Research Article Distributed Detection in Wireless Sensor...

Research ArticleDistributed Detection in Wireless Sensor Networks underByzantine Attacks

Junhai Luo and Zan Cao

School of Electronic Engineering University of Electronic Science and Technology of China Xiyuan Avenue Chengdu 611731 China

Correspondence should be addressed to Junhai Luo junhai luouestceducn

Received 21 May 2015 Revised 21 August 2015 Accepted 28 September 2015

Academic Editor Lucas Vespa

Copyright copy 2015 J Luo and Z Cao 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

Distributed detection in wireless sensor networks (WSNs) under Byzantine attacks is studied in this paper A new kind of Byzantineattacks neighborhood malicious Byzantine attacks (NMBA) is proposed In this type of Byzantine attacks part of sensors isconquered and reprogrammed by an intelligent adversaryThese sensors then are conducted to send false information to the fusioncenter (FC) in order to confuse it We see that the attacking performance of NMBA is very close to that of collaborative maliciousByzantine attacks (CMBA) and outperforms independentmalicious Byzantine attacks (IMBA)Decision fusion becomes impossiblewhen attacking power which is the fraction of compromised sensors inWSNs exceeds a specific value A closed-form expression forthe value is derived For mitigating attacking effect brought by NMBA a strategy for estimating the attacking power is proposedFurthermore a scheme to identify Byzantine attackers is presented Two kinds of discrepancy distance are constructed in thispaper to help in identifying Byzantine attackers We prove that most of Byzantine attackers are identified and performance of theidentifying scheme is proved to be excellent A data fusion scheme based on both dynamic threshold and the identifying scheme isanalyzed in this paper Numerical results are also provided to support the schemes and approaches

1 Introduction

Wireless sensor networks (WSNs) consist of a large numberof tiny power-limited sensors that are densely and spatiallydeployed to monitor physical phenomena When detectinga target in the region of interest (ROI) all the sensorsin network report their findings to the fusion center (FC)where a global-final decision is made For the advantage ofeasy deployment and fast self-organization WSNs have beenwidely used [1] Due to the increasing importance of beingused in bothmilitary and civilian applications it is imperativeto incorporate secure localization and detection into WSNsHowever limited by both the processing capability and powersupply of sensor nodes secure detection in WSNs has beena challenging task WSNs are also vulnerable to tamperingA serious threat to WSNs is Byzantine attacks where someauthenticated sensor nodes have been fully controlled by anintelligent adversary These compromised sensors are dis-patched to disrupt or confuse the FCWhile Byzantine attacksmay in general refer tomany types of Byzantine behaviors [23] our focus in this paper is on Byzantine attacks in terms of

data-falsification In this type of attack compromised nodesare reprogrammed and then forced to send falsified data tothe FC in order to undermine the inference performance ofnetwork The main goal of Byzantine attackers is to havocperformance of the FC as much as possible so that decider atthe FC is unable to utilize sensorsrsquo information to determinethe presence of target correctly

An important task that WSNs need to perform is targetdetection which is imperative for an accurate tracking oftarget In the context of Byzantine attackers attempting todisrupt the network a reliable algorithm of detection needsto be introduced In [4 5] several algorithms have beendeveloped for secure detection and localization inWSNsThetechniques based on direction of arrival (DOA) and time ofarrival (TOA) (or time-difference-of-arrival (TDOA)) havebeen investigated in [4] and [5] respectively However theTDOA is not suitable for detecting target because sensornodes are narrow band and lack accurate synchronizationSeveral researchers have focused on developing techniquesthat do not suffer from imperfect time synchronization Forexample scheme of measuring intensity of signal energy is

Hindawi Publishing CorporationInternational Journal of Distributed Sensor NetworksVolume 2015 Article ID 381642 18 pageshttpdxdoiorg1011552015381642

2 International Journal of Distributed Sensor Networks

used to detect target Therefore convenience of the energy-based method gives feasibility to detect and localize a targetthrough detecting energy In [6 7] each cognitive radioadopts an effective detection scheme based harvesting energytomake its decision and determinewhether there is a licenseduser requesting to occupy the spectrum band Each cognitiveradio sensor detects intensity of signal and measures theenergy received If the measured energy is greater thana properly predefined threshold the current spectrum isbusy Otherwise the current spectrum is idle We adopt thisscheme in this paper to detect whether there is a target in theROI

Marano et al have investigated distributed detection inthe presence of Byzantine attacks in [8 9] where Byzantineattackers are assumed to have a complete knowledge aboutthe true hypotheses In their system model the authorsassumed that the FC did not know which sensor node wasByzantine attacker Though the FC did not know which sen-sor node was Byzantine attacker it knew average percentageof compromised sensor nodes or upper bound of the averageVempaty et al have also analyzed the problem of distributeddetection under Byzantine attacks in [10] However theydid not assume that the Byzantine sensors had a completeknowledge about true hypotheses Instead they assumed thatthe Byzantine sensors made decisions about the presence oftarget through their own observations In other words eachByzantine sensor potentially flips the local decision made atthe nodeThe performance of network has also been analyzedin the context of presence of independent and collaborativeByzantine attacks respectively in [10] In addition to theanalysis of distributed detection in the presence of Byzantineattacks an adaptive learning scheme that mitigated the effectbrought by Byzantine attackers has been proposed by Vem-paty et al in [10] The scheme identifies Byzantine sensorsthrough observing the ratio value of deviations between theestimated behavior of ith sensor and the expected behaviorof an Honest sensor to the estimated behavior of ith sensorand the expected behavior of a Byzantine sensor If the valueis greater than 1 the FC declares the sensor to be ByzantineOtherwise the FC tags the sensor as Honest In order tomaximize the performance of decision fusion the FC thenremoves those decisions that are from sensors tagged asByzantine when the FC makes a global decision at the nexttime

In literature [10 11] analysis of performance of thenetwork under independent and collaborative maliciousByzantine attacks has been performed respectively When atarget enters into themonitoring region the ith compromisedsensor node has a detection and makes a localoriginaldecision In the case of independent malicious Byzantineattacks (IMBA) the local decision of ith compromised sensorcompletely depends on its own observation In the case ofcollaborative Byzantine attacks (CMBA) the local decision ofith compromised sensor depends on not only its own obser-vation but also decisions from the remaining compromisedsensors In the scenario of IMBA though Byzantine sensorcanmakemultiple observations in a proper time window andaverage these observations in order to reduce the effect ofadditive noise it is not effective significantly when Byzantine

sensor nodes stay far away from a target in the ROI Inaddition a sensor that adopts IMBA has no way to conquera certain degree of blind flipping of its own decision In thesituation of CMBA each Byzantine sensor communicateswith the left compromised sensors and refers to their deci-sions before determining its own local decision AlthoughCMBAproduces remarkable improvement in attacking effectmuch energy has been consumed in communication amongByzantine sensors especially when Byzantine sensor nodesare deployed sparsely In addition to analysis of performanceof distributed detection in the presence of Byzantine attacksthe blinding attacking power 120572 has been obtained Theblinding attacking power (120572blind) is equal to 05 and 035 inthe case of IMBA and CMBA respectively [11] Motivated bythis we propose a new Byzantine attacks model named afterneighborhood malicious Byzantine attacks (NMBA) NMBAis such a kind of attacks model where each compromisedsensor node in the network determines whether there is atarget or not in the ROI depending on not only its own localdecision but also a certain amount of decisions coming fromHonest sensors which are nearest around the compromisedsensor Then each Byzantine sensor node employs a majoritystrategy among decisions to make a final local decision Atlast the Byzantine sensor node flips the final local decisionconfidently and sends false decision to the FC

Attacking power 120572 which is also termed indicator of thevulnerability of the sensor networks is a crucial performancemetric [12 13] We assume that there are a large numberof sensor nodes deployed in the ROI According to thelaw of large numbers 120572 is equal to the ratio of number ofcompromised sensors to the total number of sensors in thenetwork (120572 isin [0 1]) In practice although the FC knows thepresence of Byzantine sensors in the network decider at theFC can hardly determine the exact attacking power [14] Ifthe decider knows the attacking power it is convenient forthe FC to adopt a robust strategy to mitigate the negativeeffect caused by Byzantine attackers [15 16] In this paperwe propose a simple and effective scheme to determine theattacking power in the perspective of decider at the FC Afterthe attacking power is estimated two kinds of discrepancydistance which are used to help in identifying Byzantinesensors are constructed in this paper

An effective scheme of decision fusion plays an importantrole in the FC [17ndash19] Many literatures focused on themitigation of Byzantine attacks and developed algorithms todesign a static and identical threshold for decision making atthe FC [11 14] Several authors have proposed online learningof normal trajectory patterns for detection in trajectory in[20] In this paper we propose an effective scheme based onboth dynamic threshold and identifying Byzantine attackersfor decision fusion at the FC

The paper is organized as follows In Section 2 wedescribe our system model including detection model andNMBA attackingmodel In order to formulate the problem ofdistributed detection in WSNs clearly we divide the processof decision fusion into three hierarchies or stages in thissection The attacks model of NMBA is proposed at thefirst stage which is different from independent Byzantineattacks and collaborative Byzantine attacksThe performance

International Journal of Distributed Sensor Networks 3

metric is also presented In Section 3 we determine theoptimal attacking strategy in the perspective of Byzantineattackers and closed-form expression for blinding region isderived Comparison among IMBA CMBA and NMBA isalso performed and numerical results are provided at thesame time From the perspective of network designer wepropose a fusion schemebased ondynamic threshold tomakea reliable global decision and analyze how the FC identifiesByzantine attackers to enhance the fusion performance inSection 4 The attacking power is also estimated Finally wepresent our conclusion in Section 5

2 System Model

21 Detection Model A network with119873 sensor nodes whichare spatially deployed in the ROI is considered All sensornodes in this network are independent on functionality Eachsensor makes a decision independently after detection Asillustrated in Figure 1 the sensor nodes which are denoted assymbol of plus are shown to be deployed on a regular gridand intensity of energy attenuated as the distance from thetarget that is represented as blue star increases It is worthmentioning that the detection scheme based on harvestingenergy is capable of handling any kind of deployment as longas the location information of each sensor node is available atthe FC The uniform sensor deployment shown in Figure 1 isonly a special case In any one kind of deployment119873 sensornodes can correctly detect a target when the target intrudesat the position 120579 = (119909

119905 119910119905) where 119909

119905and 119910

119905denote the coor-

dinate of this target location in 2D Cartesian We introducean isotropic intensity of signal attenuation model as follows

1198862

119894= 1198750(

1198890

119889E119894)

119899

(1)

where 119886119894is the signal amplitude received at ith sensor and 119875

0

is the emitted power measured at a reference distance 1198890 119899 is

the power decay exponent and 119889E119894 is the Euclidean distancebetween the target and ith sensor

119889E119894 = radic(119909119905minus 119909119894)2

+ (119910119905minus 119910119894)2

119894 = 1 2 119873 (2)

in which (119909119894 119910119894) are the coordinate of ith sensor For

simplicity but without loss of generality in this paper we let119899 = 2 119889

0= 1 [10] As a result (1) can be expressed as

1198862

119894=

1198750

(119889E119894)2 119894 = 1 2 119873 (3)

Equation (3) is a quite general model for signal attenuationof electromagnetic wave that propagates isotropically in freespace However when the signal of energy arrives at ithsensor it has been contaminated by additive white Gaussiannoise in practice Therefore the signal amplitude receivedat ith sensor is expressed as 119903

119894= 119886119894+ 119899119894 in which 119899

119894is

Gaussian noise which follows standard normal distributionHere we assume that all sensors in the network have theidentical additive white Gaussian noise that is 119899

119894sim 119873(120583 120590

2)

119894 = 1 2 119873

50 100 150 2000X-coordinate (m)

0

20

40

60

80

100

120

140

160

180

200

Y-c

oord

inat

e (m

)

Figure 1 The sensors are deployed in a regular grid Each sensorindependently harvests the energy propagated from target

Each sensor node needs to quantize the received signal ofenergy because of its limitations of bandwidth and energy andsends quantized binary measurements to the FC Thresholdof quantizers is adopted in this work for its simplicity of botheasy implementation and analysis as follows

119889119894=

1 119903119894gt 120589119894

0 119903119894lt 120589119894

(4)

where 119889119894and 120589119894are local decisions made by ith sensor after

quantizing the received signal and a predefined thresholdadopted by ith sensor respectively In this paper we assumethat all of the sensors share the identical threshold that is120589119894= 120589 119894 = 1 2 119873In this work the classical distribution detection model

is taken into account where two hypotheses are consideredEach sensor solves hypothesis testing problem and makesa local decision on either hypothesis 119867

0(target is absent)

or 1198671(target is present) We consider the scenario that the

adversary knows the complete information about the locationof sensors and is capable of attacking all the sensors simul-taneously Due to the constraint of budget the Byzantineattackers conquer only a part of nodes in the network todeteriorate capability of inference performance of networkThese Byzantine sensors transmit false decision to the FC inorder to deteriorate inference performance of the networkWe assume that the channel between the FC and local sensorsis error-free The original or local one-bit decision generatedat ith sensor node is denoted as

119889119894isin 0 1 119894 = 1 2 119873

Then the ith sensor reports one-bit decision 119889119894to the FC

where 119889119894=119889119894if ith sensor is Honest For a Byzantine sensor

the local original decision 119889119894need not be equal to

119889119894in our

attacks modelLet 119873

119867and 119873

119861be the number of Honest and Byzantine

sensors respectively The total number of sensors can beexpressed as 119873 = 119873

119867+ 119873119861and the number of Byzantine

sensor nodes 119873119861is equal to 120572 sdot 119873 In the perspective of


Byzantine attackers conquering 119873 sensors is not a wisestrategy for the adversary itself at the risk of exposed activityThe main goal of adversary is to compromise a fraction ofsensors to degrade the performance of the FC instead ofcapturing the network with a huge cost Therefore we have119873119861lt 119873 We use 119875119867

119889(119894) = Pr( 119889

119894= 1 | 119867

1 119867) and 119875119867fa (119894) =

Pr( 119889119894= 1 | 119867

0 119867) to denote the probability of detection and

false-alarm of ith sensor respectively We use 119867 to presenta sensor node to be Honest and 119894 isin 1 2 119873

119867 The

detection probability of ith sensor can be expressed as

119875119867

119889(119894) = Pr ( 119889

119894= 1 | 119867

1 119867) = Pr (119886

119894+ 119899119894gt 120589119894)

= 119876(

120589119894minus 119886119894minus 120583

120590

)

(5)

Similarly the false-alarm probability of 119894th sensor can beexpressed as

119875119867

fa (119894) = Pr (119899119894gt 120589119894) = 119876(

120589119894minus 120583

120590

) (6)

where119876(sdot) is the complementary distribution function of thestandard Gaussian

119876 (119909) = int

infin

119909

1

radic2120587

119890minus11990522119889119905 (7)

When a target intrudes into the ROI each sensor node startsto sense and record the energy propagated from the targetusing detection scheme based on harvesting energy [21]We let each sensor perform 119870 observations in a small timewindow 119879 where target is assumed to be static This is areasonable assumption For example if the sampling rate ofeach sensor is 6000Hz a target with a speed of 100 kmhonly moves 025m during 119879 = 54 sampling intervals [22]The jth observation at ith sensor node can be expressed as119889119894119895 119894 isin 1 2 119873 and 119895 isin 1 2 119870 A localoriginal

decision matrix D = [d1d2

d119873]T is generated where

d119894= (

11988911989411198891198942

119889119894119870)T is the vector of localoriginal decision

at the ith sensor node And 119889119894119895isin 0 1 119894 isin 1 2 119873 and

119895 isin 1 2 119870 The FC receives119873 vectors of decisions fromlocal sensorsThen a decisionmatrixD = [d

1 d2 d

119873]T is

formulated at the FC that isD = (119889119894119895)119873times119870

where 119889119894119895isin 0 1

119894 isin 1 2 119873 and 119895 isin 1 2 119870 The localoriginaldecision matrix D is equal to D if there is no presence ofByzantine attackers

In order to formulate the problem in the process ofdecision fusion we divide the process into three hierar-chiesstages As illustrated in Figure 2 ith sensor makes alocaloriginal vector of d

119894and sends the vector d

119894into the FC

after d119894is probably ldquoattackedrdquo at the first stage A decision

matrix D is formulated from which the vector of globaldecision z = (119911

1 1199112 119911

119870) is mapped at the second stage At

the last stage a global-final decision 119911 is mapped from vectorz at the FC

22 ByzantineAttacksModel In the attacksmodel ofNMBAthe ith Byzantine sensor has exactly119872

119894minus 1 (119894 = 1 2 119873

119861)

Natural state

The fusion center

d1 d2 dNminus1 dN

d1 d2 dNminus1 dN

middot middot middot

DNtimesK rarr z

S1 S2 SNminus1 SN

z rarr z

Figure 2Model of three hierarchies d119894is the vector of local decision

made by ith sensor 119878119894 d119894is the vector of decision sent to the FC

119894 = 1 2 119873 D119873times119870

is the decision matrix formulated at the FC zis global decision vector and 119911 is global-final decision

neighbors to consult and 119872119894le 119873119867 In order to facilitate

analysis we assume that the scenario of many Byzantinesensors flocking together does not happenNamely thewholeByzantine sensor nodes are deployed sparsely by intelligentadversary in the ROI In the case of 119873 sensors deployedon a regular grid NMBA has several neighborhood typesincluding diamond type and square type For each Byzantinesensor its neighborhood nodes are those sensors that arethe nearest and Honest around it in specific neighborhoodtype We assume that each Byzantine sensor knows theidentifications of the remaining compromised sensors EachByzantine sensor consults all of its neighborhood nodes tomake a wise and tricky decision In Figure 3 the type ofsquare neighborhood is presented and the case of119872

119894= 119872 =

9 is considered Clearly each Byzantine sensor node consultsits eight neighbors and makes a decision based on decisionsfrom its neighbors

We make the conditional iid assumption under whichobservations from sensors are conditionally independent andidentically distributed The jth observation at ith sensor thenhas the distributions

1198670 V119894119895(119896) = Pr (119889

119894119895= 119896 | 119867

0)

= (1 minus 120572)Pr (119889119894119895= 119896 | 119867

0 119867)

+ 120572Pr (119889119894119895= 119896 | 119867

0 119861)

1198671 119906119894119895(119896) = Pr (119889

119894119895= 119896 | 119867

1)

= (1 minus 120572)Pr (119889119894119895= 119896 | 119867

1 119867)

+ 120572Pr (119889119894119895= 119896 | 119867

1 119861)

119894 isin 1 2 119873 119895 isin 1 2 119870 119896 isin 0 1

(8)


0

20

40

60

80

100

120

140

160

180

200

Y-c

oord

inat

e (m

)

50 100 150 2000X-coordinate (m)

Figure 3 Square type of NMBA in the case of 119872 The blue star isa intruding target and symbol plus is denoted as sensor Byzantinesensors are denoted as plus symbol covered with diamond EachByzantine sensor has 9 decisions after consulting its 8 neighborhoodnodes

If ith sensor is Honest its observation 119896 isin 0 1 follows dis-tributions 119901 and 119902 under hypotheses119867

0and119867

1 respectively

Therefore we have

1198670 Pr ( 119889

119894119895= 119896 | 119867

0 119867) = 119902

119894119895(119896)

1198671 Pr ( 119889

119894119895= 119896 | 119867

1 119867) = 119901

119894119895(119896)

119894 = 1 2 119873119867

(9)

According to (5) (6) and (7) we get

119901119894119895(1) = 119876(

120589 minus 119886119894minus 120583

120590

)

119901119894119895(0) = 1 minus 119901

119894119895(1)

119902119894119895(1) = 119876(

120589 minus 120583

120590

)

119902119894119895(0) = 1 minus 119902

119894119895(1)

(10)

Similarly we have distributions 119909 and 119910 under the samehypotheses for Byzantine sensor as follows

1198670 Pr ( 119889

119894119895= 119896 | 119867

0 119861) = 119910

119894119895(119896)

1198671 Pr ( 119889

119894119895= 119896 | 119867

1 119861) = 119909

119894119895(119896)

119894 = 1 2 119873119861

(11)

In the attacks model of NMBA the ith Byzantine sensormakes an initial decision 119888

1198940independently and gets the119872

119894minus1

decisions from its neighborhood sensors As a result a set ofdecisions 119888

119894119897 119897 isin 0 1 119872

119894minus 1 119888

119894119897isin 0 1 is obtained

where the 119888119894119897represents the decision from the lth neighbor of

ith Byzantine sensor Then the ith Byzantine sensor makesits local or original decision using a majority strategy that

is the original local decision 119889119894119895= IF(sum119872119894minus1

119897=0119888119894119897gt 120578119894) where

IF(sdot) and 120578119894are indicator function and threshold adopted by

the ith Byzantine sensor respectively Therefore we have thefollowing equations

119910119894119895(119896) =

119872119894

sum

119898=120578119894

sum

120587isinΓ

119898

prod

119894=1

119902120587(119894)119895

(119896)

119872119894

prod

119894=119898+1

(1 minus 119902120587(119894)119895

(119896))

119909119894119895(119896) =

119872119894

sum

119898=120578119894

sum

120587isinΓ

119898

prod

119894=1

119901120587(119894)119895

(119896)

119872119894

prod

119894=119898+1

(1 minus 119901120587(119894)119895

(119896))

(12)

where119872119894isin 1 2 119873

119867 1 le 120578 le 119872

119894 and Γ denotes the set

of all permutations of the 119872119894sensors After using majority

strategy to make a local decision the ith Byzantine sensorflips confidently its decision with probability of 119875flip = 1Specifically we have

Pr (119889119894119895= 119896 |

119889119894119895= 119897 119861) =

1 when 119897 = 119896

0 when 119897 = 119896

119896 119897 isin 0 1

(13)

Thus we get

Pr (119889119894119895= 119896 | 119867

0 119861) = sum

119897 =119896

Pr ( 119889119894119895= 119897 | 119867

0 119861)

sdot Pr (119889119894119895= 119896 |

119889119894119895= 119897119867

0 119861)

+sum

119897=119896

Pr ( 119889119894119895= 119897 | 119867

0 119861)

sdot Pr (119889119894119895= 119896 |

119889119894119895= 119897119867

0 119861)

Pr (119889119894119895= 119896 | 119867

1 119861) = sum

119897 =119896

Pr ( 119889119894119895= 119897 | 119867

1 119861)

sdot Pr (119889119894119895= 119896 |

119889119894119895= 119897119867

1 119861)

+sum

119897=119896

Pr ( 119889119894119895= 119897 | 119867

1 119861)

sdot Pr (119889119894119895= 119896 |

119889119894119895= 119897119867

1 119861)

(14)

Therefore we have

Pr (119889119894119895= 0 | 119867

0 119861) = Pr ( 119889

119894119895= 1 | 119867

0 119861) = 119910

119894119895(1)

Pr (119889119894119895= 1 | 119867

0 119861) = Pr ( 119889

119894119895= 0 | 119867

0 119861) = 119910

119894119895(0)

Pr (119889119894119895= 0 | 119867

1 119861) = Pr ( 119889

119894119895= 1 | 119867

1 119861) = 119909

119894119895(1)

Pr (119889119894119895= 1 | 119867

1 119861) = Pr ( 119889

119894119895= 0 | 119867

1 119861) = 119909

119894119895(0)

(15)


Substituting (9) and (15) in (8) and after simplification weobtain

V119894119895(119896) = Pr (119889

119894119895= 119896 | 119867

0)

= (1 minus 120572) [119896119902119894119895(119896) + (1 minus 119896) (1 minus 119902

119894119895(119896))]

+ 120572119910119894119895(1 minus 119896)

119906119894119895(119896) = Pr (119889

119894119895= 119896 | 119867

1)


119894119895(119896))]

+ 120572119909119894119895(1 minus 119896)

(16)

23 Performance Metric In the perspective of Byzantineattackers the primary objective is to deteriorate the inferenceperformance of FC as much as possible On the contrary theFC wants to make inference performance as much highlyexcellent as possible in order to guarantee valid detectionIn this paper we adopt Kullback-Leibler divergence (KLD)as the network performance that characterizes inferenceperformance at the FC KLD is very important in probabilitytheory and is widely employed as information-theoreticdistance measure to characterize detection performance [2324] The KLD between the distributions V

119894119895(119896) = Pr(119889

119894119895=

119896 | 1198670) and 119906

119894119895(119896) = Pr(119889

119894119895= 119896 | 119867

1) for ith sensor can

be expressed as

KLD (119906119894119895 V119894119895) = sum

119896isin01

119906119894119895(119896) log

119906119894119895(119896)

V119894119895(119896)

(17)

The FC receives ith sensorrsquos decisions V119894119895(119896) and 119906

119894119895(119896) under

1198670and 119867

1 respectively In the perspective of Byzantine

attackers they try to minimize the KLD as much as possibleso that the FC can hardly make a right decision between1198670and 119867

1 On the other hand network designer wants

to maximize KLD of each sensorrsquos decision to mitigate thenegative effect caused by Byzantine attackers In the nextsection we explore the optimal strategy of Byzantine attacksthat impair the detection performance as much as possible byminimizing KLD

3 Optimal Strategy for Byzantine Attackers

31 Optimal Strategy for Byzantine Attacks As explored inSection 2 the Byzantine attackers attempt to make the nodesthat have been compromised have small KL divergenceas much as possible Byzantine attackers have the optimalsuperiority on degrading inference performance of FC whenKLD is equal to zero In the case of KLD = 0 the FC cannotdistinguish the distributions under119867

0or1198671 In other words

the data from sensors conveys no information We refer tothis case as the FC being blinded completely when

119906119894119895(119896) = Pr (119889

119894119895= 119896 | 119867

1) = Pr (119889

119894119895= 119896 | 119867

0)

= V119894119895(119896)

(18)

Substituting (16) in (18) and after simplification the conditionto make KLD(V

119894119895(119896) 119906

119894119895(119896)) = 0 is equivalent to

120572 =

119901119894119895(1) minus 119902

119894119895(1)

(119901119894119895(1) minus 119902

119894119895(1)) + (119909

119894119895(1) minus 119910

119894119895(1))

=

119901119894119895(1) minus 119902

119894119895(1)

(119901119894119895(1) minus 119902

119894119895(1)) + 119891 (119872

119894 120578119894 119901119894119895(1) 119902119894119895(1))

(19)

where the close-form expression of function 119891(sdot) is denotedas the following equation

119891 (119872119894 120578119894 119901119894119895(1) 119902119894119895(1))

=

119872119894

sum

119898=120578119894

sum

120587isinΓ

119898

prod

119894=1

119901120587(119894)119895

(1)

119872119894

prod

119894=119898+1

(1 minus 119901120587(119894)119895

(1))

minus

119872119894

sum

119898=120578119894

sum

120587isinΓ

119898

prod

119894=1

119902120587(119894)119895

(1)

119872119894

prod

119894=119898+1

(1 minus 119902120587(119894)119895

(1))

(20)

where119872119894and 120578119894are the number of neighborhood nodes and

threshold adopted by ith Byzantine sensor respectivelyAs mentioned above the KL distance between V

119894119895(119896) and

119906119894119895(119896) is equal to zero that is KLD(V

119894119895 119906119894119895) = 0 if and only if

V119894119895(119896) = 119906

119894119895(119896) The FC is incapable of distinguishing the two

distributions under 1198670and 119867

1when KLD is equal to zero

The attackers then project interests in theminimumattackingpower that can just make the ability of inference of the FCdestroyedThus theminimumattacking power in the contextof NMBA is denoted as

120572blind = min 120572 120572 that make KLD (119906119894119895 V119894119895) = 0 (21)

For the sake of minimizing 120572 to reach 120572blind we have the fol-lowing equation depending onoperating point (119901

119894119895(1) 119902119894119895(1))

120572blind

=

119901119894119895(1) minus 119902

119894119895(1)

(119901119894119895(1) minus 119902

119894119895(1)) +max 119891 (119872

119894 120578119894 119901119894119895(1) 119902119894119895(1))

(22)

Because of 0 lt 119901119894119895(1) minus 119902

119894119895(1) le 1 we have the following

inequality

120572blind le1

1 +max 119891 (119872119894 120578119894 119901119894119895(1) 119902119894119895(1))

(23)

To prove inequality (23) we apply the monotonic propertyof the function of 119909(119909 + 1) Due to the function possessingdifferentiability we have the following inequality

119889

119889119909

(

119909

119909 + 1

) =

1

(119909 + 1)2gt 0 (24)

Therefore 119909(119909 + 1) is a monotonically increasing functionwhen 0 le 119909 le 1 As a result inequality (23) is certified Aftercertifying (23) we have the following equation

120578119894 opt = ceil(

119872119894

2

) (25)


2 3 4 5 6 7 8 91120578

0

01

02

03

04

05

06

07

08

09

120572

Figure 4 Attacking power 120572 versus 120578 when119872 = 9 120578 varying from1 to 9 Byzantine sensors have the smallest 120572 when 120578 = 5

where ceil(sdot) is the ceiling function Therefore function 119891(sdot)in (19) reaches the maximum value point only when 120578

119894=

120578119894 opt Therefore for a pair of fixed operating points (119901

119894119895(1)

119902119894119895(1)) we have

max119891 = 119891(119872119894 ceil(

119872119894

2

) 119901119894119895(1) 119902119894119895(1)) (26)

and (22) can be represented as

120572blind =119901119894119895(1) minus 119902

119894119895(1)

(119901119894119895(1) minus 119902

119894119895(1)) +max119891

(27)

When the intelligent adversary poses attacking power 120572

which is greater than 120572blind the FC is trapped into a dilemmawhere the decider cannot utilize any information fromsensors to make a correct decisionThe situation refers to thefusion centerrsquos entering into the blinding region where the FCis blinded completely

32 Numerical Results In this subsection numerical resultsare presented to support our analysis of optimal attackingstrategy for Byzantine attackers We consider the networkwhere 119873 = 100 sensors are deployed on regular grid in theROI under the Byzantine attacks model of NMBAWe set thepower at the reference point (119889

0= 1) as 200 and the signal

amplitude arriving at the local sensor is contaminated byAWGNwith mean value 120583 = 0 and standard deviation 120590 = 3A target intrudes at position 120579 = (90 90) We consider thesquare type of NMBA where all the Byzantine sensors adoptidentical number of neighborhoodnodes and optimal thresh-old that is119872 = 119872

