is [1 2 19] The communication topologyof information flow between networked vehicles is describedby a directed graphG = (C119892E119892R119892) whereC119892 isin 1 119873Σis the finite set of networked vehicles E119892 sube C119892 timesC119892 is the setof edgesR119892 = (119903119894) isin R119899 is the set of action radius and 119894 isin C119892

represents the sender The edge 119890[119894rarr119895] = (119894 119895) isin E119892 indicatesthat node 119895 can receive the information from node 119894 that is(119894 119895) isin E119892 rArr 119889[119894rarr119895] le 119903119894 Specifically we define the set ofneighbors as follows

Definition 1 Let 119909[119894]119901 isin R3 be the position of node 120582 isin C119892the neighbor set of node 120582 at time 119905 can be defined as

N[120582]119905 = 119895 isin C119892 119903inf le 10038171003817100381710038171003817119909[119895]119901 (119905) minus 119909[120582]119901 (119905)10038171003817100381710038171003817 le 119903sup 120582= 119895 C119892 = 1 2 119873Σ (1)

where 119903sup and 119903inf denote the supremum and infimum radiiof available communication range respectively Note that the

Mathematical Problems in Engineering 3

Random neighborsCluster member

Objective node Cooperative packet

Receive-search areaTo-be-sent packet

0001

02

03 00 01 02 03

12

1

2

0

RTS

RSA

zalt ylat

xlon

yg

Og

zg

xg

CM1

CM4

CM3

CH

Cag

CM5

CM2

GM6CM6

t(k)

GM1

GM2

GM3

GM4

GM5

t(kminus N

+ 1) S

t(kminus1)

GELz

t(kminusN)

Figure 1 Remote sensing neighbors for networked navigation system

relative distances between node 120582 and N[120582]119905 are changeable

with time 119905 thus the member of N[120582]119905 may also be changed

dynamically Besides the effective area for cooperative neigh-bors can be defined by 119903sup and 119903inf namely receive-searcharea (RSA) as depicted in Figure 1

The model of RSA is defined as follows

Definition 2 Given a generic node 120582 isin C119892 its geographicneighbor 119895 isin C119892 is defined as in the RSA of node 120582 if node119895 is in the hollow sphere between 119903inf and 119903sup which aredetermined by RSS method as

lg (119903sup) = (PLdB minus 120574sup)(10120582) minus 3244120582 minus lg (119891MHz) lg (119903inf) = (PLdB minus 120574inf)(10120582) minus 3244120582 minus lg (119891MHz)

(2)

where 120574sup and 120574inf denote the received SNRs correspondingto 119903sup and 119903inf respectively PLdB is the propagation loss(dB) in free space propagation model [12] 120582 is the path lossexponent 2 le 120582 le 4 and 119891MHz is the electromagneticfrequency

22 NNS Model and Problem Formulation In this paper theNNS is assumed as the essential navigation equipment of ageneric vehicle 120582 isin C119892 The dynamics of node S can bemodeled as the discrete-time linear stochastic system

119909119905+1 = Φ119905119909119905 + Γ119905119908119905 (3)

where 119909119905 denotes system state Φ119905 and Γ119905 denote the systemmatrices which can be used to represent the motions such asthe velocity model and coordinate turn model The processnoise119908119905 is zero-mean white Gaussian with covariancematrix119876119905

A set of cooperative UAVs that provide two kinds ofobservations (i) navigational states of the observed neigh-bors denoted by xN

119905 = [119894]119905 |[119894]119905 119894 isin N[120582]119905 [119894]119905 = ([119894]119909119905 [119894]119910119905[119894]119911119905) and (ii) relative measurements (RM) between 120582 and the

neighbors 119894 isin N[120582]119905 denoted by zN

119905 = 119911[119894]119905 |119911[119894]119905 119894 isin N[120582]119905

including relative distance and relative angle are assumedto be calculated by RSS and AOA respectively The detaileddeductions can be found in [12 16] with the basic equationsas

119911[119894]RSS = 120572 minus 10120582 lg 10038161003816100381610038161003816[119894] minus [120582]10038161003816100381610038161003816 + 120578[119894]shad119911[119894]AOA120579 = arctan([119894]119910 minus [120582]119910[119894]119909 minus [120582]119909

) + 120578[119894]AOA120579119911[119894]AOA120601 = arctan( [119894]119911 minus [120582]11991110038171003817100381710038171003817[119894]119909119910 minus [120582]119909119910

10038171003817100381710038171003817) + 120578[119894]AOA120601119894 = 1 2 ℓ

(4)

Compared to the previous DSE problems [5 20 25] atypical feature here is that the observations in xN

119905 and zN119905

are mutually complementary that is each local observationof node 120582 can only be derived through the fusion of multiple


119909[119894]119905 and 119911[119894]119905 (119894 isin N[120582]119905 ) The measure noise covariance 119877[119894]

119904 inCartesian coordinates is

119877[119894]119904 = 120601 (119911[119894]RSS 119911[119894]AOA120579 119911[119894]AOA120601 120578[119894]shad 120578[119894]AOA120579 120578[119894]AOA120601) (5)

where 119911[119894]RSS 119911[119894]AOA120579 119911[119894]AOA120601 120578[119894]shad 120578[119894]AOA120579 and 120578[119894]AOA120601 aredefined in (4) In LOS environment the measurements (4)are corrupted by the noise 120576[119894]119905 which can bemodeled as zero-mean white Gaussian noise that is 120576[119894]119905 sim 119873(120576[119894]119905 0 119877[119894]

1199051) InNLOS environment the measurement is corrupted by twokinds of error sources Gaussian noise 120576[119894]119905 and measurementbias error 119890[119894]119905 The bias error can be modeled as a biaseddistribution that is 119890[119894]119905 sim 119873(119890[119894]119905 120583[119894]119905 Σ[119894]

119905 ) It is also assumedthat 120576[119894]119905 and 119890[119894]119905 are independent Thus the CM noise in (4)can be represented by

119906[119894]119905 = 120576[119894]119905 LOS condition

120576[119894]119905 + 119890[119894]119905 NLOS condition(6)

For the sake of notations the measurement equation (4)can be represented in a uniform form

119891 (xN119905 zN

119905 ) ≜ 119911[119894]119905 = 120601[119894]119905 119909119905 + 119906[119894]119905 (120574[119894]119905 ) 119894 = 1 2 ℓ (7)

where 120601[119894]119905 is the nonlinearmeasurement function dependingon specific RM methods the measurement noise 119906[119894]119905 (120574[119894]119905 ) isassumed to be zero-mean white Gaussian with covariancematrix119877[119894]

1199051 in the LOS condition that is 119906[119894]1199051 isin 119873(119906[119894]119905 0 119877[119894]1199051)

and it is assumed to be white Gaussian with mean 120583[119894]119905 andcovariance matrix 119877[119894]

1199052 in the NLOS condition that is 119906[119894]1199052 isin119873(119906[119894]119905 120583[119894]119905 119877[119894]1199052) 120574[119894]119905 is introduced to indicate the transition

of LOS and NLOS measurement errors that is 120574[119894]119905 = 1represents the LOS condition and 120574[119894]119905 = 2 represents theNLOS condition The TPM of 120574[119894]119905 is assumed to be Π(2) =[120587(2)

119901119902 ]2times2 with 120587(2)119901119902 = Pr120574[119894]119905 = 119902 | 120574[119894]119905minus1 = 119901 It should be

pointed out that the two-state Markov chain 120574[119894]119905 is dependenton the cooperative UAV since the objective might be in LOScondition for one UAV but in NLOS condition for anotherone

Under the above mixed LOSNLOS propagation envi-ronment transmission delays (TDs) may decay the accuracyof relative measurement To mitigate this disadvantage theobserved CM with TDs is modeled as

119910[119894]119905 = ℏsum

120591=0

120581[119894]119905120591119911[119894]119905minus120591 = 120581[119894]119905120591120601[119894]119905minus120591119909119905minus120591 + 120581[119894]119905120591119906[119894]119905minus120591 (120574[119894]119905minus120591) 119894 = 1 2 ℓ

(8)

with 120581[119894]119905120591 defined as a binary random variable indicating thearrival of the observation packet for state 119909(119905 minus 120591) that is

120581[119894]119905120591 = 1 if 119911[119894]119905minus120591 is received at the instant 1199050 otherwise 119894 isin 1 2 ℓ (9)

Assumption 3 Random delay 120591(119905) is bounded with 0 lt120591(119905) le ℏ where ℏ is given as the length of memory bufferand its probability distributions are Prob(120591(119905) = 120591) = 120587120591120591 = 1 ℏ If the CMs transmitted to the objective with adelay larger than ℏ they will be considered as the ones lostcompletely Thus the property 0 le sumℏ

120591=1 120587120591 le 1 is satisfiedAlso 120591(119905) is independent of 119909(0) 119908(0) and 119906(0)

Obviously Prob(120581[119894]119905120591 = 1) = 120587120591 120591 = 1 ℏUnder Assumption 3 we know that the possible receivedobservations at time 119905 (119905 ge ℏ) are

[119894]119905 = [(120581[119894]1199050119911[119894]119905 )119879 (120581[119894]1199051119911[119894]119905minus1)119879 sdot sdot sdot (120581[119894]

119905ℏ119911[119894]119905minusℏ)119879]119879 (10)

In the real-timenetworked systems the state119909(119905) can onlybe observed atmost one time and thus 120581[119894]119905120591 120591 = 1 ℏ mustsatisfy the following property

120581[119894]119905+119895119895 times 120581[119894]119905+119896119896 = 0 119895 = 119896 (11)

When 119905 lt ℏ the observation of (10) is written as

[119894]119905 = [(120581[119894]1199050119911[119894]119905 )119879 sdot sdot sdot (120581[119894]119905119905119911[119894]119905 )119879 0 sdot sdot sdot 0]119879 (12)

where we set [119894]119905 equiv 0 for 119905 lt 120591 le ℏ Then the estimation

problems considered in this paper can be stated as follows

Problem 119864119905 For a generic node 120582 isin C119892 given the set ofnavigation knowledge from the neighbors [119894]119904 (119894 isin N[120582]

119905 )with the stamp 119904 (119904 gt 0) the corresponding relative measures[119894]119896

(119896 isin [119905 minus Ξ[119894] 119905]) subject to NLOS errors and randomdelay with the information 120581[119894]

119896120591 find a linear minimummean

square error filter [120582](119905 | 119905) for navigational state of the target119909(119905) such that

119869[120582] = 119864 10038171003817100381710038171003817119909 (119905) minus [120582] (119905 | 119905)10038171003817100381710038171003817 | [119894]119904 10038161003816100381610038161003816119904ge0 [119894]119896

10038161003816100381610038161003816119905119896=119905minusΞ[119894] 120581[119894]11989612059110038161003816100381610038161003816119905119896=119905minusΞ[119894] 119894 isinN

[120582]119905 = 1 2 ℓ (13)

is minimized while the filter gain is stochastic

3 Delay-Aware FIR Estimator with DelayedComplementary Observations

In this section we aim to establish a unified frameworkto solve the addressed optimal state estimation problemin the simultaneous presence of network-induced delayand complementary property under the mixed LOSNLOSenvironment while the observations are decoded from thepackets of ℓ spatially distributed nodes

31 Delayed Observation Reorganization In our research theremote sensing observations (ie 119911[119894]119905 119909[119894]119905 ) are impactedby delay and complementary issues and thus should not beadopted directly by the estimators In this subsection wefocus on the delay issue more specifically we propose apreestimating scheme including two steps to reorganize thedelay observations without dimensional augment


Step 1 We rearrange the complementary observations119910[119894](119896)|119905119905minusΞ[119894]

119894 isin N[120582]119905 which are received in horizon[119905 minus Ξ[120582] 119905] that is (8) and rewrite them into two parts with

the batch form as

ℏ (119904) ≜ [119910[119895]1198790 (119904)100381610038161003816100381610038161003816ℓ119895=1 119910[119895]119879

ℏ (119904 + ℏ)100381610038161003816100381610038161003816ℓ119895=1]119879 119905 minus Ξ[119904] minus ℏ le 119904 le 119905 minus ℏ

119905minus119904 (119904) ≜ [119910[119895]1198790 (119904)100381610038161003816100381610038161003816ℓ119895=1 119910[119895]119879

119905minus119904 (119905)100381610038161003816100381610038161003816ℓ119895=1]119879 119905 minus ℏ lt 119904 le 119905

(14)

Then reorganized observation (sdot) is delay-free andsatisfies

ℏ (119904) ≜ ℏ (119904) 119909 (119904) + Vℏ (119904 120574[119894]119904119904+ℏ) 119905 minus Ξ[119904] minus ℏ le 119904 le 119905 minus ℏ

119905minus119904 (119904) ≜ 119905minus119904 (119904) 119909 (119904) + V119905minus119904 (119904 120574[119894]119904119905 ) 119905 minus ℏ lt 119904 le 119905(15)

where

ℏ (119904) = [(120581[119894]1199040120601[119894]119904 10038161003816100381610038161003816ℓ119894=1)119879 (120581[119894]119904+ℏℏ

120601[119894]119904+ℎ

10038161003816100381610038161003816ℓ119894=1)119879]119879 119905minus119904 (119904) = [(120581[119894]1199040120601[119894]119904 10038161003816100381610038161003816ℓ119894=1)119879 (120581[119894]119905119905minus119904120601[119894]119905 10038161003816100381610038161003816ℓ119894=1)119879]119879 Vℏ (119904 120574[119894]119904119904+ℏ)= [(120581[119894]1199040119906[119894]119904 (120574[119894]119904 )10038161003816100381610038161003816ℓ119894=1)119879 (120581[119894]

119904+ℏℏ119906[119894]119904+ℏ(120574[119894]

119904+ℏ)10038161003816100381610038161003816ℓ119894=1)119879]119879

V119905minus119904 (119904 120574[119894]119904119905 )= [(120581[119894]1199040119906[119894]119904 (120574[119894]119904 )10038161003816100381610038161003816ℓ119894=1)119879 (120581[119894]119905119905minus119904119906[119894]119905 (120574[119894]119905 )10038161003816100381610038161003816ℓ119894=1)119879]119879

(16)

with Vℏ(119904) and V119905minus119904(119904) being white noises of zero means andcovariance matrices

ℏ (119904) = diag 120581[119894]1199040119877[119894]10038161003816100381610038161003816ℓ119894=1 120581[119894]119904+ℏℏ119877[119894]10038161003816100381610038161003816ℓ119894=1 119905minus119904 (119904) = diag 120581[119894]1199040119877[119894]10038161003816100381610038161003816ℓ119894=1 120581[119894]119904+ℏℏ119877[119894]10038161003816100381610038161003816ℓ119894=1

(17)

Moreover we can follow the result in [1] that the followinglemma is true

Lemma 4 For the given time horizon [119905 minus Ξ[120582] 119905] the linearspace L1 ≜ 119910(119896)119905

119896=119905minusΞ[120582] generated by 119910(119896)119905

119896=119905minusΞ[120582]is

equivalent to the linear space of

L2 ≜ ℏ (119904)119905minusℏminus1119904=119905minusΞ[120582]minusℏ 119905minus119904 (119904)119905119904=119905minusℏ (18)

which implies thatL2 contains the same information asL1

Step 2 We further fuse the complementary observations (15)of the same stamp 119904 isin [119905 minus Ξ[120582] minus ℏ 119905) into one completemeasurement [120582]

119898 (119904) (119898 isin ℏ 119905 minus 119904) prior to the localestimation process and thus

[120582]119898 (119904) = 119877 sdot ℓsum

119894=1

119892 ([119894]119898 (119904) 119909[119894]119898 (119904))

119898 isin ℎ 119905 minus 119904 119904 = 119905 minus Ξ[120582] minus ℏ 119905(19)

where 119877 denotes the weighted matrix 119892([119894]119898 (119904) 119909[119894]119898 (119904)) is

the relative measure function Thus the local measurements[120582]119898 (119904) isin R119899119910 in (19) can be rewritten in the reorganized

linear spaceL2 as

[120582]119898 (119904) = 120599[120582]119904 [120582]

119898 (119904) 119909 (119904) + V[120582]119898 (119904 120574[120582]119898 ) (20)

where 120599[120582]119904 represents the failed fusion of [120582](119904) at time stamp119904 = 119905minusΞ[120582]minusℏ 119905minus1 which takes the value of 0 or 1 and ismodeled as Bernoulli distributed white sequences satisfyingProb120599[120582]119904 = 1 = 120599[120582] 0 lt 120599[120582] le 1Remark 5 Note that 120599[120582]119904 (119904 = 119905 minus Ξ[120582] minus ℏ 119905 minus 1) is intro-duced to represent the failure of the preestimating algorithm(19) at time 119904 with the reason that the numbers of reorganizedCMs ℏ|119905(119904) or 119905minus119904|119905(119904) are less than a lower limit [9 12]Thusthrough preestimating transformation the remote sensingCMs in (14) with high dimension are transformed into thedata missing in (20) with lower dimension