119894and 120578 = 120578

119894 opt 119894 = 1 2 119873119861 Attacking

power 120572 is observed in the case of square type of NMBA over119872 isin 1 2 119873

119867 and 120578 isin 1 2 119872 In the blinding

region the FC is incapable of utilizing any information tomake a decision to determine whether there is a target in theROI Byzantine users try to pay as much low cost as possiblewhen the FC has been completely blinded by them Attackingpower 120572 in blinding region is a convex function of threshold 120578

0

05

1

15

2

25

3

35

4

45

5

120578op

t

2 3 4 5 6 7 8 91M

Figure 5 Optimal threshold 120578 versus119872119872 varying from 1 to 9Theoptimal threshold increases with the increment of119872 and it is alwaysequal to ceil(1198722)

under theNMBA attackingmodelTherefore it is possible forByzantine attackers to adopt the least attacking power to blindthe FC From the numerical results presented in Figure 4 wesee that the attacking power 120572 reaches minimum value onlywhen 120578 = 5 in the case of NMBAwhen119872 = 9Theminimumvalue of 120572 is equal to 03844 The result is intuitive and alsoindicates that the optimal threshold associates with the num-ber of neighborhood nodes119872The other cases of NMBA arepresented in Table 1 We observe that the minimum value of120572blind is just obtained at the point where the optimal threshold120578 is got In Figure 5 we present that optimal thresholdincreases with the increase in the value of119872 We observe that120578opt is always equivalent to 1198722 when 119872 is even and 120578opt isequal to the maximum integer that is not larger than 1198722

when 119872 is odd It is clear that optimal threshold Byzantineattacker associates with the number of its neighborhood InFigures 6 and 7 it is shown that 120572blind is the monotonicallydecreasing function of 119872 in the condition that 120578 is alwaysequal to 120578opt As illustrated in Figures 5 and 6 IMBA is aparticular case when 119872 = 1 Under condition of 119872 = 1each Byzantine sensormakes a judgement only depending onits own observation and flips its decision confidently prior tosending to the FC When119872 = 2 each Byzantine sensor con-sults with two neighborhood nodes around it before makinga decision and the minimum attacking power in the blindingregion is equal to 05 When 119872 is equal to 2 the attackingeffect brought byNMBAmatcheswith IMBA120572blind ofNMBAdecreasesmonotonicallywith the increase of119872 and it reaches03844 when119872 increases to 9 120572blind decreases to 03752 when119872 increases to 25 The exact value of 120572blind corresponding to119872 and 120578opt is presented in Table 2 In addition as we can seefromFigures 6 and 7 the number of neighborhoodnodes thateach Byzantine sensor node consults including itself shouldbe odd for better attacking efficiency To further analyze theattacks of NMBA we compare it with IMBA and CMBA interms of KL distance about attacking power As we discussedabove IMBA is a special case of NMBA when 119872 = 1


Table 1 120572 at (119872 120578)119872 isin 1 2 9 and 120578 le 119872

119872120572

120578

1 2 3 4 5 6 7 8 91 050002 05000 050003 05434 04310 054344 05952 04310 04310 059525 06469 04509 04042 04509 064696 06959 04785 04042 04042 04785 069597 07410 05100 04145 03913 04145 05100 074108 07814 05438 04298 03913 03913 04298 05438 078149 08171 05790 04484 03970 03844 03970 04484 05790 08171

119871 out of 119873119861fusion rule has been used for CMBA among

the Byzantine sensors And Byzantine sensors collaboratetogether tomake decisions about the presence of target If theattacking power is greater than 05 the FC can be blindedcompletely by any kinds of attacking strategies adopted byByzantine attackers Therefore our analysis about detectionperformance is under the condition of 120572 le 05 In Figures 8and 9 we plot the KLD of three attacking models against 120572 inthe case119872 = 9 and119872 = 25 respectively From the numericalresults presented in Figures 8 and 9 we can see that the KLDof NMBA is very close to CMBA and outperforms IMBAsignificantlyThough the performance of NMBA is very closeto CMBA the difference of KL distance between them varieswith attacking power In Figures 8 and 9 KL distance ofCMBA is greater than that of NMBA in the region of 120572 isin

[0 009] and 120572 isin [0 025] respectively However CMBAperforms better when 120572meets the condition of 120572 ge 009 and120572 ge 0025 in Figures 8 and 9 respectively For CMBA theblinding point C which is marked by black-square in Figures8 and 9 is determined by the total number of Byzantine sensornodes instead of119872 For NMBA the blinding point B whichis marked by blue-square in Figures 8 and 9 is determinedby the number of neighborhood nodes For NMBA 119872 islarger and 120572blind is smaller Furthermore 120572blind = 03844 and120572blind = 03751 when119872 is equal to 9 and 25 respectively

4 Fusion Center Decision Strategy

In this section let us solve the problem of identifyingByzantine attackers to enhance inference performance of theFC In order to explain the problem well we have definedthree types of decision which include localoriginal decisionglobal decision and global-final decision As we discussedin Section 2 a decision matrix D119905

119873times119870= (119889

119894119895(119905))119873times119870

isformulated at the FC after 119870 observations in a time window119879 at tth global-final decision making 119905 isin 119873

+ We let theFC make a corresponding global decision 119911

119895(119905) over vector

of decision d119905119895at jth observation And a vector of global

decision z(119905) = (1199111(119905) 1199112(119905) 119911

119870(119905)) is formulated over

d1199051 d1199052 d119905

119870 119911119895(119905) isin 0 1 for any 119905 isin 119873+ 119895 = 1 2 119870

Majority vote which is simple and effective scheme is adoptedto make a global-final decision 119911(119905) isin 0 1 over the vectorz(119905) where ldquo1rdquo represents the presence of target and ldquo0rdquo

0

005

01

015

02

025

03

035

04

045

05

120572bl

ind

2 3 4 5 6 7 8 91M

Figure 6 120572blind versus119872119872 varying from 1 to 9 in condition of 120578 =120578opt 120572blind monotonically decreases with119872 It is clear that Byzantineattacker more easily blinds the FC when each Byzantine sensor hasmore neighborhood nodes

0

005

01

015

02

025

03

035

04

045

05

120572bl

ind

3 5 7 9 11 13 15 17 19 21 23 251M



Table 2 120572blind at (119872 120578opt)119872 isin 1 2 9 and 120578opt = ceil(1198722)

119872120578opt 11 21 32 42 53 63 74 84 95120572blind 05000 05000 04310 04310 04042 04042 03913 03913 03844

IMBANMBACMBA

C B A0

02

04

06

08

1

12

14

KLD

(120572)

01 02 03 04 050Attacking power 120572

Figure 8 KLD decreases monotonically with attacking power 120572when 119872 = 9 KLD of IMBA decreases to 0 when 120572 = 05 KLD ofNMBA decreases to 0 when 120572 = 03844 in the condition of 120578 = 120578optKLD of CMBA decreases to 0 when 120572 = 035

represents the absence of target respectively However weproject our focus on designing a rule to construct vectorof global decision z(119905) We propose a scheme of makingglobal decision that the FC can adaptively adjust its threshold120578fc(119905 119895) for d119905119895 to make a corresponding global decision 119911

119895(119905)

The information of elements in decision matrixD119905119873times119870

is alsoutilized to estimate probability of miss detection and falsealarm respectively In order to evaluate the fusion schemeperformance metric introduced in this work is the accuracyof fusion scheme in terms of identifying Byzantine sensors

We define an intuitive distance between the global-finaldecision and localoriginal decisions as

10038161003816100381610038161003816d119905119895

10038161003816100381610038161003816=

119873

sum

119894=1

10038161003816100381610038161003816119889119894119895(119905) minus 119911 (119905)

10038161003816100381610038161003816 (28)

Similarly another intuitive distance is also defined as thefollowing equation

10038161003816100381610038161003816d119905119894

10038161003816100381610038161003816=

119870

sum

119895=1

10038161003816100381610038161003816119889119894119895(119905) minus 119911 (119905)

10038161003816100381610038161003816 (29)

In (29) |d119905119894| measures the degree of discrepancy between

the global-final decision and 119870 localoriginal decisions thatare from ith sensor Similarly |d119905

119895| in (28) measures the

degree of discrepancy between the global-final decision and119873 localoriginal decisions generated at jth observation The

IMBANMBACMBA

C B A01 02 03 04 050

Attacking power 120572

0

02

04

06

08

1

12

14

KLD

(120572)

Figure 9 KLD decreases monotonically with attacking power 120572when119872 = 25 KLD of IMBA decreases to 0 when 120572 = 05 KLD ofNMBA decreases to 0 when 120572 = 03752 in the condition of 120578 = 120578optKLD of CMBA decreases to 0 when 120572 = 035

distance |d119905119894| is larger and the ith sensor is closer to behavior

of Byzantine On the contrary the distance of |d119905119894| is smaller

and the probability of ith sensor being tagged as Byzantineattacker by the FC is lower Similarly when the distance of|d119905119895| is larger the decision generated at jth observation is

worse The FC regards the jth observation as excellent andwe believe that the network performs well when |d119905

119895| is small

For simplicity we let probability of miss detection equalprobability of false alarm that is 119875

119898= 119875fa in the context of

the attacks model of NMBA

41 Scheme of Identifying Byzantine Attackers If one sensorrsquoslocaloriginal decision is different from global-final decisionit is labeled as Byzantine It is worth mentioning that Honestsensors are classified probably into Byzantine group whenthey behave like Byzantine Similarly the Byzantine attackerscan also be probably labeled as Honest We use 119873

0(119905 119895) and

1198731(119905 119895) to denote the total effective number of elements ldquo0rdquo

and ldquo1rdquo in decision matrix respectively 1198730(119905 119895) and 119873

1(119905 119895)

vary with the order of 119895 and take part in making global deci-sion Here we represent (28) and (29) again as follows

10038161003816100381610038161003816d119905119895

10038161003816100381610038161003816= sum

119894isin119868(119905119895)

10038161003816100381610038161003816119889119894119895(119905) minus 119911 (119905)

10038161003816100381610038161003816

10038161003816100381610038161003816d119905119894

10038161003816100381610038161003816=

119870

sum

119896=1

1003816100381610038161003816119889119894119896(119905) minus 119911 (119905)

1003816100381610038161003816 119894 isin 119868 (119905 119895)

(30)

where 119868(119905 119895) is defined at (37)


Proposition 1 Suppose the network with119873 sensors includingByzantine and Honest where the probability of miss detectionand false-alarm is zero that is 119875

119898= 119875119891119886

= 0 and Byzantineattackers always know the true natural state Attacking powerbrought by the attackers is estimated through the followingexpression

(119905) =

min 1198730(119905) 119873

1(119905)

119873

(31)

where1198730(119905) and119873

1(119905) are denoted as the total number of ldquo0rdquo

and ldquo1rdquo respectively

Proof See Appendix A

Based on Proposition 1 and constrained by its conditionswe have the following equation for estimating the attackingpower at the global decision making

(119905 119895) =

min 1198730(119905 119895) 119873

1(119905 119895)

1198730(119905 119895) + 119873

1(119905 119895)

(32)

Taking the scenario of Byzantine attackers without knowl-edge about true natural state into consideration in practicalapplication we have the following

Proposition 2 Suppose the network of size119873 containing bothHonest sensors and Byzantine sensor nodes where decisionmatrix D119905

119873times119870is made at tth global-final decision after 119873 sen-

sors taking119870 observations in the time window 119879 The networkis not completely blinded by adversary The estimation of leastattacking power has a lower bound which can be expressed as

(119905 119895)

ge

min 1198730(119905 119895) 119873

1(119905 119895) minus 119870 sdot [max

119896(1003816100381610038161003816d119905119896

1003816100381610038161003816) minusmin

119896(1003816100381610038161003816d119905119896

1003816100381610038161003816)]

119899 (119905 119895) sdot 119870

119896 isin 1 2 119870

(33)

It is clear that (119905 0) = 0 and 119899(119905 0) = 119873 119905 isin 119873+ In (33)

[max119896(|d119905119896|) minus min

119896(|d119905119896|)] is used to compute the maximum

distance of decisions between two certain vectors of observationwhich are from two sensors tagged as Honest at the (119895 minus 1)thglobal decision making

Proof See Appendix B

Here we let

(119905 119895)

=

min 1198730(119905 119895) 119873

1(119905 119895) minus 119870 sdot [max

119895(

10038161003816100381610038161003816d119905119895

10038161003816100381610038161003816) minusmin

119895(

10038161003816100381610038161003816d119905119895

10038161003816100381610038161003816)]

119899 (119905 119895) sdot 119870

(34)

during the process of identifying Byzantine sensors Afterestimating the attacking power (119905 119895) the FC approximatelyknows how many sensors have been compromised by anadversary among 119899(119905 119895) sensors that participate in the jthglobal decision makingThe FC then takes an action to iden-tify (119905 119895) sdot 119899(119905 119895) Byzantine sensors to help the next globaldecision making 119871(119905 119895) is defined as the set of identities of

sensors that are tagged asHonest at the (119895minus1)thThese sensorswill participate in the jth global decision making It is clearthat 119871(119905 0) = 119894 119894 = 1 2 119873 119905 isin 119873

+ We definea sequence 119868

119873(119905 119895) over |d119905

119894| | 119894 isin 119871(119905 119895 minus 1) in which

|d119905119894|119898 is used for presenting the element with order 119898 119898 isin

1 2 119899(119905 119895) If |d119905119894|119898gt |d119905119897|119899 then 119898 gt 119899 The identity 119894

is greater than 119897 when 119898 gt 119899 if |d119905119894|119898= |d119905119897|119899 or 119894 lt 119897 when

119898 lt 119899 otherwise 119894 119897 isin 119871(119905 119895 minus 1) 119898 119899 isin 1 2 119899(119905 119895)The sequence can be expressed as

119868119873(119905 119895)

=

10038161003816100381610038161003816d119905119894

10038161003816100381610038161003816

119899(119905119895)

10038161003816100381610038161003816d119905119894

10038161003816100381610038161003816

119899(119905119895)minus1

10038161003816100381610038161003816d119905119894

10038161003816100381610038161003816

1

| 119894 isin 119871 (119905 119895 minus 1)

(35)

where forall119898 isin 1 2 119899(119905 119895) |d119905119894|119898 matches only a specific

identity There are 119899(119905 119895) identities and |119868119873(119905 119895)| = 119899(119905 119895) In

order to find the sequence of d119905119894 we define a function ID(sdot)

over 119868119873(119905 119895)

119899 = ID (d119905119894)

= sum

119897isin119871(119905119895minus1)

IF (10038161003816100381610038161003816d119905119897

10038161003816100381610038161003816gt

10038161003816100381610038161003816d119905119894

10038161003816100381610038161003816

119899

)

+ sum

119897isin119871(119905119895minus1)

IF (10038161003816100381610038161003816d119905119897

10038161003816100381610038161003816=

10038161003816100381610038161003816d119905119894

10038161003816100381610038161003816

119899

| 119897 gt 119895)

119899 isin 1 2 119899 (119905 119895)

(36)

where IF(sdot) is an indicator function and IF(true) = 1 andIF(false) = 0

The FC have obtained the approximate knowledge aboutthe attacking power posed by the adversary at jth global deci-sion making The FC has also known each sensorrsquos sequencein 119868119873(119905 119895) Therefore sensors whose sequences are dropped

into the region from 119899(119905 119895) to 119899(119905 119895) sdot [1 minus (119905 119895)] + 1 arejudged to be Byzantine attackers The decisions from Byzan-tine attackers are removed at the (119895 + 1)th global decisionmakingThe identities of remaining sensors which are trustedby the FC at jth global decision making can be expressed as

119868 (119905 119895)

= 119894 | ID (d119905119894) lt (119905 119895) sdot 119899 (119905 119895) 119894 isin 119871 (119905 119895 minus 1)

(37)

and 119868(119905 119895) sube 119871(119905 119895minus1)The attacking power at tth global-finaldecision making is estimated by the following equation

(119905) =

119873 minus 119899 (119905 119870)

119873

(38)

42 Rule of Decision Fusion Based on DynamicThreshold Aswe described in Section 2 the process of making a global-final decision is divided into three stages A localorigi-nal decision matrix D119905

119873times119870is generated at the first stage A

decision matrix D119905119873times119870

is formulated after D119905119873times119870

being pro-bably attacked A vector of global decision z(119905) = (119911

1(119905)

1199112(119905) 119911

119870(119905)) is computed and obtained over vectors of

decision d1199051 d1199052 d119905

119870 through applying a policy of fusion


at the middle stage At last the decider at the FC has aglobal-final decision 119911(119905) In this paper we put our focuson proposing an effective method of decision fusion to getan excellent vector of global decision z(119905) which can easilyhelp to make a global-final decision 119911(119905) at the last stageThe rule of decision fusion based on dynamic threshold isthat threshold (120578fc(119905 119895)) for jth global decision making varieswith 119895 The number of sensors which participate in jth globaldecision making is denoted as 119899(119905 119895) In addition 119899(119905 119895)always associates with 119899(119905 119895 minus 1) tightly which is expressedas

119899 (119905 119895) = 119899 (119905 119895 minus 1) sdot [1 minus (119905 119895 minus 1)] (39)

Here 119902out of119898 fusion rule in [13] has beenused for designingthreshold 120578fc(119905 119895) which is given by

120578fc (119905 119895)

= floor(119886 sdot 119899 (119905 119895) + radic119887 sdot 119899 (119905 119895)119876minus1 (1 minus 120573)) (40)

where floor(sdot) is a floor function and 119876minus1(sdot) is the inverse

function of 119876(sdot) which has been introduced in Section 2 120573is a predefined constraint of miss detection 119886 and 119887 are givenby

119886 = [1 minus (119905 119895)] sdot 119901119894119895(1) + (119905 119895) sdot 119909

119894119895(1) (41)

119887 = [1 minus (119905 119895)] sdot 119901119894119895(1) sdot (1 minus 119901

119894119895(1)) + (119905 119895)

sdot 119910119894119895(1) sdot (1 minus 119910

119894119895(1))

(42)

respectively Therefore we get the jth global decision 119911119895(119905)

119911119895(119905) = IF(sum

119894

119889119894119895(119905) gt 120578fc (119905 119895) | 119894 isin 119871 (119905 119895 minus 1)) (43)

In order to evaluate the identifying scheme we define 120574119867119867

120574119867119861

120574119861119867

and 120574119861119861

as the accuracy of identifying Byzantineattackers 120574

119867119867and 120574119867119861

are formulated as accuracy of Honestsensors tagged as Honest and Byzantine respectively Simi-larly 120574

119861119867and 120574119861119861

are defined as accuracy of Byzantine sensorsidentified as Honest and Byzantine respectively We have theequations as follows

120574119867119861

=

119873119867119861

119873

120574119867119867

=

119899 (119905 119870) minus 119873119861119867

119873

120574119861119867

=

119873119861119867

119873

120574119861119861

=

119873 minus 119899 (119905 119870) minus 119873119867119861

119873

(44)

Therein 119873119867119861

is the number of Honest sensors identified asByzantine and the number of Byzantine sensors identified asHonest is denoted as 119873

119861119867 119873119867and 119873

119861have been described

in Section 2

43 Numerical Results In this subsection we analyze theperformance of scheme of identifying Byzantine attackersthrough numerical results There are also 100 sensor nodesthat contain Byzantine and Honest nodes which are con-sidered in our simulation Meanwhile global-final decisionmaking is performed 100 times in our simulationThe powerat the reference point (119889

0= 1) is set as 200 and the signal

amplitude arriving at the local sensor has been contaminatedby AWGN with mean value 120583 = 0 and standard deviation120590 = 3 A target intrudes at position 120579 = (90 90) The trueattacking power is fixed to 03 and 120573 = 02

In Figures 10 11 12 and 13 we show that 119899(119905 119895) decreasesfollowing the increase of 119895 and has stable tendency to a fixedlevel The sum of 119873

0(119905 119895) and 119873

1(119905 119895) is always equal to

119870 sdot 119899(119905 119895) and 1198731(119905 119895) is greater than 119873

0(119905 119895) 119873

0(119905 119895) and

1198731(119905 119895) verge to119870sdot119873

119861and119870sdot119873

119867 respectively And (119905 119895) rarr

0when 119895 exceeds a certain number and (119905 119895) = 0when 119895 ge 7

in the scheme of identifying Byzantine attackers Thresholdfor global decision making varies with 119895 and 120578fc(119905 119895) vergesto constant value when (119905 119895) is equal to zero The attackingpower is estimated after each global decision making Part ofsensors is judged to be Byzantine from 119899(119905 119895) sensorsThe leftsensors then participate in next global decisionmakingWiththe conducting of global decision making there are no sen-sors tagged as Byzantine until jth reaches a certain numberAt the FC the total number of sensors tagged as Byzantine iscomputed as119873minus119899(119905 119870) We plot 120574

119867119867 120574119861119861 120574119867119861

and 120574119861119867

ver-sus 119905 in Figures 14 and 15 under different cases As illustratedin Figures 14 and 15 120574

119861119861is close to 03 which is black line with

cross And 120574119867119867

is close to 07 which is denoted as blue linewith cross 120574

119861119867and 120574119867119861

are presented as black line with starand blue line with star in the Figures 14 and 15 respectively Itis clear that few Byzantine sensors are judged to be Honestby this identifying scheme In other words the Byzantineattackers are almost found out We also can find that thescheme performs well at identifying Byzantine attackers

5 Conclusion

We have divided the process of data fusion into three hier-archiesstages in this paper In addition two problems havebeen put forward at the first and second stage respectivelyAt the first stage we proposed neighborhood maliciousByzantine attacks model Distributed detection under neigh-borhood malicious Byzantine attacks has been taken intoconsideration We have shown that NMBA is an intelligentstrategy adopted by an adversary The attacking performanceof NMBA has also been analyzed We have also provedthat attacking effect of NMBA matches with CMBArsquos whenthe attacking power is lower 03844 and NMBA alwaysoutperforms IMBA in any case It has been analyzed thatdata fusion is incapable when the attacking power enters theblinding region and the closed-form expression for blindingregion has been derived At the second stage a data fusionrule based on dynamic threshold has been put forward andwe have proposed an effective way of identifying Byzantineattackers Consequently we have shown that most Byzantineattackers are identified through the scheme and significantimprovement of performance of this scheme in terms of


0

10

20

30

40

50

60

70

80

90

100n(tj)

2 3 4 5 6 7 8 9 101j

(a) 119899(119905 119895) decreases with the increment of 119895 and converges to a stablelevel when 119895 ge 6

0

100

200

300

400

500

600

N1(tj)

2 3 4 5 6 7 8 9 101j

(b) 1198731(119905 119895) decreases with 119895 and converges to a stable level when 119895 ge 6

0

50

100

150

200

250

300

350

400

450

N0(tj)

2 3 4 5 6 7 8 9 101j

(c) 1198730(119905 119895) decreases with 119895 and converges to a stable level when 119895 ge 6

0

10

20

30

40

50

60

70

2 3 4 5 6 7 8 9 101j

120578fc(tj)

(d) 120578fc(119905 119895) decreases with 119895 and converges to stable level when 119895 ge 6

0

005

01

015

02

025

03

035

120572(tj)

2 3 4 5 6 7 8 9 101j

(e) 120572(119905 119895) versus 119895The attacking power120572 estimated at jth global decisionmaking decreases with 119895 120572(119905 119895) converges to 0 when 119895 ge 6 It is clear thatmore and more Byzantine sensors are identified with increase of 119895

Figure 10 119899(119905 119895)1198730(119905 119895)119873

1(119905 119895) 120572(119905 119895) and 120578fc(119905 119895) are plotted versus 119895 when119873 = 100119870 = 10119872 = 9


2 3 4 5 6 7 8 9 101j

0

10

20

30

40

50

60

70

80

90

100n(tj)


2 3 4 5 6 7 8 9 101j

0

100

200

300

400

500

600

N1(tj)


0

50

100

150

200

250

300

350

400

450

N0(tj)

2 3 4 5 6 7 8 9 101j


2 3 4 5 6 7 8 9 101j

0

10

20

30

40

50

60

70

120578fc(tj)


0

005

01

015

02

025

03

035

120572(tj)

2 3 4 5 6 7 8 9 101j


Figure 11 119899(119905 119895)1198730(119905 119895)119873



2 3 4 5 6 7 8 9 10 11 12 13 14 151j

0

10

20

30

40

50

60

70

80

90

100n(tj)


0

100

200

300

400

500

600

700

800

900

N1(tj)

2 3 4 5 6 7 8 9 10 11 12 13 14 151j


2 3 4 5 6 7 8 9 10 11 12 13 14 151j

0

100

200

300

400

500

600

700

N0(tj)


0

10

20

30

40

50

60

70

2 3 4 5 6 7 8 9 10 11 12 13 14 151j

120578fc(tj)


2 3 4 5 6 7 8 9 10 11 12 13 14 151j

0

005

01

015

02

025

03

035

120572(tj)


Figure 12 119899(119905 119895)1198730(119905 119895)119873



0

10

20

30

40

50

60

70

80

90

100n(tj)

2 3 4 5 6 7 8 9 10 11 12 13 14 151j


0

100

200

300

400

500

600

700

800

900

N1(tj)

2 3 4 5 6 7 8 9 10 11 12 13 14 151j


0

100

200

300

400

500

600

700

N0(tj)

2 3 4 5 6 7 8 9 10 11 12 13 14 151j


0

10

20

30

40

50

60

70

2 3 4 5 6 7 8 9 10 11 12 13 14 151j

120578fc(tj)


0

005

01

015

02

025

03

035

04

120572(tj)

2 3 4 5 6 7 8 9 10 11 12 13 14 151j


Figure 13 119899(119905 119895)1198730(119905 119895)119873



0

01

02

03

04

05

06

07

20 40 60 80 1000(a) 119870 = 10119872 = 9

0

01

02

03

04

05

06

07

20 40 60 80 1000(b) 119870 = 10119872 = 25

Figure 14 120574119867119867

120574119861119861 120574119867119861 and 120574

119861119867versus 119905 120574

119867119867 120574119861119861 120574119867119861 and 120574

119861119867vibrate with 119905 in different cases of119872 and119870 120574

119861119861is very close to 03 and 120574

119867119867

vibrates under 07 120574119861119867

is very close to 0 and 120574119867119861

vibrates above 0

20 40 60 80 10000

01

02

03

04

05

06

07

(a) 119870 = 15119872 = 920 40 60 80 1000

0

01

02

03

04

05

06

07

(b) 119870 = 15119872 = 25

Figure 15 120574119867119867

120574119861119861 120574119867119861 and 120574

119861119867versus 119905 120574

119867119867 120574119861119861 120574119867119861 and 120574



119867119867

vibrates under 07 120574119861119867


vibrates above 0

accuracy is obtained Based on the scheme of identifyingByzantine attackers a data fusion scheme with dynamicthreshold has been explored at this stage

Appendices

A Proof for Proposition 1

If Byzantine attackers always know the true hypothesis eachdecision is flipped by Byzantine attacker with probability of1 prior to sending it to the FC [8 9] Because of 119875

119898=

119875fa = 0 each Honest sensor can detect the natural statecorrectly It is clear that the attacking power is119873

0(119905)119873when

global-final decision is ldquo1rdquo Similarly the attacking power is1198731(119905)119873 when global-final decision is ldquo0rdquo In other words

min1198730(119905)1198731(119905) is always the number of decisions from

Byzantine sensors Therefore the attacking power can beestimated through expression

(119905) =

min 1198730(119905) 119873

1(119905)

119873

(A1)

B Proof for Proposition 2

The way that attacking power is estimated through the ratiobetween the number of decisions that are different fromglobal-final decision to the total number of decisions isadopted Under the condition that the network has notbeen blinded completely several Byzantine sensor nodesmay have made mistakes and then behaved like Honestsensors Similarly some Honest sensors may have behaved


like Byzantine for example situation of 119875119898

= 119875fa = 0Therefore we have

min 1198730(119905 119895) 119873

1(119905 119895)

119899 (119905 119895) sdot 119870

= 120572 (119905 119895) +

119873119861

01(119905 119895)

119899 (119905 119895) sdot 119870

+

119873119867

01(119905 119895)

119899 (119905 119895) sdot 119870

(B1)

In (B1)11987311986101(119905 119895) is denoted as the total number of decisions

that come from Byzantine attackers and are identical withthe FCrsquos global-final decision 119873119867

01(119905 119895) in (B1) is denoted as

the total number of decisions that come fromHonest sensorsand are different from the FCrsquos global-final decision 120572(119905 119895) isthe ratio of the number of decisions that are from Byzantineattackerswhich have attacked successfully to the total numberof decisions Equation (B1) can be represented as

min 1198730(119905 119895) 119873

1(119905 119895)

119899 (119905 119895) sdot 119870

= 120572 (119905 119895) +

119873119861

01(119905 119895)

119899 (119905 119895) sdot 119870

+ 119902119894119895(1)

(B2)

in the absence of target or

min 1198730(119905 119895) 119873

1(119905 119895)

119899 (119905 119895) sdot 119870

= 120572 (119905 119895) +

119873119861

01(119905 119895)

119899 (119905 119895) sdot 119870

+ 1 minus 119901119894119895(1)

(B3)

in the presence of target where 119901119894119895(1) and 119902

119894119895(1) are the

estimation of probability of detection and false alarm respec-tively Therefore they are given by

119902119894119895(1) =

119873119867

01(119905 119895)

119899 (119905 119895) sdot 119870

or 119901119894119895(1) = 1 minus

119873119867

01(119905 119895)

119899 (119905 119895) sdot 119870

(B4)

In the perspective of decider at the FC it is unknown whichsensor is Byzantine However all the Byzantine attackerspossess high efficiency of attacking according to (14) in thecontext of adopting strategy of NMBATherefore we have

(119905 119895) =

min 1198730(119905 119895) 119873

1(119905 119895)

119899 (119905 119895) sdot 119870

minus 119902119894119895(1) (B5)

or

(119905 119895) =

min 1198730(119905 119895) 119873

1(119905 119895)

119899 (119905 119895) sdot 119870

+ 119901119894119895(1) minus 1 (B6)

Because of

119901119894119895(1) ge 1 minus

max119896(1003816100381610038161003816d119905119896

1003816100381610038161003816) minusmin

119896(1003816100381610038161003816d119905119896

1003816100381610038161003816)