Besides 119877 and 119892([119894]119898 (119904) 119909[119894]119898 (119904)) in (19) will be deduced

through a complementary fusion strategy with the details asfollows

32 Complementary Fusion Strategy Let [119894]119898 (119904) 119894 = 1 2 ℓ be the set of complementary observations generated at

instant 119904 the relative measures between objective 120582 and itsneighbors 119894 isin N[120582]

119904 that is relative distance and relativeangle are calculated through RSS and AOA respectively asdepicted in (4) and the measure noise covariance [119894]

119898 (119904) in(17)

Hereafter the fused measurement covariance 119877 and thefused measurement variable 119910[120582]

119898 (119904) (119898 isin ℏ 119905 minus 119904) can beexpressed as

119877 = ( ℓsum119894=1

([119894]

119898 (119904))minus1)minus1 119898 isin ℏ 119905 minus 119904 (21)

119910[120582]119898 (119904)= 119877sdot ℓsum119894=1

(([119894]

119898 (119904))minus1 ([119894]119898 (119904) + 119909[ℓ]119898 (119904) minus 119909[119894]119898 (119904)))

(22)

Besides the sensitivity matrix of the fused measurementvariable (22) can be written as

119867 = 119877 sdot ℓsum119894=1

[([119894]

119898 (119904))minus1119867[119894]] (23)


hereinto 119867[119894] is the sensitivity matrix corresponding to[119894]119898 (119904)

119867[119894] = [minus1198683times3 03times3] (24)

Note that the estimation error in this research maynot always fall within the confidence level of the predictedcovariance boundary To avoid this problematic scenario weintroduce a new weighting factor 120595119894119910 as follows

Definition 6 Given the CMs [119894]119898 (119904)119898 isin ℏ 119905 minus 119904 sensitivity

matrix 119867[119894] and the inverse measurement noise variance([119894]

119898 (119904))minus1 the weighting factor120595119894119910 is applied before they arefused more specifically it is defined as

120595119894119910 = 9848582119894984858119879984858 (25)

where sumℓ119894=1 120595119894119910 = 1 984858 = [9848581 9848582 sdot sdot sdot 984858ℓ]119879

Theweighting factor120595119894119910 is a function of (4) For instance

RSS vector is defined as P119898rss = [[1]119898rss sdot sdot sdot [ℓ]

119898rss]119879 119898 isinℏ 119905 minus 119904 In the proposed weight strategy the weightingfactor of each CM is assigned based on the relative measuresbetween 119894 isin N[120582]

119904 and 120582 Lower weight is assigned to theneighbors 119894 isin N[120582]

119904 closer to 120582 In other words weightingfactor 120595119894119910 is selected to be proportional to the relativemeasures between the neighbors The last measure [ℓ]

119898rsswhich is equivalent to max(P119898rss) has highest 120595119894119910 Definethe terms ΔP119898rss and 984858 as

ΔP[119894]119898rss = [ℓ]

119898rss minus [119894]119898rss 119898 isin ℏ 119905 minus 119904

984858119894 = 1 minus ΔP[119894]119898rss10038171003817100381710038171003817ΔP[119894]119898rss

10038171003817100381710038171003817 119898 isin ℏ 119905 minus 119904 (26)

where ΔP119898rss = [ΔP[1]119898rss ΔP[2]

119898rss ΔP[ℓ]119898rss]

Thus the newly fused measurement noise covariance in(21) can be rewritten as

119877 = ℓ sdot ( ℓsum119894=1

120595119894119910 ([119894]


Furthermore the sensitivity matrix in (23) can be rewrit-ten as

119867 = ( ℓsum119894=1

120595119894119910 ([119894]

119898 (119904))minus1)minus1

sdot ℓsum119894=1

[120595119894119910 ([119894]

119898 (119904))minus1119867[119894]] 119898 isin ℏ 119905 minus 119904 (28)

Meanwhile the newly fused measurement vector in (22)can be rewritten as

119910[120582]119898 (119904) = 119877 sdot ℓsum

119894=1

119892 ([119894]119898 (119904) 119909[119894]119898 (119904)) = ℓ

sdot ( ℓsum119894=1

120595119894119910 ([119894]

119898 (119904))minus1)minus1ℓsum119894=1

1ℓsdot [120595119894119910 ([119894]

119898 (119904))minus1 ([119894]119898 (119904) + 119909[ℓ]119898 (119904) minus 119909[119894]119898 (119904))]

119898 isin ℏ 119905 minus 119904

(29)

where119892 ([119894]119898 (119904) 119909[119894]119898 (119904))

= 1ℓsdot [120595119894119910 ([119894]

119898 (119904))minus1 ([119894]119898 (119904) + 119909[ℓ]119898 (119904) minus 119909[119894]119898 (119904))]

119898 isin ℏ 119905 minus 119904 (30)

Note that the resulting problem is solved by a robustrecursive scheme which allows one to determine approxi-mate estimates with a reduced computational condition Thedetails are presented in the following subsection

33 Delay-Aware FIR Estimation Algorithm In this subsec-tion we proposed a novel unbiased FIR estimator namelydelay-aware FIR (DFIR) estimation algorithm to realizethe network-based state estimation of a generic node 120582 isinC119892 based on the cooperative observation of its neighbor-hood N[120582]

119904 Note that several unique network-induced con-straints are involved including complementary observationsMarkov jump parameters and random delay hereinto par-tial constraints have been transformed in Sections 31 and 32which may simplify the design procedure of DFIR

Prior to themain result of this section let us recall a usefullemma that will be useful in DFIR algorithm

Lemma 7 (see [24]) Suppose that the general trace optimiza-tion (GTO) problem is given as

minH

tr [(HA minusB)C (HA minusB)119879 +HDH119879]

subject to HE = F (31)

where A B C D E and F denote the constant matriceswith appropriate dimensions C = C119879 gt 0 andD = D119879 gt 0tr(J) is the trace of the main diagonal ofJ

Thus the unique solution to the GTO problem can beobtained asH

= [F B] [[(E119879Λminus1E)minus1 E119879Λminus1

CA119879Λminus1 [119868 minusE (E119879Λminus1E)minus1 E119879Λminus1]]] (32)

where Λ = ACA119879 +D


On the recent finite-time horizon [119905 minus Ξ[120582] minus ℏ 119905 minus 1] ofreorganized linear space L2 the receding horizon filter for(3) and (20) can be written as

[120582]119905|119905minus1 = 119905minus1sum119896=119905minusΞ[120582]minusℏ

119905minus119896120574(119904)[120582]ℏ (119904) = H120574(119904)Y

[120582]

119905minus1 (33)

where H120574(119904) is the gain matrix Y[120582]

119905minus1 is the augmentedmeasurement vector and 120574(119904) is the set of switching factorson the horizon [119905 minus Ξ[120582] minus ℏ 119905 minus 1] defined by (34)ndash(36)respectively as

H120574(119904) ≜ [Ξ[120582]+ℏ120574(119904) Ξ[120582]+ℏminus1120574(119904) sdot sdot sdot 1120574(119904)] (34)

Y[120582]

119905minus1 ≜ [[120582]119879

119905minusΞ[120582]minusℏ[120582]119879

119905minusΞ[120582]minusℏ+1sdot sdot sdot [120582]119879

119905minus1 ]119879 (35)

120574 (119904) ≜ 120574[120582]119905minusΞ[120582]minusℏ

120574[120582]119905minusΞ[120582]minusℏ+1

120574[120582]119905minus1 (36)

For ease of notation only we express the horizon size ofestimator Ξ[120582] + ℏ as Ξ that is Ξ ≜ Ξ[120582] + ℏ Y[120582]

119905minus1 can berewritten in the batch form as

Y[120582]

119905minus1 = FΞ120574(119904)119909119905minusΞ +HΞ120574(119904)W119905minus1 +V119905minus1120574(119904) (37)

where

FΞ120574(119904) ≜[[[[[[[[[

119905minusΞ120574[120582]119905minusΞ

ϝ119905minus1119905minusΞ

119905minusΞ+1120574[120582]119905minusΞ+1

ϝ119905minus1119905minus2119905minus1120574[120582]119905minus1Φminus1

119905minus1

]]]]]]]]]

W119905minus1 ≜[[[[[[[[

119908119905minusΞ119908119905minusΞ+1119908119905minus1

]]]]]]]]

V119905minus1120574(119904) ≜[[[[[[[[

V119905minusΞ120574[120582]119905minusΞ

V119905minusΞ+1120574[120582]119905minusΞ+1

V119905minus1120574[120582]119905minus1

]]]]]]]]

HΞ120574(119904) ≜[[[[[[[[[

119905minusΞ120574[120582]119905minusΞ

Φminus1

119905minusΞΓ119905minusΞ 119905minusΞ120574[120582]

119905minusΞ

ϝ119905minusΞ+1119905minusΞ

Γ119905minusΞ sdot sdot sdot 119905minusΞ120574[120582]119905minusΞ

ϝ119905minus1119905minusΞΓ119905minusΞ0 119905minusΞ+1120574[120582]

119905minusΞ+1

Φminus1

119905minusΞ+1Γ119905minusΞ+1 sdot sdot sdot 119905minusΞ+1120574[120582]

119905minusΞ+1

ϝ119905minus1119905minusΞ+1

Γ119905minusΞ+1 d

0 0 sdot sdot sdot 119905minus1120574[120582]119905minus1Φminus1

119905minus1Γ119905minus1

]]]]]]]]]

QΞ ≜ [[diag(Ξ⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞119876 119876 119876)]]

RΞ ≜ [[diag(Ξ⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞ )]]

ϝ119905minusℎ119905minus119892 ≜ ℎprod119894=119892

Φminus1119905minus119894 = Φminus1

119905minus119892Φminus1119905minus119892+1 sdot sdot sdot Φminus1

119905minusℎ

(38)


The noise terms HΞ120574(119904)W119905minus1 + V119905minus1120574(119904) in (37) can beexpressed as zero mean with covariance ΘΞ120574(119904) by

Θ119894120574(119904) =H119894120574(119904) [diag( 119894⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞119876 119876 119876)]H119879119894120574(119904)

+ [[diag(119894⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞ )]]

(39)

Substituting (37) into (33) and taking the expectation onboth sides of (33) the following relations can be obtained

119905|119905minus1 =HY[120582]

119905minus1

=HFΞ120574(119904)119909119905minusΞ+H (HΞ120574(119904)W119905minus1 +V119905minus1120574(119904))

119864 [119905|119905minus1] =HFΞ120574(119904)119864 [119909119905minusΞ] (40)

with the horizon initial conditions

119905minusΞ = 120603119905minusΞ = 119864 [119909119905minusΞ] 119905minusΞ = 119905minusΞ = cov 119909119905minusΞ 119909119905minusΞ (41)

determined by Lyapunov equation on the horizon 119896 isin[1199050 119905119905minusΞ) as120603119896+1 = Φ1198961206031198961206031199050

= 1206030119896+1 = Φ119896119896 + Φ119879

119896 + 1198961199050

= 0(42)

Consider the unbiased condition that is 119864[119905|119905minus1] =119864[119909119905minusΞ] and irrespective of the initial states constraint (43)is required

HFΞ120574(119904) = 119868 (43)

Thus the objective now turns to obtain the optimal gainmatrix H

119900with the unbiasedness constraint (43) such that

the estimation error 119890119905 of the estimate 119905|119905minus1 has the followingminimum variance

H119900 = argmin

H

119864 [119890119879119905 119890119905]= argmin

H

119864 tr [119909119905 minus 119905|119905minus1]119879 [119909119905 minus 119905|119905minus1]= argmin

H

(HΞ120574(119904)W119905minus1 +V119905minus1120574(119904))119879sdotH119879

H (HΞ120574(119904)W119905minus1 +V119905minus1120574(119904))

(44)

By using Lemma 7 the gain matrix H can be obtained bythe following correspondences

Alarr997888HΞBlarr997888 119874Clarr997888 QΞDlarr997888RΞElarr997888 FΞ120574(119904)Flarr997888 119868

(45)

Thus when Φ119896 119896120574[120582]119896 (119896 isin [119905 minus Ξ 119905 minus 1]) is observable

and Ξ[120582] + ℏ ge 119899 the DFIR for the reorganized system (20)can be given as

[120582]119905|119905minus1 =H119900Y

[120582]

119905minus1

= [119868 0] [[(F119879

Ξ120574(119904)Θminus1

Ξ120574(119904)FΞ120574(119904))minus1 F119879

Ξ120574(119904)Θminus1

Ξ120574(119904)lowast ]] Y[120582]

119905minus1

= (F119879Ξ120574(119904)

Θminus1

Ξ120574(119904)FΞ120574(119904))minus1 F119879

Ξ120574(119904)Θminus1

Ξ120574(119904)Y

[120582]

119905minus1(46)

Remark 8 Thebatch formof RHE (44) requires the inversioncomputation of matrices Θminus1

Ξ120574(119904)and F119879

Ξ120574(119904)Θminus1

Ξ120574(119904)FΞ120574(119904)

whose dimensionwill become large as the horizon lengthΞ[120582]

increases besides the singularity problem will be encoun-tered due to the impacts of LOSNLOS condition

Remark 9 The FIR filter uses finite measurements of therecent time interval The length of the interval is calledhorizon size which is also referred to as the window size ormemory size In (13) the matrices with subscript ldquoΞ[120582]rdquo areobtained using the information on the recent time interval[119905minusΞ[120582]minusℏ 119905minus1] where the horizon size is Ξ[120582]+ℏThe largematrices defined in (13) are time-varying

4 Adaptive Finite-Horizon Group Estimator

In this section we introduce an adaptive horizon group(AHG) approach to manage the horizon size in DFIR esti-mator Different from the previous horizon group shift policy[26] AHG policy is designed to adjust the horizon size ofDFIR algorithm according to the dynamic changed cooper-ative neighbors and the concerned network constraints TheAHG algorithm was combined with the proposed DFIR esti-mator and obtained AFGE estimator The detailed algorithmof estimating with AHG is presented below

The first step of estimation with the DFIR is to constructa group of horizon sizes (ie horizon group) The horizongroup at time 119904 is denoted byG119904 and is defined as

G119904 = Ξ | Ξ = 2Ξ119888119904 + (120577 times 120591[119894]119904 minus ℏ) 120591[119894]119904

= 0 1 2 ℏ (47)


Cooperativeneighbors

Horizon group

Random delay

Buffer

DFIRestimator

EvaluationHorizon select

Estimation selectionMixed

LOSNLOSmediums

z[i]t = 120601[i]t xt + u[i]

t (120574[i]t )

i = 1 2 985747

Ξcs+1

119970s

xs

x1s x2

smiddot middot middot

x987407+1s

Ξ1s

Ξ2s

Ξ987407+1s

120585 (xis (Ξ

is))

arg maxx119904(Ξ

119894119904)120585 (xs (Ξi

s))

y[i]t ≜ 120581[i]

t120591[z[i]t ]

x[i]t |iisinT[119904]

y[i]

tminusΞt|iisin119977[119904]

Figure 2 Adaptive finite-horizon group estimation scheme with AHG

where Ξ is the horizon size Ξ119888119904 is the center horizon size at

time 119904 120577 is the interval between horizon sizes and ℏ + 1 isthe total number of horizon sizes in a horizon group Notethat 120591[119894]119904 impliedly reflected the possible transmission delayfrom the cooperative neighbor 119894 isin N[120582]

119904 and ℏ denotes thetolerable maximum transmission delay which is defined inAssumption 3

The second step is to execute the DFIR estimator repeat-edly with the horizon sizes in G119904 which produces a groupof state estimates 1119904 2119904 ℏ+1119904 The reliable estimate isthen selected by maximizing the likelihood function of119910[120582]

119898 (119904) 119892119884(119910[120582]119898 (119904) | 119909 = 119904(Ξ)) where 119904(Ξ) is the state

estimate obtained from (46)with the horizon size Ξ Note thatthe density function of V[120582]119898 (119904 120574[120582]119898 ) is 119892119881(V[120582]119898 (119904 120574[120582]119898 )) where

119892119881 (V[120582]119898 (119904 120574[120582]119898 )) = 1radic(2120587)119902 det (119877)

sdot exp minus12 (V[120582]119898 (119904 120574[120582]119898 ))119879 119877minus1V[120582]119898 (119904 120574[120582]119898 )

(48)

The state estimates are then evaluated based on theirlikelihoods which can be computed as follows

120585 (119894119904 (Ξ)) = 119901 (119910[120582]119898 (119904) | 119894119904 (Ξ)) = 1

radic(2120587)119902 det (119877)times expminus12 (119910[120582]

119898 (119904) minus 120599[120582]119904 [120582]

119898 (119904) 119894119904 (Ξ))119879sdot 119877minus1 (119910[120582]

119898 (119904) minus 120599[120582]119904 [120582]

119898 (119904) 119894119904 (Ξ)) (49)

where 120585(119894119904(Ξ)) is the likelihood of the state estimate 119894119904(Ξ)and [120582]

119898 (119904) is the measurement matrix defined in (20)

J (119904 (Ξ)) = 12 [(119910[120582]119898 (119904) minus 120599[120582]119904 [120582]

119898 (119904) 119894119904 (Ξ))119879

sdot 119877minus1 (119910[120582]119898 (119904) minus 120599[120582]119904 [120582]

119898 (119904) 119894119904 (Ξ))] (50)

Thus a reliable estimate is selected by minimizingJ(119904(Ξ)) We use J(119904(Ξ)) as a measure of accuracy of theestimation result The smaller the cost J(119904(Ξ)) is the morereliable the estimated state is

The final step is to change the horizon group whichis called the AHG If we determine Ξ119888

119904 G119904 is determinedautomatically using (47) The horizon size that produces areliable estimate at time 119904 is used as the center horizon sizeat time 119904 + 1 Let 119904(Ξ119894

119904) be the state estimate obtained fromthe FIR estimator using the horizon size Ξ119894

119904 The AHG canthen be described by the following equation

Ξ119888119904+1 = argmax

Ξ119888119904

120585 (119904 (Ξ119894119904)) (51)

In this way the horizon group is moved (or shifted)toward the horizon size that produces good estimationperformance In the AHG algorithm the horizon size variesover time Thus we impose the lower bound Ξmin and theupper bound Ξmax on the horizon size We now summarizethe estimation algorithm with DFIR as depicted in Figure 2whereas the detailed algorithm is presented as follows

Algorithm 10 (AFGE estimator) Select design parameters oftheAHGalgorithm such as ℏ 120577 Ξmin and Ξmax Set the initial


value of the center horizon size as Ξ119888119904 = Ξmin For 119904 = Ξmin +1 Ξmin + 2 perform the following

(a) Construct horizon group G119904 that contains ℏ horizonsizes using definition (9) as follows

G119904 = 2Ξ119888119904 minus ℏ 2Ξ119888

119904 + 120577 minus ℏ 2Ξ119888119904 + 120577 times (ℏ minus 1)

minus ℏ 2Ξ119888119904 + 120577 times ℏ minus ℏ = Ξ1

119904 Ξ2119904 Ξ119894

119904 Ξℏ+1119904 (52)

(b) Using the horizon sizes Ξ119894119904 (119894 = 1 2 ℏ + 1)

contained inG119904 perform the followingfor 119894 = 1 to ℏ + 1 do

if Ξ119894119904 lt 119904 and Ξmin lt Ξ119894

119904 lt Ξmax then

(i) Obtain 119904(Ξ119894119904) by performing DFIR estima-

tor (Algorithm 10) using Ξ119894119904 as follows

[120582]119904|119904minus1 (Ξ119894119904) = (F119879

Ξ119894119904+ℏ120574(119904)Θminus1

Ξ119894119904+ℏ120574(119904)FΞ119894119904+ℏ120574(119904)

)minus1sdotF119879

Ξ119894119904+ℏ120574(119904)Θminus1

Ξ119894119904+ℏ120574(119904)Y

[120582]

119904minus1 (Ξ119894119904) (53)

where FΞ119894119904+ℏ120574(119904) ΘΞ119894119904+ℏ120574(119904)

and Y[120582]

119904minus1(Ξ119894119904)

can be obtained using the horizon size Ξ119894119904

and definition (37)(ii) Compute the likelihood of 119904(Ξ119894

119904) using(49) as

120585 (119894119904 (Ξ119894119904)) = 1


119898 (119904) minus 120599[120582]119904 [120582]

119898 (119904) 119894119904 (Ξ119894119904))119879

sdot 119877minus1 (119910[120582]119898 (119904) minus 120599[120582]119904 [120582]

119898 (119904) 119894119904 (Ξ119894119904))

(54)

endif

endfor(c) The output (ie state estimate) of DFIR at time 119904

denoted by 119904 is determined as

119904 = arg max119904(Ξ119894119904)120585 (119904 (Ξ119894

119904)) (55)

(d) The center size at time 119904 + 1 denoted by Ξ119888119904+1 is

determined as follows

Ξ119888119904+1 = argmax

Ξ119888119904

120585 (119904 (Ξ119894119904)) (56)

5 An Application to Clustered UAV Systems

51 Simulation Configurations and Metrics In this sectionwe illustrate the results proposed in Sections 3 and 4 by

numerical simulationsThe position and velocity of a genericUAV evolve according to the state-space model as follows

[119909119901 (119905 + 1)119909V (119905 + 1)] = [1198683 120575119905 sdot 119868303times3 1198683 ][119909119901 (119905)119909V (119905)]+ [05 sdot 1205752119905 sdot 13times1120575119905 sdot 13times1 ]119908 (119905)

(57)

where 119909119901(119905) and 119909V(119905) denote the position and velocity of agenericUAVat time 119905 respectively 120575119905 represents the samplingperiod and 120575119905 = 01119904 and 119908(119905) is a zero-mean white noisewith variance 119876119908 119876119908 = 0011198686 the initial values are 0|0 =[120 30 0 0 0 0]119879 and 1198750|0 = diag252I3 012I3

Consider a fleet of119873Σ = 75UAVs deployed schematicallyover a field of [minus5 5] times [minus5 5] times [0 1] kmThe initial networktopology is depicted in Figure 3(a) The objective UAV is setto track a trajectory as in Figure 3(b) and in flight for 1000 sec

The objective UAV updates its state essentially based oncooperative packets of its neighbors which include the timestamp 119904 and the navigational state of the sender [119894]119905 =[[119894]119879119901 (119905) [119894]119879V (119905)]119879 The relative position and relative angleare measured by RSS and AOA methods respectively withthe observation models as in (4)

For RSS measurements the transmission power is takenas 119870 = 20 dBm the path loss exponent is 120582 = 21 and 120572 =3244The standard deviation formeasurement noise of AOAis 01 degrees The variances of LOS and NLOS measurementnoises are taken as 119877[119894]

1199051 = 64 and 119877[119894]1199052 = 100 respectively

The bias of the NLOS measurement error is assumed to be120583[119894]119905 = minus5 The occurrence probability of NLOS measurementerror is taken as119875NLOS = 05 for each neighborThe transitionprobabilities of Markov chain 120582[119894]119905 are set as 120587(2)

11 = 120587(2)22 = 09

and 120587(2)12 = 120587(2)

21 = 01 To reduce the computational cost notall the cooperative measurements are used instead only ℓ =5 strongest received signals are adopted for state update of theobjective UAV

As to the random delay it is assumed that 120591(119905) isin0 1 2 3 120591(119905) is a prior known to the estimator and its pathscan be described by 120581[119894]119905120591 (119894 = 1 2 ℓ) with the restriction120581[119894]119905+1205911 1205911 times120581[119894]119905+1205912 1205912 = 0 if 1205911 = 1205912 Hereinto it is assumed that 120591(119905)is of the probabilities

Prob 120591 (119905) = 0 = 04Prob 120591 (119905) = 1 = 03Prob 120591 (119905) = 2 = 02Prob 120591 (119905) = 3 = 01

(58)

A distributed fusion receding horizon estimator (DRHE)in [25] is adopted to compare with DFIR in Section 3 and theAFGE in Section 4 Let RMSE denote the root mean squareerror for the estimation of 119909[119894]119905 that is ((1M) sum119872

119895=1(119909[119895]119905 minus[119895]119905|119905)2)05 whereM = 500 is theMonte Carlo simulation runs

52 Simulation Results and Analysis In this subsectionseveral key properties of the proposed AFGE approach that


05000

0

5000

0200400600800

1000

Sink1

Normal1Normal2

Normal3Normal4

Z(m

)

Y (m)

Initial network topology (Nnode = 75)

X (m)

minus5000

minus5000

(a)

Ideal trajectory of objective UAV

0 1000 2000 3000 4000 5000 6000

0

2000

4000

60000

200400600800

1

23 4

5 67 8

9

10

11

Z(m

)

Y (m)

X (m)

(b)

Figure 3 Initial settings of network topology and ideal trajectory (a) Initial topology of the networked vehicles (b) Ideal trajectory ofobjective vehicle

0 5 10 15 20 25 30 35 400

05

1

15

2

25

3

Horizon size

RMSE

10 12 14

01020304

AFGE estimatorDRHE estimator

DFIR estimator

Figure 4 RMSEs comparison of DFIR DRHE and AFGE estima-tors

is AHG scheme and estimation errors are tested on thenetworked UAVs with the models in Section 51 The designparameters of the AFGE algorithm are taken as ℏ = 3 120577 = 1Ξmin = 2 and Ξmax = 40

First we verify that AHG improves the estimation per-formance of the DFIR estimator by adapting the horizonsize To this end we compare the AFGE with DFIR andDRHE While the DFIR uses a constant horizon size theAFGE uses the adaptive horizon size Figure 4 depicts theRMSE comparisons among DFIR DRHE and AFGE fordiverse horizon sizes in the rectilinear orbit determinationproblem In addition this figure shows the RMSE of theAFGE estimatorTheDFIR estimator produces theminimumRMSE of 0216 when the horizon size is 12 whereas theminimum RMSE of DRHE is 0320 with the horizon size of

13 The RMSE of the AFGE is 0196 which is smaller thanthat of DFIR and DRHE AFGE outperforms the DFIR usingthe best constant horizon size Thus it is verified that AHGimproves the estimation performance of the DFIR

Next we consider the UAV navigation scenario with theinitial settings as in Section 51 Besides we set the horizonsize of DFIR to Ξ[120582] = 12 according to the analysis inFigure 4The simulation results can be found in Figures 5 and6

Hereinto Figure 5(a) reports the localization resultsbased on RSSAOA relative measurement and the CPs of itsneighbors while the related parameters are recorded in Fig-ures 5(b) and 5(c) It is clear fromFigure 5(b) that the numberof received CPs in each step is greater than four whichmeansthe observations are sufficient for the geometric localizationin three-dimension space [9] One path of 120591(119905) is given inFigure 5(c) based on which the optimal filter can be obtaineddirectly by using the scheme in Sections 3 and 4 As expectedthe localization result in Figure 5(a) demonstrates the validityof the proposed AFGE algorithm and navigation scheme

More specifically Figures 6(a) and 6(b) present thecomparison of estimation results on location and velocityerrors respectively The estimators of DFIR in Section 3 andDRHE [25] are chosen as the comparison with our proposedAFGE algorithm

As shown in the simulation results the convergencerate of the DRHE is faster than that of DFIR due to thefact that DRHE adopts a distributed fusion structure in thedesign of estimator however the convergence error of theDFIR is smaller than that of DRHE which is partly due toits reorganization of delayed observations and is also dueto the design of DFIR considering the Markov jumping ofLOSNLOS errors Furthermore the performance of AFGEoutperforms DRHE and DFIR the reason can be owed to itsreorganization scheme for delayed observation on one hand


Localization results

Real trajectoryDRHE resultDFIR result

AFGE resultStart point

120

12002

12004

12006

29993030013002

300330043005

0

500

1000

120120005

12001

299953030005

0

100minus500

minus100

Pal

t(m

)

Plat ( ∘)

Plon (∘)

(a)

0 200 400 600 800 1000

4

6

8

10

12

14

Num

ber o

f obs

erve

d no

des

Time (sec)

Change of ONsNumber of ONs

Observed neighbors with Nnode = 75

(b)

0 200 400 600 800 1000

0

1

2

3

Time (sec)

0 200 400 600 800 1000

0

05

1

15

Time (sec)

minus05

TME120574

(t)

RD120591

(c)

Figure 5 Localization results comparison and the related parameters (a) Localization trajectories based on DRHE DFIR and AFGE (b)Number of corresponding observable neighbors (c) Random delay 120591(119905) isin 0 1 2 3 and the transition of NOSNLOS errors 120574(119905) isin 0 1

and considering the Markov jumping of LOSNLOS errorson the other hand Besides the proposed AFGE also benefitsfrom the AHG policy which can manage the horizon sizethus acquiring the fastest convergence rate with the smallestestimation error

6 Conclusions and DiscussionsIn this paper an adaptive horizon estimator has beenaddressed to solve the state estimation of NNS with the fea-tures of remote sensing complementary observations (RSOs)under mixed LOSNLOS environments An adaptive finite-horizon group estimator (AFGE) has been proposed wherethe horizon size can be adjusted in real time according to

stochastic parameters and randomdelays First a delay-awareFIR (DFIR) estimator has been derived with observationreorganization and complementary fusion strategies Secondan adaptive horizon group (AHG) policy has been proposedto manage the horizon size The AFGE algorithm is thusrealized by combining AHG policy and DFIR estimator Anumerical simulation example has been used to demonstratethe potential of the proposed approach to robust estimationin terms of both performances and computational tractabil-ity

Future research will focus on other typical issues of NNSin terms ofmultirate fusion problem and cooperative attitudeestimation problem


Position estimation error

0 500 1000

0

50

100

150

200

250

Time (sec)

minus50

minus100

minus150

minus200

minus250

ΔP

lon

(m)

0 500 1000

0

50

100

150

200

Time (sec)

minus50

minus100

minus150

ΔP

lat

(m)

Time (sec)0 500 1000

0

200

400

600

800

1000

1200

minus200

minus400

ΔP

alt

(m)

DRHEDFIRAFGE

(a)

0 500 1000

0

1

2

3

4

0 500 1000

0

1

2

3

Time (sec) Time (sec)Time (sec)

Velocity estimation error

0 500 1000

0

5

10

15

20

25

minus5

minus1

minus2

minus3

minus4

minus5

minus1

minus2

ΔV

lon

(ms

)

ΔV

lat

(ms

)

ΔV

alt

(ms

)

DRHEDFIRAFGE

(b)

Figure 6 Performance comparison of estimation results between DRHE DFIR and AFGE (a) Localization results (b) Velocity results


Competing Interests

The authors declare that they have no competing interests

Acknowledgments

This work was supported in part by the National NaturalScience Foundation of China under Grant nos 61473306and 61203168 and partly by the Defense Advanced ResearchProject of China under Grant no 9140A09040614JB14001

References

[1] C Gao G Zhao J Lu and S Pan ldquoDecentralised moving-horizon state estimation for a class of networked spatial-navigation systems with random parametric uncertainties andcommunication link failuresrdquo IET Control Theory amp Applica-tions vol 9 no 18 pp 2666ndash2677 2015

[2] C Gao G R Zhao J H Lu and S Pan ldquoA grid-based coopera-tive QoS routing protocol with fadingmemory optimization fornavigation carrier ad hoc networksrdquo Computer Networks vol76 pp 294ndash316 2015

[3] B Allotta R Costanzi E Meli L Pugi A Ridolfi and GVettori ldquoCooperative localization of a team of AUVs by atetrahedral configurationrdquo Robotics and Autonomous Systemsvol 62 no 8 pp 1228ndash1237 2014

[4] M Bibuli G Bruzzone M Caccia A Ranieri and E ZereikldquoMulti-vehicle cooperative path-following guidance system fordiver operation supportrdquo in Proceedings of the 10th IFACConfer-ence on Manoeuvring and Control of Marine Craft (MCMC rsquo15)pp 75ndash80 Copenhagen Denmark August 2015

[5] Z X Jiang and B T Cui ldquoEstimation of spatially distributedprocesses using mobile sensor networks with missing measure-mentsrdquoChinese Physics B vol 24 no 2 Article ID 020702 2015

[6] C Toth and G Jozkow ldquoRemote sensing platforms and sensorsa surveyrdquo ISPRS Journal of Photogrammetry andRemote Sensingvol 115 pp 22ndash36 2016

[7] SM Azizi and K Khorasani ldquoNetworked estimation of relativesensing multiagent systems using reconfigurable sets of subob-serversrdquo IEEE Transactions on Control Systems Technology vol22 no 6 pp 2188ndash2204 2014

[8] T A Wettergren ldquoPerformance of search via track-before-detect for distributed sensor networksrdquo IEEE Transactions onAerospace and Electronic Systems vol 44 no 1 pp 314ndash3252008

[9] A F H Goetz J BWellman andW L Barnes ldquoOptical remotesensing of the earthrdquo Proceedings of the IEEE vol 73 no 6 pp950ndash969 1985

[10] F-Q Wen G Zhang and D Ben ldquoDirection-of-arrival estima-tion for co-located multiple-input multiple-output radar usingstructural sparsity Bayesian learningrdquoChinese Physics B vol 24no 11 Article ID 110201 2015

[11] I Guvenc and C-C Chong ldquoA survey on TOA based wirelesslocalization andNLOSmitigation techniquesrdquo IEEE Communi-cations Surveys and Tutorials vol 11 no 3 pp 107ndash124 2009

[12] K Yu and Y J Guo ldquoStatistical NLOS identification basedon AOA TOA and signal strengthrdquo IEEE Transactions onVehicular Technology vol 58 no 1 pp 274ndash286 2009