119899 (119905 119895)

(B7)

or

119902119894119895(1) le

max119896(1003816100381610038161003816d119905119896

1003816100381610038161003816) minusmin

119896(1003816100381610038161003816d119905119896

1003816100381610038161003816)

119899 (119905 119895)

(B8)

we get

(119905 119895)

ge

min 1198730(119905 119895) 119873

1(119905 119895) minus 119870 sdot [max

119895(

10038161003816100381610038161003816d119905119895

10038161003816100381610038161003816) minusmin

119895(

10038161003816100381610038161003816d119905119895

10038161003816100381610038161003816)]

119899 (119905 119895) sdot 119870

(B9)

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The authors specifically acknowledge the financial supportthrough the National Science Foundation of China (Grantno 61001086) the Fundamental Research Funds for theCentral University (Grant no ZYGX2011X004) and theproject Sponsored by Academic Training Funds Universityof Electronic Science and Technology of China (Grant no201506075013)

References

[1] J-F Chamberland and V V Veeravalli ldquoWireless sensors in dis-tributed detection applicationsrdquo IEEE Signal Processing Maga-zine vol 24 no 3 pp 16ndash25 2007

[2] L Lamport R Shostak and M Pease ldquoThe byzantine generalsproblemrdquo ACM Transactions on Programming Languages andSystems vol 4 no 3 pp 382ndash401 1982

[3] H Chen V P Jilkov and X R Li ldquoOptimizing decision fusionin the presence of byzantine datardquo in Proceedings of the 17thInternational Conference on Information Fusion pp 1ndash8 July2014

[4] P Stoica andANehorai ldquoPerformance study of conditional andunconditional direction-of-arrival estimationrdquo IEEE Transac-tions on Acoustics Speech and Signal Processing vol 38 no 10pp 1783ndash1795 1990

[5] D Liu K Liu Y Ma and J Yu ldquoJoint TOA and DOA localiza-tion in indoor environment using virtual stationsrdquo IEEE Com-munications Letters vol 18 no 8 pp 1423ndash1426 2014

[6] S Althunibat and F Granelli ldquoAn objection-based collaborativespectrum sensing for cognitive radio networksrdquo IEEE Commu-nications Letters vol 18 no 8 pp 1291ndash1294 2014

[7] X Lu P Wang D Niyato and E Hossain ldquoDynamic spectrumaccess in cognitive radio networks with RF energy harvestingrdquoIEEE Wireless Communications vol 21 no 3 pp 102ndash110 2014

[8] S Marano V Matta and L Tong ldquoDistributed detection in thepresence of Byzantine attacksrdquo IEEE Transactions on Signal Pro-cessing vol 57 no 1 pp 16ndash29 2009

[9] S Marano V Matta and L Tong ldquoDistributed inferencein the presence of byzantine sensorsrdquo in Proceedings of the40th Asilomar Conference on Signals Systems and Computers(ACSSC rsquo06) pp 281ndash284 Pacific Grove Calif USA November2006


[10] A Vempaty O Ozdemir K Agrawal H Chen and P K Varsh-ney ldquoLocalization in wireless sensor networks byzantines andmitigation techniquesrdquo IEEE Transactions on Signal Processingvol 61 no 6 pp 1495ndash1508 2013

[11] A S Rawat P Anand H Chen and P K Varshney ldquoCollab-orative spectrum sensing in the presence of byzantine attacksin cognitive radio networksrdquo IEEE Transactions on SignalProcessing vol 59 no 2 pp 774ndash786 2011

[12] X F He H Y Dai and P Ning ldquoA Byzantine attack defenderthe conditional frequency checkrdquo in Proceedings of the IEEEInternational Symposium on Information Theory (ISIT rsquo12) pp975ndash979 IEEE Cambridge Mass USA July 2012

[13] M Abdelhakim L E Lightfoot J Ren and T Li ldquoDistributeddetection in mobile access wireless sensor networks underbyzantine attacksrdquo IEEE Transactions on Parallel and Distri-buted Systems vol 25 no 4 pp 950ndash959 2013

[14] E Soltanmohammadi M Orooji and M Naraghi-PourldquoDecentralized hypothesis testing in wireless sensor networksin the presence of misbehaving nodesrdquo IEEE Transactions onInformation Forensics and Security vol 8 no 1 pp 205ndash2152012

[15] B Kailkhura S Brahma Y S Han and P K Varshney ldquoDis-tributed detection in tree topologies with byzantinesrdquo IEEETransactions on Signal Processing vol 62 no 12 pp 3208ndash32192014

[16] S Talarico N A Schmid M Alkhweldi and M C ValentildquoDistributed estimation of a parametric field algorithms andperformance analysisrdquo IEEE Transactions on Signal Processingvol 62 no 5 pp 1041ndash1053 2014

[17] P Zhang J Y Koh S Lin and I Nevat ldquoDistributed eventdetection under byzantine attack in wireless sensor networksrdquoin Proceedings of the 9th IEEE International Conference onIntelligent Sensors Sensor Networks and Information Processing(ISSNIP rsquo14) pp 1ndash6 IEEE Singapore April 2014

[18] Q Liu X Wang and N S V Rao ldquoFusion of state estimatesover long-haul sensor networks with random loss and delayrdquoIEEEACM Transactions on Networking vol 23 no 2 pp 644ndash656 2015

[19] M H El-Ayadi ldquoNonstochastic adaptive decision fusion in dis-tributed-detection systemsrdquo IEEE Transactions on Aerospaceand Electronic Systems vol 38 no 4 pp 1158ndash1171 2002

[20] R Laxhammar and G Falkman ldquoOnline learning and sequen-tial anomaly detection in trajectoriesrdquo IEEE Transactions onPattern Analysis and Machine Intelligence vol 36 no 6 pp1158ndash1173 2013

[21] Y Chen J Yang W Trappe and R P Martin ldquoDetecting andlocalizing identity-based attacks in wireless and sensor net-worksrdquo IEEE Transactions on Vehicular Technology vol 59 no5 pp 2418ndash2434 2010

[22] R Niu and P K Varshney ldquoTarget location estimation in sensornetworks with quantized datardquo IEEE Transactions on SignalProcessing vol 54 no 12 pp 4519ndash4528 2006

[23] T M Cover and J A Thomas Elements of Information TheoryJohn Wiley amp Sons New York NY USA 1991

[24] S Kullback InformationTheory and Statistics Dover NewYorkNY USA 1997

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014


Active and Passive Electronic Components

Control Scienceand Engineering

Journal of



RotatingMachinery


Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design



Shock and Vibration


Civil EngineeringAdvances in

Acoustics and VibrationAdvances in



Electrical and Computer Engineering

Journal of

Advances inOptoElectronics


Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of


Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Chemical EngineeringInternational Journal of Antennas and

Propagation




Navigation and Observation



DistributedSensor Networks



used to detect target Therefore convenience of the energy-based method gives feasibility to detect and localize a targetthrough detecting energy In [6 7] each cognitive radioadopts an effective detection scheme based harvesting energytomake its decision and determinewhether there is a licenseduser requesting to occupy the spectrum band Each cognitiveradio sensor detects intensity of signal and measures theenergy received If the measured energy is greater thana properly predefined threshold the current spectrum isbusy Otherwise the current spectrum is idle We adopt thisscheme in this paper to detect whether there is a target in theROI

Marano et al have investigated distributed detection inthe presence of Byzantine attacks in [8 9] where Byzantineattackers are assumed to have a complete knowledge aboutthe true hypotheses In their system model the authorsassumed that the FC did not know which sensor node wasByzantine attacker Though the FC did not know which sen-sor node was Byzantine attacker it knew average percentageof compromised sensor nodes or upper bound of the averageVempaty et al have also analyzed the problem of distributeddetection under Byzantine attacks in [10] However theydid not assume that the Byzantine sensors had a completeknowledge about true hypotheses Instead they assumed thatthe Byzantine sensors made decisions about the presence oftarget through their own observations In other words eachByzantine sensor potentially flips the local decision made atthe nodeThe performance of network has also been analyzedin the context of presence of independent and collaborativeByzantine attacks respectively in [10] In addition to theanalysis of distributed detection in the presence of Byzantineattacks an adaptive learning scheme that mitigated the effectbrought by Byzantine attackers has been proposed by Vem-paty et al in [10] The scheme identifies Byzantine sensorsthrough observing the ratio value of deviations between theestimated behavior of ith sensor and the expected behaviorof an Honest sensor to the estimated behavior of ith sensorand the expected behavior of a Byzantine sensor If the valueis greater than 1 the FC declares the sensor to be ByzantineOtherwise the FC tags the sensor as Honest In order tomaximize the performance of decision fusion the FC thenremoves those decisions that are from sensors tagged asByzantine when the FC makes a global decision at the nexttime

In literature [10 11] analysis of performance of thenetwork under independent and collaborative maliciousByzantine attacks has been performed respectively When atarget enters into themonitoring region the ith compromisedsensor node has a detection and makes a localoriginaldecision In the case of independent malicious Byzantineattacks (IMBA) the local decision of ith compromised sensorcompletely depends on its own observation In the case ofcollaborative Byzantine attacks (CMBA) the local decision ofith compromised sensor depends on not only its own obser-vation but also decisions from the remaining compromisedsensors In the scenario of IMBA though Byzantine sensorcanmakemultiple observations in a proper time window andaverage these observations in order to reduce the effect ofadditive noise it is not effective significantly when Byzantine

sensor nodes stay far away from a target in the ROI Inaddition a sensor that adopts IMBA has no way to conquera certain degree of blind flipping of its own decision In thesituation of CMBA each Byzantine sensor communicateswith the left compromised sensors and refers to their deci-sions before determining its own local decision AlthoughCMBAproduces remarkable improvement in attacking effectmuch energy has been consumed in communication amongByzantine sensors especially when Byzantine sensor nodesare deployed sparsely In addition to analysis of performanceof distributed detection in the presence of Byzantine attacksthe blinding attacking power 120572 has been obtained Theblinding attacking power (120572blind) is equal to 05 and 035 inthe case of IMBA and CMBA respectively [11] Motivated bythis we propose a new Byzantine attacks model named afterneighborhood malicious Byzantine attacks (NMBA) NMBAis such a kind of attacks model where each compromisedsensor node in the network determines whether there is atarget or not in the ROI depending on not only its own localdecision but also a certain amount of decisions coming fromHonest sensors which are nearest around the compromisedsensor Then each Byzantine sensor node employs a majoritystrategy among decisions to make a final local decision Atlast the Byzantine sensor node flips the final local decisionconfidently and sends false decision to the FC

Attacking power 120572 which is also termed indicator of thevulnerability of the sensor networks is a crucial performancemetric [12 13] We assume that there are a large numberof sensor nodes deployed in the ROI According to thelaw of large numbers 120572 is equal to the ratio of number ofcompromised sensors to the total number of sensors in thenetwork (120572 isin [0 1]) In practice although the FC knows thepresence of Byzantine sensors in the network decider at theFC can hardly determine the exact attacking power [14] Ifthe decider knows the attacking power it is convenient forthe FC to adopt a robust strategy to mitigate the negativeeffect caused by Byzantine attackers [15 16] In this paperwe propose a simple and effective scheme to determine theattacking power in the perspective of decider at the FC Afterthe attacking power is estimated two kinds of discrepancydistance which are used to help in identifying Byzantinesensors are constructed in this paper

An effective scheme of decision fusion plays an importantrole in the FC [17ndash19] Many literatures focused on themitigation of Byzantine attacks and developed algorithms todesign a static and identical threshold for decision making atthe FC [11 14] Several authors have proposed online learningof normal trajectory patterns for detection in trajectory in[20] In this paper we propose an effective scheme based onboth dynamic threshold and identifying Byzantine attackersfor decision fusion at the FC

The paper is organized as follows In Section 2 wedescribe our system model including detection model andNMBA attackingmodel In order to formulate the problem ofdistributed detection in WSNs clearly we divide the processof decision fusion into three hierarchies or stages in thissection The attacks model of NMBA is proposed at thefirst stage which is different from independent Byzantineattacks and collaborative Byzantine attacksThe performance



2 System Model


119905 119910119905) where 119909

119905and 119910



1198862

119894= 1198750(

1198890

119889E119894)

119899

(1)


0



119889E119894 = radic(119909119905minus 119909119894)2

+ (119910119905minus 119910119894)2

119894 = 1 2 119873 (2)




1198862

119894=

1198750

(119889E119894)2 119894 = 1 2 119873 (3)


119894= 119886119894+ 119899119894 in which 119899

119894is


119894sim 119873(120583 120590

2)

119894 = 1 2 119873

50 100 150 2000X-coordinate (m)

0

20

40

60

80

100

120

140

160

180

200

Y-c

oord

inat

e (m

)



119889119894=

1 119903119894gt 120589119894

0 119903119894lt 120589119894

(4)




0(target is absent)



119889119894isin 0 1 119894 = 1 2 119873




119889119894in our


119867and 119873







119889(119894) = Pr( 119889

119894= 1 | 119867

1 119867) and 119875119867fa (119894) =

Pr( 119889119894= 1 | 119867



119867 The


119875119867

119889(119894) = Pr ( 119889

119894= 1 | 119867

1 119867) = Pr (119886

119894+ 119899119894gt 120589119894)

= 119876(

120589119894minus 119886119894minus 120583

120590

)

(5)


119875119867

fa (119894) = Pr (119899119894gt 120589119894) = 119876(

120589119894minus 120583

120590

) (6)


119876 (119909) = int

infin

119909

1

radic2120587

119890minus11990522119889119905 (7)




d119894= (

11988911989411198891198942




1 d2 d

119873]T is


where 119889119894119895isin 0 1




119894into the FC



1 1199112 119911




119894minus 1 (119894 = 1 2 119873

119861)

Natural state

The fusion center

d1 d2 dNminus1 dN

d1 d2 dNminus1 dN


DNtimesK rarr z

S1 S2 SNminus1 SN

z rarr z



119894 = 1 2 119873 D119873times119870




119894= 119872 =



1198670 V119894119895(119896) = Pr (119889

119894119895= 119896 | 119867

0)

= (1 minus 120572)Pr (119889119894119895= 119896 | 119867

0 119867)

+ 120572Pr (119889119894119895= 119896 | 119867

0 119861)

1198671 119906119894119895(119896) = Pr (119889

119894119895= 119896 | 119867

1)

= (1 minus 120572)Pr (119889119894119895= 119896 | 119867

1 119867)

+ 120572Pr (119889119894119895= 119896 | 119867

1 119861)

119894 isin 1 2 119873 119895 isin 1 2 119870 119896 isin 0 1

(8)


0

20

40

60

80

100

120

140

160

180

200

Y-c

oord

inat

e (m

)

50 100 150 2000X-coordinate (m)



0and119867

1 respectively

Therefore we have

1198670 Pr ( 119889

119894119895= 119896 | 119867

0 119867) = 119902

119894119895(119896)

1198671 Pr ( 119889

119894119895= 119896 | 119867

1 119867) = 119901

119894119895(119896)

119894 = 1 2 119873119867

(9)


119901119894119895(1) = 119876(

120589 minus 119886119894minus 120583

120590

)

119901119894119895(0) = 1 minus 119901

119894119895(1)

119902119894119895(1) = 119876(

120589 minus 120583

120590

)

119902119894119895(0) = 1 minus 119902

119894119895(1)

(10)


1198670 Pr ( 119889

119894119895= 119896 | 119867

0 119861) = 119910

119894119895(119896)

1198671 Pr ( 119889

119894119895= 119896 | 119867

1 119861) = 119909

119894119895(119896)

119894 = 1 2 119873119861

(11)



119894minus1


119894119897 119897 isin 0 1 119872

119894minus 1 119888





119897=0119888119894119897gt 120578119894) where



119910119894119895(119896) =

119872119894

sum

119898=120578119894

sum

120587isinΓ

119898

prod

119894=1

119902120587(119894)119895

(119896)

119872119894

prod

119894=119898+1

(1 minus 119902120587(119894)119895

(119896))

119909119894119895(119896) =

119872119894

sum

119898=120578119894

sum

120587isinΓ

119898

prod

119894=1

119901120587(119894)119895

(119896)

119872119894

prod

119894=119898+1

(1 minus 119901120587(119894)119895

(119896))

(12)

where119872119894isin 1 2 119873

119867 1 le 120578 le 119872




Pr (119889119894119895= 119896 |

119889119894119895= 119897 119861) =

1 when 119897 = 119896

0 when 119897 = 119896

119896 119897 isin 0 1

(13)

Thus we get

Pr (119889119894119895= 119896 | 119867

0 119861) = sum

119897 =119896

Pr ( 119889119894119895= 119897 | 119867

0 119861)

sdot Pr (119889119894119895= 119896 |

119889119894119895= 119897119867

0 119861)

+sum

119897=119896

Pr ( 119889119894119895= 119897 | 119867

0 119861)

sdot Pr (119889119894119895= 119896 |

119889119894119895= 119897119867

0 119861)

Pr (119889119894119895= 119896 | 119867

1 119861) = sum

119897 =119896

Pr ( 119889119894119895= 119897 | 119867

1 119861)

sdot Pr (119889119894119895= 119896 |

119889119894119895= 119897119867

1 119861)

+sum

119897=119896

Pr ( 119889119894119895= 119897 | 119867

1 119861)

sdot Pr (119889119894119895= 119896 |

119889119894119895= 119897119867

1 119861)

(14)

Therefore we have

Pr (119889119894119895= 0 | 119867

0 119861) = Pr ( 119889

119894119895= 1 | 119867

0 119861) = 119910

119894119895(1)

Pr (119889119894119895= 1 | 119867

0 119861) = Pr ( 119889

119894119895= 0 | 119867

0 119861) = 119910

119894119895(0)

Pr (119889119894119895= 0 | 119867

1 119861) = Pr ( 119889

119894119895= 1 | 119867

1 119861) = 119909

119894119895(1)

Pr (119889119894119895= 1 | 119867

1 119861) = Pr ( 119889

119894119895= 0 | 119867

1 119861) = 119909

119894119895(0)

(15)



V119894119895(119896) = Pr (119889

119894119895= 119896 | 119867

0)


119894119895(119896))]

+ 120572119910119894119895(1 minus 119896)

119906119894119895(119896) = Pr (119889

119894119895= 119896 | 119867

1)


119894119895(119896))]

+ 120572119909119894119895(1 minus 119896)

(16)


119894119895(119896) = Pr(119889

119894119895=

119896 | 1198670) and 119906

119894119895(119896) = Pr(119889

119894119895= 119896 | 119867


be expressed as

KLD (119906119894119895 V119894119895) = sum

119896isin01

119906119894119895(119896) log

119906119894119895(119896)

V119894119895(119896)

(17)


119894119895(119896) under

1198670and 119867









119906119894119895(119896) = Pr (119889

119894119895= 119896 | 119867

1) = Pr (119889

119894119895= 119896 | 119867

0)

= V119894119895(119896)

(18)


119894119895(119896) 119906

119894119895(119896)) = 0 is equivalent to

120572 =

119901119894119895(1) minus 119902

119894119895(1)

(119901119894119895(1) minus 119902

119894119895(1)) + (119909

119894119895(1) minus 119910

119894119895(1))

=

119901119894119895(1) minus 119902

119894119895(1)

(119901119894119895(1) minus 119902

119894119895(1)) + 119891 (119872

119894 120578119894 119901119894119895(1) 119902119894119895(1))

(19)


119891 (119872119894 120578119894 119901119894119895(1) 119902119894119895(1))

=

119872119894

sum

119898=120578119894

sum

120587isinΓ

119898

prod

119894=1

119901120587(119894)119895

(1)

119872119894

prod

119894=119898+1

(1 minus 119901120587(119894)119895

(1))

minus

119872119894

sum

119898=120578119894

sum

120587isinΓ

119898

prod

119894=1

119902120587(119894)119895

(1)

119872119894

prod

119894=119898+1

(1 minus 119902120587(119894)119895

(1))

(20)



119894119895(119896) and


119894119895 119906119894119895) = 0 if and only if

V119894119895(119896) = 119906







119894119895(1) 119902119894119895(1))

120572blind

=

119901119894119895(1) minus 119902

119894119895(1)

(119901119894119895(1) minus 119902

119894119895(1)) +max 119891 (119872

119894 120578119894 119901119894119895(1) 119902119894119895(1))

(22)



inequality

120572blind le1

1 +max 119891 (119872119894 120578119894 119901119894119895(1) 119902119894119895(1))

(23)


119889

119889119909

(

119909

119909 + 1

) =

1

(119909 + 1)2gt 0 (24)


120578119894 opt = ceil(

119872119894

2

) (25)


2 3 4 5 6 7 8 91120578

0

01

02

03

04

05

06

07

08

09

120572



119894=


119894119895(1)

119902119894119895(1)) we have

max119891 = 119891(119872119894 ceil(

119872119894

2

) 119901119894119895(1) 119902119894119895(1)) (26)


120572blind =119901119894119895(1) minus 119902

119894119895(1)

(119901119894119895(1) minus 119902

119894119895(1)) +max119891

(27)






119894and 120578 = 120578

119894 opt 119894 = 1 2 119873119861 Attacking




0

05

1

15

2

25

3

35

4

45

5

120578op

t

2 3 4 5 6 7 8 91M






119872120572

120578

1 2 3 4 5 6 7 8 91 050002 05000 050003 05434 04310 054344 05952 04310 04310 059525 06469 04509 04042 04509 064696 06959 04785 04042 04042 04785 069597 07410 05100 04145 03913 04145 05100 074108 07814 05438 04298 03913 03913 04298 05438 078149 08171 05790 04484 03970 03844 03970 04484 05790 08171






119873times119870= (119889

119894119895(119905))119873times119870



119895(119905) over vector


decision z(119905) = (1199111(119905) 1199112(119905) 119911


d1199051 d1199052 d119905

119870 119911119895(119905) isin 0 1 for any 119905 isin 119873+ 119895 = 1 2 119870


0

005

01

015

02

025

03

035

04

045

05

120572bl

ind

2 3 4 5 6 7 8 91M


0

005

01

015

02

025

03

035

04

045

05

120572bl

ind

3 5 7 9 11 13 15 17 19 21 23 251M




119872120578opt 11 21 32 42 53 63 74 84 95120572blind 05000 05000 04310 04310 04042 04042 03913 03913 03844

IMBANMBACMBA

C B A0

02

04

06

08

1

12

14

KLD

(120572)

01 02 03 04 050Attacking power 120572



119895(119905)




10038161003816100381610038161003816d119905119895

10038161003816100381610038161003816=

119873

sum

119894=1

10038161003816100381610038161003816119889119894119895(119905) minus 119911 (119905)

10038161003816100381610038161003816 (28)


10038161003816100381610038161003816d119905119894

10038161003816100381610038161003816=

119870

sum

119895=1

10038161003816100381610038161003816119889119894119895(119905) minus 119911 (119905)

10038161003816100381610038161003816 (29)





IMBANMBACMBA

C B A01 02 03 04 050


0

02

04

06

08

1

12

14

KLD

(120572)






119895| is small





0(119905 119895) and



1(119905 119895)


10038161003816100381610038161003816d119905119895

10038161003816100381610038161003816= sum

119894isin119868(119905119895)

10038161003816100381610038161003816119889119894119895(119905) minus 119911 (119905)

10038161003816100381610038161003816

10038161003816100381610038161003816d119905119894

10038161003816100381610038161003816=

119870

sum

119896=1

1003816100381610038161003816119889119894119896(119905) minus 119911 (119905)

1003816100381610038161003816 119894 isin 119868 (119905 119895)

(30)




119898= 119875119891119886


(119905) =

min 1198730(119905) 119873

1(119905)

119873

(31)

where1198730(119905) and119873





(119905 119895) =

min 1198730(119905 119895) 119873

1(119905 119895)

1198730(119905 119895) + 119873

1(119905 119895)

(32)





(119905 119895)

ge

min 1198730(119905 119895) 119873

1(119905 119895) minus 119870 sdot [max

119896(1003816100381610038161003816d119905119896

1003816100381610038161003816) minusmin

119896(1003816100381610038161003816d119905119896

1003816100381610038161003816)]

119899 (119905 119895) sdot 119870

119896 isin 1 2 119870

(33)


[max119896(|d119905119896|) minus min




Here we let

(119905 119895)

=

min 1198730(119905 119895) 119873

1(119905 119895) minus 119870 sdot [max

119895(

10038161003816100381610038161003816d119905119895

10038161003816100381610038161003816) minusmin

119895(

10038161003816100381610038161003816d119905119895

10038161003816100381610038161003816)]

119899 (119905 119895) sdot 119870

(34)




119873(119905 119895) over |d119905






119868119873(119905 119895)

=

10038161003816100381610038161003816d119905119894

10038161003816100381610038161003816

119899(119905119895)

10038161003816100381610038161003816d119905119894

10038161003816100381610038161003816

119899(119905119895)minus1

10038161003816100381610038161003816d119905119894

10038161003816100381610038161003816

1

| 119894 isin 119871 (119905 119895 minus 1)

(35)




over 119868119873(119905 119895)

119899 = ID (d119905119894)

= sum

119897isin119871(119905119895minus1)

IF (10038161003816100381610038161003816d119905119897

10038161003816100381610038161003816gt

10038161003816100381610038161003816d119905119894

10038161003816100381610038161003816

119899

)

+ sum

119897isin119871(119905119895minus1)

IF (10038161003816100381610038161003816d119905119897

10038161003816100381610038161003816=

10038161003816100381610038161003816d119905119894

10038161003816100381610038161003816

119899

| 119897 gt 119895)

119899 isin 1 2 119899 (119905 119895)

(36)




119868 (119905 119895)


(37)


(119905) =

119873 minus 119899 (119905 119870)

119873

(38)






1(119905)

1199112(119905) 119911


decision d1199051 d1199052 d119905






120578fc (119905 119895)




119886 = [1 minus (119905 119895)] sdot 119901119894119895(1) + (119905 119895) sdot 119909

119894119895(1) (41)


119894119895(1)) + (119905 119895)

sdot 119910119894119895(1) sdot (1 minus 119910

119894119895(1))

(42)


119911119895(119905) = IF(sum

119894

119889119894119895(119905) gt 120578fc (119905 119895) | 119894 isin 119871 (119905 119895 minus 1)) (43)


120574119867119861

120574119861119867

and 120574119861119861


119867119867and 120574119867119861


119861119867and 120574119861119861


120574119867119861

=

119873119867119861

119873

120574119867119867

=

119899 (119905 119870) minus 119873119861119867

119873

120574119861119867

=

119873119861119867

119873

120574119861119861

=

119873 minus 119899 (119905 119870) minus 119873119867119861

119873

(44)

Therein 119873119867119861


119861119867 119873119867and 119873


in Section 2





0(119905 119895) and 119873



0(119905 119895) 119873

0(119905 119895) and

1198731(119905 119895) verge to119870sdot119873

119861and119870sdot119873




119867119867 120574119861119861 120574119867119861

and 120574119861119867



cross And 120574119867119867


119861119867and 120574119867119861


5 Conclusion



0

10

20

30

40

50

60

70

80

90

100n(tj)

2 3 4 5 6 7 8 9 101j


0

100

200

300

400

500

600

N1(tj)

2 3 4 5 6 7 8 9 101j


0

50

100

150

200

250

300

350

400

450

N0(tj)

2 3 4 5 6 7 8 9 101j


0

10

20

30

40

50

60

70

2 3 4 5 6 7 8 9 101j

120578fc(tj)


0

005

01

015

02

025

03

035

120572(tj)

2 3 4 5 6 7 8 9 101j


Figure 10 119899(119905 119895)1198730(119905 119895)119873



2 3 4 5 6 7 8 9 101j

0

10

20

30

40

50

60

70

80

90

100n(tj)


2 3 4 5 6 7 8 9 101j

0

100

200

300

400

500

600

N1(tj)


0

50

100

150

200

250

300

350

400

450

N0(tj)

2 3 4 5 6 7 8 9 101j


2 3 4 5 6 7 8 9 101j

0

10

20

30

40

50

60

70

120578fc(tj)


0

005

01

015

02

025

03

035

120572(tj)

2 3 4 5 6 7 8 9 101j


Figure 11 119899(119905 119895)1198730(119905 119895)119873



2 3 4 5 6 7 8 9 10 11 12 13 14 151j

0

10

20

30

40

50

60

70

80

90

100n(tj)


0

100

200

300

400

500

600

700

800

900

N1(tj)

2 3 4 5 6 7 8 9 10 11 12 13 14 151j


2 3 4 5 6 7 8 9 10 11 12 13 14 151j

0

100

200

300

400

500

600

700

N0(tj)


0

10

20

30

40

50

60

70

2 3 4 5 6 7 8 9 10 11 12 13 14 151j

120578fc(tj)


2 3 4 5 6 7 8 9 10 11 12 13 14 151j

0

005

01

015

02

025

03

035

120572(tj)


Figure 12 119899(119905 119895)1198730(119905 119895)119873



0

10

20

30

40

50

60

70

80

90

100n(tj)

2 3 4 5 6 7 8 9 10 11 12 13 14 151j


0

100

200

300

400

500

600

700

800

900

N1(tj)

2 3 4 5 6 7 8 9 10 11 12 13 14 151j


0

100

200

300

400

500

600

700

N0(tj)

2 3 4 5 6 7 8 9 10 11 12 13 14 151j


0

10

20

30

40

50

60

70

2 3 4 5 6 7 8 9 10 11 12 13 14 151j

120578fc(tj)


0

005

01

015

02

025

03

035

04

120572(tj)

2 3 4 5 6 7 8 9 10 11 12 13 14 151j


Figure 13 119899(119905 119895)1198730(119905 119895)119873



0

01

02

03

04

05

06

07

20 40 60 80 1000(a) 119870 = 10119872 = 9

0

01

02

03

04

05

06

07

20 40 60 80 1000(b) 119870 = 10119872 = 25

Figure 14 120574119867119867

120574119861119861 120574119867119861 and 120574

119861119867versus 119905 120574

119867119867 120574119861119861 120574119867119861 and 120574



119867119867

vibrates under 07 120574119861119867


vibrates above 0

20 40 60 80 10000

01

02

03

04

05

06

07

(a) 119870 = 15119872 = 920 40 60 80 1000

0

01

02

03

04

05

06

07

(b) 119870 = 15119872 = 25

Figure 15 120574119867119867

120574119861119861 120574119867119861 and 120574

119861119867versus 119905 120574

119867119867 120574119861119861 120574119867119861 and 120574



119867119867

vibrates under 07 120574119861119867


vibrates above 0


Appendices



119898=


0(119905)119873when




(119905) =

min 1198730(119905) 119873

1(119905)