[13] MHeidari N A Alsindi and K Pahlavan ldquoUDP identificationand error mitigation in ToA-based indoor localization systems

using neural network architecturerdquo IEEE Transactions onWire-less Communications vol 8 no 7 pp 3597ndash3607 2009

[14] F Yin C Fritsche F Gustafsson and A M Zoubir ldquoEM-and JMAP-ML based joint estimation algorithms for robustwireless geolocation in mixed LOSNLOS environmentsrdquo IEEETransactions on Signal Processing vol 62 no 1 pp 168ndash1822014

[15] C Morelli M Nicoli V Rampa and U Spagnolini ldquoHiddenMarkov models for radio localization in mixed LOSNLOSconditionsrdquo IEEE Transactions on Signal Processing vol 55 no4 pp 1525ndash1542 2007

[16] L Yi S G Razul Z Lin and C M See ldquoIndividual AOAmeasurement detection algorithm for target tracking in mixedLOSNLOS environmentsrdquo in Proceedings of the IEEE Inter-national Conference on Acoustics Speech and Signal Processing(ICASSP rsquo13) pp 3924ndash3928 Vancouver Canada May 2013

[17] L Yi S G Razul Z Lin and C M See ldquoTarget trackingin mixed LOSNLOS environments based on individual mea-surement estimation and LOS detectionrdquo IEEE Transactions onWireless Communications vol 13 no 1 pp 99ndash111 2014

[18] M C Vanderveen C B Papadias and A Paulraj ldquoJoint angleand delay estimation (JADE) for multipath signals arriving atan antenna arrayrdquo IEEE Communications Letters vol 1 no 1pp 12ndash14 1997

[19] J H Lu C Gao G R Zhao and S Liu ldquoDecentralized state esti-mation for networked navigation systems with communicationdelay and packet loss the receding horizon caserdquo in Proceedingsof the 17th IFAC Symposium on System Identification pp 1094ndash1099 Beijing China October 2015

[20] A M Syed and R Saravanakumar ldquoAugmented Lyapunovapproach to119867infin state estimation of static neural networks withdiscrete and distributed time-varying delaysrdquoChinese Physics Bvol 24 Article ID 050201 2015

[21] C Gao G R Zhao S Liu and J H Lu ldquoDecentralizednavigational state estimation for networked multiple vehiclessubject to limited data raterdquo in Proceedings of the 17th IFACSymposium on System Identification pp 1088ndash1093 BeijingChina October 2015

[22] B Q Xue S Y Li and Q M Zhu ldquoMoving horizon stateestimation for networked control system with multiple packetdropoutsrdquo IEEE Transactions on Automatic Control vol 57 no9 pp 2360ndash2366 2012

[23] G C Goodwin H Haimovich D E Quevedo and J S WelshldquoA moving horizon approach to networked control systemdesignrdquo IEEE Transactions on Automatic Control vol 49 no 9pp 1427ndash1445 2004

[24] W H Kwon P S Kim and P Park ldquoA receding horizonKalman FIR filter for discrete time-invariant systemsrdquo IEEETransactions on Automatic Control vol 44 no 9 pp 1787ndash17911999

[25] I Y Song and V Shin ldquoDistributed mixed continuous-discretereceding horizon filter for multisensory uncertain active sus-pension systems with measurement delaysrdquo IET ControlTheoryamp Applications vol 7 no 15 pp 1922ndash1931 2013

[26] J M Pak C K Ahn C J Lee P Shi M T Lim andM K SongldquoFuzzy horizon group shift FIR filtering for nonlinear systemswith Takagi-SugenomodelrdquoNeurocomputing vol 174 pp 1013ndash1020 2016

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of


Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of


Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of


Mathematical PhysicsAdvances in

Complex AnalysisJournal of


OptimizationJournal of


CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of


Operations ResearchAdvances in

Journal of


Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences


The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Algebra

Discrete Dynamics in Nature and Society



Decision SciencesAdvances in

Discrete MathematicsJournal of


Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


the researches on the complementary property of wirelessnetworks are quite few let alone the packet-based RSO ofNNS that is the navigational knowledge of the sender andthe relative measurements

Non-line-of-sight (NLOS) propagation might take placein a wireless network whenever there are obstacles hin-dering the radio signal path between the objective and itscooperative neighbors especially in an urban region or theindoor environment Whenever measured by received signalstrength (RSS) or angle of arrival (AOA) NLOS propagationoften induces transmission delay or dropout of the packetsand thus large localization errorsmight occur if not addressed[11] Several strategies have been proposed for mitigating thebad effect of NLOS errors such as statistical identification[12] hypothesis test [13] and robust parameter estimation[14] In [15] the switching behavior of NLOS and the line-of-sight (LOS) has been described by a hidden MarkovModel and an estimator has been designed by combiningKalman filter with the interacting multiple model (IMM)approach These results are further extended to nonlinearAOA measurement by using the MUSIC method in [16]Similar approach has also been used to observe a mobileobjective in mixed LOSNLOS environments by using indi-vidual measurement and LOS detection [17] However littleresearch has been investigated to mitigate the delay anddropout problems caused by NLOS propagation In [18] ajoint AOA and delay estimation method has been proposedfor space-time coherent distributed signals based on searchThe problem of joint AOARSS based state estimation forNNS in mixed LOS and NLOS environment has not beenstudied

In reality almost all networked systems can record theresults over a recent finite horizon for such time-varyingsystems one would be more interested in the optimalestimation based on the finite recent data When it comesto the filter design issue finite-horizon estimation is aneffective approach and gives rise to many available methodsin terms of moving horizon estimation [1 19] finite-horizonHinfin fault estimation [20] and finite-horizon FIR estimation[21] However finite-horizon estimation approach has itsown limitations especially on expensively computationalburden [22] with the reasons in terms of the adoption offinite recent measurements the inversion computation [23]and the singularity problem [24] One of the improvementmethods is to manage the horizon size (window length) andseveral approaches [25] have been proposed Although theexisting approaches are effective with linear time-invariantsystems they can be less effective with wireless networkedsystems with the reason that the constant (ie fixed) horizonsize of the existing methods may be not able to handlethe network-induced constraints owing to mixed LOSNLOSenvironmentThus it is desirable to adapt the horizon size infinite-horizon estimation and this motivates us to investigatethis issue

Summarizing the above analysis it can be concludedthat there is a great need to examine how the mixedLOSNLOS environments affect the performance of thedistributed estimation over a sensor network with RSOdata To address the effects of NLOS errors and random

delays a jump Markov system has been modeled with theswitching parameters corresponding to LOSNLOS errorsAn adaptive finite-horizon group estimator (AFGE) has beenproposed where the horizon size can be adjusted in real timeaccording to stochastic parameters and random delays Anumerical example is proposed to verify the effectiveness ofthe proposed AFGE scheme The main contributions of thisresearch can be highlighted as follows (i) The observationmodel of NNS is a special one that has complementaryproperty and is affected by switching LOSNLOS errors andrandom transmission delay (ii) An observation reorganiza-tion scheme and a complementary fusion strategy have beenintroduced to DFIR estimation procedure to mitigate theimpacts of delay and complementary property respectively(iii) The proposed AFGE scheme contains an AHG policywhich is proposed to manage the horizon size

The remainder of this paper is organized as follows InSection 2 the graph-based network background is intro-duced while the NNS model and some preliminaries arebriefly outlined Section 3 describes the derivation of a newDFIR estimator Section 4 proposes the AHG method andcompletes the AFGE In Section 5 an application of clusteredUAVs is given for the designedAFGE Finally the conclusionsare drawn in Section 6

Notation 119906119866 stands for the Euclidean norm of a genericvector 119906 1199062119866 = 119906119879119866119906 where119866 is a symmetric positive semi-definite matrix Given a generic vector 119906119905 119906119905119905minus119873 = 119906119905minus119873119905 ≜col(119906119905minus119873 119906119905minus119873+1 119906119905) a generic matrix 119860119905

119905minus119873 = 119860 119905minus119873119905 ≜col(119860 119905minus119873 119860 119905minus119873+1 119860 119905) 119905 = 119873119873 + 1 L1199101 119910119899denotes the linear space spanned by 1199101 1199101198992 Preliminaries and Problem FormulationIn this section we introduce some preliminaries related to thedescription of communication network and then present theproblem setup

21 Graph-BasedNetworkDescription Suppose that119873Σ vehi-cles form into an Ad hoc network where every vehicle acts asa networked node and carries out the uniform communica-tion protocol that is [1 2 19] The communication topologyof information flow between networked vehicles is describedby a directed graphG = (C119892E119892R119892) whereC119892 isin 1 119873Σis the finite set of networked vehicles E119892 sube C119892 timesC119892 is the setof edgesR119892 = (119903119894) isin R119899 is the set of action radius and 119894 isin C119892

represents the sender The edge 119890[119894rarr119895] = (119894 119895) isin E119892 indicatesthat node 119895 can receive the information from node 119894 that is(119894 119895) isin E119892 rArr 119889[119894rarr119895] le 119903119894 Specifically we define the set ofneighbors as follows

Definition 1 Let 119909[119894]119901 isin R3 be the position of node 120582 isin C119892the neighbor set of node 120582 at time 119905 can be defined as

N[120582]119905 = 119895 isin C119892 119903inf le 10038171003817100381710038171003817119909[119895]119901 (119905) minus 119909[120582]119901 (119905)10038171003817100381710038171003817 le 119903sup 120582= 119895 C119892 = 1 2 119873Σ (1)

where 119903sup and 119903inf denote the supremum and infimum radiiof available communication range respectively Note that the





0001

02

03 00 01 02 03

12

1

2

0

RTS

RSA

zalt ylat

xlon

yg

Og

zg

xg

CM1

CM4

CM3

CH

Cag

CM5

CM2

GM6CM6

t(k)

GM1

GM2

GM3

GM4

GM5

t(kminus N

+ 1) S

t(kminus1)

GELz

t(kminusN)








(2)



119909119905+1 = Φ119905119909119905 + Γ119905119908119905 (3)





119905 = 119911[119894]119905 |119911[119894]119905 119894 isin N[120582]119905




10038171003817100381710038171003817) + 120578[119894]AOA120601119894 = 1 2 ℓ

(4)


119905 and zN119905









119906[119894]119905 = 120576[119894]119905 LOS condition

120576[119894]119905 + 119890[119894]119905 NLOS condition(6)


119891 (xN119905 zN

119905 ) ≜ 119911[119894]119905 = 120601[119894]119905 119909119905 + 119906[119894]119905 (120574[119894]119905 ) 119894 = 1 2 ℓ (7)









119910[119894]119905 = ℏsum

120591=0


(8)







119905ℏ119911[119894]119905minusℏ)119879]119879 (10)


120581[119894]119905+119895119895 times 120581[119894]119905+119896119896 = 0 119895 = 119896 (11)










119869[120582] = 119864 10038171003817100381710038171003817119909 (119905) minus [120582] (119905 | 119905)10038171003817100381710038171003817 | [119894]119904 10038161003816100381610038161003816119904ge0 [119894]119896


[120582]119905 = 1 2 ℓ (13)








the batch form as

ℏ (119904) ≜ [119910[119895]1198790 (119904)100381610038161003816100381610038161003816ℓ119895=1 119910[119895]119879


119905minus119904 (119904) ≜ [119910[119895]1198790 (119904)100381610038161003816100381610038161003816ℓ119895=1 119910[119895]119879


(14)




where

ℏ (119904) = [(120581[119894]1199040120601[119894]119904 10038161003816100381610038161003816ℓ119894=1)119879 (120581[119894]119904+ℏℏ

120601[119894]119904+ℎ


119904+ℏℏ119906[119894]119904+ℏ(120574[119894]

119904+ℏ)10038161003816100381610038161003816ℓ119894=1)119879]119879

V119905minus119904 (119904 120574[119894]119904119905 )= [(120581[119894]1199040119906[119894]119904 (120574[119894]119904 )10038161003816100381610038161003816ℓ119894=1)119879 (120581[119894]119905119905minus119904119906[119894]119905 (120574[119894]119905 )10038161003816100381610038161003816ℓ119894=1)119879]119879

(16)



(17)




119896=119905minusΞ[120582]is






[120582]119898 (119904) = 119877 sdot ℓsum

119894=1

119892 ([119894]119898 (119904) 119909[119894]119898 (119904))




linear spaceL2 as

[120582]119898 (119904) = 120599[120582]119904 [120582]

119898 (119904) 119909 (119904) + V[120582]119898 (119904 120574[120582]119898 ) (20)







119898 (119904) in(17)



119877 = ( ℓsum119894=1

([119894]


119910[120582]119898 (119904)= 119877sdot ℓsum119894=1

(([119894]

119898 (119904))minus1 ([119894]119898 (119904) + 119909[ℓ]119898 (119904) minus 119909[119894]119898 (119904)))

(22)


119867 = 119877 sdot ℓsum119894=1

[([119894]

119898 (119904))minus1119867[119894]] (23)








120595119894119910 = 9848582119894984858119879984858 (25)








ΔP[119894]119898rss = [ℓ]



10038171003817100381710038171003817 119898 isin ℏ 119905 minus 119904 (26)


119898rss ΔP[ℓ]119898rss]


119877 = ℓ sdot ( ℓsum119894=1

120595119894119910 ([119894]



119867 = ( ℓsum119894=1

120595119894119910 ([119894]

119898 (119904))minus1)minus1

sdot ℓsum119894=1

[120595119894119910 ([119894]



119910[120582]119898 (119904) = 119877 sdot ℓsum

119894=1

119892 ([119894]119898 (119904) 119909[119894]119898 (119904)) = ℓ


120595119894119910 ([119894]

119898 (119904))minus1)minus1ℓsum119894=1

1ℓsdot [120595119894119910 ([119894]

119898 (119904))minus1 ([119894]119898 (119904) + 119909[ℓ]119898 (119904) minus 119909[119894]119898 (119904))]

119898 isin ℏ 119905 minus 119904

(29)

where119892 ([119894]119898 (119904) 119909[119894]119898 (119904))

= 1ℓsdot [120595119894119910 ([119894]

119898 (119904))minus1 ([119894]119898 (119904) + 119909[ℓ]119898 (119904) minus 119909[119894]119898 (119904))]

119898 isin ℏ 119905 minus 119904 (30)






minH











119905minus119896120574(119904)[120582]ℏ (119904) = H120574(119904)Y

[120582]

119905minus1 (33)




Y[120582]

119905minus1 ≜ [[120582]119879

119905minusΞ[120582]minusℏ[120582]119879


119905minus1 ]119879 (35)

120574 (119904) ≜ 120574[120582]119905minusΞ[120582]minusℏ

120574[120582]119905minusΞ[120582]minusℏ+1

120574[120582]119905minus1 (36)



Y[120582]


where

FΞ120574(119904) ≜[[[[[[[[[

119905minusΞ120574[120582]119905minusΞ


119905minusΞ+1120574[120582]119905minusΞ+1


119905minus1

]]]]]]]]]

W119905minus1 ≜[[[[[[[[


]]]]]]]]

V119905minus1120574(119904) ≜[[[[[[[[

V119905minusΞ120574[120582]119905minusΞ

V119905minusΞ+1120574[120582]119905minusΞ+1

V119905minus1120574[120582]119905minus1

]]]]]]]]

HΞ120574(119904) ≜[[[[[[[[[

119905minusΞ120574[120582]119905minusΞ

Φminus1


119905minusΞ




119905minusΞ+1

Φminus1


119905minusΞ+1


Γ119905minusΞ+1 d



]]]]]]]]]

QΞ ≜ [[diag(Ξ⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞119876 119876 119876)]]

RΞ ≜ [[diag(Ξ⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞ )]]




119905minusℎ

(38)



Θ119894120574(119904) =H119894120574(119904) [diag( 119894⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞119876 119876 119876)]H119879119894120574(119904)

+ [[diag(119894⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞ )]]

(39)


119905|119905minus1 =HY[120582]

119905minus1


119864 [119905|119905minus1] =HFΞ120574(119904)119864 [119909119905minusΞ] (40)




= 1206030119896+1 = Φ119896119896 + Φ119879

119896 + 1198961199050

= 0(42)


HFΞ120574(119904) = 119868 (43)




H119900 = argmin

H

119864 [119890119879119905 119890119905]= argmin

H


H



(44)



(45)



[120582]119905|119905minus1 =H119900Y

[120582]

119905minus1

= [119868 0] [[(F119879

Ξ120574(119904)Θminus1

Ξ120574(119904)FΞ120574(119904))minus1 F119879

Ξ120574(119904)Θminus1

Ξ120574(119904)lowast ]] Y[120582]

119905minus1

= (F119879Ξ120574(119904)

Θminus1

Ξ120574(119904)FΞ120574(119904))minus1 F119879

Ξ120574(119904)Θminus1

Ξ120574(119904)Y

[120582]

119905minus1(46)


Ξ120574(119904)and F119879

Ξ120574(119904)Θminus1

Ξ120574(119904)FΞ120574(119904)








= 0 1 2 ℏ (47)



Horizon group

Random delay

Buffer

DFIRestimator



LOSNLOSmediums

z[i]t = 120601[i]t xt + u[i]

t (120574[i]t )

i = 1 2 985747

Ξcs+1

119970s

xs

x1s x2


x987407+1s

Ξ1s

Ξ2s

Ξ987407+1s

120585 (xis (Ξ

is))

arg maxx119904(Ξ

119894119904)120585 (xs (Ξi

s))

y[i]t ≜ 120581[i]

t120591[z[i]t ]


y[i]









119892119881 (V[120582]119898 (119904 120574[120582]119898 )) = 1radic(2120587)119902 det (119877)


(48)


120585 (119894119904 (Ξ)) = 119901 (119910[120582]119898 (119904) | 119894119904 (Ξ)) = 1


119898 (119904) minus 120599[120582]119904 [120582]

119898 (119904) 119894119904 (Ξ))119879sdot 119877minus1 (119910[120582]

119898 (119904) minus 120599[120582]119904 [120582]

119898 (119904) 119894119904 (Ξ)) (49)



J (119904 (Ξ)) = 12 [(119910[120582]119898 (119904) minus 120599[120582]119904 [120582]

119898 (119904) 119894119904 (Ξ))119879

sdot 119877minus1 (119910[120582]119898 (119904) minus 120599[120582]119904 [120582]

119898 (119904) 119894119904 (Ξ))] (50)






Ξ119888119904+1 = argmax

Ξ119888119904

120585 (119904 (Ξ119894119904)) (51)






G119904 = 2Ξ119888119904 minus ℏ 2Ξ119888



119904 Ξ2119904 Ξ119894

119904 Ξℏ+1119904 (52)







[120582]119904|119904minus1 (Ξ119894119904) = (F119879

Ξ119894119904+ℏ120574(119904)Θminus1

Ξ119894119904+ℏ120574(119904)FΞ119894119904+ℏ120574(119904)

)minus1sdotF119879

Ξ119894119904+ℏ120574(119904)Θminus1

Ξ119894119904+ℏ120574(119904)Y

[120582]

119904minus1 (Ξ119894119904) (53)

where FΞ119894119904+ℏ120574(119904) ΘΞ119894119904+ℏ120574(119904)

and Y[120582]

119904minus1(Ξ119894119904)



119904) using(49) as

120585 (119894119904 (Ξ119894119904)) = 1


119898 (119904) minus 120599[120582]119904 [120582]

119898 (119904) 119894119904 (Ξ119894119904))119879

sdot 119877minus1 (119910[120582]119898 (119904) minus 120599[120582]119904 [120582]

119898 (119904) 119894119904 (Ξ119894119904))

(54)

endif



119904 = arg max119904(Ξ119894119904)120585 (119904 (Ξ119894

119904)) (55)



Ξ119888119904+1 = argmax

Ξ119888119904

120585 (119904 (Ξ119894119904)) (56)





(57)







11 = 120587(2)22 = 09

and 120587(2)12 = 120587(2)




(58)





05000

0

5000

0200400600800

1000

Sink1

Normal1Normal2

Normal3Normal4

Z(m

)

Y (m)


X (m)

minus5000

minus5000

(a)


0 1000 2000 3000 4000 5000 6000

0

2000

4000

60000

200400600800

1

23 4

5 67 8

9

10

11

Z(m

)

Y (m)

X (m)

(b)


0 5 10 15 20 25 30 35 400

05

1

15

2

25

3

Horizon size

RMSE

10 12 14

01020304


DFIR estimator













120

12002

12004

12006

29993030013002

300330043005

0

500

1000

120120005

12001

299953030005

0

100minus500

minus100

Pal

t(m

)

Plat ( ∘)

Plon (∘)

(a)

0 200 400 600 800 1000

4

6

8

10

12

14

Num

ber o

f obs

erve

d no

des

Time (sec)



(b)

0 200 400 600 800 1000

0

1

2

3

Time (sec)

0 200 400 600 800 1000

0

05

1

15

Time (sec)

minus05

TME120574

(t)

RD120591

(c)








0 500 1000

0

50

100

150

200

250

Time (sec)

minus50

minus100

minus150

minus200

minus250

ΔP

lon

(m)

0 500 1000

0

50

100

150

200

Time (sec)

minus50

minus100

minus150

ΔP

lat

(m)

Time (sec)0 500 1000

0

200

400

600

800

1000

1200

minus200

minus400

ΔP

alt

(m)

DRHEDFIRAFGE

(a)

0 500 1000

0

1

2

3

4

0 500 1000

0

1

2

3



0 500 1000

0

5

10

15

20

25

minus5

minus1

minus2

minus3

minus4

minus5

minus1

minus2

ΔV

lon

(ms

)

ΔV

lat

(ms

)

ΔV

alt

(ms

)

DRHEDFIRAFGE

(b)



Competing Interests


Acknowledgments


References



































Volume 2014




Journal of











Journal of


Function Spaces






Algebra













0001

02

03 00 01 02 03

12

1

2

0

RTS

RSA

zalt ylat

xlon

yg

Og

zg

xg

CM1

CM4

CM3

CH

Cag

CM5

CM2

GM6CM6

t(k)

GM1

GM2

GM3

GM4

GM5

t(kminus N

+ 1) S

t(kminus1)

GELz

t(kminusN)








(2)



119909119905+1 = Φ119905119909119905 + Γ119905119908119905 (3)





119905 = 119911[119894]119905 |119911[119894]119905 119894 isin N[120582]119905




10038171003817100381710038171003817) + 120578[119894]AOA120601119894 = 1 2 ℓ

(4)


119905 and zN119905









119906[119894]119905 = 120576[119894]119905 LOS condition

120576[119894]119905 + 119890[119894]119905 NLOS condition(6)


119891 (xN119905 zN

119905 ) ≜ 119911[119894]119905 = 120601[119894]119905 119909119905 + 119906[119894]119905 (120574[119894]119905 ) 119894 = 1 2 ℓ (7)









119910[119894]119905 = ℏsum

120591=0


(8)







119905ℏ119911[119894]119905minusℏ)119879]119879 (10)


120581[119894]119905+119895119895 times 120581[119894]119905+119896119896 = 0 119895 = 119896 (11)










119869[120582] = 119864 10038171003817100381710038171003817119909 (119905) minus [120582] (119905 | 119905)10038171003817100381710038171003817 | [119894]119904 10038161003816100381610038161003816119904ge0 [119894]119896


[120582]119905 = 1 2 ℓ (13)








the batch form as

ℏ (119904) ≜ [119910[119895]1198790 (119904)100381610038161003816100381610038161003816ℓ119895=1 119910[119895]119879


119905minus119904 (119904) ≜ [119910[119895]1198790 (119904)100381610038161003816100381610038161003816ℓ119895=1 119910[119895]119879


(14)




where

ℏ (119904) = [(120581[119894]1199040120601[119894]119904 10038161003816100381610038161003816ℓ119894=1)119879 (120581[119894]119904+ℏℏ

120601[119894]119904+ℎ


119904+ℏℏ119906[119894]119904+ℏ(120574[119894]

119904+ℏ)10038161003816100381610038161003816ℓ119894=1)119879]119879

V119905minus119904 (119904 120574[119894]119904119905 )= [(120581[119894]1199040119906[119894]119904 (120574[119894]119904 )10038161003816100381610038161003816ℓ119894=1)119879 (120581[119894]119905119905minus119904119906[119894]119905 (120574[119894]119905 )10038161003816100381610038161003816ℓ119894=1)119879]119879

(16)



(17)




119896=119905minusΞ[120582]is






[120582]119898 (119904) = 119877 sdot ℓsum

119894=1

119892 ([119894]119898 (119904) 119909[119894]119898 (119904))




linear spaceL2 as

[120582]119898 (119904) = 120599[120582]119904 [120582]

119898 (119904) 119909 (119904) + V[120582]119898 (119904 120574[120582]119898 ) (20)







119898 (119904) in(17)



119877 = ( ℓsum119894=1

([119894]


119910[120582]119898 (119904)= 119877sdot ℓsum119894=1

(([119894]

119898 (119904))minus1 ([119894]119898 (119904) + 119909[ℓ]119898 (119904) minus 119909[119894]119898 (119904)))

(22)


119867 = 119877 sdot ℓsum119894=1

[([119894]

119898 (119904))minus1119867[119894]] (23)








120595119894119910 = 9848582119894984858119879984858 (25)








ΔP[119894]119898rss = [ℓ]



10038171003817100381710038171003817 119898 isin ℏ 119905 minus 119904 (26)


119898rss ΔP[ℓ]119898rss]


119877 = ℓ sdot ( ℓsum119894=1

120595119894119910 ([119894]



119867 = ( ℓsum119894=1

120595119894119910 ([119894]

119898 (119904))minus1)minus1

sdot ℓsum119894=1

[120595119894119910 ([119894]



119910[120582]119898 (119904) = 119877 sdot ℓsum

119894=1

119892 ([119894]119898 (119904) 119909[119894]119898 (119904)) = ℓ


120595119894119910 ([119894]

119898 (119904))minus1)minus1ℓsum119894=1

1ℓsdot [120595119894119910 ([119894]

119898 (119904))minus1 ([119894]119898 (119904) + 119909[ℓ]119898 (119904) minus 119909[119894]119898 (119904))]

119898 isin ℏ 119905 minus 119904

(29)

where119892 ([119894]119898 (119904) 119909[119894]119898 (119904))

= 1ℓsdot [120595119894119910 ([119894]

119898 (119904))minus1 ([119894]119898 (119904) + 119909[ℓ]119898 (119904) minus 119909[119894]119898 (119904))]

119898 isin ℏ 119905 minus 119904 (30)






minH











119905minus119896120574(119904)[120582]ℏ (119904) = H120574(119904)Y

[120582]

119905minus1 (33)




Y[120582]

119905minus1 ≜ [[120582]119879

119905minusΞ[120582]minusℏ[120582]119879


119905minus1 ]119879 (35)

120574 (119904) ≜ 120574[120582]119905minusΞ[120582]minusℏ

120574[120582]119905minusΞ[120582]minusℏ+1

120574[120582]119905minus1 (36)



Y[120582]


where

FΞ120574(119904) ≜[[[[[[[[[

119905minusΞ120574[120582]119905minusΞ


119905minusΞ+1120574[120582]119905minusΞ+1


119905minus1

]]]]]]]]]

W119905minus1 ≜[[[[[[[[


]]]]]]]]

V119905minus1120574(119904) ≜[[[[[[[[

V119905minusΞ120574[120582]119905minusΞ

V119905minusΞ+1120574[120582]119905minusΞ+1

V119905minus1120574[120582]119905minus1

]]]]]]]]

HΞ120574(119904) ≜[[[[[[[[[

119905minusΞ120574[120582]119905minusΞ

Φminus1


119905minusΞ




119905minusΞ+1

Φminus1


119905minusΞ+1


Γ119905minusΞ+1 d



]]]]]]]]]

QΞ ≜ [[diag(Ξ⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞119876 119876 119876)]]

RΞ ≜ [[diag(Ξ⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞ )]]




119905minusℎ

(38)



Θ119894120574(119904) =H119894120574(119904) [diag( 119894⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞119876 119876 119876)]H119879119894120574(119904)

+ [[diag(119894⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞ )]]

(39)


119905|119905minus1 =HY[120582]

119905minus1


119864 [119905|119905minus1] =HFΞ120574(119904)119864 [119909119905minusΞ] (40)




= 1206030119896+1 = Φ119896119896 + Φ119879

119896 + 1198961199050

= 0(42)


HFΞ120574(119904) = 119868 (43)




H119900 = argmin

H

119864 [119890119879119905 119890119905]= argmin

H


H



(44)



(45)



[120582]119905|119905minus1 =H119900Y

[120582]

119905minus1

= [119868 0] [[(F119879

Ξ120574(119904)Θminus1

Ξ120574(119904)FΞ120574(119904))minus1 F119879

Ξ120574(119904)Θminus1

Ξ120574(119904)lowast ]] Y[120582]

119905minus1

= (F119879Ξ120574(119904)

Θminus1

Ξ120574(119904)FΞ120574(119904))minus1 F119879

Ξ120574(119904)Θminus1

Ξ120574(119904)Y

[120582]

119905minus1(46)


Ξ120574(119904)and F119879

Ξ120574(119904)Θminus1

Ξ120574(119904)FΞ120574(119904)








= 0 1 2 ℏ (47)



Horizon group

Random delay

Buffer

DFIRestimator



LOSNLOSmediums

z[i]t = 120601[i]t xt + u[i]

t (120574[i]t )

i = 1 2 985747

Ξcs+1

119970s

xs

x1s x2


x987407+1s

Ξ1s

Ξ2s

Ξ987407+1s

120585 (xis (Ξ

is))

arg maxx119904(Ξ

119894119904)120585 (xs (Ξi

s))

y[i]t ≜ 120581[i]

t120591[z[i]t ]


y[i]









119892119881 (V[120582]119898 (119904 120574[120582]119898 )) = 1radic(2120587)119902 det (119877)


(48)


120585 (119894119904 (Ξ)) = 119901 (119910[120582]119898 (119904) | 119894119904 (Ξ)) = 1


119898 (119904) minus 120599[120582]119904 [120582]

119898 (119904) 119894119904 (Ξ))119879sdot 119877minus1 (119910[120582]

119898 (119904) minus 120599[120582]119904 [120582]

119898 (119904) 119894119904 (Ξ)) (49)



J (119904 (Ξ)) = 12 [(119910[120582]119898 (119904) minus 120599[120582]119904 [120582]

119898 (119904) 119894119904 (Ξ))119879

sdot 119877minus1 (119910[120582]119898 (119904) minus 120599[120582]119904 [120582]

119898 (119904) 119894119904 (Ξ))] (50)






Ξ119888119904+1 = argmax

Ξ119888119904

120585 (119904 (Ξ119894119904)) (51)






G119904 = 2Ξ119888119904 minus ℏ 2Ξ119888



119904 Ξ2119904 Ξ119894

119904 Ξℏ+1119904 (52)







[120582]119904|119904minus1 (Ξ119894119904) = (F119879

Ξ119894119904+ℏ120574(119904)Θminus1

Ξ119894119904+ℏ120574(119904)FΞ119894119904+ℏ120574(119904)

)minus1sdotF119879

Ξ119894119904+ℏ120574(119904)Θminus1

Ξ119894119904+ℏ120574(119904)Y

[120582]

119904minus1 (Ξ119894119904) (53)

where FΞ119894119904+ℏ120574(119904) ΘΞ119894119904+ℏ120574(119904)

and Y[120582]

119904minus1(Ξ119894119904)



119904) using(49) as

120585 (119894119904 (Ξ119894119904)) = 1


119898 (119904) minus 120599[120582]119904 [120582]

119898 (119904) 119894119904 (Ξ119894119904))119879

sdot 119877minus1 (119910[120582]119898 (119904) minus 120599[120582]119904 [120582]

119898 (119904) 119894119904 (Ξ119894119904))

(54)

endif



119904 = arg max119904(Ξ119894119904)120585 (119904 (Ξ119894

119904)) (55)



Ξ119888119904+1 = argmax

Ξ119888119904

120585 (119904 (Ξ119894119904)) (56)





(57)







11 = 120587(2)22 = 09

and 120587(2)12 = 120587(2)




(58)





05000

0

5000

0200400600800

1000

Sink1

Normal1Normal2

Normal3Normal4

Z(m

)

Y (m)


X (m)

minus5000

minus5000

(a)


0 1000 2000 3000 4000 5000 6000

0

2000

4000

60000

200400600800

1

23 4

5 67 8

9

10

11

Z(m

)

Y (m)

X (m)

(b)


0 5 10 15 20 25 30 35 400

05

1

15

2

25

3

Horizon size

RMSE

10 12 14

01020304


DFIR estimator













120

12002

12004

12006

29993030013002

300330043005

0

500

1000

120120005

12001

299953030005

0

100minus500

minus100

Pal

t(m

)

Plat ( ∘)

Plon (∘)

(a)

0 200 400 600 800 1000

4

6

8

10

12

14

Num

ber o

f obs

erve

d no

des

Time (sec)



(b)

0 200 400 600 800 1000

0

1

2

3

Time (sec)

0 200 400 600 800 1000

0

05

1

15

Time (sec)

minus05

TME120574

(t)

RD120591

(c)








0 500 1000

0

50

100

150

200

250

Time (sec)

minus50

minus100

minus150

minus200

minus250

ΔP

lon

(m)

0 500 1000

0

50

100

150

200

Time (sec)

minus50

minus100

minus150

ΔP

lat

(m)

Time (sec)0 500 1000

0

200

400

600

800

1000

1200

minus200

minus400

ΔP

alt

(m)

DRHEDFIRAFGE

(a)

0 500 1000

0

1

2

3

4

0 500 1000

0

1

2

3



0 500 1000

0

5

10

15

20

25

minus5

minus1

minus2

minus3

minus4

minus5

minus1

minus2

ΔV

lon

(ms

)