119873

(A1)






min 1198730(119905 119895) 119873

1(119905 119895)

119899 (119905 119895) sdot 119870

= 120572 (119905 119895) +

119873119861

01(119905 119895)

119899 (119905 119895) sdot 119870

+

119873119867

01(119905 119895)

119899 (119905 119895) sdot 119870

(B1)



01(119905 119895) in (B1) is denoted as


min 1198730(119905 119895) 119873

1(119905 119895)

119899 (119905 119895) sdot 119870

= 120572 (119905 119895) +

119873119861

01(119905 119895)

119899 (119905 119895) sdot 119870

+ 119902119894119895(1)

(B2)


min 1198730(119905 119895) 119873

1(119905 119895)

119899 (119905 119895) sdot 119870

= 120572 (119905 119895) +

119873119861

01(119905 119895)

119899 (119905 119895) sdot 119870

+ 1 minus 119901119894119895(1)

(B3)


119894119895(1) are the


119902119894119895(1) =

119873119867

01(119905 119895)

119899 (119905 119895) sdot 119870

or 119901119894119895(1) = 1 minus

119873119867

01(119905 119895)

119899 (119905 119895) sdot 119870

(B4)


(119905 119895) =

min 1198730(119905 119895) 119873

1(119905 119895)

119899 (119905 119895) sdot 119870

minus 119902119894119895(1) (B5)

or

(119905 119895) =

min 1198730(119905 119895) 119873

1(119905 119895)

119899 (119905 119895) sdot 119870

+ 119901119894119895(1) minus 1 (B6)

Because of

119901119894119895(1) ge 1 minus

max119896(1003816100381610038161003816d119905119896

1003816100381610038161003816) minusmin

119896(1003816100381610038161003816d119905119896

1003816100381610038161003816)

119899 (119905 119895)

(B7)

or

119902119894119895(1) le

max119896(1003816100381610038161003816d119905119896

1003816100381610038161003816) minusmin

119896(1003816100381610038161003816d119905119896

1003816100381610038161003816)

119899 (119905 119895)

(B8)

we get

(119905 119895)

ge

min 1198730(119905 119895) 119873

1(119905 119895) minus 119870 sdot [max

119895(

10038161003816100381610038161003816d119905119895

10038161003816100381610038161003816) minusmin

119895(

10038161003816100381610038161003816d119905119895

10038161003816100381610038161003816)]

119899 (119905 119895) sdot 119870

(B9)



Acknowledgments


References




























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











2 System Model


119905 119910119905) where 119909

119905and 119910



1198862

119894= 1198750(

1198890

119889E119894)

119899

(1)


0



119889E119894 = radic(119909119905minus 119909119894)2

+ (119910119905minus 119910119894)2

119894 = 1 2 119873 (2)




1198862

119894=

1198750

(119889E119894)2 119894 = 1 2 119873 (3)


119894= 119886119894+ 119899119894 in which 119899

119894is


119894sim 119873(120583 120590

2)

119894 = 1 2 119873

50 100 150 2000X-coordinate (m)

0

20

40

60

80

100

120

140

160

180

200

Y-c

oord

inat

e (m

)



119889119894=

1 119903119894gt 120589119894

0 119903119894lt 120589119894

(4)




0(target is absent)



119889119894isin 0 1 119894 = 1 2 119873




119889119894in our


119867and 119873







119889(119894) = Pr( 119889

119894= 1 | 119867

1 119867) and 119875119867fa (119894) =

Pr( 119889119894= 1 | 119867



119867 The


119875119867

119889(119894) = Pr ( 119889

119894= 1 | 119867

1 119867) = Pr (119886

119894+ 119899119894gt 120589119894)

= 119876(

120589119894minus 119886119894minus 120583

120590

)

(5)


119875119867

fa (119894) = Pr (119899119894gt 120589119894) = 119876(

120589119894minus 120583

120590

) (6)


119876 (119909) = int

infin

119909

1

radic2120587

119890minus11990522119889119905 (7)




d119894= (

11988911989411198891198942




1 d2 d

119873]T is


where 119889119894119895isin 0 1




119894into the FC



1 1199112 119911




119894minus 1 (119894 = 1 2 119873

119861)

Natural state

The fusion center

d1 d2 dNminus1 dN

d1 d2 dNminus1 dN


DNtimesK rarr z

S1 S2 SNminus1 SN

z rarr z



119894 = 1 2 119873 D119873times119870




119894= 119872 =



1198670 V119894119895(119896) = Pr (119889

119894119895= 119896 | 119867

0)

= (1 minus 120572)Pr (119889119894119895= 119896 | 119867

0 119867)

+ 120572Pr (119889119894119895= 119896 | 119867

0 119861)

1198671 119906119894119895(119896) = Pr (119889

119894119895= 119896 | 119867

1)

= (1 minus 120572)Pr (119889119894119895= 119896 | 119867

1 119867)

+ 120572Pr (119889119894119895= 119896 | 119867

1 119861)

119894 isin 1 2 119873 119895 isin 1 2 119870 119896 isin 0 1

(8)


0

20

40

60

80

100

120

140

160

180

200

Y-c

oord

inat

e (m

)

50 100 150 2000X-coordinate (m)



0and119867

1 respectively

Therefore we have

1198670 Pr ( 119889

119894119895= 119896 | 119867

0 119867) = 119902

119894119895(119896)

1198671 Pr ( 119889

119894119895= 119896 | 119867

1 119867) = 119901

119894119895(119896)

119894 = 1 2 119873119867

(9)


119901119894119895(1) = 119876(

120589 minus 119886119894minus 120583

120590

)

119901119894119895(0) = 1 minus 119901

119894119895(1)

119902119894119895(1) = 119876(

120589 minus 120583

120590

)

119902119894119895(0) = 1 minus 119902

119894119895(1)

(10)


1198670 Pr ( 119889

119894119895= 119896 | 119867

0 119861) = 119910

119894119895(119896)

1198671 Pr ( 119889

119894119895= 119896 | 119867

1 119861) = 119909

119894119895(119896)

119894 = 1 2 119873119861

(11)



119894minus1


119894119897 119897 isin 0 1 119872

119894minus 1 119888





119897=0119888119894119897gt 120578119894) where



119910119894119895(119896) =

119872119894

sum

119898=120578119894

sum

120587isinΓ

119898

prod

119894=1

119902120587(119894)119895

(119896)

119872119894

prod

119894=119898+1

(1 minus 119902120587(119894)119895

(119896))

119909119894119895(119896) =

119872119894

sum

119898=120578119894

sum

120587isinΓ

119898

prod

119894=1

119901120587(119894)119895

(119896)

119872119894

prod

119894=119898+1

(1 minus 119901120587(119894)119895

(119896))

(12)

where119872119894isin 1 2 119873

119867 1 le 120578 le 119872




Pr (119889119894119895= 119896 |

119889119894119895= 119897 119861) =

1 when 119897 = 119896

0 when 119897 = 119896

119896 119897 isin 0 1

(13)

Thus we get

Pr (119889119894119895= 119896 | 119867

0 119861) = sum

119897 =119896

Pr ( 119889119894119895= 119897 | 119867

0 119861)

sdot Pr (119889119894119895= 119896 |

119889119894119895= 119897119867

0 119861)

+sum

119897=119896

Pr ( 119889119894119895= 119897 | 119867

0 119861)

sdot Pr (119889119894119895= 119896 |

119889119894119895= 119897119867

0 119861)

Pr (119889119894119895= 119896 | 119867

1 119861) = sum

119897 =119896

Pr ( 119889119894119895= 119897 | 119867

1 119861)

sdot Pr (119889119894119895= 119896 |

119889119894119895= 119897119867

1 119861)

+sum

119897=119896

Pr ( 119889119894119895= 119897 | 119867

1 119861)

sdot Pr (119889119894119895= 119896 |

119889119894119895= 119897119867

1 119861)

(14)

Therefore we have

Pr (119889119894119895= 0 | 119867

0 119861) = Pr ( 119889

119894119895= 1 | 119867

0 119861) = 119910

119894119895(1)

Pr (119889119894119895= 1 | 119867

0 119861) = Pr ( 119889

119894119895= 0 | 119867

0 119861) = 119910

119894119895(0)

Pr (119889119894119895= 0 | 119867

1 119861) = Pr ( 119889

119894119895= 1 | 119867

1 119861) = 119909

119894119895(1)

Pr (119889119894119895= 1 | 119867

1 119861) = Pr ( 119889

119894119895= 0 | 119867

1 119861) = 119909

119894119895(0)

(15)



V119894119895(119896) = Pr (119889

119894119895= 119896 | 119867

0)


119894119895(119896))]

+ 120572119910119894119895(1 minus 119896)

119906119894119895(119896) = Pr (119889

119894119895= 119896 | 119867

1)


119894119895(119896))]

+ 120572119909119894119895(1 minus 119896)

(16)


119894119895(119896) = Pr(119889

119894119895=

119896 | 1198670) and 119906

119894119895(119896) = Pr(119889

119894119895= 119896 | 119867


be expressed as

KLD (119906119894119895 V119894119895) = sum

119896isin01

119906119894119895(119896) log

119906119894119895(119896)

V119894119895(119896)

(17)


119894119895(119896) under

1198670and 119867









119906119894119895(119896) = Pr (119889

119894119895= 119896 | 119867

1) = Pr (119889

119894119895= 119896 | 119867

0)

= V119894119895(119896)

(18)


119894119895(119896) 119906

119894119895(119896)) = 0 is equivalent to

120572 =

119901119894119895(1) minus 119902

119894119895(1)

(119901119894119895(1) minus 119902

119894119895(1)) + (119909

119894119895(1) minus 119910

119894119895(1))

=

119901119894119895(1) minus 119902

119894119895(1)

(119901119894119895(1) minus 119902

119894119895(1)) + 119891 (119872

119894 120578119894 119901119894119895(1) 119902119894119895(1))

(19)


119891 (119872119894 120578119894 119901119894119895(1) 119902119894119895(1))

=

119872119894

sum

119898=120578119894

sum

120587isinΓ

119898

prod

119894=1

119901120587(119894)119895

(1)

119872119894

prod

119894=119898+1

(1 minus 119901120587(119894)119895

(1))

minus

119872119894

sum

119898=120578119894

sum

120587isinΓ

119898

prod

119894=1

119902120587(119894)119895

(1)

119872119894

prod

119894=119898+1

(1 minus 119902120587(119894)119895

(1))

(20)



119894119895(119896) and


119894119895 119906119894119895) = 0 if and only if

V119894119895(119896) = 119906







119894119895(1) 119902119894119895(1))

120572blind

=

119901119894119895(1) minus 119902

119894119895(1)

(119901119894119895(1) minus 119902

119894119895(1)) +max 119891 (119872

119894 120578119894 119901119894119895(1) 119902119894119895(1))

(22)



inequality

120572blind le1

1 +max 119891 (119872119894 120578119894 119901119894119895(1) 119902119894119895(1))

(23)


119889

119889119909

(

119909

119909 + 1

) =

1

(119909 + 1)2gt 0 (24)


120578119894 opt = ceil(

119872119894

2

) (25)


2 3 4 5 6 7 8 91120578

0

01

02

03

04

05

06

07

08

09

120572



119894=


119894119895(1)

119902119894119895(1)) we have

max119891 = 119891(119872119894 ceil(

119872119894

2

) 119901119894119895(1) 119902119894119895(1)) (26)


120572blind =119901119894119895(1) minus 119902

119894119895(1)

(119901119894119895(1) minus 119902

119894119895(1)) +max119891

(27)






119894and 120578 = 120578

119894 opt 119894 = 1 2 119873119861 Attacking




0

05

1

15

2

25

3

35

4

45

5

120578op

t

2 3 4 5 6 7 8 91M






119872120572

120578

1 2 3 4 5 6 7 8 91 050002 05000 050003 05434 04310 054344 05952 04310 04310 059525 06469 04509 04042 04509 064696 06959 04785 04042 04042 04785 069597 07410 05100 04145 03913 04145 05100 074108 07814 05438 04298 03913 03913 04298 05438 078149 08171 05790 04484 03970 03844 03970 04484 05790 08171






119873times119870= (119889

119894119895(119905))119873times119870



119895(119905) over vector


decision z(119905) = (1199111(119905) 1199112(119905) 119911


d1199051 d1199052 d119905

119870 119911119895(119905) isin 0 1 for any 119905 isin 119873+ 119895 = 1 2 119870


0

005

01

015

02

025

03

035

04

045

05

120572bl

ind

2 3 4 5 6 7 8 91M


0

005

01

015

02

025

03

035

04

045

05

120572bl

ind

3 5 7 9 11 13 15 17 19 21 23 251M




119872120578opt 11 21 32 42 53 63 74 84 95120572blind 05000 05000 04310 04310 04042 04042 03913 03913 03844

IMBANMBACMBA

C B A0

02

04

06

08

1

12

14

KLD

(120572)

01 02 03 04 050Attacking power 120572



119895(119905)




10038161003816100381610038161003816d119905119895

10038161003816100381610038161003816=

119873

sum

119894=1

10038161003816100381610038161003816119889119894119895(119905) minus 119911 (119905)

10038161003816100381610038161003816 (28)


10038161003816100381610038161003816d119905119894

10038161003816100381610038161003816=

119870

sum

119895=1

10038161003816100381610038161003816119889119894119895(119905) minus 119911 (119905)

10038161003816100381610038161003816 (29)





IMBANMBACMBA

C B A01 02 03 04 050


0

02

04

06

08

1

12

14

KLD

(120572)






119895| is small





0(119905 119895) and



1(119905 119895)


10038161003816100381610038161003816d119905119895

10038161003816100381610038161003816= sum

119894isin119868(119905119895)

10038161003816100381610038161003816119889119894119895(119905) minus 119911 (119905)

10038161003816100381610038161003816

10038161003816100381610038161003816d119905119894

10038161003816100381610038161003816=

119870

sum

119896=1

1003816100381610038161003816119889119894119896(119905) minus 119911 (119905)

1003816100381610038161003816 119894 isin 119868 (119905 119895)

(30)




119898= 119875119891119886


(119905) =

min 1198730(119905) 119873

1(119905)

119873

(31)

where1198730(119905) and119873





(119905 119895) =

min 1198730(119905 119895) 119873

1(119905 119895)

1198730(119905 119895) + 119873

1(119905 119895)

(32)





(119905 119895)

ge

min 1198730(119905 119895) 119873

1(119905 119895) minus 119870 sdot [max

119896(1003816100381610038161003816d119905119896

1003816100381610038161003816) minusmin

119896(1003816100381610038161003816d119905119896

1003816100381610038161003816)]

119899 (119905 119895) sdot 119870

119896 isin 1 2 119870

(33)


[max119896(|d119905119896|) minus min




Here we let

(119905 119895)

=

min 1198730(119905 119895) 119873

1(119905 119895) minus 119870 sdot [max

119895(

10038161003816100381610038161003816d119905119895

10038161003816100381610038161003816) minusmin

119895(

10038161003816100381610038161003816d119905119895

10038161003816100381610038161003816)]

119899 (119905 119895) sdot 119870

(34)




119873(119905 119895) over |d119905






119868119873(119905 119895)

=

10038161003816100381610038161003816d119905119894

10038161003816100381610038161003816

119899(119905119895)

10038161003816100381610038161003816d119905119894

10038161003816100381610038161003816

119899(119905119895)minus1

10038161003816100381610038161003816d119905119894

10038161003816100381610038161003816

1

| 119894 isin 119871 (119905 119895 minus 1)

(35)




over 119868119873(119905 119895)

119899 = ID (d119905119894)

= sum

119897isin119871(119905119895minus1)

IF (10038161003816100381610038161003816d119905119897

10038161003816100381610038161003816gt

10038161003816100381610038161003816d119905119894

10038161003816100381610038161003816

119899

)

+ sum

119897isin119871(119905119895minus1)

IF (10038161003816100381610038161003816d119905119897

10038161003816100381610038161003816=

10038161003816100381610038161003816d119905119894

10038161003816100381610038161003816

119899

| 119897 gt 119895)

119899 isin 1 2 119899 (119905 119895)

(36)




119868 (119905 119895)


(37)


(119905) =

119873 minus 119899 (119905 119870)

119873

(38)






1(119905)

1199112(119905) 119911


decision d1199051 d1199052 d119905






120578fc (119905 119895)




119886 = [1 minus (119905 119895)] sdot 119901119894119895(1) + (119905 119895) sdot 119909

119894119895(1) (41)


119894119895(1)) + (119905 119895)

sdot 119910119894119895(1) sdot (1 minus 119910

119894119895(1))

(42)


119911119895(119905) = IF(sum

119894

119889119894119895(119905) gt 120578fc (119905 119895) | 119894 isin 119871 (119905 119895 minus 1)) (43)


120574119867119861

120574119861119867

and 120574119861119861


119867119867and 120574119867119861


119861119867and 120574119861119861


120574119867119861

=

119873119867119861

119873

120574119867119867

=

119899 (119905 119870) minus 119873119861119867

119873

120574119861119867

=

119873119861119867

119873

120574119861119861

=

119873 minus 119899 (119905 119870) minus 119873119867119861

119873

(44)

Therein 119873119867119861


119861119867 119873119867and 119873


in Section 2





0(119905 119895) and 119873



0(119905 119895) 119873

0(119905 119895) and

1198731(119905 119895) verge to119870sdot119873

119861and119870sdot119873




119867119867 120574119861119861 120574119867119861

and 120574119861119867



cross And 120574119867119867


119861119867and 120574119867119861


5 Conclusion



0

10

20

30

40

50

60

70

80

90

100n(tj)

2 3 4 5 6 7 8 9 101j


0

100

200

300

400

500

600

N1(tj)

2 3 4 5 6 7 8 9 101j


0

50

100

150

200

250

300

350

400

450

N0(tj)

2 3 4 5 6 7 8 9 101j


0

10

20

30

40

50

60

70

2 3 4 5 6 7 8 9 101j

120578fc(tj)


0

005

01

015

02

025

03

035

120572(tj)

2 3 4 5 6 7 8 9 101j


Figure 10 119899(119905 119895)1198730(119905 119895)119873



2 3 4 5 6 7 8 9 101j

0

10

20

30

40

50

60

70

80

90

100n(tj)


2 3 4 5 6 7 8 9 101j

0

100

200

300

400

500

600

N1(tj)


0

50

100

150

200

250

300

350

400

450

N0(tj)

2 3 4 5 6 7 8 9 101j


2 3 4 5 6 7 8 9 101j

0

10

20

30

40

50

60

70

120578fc(tj)


0

005

01

015

02

025

03

035

120572(tj)

2 3 4 5 6 7 8 9 101j


Figure 11 119899(119905 119895)1198730(119905 119895)119873



2 3 4 5 6 7 8 9 10 11 12 13 14 151j

0

10

20

30

40

50

60

70

80

90

100n(tj)


0

100

200

300

400

500

600

700

800

900

N1(tj)

2 3 4 5 6 7 8 9 10 11 12 13 14 151j


2 3 4 5 6 7 8 9 10 11 12 13 14 151j

0

100

200

300

400

500

600

700

N0(tj)


0

10

20

30

40

50

60

70

2 3 4 5 6 7 8 9 10 11 12 13 14 151j

120578fc(tj)


2 3 4 5 6 7 8 9 10 11 12 13 14 151j

0

005

01

015

02

025

03

035

120572(tj)


Figure 12 119899(119905 119895)1198730(119905 119895)119873



0

10

20

30

40

50

60

70

80

90

100n(tj)

2 3 4 5 6 7 8 9 10 11 12 13 14 151j


0

100

200

300

400

500

600

700

800

900

N1(tj)

2 3 4 5 6 7 8 9 10 11 12 13 14 151j


0

100

200

300

400

500

600

700

N0(tj)

2 3 4 5 6 7 8 9 10 11 12 13 14 151j


0

10

20

30

40

50

60

70

2 3 4 5 6 7 8 9 10 11 12 13 14 151j

120578fc(tj)


0

005

01

015

02

025

03

035

04

120572(tj)

2 3 4 5 6 7 8 9 10 11 12 13 14 151j


Figure 13 119899(119905 119895)1198730(119905 119895)119873



0

01

02

03

04

05

06

07

20 40 60 80 1000(a) 119870 = 10119872 = 9

0

01

02

03

04

05

06

07

20 40 60 80 1000(b) 119870 = 10119872 = 25

Figure 14 120574119867119867

120574119861119861 120574119867119861 and 120574

119861119867versus 119905 120574

119867119867 120574119861119861 120574119867119861 and 120574



119867119867

vibrates under 07 120574119861119867


vibrates above 0

20 40 60 80 10000

01

02

03

04

05

06

07

(a) 119870 = 15119872 = 920 40 60 80 1000

0

01

02

03

04

05

06

07

(b) 119870 = 15119872 = 25

Figure 15 120574119867119867

120574119861119861 120574119867119861 and 120574

119861119867versus 119905 120574

119867119867 120574119861119861 120574119867119861 and 120574



119867119867

vibrates under 07 120574119861119867


vibrates above 0


Appendices



119898=


0(119905)119873when




(119905) =

min 1198730(119905) 119873

1(119905)

119873

(A1)






min 1198730(119905 119895) 119873

1(119905 119895)

119899 (119905 119895) sdot 119870

= 120572 (119905 119895) +

119873119861

01(119905 119895)

119899 (119905 119895) sdot 119870

+

119873119867

01(119905 119895)

119899 (119905 119895) sdot 119870

(B1)



01(119905 119895) in (B1) is denoted as


min 1198730(119905 119895) 119873

1(119905 119895)

119899 (119905 119895) sdot 119870

= 120572 (119905 119895) +

119873119861

01(119905 119895)

119899 (119905 119895) sdot 119870

+ 119902119894119895(1)

(B2)


min 1198730(119905 119895) 119873

1(119905 119895)

119899 (119905 119895) sdot 119870

= 120572 (119905 119895) +

119873119861

01(119905 119895)

119899 (119905 119895) sdot 119870

+ 1 minus 119901119894119895(1)

(B3)


119894119895(1) are the


119902119894119895(1) =

119873119867

01(119905 119895)

119899 (119905 119895) sdot 119870

or 119901119894119895(1) = 1 minus

119873119867

01(119905 119895)

119899 (119905 119895) sdot 119870

(B4)


(119905 119895) =

min 1198730(119905 119895) 119873

1(119905 119895)

119899 (119905 119895) sdot 119870

minus 119902119894119895(1) (B5)

or

(119905 119895) =

min 1198730(119905 119895) 119873

1(119905 119895)

119899 (119905 119895) sdot 119870

+ 119901119894119895(1) minus 1 (B6)

Because of

119901119894119895(1) ge 1 minus

max119896(1003816100381610038161003816d119905119896

1003816100381610038161003816) minusmin

119896(1003816100381610038161003816d119905119896

1003816100381610038161003816)

119899 (119905 119895)

(B7)

or

119902119894119895(1) le

max119896(1003816100381610038161003816d119905119896

1003816100381610038161003816) minusmin

119896(1003816100381610038161003816d119905119896

1003816100381610038161003816)

119899 (119905 119895)

(B8)

we get

(119905 119895)

ge

min 1198730(119905 119895) 119873

1(119905 119895) minus 119870 sdot [max

119895(

10038161003816100381610038161003816d119905119895

10038161003816100381610038161003816) minusmin

119895(

10038161003816100381610038161003816d119905119895

10038161003816100381610038161003816)]

119899 (119905 119895) sdot 119870

(B9)



Acknowledgments


References




























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











119889(119894) = Pr( 119889

119894= 1 | 119867

1 119867) and 119875119867fa (119894) =

Pr( 119889119894= 1 | 119867



119867 The


119875119867

119889(119894) = Pr ( 119889

119894= 1 | 119867

1 119867) = Pr (119886

119894+ 119899119894gt 120589119894)

= 119876(

120589119894minus 119886119894minus 120583

120590

)

(5)


119875119867

fa (119894) = Pr (119899119894gt 120589119894) = 119876(

120589119894minus 120583

120590

) (6)


119876 (119909) = int

infin

119909

1

radic2120587

119890minus11990522119889119905 (7)




d119894= (

11988911989411198891198942




1 d2 d

119873]T is


where 119889119894119895isin 0 1




119894into the FC



1 1199112 119911




119894minus 1 (119894 = 1 2 119873

119861)

Natural state

The fusion center

d1 d2 dNminus1 dN

d1 d2 dNminus1 dN


DNtimesK rarr z

S1 S2 SNminus1 SN

z rarr z



119894 = 1 2 119873 D119873times119870




119894= 119872 =



1198670 V119894119895(119896) = Pr (119889

119894119895= 119896 | 119867

0)

= (1 minus 120572)Pr (119889119894119895= 119896 | 119867

0 119867)

+ 120572Pr (119889119894119895= 119896 | 119867

0 119861)

1198671 119906119894119895(119896) = Pr (119889

119894119895= 119896 | 119867

1)

= (1 minus 120572)Pr (119889119894119895= 119896 | 119867

1 119867)

+ 120572Pr (119889119894119895= 119896 | 119867

1 119861)

119894 isin 1 2 119873 119895 isin 1 2 119870 119896 isin 0 1

(8)


0

20

40

60

80

100

120

140

160

180

200

Y-c

oord

inat

e (m

)

50 100 150 2000X-coordinate (m)



0and119867

1 respectively

Therefore we have

1198670 Pr ( 119889

119894119895= 119896 | 119867

0 119867) = 119902

119894119895(119896)

1198671 Pr ( 119889

119894119895= 119896 | 119867

1 119867) = 119901

119894119895(119896)

119894 = 1 2 119873119867

(9)


119901119894119895(1) = 119876(

120589 minus 119886119894minus 120583

120590

)

119901119894119895(0) = 1 minus 119901

119894119895(1)

119902119894119895(1) = 119876(

120589 minus 120583

120590

)

119902119894119895(0) = 1 minus 119902

119894119895(1)

(10)


1198670 Pr ( 119889

119894119895= 119896 | 119867

0 119861) = 119910

119894119895(119896)

1198671 Pr ( 119889

119894119895= 119896 | 119867

1 119861) = 119909

119894119895(119896)

119894 = 1 2 119873119861

(11)



119894minus1


119894119897 119897 isin 0 1 119872

119894minus 1 119888





119897=0119888119894119897gt 120578119894) where



119910119894119895(119896) =

119872119894

sum

119898=120578119894

sum

120587isinΓ

119898

prod

119894=1

119902120587(119894)119895

(119896)

119872119894

prod

119894=119898+1

(1 minus 119902120587(119894)119895

(119896))

119909119894119895(119896) =

119872119894

sum

119898=120578119894

sum

120587isinΓ

119898

prod

119894=1

119901120587(119894)119895

(119896)

119872119894

prod

119894=119898+1

(1 minus 119901120587(119894)119895

(119896))

(12)

where119872119894isin 1 2 119873

119867 1 le 120578 le 119872




Pr (119889119894119895= 119896 |

119889119894119895= 119897 119861) =

1 when 119897 = 119896

0 when 119897 = 119896

119896 119897 isin 0 1

(13)

Thus we get

Pr (119889119894119895= 119896 | 119867

0 119861) = sum

119897 =119896

Pr ( 119889119894119895= 119897 | 119867

0 119861)

sdot Pr (119889119894119895= 119896 |

119889119894119895= 119897119867

0 119861)

+sum

119897=119896

Pr ( 119889119894119895= 119897 | 119867

0 119861)

sdot Pr (119889119894119895= 119896 |

119889119894119895= 119897119867

0 119861)

Pr (119889119894119895= 119896 | 119867

1 119861) = sum

119897 =119896

Pr ( 119889119894119895= 119897 | 119867

1 119861)

sdot Pr (119889119894119895= 119896 |

119889119894119895= 119897119867

1 119861)

+sum

119897=119896

Pr ( 119889119894119895= 119897 | 119867

1 119861)

sdot Pr (119889119894119895= 119896 |

119889119894119895= 119897119867

1 119861)

(14)

Therefore we have

Pr (119889119894119895= 0 | 119867

0 119861) = Pr ( 119889

119894119895= 1 | 119867

0 119861) = 119910

119894119895(1)

Pr (119889119894119895= 1 | 119867

0 119861) = Pr ( 119889

119894119895= 0 | 119867

0 119861) = 119910

119894119895(0)

Pr (119889119894119895= 0 | 119867

1 119861) = Pr ( 119889

119894119895= 1 | 119867

1 119861) = 119909

119894119895(1)

Pr (119889119894119895= 1 | 119867

1 119861) = Pr ( 119889

119894119895= 0 | 119867

1 119861) = 119909

119894119895(0)

(15)



V119894119895(119896) = Pr (119889

119894119895= 119896 | 119867

0)


119894119895(119896))]

+ 120572119910119894119895(1 minus 119896)

119906119894119895(119896) = Pr (119889

119894119895= 119896 | 119867

1)


119894119895(119896))]

+ 120572119909119894119895(1 minus 119896)

(16)


119894119895(119896) = Pr(119889

119894119895=

119896 | 1198670) and 119906

119894119895(119896) = Pr(119889

119894119895= 119896 | 119867


be expressed as

KLD (119906119894119895 V119894119895) = sum

119896isin01

119906119894119895(119896) log

119906119894119895(119896)

V119894119895(119896)

(17)


119894119895(119896) under

1198670and 119867









119906119894119895(119896) = Pr (119889

119894119895= 119896 | 119867

1) = Pr (119889

119894119895= 119896 | 119867

0)

= V119894119895(119896)

(18)


119894119895(119896) 119906

119894119895(119896)) = 0 is equivalent to

120572 =

119901119894119895(1) minus 119902

119894119895(1)

(119901119894119895(1) minus 119902

119894119895(1)) + (119909

119894119895(1) minus 119910

119894119895(1))

=

119901119894119895(1) minus 119902

119894119895(1)

(119901119894119895(1) minus 119902

119894119895(1)) + 119891 (119872

119894 120578119894 119901119894119895(1) 119902119894119895(1))

(19)