ΔV

lat

(ms

)

ΔV

alt

(ms

)

DRHEDFIRAFGE

(b)



Competing Interests


Acknowledgments


References



































Volume 2014




Journal of











Journal of


Function Spaces






Algebra
















119906[119894]119905 = 120576[119894]119905 LOS condition

120576[119894]119905 + 119890[119894]119905 NLOS condition(6)


119891 (xN119905 zN

119905 ) ≜ 119911[119894]119905 = 120601[119894]119905 119909119905 + 119906[119894]119905 (120574[119894]119905 ) 119894 = 1 2 ℓ (7)









119910[119894]119905 = ℏsum

120591=0


(8)







119905ℏ119911[119894]119905minusℏ)119879]119879 (10)


120581[119894]119905+119895119895 times 120581[119894]119905+119896119896 = 0 119895 = 119896 (11)










119869[120582] = 119864 10038171003817100381710038171003817119909 (119905) minus [120582] (119905 | 119905)10038171003817100381710038171003817 | [119894]119904 10038161003816100381610038161003816119904ge0 [119894]119896


[120582]119905 = 1 2 ℓ (13)








the batch form as

ℏ (119904) ≜ [119910[119895]1198790 (119904)100381610038161003816100381610038161003816ℓ119895=1 119910[119895]119879


119905minus119904 (119904) ≜ [119910[119895]1198790 (119904)100381610038161003816100381610038161003816ℓ119895=1 119910[119895]119879


(14)




where

ℏ (119904) = [(120581[119894]1199040120601[119894]119904 10038161003816100381610038161003816ℓ119894=1)119879 (120581[119894]119904+ℏℏ

120601[119894]119904+ℎ


119904+ℏℏ119906[119894]119904+ℏ(120574[119894]

119904+ℏ)10038161003816100381610038161003816ℓ119894=1)119879]119879

V119905minus119904 (119904 120574[119894]119904119905 )= [(120581[119894]1199040119906[119894]119904 (120574[119894]119904 )10038161003816100381610038161003816ℓ119894=1)119879 (120581[119894]119905119905minus119904119906[119894]119905 (120574[119894]119905 )10038161003816100381610038161003816ℓ119894=1)119879]119879

(16)



(17)




119896=119905minusΞ[120582]is






[120582]119898 (119904) = 119877 sdot ℓsum

119894=1

119892 ([119894]119898 (119904) 119909[119894]119898 (119904))




linear spaceL2 as

[120582]119898 (119904) = 120599[120582]119904 [120582]

119898 (119904) 119909 (119904) + V[120582]119898 (119904 120574[120582]119898 ) (20)







119898 (119904) in(17)



119877 = ( ℓsum119894=1

([119894]


119910[120582]119898 (119904)= 119877sdot ℓsum119894=1

(([119894]

119898 (119904))minus1 ([119894]119898 (119904) + 119909[ℓ]119898 (119904) minus 119909[119894]119898 (119904)))

(22)


119867 = 119877 sdot ℓsum119894=1

[([119894]

119898 (119904))minus1119867[119894]] (23)








120595119894119910 = 9848582119894984858119879984858 (25)








ΔP[119894]119898rss = [ℓ]



10038171003817100381710038171003817 119898 isin ℏ 119905 minus 119904 (26)


119898rss ΔP[ℓ]119898rss]


119877 = ℓ sdot ( ℓsum119894=1

120595119894119910 ([119894]



119867 = ( ℓsum119894=1

120595119894119910 ([119894]

119898 (119904))minus1)minus1

sdot ℓsum119894=1

[120595119894119910 ([119894]



119910[120582]119898 (119904) = 119877 sdot ℓsum

119894=1

119892 ([119894]119898 (119904) 119909[119894]119898 (119904)) = ℓ


120595119894119910 ([119894]

119898 (119904))minus1)minus1ℓsum119894=1

1ℓsdot [120595119894119910 ([119894]

119898 (119904))minus1 ([119894]119898 (119904) + 119909[ℓ]119898 (119904) minus 119909[119894]119898 (119904))]

119898 isin ℏ 119905 minus 119904

(29)

where119892 ([119894]119898 (119904) 119909[119894]119898 (119904))

= 1ℓsdot [120595119894119910 ([119894]

119898 (119904))minus1 ([119894]119898 (119904) + 119909[ℓ]119898 (119904) minus 119909[119894]119898 (119904))]

119898 isin ℏ 119905 minus 119904 (30)






minH











119905minus119896120574(119904)[120582]ℏ (119904) = H120574(119904)Y

[120582]

119905minus1 (33)




Y[120582]

119905minus1 ≜ [[120582]119879

119905minusΞ[120582]minusℏ[120582]119879


119905minus1 ]119879 (35)

120574 (119904) ≜ 120574[120582]119905minusΞ[120582]minusℏ

120574[120582]119905minusΞ[120582]minusℏ+1

120574[120582]119905minus1 (36)



Y[120582]


where

FΞ120574(119904) ≜[[[[[[[[[

119905minusΞ120574[120582]119905minusΞ


119905minusΞ+1120574[120582]119905minusΞ+1


119905minus1

]]]]]]]]]

W119905minus1 ≜[[[[[[[[


]]]]]]]]

V119905minus1120574(119904) ≜[[[[[[[[

V119905minusΞ120574[120582]119905minusΞ

V119905minusΞ+1120574[120582]119905minusΞ+1

V119905minus1120574[120582]119905minus1

]]]]]]]]

HΞ120574(119904) ≜[[[[[[[[[

119905minusΞ120574[120582]119905minusΞ

Φminus1


119905minusΞ




119905minusΞ+1

Φminus1


119905minusΞ+1


Γ119905minusΞ+1 d



]]]]]]]]]

QΞ ≜ [[diag(Ξ⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞119876 119876 119876)]]

RΞ ≜ [[diag(Ξ⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞ )]]




119905minusℎ

(38)



Θ119894120574(119904) =H119894120574(119904) [diag( 119894⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞119876 119876 119876)]H119879119894120574(119904)

+ [[diag(119894⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞ )]]

(39)


119905|119905minus1 =HY[120582]

119905minus1


119864 [119905|119905minus1] =HFΞ120574(119904)119864 [119909119905minusΞ] (40)




= 1206030119896+1 = Φ119896119896 + Φ119879

119896 + 1198961199050

= 0(42)


HFΞ120574(119904) = 119868 (43)




H119900 = argmin

H

119864 [119890119879119905 119890119905]= argmin

H


H



(44)



(45)



[120582]119905|119905minus1 =H119900Y

[120582]

119905minus1

= [119868 0] [[(F119879

Ξ120574(119904)Θminus1

Ξ120574(119904)FΞ120574(119904))minus1 F119879

Ξ120574(119904)Θminus1

Ξ120574(119904)lowast ]] Y[120582]

119905minus1

= (F119879Ξ120574(119904)

Θminus1

Ξ120574(119904)FΞ120574(119904))minus1 F119879

Ξ120574(119904)Θminus1

Ξ120574(119904)Y

[120582]

119905minus1(46)


Ξ120574(119904)and F119879

Ξ120574(119904)Θminus1

Ξ120574(119904)FΞ120574(119904)








= 0 1 2 ℏ (47)



Horizon group

Random delay

Buffer

DFIRestimator



LOSNLOSmediums

z[i]t = 120601[i]t xt + u[i]

t (120574[i]t )

i = 1 2 985747

Ξcs+1

119970s

xs

x1s x2


x987407+1s

Ξ1s

Ξ2s

Ξ987407+1s

120585 (xis (Ξ

is))

arg maxx119904(Ξ

119894119904)120585 (xs (Ξi

s))

y[i]t ≜ 120581[i]

t120591[z[i]t ]


y[i]









119892119881 (V[120582]119898 (119904 120574[120582]119898 )) = 1radic(2120587)119902 det (119877)


(48)


120585 (119894119904 (Ξ)) = 119901 (119910[120582]119898 (119904) | 119894119904 (Ξ)) = 1


119898 (119904) minus 120599[120582]119904 [120582]

119898 (119904) 119894119904 (Ξ))119879sdot 119877minus1 (119910[120582]

119898 (119904) minus 120599[120582]119904 [120582]

119898 (119904) 119894119904 (Ξ)) (49)



J (119904 (Ξ)) = 12 [(119910[120582]119898 (119904) minus 120599[120582]119904 [120582]

119898 (119904) 119894119904 (Ξ))119879

sdot 119877minus1 (119910[120582]119898 (119904) minus 120599[120582]119904 [120582]

119898 (119904) 119894119904 (Ξ))] (50)






Ξ119888119904+1 = argmax

Ξ119888119904

120585 (119904 (Ξ119894119904)) (51)






G119904 = 2Ξ119888119904 minus ℏ 2Ξ119888



119904 Ξ2119904 Ξ119894

119904 Ξℏ+1119904 (52)







[120582]119904|119904minus1 (Ξ119894119904) = (F119879

Ξ119894119904+ℏ120574(119904)Θminus1

Ξ119894119904+ℏ120574(119904)FΞ119894119904+ℏ120574(119904)

)minus1sdotF119879

Ξ119894119904+ℏ120574(119904)Θminus1

Ξ119894119904+ℏ120574(119904)Y

[120582]

119904minus1 (Ξ119894119904) (53)

where FΞ119894119904+ℏ120574(119904) ΘΞ119894119904+ℏ120574(119904)

and Y[120582]

119904minus1(Ξ119894119904)



119904) using(49) as

120585 (119894119904 (Ξ119894119904)) = 1


119898 (119904) minus 120599[120582]119904 [120582]

119898 (119904) 119894119904 (Ξ119894119904))119879

sdot 119877minus1 (119910[120582]119898 (119904) minus 120599[120582]119904 [120582]

119898 (119904) 119894119904 (Ξ119894119904))

(54)

endif



119904 = arg max119904(Ξ119894119904)120585 (119904 (Ξ119894

119904)) (55)



Ξ119888119904+1 = argmax

Ξ119888119904

120585 (119904 (Ξ119894119904)) (56)





(57)







11 = 120587(2)22 = 09

and 120587(2)12 = 120587(2)




(58)





05000

0

5000

0200400600800

1000

Sink1

Normal1Normal2

Normal3Normal4

Z(m

)

Y (m)


X (m)

minus5000

minus5000

(a)


0 1000 2000 3000 4000 5000 6000

0

2000

4000

60000

200400600800

1

23 4

5 67 8

9

10

11

Z(m

)

Y (m)

X (m)

(b)


0 5 10 15 20 25 30 35 400

05

1

15

2

25

3

Horizon size

RMSE

10 12 14

01020304


DFIR estimator













120

12002

12004

12006

29993030013002

300330043005

0

500

1000

120120005

12001

299953030005

0

100minus500

minus100

Pal

t(m

)

Plat ( ∘)

Plon (∘)

(a)

0 200 400 600 800 1000

4

6

8

10

12

14

Num

ber o

f obs

erve

d no

des

Time (sec)



(b)

0 200 400 600 800 1000

0

1

2

3

Time (sec)

0 200 400 600 800 1000

0

05

1

15

Time (sec)

minus05

TME120574

(t)

RD120591

(c)








0 500 1000

0

50

100

150

200

250

Time (sec)

minus50

minus100

minus150

minus200

minus250

ΔP

lon

(m)

0 500 1000

0

50

100

150

200

Time (sec)

minus50

minus100

minus150

ΔP

lat

(m)

Time (sec)0 500 1000

0

200

400

600

800

1000

1200

minus200

minus400

ΔP

alt

(m)

DRHEDFIRAFGE

(a)

0 500 1000

0

1

2

3

4

0 500 1000

0

1

2

3



0 500 1000

0

5

10

15

20

25

minus5

minus1

minus2

minus3

minus4

minus5

minus1

minus2

ΔV

lon

(ms

)

ΔV

lat

(ms

)

ΔV

alt

(ms

)

DRHEDFIRAFGE

(b)



Competing Interests


Acknowledgments


References



































Volume 2014




Journal of











Journal of


Function Spaces






Algebra












the batch form as

ℏ (119904) ≜ [119910[119895]1198790 (119904)100381610038161003816100381610038161003816ℓ119895=1 119910[119895]119879


119905minus119904 (119904) ≜ [119910[119895]1198790 (119904)100381610038161003816100381610038161003816ℓ119895=1 119910[119895]119879


(14)




where

ℏ (119904) = [(120581[119894]1199040120601[119894]119904 10038161003816100381610038161003816ℓ119894=1)119879 (120581[119894]119904+ℏℏ

120601[119894]119904+ℎ


119904+ℏℏ119906[119894]119904+ℏ(120574[119894]

119904+ℏ)10038161003816100381610038161003816ℓ119894=1)119879]119879

V119905minus119904 (119904 120574[119894]119904119905 )= [(120581[119894]1199040119906[119894]119904 (120574[119894]119904 )10038161003816100381610038161003816ℓ119894=1)119879 (120581[119894]119905119905minus119904119906[119894]119905 (120574[119894]119905 )10038161003816100381610038161003816ℓ119894=1)119879]119879

(16)



(17)




119896=119905minusΞ[120582]is






[120582]119898 (119904) = 119877 sdot ℓsum

119894=1

119892 ([119894]119898 (119904) 119909[119894]119898 (119904))




linear spaceL2 as

[120582]119898 (119904) = 120599[120582]119904 [120582]

119898 (119904) 119909 (119904) + V[120582]119898 (119904 120574[120582]119898 ) (20)







119898 (119904) in(17)



119877 = ( ℓsum119894=1

([119894]


119910[120582]119898 (119904)= 119877sdot ℓsum119894=1

(([119894]

119898 (119904))minus1 ([119894]119898 (119904) + 119909[ℓ]119898 (119904) minus 119909[119894]119898 (119904)))

(22)


119867 = 119877 sdot ℓsum119894=1

[([119894]

119898 (119904))minus1119867[119894]] (23)








120595119894119910 = 9848582119894984858119879984858 (25)








ΔP[119894]119898rss = [ℓ]



10038171003817100381710038171003817 119898 isin ℏ 119905 minus 119904 (26)


119898rss ΔP[ℓ]119898rss]


119877 = ℓ sdot ( ℓsum119894=1

120595119894119910 ([119894]



119867 = ( ℓsum119894=1

120595119894119910 ([119894]

119898 (119904))minus1)minus1

sdot ℓsum119894=1

[120595119894119910 ([119894]



119910[120582]119898 (119904) = 119877 sdot ℓsum

119894=1

119892 ([119894]119898 (119904) 119909[119894]119898 (119904)) = ℓ


120595119894119910 ([119894]

119898 (119904))minus1)minus1ℓsum119894=1

1ℓsdot [120595119894119910 ([119894]

119898 (119904))minus1 ([119894]119898 (119904) + 119909[ℓ]119898 (119904) minus 119909[119894]119898 (119904))]

119898 isin ℏ 119905 minus 119904

(29)

where119892 ([119894]119898 (119904) 119909[119894]119898 (119904))

= 1ℓsdot [120595119894119910 ([119894]

119898 (119904))minus1 ([119894]119898 (119904) + 119909[ℓ]119898 (119904) minus 119909[119894]119898 (119904))]

119898 isin ℏ 119905 minus 119904 (30)






minH











119905minus119896120574(119904)[120582]ℏ (119904) = H120574(119904)Y

[120582]

119905minus1 (33)




Y[120582]

119905minus1 ≜ [[120582]119879

119905minusΞ[120582]minusℏ[120582]119879


119905minus1 ]119879 (35)

120574 (119904) ≜ 120574[120582]119905minusΞ[120582]minusℏ

120574[120582]119905minusΞ[120582]minusℏ+1

120574[120582]119905minus1 (36)



Y[120582]


where

FΞ120574(119904) ≜[[[[[[[[[

119905minusΞ120574[120582]119905minusΞ


119905minusΞ+1120574[120582]119905minusΞ+1


119905minus1

]]]]]]]]]

W119905minus1 ≜[[[[[[[[


]]]]]]]]

V119905minus1120574(119904) ≜[[[[[[[[

V119905minusΞ120574[120582]119905minusΞ

V119905minusΞ+1120574[120582]119905minusΞ+1

V119905minus1120574[120582]119905minus1

]]]]]]]]

HΞ120574(119904) ≜[[[[[[[[[

119905minusΞ120574[120582]119905minusΞ

Φminus1


119905minusΞ




119905minusΞ+1

Φminus1


119905minusΞ+1


Γ119905minusΞ+1 d



]]]]]]]]]

QΞ ≜ [[diag(Ξ⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞119876 119876 119876)]]

RΞ ≜ [[diag(Ξ⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞ )]]




119905minusℎ

(38)



Θ119894120574(119904) =H119894120574(119904) [diag( 119894⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞119876 119876 119876)]H119879119894120574(119904)

+ [[diag(119894⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞ )]]

(39)


119905|119905minus1 =HY[120582]

119905minus1


119864 [119905|119905minus1] =HFΞ120574(119904)119864 [119909119905minusΞ] (40)




= 1206030119896+1 = Φ119896119896 + Φ119879

119896 + 1198961199050

= 0(42)


HFΞ120574(119904) = 119868 (43)




H119900 = argmin

H

119864 [119890119879119905 119890119905]= argmin

H


H



(44)



(45)



[120582]119905|119905minus1 =H119900Y

[120582]

119905minus1

= [119868 0] [[(F119879

Ξ120574(119904)Θminus1

Ξ120574(119904)FΞ120574(119904))minus1 F119879

Ξ120574(119904)Θminus1

Ξ120574(119904)lowast ]] Y[120582]

119905minus1

= (F119879Ξ120574(119904)

Θminus1

Ξ120574(119904)FΞ120574(119904))minus1 F119879

Ξ120574(119904)Θminus1

Ξ120574(119904)Y

[120582]

119905minus1(46)


Ξ120574(119904)and F119879

Ξ120574(119904)Θminus1

Ξ120574(119904)FΞ120574(119904)








= 0 1 2 ℏ (47)



Horizon group

Random delay

Buffer

DFIRestimator



LOSNLOSmediums

z[i]t = 120601[i]t xt + u[i]

t (120574[i]t )

i = 1 2 985747

Ξcs+1

119970s

xs

x1s x2


x987407+1s

Ξ1s

Ξ2s

Ξ987407+1s

120585 (xis (Ξ

is))

arg maxx119904(Ξ

119894119904)120585 (xs (Ξi

s))

y[i]t ≜ 120581[i]

t120591[z[i]t ]


y[i]









119892119881 (V[120582]119898 (119904 120574[120582]119898 )) = 1radic(2120587)119902 det (119877)


(48)


120585 (119894119904 (Ξ)) = 119901 (119910[120582]119898 (119904) | 119894119904 (Ξ)) = 1


119898 (119904) minus 120599[120582]119904 [120582]

119898 (119904) 119894119904 (Ξ))119879sdot 119877minus1 (119910[120582]

119898 (119904) minus 120599[120582]119904 [120582]

119898 (119904) 119894119904 (Ξ)) (49)



J (119904 (Ξ)) = 12 [(119910[120582]119898 (119904) minus 120599[120582]119904 [120582]

119898 (119904) 119894119904 (Ξ))119879

sdot 119877minus1 (119910[120582]119898 (119904) minus 120599[120582]119904 [120582]

119898 (119904) 119894119904 (Ξ))] (50)






Ξ119888119904+1 = argmax

Ξ119888119904

120585 (119904 (Ξ119894119904)) (51)






G119904 = 2Ξ119888119904 minus ℏ 2Ξ119888



119904 Ξ2119904 Ξ119894

119904 Ξℏ+1119904 (52)







[120582]119904|119904minus1 (Ξ119894119904) = (F119879

Ξ119894119904+ℏ120574(119904)Θminus1

Ξ119894119904+ℏ120574(119904)FΞ119894119904+ℏ120574(119904)

)minus1sdotF119879

Ξ119894119904+ℏ120574(119904)Θminus1

Ξ119894119904+ℏ120574(119904)Y

[120582]

119904minus1 (Ξ119894119904) (53)

where FΞ119894119904+ℏ120574(119904) ΘΞ119894119904+ℏ120574(119904)

and Y[120582]

119904minus1(Ξ119894119904)



119904) using(49) as

120585 (119894119904 (Ξ119894119904)) = 1


119898 (119904) minus 120599[120582]119904 [120582]

119898 (119904) 119894119904 (Ξ119894119904))119879

sdot 119877minus1 (119910[120582]119898 (119904) minus 120599[120582]119904 [120582]

119898 (119904) 119894119904 (Ξ119894119904))

(54)

endif



119904 = arg max119904(Ξ119894119904)120585 (119904 (Ξ119894

119904)) (55)



Ξ119888119904+1 = argmax

Ξ119888119904

120585 (119904 (Ξ119894119904)) (56)





(57)







11 = 120587(2)22 = 09

and 120587(2)12 = 120587(2)




(58)





05000

0

5000

0200400600800

1000

Sink1

Normal1Normal2

Normal3Normal4

Z(m

)

Y (m)


X (m)

minus5000

minus5000

(a)


0 1000 2000 3000 4000 5000 6000

0

2000

4000

60000

200400600800

1

23 4

5 67 8

9

10

11

Z(m

)

Y (m)

X (m)

(b)


0 5 10 15 20 25 30 35 400

05

1

15

2

25

3

Horizon size

RMSE

10 12 14

01020304


DFIR estimator













120

12002

12004

12006

29993030013002

300330043005

0

500

1000

120120005

12001

299953030005

0

100minus500

minus100

Pal

t(m

)

Plat ( ∘)

Plon (∘)

(a)

0 200 400 600 800 1000

4

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ber o

f obs

erve

d no

des

Time (sec)



(b)

0 200 400 600 800 1000

0

1

2

3

Time (sec)

0 200 400 600 800 1000

0

05

1

15

Time (sec)

minus05

TME120574

(t)

RD120591

(c)








0 500 1000

0

50

100

150

200

250

Time (sec)

minus50

minus100

minus150

minus200

minus250

ΔP

lon

(m)

0 500 1000

0

50

100

150

200

Time (sec)

minus50

minus100

minus150

ΔP

lat

(m)

Time (sec)0 500 1000

0

200

400

600

800

1000

1200

minus200

minus400

ΔP

alt

(m)

DRHEDFIRAFGE

(a)

0 500 1000

0

1

2

3

4

0 500 1000

0

1

2

3



0 500 1000

0

5

10

15

20

25

minus5

minus1

minus2

minus3

minus4

minus5

minus1

minus2

ΔV

lon

(ms

)

ΔV

lat

(ms

)

ΔV

alt

(ms

)

DRHEDFIRAFGE

(b)



Competing Interests


Acknowledgments


References



































Volume 2014




Journal of











Journal of


Function Spaces






Algebra
















120595119894119910 = 9848582119894984858119879984858 (25)








ΔP[119894]119898rss = [ℓ]



10038171003817100381710038171003817 119898 isin ℏ 119905 minus 119904 (26)


119898rss ΔP[ℓ]119898rss]


119877 = ℓ sdot ( ℓsum119894=1

120595119894119910 ([119894]



119867 = ( ℓsum119894=1

120595119894119910 ([119894]

119898 (119904))minus1)minus1

sdot ℓsum119894=1

[120595119894119910 ([119894]



119910[120582]119898 (119904) = 119877 sdot ℓsum

119894=1

119892 ([119894]119898 (119904) 119909[119894]119898 (119904)) = ℓ


120595119894119910 ([119894]

119898 (119904))minus1)minus1ℓsum119894=1

1ℓsdot [120595119894119910 ([119894]

119898 (119904))minus1 ([119894]119898 (119904) + 119909[ℓ]119898 (119904) minus 119909[119894]119898 (119904))]

119898 isin ℏ 119905 minus 119904

(29)

where119892 ([119894]119898 (119904) 119909[119894]119898 (119904))

= 1ℓsdot [120595119894119910 ([119894]

119898 (119904))minus1 ([119894]119898 (119904) + 119909[ℓ]119898 (119904) minus 119909[119894]119898 (119904))]

119898 isin ℏ 119905 minus 119904 (30)






minH











119905minus119896120574(119904)[120582]ℏ (119904) = H120574(119904)Y

[120582]

119905minus1 (33)




Y[120582]

119905minus1 ≜ [[120582]119879

119905minusΞ[120582]minusℏ[120582]119879


119905minus1 ]119879 (35)

120574 (119904) ≜ 120574[120582]119905minusΞ[120582]minusℏ

120574[120582]119905minusΞ[120582]minusℏ+1

120574[120582]119905minus1 (36)



Y[120582]


where

FΞ120574(119904) ≜[[[[[[[[[

119905minusΞ120574[120582]119905minusΞ


119905minusΞ+1120574[120582]119905minusΞ+1


119905minus1

]]]]]]]]]

W119905minus1 ≜[[[[[[[[


]]]]]]]]

V119905minus1120574(119904) ≜[[[[[[[[

V119905minusΞ120574[120582]119905minusΞ

V119905minusΞ+1120574[120582]119905minusΞ+1

V119905minus1120574[120582]119905minus1

]]]]]]]]

HΞ120574(119904) ≜[[[[[[[[[

119905minusΞ120574[120582]119905minusΞ

Φminus1


119905minusΞ




119905minusΞ+1

Φminus1


119905minusΞ+1


Γ119905minusΞ+1 d



]]]]]]]]]

QΞ ≜ [[diag(Ξ⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞119876 119876 119876)]]

RΞ ≜ [[diag(Ξ⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞ )]]




119905minusℎ

(38)



Θ119894120574(119904) =H119894120574(119904) [diag( 119894⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞119876 119876 119876)]H119879119894120574(119904)

+ [[diag(119894⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞ )]]

(39)


119905|119905minus1 =HY[120582]

119905minus1


119864 [119905|119905minus1] =HFΞ120574(119904)119864 [119909119905minusΞ] (40)




= 1206030119896+1 = Φ119896119896 + Φ119879

119896 + 1198961199050

= 0(42)


HFΞ120574(119904) = 119868 (43)




H119900 = argmin

H

119864 [119890119879119905 119890119905]= argmin

H


H



(44)



(45)



[120582]119905|119905minus1 =H119900Y

[120582]

119905minus1

= [119868 0] [[(F119879

Ξ120574(119904)Θminus1

Ξ120574(119904)FΞ120574(119904))minus1 F119879

Ξ120574(119904)Θminus1

Ξ120574(119904)lowast ]] Y[120582]

119905minus1

= (F119879Ξ120574(119904)

Θminus1

Ξ120574(119904)FΞ120574(119904))minus1 F119879

Ξ120574(119904)Θminus1

Ξ120574(119904)Y

[120582]

119905minus1(46)


Ξ120574(119904)and F119879

Ξ120574(119904)Θminus1

Ξ120574(119904)FΞ120574(119904)








= 0 1 2 ℏ (47)



Horizon group

Random delay

Buffer

DFIRestimator



LOSNLOSmediums

z[i]t = 120601[i]t xt + u[i]

t (120574[i]t )

i = 1 2 985747

Ξcs+1

119970s

xs

x1s x2


x987407+1s

Ξ1s

Ξ2s

Ξ987407+1s

120585 (xis (Ξ

is))

arg maxx119904(Ξ

119894119904)120585 (xs (Ξi

s))

y[i]t ≜ 120581[i]

t120591[z[i]t ]


y[i]









119892119881 (V[120582]119898 (119904 120574[120582]119898 )) = 1radic(2120587)119902 det (119877)


(48)


120585 (119894119904 (Ξ)) = 119901 (119910[120582]119898 (119904) | 119894119904 (Ξ)) = 1


119898 (119904) minus 120599[120582]119904 [120582]

119898 (119904) 119894119904 (Ξ))119879sdot 119877minus1 (119910[120582]

119898 (119904) minus 120599[120582]119904 [120582]

119898 (119904) 119894119904 (Ξ)) (49)



J (119904 (Ξ)) = 12 [(119910[120582]119898 (119904) minus 120599[120582]119904 [120582]

119898 (119904) 119894119904 (Ξ))119879

sdot 119877minus1 (119910[120582]119898 (119904) minus 120599[120582]119904 [120582]

119898 (119904) 119894119904 (Ξ))] (50)






Ξ119888119904+1 = argmax

Ξ119888119904

120585 (119904 (Ξ119894119904)) (51)






G119904 = 2Ξ119888119904 minus ℏ 2Ξ119888



119904 Ξ2119904 Ξ119894

119904 Ξℏ+1119904 (52)







[120582]119904|119904minus1 (Ξ119894119904) = (F119879

Ξ119894119904+ℏ120574(119904)Θminus1

Ξ119894119904+ℏ120574(119904)FΞ119894119904+ℏ120574(119904)

)minus1sdotF119879

Ξ119894119904+ℏ120574(119904)Θminus1

Ξ119894119904+ℏ120574(119904)Y

[120582]

119904minus1 (Ξ119894119904) (53)

where FΞ119894119904+ℏ120574(119904) ΘΞ119894119904+ℏ120574(119904)

and Y[120582]

119904minus1(Ξ119894119904)



119904) using(49) as

120585 (119894119904 (Ξ119894119904)) = 1


119898 (119904) minus 120599[120582]119904 [120582]

119898 (119904) 119894119904 (Ξ119894119904))119879

sdot 119877minus1 (119910[120582]119898 (119904) minus 120599[120582]119904 [120582]

119898 (119904) 119894119904 (Ξ119894119904))

(54)

endif



119904 = arg max119904(Ξ119894119904)120585 (119904 (Ξ119894

119904)) (55)



Ξ119888119904+1 = argmax

Ξ119888119904

120585 (119904 (Ξ119894119904)) (56)





(57)







11 = 120587(2)22 = 09

and 120587(2)12 = 120587(2)




(58)





05000

0

5000

0200400600800

1000

Sink1

Normal1Normal2

Normal3Normal4

Z(m

)

Y (m)


X (m)

minus5000

minus5000

(a)


0 1000 2000 3000 4000 5000 6000

0

2000

4000

60000

200400600800

1

23 4

5 67 8

9

10

11

Z(m

)

Y (m)

X (m)

(b)


0 5 10 15 20 25 30 35 400

05

1

15

2

25

3

Horizon size

RMSE

10 12 14

01020304


DFIR estimator













120

12002

12004

12006

29993030013002

300330043005

0

500

1000

120120005

12001

299953030005

0

100minus500

minus100

Pal

t(m

)

Plat ( ∘)

Plon (∘)

(a)

0 200 400 600 800 1000

4

6

8

10

12

14

Num

ber o

f obs

erve

d no

des

Time (sec)



(b)

0 200 400 600 800 1000

0

1

2

3

Time (sec)

0 200 400 600 800 1000

0

05

1

15

Time (sec)

minus05

TME120574

(t)

RD120591

(c)








0 500 1000

0

50

100

150

200

250

Time (sec)

minus50

minus100

minus150

minus200

minus250

ΔP

lon

(m)

0 500 1000

0

50

100

150

200

Time (sec)

minus50

minus100

minus150

ΔP

lat

(m)

Time (sec)0 500 1000

0

200

400

600

800

1000

1200

minus200

minus400

ΔP

alt

(m)

DRHEDFIRAFGE

(a)

0 500 1000

0

1

2

3

4

0 500 1000

0

1

2

3



0 500 1000

0

5

10

15

20

25

minus5

minus1

minus2

minus3

minus4

minus5

minus1

minus2

ΔV

lon

(ms

)

ΔV

lat

(ms

)

ΔV

alt

(ms

)

DRHEDFIRAFGE

(b)



Competing Interests


Acknowledgments


References



































Volume 2014




Journal of











Journal of


Function Spaces






Algebra












119905minus119896120574(119904)[120582]ℏ (119904) = H120574(119904)Y

[120582]

119905minus1 (33)




Y[120582]

119905minus1 ≜ [[120582]119879

119905minusΞ[120582]minusℏ[120582]119879


119905minus1 ]119879 (35)

120574 (119904) ≜ 120574[120582]119905minusΞ[120582]minusℏ

120574[120582]119905minusΞ[120582]minusℏ+1

120574[120582]119905minus1 (36)



Y[120582]


where

FΞ120574(119904) ≜[[[[[[[[[

119905minusΞ120574[120582]119905minusΞ


119905minusΞ+1120574[120582]119905minusΞ+1


119905minus1

]]]]]]]]]

W119905minus1 ≜[[[[[[[[


]]]]]]]]

V119905minus1120574(119904) ≜[[[[[[[[

V119905minusΞ120574[120582]119905minusΞ

V119905minusΞ+1120574[120582]119905minusΞ+1

V119905minus1120574[120582]119905minus1

]]]]]]]]

HΞ120574(119904) ≜[[[[[[[[[

119905minusΞ120574[120582]119905minusΞ

Φminus1


119905minusΞ




119905minusΞ+1

Φminus1


119905minusΞ+1


Γ119905minusΞ+1 d



]]]]]]]]]

QΞ ≜ [[diag(Ξ⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞119876 119876 119876)]]

RΞ ≜ [[diag(Ξ⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞ )]]




119905minusℎ

(38)



Θ119894120574(119904) =H119894120574(119904) [diag( 119894⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞119876 119876 119876)]H119879119894120574(119904)

+ [[diag(119894⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞ )]]

(39)


119905|119905minus1 =HY[120582]

119905minus1


119864 [119905|119905minus1] =HFΞ120574(119904)119864 [119909119905minusΞ] (40)




= 1206030119896+1 = Φ119896119896 + Φ119879

119896 + 1198961199050

= 0(42)


HFΞ120574(119904) = 119868 (43)




H119900 = argmin

H

119864 [119890119879119905 119890119905]= argmin

H


H



(44)



(45)