119891 (119872119894 120578119894 119901119894119895(1) 119902119894119895(1))

=

119872119894

sum

119898=120578119894

sum

120587isinΓ

119898

prod

119894=1

119901120587(119894)119895

(1)

119872119894

prod

119894=119898+1

(1 minus 119901120587(119894)119895

(1))

minus

119872119894

sum

119898=120578119894

sum

120587isinΓ

119898

prod

119894=1

119902120587(119894)119895

(1)

119872119894

prod

119894=119898+1

(1 minus 119902120587(119894)119895

(1))

(20)



119894119895(119896) and


119894119895 119906119894119895) = 0 if and only if

V119894119895(119896) = 119906







119894119895(1) 119902119894119895(1))

120572blind

=

119901119894119895(1) minus 119902

119894119895(1)

(119901119894119895(1) minus 119902

119894119895(1)) +max 119891 (119872

119894 120578119894 119901119894119895(1) 119902119894119895(1))

(22)



inequality

120572blind le1

1 +max 119891 (119872119894 120578119894 119901119894119895(1) 119902119894119895(1))

(23)


119889

119889119909

(

119909

119909 + 1

) =

1

(119909 + 1)2gt 0 (24)


120578119894 opt = ceil(

119872119894

2

) (25)


2 3 4 5 6 7 8 91120578

0

01

02

03

04

05

06

07

08

09

120572



119894=


119894119895(1)

119902119894119895(1)) we have

max119891 = 119891(119872119894 ceil(

119872119894

2

) 119901119894119895(1) 119902119894119895(1)) (26)


120572blind =119901119894119895(1) minus 119902

119894119895(1)

(119901119894119895(1) minus 119902

119894119895(1)) +max119891

(27)






119894and 120578 = 120578

119894 opt 119894 = 1 2 119873119861 Attacking




0

05

1

15

2

25

3

35

4

45

5

120578op

t

2 3 4 5 6 7 8 91M






119872120572

120578

1 2 3 4 5 6 7 8 91 050002 05000 050003 05434 04310 054344 05952 04310 04310 059525 06469 04509 04042 04509 064696 06959 04785 04042 04042 04785 069597 07410 05100 04145 03913 04145 05100 074108 07814 05438 04298 03913 03913 04298 05438 078149 08171 05790 04484 03970 03844 03970 04484 05790 08171






119873times119870= (119889

119894119895(119905))119873times119870



119895(119905) over vector


decision z(119905) = (1199111(119905) 1199112(119905) 119911


d1199051 d1199052 d119905

119870 119911119895(119905) isin 0 1 for any 119905 isin 119873+ 119895 = 1 2 119870


0

005

01

015

02

025

03

035

04

045

05

120572bl

ind

2 3 4 5 6 7 8 91M


0

005

01

015

02

025

03

035

04

045

05

120572bl

ind

3 5 7 9 11 13 15 17 19 21 23 251M




119872120578opt 11 21 32 42 53 63 74 84 95120572blind 05000 05000 04310 04310 04042 04042 03913 03913 03844

IMBANMBACMBA

C B A0

02

04

06

08

1

12

14

KLD

(120572)

01 02 03 04 050Attacking power 120572



119895(119905)




10038161003816100381610038161003816d119905119895

10038161003816100381610038161003816=

119873

sum

119894=1

10038161003816100381610038161003816119889119894119895(119905) minus 119911 (119905)

10038161003816100381610038161003816 (28)


10038161003816100381610038161003816d119905119894

10038161003816100381610038161003816=

119870

sum

119895=1

10038161003816100381610038161003816119889119894119895(119905) minus 119911 (119905)

10038161003816100381610038161003816 (29)





IMBANMBACMBA

C B A01 02 03 04 050


0

02

04

06

08

1

12

14

KLD

(120572)






119895| is small





0(119905 119895) and



1(119905 119895)


10038161003816100381610038161003816d119905119895

10038161003816100381610038161003816= sum

119894isin119868(119905119895)

10038161003816100381610038161003816119889119894119895(119905) minus 119911 (119905)

10038161003816100381610038161003816

10038161003816100381610038161003816d119905119894

10038161003816100381610038161003816=

119870

sum

119896=1

1003816100381610038161003816119889119894119896(119905) minus 119911 (119905)

1003816100381610038161003816 119894 isin 119868 (119905 119895)

(30)




119898= 119875119891119886


(119905) =

min 1198730(119905) 119873

1(119905)

119873

(31)

where1198730(119905) and119873





(119905 119895) =

min 1198730(119905 119895) 119873

1(119905 119895)

1198730(119905 119895) + 119873

1(119905 119895)

(32)





(119905 119895)

ge

min 1198730(119905 119895) 119873

1(119905 119895) minus 119870 sdot [max

119896(1003816100381610038161003816d119905119896

1003816100381610038161003816) minusmin

119896(1003816100381610038161003816d119905119896

1003816100381610038161003816)]

119899 (119905 119895) sdot 119870

119896 isin 1 2 119870

(33)


[max119896(|d119905119896|) minus min




Here we let

(119905 119895)

=

min 1198730(119905 119895) 119873

1(119905 119895) minus 119870 sdot [max

119895(

10038161003816100381610038161003816d119905119895

10038161003816100381610038161003816) minusmin

119895(

10038161003816100381610038161003816d119905119895

10038161003816100381610038161003816)]

119899 (119905 119895) sdot 119870

(34)




119873(119905 119895) over |d119905






119868119873(119905 119895)

=

10038161003816100381610038161003816d119905119894

10038161003816100381610038161003816

119899(119905119895)

10038161003816100381610038161003816d119905119894

10038161003816100381610038161003816

119899(119905119895)minus1

10038161003816100381610038161003816d119905119894

10038161003816100381610038161003816

1

| 119894 isin 119871 (119905 119895 minus 1)

(35)




over 119868119873(119905 119895)

119899 = ID (d119905119894)

= sum

119897isin119871(119905119895minus1)

IF (10038161003816100381610038161003816d119905119897

10038161003816100381610038161003816gt

10038161003816100381610038161003816d119905119894

10038161003816100381610038161003816

119899

)

+ sum

119897isin119871(119905119895minus1)

IF (10038161003816100381610038161003816d119905119897

10038161003816100381610038161003816=

10038161003816100381610038161003816d119905119894

10038161003816100381610038161003816

119899

| 119897 gt 119895)

119899 isin 1 2 119899 (119905 119895)

(36)




119868 (119905 119895)


(37)


(119905) =

119873 minus 119899 (119905 119870)

119873

(38)






1(119905)

1199112(119905) 119911


decision d1199051 d1199052 d119905






120578fc (119905 119895)




119886 = [1 minus (119905 119895)] sdot 119901119894119895(1) + (119905 119895) sdot 119909

119894119895(1) (41)


119894119895(1)) + (119905 119895)

sdot 119910119894119895(1) sdot (1 minus 119910

119894119895(1))

(42)


119911119895(119905) = IF(sum

119894

119889119894119895(119905) gt 120578fc (119905 119895) | 119894 isin 119871 (119905 119895 minus 1)) (43)


120574119867119861

120574119861119867

and 120574119861119861


119867119867and 120574119867119861


119861119867and 120574119861119861


120574119867119861

=

119873119867119861

119873

120574119867119867

=

119899 (119905 119870) minus 119873119861119867

119873

120574119861119867

=

119873119861119867

119873

120574119861119861

=

119873 minus 119899 (119905 119870) minus 119873119867119861

119873

(44)

Therein 119873119867119861


119861119867 119873119867and 119873


in Section 2





0(119905 119895) and 119873



0(119905 119895) 119873

0(119905 119895) and

1198731(119905 119895) verge to119870sdot119873

119861and119870sdot119873




119867119867 120574119861119861 120574119867119861

and 120574119861119867



cross And 120574119867119867


119861119867and 120574119867119861


5 Conclusion



0

10

20

30

40

50

60

70

80

90

100n(tj)

2 3 4 5 6 7 8 9 101j


0

100

200

300

400

500

600

N1(tj)

2 3 4 5 6 7 8 9 101j


0

50

100

150

200

250

300

350

400

450

N0(tj)

2 3 4 5 6 7 8 9 101j


0

10

20

30

40

50

60

70

2 3 4 5 6 7 8 9 101j

120578fc(tj)


0

005

01

015

02

025

03

035

120572(tj)

2 3 4 5 6 7 8 9 101j


Figure 10 119899(119905 119895)1198730(119905 119895)119873



2 3 4 5 6 7 8 9 101j

0

10

20

30

40

50

60

70

80

90

100n(tj)


2 3 4 5 6 7 8 9 101j

0

100

200

300

400

500

600

N1(tj)


0

50

100

150

200

250

300

350

400

450

N0(tj)

2 3 4 5 6 7 8 9 101j


2 3 4 5 6 7 8 9 101j

0

10

20

30

40

50

60

70

120578fc(tj)


0

005

01

015

02

025

03

035

120572(tj)

2 3 4 5 6 7 8 9 101j


Figure 11 119899(119905 119895)1198730(119905 119895)119873



2 3 4 5 6 7 8 9 10 11 12 13 14 151j

0

10

20

30

40

50

60

70

80

90

100n(tj)


0

100

200

300

400

500

600

700

800

900

N1(tj)

2 3 4 5 6 7 8 9 10 11 12 13 14 151j


2 3 4 5 6 7 8 9 10 11 12 13 14 151j

0

100

200

300

400

500

600

700

N0(tj)


0

10

20

30

40

50

60

70

2 3 4 5 6 7 8 9 10 11 12 13 14 151j

120578fc(tj)


2 3 4 5 6 7 8 9 10 11 12 13 14 151j

0

005

01

015

02

025

03

035

120572(tj)


Figure 12 119899(119905 119895)1198730(119905 119895)119873



0

10

20

30

40

50

60

70

80

90

100n(tj)

2 3 4 5 6 7 8 9 10 11 12 13 14 151j


0

100

200

300

400

500

600

700

800

900

N1(tj)

2 3 4 5 6 7 8 9 10 11 12 13 14 151j


0

100

200

300

400

500

600

700

N0(tj)

2 3 4 5 6 7 8 9 10 11 12 13 14 151j


0

10

20

30

40

50

60

70

2 3 4 5 6 7 8 9 10 11 12 13 14 151j

120578fc(tj)


0

005

01

015

02

025

03

035

04

120572(tj)

2 3 4 5 6 7 8 9 10 11 12 13 14 151j


Figure 13 119899(119905 119895)1198730(119905 119895)119873



0

01

02

03

04

05

06

07

20 40 60 80 1000(a) 119870 = 10119872 = 9

0

01

02

03

04

05

06

07

20 40 60 80 1000(b) 119870 = 10119872 = 25

Figure 14 120574119867119867

120574119861119861 120574119867119861 and 120574

119861119867versus 119905 120574

119867119867 120574119861119861 120574119867119861 and 120574



119867119867

vibrates under 07 120574119861119867


vibrates above 0

20 40 60 80 10000

01

02

03

04

05

06

07

(a) 119870 = 15119872 = 920 40 60 80 1000

0

01

02

03

04

05

06

07

(b) 119870 = 15119872 = 25

Figure 15 120574119867119867

120574119861119861 120574119867119861 and 120574

119861119867versus 119905 120574

119867119867 120574119861119861 120574119867119861 and 120574



119867119867

vibrates under 07 120574119861119867


vibrates above 0


Appendices



119898=


0(119905)119873when




(119905) =

min 1198730(119905) 119873

1(119905)

119873

(A1)






min 1198730(119905 119895) 119873

1(119905 119895)

119899 (119905 119895) sdot 119870

= 120572 (119905 119895) +

119873119861

01(119905 119895)

119899 (119905 119895) sdot 119870

+

119873119867

01(119905 119895)

119899 (119905 119895) sdot 119870

(B1)



01(119905 119895) in (B1) is denoted as


min 1198730(119905 119895) 119873

1(119905 119895)

119899 (119905 119895) sdot 119870

= 120572 (119905 119895) +

119873119861

01(119905 119895)

119899 (119905 119895) sdot 119870

+ 119902119894119895(1)

(B2)


min 1198730(119905 119895) 119873

1(119905 119895)

119899 (119905 119895) sdot 119870

= 120572 (119905 119895) +

119873119861

01(119905 119895)

119899 (119905 119895) sdot 119870

+ 1 minus 119901119894119895(1)

(B3)


119894119895(1) are the


119902119894119895(1) =

119873119867

01(119905 119895)

119899 (119905 119895) sdot 119870

or 119901119894119895(1) = 1 minus

119873119867

01(119905 119895)

119899 (119905 119895) sdot 119870

(B4)


(119905 119895) =

min 1198730(119905 119895) 119873

1(119905 119895)

119899 (119905 119895) sdot 119870

minus 119902119894119895(1) (B5)

or

(119905 119895) =

min 1198730(119905 119895) 119873

1(119905 119895)

119899 (119905 119895) sdot 119870

+ 119901119894119895(1) minus 1 (B6)

Because of

119901119894119895(1) ge 1 minus

max119896(1003816100381610038161003816d119905119896

1003816100381610038161003816) minusmin

119896(1003816100381610038161003816d119905119896

1003816100381610038161003816)

119899 (119905 119895)

(B7)

or

119902119894119895(1) le

max119896(1003816100381610038161003816d119905119896

1003816100381610038161003816) minusmin

119896(1003816100381610038161003816d119905119896

1003816100381610038161003816)

119899 (119905 119895)

(B8)

we get

(119905 119895)

ge

min 1198730(119905 119895) 119873

1(119905 119895) minus 119870 sdot [max

119895(

10038161003816100381610038161003816d119905119895

10038161003816100381610038161003816) minusmin

119895(

10038161003816100381610038161003816d119905119895

10038161003816100381610038161003816)]

119899 (119905 119895) sdot 119870

(B9)



Acknowledgments


References




























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










0

20

40

60

80

100

120

140

160

180

200

Y-c

oord

inat

e (m

)

50 100 150 2000X-coordinate (m)



0and119867

1 respectively

Therefore we have

1198670 Pr ( 119889

119894119895= 119896 | 119867

0 119867) = 119902

119894119895(119896)

1198671 Pr ( 119889

119894119895= 119896 | 119867

1 119867) = 119901

119894119895(119896)

119894 = 1 2 119873119867

(9)


119901119894119895(1) = 119876(

120589 minus 119886119894minus 120583

120590

)

119901119894119895(0) = 1 minus 119901

119894119895(1)

119902119894119895(1) = 119876(

120589 minus 120583

120590

)

119902119894119895(0) = 1 minus 119902

119894119895(1)

(10)


1198670 Pr ( 119889

119894119895= 119896 | 119867

0 119861) = 119910

119894119895(119896)

1198671 Pr ( 119889

119894119895= 119896 | 119867

1 119861) = 119909

119894119895(119896)

119894 = 1 2 119873119861

(11)



119894minus1


119894119897 119897 isin 0 1 119872

119894minus 1 119888





119897=0119888119894119897gt 120578119894) where



119910119894119895(119896) =

119872119894

sum

119898=120578119894

sum

120587isinΓ

119898

prod

119894=1

119902120587(119894)119895

(119896)

119872119894

prod

119894=119898+1

(1 minus 119902120587(119894)119895

(119896))

119909119894119895(119896) =

119872119894

sum

119898=120578119894

sum

120587isinΓ

119898

prod

119894=1

119901120587(119894)119895

(119896)

119872119894

prod

119894=119898+1

(1 minus 119901120587(119894)119895

(119896))

(12)

where119872119894isin 1 2 119873

119867 1 le 120578 le 119872




Pr (119889119894119895= 119896 |

119889119894119895= 119897 119861) =

1 when 119897 = 119896

0 when 119897 = 119896

119896 119897 isin 0 1

(13)

Thus we get

Pr (119889119894119895= 119896 | 119867

0 119861) = sum

119897 =119896

Pr ( 119889119894119895= 119897 | 119867

0 119861)

sdot Pr (119889119894119895= 119896 |

119889119894119895= 119897119867

0 119861)

+sum

119897=119896

Pr ( 119889119894119895= 119897 | 119867

0 119861)

sdot Pr (119889119894119895= 119896 |

119889119894119895= 119897119867

0 119861)

Pr (119889119894119895= 119896 | 119867

1 119861) = sum

119897 =119896

Pr ( 119889119894119895= 119897 | 119867

1 119861)

sdot Pr (119889119894119895= 119896 |

119889119894119895= 119897119867

1 119861)

+sum

119897=119896

Pr ( 119889119894119895= 119897 | 119867

1 119861)

sdot Pr (119889119894119895= 119896 |

119889119894119895= 119897119867

1 119861)

(14)

Therefore we have

Pr (119889119894119895= 0 | 119867

0 119861) = Pr ( 119889

119894119895= 1 | 119867

0 119861) = 119910

119894119895(1)

Pr (119889119894119895= 1 | 119867

0 119861) = Pr ( 119889

119894119895= 0 | 119867

0 119861) = 119910

119894119895(0)

Pr (119889119894119895= 0 | 119867

1 119861) = Pr ( 119889

119894119895= 1 | 119867

1 119861) = 119909

119894119895(1)

Pr (119889119894119895= 1 | 119867

1 119861) = Pr ( 119889

119894119895= 0 | 119867

1 119861) = 119909

119894119895(0)

(15)



V119894119895(119896) = Pr (119889

119894119895= 119896 | 119867

0)


119894119895(119896))]

+ 120572119910119894119895(1 minus 119896)

119906119894119895(119896) = Pr (119889

119894119895= 119896 | 119867

1)


119894119895(119896))]

+ 120572119909119894119895(1 minus 119896)

(16)


119894119895(119896) = Pr(119889

119894119895=

119896 | 1198670) and 119906

119894119895(119896) = Pr(119889

119894119895= 119896 | 119867


be expressed as

KLD (119906119894119895 V119894119895) = sum

119896isin01

119906119894119895(119896) log

119906119894119895(119896)

V119894119895(119896)

(17)


119894119895(119896) under

1198670and 119867









119906119894119895(119896) = Pr (119889

119894119895= 119896 | 119867

1) = Pr (119889

119894119895= 119896 | 119867

0)

= V119894119895(119896)

(18)


119894119895(119896) 119906

119894119895(119896)) = 0 is equivalent to

120572 =

119901119894119895(1) minus 119902

119894119895(1)

(119901119894119895(1) minus 119902

119894119895(1)) + (119909

119894119895(1) minus 119910

119894119895(1))

=

119901119894119895(1) minus 119902

119894119895(1)

(119901119894119895(1) minus 119902

119894119895(1)) + 119891 (119872

119894 120578119894 119901119894119895(1) 119902119894119895(1))

(19)


119891 (119872119894 120578119894 119901119894119895(1) 119902119894119895(1))

=

119872119894

sum

119898=120578119894

sum

120587isinΓ

119898

prod

119894=1

119901120587(119894)119895

(1)

119872119894

prod

119894=119898+1

(1 minus 119901120587(119894)119895

(1))

minus

119872119894

sum

119898=120578119894

sum

120587isinΓ

119898

prod

119894=1

119902120587(119894)119895

(1)

119872119894

prod

119894=119898+1

(1 minus 119902120587(119894)119895

(1))

(20)



119894119895(119896) and


119894119895 119906119894119895) = 0 if and only if

V119894119895(119896) = 119906







119894119895(1) 119902119894119895(1))

120572blind

=

119901119894119895(1) minus 119902

119894119895(1)

(119901119894119895(1) minus 119902

119894119895(1)) +max 119891 (119872

119894 120578119894 119901119894119895(1) 119902119894119895(1))

(22)



inequality

120572blind le1

1 +max 119891 (119872119894 120578119894 119901119894119895(1) 119902119894119895(1))

(23)


119889

119889119909

(

119909

119909 + 1

) =

1

(119909 + 1)2gt 0 (24)


120578119894 opt = ceil(

119872119894

2

) (25)


2 3 4 5 6 7 8 91120578

0

01

02

03

04

05

06

07

08

09

120572



119894=


119894119895(1)

119902119894119895(1)) we have

max119891 = 119891(119872119894 ceil(

119872119894

2

) 119901119894119895(1) 119902119894119895(1)) (26)


120572blind =119901119894119895(1) minus 119902

119894119895(1)

(119901119894119895(1) minus 119902

119894119895(1)) +max119891

(27)






119894and 120578 = 120578

119894 opt 119894 = 1 2 119873119861 Attacking




0

05

1

15

2

25

3

35

4

45

5

120578op

t

2 3 4 5 6 7 8 91M






119872120572

120578

1 2 3 4 5 6 7 8 91 050002 05000 050003 05434 04310 054344 05952 04310 04310 059525 06469 04509 04042 04509 064696 06959 04785 04042 04042 04785 069597 07410 05100 04145 03913 04145 05100 074108 07814 05438 04298 03913 03913 04298 05438 078149 08171 05790 04484 03970 03844 03970 04484 05790 08171






119873times119870= (119889

119894119895(119905))119873times119870



119895(119905) over vector


decision z(119905) = (1199111(119905) 1199112(119905) 119911


d1199051 d1199052 d119905

119870 119911119895(119905) isin 0 1 for any 119905 isin 119873+ 119895 = 1 2 119870


0

005

01

015

02

025

03

035

04

045

05

120572bl

ind

2 3 4 5 6 7 8 91M


0

005

01

015

02

025

03

035

04

045

05

120572bl

ind

3 5 7 9 11 13 15 17 19 21 23 251M




119872120578opt 11 21 32 42 53 63 74 84 95120572blind 05000 05000 04310 04310 04042 04042 03913 03913 03844

IMBANMBACMBA

C B A0

02

04

06

08

1

12

14

KLD

(120572)

01 02 03 04 050Attacking power 120572



119895(119905)




10038161003816100381610038161003816d119905119895

10038161003816100381610038161003816=

119873

sum

119894=1

10038161003816100381610038161003816119889119894119895(119905) minus 119911 (119905)

10038161003816100381610038161003816 (28)


10038161003816100381610038161003816d119905119894

10038161003816100381610038161003816=

119870

sum

119895=1

10038161003816100381610038161003816119889119894119895(119905) minus 119911 (119905)

10038161003816100381610038161003816 (29)





IMBANMBACMBA

C B A01 02 03 04 050


0

02

04

06

08

1

12

14

KLD

(120572)






119895| is small





0(119905 119895) and



1(119905 119895)


10038161003816100381610038161003816d119905119895

10038161003816100381610038161003816= sum

119894isin119868(119905119895)

10038161003816100381610038161003816119889119894119895(119905) minus 119911 (119905)

10038161003816100381610038161003816

10038161003816100381610038161003816d119905119894

10038161003816100381610038161003816=

119870

sum

119896=1

1003816100381610038161003816119889119894119896(119905) minus 119911 (119905)

1003816100381610038161003816 119894 isin 119868 (119905 119895)

(30)




119898= 119875119891119886


(119905) =

min 1198730(119905) 119873

1(119905)

119873

(31)

where1198730(119905) and119873





(119905 119895) =

min 1198730(119905 119895) 119873

1(119905 119895)

1198730(119905 119895) + 119873

1(119905 119895)

(32)





(119905 119895)

ge

min 1198730(119905 119895) 119873

1(119905 119895) minus 119870 sdot [max

119896(1003816100381610038161003816d119905119896

1003816100381610038161003816) minusmin

119896(1003816100381610038161003816d119905119896

1003816100381610038161003816)]

119899 (119905 119895) sdot 119870

119896 isin 1 2 119870

(33)


[max119896(|d119905119896|) minus min




Here we let

(119905 119895)

=

min 1198730(119905 119895) 119873

1(119905 119895) minus 119870 sdot [max

119895(

10038161003816100381610038161003816d119905119895

10038161003816100381610038161003816) minusmin

119895(

10038161003816100381610038161003816d119905119895

10038161003816100381610038161003816)]

119899 (119905 119895) sdot 119870

(34)




119873(119905 119895) over |d119905






119868119873(119905 119895)

=

10038161003816100381610038161003816d119905119894

10038161003816100381610038161003816

119899(119905119895)

10038161003816100381610038161003816d119905119894

10038161003816100381610038161003816

119899(119905119895)minus1

10038161003816100381610038161003816d119905119894

10038161003816100381610038161003816

1

| 119894 isin 119871 (119905 119895 minus 1)

(35)




over 119868119873(119905 119895)

119899 = ID (d119905119894)

= sum

119897isin119871(119905119895minus1)

IF (10038161003816100381610038161003816d119905119897

10038161003816100381610038161003816gt

10038161003816100381610038161003816d119905119894

10038161003816100381610038161003816

119899

)

+ sum

119897isin119871(119905119895minus1)

IF (10038161003816100381610038161003816d119905119897

10038161003816100381610038161003816=

10038161003816100381610038161003816d119905119894

10038161003816100381610038161003816

119899

| 119897 gt 119895)

119899 isin 1 2 119899 (119905 119895)

(36)




119868 (119905 119895)


(37)


(119905) =

119873 minus 119899 (119905 119870)

119873

(38)






1(119905)

1199112(119905) 119911


decision d1199051 d1199052 d119905






120578fc (119905 119895)




119886 = [1 minus (119905 119895)] sdot 119901119894119895(1) + (119905 119895) sdot 119909

119894119895(1) (41)


119894119895(1)) + (119905 119895)

sdot 119910119894119895(1) sdot (1 minus 119910

119894119895(1))

(42)


119911119895(119905) = IF(sum

119894

119889119894119895(119905) gt 120578fc (119905 119895) | 119894 isin 119871 (119905 119895 minus 1)) (43)


120574119867119861

120574119861119867

and 120574119861119861


119867119867and 120574119867119861


119861119867and 120574119861119861


120574119867119861

=

119873119867119861

119873

120574119867119867

=

119899 (119905 119870) minus 119873119861119867

119873

120574119861119867

=

119873119861119867

119873

120574119861119861

=

119873 minus 119899 (119905 119870) minus 119873119867119861

119873

(44)

Therein 119873119867119861


119861119867 119873119867and 119873


in Section 2





0(119905 119895) and 119873



0(119905 119895) 119873

0(119905 119895) and

1198731(119905 119895) verge to119870sdot119873

119861and119870sdot119873




119867119867 120574119861119861 120574119867119861

and 120574119861119867



cross And 120574119867119867


119861119867and 120574119867119861


5 Conclusion



0

10

20

30

40

50

60

70

80

90

100n(tj)

2 3 4 5 6 7 8 9 101j


0

100

200

300

400

500

600

N1(tj)

2 3 4 5 6 7 8 9 101j


0

50

100

150

200

250

300

350

400

450

N0(tj)

2 3 4 5 6 7 8 9 101j


0

10

20

30

40

50

60

70

2 3 4 5 6 7 8 9 101j

120578fc(tj)


0

005

01

015

02

025

03

035

120572(tj)

2 3 4 5 6 7 8 9 101j


Figure 10 119899(119905 119895)1198730(119905 119895)119873



2 3 4 5 6 7 8 9 101j

0

10

20

30

40

50

60

70

80

90

100n(tj)


2 3 4 5 6 7 8 9 101j

0

100

200

300

400

500

600

N1(tj)


0

50

100

150

200

250

300

350

400

450

N0(tj)

2 3 4 5 6 7 8 9 101j


2 3 4 5 6 7 8 9 101j

0

10

20

30

40

50

60

70

120578fc(tj)


0

005

01

015

02

025

03

035

120572(tj)

2 3 4 5 6 7 8 9 101j


Figure 11 119899(119905 119895)1198730(119905 119895)119873



2 3 4 5 6 7 8 9 10 11 12 13 14 151j

0

10

20

30

40

50

60

70

80

90

100n(tj)


0

100

200

300

400

500

600

700

800

900

N1(tj)

2 3 4 5 6 7 8 9 10 11 12 13 14 151j


2 3 4 5 6 7 8 9 10 11 12 13 14 151j

0

100

200

300

400

500

600

700

N0(tj)


0

10

20

30

40

50

60

70

2 3 4 5 6 7 8 9 10 11 12 13 14 151j

120578fc(tj)


2 3 4 5 6 7 8 9 10 11 12 13 14 151j

0

005

01

015

02

025

03

035

120572(tj)


Figure 12 119899(119905 119895)1198730(119905 119895)119873



0

10

20

30

40

50

60

70

80

90

100n(tj)

2 3 4 5 6 7 8 9 10 11 12 13 14 151j


0

100

200

300

400

500

600

700

800

900

N1(tj)

2 3 4 5 6 7 8 9 10 11 12 13 14 151j


0

100

200

300

400

500

600

700

N0(tj)

2 3 4 5 6 7 8 9 10 11 12 13 14 151j


0

10

20

30

40

50

60

70

2 3 4 5 6 7 8 9 10 11 12 13 14 151j

120578fc(tj)


0

005

01

015

02

025

03

035

04

120572(tj)

2 3 4 5 6 7 8 9 10 11 12 13 14 151j


Figure 13 119899(119905 119895)1198730(119905 119895)119873



0

01

02

03

04

05

06

07

20 40 60 80 1000(a) 119870 = 10119872 = 9

0

01

02

03

04

05

06

07

20 40 60 80 1000(b) 119870 = 10119872 = 25

Figure 14 120574119867119867

120574119861119861 120574119867119861 and 120574

119861119867versus 119905 120574

119867119867 120574119861119861 120574119867119861 and 120574



119867119867

vibrates under 07 120574119861119867


vibrates above 0

20 40 60 80 10000

01

02

03

04

05

06

07

(a) 119870 = 15119872 = 920 40 60 80 1000

0

01

02

03

04

05

06

07

(b) 119870 = 15119872 = 25

Figure 15 120574119867119867

120574119861119861 120574119867119861 and 120574

119861119867versus 119905 120574

119867119867 120574119861119861 120574119867119861 and 120574



119867119867

vibrates under 07 120574119861119867


vibrates above 0


Appendices



119898=


0(119905)119873when




(119905) =

min 1198730(119905) 119873

1(119905)

119873

(A1)






min 1198730(119905 119895) 119873

1(119905 119895)

119899 (119905 119895) sdot 119870

= 120572 (119905 119895) +

119873119861

01(119905 119895)

119899 (119905 119895) sdot 119870

+

119873119867

01(119905 119895)

119899 (119905 119895) sdot 119870

(B1)



01(119905 119895) in (B1) is denoted as


min 1198730(119905 119895) 119873

1(119905 119895)