[120582]119905|119905minus1 =H119900Y

[120582]

119905minus1

= [119868 0] [[(F119879

Ξ120574(119904)Θminus1

Ξ120574(119904)FΞ120574(119904))minus1 F119879

Ξ120574(119904)Θminus1

Ξ120574(119904)lowast ]] Y[120582]

119905minus1

= (F119879Ξ120574(119904)

Θminus1

Ξ120574(119904)FΞ120574(119904))minus1 F119879

Ξ120574(119904)Θminus1

Ξ120574(119904)Y

[120582]

119905minus1(46)


Ξ120574(119904)and F119879

Ξ120574(119904)Θminus1

Ξ120574(119904)FΞ120574(119904)








= 0 1 2 ℏ (47)



Horizon group

Random delay

Buffer

DFIRestimator



LOSNLOSmediums

z[i]t = 120601[i]t xt + u[i]

t (120574[i]t )

i = 1 2 985747

Ξcs+1

119970s

xs

x1s x2


x987407+1s

Ξ1s

Ξ2s

Ξ987407+1s

120585 (xis (Ξ

is))

arg maxx119904(Ξ

119894119904)120585 (xs (Ξi

s))

y[i]t ≜ 120581[i]

t120591[z[i]t ]


y[i]









119892119881 (V[120582]119898 (119904 120574[120582]119898 )) = 1radic(2120587)119902 det (119877)


(48)


120585 (119894119904 (Ξ)) = 119901 (119910[120582]119898 (119904) | 119894119904 (Ξ)) = 1


119898 (119904) minus 120599[120582]119904 [120582]

119898 (119904) 119894119904 (Ξ))119879sdot 119877minus1 (119910[120582]

119898 (119904) minus 120599[120582]119904 [120582]

119898 (119904) 119894119904 (Ξ)) (49)



J (119904 (Ξ)) = 12 [(119910[120582]119898 (119904) minus 120599[120582]119904 [120582]

119898 (119904) 119894119904 (Ξ))119879

sdot 119877minus1 (119910[120582]119898 (119904) minus 120599[120582]119904 [120582]

119898 (119904) 119894119904 (Ξ))] (50)






Ξ119888119904+1 = argmax

Ξ119888119904

120585 (119904 (Ξ119894119904)) (51)






G119904 = 2Ξ119888119904 minus ℏ 2Ξ119888



119904 Ξ2119904 Ξ119894

119904 Ξℏ+1119904 (52)







[120582]119904|119904minus1 (Ξ119894119904) = (F119879

Ξ119894119904+ℏ120574(119904)Θminus1

Ξ119894119904+ℏ120574(119904)FΞ119894119904+ℏ120574(119904)

)minus1sdotF119879

Ξ119894119904+ℏ120574(119904)Θminus1

Ξ119894119904+ℏ120574(119904)Y

[120582]

119904minus1 (Ξ119894119904) (53)

where FΞ119894119904+ℏ120574(119904) ΘΞ119894119904+ℏ120574(119904)

and Y[120582]

119904minus1(Ξ119894119904)



119904) using(49) as

120585 (119894119904 (Ξ119894119904)) = 1


119898 (119904) minus 120599[120582]119904 [120582]

119898 (119904) 119894119904 (Ξ119894119904))119879

sdot 119877minus1 (119910[120582]119898 (119904) minus 120599[120582]119904 [120582]

119898 (119904) 119894119904 (Ξ119894119904))

(54)

endif



119904 = arg max119904(Ξ119894119904)120585 (119904 (Ξ119894

119904)) (55)



Ξ119888119904+1 = argmax

Ξ119888119904

120585 (119904 (Ξ119894119904)) (56)





(57)







11 = 120587(2)22 = 09

and 120587(2)12 = 120587(2)




(58)





05000

0

5000

0200400600800

1000

Sink1

Normal1Normal2

Normal3Normal4

Z(m

)

Y (m)


X (m)

minus5000

minus5000

(a)


0 1000 2000 3000 4000 5000 6000

0

2000

4000

60000

200400600800

1

23 4

5 67 8

9

10

11

Z(m

)

Y (m)

X (m)

(b)


0 5 10 15 20 25 30 35 400

05

1

15

2

25

3

Horizon size

RMSE

10 12 14

01020304


DFIR estimator













120

12002

12004

12006

29993030013002

300330043005

0

500

1000

120120005

12001

299953030005

0

100minus500

minus100

Pal

t(m

)

Plat ( ∘)

Plon (∘)

(a)

0 200 400 600 800 1000

4

6

8

10

12

14

Num

ber o

f obs

erve

d no

des

Time (sec)



(b)

0 200 400 600 800 1000

0

1

2

3

Time (sec)

0 200 400 600 800 1000

0

05

1

15

Time (sec)

minus05

TME120574

(t)

RD120591

(c)








0 500 1000

0

50

100

150

200

250

Time (sec)

minus50

minus100

minus150

minus200

minus250

ΔP

lon

(m)

0 500 1000

0

50

100

150

200

Time (sec)

minus50

minus100

minus150

ΔP

lat

(m)

Time (sec)0 500 1000

0

200

400

600

800

1000

1200

minus200

minus400

ΔP

alt

(m)

DRHEDFIRAFGE

(a)

0 500 1000

0

1

2

3

4

0 500 1000

0

1

2

3



0 500 1000

0

5

10

15

20

25

minus5

minus1

minus2

minus3

minus4

minus5

minus1

minus2

ΔV

lon

(ms

)

ΔV

lat

(ms

)

ΔV

alt

(ms

)

DRHEDFIRAFGE

(b)



Competing Interests


Acknowledgments


References



































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











Θ119894120574(119904) =H119894120574(119904) [diag( 119894⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞119876 119876 119876)]H119879119894120574(119904)

+ [[diag(119894⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞ )]]

(39)


119905|119905minus1 =HY[120582]

119905minus1


119864 [119905|119905minus1] =HFΞ120574(119904)119864 [119909119905minusΞ] (40)




= 1206030119896+1 = Φ119896119896 + Φ119879

119896 + 1198961199050

= 0(42)


HFΞ120574(119904) = 119868 (43)




H119900 = argmin

H

119864 [119890119879119905 119890119905]= argmin

H


H



(44)



(45)



[120582]119905|119905minus1 =H119900Y

[120582]

119905minus1

= [119868 0] [[(F119879

Ξ120574(119904)Θminus1

Ξ120574(119904)FΞ120574(119904))minus1 F119879

Ξ120574(119904)Θminus1

Ξ120574(119904)lowast ]] Y[120582]

119905minus1

= (F119879Ξ120574(119904)

Θminus1

Ξ120574(119904)FΞ120574(119904))minus1 F119879

Ξ120574(119904)Θminus1

Ξ120574(119904)Y

[120582]

119905minus1(46)


Ξ120574(119904)and F119879

Ξ120574(119904)Θminus1

Ξ120574(119904)FΞ120574(119904)








= 0 1 2 ℏ (47)



Horizon group

Random delay

Buffer

DFIRestimator



LOSNLOSmediums

z[i]t = 120601[i]t xt + u[i]

t (120574[i]t )

i = 1 2 985747

Ξcs+1

119970s

xs

x1s x2


x987407+1s

Ξ1s

Ξ2s

Ξ987407+1s

120585 (xis (Ξ

is))

arg maxx119904(Ξ

119894119904)120585 (xs (Ξi

s))

y[i]t ≜ 120581[i]

t120591[z[i]t ]


y[i]









119892119881 (V[120582]119898 (119904 120574[120582]119898 )) = 1radic(2120587)119902 det (119877)


(48)


120585 (119894119904 (Ξ)) = 119901 (119910[120582]119898 (119904) | 119894119904 (Ξ)) = 1


119898 (119904) minus 120599[120582]119904 [120582]

119898 (119904) 119894119904 (Ξ))119879sdot 119877minus1 (119910[120582]

119898 (119904) minus 120599[120582]119904 [120582]

119898 (119904) 119894119904 (Ξ)) (49)



J (119904 (Ξ)) = 12 [(119910[120582]119898 (119904) minus 120599[120582]119904 [120582]

119898 (119904) 119894119904 (Ξ))119879

sdot 119877minus1 (119910[120582]119898 (119904) minus 120599[120582]119904 [120582]

119898 (119904) 119894119904 (Ξ))] (50)






Ξ119888119904+1 = argmax

Ξ119888119904

120585 (119904 (Ξ119894119904)) (51)






G119904 = 2Ξ119888119904 minus ℏ 2Ξ119888



119904 Ξ2119904 Ξ119894

119904 Ξℏ+1119904 (52)







[120582]119904|119904minus1 (Ξ119894119904) = (F119879

Ξ119894119904+ℏ120574(119904)Θminus1

Ξ119894119904+ℏ120574(119904)FΞ119894119904+ℏ120574(119904)

)minus1sdotF119879

Ξ119894119904+ℏ120574(119904)Θminus1

Ξ119894119904+ℏ120574(119904)Y

[120582]

119904minus1 (Ξ119894119904) (53)

where FΞ119894119904+ℏ120574(119904) ΘΞ119894119904+ℏ120574(119904)

and Y[120582]

119904minus1(Ξ119894119904)



119904) using(49) as

120585 (119894119904 (Ξ119894119904)) = 1


119898 (119904) minus 120599[120582]119904 [120582]

119898 (119904) 119894119904 (Ξ119894119904))119879

sdot 119877minus1 (119910[120582]119898 (119904) minus 120599[120582]119904 [120582]

119898 (119904) 119894119904 (Ξ119894119904))

(54)

endif



119904 = arg max119904(Ξ119894119904)120585 (119904 (Ξ119894

119904)) (55)



Ξ119888119904+1 = argmax

Ξ119888119904

120585 (119904 (Ξ119894119904)) (56)





(57)







11 = 120587(2)22 = 09

and 120587(2)12 = 120587(2)




(58)





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DFIR estimator













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)

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lat

(ms

)

ΔV

alt

(ms

)

DRHEDFIRAFGE

(b)



Competing Interests


Acknowledgments


References



































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











Horizon group

Random delay

Buffer

DFIRestimator



LOSNLOSmediums

z[i]t = 120601[i]t xt + u[i]

t (120574[i]t )

i = 1 2 985747

Ξcs+1

119970s

xs

x1s x2


x987407+1s

Ξ1s

Ξ2s

Ξ987407+1s

120585 (xis (Ξ

is))

arg maxx119904(Ξ

119894119904)120585 (xs (Ξi

s))

y[i]t ≜ 120581[i]

t120591[z[i]t ]


y[i]









119892119881 (V[120582]119898 (119904 120574[120582]119898 )) = 1radic(2120587)119902 det (119877)


(48)


120585 (119894119904 (Ξ)) = 119901 (119910[120582]119898 (119904) | 119894119904 (Ξ)) = 1


119898 (119904) minus 120599[120582]119904 [120582]

119898 (119904) 119894119904 (Ξ))119879sdot 119877minus1 (119910[120582]

119898 (119904) minus 120599[120582]119904 [120582]

119898 (119904) 119894119904 (Ξ)) (49)



J (119904 (Ξ)) = 12 [(119910[120582]119898 (119904) minus 120599[120582]119904 [120582]

119898 (119904) 119894119904 (Ξ))119879

sdot 119877minus1 (119910[120582]119898 (119904) minus 120599[120582]119904 [120582]

119898 (119904) 119894119904 (Ξ))] (50)






Ξ119888119904+1 = argmax

Ξ119888119904

120585 (119904 (Ξ119894119904)) (51)






G119904 = 2Ξ119888119904 minus ℏ 2Ξ119888



119904 Ξ2119904 Ξ119894

119904 Ξℏ+1119904 (52)







[120582]119904|119904minus1 (Ξ119894119904) = (F119879

Ξ119894119904+ℏ120574(119904)Θminus1

Ξ119894119904+ℏ120574(119904)FΞ119894119904+ℏ120574(119904)

)minus1sdotF119879

Ξ119894119904+ℏ120574(119904)Θminus1

Ξ119894119904+ℏ120574(119904)Y

[120582]

119904minus1 (Ξ119894119904) (53)

where FΞ119894119904+ℏ120574(119904) ΘΞ119894119904+ℏ120574(119904)

and Y[120582]

119904minus1(Ξ119894119904)



119904) using(49) as

120585 (119894119904 (Ξ119894119904)) = 1


119898 (119904) minus 120599[120582]119904 [120582]

119898 (119904) 119894119904 (Ξ119894119904))119879

sdot 119877minus1 (119910[120582]119898 (119904) minus 120599[120582]119904 [120582]

119898 (119904) 119894119904 (Ξ119894119904))

(54)

endif



119904 = arg max119904(Ξ119894119904)120585 (119904 (Ξ119894

119904)) (55)



Ξ119888119904+1 = argmax

Ξ119888119904

120585 (119904 (Ξ119894119904)) (56)





(57)







11 = 120587(2)22 = 09

and 120587(2)12 = 120587(2)




(58)





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0200400600800

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Normal3Normal4

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)

Y (m)


X (m)

minus5000

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(a)


0 1000 2000 3000 4000 5000 6000

0

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200400600800

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5 67 8

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Y (m)

X (m)

(b)


0 5 10 15 20 25 30 35 400

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10 12 14

01020304


DFIR estimator













120

12002

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29993030013002

300330043005

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(a)

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(b)

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TME120574

(t)

RD120591

(c)








0 500 1000

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(m)

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(m)

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(a)

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0 500 1000

0

5

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ΔV

lon

(ms

)

ΔV

lat

(ms

)

ΔV

alt

(ms

)

DRHEDFIRAFGE

(b)



Competing Interests


Acknowledgments


References



































Volume 2014




Journal of











Journal of


Function Spaces






Algebra












G119904 = 2Ξ119888119904 minus ℏ 2Ξ119888



119904 Ξ2119904 Ξ119894

119904 Ξℏ+1119904 (52)







[120582]119904|119904minus1 (Ξ119894119904) = (F119879

Ξ119894119904+ℏ120574(119904)Θminus1

Ξ119894119904+ℏ120574(119904)FΞ119894119904+ℏ120574(119904)

)minus1sdotF119879

Ξ119894119904+ℏ120574(119904)Θminus1

Ξ119894119904+ℏ120574(119904)Y

[120582]

119904minus1 (Ξ119894119904) (53)

where FΞ119894119904+ℏ120574(119904) ΘΞ119894119904+ℏ120574(119904)

and Y[120582]

119904minus1(Ξ119894119904)



119904) using(49) as

120585 (119894119904 (Ξ119894119904)) = 1


119898 (119904) minus 120599[120582]119904 [120582]

119898 (119904) 119894119904 (Ξ119894119904))119879

sdot 119877minus1 (119910[120582]119898 (119904) minus 120599[120582]119904 [120582]

119898 (119904) 119894119904 (Ξ119894119904))

(54)

endif



119904 = arg max119904(Ξ119894119904)120585 (119904 (Ξ119894

119904)) (55)



Ξ119888119904+1 = argmax

Ξ119888119904

120585 (119904 (Ξ119894119904)) (56)





(57)







11 = 120587(2)22 = 09

and 120587(2)12 = 120587(2)




(58)





05000

0

5000

0200400600800

1000

Sink1

Normal1Normal2

Normal3Normal4

Z(m

)

Y (m)


X (m)

minus5000

minus5000

(a)


0 1000 2000 3000 4000 5000 6000

0

2000

4000

60000

200400600800

1

23 4

5 67 8

9

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Z(m

)

Y (m)

X (m)

(b)


0 5 10 15 20 25 30 35 400

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RMSE

10 12 14

01020304


DFIR estimator













120

12002

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(c)








0 500 1000

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(ms

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lat

(ms

)

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alt

(ms

)

DRHEDFIRAFGE

(b)



Competing Interests


Acknowledgments


References



































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










05000

0

5000

0200400600800

1000

Sink1

Normal1Normal2

Normal3Normal4

Z(m

)

Y (m)


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(a)


0 1000 2000 3000 4000 5000 6000

0

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(b)


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10 12 14

01020304


DFIR estimator













120

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(c)








0 500 1000

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ΔV

lon

(ms

)

ΔV

lat

(ms

)

ΔV

alt

(ms

)

DRHEDFIRAFGE

(b)



Competing Interests


Acknowledgments


References



































Volume 2014




Journal of











Journal of


Function Spaces






Algebra













120

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lon

(ms

)

ΔV

lat

(ms

)

ΔV

alt

(ms

)

DRHEDFIRAFGE

(b)



Competing Interests


Acknowledgments


References



































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











0 500 1000

0

50

100

150

200

250

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minus50

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(m)

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lon

(ms

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ΔV

lat

(ms

)

ΔV

alt

(ms

)

DRHEDFIRAFGE

(b)



Competing Interests


Acknowledgments


References



































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










Competing Interests


Acknowledgments


References



































Volume 2014




Journal of











Journal of


Function Spaces






Algebra
















Volume 2014




Journal of











Journal of


Function Spaces






Algebra









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