119899 (119905 119895) sdot 119870

= 120572 (119905 119895) +

119873119861

01(119905 119895)

119899 (119905 119895) sdot 119870

+ 119902119894119895(1)

(B2)


min 1198730(119905 119895) 119873

1(119905 119895)

119899 (119905 119895) sdot 119870

= 120572 (119905 119895) +

119873119861

01(119905 119895)

119899 (119905 119895) sdot 119870

+ 1 minus 119901119894119895(1)

(B3)


119894119895(1) are the


119902119894119895(1) =

119873119867

01(119905 119895)

119899 (119905 119895) sdot 119870

or 119901119894119895(1) = 1 minus

119873119867

01(119905 119895)

119899 (119905 119895) sdot 119870

(B4)


(119905 119895) =

min 1198730(119905 119895) 119873

1(119905 119895)

119899 (119905 119895) sdot 119870

minus 119902119894119895(1) (B5)

or

(119905 119895) =

min 1198730(119905 119895) 119873

1(119905 119895)

119899 (119905 119895) sdot 119870

+ 119901119894119895(1) minus 1 (B6)

Because of

119901119894119895(1) ge 1 minus

max119896(1003816100381610038161003816d119905119896

1003816100381610038161003816) minusmin

119896(1003816100381610038161003816d119905119896

1003816100381610038161003816)

119899 (119905 119895)

(B7)

or

119902119894119895(1) le

max119896(1003816100381610038161003816d119905119896

1003816100381610038161003816) minusmin

119896(1003816100381610038161003816d119905119896

1003816100381610038161003816)

119899 (119905 119895)

(B8)

we get

(119905 119895)

ge

min 1198730(119905 119895) 119873

1(119905 119895) minus 119870 sdot [max

119895(

10038161003816100381610038161003816d119905119895

10038161003816100381610038161003816) minusmin

119895(

10038161003816100381610038161003816d119905119895

10038161003816100381610038161003816)]

119899 (119905 119895) sdot 119870

(B9)



Acknowledgments


References




























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











V119894119895(119896) = Pr (119889

119894119895= 119896 | 119867

0)


119894119895(119896))]

+ 120572119910119894119895(1 minus 119896)

119906119894119895(119896) = Pr (119889

119894119895= 119896 | 119867

1)


119894119895(119896))]

+ 120572119909119894119895(1 minus 119896)

(16)


119894119895(119896) = Pr(119889

119894119895=

119896 | 1198670) and 119906

119894119895(119896) = Pr(119889

119894119895= 119896 | 119867


be expressed as

KLD (119906119894119895 V119894119895) = sum

119896isin01

119906119894119895(119896) log

119906119894119895(119896)

V119894119895(119896)

(17)


119894119895(119896) under

1198670and 119867









119906119894119895(119896) = Pr (119889

119894119895= 119896 | 119867

1) = Pr (119889

119894119895= 119896 | 119867

0)

= V119894119895(119896)

(18)


119894119895(119896) 119906

119894119895(119896)) = 0 is equivalent to

120572 =

119901119894119895(1) minus 119902

119894119895(1)

(119901119894119895(1) minus 119902

119894119895(1)) + (119909

119894119895(1) minus 119910

119894119895(1))

=

119901119894119895(1) minus 119902

119894119895(1)

(119901119894119895(1) minus 119902

119894119895(1)) + 119891 (119872

119894 120578119894 119901119894119895(1) 119902119894119895(1))

(19)


119891 (119872119894 120578119894 119901119894119895(1) 119902119894119895(1))

=

119872119894

sum

119898=120578119894

sum

120587isinΓ

119898

prod

119894=1

119901120587(119894)119895

(1)

119872119894

prod

119894=119898+1

(1 minus 119901120587(119894)119895

(1))

minus

119872119894

sum

119898=120578119894

sum

120587isinΓ

119898

prod

119894=1

119902120587(119894)119895

(1)

119872119894

prod

119894=119898+1

(1 minus 119902120587(119894)119895

(1))

(20)



119894119895(119896) and


119894119895 119906119894119895) = 0 if and only if

V119894119895(119896) = 119906







119894119895(1) 119902119894119895(1))

120572blind

=

119901119894119895(1) minus 119902

119894119895(1)

(119901119894119895(1) minus 119902

119894119895(1)) +max 119891 (119872

119894 120578119894 119901119894119895(1) 119902119894119895(1))

(22)



inequality

120572blind le1

1 +max 119891 (119872119894 120578119894 119901119894119895(1) 119902119894119895(1))

(23)


119889

119889119909

(

119909

119909 + 1

) =

1

(119909 + 1)2gt 0 (24)


120578119894 opt = ceil(

119872119894

2

) (25)


2 3 4 5 6 7 8 91120578

0

01

02

03

04

05

06

07

08

09

120572



119894=


119894119895(1)

119902119894119895(1)) we have

max119891 = 119891(119872119894 ceil(

119872119894

2

) 119901119894119895(1) 119902119894119895(1)) (26)


120572blind =119901119894119895(1) minus 119902

119894119895(1)

(119901119894119895(1) minus 119902

119894119895(1)) +max119891

(27)






119894and 120578 = 120578

119894 opt 119894 = 1 2 119873119861 Attacking




0

05

1

15

2

25

3

35

4

45

5

120578op

t

2 3 4 5 6 7 8 91M






119872120572

120578

1 2 3 4 5 6 7 8 91 050002 05000 050003 05434 04310 054344 05952 04310 04310 059525 06469 04509 04042 04509 064696 06959 04785 04042 04042 04785 069597 07410 05100 04145 03913 04145 05100 074108 07814 05438 04298 03913 03913 04298 05438 078149 08171 05790 04484 03970 03844 03970 04484 05790 08171






119873times119870= (119889

119894119895(119905))119873times119870



119895(119905) over vector


decision z(119905) = (1199111(119905) 1199112(119905) 119911


d1199051 d1199052 d119905

119870 119911119895(119905) isin 0 1 for any 119905 isin 119873+ 119895 = 1 2 119870


0

005

01

015

02

025

03

035

04

045

05

120572bl

ind

2 3 4 5 6 7 8 91M


0

005

01

015

02

025

03

035

04

045

05

120572bl

ind

3 5 7 9 11 13 15 17 19 21 23 251M




119872120578opt 11 21 32 42 53 63 74 84 95120572blind 05000 05000 04310 04310 04042 04042 03913 03913 03844

IMBANMBACMBA

C B A0

02

04

06

08

1

12

14

KLD

(120572)

01 02 03 04 050Attacking power 120572



119895(119905)




10038161003816100381610038161003816d119905119895

10038161003816100381610038161003816=

119873

sum

119894=1

10038161003816100381610038161003816119889119894119895(119905) minus 119911 (119905)

10038161003816100381610038161003816 (28)


10038161003816100381610038161003816d119905119894

10038161003816100381610038161003816=

119870

sum

119895=1

10038161003816100381610038161003816119889119894119895(119905) minus 119911 (119905)

10038161003816100381610038161003816 (29)





IMBANMBACMBA

C B A01 02 03 04 050


0

02

04

06

08

1

12

14

KLD

(120572)






119895| is small





0(119905 119895) and



1(119905 119895)


10038161003816100381610038161003816d119905119895

10038161003816100381610038161003816= sum

119894isin119868(119905119895)

10038161003816100381610038161003816119889119894119895(119905) minus 119911 (119905)

10038161003816100381610038161003816

10038161003816100381610038161003816d119905119894

10038161003816100381610038161003816=

119870

sum

119896=1

1003816100381610038161003816119889119894119896(119905) minus 119911 (119905)

1003816100381610038161003816 119894 isin 119868 (119905 119895)

(30)




119898= 119875119891119886


(119905) =

min 1198730(119905) 119873

1(119905)

119873

(31)

where1198730(119905) and119873





(119905 119895) =

min 1198730(119905 119895) 119873

1(119905 119895)

1198730(119905 119895) + 119873

1(119905 119895)

(32)





(119905 119895)

ge

min 1198730(119905 119895) 119873

1(119905 119895) minus 119870 sdot [max

119896(1003816100381610038161003816d119905119896

1003816100381610038161003816) minusmin

119896(1003816100381610038161003816d119905119896

1003816100381610038161003816)]

119899 (119905 119895) sdot 119870

119896 isin 1 2 119870

(33)


[max119896(|d119905119896|) minus min




Here we let

(119905 119895)

=

min 1198730(119905 119895) 119873

1(119905 119895) minus 119870 sdot [max

119895(

10038161003816100381610038161003816d119905119895

10038161003816100381610038161003816) minusmin

119895(

10038161003816100381610038161003816d119905119895

10038161003816100381610038161003816)]

119899 (119905 119895) sdot 119870

(34)




119873(119905 119895) over |d119905






119868119873(119905 119895)

=

10038161003816100381610038161003816d119905119894

10038161003816100381610038161003816

119899(119905119895)

10038161003816100381610038161003816d119905119894

10038161003816100381610038161003816

119899(119905119895)minus1

10038161003816100381610038161003816d119905119894

10038161003816100381610038161003816

1

| 119894 isin 119871 (119905 119895 minus 1)

(35)




over 119868119873(119905 119895)

119899 = ID (d119905119894)

= sum

119897isin119871(119905119895minus1)

IF (10038161003816100381610038161003816d119905119897

10038161003816100381610038161003816gt

10038161003816100381610038161003816d119905119894

10038161003816100381610038161003816

119899

)

+ sum

119897isin119871(119905119895minus1)

IF (10038161003816100381610038161003816d119905119897

10038161003816100381610038161003816=

10038161003816100381610038161003816d119905119894

10038161003816100381610038161003816

119899

| 119897 gt 119895)

119899 isin 1 2 119899 (119905 119895)

(36)




119868 (119905 119895)


(37)


(119905) =

119873 minus 119899 (119905 119870)

119873

(38)






1(119905)

1199112(119905) 119911


decision d1199051 d1199052 d119905






120578fc (119905 119895)




119886 = [1 minus (119905 119895)] sdot 119901119894119895(1) + (119905 119895) sdot 119909

119894119895(1) (41)


119894119895(1)) + (119905 119895)

sdot 119910119894119895(1) sdot (1 minus 119910

119894119895(1))

(42)


119911119895(119905) = IF(sum

119894

119889119894119895(119905) gt 120578fc (119905 119895) | 119894 isin 119871 (119905 119895 minus 1)) (43)


120574119867119861

120574119861119867

and 120574119861119861


119867119867and 120574119867119861


119861119867and 120574119861119861


120574119867119861

=

119873119867119861

119873

120574119867119867

=

119899 (119905 119870) minus 119873119861119867

119873

120574119861119867

=

119873119861119867

119873

120574119861119861

=

119873 minus 119899 (119905 119870) minus 119873119867119861

119873

(44)

Therein 119873119867119861


119861119867 119873119867and 119873


in Section 2





0(119905 119895) and 119873



0(119905 119895) 119873

0(119905 119895) and

1198731(119905 119895) verge to119870sdot119873

119861and119870sdot119873




119867119867 120574119861119861 120574119867119861

and 120574119861119867



cross And 120574119867119867


119861119867and 120574119867119861


5 Conclusion



0

10

20

30

40

50

60

70

80

90

100n(tj)

2 3 4 5 6 7 8 9 101j


0

100

200

300

400

500

600

N1(tj)

2 3 4 5 6 7 8 9 101j


0

50

100

150

200

250

300

350

400

450

N0(tj)

2 3 4 5 6 7 8 9 101j


0

10

20

30

40

50

60

70

2 3 4 5 6 7 8 9 101j

120578fc(tj)


0

005

01

015

02

025

03

035

120572(tj)

2 3 4 5 6 7 8 9 101j


Figure 10 119899(119905 119895)1198730(119905 119895)119873



2 3 4 5 6 7 8 9 101j

0

10

20

30

40

50

60

70

80

90

100n(tj)


2 3 4 5 6 7 8 9 101j

0

100

200

300

400

500

600

N1(tj)


0

50

100

150

200

250

300

350

400

450

N0(tj)

2 3 4 5 6 7 8 9 101j


2 3 4 5 6 7 8 9 101j

0

10

20

30

40

50

60

70

120578fc(tj)


0

005

01

015

02

025

03

035

120572(tj)

2 3 4 5 6 7 8 9 101j


Figure 11 119899(119905 119895)1198730(119905 119895)119873



2 3 4 5 6 7 8 9 10 11 12 13 14 151j

0

10

20

30

40

50

60

70

80

90

100n(tj)


0

100

200

300

400

500

600

700

800

900

N1(tj)

2 3 4 5 6 7 8 9 10 11 12 13 14 151j


2 3 4 5 6 7 8 9 10 11 12 13 14 151j

0

100

200

300

400

500

600

700

N0(tj)


0

10

20

30

40

50

60

70

2 3 4 5 6 7 8 9 10 11 12 13 14 151j

120578fc(tj)


2 3 4 5 6 7 8 9 10 11 12 13 14 151j

0

005

01

015

02

025

03

035

120572(tj)


Figure 12 119899(119905 119895)1198730(119905 119895)119873



0

10

20

30

40

50

60

70

80

90

100n(tj)

2 3 4 5 6 7 8 9 10 11 12 13 14 151j


0

100

200

300

400

500

600

700

800

900

N1(tj)

2 3 4 5 6 7 8 9 10 11 12 13 14 151j


0

100

200

300

400

500

600

700

N0(tj)

2 3 4 5 6 7 8 9 10 11 12 13 14 151j


0

10

20

30

40

50

60

70

2 3 4 5 6 7 8 9 10 11 12 13 14 151j

120578fc(tj)


0

005

01

015

02

025

03

035

04

120572(tj)

2 3 4 5 6 7 8 9 10 11 12 13 14 151j


Figure 13 119899(119905 119895)1198730(119905 119895)119873



0

01

02

03

04

05

06

07

20 40 60 80 1000(a) 119870 = 10119872 = 9

0

01

02

03

04

05

06

07

20 40 60 80 1000(b) 119870 = 10119872 = 25

Figure 14 120574119867119867

120574119861119861 120574119867119861 and 120574

119861119867versus 119905 120574

119867119867 120574119861119861 120574119867119861 and 120574



119867119867

vibrates under 07 120574119861119867


vibrates above 0

20 40 60 80 10000

01

02

03

04

05

06

07

(a) 119870 = 15119872 = 920 40 60 80 1000

0

01

02

03

04

05

06

07

(b) 119870 = 15119872 = 25

Figure 15 120574119867119867

120574119861119861 120574119867119861 and 120574

119861119867versus 119905 120574

119867119867 120574119861119861 120574119867119861 and 120574



119867119867

vibrates under 07 120574119861119867


vibrates above 0


Appendices



119898=


0(119905)119873when




(119905) =

min 1198730(119905) 119873

1(119905)

119873

(A1)






min 1198730(119905 119895) 119873

1(119905 119895)

119899 (119905 119895) sdot 119870

= 120572 (119905 119895) +

119873119861

01(119905 119895)

119899 (119905 119895) sdot 119870

+

119873119867

01(119905 119895)

119899 (119905 119895) sdot 119870

(B1)



01(119905 119895) in (B1) is denoted as


min 1198730(119905 119895) 119873

1(119905 119895)

119899 (119905 119895) sdot 119870

= 120572 (119905 119895) +

119873119861

01(119905 119895)

119899 (119905 119895) sdot 119870

+ 119902119894119895(1)

(B2)


min 1198730(119905 119895) 119873

1(119905 119895)

119899 (119905 119895) sdot 119870

= 120572 (119905 119895) +

119873119861

01(119905 119895)

119899 (119905 119895) sdot 119870

+ 1 minus 119901119894119895(1)

(B3)


119894119895(1) are the


119902119894119895(1) =

119873119867

01(119905 119895)

119899 (119905 119895) sdot 119870

or 119901119894119895(1) = 1 minus

119873119867

01(119905 119895)

119899 (119905 119895) sdot 119870

(B4)


(119905 119895) =

min 1198730(119905 119895) 119873

1(119905 119895)

119899 (119905 119895) sdot 119870

minus 119902119894119895(1) (B5)

or

(119905 119895) =

min 1198730(119905 119895) 119873

1(119905 119895)

119899 (119905 119895) sdot 119870

+ 119901119894119895(1) minus 1 (B6)

Because of

119901119894119895(1) ge 1 minus

max119896(1003816100381610038161003816d119905119896

1003816100381610038161003816) minusmin

119896(1003816100381610038161003816d119905119896

1003816100381610038161003816)

119899 (119905 119895)

(B7)

or

119902119894119895(1) le

max119896(1003816100381610038161003816d119905119896

1003816100381610038161003816) minusmin

119896(1003816100381610038161003816d119905119896

1003816100381610038161003816)

119899 (119905 119895)

(B8)

we get

(119905 119895)

ge

min 1198730(119905 119895) 119873

1(119905 119895) minus 119870 sdot [max

119895(

10038161003816100381610038161003816d119905119895

10038161003816100381610038161003816) minusmin

119895(

10038161003816100381610038161003816d119905119895

10038161003816100381610038161003816)]

119899 (119905 119895) sdot 119870

(B9)



Acknowledgments


References




























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










2 3 4 5 6 7 8 91120578

0

01

02

03

04

05

06

07

08

09

120572



119894=


119894119895(1)

119902119894119895(1)) we have

max119891 = 119891(119872119894 ceil(

119872119894

2

) 119901119894119895(1) 119902119894119895(1)) (26)


120572blind =119901119894119895(1) minus 119902

119894119895(1)

(119901119894119895(1) minus 119902

119894119895(1)) +max119891

(27)






119894and 120578 = 120578

119894 opt 119894 = 1 2 119873119861 Attacking




0

05

1

15

2

25

3

35

4

45

5

120578op

t

2 3 4 5 6 7 8 91M






119872120572

120578

1 2 3 4 5 6 7 8 91 050002 05000 050003 05434 04310 054344 05952 04310 04310 059525 06469 04509 04042 04509 064696 06959 04785 04042 04042 04785 069597 07410 05100 04145 03913 04145 05100 074108 07814 05438 04298 03913 03913 04298 05438 078149 08171 05790 04484 03970 03844 03970 04484 05790 08171






119873times119870= (119889

119894119895(119905))119873times119870



119895(119905) over vector


decision z(119905) = (1199111(119905) 1199112(119905) 119911


d1199051 d1199052 d119905

119870 119911119895(119905) isin 0 1 for any 119905 isin 119873+ 119895 = 1 2 119870


0

005

01

015

02

025

03

035

04

045

05

120572bl

ind

2 3 4 5 6 7 8 91M


0

005

01

015

02

025

03

035

04

045

05

120572bl

ind

3 5 7 9 11 13 15 17 19 21 23 251M




119872120578opt 11 21 32 42 53 63 74 84 95120572blind 05000 05000 04310 04310 04042 04042 03913 03913 03844

IMBANMBACMBA

C B A0

02

04

06

08

1

12

14

KLD

(120572)

01 02 03 04 050Attacking power 120572



119895(119905)




10038161003816100381610038161003816d119905119895

10038161003816100381610038161003816=

119873

sum

119894=1

10038161003816100381610038161003816119889119894119895(119905) minus 119911 (119905)

10038161003816100381610038161003816 (28)


10038161003816100381610038161003816d119905119894

10038161003816100381610038161003816=

119870

sum

119895=1

10038161003816100381610038161003816119889119894119895(119905) minus 119911 (119905)

10038161003816100381610038161003816 (29)





IMBANMBACMBA

C B A01 02 03 04 050


0

02

04

06

08

1

12

14

KLD

(120572)






119895| is small





0(119905 119895) and



1(119905 119895)


10038161003816100381610038161003816d119905119895

10038161003816100381610038161003816= sum

119894isin119868(119905119895)

10038161003816100381610038161003816119889119894119895(119905) minus 119911 (119905)

10038161003816100381610038161003816

10038161003816100381610038161003816d119905119894

10038161003816100381610038161003816=

119870

sum

119896=1

1003816100381610038161003816119889119894119896(119905) minus 119911 (119905)

1003816100381610038161003816 119894 isin 119868 (119905 119895)

(30)




119898= 119875119891119886


(119905) =

min 1198730(119905) 119873

1(119905)

119873

(31)

where1198730(119905) and119873





(119905 119895) =

min 1198730(119905 119895) 119873

1(119905 119895)

1198730(119905 119895) + 119873

1(119905 119895)

(32)





(119905 119895)

ge

min 1198730(119905 119895) 119873

1(119905 119895) minus 119870 sdot [max

119896(1003816100381610038161003816d119905119896

1003816100381610038161003816) minusmin

119896(1003816100381610038161003816d119905119896

1003816100381610038161003816)]

119899 (119905 119895) sdot 119870

119896 isin 1 2 119870

(33)


[max119896(|d119905119896|) minus min




Here we let

(119905 119895)

=

min 1198730(119905 119895) 119873

1(119905 119895) minus 119870 sdot [max

119895(

10038161003816100381610038161003816d119905119895

10038161003816100381610038161003816) minusmin

119895(

10038161003816100381610038161003816d119905119895

10038161003816100381610038161003816)]

119899 (119905 119895) sdot 119870

(34)




119873(119905 119895) over |d119905






119868119873(119905 119895)

=

10038161003816100381610038161003816d119905119894

10038161003816100381610038161003816

119899(119905119895)

10038161003816100381610038161003816d119905119894

10038161003816100381610038161003816

119899(119905119895)minus1

10038161003816100381610038161003816d119905119894

10038161003816100381610038161003816

1

| 119894 isin 119871 (119905 119895 minus 1)

(35)




over 119868119873(119905 119895)

119899 = ID (d119905119894)

= sum

119897isin119871(119905119895minus1)

IF (10038161003816100381610038161003816d119905119897

10038161003816100381610038161003816gt

10038161003816100381610038161003816d119905119894

10038161003816100381610038161003816

119899

)

+ sum

119897isin119871(119905119895minus1)

IF (10038161003816100381610038161003816d119905119897

10038161003816100381610038161003816=

10038161003816100381610038161003816d119905119894

10038161003816100381610038161003816

119899

| 119897 gt 119895)

119899 isin 1 2 119899 (119905 119895)

(36)




119868 (119905 119895)


(37)


(119905) =

119873 minus 119899 (119905 119870)

119873

(38)






1(119905)

1199112(119905) 119911


decision d1199051 d1199052 d119905






120578fc (119905 119895)




119886 = [1 minus (119905 119895)] sdot 119901119894119895(1) + (119905 119895) sdot 119909

119894119895(1) (41)


119894119895(1)) + (119905 119895)

sdot 119910119894119895(1) sdot (1 minus 119910

119894119895(1))

(42)


119911119895(119905) = IF(sum

119894

119889119894119895(119905) gt 120578fc (119905 119895) | 119894 isin 119871 (119905 119895 minus 1)) (43)


120574119867119861

120574119861119867

and 120574119861119861


119867119867and 120574119867119861


119861119867and 120574119861119861


120574119867119861

=

119873119867119861

119873

120574119867119867

=

119899 (119905 119870) minus 119873119861119867

119873

120574119861119867

=

119873119861119867

119873

120574119861119861

=

119873 minus 119899 (119905 119870) minus 119873119867119861

119873

(44)

Therein 119873119867119861


119861119867 119873119867and 119873


in Section 2





0(119905 119895) and 119873



0(119905 119895) 119873

0(119905 119895) and

1198731(119905 119895) verge to119870sdot119873

119861and119870sdot119873




119867119867 120574119861119861 120574119867119861

and 120574119861119867



cross And 120574119867119867


119861119867and 120574119867119861


5 Conclusion



0

10

20

30

40

50

60

70

80

90

100n(tj)

2 3 4 5 6 7 8 9 101j


0

100

200

300

400

500

600

N1(tj)

2 3 4 5 6 7 8 9 101j


0

50

100

150

200

250

300

350

400

450

N0(tj)

2 3 4 5 6 7 8 9 101j


0

10

20

30

40

50

60

70

2 3 4 5 6 7 8 9 101j

120578fc(tj)


0

005

01

015

02

025

03

035

120572(tj)

2 3 4 5 6 7 8 9 101j


Figure 10 119899(119905 119895)1198730(119905 119895)119873



2 3 4 5 6 7 8 9 101j

0

10

20

30

40

50

60

70

80

90

100n(tj)


2 3 4 5 6 7 8 9 101j

0

100

200

300

400

500

600

N1(tj)


0

50

100

150

200

250

300

350

400

450

N0(tj)

2 3 4 5 6 7 8 9 101j


2 3 4 5 6 7 8 9 101j

0

10

20

30

40

50

60

70

120578fc(tj)


0

005

01

015

02

025

03

035

120572(tj)

2 3 4 5 6 7 8 9 101j


Figure 11 119899(119905 119895)1198730(119905 119895)119873



2 3 4 5 6 7 8 9 10 11 12 13 14 151j

0

10

20

30

40

50

60

70

80

90

100n(tj)


0

100

200

300

400

500

600

700

800

900

N1(tj)

2 3 4 5 6 7 8 9 10 11 12 13 14 151j


2 3 4 5 6 7 8 9 10 11 12 13 14 151j

0

100

200

300

400

500

600

700

N0(tj)


0

10

20

30

40

50

60

70

2 3 4 5 6 7 8 9 10 11 12 13 14 151j

120578fc(tj)


2 3 4 5 6 7 8 9 10 11 12 13 14 151j

0

005

01

015

02

025

03

035

120572(tj)


Figure 12 119899(119905 119895)1198730(119905 119895)119873



0

10

20

30

40

50

60

70

80

90

100n(tj)

2 3 4 5 6 7 8 9 10 11 12 13 14 151j


0

100

200

300

400

500

600

700

800

900

N1(tj)

2 3 4 5 6 7 8 9 10 11 12 13 14 151j


0

100

200

300

400

500

600

700

N0(tj)

2 3 4 5 6 7 8 9 10 11 12 13 14 151j


0

10

20

30

40

50

60

70

2 3 4 5 6 7 8 9 10 11 12 13 14 151j

120578fc(tj)


0

005

01

015

02

025

03

035

04

120572(tj)

2 3 4 5 6 7 8 9 10 11 12 13 14 151j


Figure 13 119899(119905 119895)1198730(119905 119895)119873



0

01

02

03

04

05

06

07

20 40 60 80 1000(a) 119870 = 10119872 = 9

0

01

02

03

04

05

06

07

20 40 60 80 1000(b) 119870 = 10119872 = 25

Figure 14 120574119867119867

120574119861119861 120574119867119861 and 120574

119861119867versus 119905 120574

119867119867 120574119861119861 120574119867119861 and 120574



119867119867

vibrates under 07 120574119861119867


vibrates above 0

20 40 60 80 10000

01

02

03

04

05

06

07

(a) 119870 = 15119872 = 920 40 60 80 1000

0

01

02

03

04

05

06

07

(b) 119870 = 15119872 = 25

Figure 15 120574119867119867

120574119861119861 120574119867119861 and 120574

119861119867versus 119905 120574

119867119867 120574119861119861 120574119867119861 and 120574



119867119867

vibrates under 07 120574119861119867


vibrates above 0


Appendices



119898=


0(119905)119873when




(119905) =

min 1198730(119905) 119873

1(119905)

119873

(A1)






min 1198730(119905 119895) 119873

1(119905 119895)

119899 (119905 119895) sdot 119870

= 120572 (119905 119895) +

119873119861

01(119905 119895)

119899 (119905 119895) sdot 119870

+

119873119867

01(119905 119895)

119899 (119905 119895) sdot 119870

(B1)



01(119905 119895) in (B1) is denoted as


min 1198730(119905 119895) 119873

1(119905 119895)

119899 (119905 119895) sdot 119870

= 120572 (119905 119895) +

119873119861

01(119905 119895)

119899 (119905 119895) sdot 119870

+ 119902119894119895(1)

(B2)


min 1198730(119905 119895) 119873

1(119905 119895)

119899 (119905 119895) sdot 119870

= 120572 (119905 119895) +

119873119861

01(119905 119895)

119899 (119905 119895) sdot 119870

+ 1 minus 119901119894119895(1)

(B3)


119894119895(1) are the


119902119894119895(1) =

119873119867

01(119905 119895)

119899 (119905 119895) sdot 119870

or 119901119894119895(1) = 1 minus

119873119867

01(119905 119895)

119899 (119905 119895) sdot 119870

(B4)


(119905 119895) =

min 1198730(119905 119895) 119873

1(119905 119895)

119899 (119905 119895) sdot 119870

minus 119902119894119895(1) (B5)

or

(119905 119895) =

min 1198730(119905 119895) 119873

1(119905 119895)

119899 (119905 119895) sdot 119870

+ 119901119894119895(1) minus 1 (B6)

Because of

119901119894119895(1) ge 1 minus

max119896(1003816100381610038161003816d119905119896

1003816100381610038161003816) minusmin

119896(1003816100381610038161003816d119905119896

1003816100381610038161003816)

119899 (119905 119895)

(B7)

or

119902119894119895(1) le

max119896(1003816100381610038161003816d119905119896

1003816100381610038161003816) minusmin

119896(1003816100381610038161003816d119905119896

1003816100381610038161003816)

119899 (119905 119895)

(B8)

we get

(119905 119895)

ge

min 1198730(119905 119895) 119873

1(119905 119895) minus 119870 sdot [max

119895(

10038161003816100381610038161003816d119905119895

10038161003816100381610038161003816) minusmin

119895(

10038161003816100381610038161003816d119905119895

10038161003816100381610038161003816)]

119899 (119905 119895) sdot 119870

(B9)



Acknowledgments


References




























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











119872120572

120578

1 2 3 4 5 6 7 8 91 050002 05000 050003 05434 04310 054344 05952 04310 04310 059525 06469 04509 04042 04509 064696 06959 04785 04042 04042 04785 069597 07410 05100 04145 03913 04145 05100 074108 07814 05438 04298 03913 03913 04298 05438 078149 08171 05790 04484 03970 03844 03970 04484 05790 08171






119873times119870= (119889

119894119895(119905))119873times119870



119895(119905) over vector


decision z(119905) = (1199111(119905) 1199112(119905) 119911


d1199051 d1199052 d119905

119870 119911119895(119905) isin 0 1 for any 119905 isin 119873+ 119895 = 1 2 119870


0

005

01

015

02

025

03

035

04

045

05

120572bl

ind

2 3 4 5 6 7 8 91M


0

005

01

015

02

025

03

035

04

045

05

120572bl

ind

3 5 7 9 11 13 15 17 19 21 23 251M




119872120578opt 11 21 32 42 53 63 74 84 95120572blind 05000 05000 04310 04310 04042 04042 03913 03913 03844

IMBANMBACMBA

C B A0

02

04

06

08

1

12

14

KLD

(120572)

01 02 03 04 050Attacking power 120572



119895(119905)




10038161003816100381610038161003816d119905119895

10038161003816100381610038161003816=

119873

sum

119894=1

10038161003816100381610038161003816119889119894119895(119905) minus 119911 (119905)

10038161003816100381610038161003816 (28)


10038161003816100381610038161003816d119905119894

10038161003816100381610038161003816=

119870

sum

119895=1

10038161003816100381610038161003816119889119894119895(119905) minus 119911 (119905)

10038161003816100381610038161003816 (29)





IMBANMBACMBA

C B A01 02 03 04 050


0

02

04

06

08

1

12

14

KLD

(120572)






119895| is small





0(119905 119895) and



1(119905 119895)


10038161003816100381610038161003816d119905119895

10038161003816100381610038161003816= sum

119894isin119868(119905119895)

10038161003816100381610038161003816119889119894119895(119905) minus 119911 (119905)

10038161003816100381610038161003816

10038161003816100381610038161003816d119905119894

10038161003816100381610038161003816=

119870

sum

119896=1

1003816100381610038161003816119889119894119896(119905) minus 119911 (119905)

1003816100381610038161003816 119894 isin 119868 (119905 119895)

(30)




119898= 119875119891119886


(119905) =

min 1198730(119905) 119873

1(119905)

119873

(31)

where1198730(119905) and119873





(119905 119895) =

min 1198730(119905 119895) 119873

1(119905 119895)

1198730(119905 119895) + 119873

1(119905 119895)

(32)





(119905 119895)

ge

min 1198730(119905 119895) 119873

1(119905 119895) minus 119870 sdot [max

119896(1003816100381610038161003816d119905119896

1003816100381610038161003816) minusmin

119896(1003816100381610038161003816d119905119896

1003816100381610038161003816)]

119899 (119905 119895) sdot 119870

119896 isin 1 2 119870

(33)


[max119896(|d119905119896|) minus min




Here we let

(119905 119895)

=

min 1198730(119905 119895) 119873

1(119905 119895) minus 119870 sdot [max

119895(

10038161003816100381610038161003816d119905119895

10038161003816100381610038161003816) minusmin

119895(

10038161003816100381610038161003816d119905119895

10038161003816100381610038161003816)]

119899 (119905 119895) sdot 119870

(34)




119873(119905 119895) over |d119905






119868119873(119905 119895)

=

10038161003816100381610038161003816d119905119894

10038161003816100381610038161003816

119899(119905119895)

10038161003816100381610038161003816d119905119894

10038161003816100381610038161003816

119899(119905119895)minus1

10038161003816100381610038161003816d119905119894

10038161003816100381610038161003816

1

| 119894 isin 119871 (119905 119895 minus 1)

(35)




over 119868119873(119905 119895)

119899 = ID (d119905119894)

= sum

119897isin119871(119905119895minus1)

IF (10038161003816100381610038161003816d119905119897

10038161003816100381610038161003816gt

10038161003816100381610038161003816d119905119894

10038161003816100381610038161003816

119899

)

+ sum

119897isin119871(119905119895minus1)

IF (10038161003816100381610038161003816d119905119897

10038161003816100381610038161003816=

10038161003816100381610038161003816d119905119894

10038161003816100381610038161003816

119899

| 119897 gt 119895)

119899 isin 1 2 119899 (119905 119895)

(36)




119868 (119905 119895)


(37)


(119905) =

119873 minus 119899 (119905 119870)

119873

(38)






1(119905)

1199112(119905) 119911


decision d1199051 d1199052 d119905






120578fc (119905 119895)




119886 = [1 minus (119905 119895)] sdot 119901119894119895(1) + (119905 119895) sdot 119909

119894119895(1) (41)


119894119895(1)) + (119905 119895)

sdot 119910119894119895(1) sdot (1 minus 119910

119894119895(1))

(42)


119911119895(119905) = IF(sum

119894

119889119894119895(119905) gt 120578fc (119905 119895) | 119894 isin 119871 (119905 119895 minus 1)) (43)


120574119867119861

120574119861119867

and 120574119861119861


119867119867and 120574119867119861


119861119867and 120574119861119861


120574119867119861

=

119873119867119861

119873

120574119867119867

=

119899 (119905 119870) minus 119873119861119867

119873

120574119861119867

=

119873119861119867

119873

120574119861119861

=

119873 minus 119899 (119905 119870) minus 119873119867119861

119873

(44)

Therein 119873119867119861


119861119867 119873119867and 119873


in Section 2





0(119905 119895) and 119873



0(119905 119895) 119873

0(119905 119895) and

1198731(119905 119895) verge to119870sdot119873

119861and119870sdot119873




119867119867 120574119861119861 120574119867119861

and 120574119861119867



cross And 120574119867119867


119861119867and 120574119867119861


5 Conclusion



0

10

20

30

40

50

60

70

80

90

100n(tj)

2 3 4 5 6 7 8 9 101j


0

100

200

300

400

500

600

N1(tj)

2 3 4 5 6 7 8 9 101j


0

50

100

150

200

250

300

350

400

450

N0(tj)

2 3 4 5 6 7 8 9 101j


0

10

20

30

40

50

60

70

2 3 4 5 6 7 8 9 101j

120578fc(tj)


0

005

01

015

02

025

03

035

120572(tj)

2 3 4 5 6 7 8 9 101j


Figure 10 119899(119905 119895)1198730(119905 119895)119873



2 3 4 5 6 7 8 9 101j

0

10

20

30

40

50

60

70

80

90

100n(tj)


2 3 4 5 6 7 8 9 101j

0

100

200

300

400

500

600

N1(tj)


0

50

100

150

200

250

300

350

400

450

N0(tj)

2 3 4 5 6 7 8 9 101j


2 3 4 5 6 7 8 9 101j

0

10

20

30

40

50

60

70

120578fc(tj)


0

005

01

015

02

025

03

035

120572(tj)

2 3 4 5 6 7 8 9 101j


Figure 11 119899(119905 119895)1198730(119905 119895)119873



2 3 4 5 6 7 8 9 10 11 12 13 14 151j

0

10

20

30

40

50

60

70

80

90

100n(tj)


0

100

200

300

400

500

600

700

800

900

N1(tj)

2 3 4 5 6 7 8 9 10 11 12 13 14 151j


2 3 4 5 6 7 8 9 10 11 12 13 14 151j

0

100

200

300

400

500

600

700

N0(tj)


0

10

20

30

40

50

60

70

2 3 4 5 6 7 8 9 10 11 12 13 14 151j

120578fc(tj)


2 3 4 5 6 7 8 9 10 11 12 13 14 151j

0

005

01

015

02

025

03

035

120572(tj)


Figure 12 119899(119905 119895)1198730(119905 119895)119873



0

10

20

30

40

50

60

70

80

90

100n(tj)

2 3 4 5 6 7 8 9 10 11 12 13 14 151j


0

100

200

300

400

500

600

700

800

900

N1(tj)

2 3 4 5 6 7 8 9 10 11 12 13 14 151j


0

100

200

300

400

500

600

700

N0(tj)

2 3 4 5 6 7 8 9 10 11 12 13 14 151j


0

10

20

30

40

50

60

70

2 3 4 5 6 7 8 9 10 11 12 13 14 151j

120578fc(tj)


0

005

01

015

02

025

03

035

04

120572(tj)

2 3 4 5 6 7 8 9 10 11 12 13 14 151j


Figure 13 119899(119905 119895)1198730(119905 119895)119873



0

01

02

03

04

05

06

07

20 40 60 80 1000(a) 119870 = 10119872 = 9

0

01

02

03

04

05

06

07

20 40 60 80 1000(b) 119870 = 10119872 = 25

Figure 14 120574119867119867

120574119861119861 120574119867119861 and 120574

119861119867versus 119905 120574

119867119867 120574119861119861 120574119867119861 and 120574



119867119867

vibrates under 07 120574119861119867


vibrates above 0

20 40 60 80 10000

01

02

03

04

05

06

07

(a) 119870 = 15119872 = 920 40 60 80 1000

0

01

02

03

04

05

06

07

(b) 119870 = 15119872 = 25

Figure 15 120574119867119867

120574119861119861 120574119867119861 and 120574

119861119867versus 119905 120574

119867119867 120574119861119861 120574119867119861 and 120574



119867119867

vibrates under 07 120574119861119867


vibrates above 0


Appendices



119898=


0(119905)119873when




(119905) =

min 1198730(119905) 119873

1(119905)

119873

(A1)






min 1198730(119905 119895) 119873

1(119905 119895)

119899 (119905 119895) sdot 119870

= 120572 (119905 119895) +

119873119861

01(119905 119895)

119899 (119905 119895) sdot 119870

+

119873119867

01(119905 119895)

119899 (119905 119895) sdot 119870

(B1)



01(119905 119895) in (B1) is denoted as


min 1198730(119905 119895) 119873

1(119905 119895)

119899 (119905 119895) sdot 119870

= 120572 (119905 119895) +

119873119861

01(119905 119895)

119899 (119905 119895) sdot 119870

+ 119902119894119895(1)

(B2)


min 1198730(119905 119895) 119873

1(119905 119895)

119899 (119905 119895) sdot 119870

= 120572 (119905 119895) +

119873119861

01(119905 119895)

119899 (119905 119895) sdot 119870

+ 1 minus 119901119894119895(1)

(B3)


119894119895(1) are the


119902119894119895(1) =

119873119867

01(119905 119895)

119899 (119905 119895) sdot 119870

or 119901119894119895(1) = 1 minus

119873119867

01(119905 119895)

119899 (119905 119895) sdot 119870

(B4)


(119905 119895) =

min 1198730(119905 119895) 119873

1(119905 119895)

119899 (119905 119895) sdot 119870

minus 119902119894119895(1) (B5)

or

(119905 119895) =

min 1198730(119905 119895) 119873

1(119905 119895)

119899 (119905 119895) sdot 119870

+ 119901119894119895(1) minus 1 (B6)

Because of

119901119894119895(1) ge 1 minus

max119896(1003816100381610038161003816d119905119896

1003816100381610038161003816) minusmin

119896(1003816100381610038161003816d119905119896

1003816100381610038161003816)

119899 (119905 119895)

(B7)

or

119902119894119895(1) le

max119896(1003816100381610038161003816d119905119896

1003816100381610038161003816) minusmin

119896(1003816100381610038161003816d119905119896

1003816100381610038161003816)

119899 (119905 119895)

(B8)

we get

(119905 119895)

ge

min 1198730(119905 119895) 119873

1(119905 119895) minus 119870 sdot [max

119895(

10038161003816100381610038161003816d119905119895

10038161003816100381610038161003816) minusmin

119895(

10038161003816100381610038161003816d119905119895

10038161003816100381610038161003816)]

119899 (119905 119895) sdot 119870

(B9)



Acknowledgments


References




























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











119872120578opt 11 21 32 42 53 63 74 84 95120572blind 05000 05000 04310 04310 04042 04042 03913 03913 03844

IMBANMBACMBA

C B A0

02

04

06

08

1

12

14

KLD

(120572)

01 02 03 04 050Attacking power 120572



119895(119905)




10038161003816100381610038161003816d119905119895

10038161003816100381610038161003816=

119873

sum

119894=1

10038161003816100381610038161003816119889119894119895(119905) minus 119911 (119905)

10038161003816100381610038161003816 (28)


10038161003816100381610038161003816d119905119894

10038161003816100381610038161003816=

119870

sum

119895=1

10038161003816100381610038161003816119889119894119895(119905) minus 119911 (119905)

10038161003816100381610038161003816 (29)





IMBANMBACMBA

C B A01 02 03 04 050


0

02

04

06

08

1

12

14

KLD

(120572)






119895| is small





0(119905 119895) and



1(119905 119895)


10038161003816100381610038161003816d119905119895

10038161003816100381610038161003816= sum

119894isin119868(119905119895)

10038161003816100381610038161003816119889119894119895(119905) minus 119911 (119905)

10038161003816100381610038161003816

10038161003816100381610038161003816d119905119894

10038161003816100381610038161003816=

119870

sum

119896=1

1003816100381610038161003816119889119894119896(119905) minus 119911 (119905)

1003816100381610038161003816 119894 isin 119868 (119905 119895)

(30)




119898= 119875119891119886


(119905) =

min 1198730(119905) 119873

1(119905)

119873

(31)

where1198730(119905) and119873





(119905 119895) =

min 1198730(119905 119895) 119873

1(119905 119895)

1198730(119905 119895) + 119873

1(119905 119895)

(32)





(119905 119895)

ge

min 1198730(119905 119895) 119873

1(119905 119895) minus 119870 sdot [max

119896(1003816100381610038161003816d119905119896

1003816100381610038161003816) minusmin

119896(1003816100381610038161003816d119905119896

1003816100381610038161003816)]

119899 (119905 119895) sdot 119870

119896 isin 1 2 119870

(33)


[max119896(|d119905119896|) minus min




Here we let

(119905 119895)

=

min 1198730(119905 119895) 119873

1(119905 119895) minus 119870 sdot [max

119895(

10038161003816100381610038161003816d119905119895

10038161003816100381610038161003816) minusmin

119895(

10038161003816100381610038161003816d119905119895

10038161003816100381610038161003816)]

119899 (119905 119895) sdot 119870

(34)




119873(119905 119895) over |d119905






119868119873(119905 119895)

=

10038161003816100381610038161003816d119905119894

10038161003816100381610038161003816

119899(119905119895)

10038161003816100381610038161003816d119905119894

10038161003816100381610038161003816

119899(119905119895)minus1

10038161003816100381610038161003816d119905119894

10038161003816100381610038161003816

1

| 119894 isin 119871 (119905 119895 minus 1)

(35)




over 119868119873(119905 119895)

119899 = ID (d119905119894)

= sum

119897isin119871(119905119895minus1)

IF (10038161003816100381610038161003816d119905119897

10038161003816100381610038161003816gt

10038161003816100381610038161003816d119905119894

10038161003816100381610038161003816

119899

)

+ sum

119897isin119871(119905119895minus1)

IF (10038161003816100381610038161003816d119905119897

10038161003816100381610038161003816=

10038161003816100381610038161003816d119905119894

10038161003816100381610038161003816

119899

| 119897 gt 119895)

119899 isin 1 2 119899 (119905 119895)

(36)




119868 (119905 119895)


(37)


(119905) =

119873 minus 119899 (119905 119870)

119873

(38)






1(119905)

1199112(119905) 119911


decision d1199051 d1199052 d119905






120578fc (119905 119895)




119886 = [1 minus (119905 119895)] sdot 119901119894119895(1) + (119905 119895) sdot 119909

119894119895(1) (41)


119894119895(1)) + (119905 119895)

sdot 119910119894119895(1) sdot (1 minus 119910

119894119895(1))

(42)


119911119895(119905) = IF(sum

119894

119889119894119895(119905) gt 120578fc (119905 119895) | 119894 isin 119871 (119905 119895 minus 1)) (43)


120574119867119861

120574119861119867

and 120574119861119861


119867119867and 120574119867119861


119861119867and 120574119861119861


120574119867119861

=

119873119867119861

119873

120574119867119867

=

119899 (119905 119870) minus 119873119861119867

119873

120574119861119867

=

119873119861119867

119873

120574119861119861

=

119873 minus 119899 (119905 119870) minus 119873119867119861

119873

(44)

Therein 119873119867119861


119861119867 119873119867and 119873


in Section 2





0(119905 119895) and 119873



0(119905 119895) 119873

0(119905 119895) and

1198731(119905 119895) verge to119870sdot119873

119861and119870sdot119873




119867119867 120574119861119861 120574119867119861

and 120574119861119867



cross And 120574119867119867


119861119867and 120574119867119861


5 Conclusion



0

10

20

30

40

50

60

70

80

90

100n(tj)

2 3 4 5 6 7 8 9 101j


0

100

200

300

400

500

600

N1(tj)

2 3 4 5 6 7 8 9 101j


0

50

100

150

200

250

300

350

400

450

N0(tj)

2 3 4 5 6 7 8 9 101j


0

10

20

30

40

50

60

70

2 3 4 5 6 7 8 9 101j

120578fc(tj)


0

005

01

015

02

025

03

035

120572(tj)

2 3 4 5 6 7 8 9 101j


Figure 10 119899(119905 119895)1198730(119905 119895)119873



2 3 4 5 6 7 8 9 101j

0

10

20

30

40

50

60

70

80

90

100n(tj)


2 3 4 5 6 7 8 9 101j

0

100

200

300

400

500

600

N1(tj)


0

50

100

150

200

250

300

350

400

450

N0(tj)

2 3 4 5 6 7 8 9 101j


2 3 4 5 6 7 8 9 101j

0

10

20

30

40

50

60

70

120578fc(tj)


0

005

01

015

02

025

03

035

120572(tj)

2 3 4 5 6 7 8 9 101j


Figure 11 119899(119905 119895)1198730(119905 119895)119873



2 3 4 5 6 7 8 9 10 11 12 13 14 151j

0

10

20

30

40

50

60

70

80

90

100n(tj)


0

100

200

300

400

500

600

700

800

900

N1(tj)

2 3 4 5 6 7 8 9 10 11 12 13 14 151j


2 3 4 5 6 7 8 9 10 11 12 13 14 151j

0

100

200

300

400

500

600

700

N0(tj)


0

10

20

30

40

50

60

70

2 3 4 5 6 7 8 9 10 11 12 13 14 151j

120578fc(tj)


2 3 4 5 6 7 8 9 10 11 12 13 14 151j

0

005

01

015

02

025

03

035

120572(tj)


Figure 12 119899(119905 119895)1198730(119905 119895)119873



0

10

20

30

40

50

60

70

80

90

100n(tj)

2 3 4 5 6 7 8 9 10 11 12 13 14 151j


0

100

200

300

400

500

600

700

800

900

N1(tj)

2 3 4 5 6 7 8 9 10 11 12 13 14 151j


0

100

200

300

400

500

600

700

N0(tj)

2 3 4 5 6 7 8 9 10 11 12 13 14 151j


0

10

20

30

40

50

60

70

2 3 4 5 6 7 8 9 10 11 12 13 14 151j

120578fc(tj)


0

005

01

015

02

025

03

035

04

120572(tj)

2 3 4 5 6 7 8 9 10 11 12 13 14 151j


Figure 13 119899(119905 119895)1198730(119905 119895)119873



0

01

02

03

04

05

06

07

20 40 60 80 1000(a) 119870 = 10119872 = 9

0

01

02

03

04

05

06

07

20 40 60 80 1000(b) 119870 = 10119872 = 25

Figure 14 120574119867119867

120574119861119861 120574119867119861 and 120574

119861119867versus 119905 120574

119867119867 120574119861119861 120574119867119861 and 120574



119867119867

vibrates under 07 120574119861119867


vibrates above 0

20 40 60 80 10000

01

02

03

04

05

06

07

(a) 119870 = 15119872 = 920 40 60 80 1000

0

01

02

03

04

05

06

07

(b) 119870 = 15119872 = 25

Figure 15 120574119867119867

120574119861119861 120574119867119861 and 120574

119861119867versus 119905 120574

119867119867 120574119861119861 120574119867119861 and 120574



119867119867

vibrates under 07 120574119861119867


vibrates above 0


Appendices



119898=


0(119905)119873when




(119905) =

min 1198730(119905) 119873

1(119905)

119873

(A1)






min 1198730(119905 119895) 119873

1(119905 119895)

119899 (119905 119895) sdot 119870

= 120572 (119905 119895) +

119873119861

01(119905 119895)

119899 (119905 119895) sdot 119870

+

119873119867

01(119905 119895)

119899 (119905 119895) sdot 119870

(B1)



01(119905 119895) in (B1) is denoted as


min 1198730(119905 119895) 119873

1(119905 119895)

119899 (119905 119895) sdot 119870

= 120572 (119905 119895) +

119873119861

01(119905 119895)

119899 (119905 119895) sdot 119870

+ 119902119894119895(1)

(B2)


min 1198730(119905 119895) 119873

1(119905 119895)

119899 (119905 119895) sdot 119870

= 120572 (119905 119895) +

119873119861

01(119905 119895)

119899 (119905 119895) sdot 119870

+ 1 minus 119901119894119895(1)

(B3)


119894119895(1) are the


119902119894119895(1) =

119873119867

01(119905 119895)

119899 (119905 119895) sdot 119870

or 119901119894119895(1) = 1 minus

119873119867

01(119905 119895)

119899 (119905 119895) sdot 119870

(B4)


(119905 119895) =

min 1198730(119905 119895) 119873

1(119905 119895)

119899 (119905 119895) sdot 119870

minus 119902119894119895(1) (B5)

or

(119905 119895) =

min 1198730(119905 119895) 119873

1(119905 119895)

119899 (119905 119895) sdot 119870

+ 119901119894119895(1) minus 1 (B6)

Because of

119901119894119895(1) ge 1 minus

max119896(1003816100381610038161003816d119905119896

1003816100381610038161003816) minusmin

119896(1003816100381610038161003816d119905119896

1003816100381610038161003816)

119899 (119905 119895)

(B7)

or

119902119894119895(1) le

max119896(1003816100381610038161003816d119905119896

1003816100381610038161003816) minusmin

119896(1003816100381610038161003816d119905119896

1003816100381610038161003816)

119899 (119905 119895)

(B8)

we get

(119905 119895)

ge

min 1198730(119905 119895) 119873

1(119905 119895) minus 119870 sdot [max

119895(

10038161003816100381610038161003816d119905119895

10038161003816100381610038161003816) minusmin

119895(

10038161003816100381610038161003816d119905119895

10038161003816100381610038161003816)]

119899 (119905 119895) sdot 119870

(B9)



Acknowledgments


References




























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











119898= 119875119891119886


(119905) =

min 1198730(119905) 119873

1(119905)

119873

(31)

where1198730(119905) and119873





(119905 119895) =

min 1198730(119905 119895) 119873

1(119905 119895)

1198730(119905 119895) + 119873

1(119905 119895)

(32)





(119905 119895)

ge

min 1198730(119905 119895) 119873

1(119905 119895) minus 119870 sdot [max

119896(1003816100381610038161003816d119905119896

1003816100381610038161003816) minusmin

119896(1003816100381610038161003816d119905119896

1003816100381610038161003816)]

119899 (119905 119895) sdot 119870

119896 isin 1 2 119870

(33)


[max119896(|d119905119896|) minus min




Here we let

(119905 119895)

=

min 1198730(119905 119895) 119873

1(119905 119895) minus 119870 sdot [max

119895(

10038161003816100381610038161003816d119905119895

10038161003816100381610038161003816) minusmin

119895(

10038161003816100381610038161003816d119905119895

10038161003816100381610038161003816)]

119899 (119905 119895) sdot 119870

(34)




119873(119905 119895) over |d119905






119868119873(119905 119895)

=

10038161003816100381610038161003816d119905119894

10038161003816100381610038161003816

119899(119905119895)

10038161003816100381610038161003816d119905119894

10038161003816100381610038161003816

119899(119905119895)minus1

10038161003816100381610038161003816d119905119894

10038161003816100381610038161003816

1

| 119894 isin 119871 (119905 119895 minus 1)

(35)




over 119868119873(119905 119895)

119899 = ID (d119905119894)

= sum

119897isin119871(119905119895minus1)

IF (10038161003816100381610038161003816d119905119897

10038161003816100381610038161003816gt

10038161003816100381610038161003816d119905119894

10038161003816100381610038161003816

119899

)

+ sum

119897isin119871(119905119895minus1)

IF (10038161003816100381610038161003816d119905119897

10038161003816100381610038161003816=

10038161003816100381610038161003816d119905119894

10038161003816100381610038161003816

119899

| 119897 gt 119895)

119899 isin 1 2 119899 (119905 119895)

(36)




119868 (119905 119895)


(37)


(119905) =

119873 minus 119899 (119905 119870)

119873

(38)






1(119905)

1199112(119905) 119911


decision d1199051 d1199052 d119905






120578fc (119905 119895)




119886 = [1 minus (119905 119895)] sdot 119901119894119895(1) + (119905 119895) sdot 119909

119894119895(1) (41)


119894119895(1)) + (119905 119895)

sdot 119910119894119895(1) sdot (1 minus 119910

119894119895(1))

(42)


119911119895(119905) = IF(sum

119894

119889119894119895(119905) gt 120578fc (119905 119895) | 119894 isin 119871 (119905 119895 minus 1)) (43)


120574119867119861

120574119861119867

and 120574119861119861


119867119867and 120574119867119861


119861119867and 120574119861119861


120574119867119861

=

119873119867119861

119873

120574119867119867

=

119899 (119905 119870) minus 119873119861119867

119873

120574119861119867

=

119873119861119867

119873

120574119861119861

=

119873 minus 119899 (119905 119870) minus 119873119867119861

119873

(44)

Therein 119873119867119861


119861119867 119873119867and 119873


in Section 2





0(119905 119895) and 119873



0(119905 119895) 119873

0(119905 119895) and

1198731(119905 119895) verge to119870sdot119873

119861and119870sdot119873




119867119867 120574119861119861 120574119867119861

and 120574119861119867



cross And 120574119867119867


119861119867and 120574119867119861


5 Conclusion



0

10

20

30

40

50

60

70

80

90

100n(tj)

2 3 4 5 6 7 8 9 101j


0

100

200

300

400

500

600

N1(tj)

2 3 4 5 6 7 8 9 101j


0

50

100

150

200

250

300

350

400

450

N0(tj)

2 3 4 5 6 7 8 9 101j


0

10

20

30

40

50

60

70

2 3 4 5 6 7 8 9 101j

120578fc(tj)


0

005

01

015

02

025

03

035

120572(tj)

2 3 4 5 6 7 8 9 101j


Figure 10 119899(119905 119895)1198730(119905 119895)119873



2 3 4 5 6 7 8 9 101j

0

10

20

30

40

50

60

70

80

90

100n(tj)


2 3 4 5 6 7 8 9 101j

0

100

200

300

400

500

600

N1(tj)


0

50

100

150

200

250

300

350

400

450

N0(tj)

2 3 4 5 6 7 8 9 101j


2 3 4 5 6 7 8 9 101j

0

10

20

30

40

50

60

70

120578fc(tj)


0

005

01

015

02

025

03

035

120572(tj)

2 3 4 5 6 7 8 9 101j


Figure 11 119899(119905 119895)1198730(119905 119895)119873



2 3 4 5 6 7 8 9 10 11 12 13 14 151j

0

10

20

30

40

50

60

70

80

90

100n(tj)


0

100

200

300

400

500

600

700

800

900

N1(tj)

2 3 4 5 6 7 8 9 10 11 12 13 14 151j


2 3 4 5 6 7 8 9 10 11 12 13 14 151j

0

100

200

300

400

500

600

700

N0(tj)


0

10

20

30

40

50

60

70

2 3 4 5 6 7 8 9 10 11 12 13 14 151j

120578fc(tj)


2 3 4 5 6 7 8 9 10 11 12 13 14 151j

0

005

01

015

02

025

03

035

120572(tj)


Figure 12 119899(119905 119895)1198730(119905 119895)119873



0

10

20

30

40

50

60

70

80

90

100n(tj)

2 3 4 5 6 7 8 9 10 11 12 13 14 151j


0

100

200

300

400

500

600

700

800

900

N1(tj)

2 3 4 5 6 7 8 9 10 11 12 13 14 151j


0

100

200

300

400

500

600

700

N0(tj)

2 3 4 5 6 7 8 9 10 11 12 13 14 151j


0

10

20

30

40

50

60

70

2 3 4 5 6 7 8 9 10 11 12 13 14 151j

120578fc(tj)


0

005

01

015

02

025

03

035

04

120572(tj)

2 3 4 5 6 7 8 9 10 11 12 13 14 151j


Figure 13 119899(119905 119895)1198730(119905 119895)119873



0

01

02

03

04

05

06

07

20 40 60 80 1000(a) 119870 = 10119872 = 9

0

01

02

03

04

05

06

07

20 40 60 80 1000(b) 119870 = 10119872 = 25

Figure 14 120574119867119867

120574119861119861 120574119867119861 and 120574

119861119867versus 119905 120574

119867119867 120574119861119861 120574119867119861 and 120574



119867119867

vibrates under 07 120574119861119867


vibrates above 0

20 40 60 80 10000

01

02

03

04

05

06

07

(a) 119870 = 15119872 = 920 40 60 80 1000

0

01

02

03

04

05

06

07

(b) 119870 = 15119872 = 25

Figure 15 120574119867119867

120574119861119861 120574119867119861 and 120574

119861119867versus 119905 120574

119867119867 120574119861119861 120574119867119861 and 120574



119867119867

vibrates under 07 120574119861119867


vibrates above 0


Appendices



119898=


0(119905)119873when




(119905) =

min 1198730(119905) 119873

1(119905)

119873

(A1)






min 1198730(119905 119895) 119873

1(119905 119895)

119899 (119905 119895) sdot 119870

= 120572 (119905 119895) +

119873119861

01(119905 119895)

119899 (119905 119895) sdot 119870

+

119873119867

01(119905 119895)

119899 (119905 119895) sdot 119870

(B1)



01(119905 119895) in (B1) is denoted as


min 1198730(119905 119895) 119873

1(119905 119895)

119899 (119905 119895) sdot 119870

= 120572 (119905 119895) +

119873119861

01(119905 119895)

119899 (119905 119895) sdot 119870

+ 119902119894119895(1)

(B2)


min 1198730(119905 119895) 119873

1(119905 119895)

119899 (119905 119895) sdot 119870

= 120572 (119905 119895) +

119873119861

01(119905 119895)

119899 (119905 119895) sdot 119870

+ 1 minus 119901119894119895(1)

(B3)


119894119895(1) are the


119902119894119895(1) =

119873119867

01(119905 119895)

119899 (119905 119895) sdot 119870

or 119901119894119895(1) = 1 minus

119873119867

01(119905 119895)

119899 (119905 119895) sdot 119870

(B4)


(119905 119895) =

min 1198730(119905 119895) 119873

1(119905 119895)

119899 (119905 119895) sdot 119870

minus 119902119894119895(1) (B5)

or

(119905 119895) =

min 1198730(119905 119895) 119873

1(119905 119895)

119899 (119905 119895) sdot 119870

+ 119901119894119895(1) minus 1 (B6)

Because of

119901119894119895(1) ge 1 minus

max119896(1003816100381610038161003816d119905119896

1003816100381610038161003816) minusmin

119896(1003816100381610038161003816d119905119896

1003816100381610038161003816)

119899 (119905 119895)

(B7)

or

119902119894119895(1) le

max119896(1003816100381610038161003816d119905119896

1003816100381610038161003816) minusmin

119896(1003816100381610038161003816d119905119896

1003816100381610038161003816)

119899 (119905 119895)

(B8)

we get

(119905 119895)

ge

min 1198730(119905 119895) 119873

1(119905 119895) minus 119870 sdot [max

119895(

10038161003816100381610038161003816d119905119895

10038161003816100381610038161003816) minusmin

119895(

10038161003816100381610038161003816d119905119895

10038161003816100381610038161003816)]

119899 (119905 119895) sdot 119870

(B9)



Acknowledgments


References




























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation













120578fc (119905 119895)




119886 = [1 minus (119905 119895)] sdot 119901119894119895(1) + (119905 119895) sdot 119909

119894119895(1) (41)


119894119895(1)) + (119905 119895)

sdot 119910119894119895(1) sdot (1 minus 119910

119894119895(1))

(42)


119911119895(119905) = IF(sum

119894

119889119894119895(119905) gt 120578fc (119905 119895) | 119894 isin 119871 (119905 119895 minus 1)) (43)


120574119867119861

120574119861119867

and 120574119861119861


119867119867and 120574119867119861


119861119867and 120574119861119861


120574119867119861

=

119873119867119861

119873

120574119867119867

=

119899 (119905 119870) minus 119873119861119867

119873

120574119861119867

=

119873119861119867

119873

120574119861119861

=

119873 minus 119899 (119905 119870) minus 119873119867119861

119873

(44)

Therein 119873119867119861


119861119867 119873119867and 119873


in Section 2





0(119905 119895) and 119873



0(119905 119895) 119873

0(119905 119895) and

1198731(119905 119895) verge to119870sdot119873

119861and119870sdot119873




119867119867 120574119861119861 120574119867119861

and 120574119861119867



cross And 120574119867119867


119861119867and 120574119867119861


5 Conclusion



0

10

20

30

40

50

60

70

80

90

100n(tj)

2 3 4 5 6 7 8 9 101j


0

100

200

300

400

500

600

N1(tj)

2 3 4 5 6 7 8 9 101j


0

50

100

150

200

250

300

350

400

450

N0(tj)

2 3 4 5 6 7 8 9 101j


0

10

20

30

40

50

60

70

2 3 4 5 6 7 8 9 101j

120578fc(tj)


0

005

01

015

02

025

03

035

120572(tj)

2 3 4 5 6 7 8 9 101j


Figure 10 119899(119905 119895)1198730(119905 119895)119873



2 3 4 5 6 7 8 9 101j

0

10

20

30

40

50

60

70

80

90

100n(tj)


2 3 4 5 6 7 8 9 101j

0

100

200

300

400

500

600

N1(tj)


0

50

100

150

200

250

300

350

400

450

N0(tj)

2 3 4 5 6 7 8 9 101j


2 3 4 5 6 7 8 9 101j

0

10

20

30

40

50

60

70

120578fc(tj)


0

005

01

015

02

025

03

035

120572(tj)

2 3 4 5 6 7 8 9 101j


Figure 11 119899(119905 119895)1198730(119905 119895)119873



2 3 4 5 6 7 8 9 10 11 12 13 14 151j

0

10

20

30

40

50

60

70

80

90

100n(tj)


0

100

200

300

400

500

600

700

800

900

N1(tj)

2 3 4 5 6 7 8 9 10 11 12 13 14 151j


2 3 4 5 6 7 8 9 10 11 12 13 14 151j

0

100

200

300

400

500

600

700

N0(tj)


0

10

20

30

40

50

60

70

2 3 4 5 6 7 8 9 10 11 12 13 14 151j

120578fc(tj)


2 3 4 5 6 7 8 9 10 11 12 13 14 151j

0

005

01

015

02

025

03

035

120572(tj)


Figure 12 119899(119905 119895)1198730(119905 119895)119873



0

10

20

30

40

50

60

70

80

90

100n(tj)

2 3 4 5 6 7 8 9 10 11 12 13 14 151j


0

100

200

300

400

500

600

700

800

900

N1(tj)

2 3 4 5 6 7 8 9 10 11 12 13 14 151j


0

100

200

300

400

500

600

700

N0(tj)

2 3 4 5 6 7 8 9 10 11 12 13 14 151j


0

10

20

30

40

50

60

70

2 3 4 5 6 7 8 9 10 11 12 13 14 151j

120578fc(tj)


0

005

01

015

02

025

03

035

04

120572(tj)

2 3 4 5 6 7 8 9 10 11 12 13 14 151j


Figure 13 119899(119905 119895)1198730(119905 119895)119873



0

01

02

03

04

05

06

07

20 40 60 80 1000(a) 119870 = 10119872 = 9

0

01

02

03

04

05

06

07

20 40 60 80 1000(b) 119870 = 10119872 = 25

Figure 14 120574119867119867

120574119861119861 120574119867119861 and 120574

119861119867versus 119905 120574

119867119867 120574119861119861 120574119867119861 and 120574



119867119867

vibrates under 07 120574119861119867


vibrates above 0

20 40 60 80 10000

01

02

03

04

05

06

07

(a) 119870 = 15119872 = 920 40 60 80 1000

0

01

02

03

04

05

06

07

(b) 119870 = 15119872 = 25

Figure 15 120574119867119867

120574119861119861 120574119867119861 and 120574

119861119867versus 119905 120574

119867119867 120574119861119861 120574119867119861 and 120574



119867119867

vibrates under 07 120574119861119867


vibrates above 0


Appendices



119898=


0(119905)119873when




(119905) =

min 1198730(119905) 119873

1(119905)

119873

(A1)






min 1198730(119905 119895) 119873

1(119905 119895)

119899 (119905 119895) sdot 119870

= 120572 (119905 119895) +

119873119861

01(119905 119895)

119899 (119905 119895) sdot 119870

+

119873119867

01(119905 119895)

119899 (119905 119895) sdot 119870

(B1)



01(119905 119895) in (B1) is denoted as


min 1198730(119905 119895) 119873

1(119905 119895)

119899 (119905 119895) sdot 119870

= 120572 (119905 119895) +

119873119861

01(119905 119895)

119899 (119905 119895) sdot 119870

+ 119902119894119895(1)

(B2)


min 1198730(119905 119895) 119873

1(119905 119895)

119899 (119905 119895) sdot 119870

= 120572 (119905 119895) +

119873119861

01(119905 119895)

119899 (119905 119895) sdot 119870

+ 1 minus 119901119894119895(1)

(B3)


119894119895(1) are the


119902119894119895(1) =

119873119867

01(119905 119895)

119899 (119905 119895) sdot 119870

or 119901119894119895(1) = 1 minus

119873119867

01(119905 119895)

119899 (119905 119895) sdot 119870

(B4)


(119905 119895) =

min 1198730(119905 119895) 119873

1(119905 119895)

119899 (119905 119895) sdot 119870

minus 119902119894119895(1) (B5)

or

(119905 119895) =

min 1198730(119905 119895) 119873

1(119905 119895)

119899 (119905 119895) sdot 119870

+ 119901119894119895(1) minus 1 (B6)

Because of

119901119894119895(1) ge 1 minus

max119896(1003816100381610038161003816d119905119896

1003816100381610038161003816) minusmin

119896(1003816100381610038161003816d119905119896

1003816100381610038161003816)

119899 (119905 119895)

(B7)

or

119902119894119895(1) le

max119896(1003816100381610038161003816d119905119896

1003816100381610038161003816) minusmin

119896(1003816100381610038161003816d119905119896

1003816100381610038161003816)

119899 (119905 119895)

(B8)

we get

(119905 119895)

ge

min 1198730(119905 119895) 119873

1(119905 119895) minus 119870 sdot [max

119895(

10038161003816100381610038161003816d119905119895

10038161003816100381610038161003816) minusmin

119895(

10038161003816100381610038161003816d119905119895

10038161003816100381610038161003816)]

119899 (119905 119895) sdot 119870

(B9)



Acknowledgments


References




























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










0

10

20

30

40

50

60

70

80

90

100n(tj)

2 3 4 5 6 7 8 9 101j


0

100

200

300

400

500

600

N1(tj)

2 3 4 5 6 7 8 9 101j


0

50

100

150

200

250

300

350

400

450

N0(tj)

2 3 4 5 6 7 8 9 101j


0

10

20

30

40

50

60

70

2 3 4 5 6 7 8 9 101j

120578fc(tj)


0

005

01

015

02

025

03

035

120572(tj)

2 3 4 5 6 7 8 9 101j


Figure 10 119899(119905 119895)1198730(119905 119895)119873



2 3 4 5 6 7 8 9 101j

0

10

20

30

40

50

60

70

80

90

100n(tj)


2 3 4 5 6 7 8 9 101j

0

100

200

300

400

500

600

N1(tj)


0

50

100

150

200

250

300

350

400

450

N0(tj)

2 3 4 5 6 7 8 9 101j


2 3 4 5 6 7 8 9 101j

0

10

20

30

40

50

60

70

120578fc(tj)


0

005

01

015

02

025

03

035

120572(tj)

2 3 4 5 6 7 8 9 101j


Figure 11 119899(119905 119895)1198730(119905 119895)119873



2 3 4 5 6 7 8 9 10 11 12 13 14 151j

0

10

20

30

40

50

60

70

80

90

100n(tj)


0

100

200

300

400

500

600

700

800

900

N1(tj)

2 3 4 5 6 7 8 9 10 11 12 13 14 151j


2 3 4 5 6 7 8 9 10 11 12 13 14 151j

0

100

200

300

400

500

600

700

N0(tj)


0

10

20

30

40

50

60

70

2 3 4 5 6 7 8 9 10 11 12 13 14 151j

120578fc(tj)


2 3 4 5 6 7 8 9 10 11 12 13 14 151j

0

005

01

015

02

025

03

035

120572(tj)


Figure 12 119899(119905 119895)1198730(119905 119895)119873



0

10

20

30

40

50

60

70

80

90

100n(tj)

2 3 4 5 6 7 8 9 10 11 12 13 14 151j


0

100

200

300

400

500

600

700

800

900

N1(tj)

2 3 4 5 6 7 8 9 10 11 12 13 14 151j


0

100

200

300

400

500

600

700

N0(tj)

2 3 4 5 6 7 8 9 10 11 12 13 14 151j


0

10

20

30

40

50

60

70

2 3 4 5 6 7 8 9 10 11 12 13 14 151j

120578fc(tj)


0

005

01

015

02

025

03

035

04

120572(tj)

2 3 4 5 6 7 8 9 10 11 12 13 14 151j


Figure 13 119899(119905 119895)1198730(119905 119895)119873



0

01

02

03

04

05

06

07

20 40 60 80 1000(a) 119870 = 10119872 = 9

0

01

02

03

04

05

06

07

20 40 60 80 1000(b) 119870 = 10119872 = 25

Figure 14 120574119867119867

120574119861119861 120574119867119861 and 120574

119861119867versus 119905 120574

119867119867 120574119861119861 120574119867119861 and 120574



119867119867

vibrates under 07 120574119861119867


vibrates above 0

20 40 60 80 10000

01

02

03

04

05

06

07

(a) 119870 = 15119872 = 920 40 60 80 1000

0

01

02

03

04

05

06

07

(b) 119870 = 15119872 = 25

Figure 15 120574119867119867

120574119861119861 120574119867119861 and 120574

119861119867versus 119905 120574

119867119867 120574119861119861 120574119867119861 and 120574



119867119867

vibrates under 07 120574119861119867


vibrates above 0


Appendices



119898=


0(119905)119873when




(119905) =

min 1198730(119905) 119873

1(119905)

119873

(A1)






min 1198730(119905 119895) 119873

1(119905 119895)

119899 (119905 119895) sdot 119870

= 120572 (119905 119895) +

119873119861

01(119905 119895)

119899 (119905 119895) sdot 119870

+

119873119867

01(119905 119895)

119899 (119905 119895) sdot 119870

(B1)



01(119905 119895) in (B1) is denoted as


min 1198730(119905 119895) 119873

1(119905 119895)

119899 (119905 119895) sdot 119870

= 120572 (119905 119895) +

119873119861

01(119905 119895)

119899 (119905 119895) sdot 119870

+ 119902119894119895(1)

(B2)


min 1198730(119905 119895) 119873

1(119905 119895)

119899 (119905 119895) sdot 119870

= 120572 (119905 119895) +

119873119861

01(119905 119895)

119899 (119905 119895) sdot 119870

+ 1 minus 119901119894119895(1)

(B3)


119894119895(1) are the


119902119894119895(1) =

119873119867

01(119905 119895)

119899 (119905 119895) sdot 119870

or 119901119894119895(1) = 1 minus

119873119867

01(119905 119895)

119899 (119905 119895) sdot 119870

(B4)


(119905 119895) =

min 1198730(119905 119895) 119873

1(119905 119895)

119899 (119905 119895) sdot 119870

minus 119902119894119895(1) (B5)

or

(119905 119895) =

min 1198730(119905 119895) 119873

1(119905 119895)

119899 (119905 119895) sdot 119870

+ 119901119894119895(1) minus 1 (B6)

Because of

119901119894119895(1) ge 1 minus

max119896(1003816100381610038161003816d119905119896

1003816100381610038161003816) minusmin

119896(1003816100381610038161003816d119905119896

1003816100381610038161003816)

119899 (119905 119895)

(B7)

or

119902119894119895(1) le

max119896(1003816100381610038161003816d119905119896

1003816100381610038161003816) minusmin

119896(1003816100381610038161003816d119905119896

1003816100381610038161003816)

119899 (119905 119895)

(B8)

we get

(119905 119895)

ge

min 1198730(119905 119895) 119873

1(119905 119895) minus 119870 sdot [max

119895(

10038161003816100381610038161003816d119905119895

10038161003816100381610038161003816) minusmin

119895(

10038161003816100381610038161003816d119905119895

10038161003816100381610038161003816)]

119899 (119905 119895) sdot 119870

(B9)



Acknowledgments


References




























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










2 3 4 5 6 7 8 9 101j

0

10

20

30

40

50

60

70

80

90

100n(tj)


2 3 4 5 6 7 8 9 101j

0

100

200

300

400

500

600

N1(tj)


0

50

100

150

200

250

300

350

400

450

N0(tj)

2 3 4 5 6 7 8 9 101j


2 3 4 5 6 7 8 9 101j

0

10

20

30

40

50

60

70

120578fc(tj)


0

005

01

015

02

025

03

035

120572(tj)

2 3 4 5 6 7 8 9 101j


Figure 11 119899(119905 119895)1198730(119905 119895)119873



2 3 4 5 6 7 8 9 10 11 12 13 14 151j

0

10

20

30

40

50

60

70

80

90

100n(tj)


0

100

200

300

400

500

600

700

800

900

N1(tj)

2 3 4 5 6 7 8 9 10 11 12 13 14 151j


2 3 4 5 6 7 8 9 10 11 12 13 14 151j

0

100

200

300

400

500

600

700

N0(tj)


0

10

20

30

40

50

60

70

2 3 4 5 6 7 8 9 10 11 12 13 14 151j

120578fc(tj)


2 3 4 5 6 7 8 9 10 11 12 13 14 151j

0

005

01

015

02

025

03

035

120572(tj)


Figure 12 119899(119905 119895)1198730(119905 119895)119873



0

10

20

30

40

50

60

70

80

90

100n(tj)

2 3 4 5 6 7 8 9 10 11 12 13 14 151j


0

100

200

300

400

500

600

700

800

900

N1(tj)

2 3 4 5 6 7 8 9 10 11 12 13 14 151j


0

100

200

300

400

500

600

700

N0(tj)

2 3 4 5 6 7 8 9 10 11 12 13 14 151j


0

10

20

30

40

50

60

70

2 3 4 5 6 7 8 9 10 11 12 13 14 151j

120578fc(tj)


0

005

01

015

02

025

03

035

04

120572(tj)

2 3 4 5 6 7 8 9 10 11 12 13 14 151j


Figure 13 119899(119905 119895)1198730(119905 119895)119873



0

01

02

03

04

05

06

07

20 40 60 80 1000(a) 119870 = 10119872 = 9

0

01

02

03

04

05

06

07

20 40 60 80 1000(b) 119870 = 10119872 = 25

Figure 14 120574119867119867

120574119861119861 120574119867119861 and 120574

119861119867versus 119905 120574

119867119867 120574119861119861 120574119867119861 and 120574



119867119867

vibrates under 07 120574119861119867


vibrates above 0

20 40 60 80 10000

01

02

03

04

05

06

07

(a) 119870 = 15119872 = 920 40 60 80 1000

0

01

02

03

04

05

06

07

(b) 119870 = 15119872 = 25

Figure 15 120574119867119867

120574119861119861 120574119867119861 and 120574

119861119867versus 119905 120574

119867119867 120574119861119861 120574119867119861 and 120574



119867119867

vibrates under 07 120574119861119867


vibrates above 0


Appendices



119898=


0(119905)119873when




(119905) =

min 1198730(119905) 119873

1(119905)

119873

(A1)






min 1198730(119905 119895) 119873

1(119905 119895)

119899 (119905 119895) sdot 119870

= 120572 (119905 119895) +

119873119861

01(119905 119895)

119899 (119905 119895) sdot 119870

+

119873119867

01(119905 119895)

119899 (119905 119895) sdot 119870

(B1)



01(119905 119895) in (B1) is denoted as


min 1198730(119905 119895) 119873

1(119905 119895)

119899 (119905 119895) sdot 119870

= 120572 (119905 119895) +

119873119861

01(119905 119895)

119899 (119905 119895) sdot 119870

+ 119902119894119895(1)

(B2)


min 1198730(119905 119895) 119873

1(119905 119895)

119899 (119905 119895) sdot 119870

= 120572 (119905 119895) +

119873119861

01(119905 119895)

119899 (119905 119895) sdot 119870

+ 1 minus 119901119894119895(1)

(B3)


119894119895(1) are the


119902119894119895(1) =

119873119867

01(119905 119895)

119899 (119905 119895) sdot 119870

or 119901119894119895(1) = 1 minus

119873119867

01(119905 119895)

119899 (119905 119895) sdot 119870

(B4)


(119905 119895) =

min 1198730(119905 119895) 119873

1(119905 119895)

119899 (119905 119895) sdot 119870

minus 119902119894119895(1) (B5)

or

(119905 119895) =

min 1198730(119905 119895) 119873

1(119905 119895)

119899 (119905 119895) sdot 119870

+ 119901119894119895(1) minus 1 (B6)

Because of

119901119894119895(1) ge 1 minus

max119896(1003816100381610038161003816d119905119896

1003816100381610038161003816) minusmin

119896(1003816100381610038161003816d119905119896

1003816100381610038161003816)

119899 (119905 119895)

(B7)

or

119902119894119895(1) le

max119896(1003816100381610038161003816d119905119896

1003816100381610038161003816) minusmin

119896(1003816100381610038161003816d119905119896

1003816100381610038161003816)

119899 (119905 119895)

(B8)

we get

(119905 119895)

ge

min 1198730(119905 119895) 119873

1(119905 119895) minus 119870 sdot [max

119895(

10038161003816100381610038161003816d119905119895

10038161003816100381610038161003816) minusmin

119895(

10038161003816100381610038161003816d119905119895

10038161003816100381610038161003816)]

119899 (119905 119895) sdot 119870

(B9)



Acknowledgments


References




























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










2 3 4 5 6 7 8 9 10 11 12 13 14 151j

0

10

20

30

40

50

60

70

80

90

100n(tj)


0

100

200

300

400

500

600

700

800

900

N1(tj)

2 3 4 5 6 7 8 9 10 11 12 13 14 151j


2 3 4 5 6 7 8 9 10 11 12 13 14 151j

0

100

200

300

400

500

600

700

N0(tj)


0

10

20

30

40

50

60

70

2 3 4 5 6 7 8 9 10 11 12 13 14 151j

120578fc(tj)


2 3 4 5 6 7 8 9 10 11 12 13 14 151j

0

005

01

015

02

025

03

035

120572(tj)


Figure 12 119899(119905 119895)1198730(119905 119895)119873



0

10

20

30

40

50

60

70

80

90

100n(tj)

2 3 4 5 6 7 8 9 10 11 12 13 14 151j


0

100

200

300

400

500

600

700

800

900

N1(tj)

2 3 4 5 6 7 8 9 10 11 12 13 14 151j


0

100

200

300

400

500

600

700

N0(tj)

2 3 4 5 6 7 8 9 10 11 12 13 14 151j


0

10

20

30

40

50

60

70

2 3 4 5 6 7 8 9 10 11 12 13 14 151j

120578fc(tj)


0

005

01

015

02

025

03

035

04

120572(tj)

2 3 4 5 6 7 8 9 10 11 12 13 14 151j


Figure 13 119899(119905 119895)1198730(119905 119895)119873



0

01

02

03

04

05

06

07

20 40 60 80 1000(a) 119870 = 10119872 = 9

0

01

02

03

04

05

06

07

20 40 60 80 1000(b) 119870 = 10119872 = 25

Figure 14 120574119867119867

120574119861119861 120574119867119861 and 120574

119861119867versus 119905 120574

119867119867 120574119861119861 120574119867119861 and 120574



119867119867

vibrates under 07 120574119861119867


vibrates above 0

20 40 60 80 10000

01

02

03

04

05

06

07

(a) 119870 = 15119872 = 920 40 60 80 1000

0

01

02

03

04

05

06

07

(b) 119870 = 15119872 = 25

Figure 15 120574119867119867

120574119861119861 120574119867119861 and 120574

119861119867versus 119905 120574

119867119867 120574119861119861 120574119867119861 and 120574



119867119867

vibrates under 07 120574119861119867


vibrates above 0


Appendices



119898=


0(119905)119873when




(119905) =

min 1198730(119905) 119873

1(119905)

119873

(A1)






min 1198730(119905 119895) 119873

1(119905 119895)

119899 (119905 119895) sdot 119870

= 120572 (119905 119895) +

119873119861

01(119905 119895)

119899 (119905 119895) sdot 119870

+

119873119867

01(119905 119895)

119899 (119905 119895) sdot 119870

(B1)



01(119905 119895) in (B1) is denoted as


min 1198730(119905 119895) 119873

1(119905 119895)

119899 (119905 119895) sdot 119870

= 120572 (119905 119895) +

119873119861

01(119905 119895)

119899 (119905 119895) sdot 119870

+ 119902119894119895(1)

(B2)


min 1198730(119905 119895) 119873

1(119905 119895)

119899 (119905 119895) sdot 119870

= 120572 (119905 119895) +

119873119861

01(119905 119895)

119899 (119905 119895) sdot 119870

+ 1 minus 119901119894119895(1)

(B3)


119894119895(1) are the


119902119894119895(1) =

119873119867

01(119905 119895)

119899 (119905 119895) sdot 119870

or 119901119894119895(1) = 1 minus

119873119867

01(119905 119895)

119899 (119905 119895) sdot 119870

(B4)


(119905 119895) =

min 1198730(119905 119895) 119873

1(119905 119895)

119899 (119905 119895) sdot 119870

minus 119902119894119895(1) (B5)

or

(119905 119895) =

min 1198730(119905 119895) 119873

1(119905 119895)

119899 (119905 119895) sdot 119870

+ 119901119894119895(1) minus 1 (B6)

Because of

119901119894119895(1) ge 1 minus

max119896(1003816100381610038161003816d119905119896

1003816100381610038161003816) minusmin

119896(1003816100381610038161003816d119905119896

1003816100381610038161003816)

119899 (119905 119895)

(B7)

or

119902119894119895(1) le

max119896(1003816100381610038161003816d119905119896

1003816100381610038161003816) minusmin

119896(1003816100381610038161003816d119905119896

1003816100381610038161003816)

119899 (119905 119895)

(B8)

we get

(119905 119895)

ge

min 1198730(119905 119895) 119873

1(119905 119895) minus 119870 sdot [max

119895(

10038161003816100381610038161003816d119905119895

10038161003816100381610038161003816) minusmin

119895(

10038161003816100381610038161003816d119905119895

10038161003816100381610038161003816)]

119899 (119905 119895) sdot 119870

(B9)



Acknowledgments


References




























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










0

10

20

30

40

50

60

70

80

90

100n(tj)

2 3 4 5 6 7 8 9 10 11 12 13 14 151j


0

100

200

300

400

500

600

700

800

900

N1(tj)

2 3 4 5 6 7 8 9 10 11 12 13 14 151j


0

100

200

300

400

500

600

700

N0(tj)

2 3 4 5 6 7 8 9 10 11 12 13 14 151j


0

10

20

30

40

50

60

70

2 3 4 5 6 7 8 9 10 11 12 13 14 151j

120578fc(tj)


0

005

01

015

02

025

03

035

04

120572(tj)

2 3 4 5 6 7 8 9 10 11 12 13 14 151j


Figure 13 119899(119905 119895)1198730(119905 119895)119873



0

01

02

03

04

05

06

07

20 40 60 80 1000(a) 119870 = 10119872 = 9

0

01

02

03

04

05

06

07

20 40 60 80 1000(b) 119870 = 10119872 = 25

Figure 14 120574119867119867

120574119861119861 120574119867119861 and 120574

119861119867versus 119905 120574

119867119867 120574119861119861 120574119867119861 and 120574



119867119867

vibrates under 07 120574119861119867


vibrates above 0

20 40 60 80 10000

01

02

03

04

05

06

07

(a) 119870 = 15119872 = 920 40 60 80 1000

0

01

02

03

04

05

06

07

(b) 119870 = 15119872 = 25

Figure 15 120574119867119867

120574119861119861 120574119867119861 and 120574

119861119867versus 119905 120574

119867119867 120574119861119861 120574119867119861 and 120574



119867119867

vibrates under 07 120574119861119867


vibrates above 0


Appendices



119898=


0(119905)119873when




(119905) =

min 1198730(119905) 119873

1(119905)

119873

(A1)






min 1198730(119905 119895) 119873

1(119905 119895)

119899 (119905 119895) sdot 119870

= 120572 (119905 119895) +

119873119861

01(119905 119895)

119899 (119905 119895) sdot 119870

+

119873119867

01(119905 119895)

119899 (119905 119895) sdot 119870

(B1)



01(119905 119895) in (B1) is denoted as


min 1198730(119905 119895) 119873

1(119905 119895)

119899 (119905 119895) sdot 119870

= 120572 (119905 119895) +

119873119861

01(119905 119895)

119899 (119905 119895) sdot 119870

+ 119902119894119895(1)

(B2)


min 1198730(119905 119895) 119873

1(119905 119895)

119899 (119905 119895) sdot 119870

= 120572 (119905 119895) +

119873119861

01(119905 119895)

119899 (119905 119895) sdot 119870

+ 1 minus 119901119894119895(1)

(B3)


119894119895(1) are the


119902119894119895(1) =

119873119867

01(119905 119895)

119899 (119905 119895) sdot 119870

or 119901119894119895(1) = 1 minus

119873119867

01(119905 119895)

119899 (119905 119895) sdot 119870

(B4)


(119905 119895) =

min 1198730(119905 119895) 119873

1(119905 119895)

119899 (119905 119895) sdot 119870

minus 119902119894119895(1) (B5)

or

(119905 119895) =

min 1198730(119905 119895) 119873

1(119905 119895)

119899 (119905 119895) sdot 119870

+ 119901119894119895(1) minus 1 (B6)

Because of

119901119894119895(1) ge 1 minus

max119896(1003816100381610038161003816d119905119896

1003816100381610038161003816) minusmin

119896(1003816100381610038161003816d119905119896

1003816100381610038161003816)

119899 (119905 119895)

(B7)

or

119902119894119895(1) le

max119896(1003816100381610038161003816d119905119896

1003816100381610038161003816) minusmin

119896(1003816100381610038161003816d119905119896

1003816100381610038161003816)

119899 (119905 119895)

(B8)

we get

(119905 119895)

ge

min 1198730(119905 119895) 119873

1(119905 119895) minus 119870 sdot [max

119895(

10038161003816100381610038161003816d119905119895

10038161003816100381610038161003816) minusmin

119895(

10038161003816100381610038161003816d119905119895

10038161003816100381610038161003816)]

119899 (119905 119895) sdot 119870

(B9)



Acknowledgments


References




























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










0

01

02

03

04

05

06

07

20 40 60 80 1000(a) 119870 = 10119872 = 9

0

01

02

03

04

05

06

07

20 40 60 80 1000(b) 119870 = 10119872 = 25

Figure 14 120574119867119867

120574119861119861 120574119867119861 and 120574

119861119867versus 119905 120574

119867119867 120574119861119861 120574119867119861 and 120574



119867119867

vibrates under 07 120574119861119867


vibrates above 0

20 40 60 80 10000

01

02

03

04

05

06

07

(a) 119870 = 15119872 = 920 40 60 80 1000

0

01

02

03

04

05

06

07

(b) 119870 = 15119872 = 25

Figure 15 120574119867119867

120574119861119861 120574119867119861 and 120574

119861119867versus 119905 120574

119867119867 120574119861119861 120574119867119861 and 120574



119867119867

vibrates under 07 120574119861119867


vibrates above 0


Appendices



119898=


0(119905)119873when




(119905) =

min 1198730(119905) 119873

1(119905)

119873

(A1)






min 1198730(119905 119895) 119873

1(119905 119895)

119899 (119905 119895) sdot 119870

= 120572 (119905 119895) +

119873119861

01(119905 119895)

119899 (119905 119895) sdot 119870

+

119873119867

01(119905 119895)

119899 (119905 119895) sdot 119870

(B1)



01(119905 119895) in (B1) is denoted as


min 1198730(119905 119895) 119873

1(119905 119895)

119899 (119905 119895) sdot 119870

= 120572 (119905 119895) +

119873119861

01(119905 119895)

119899 (119905 119895) sdot 119870

+ 119902119894119895(1)

(B2)


min 1198730(119905 119895) 119873

1(119905 119895)

119899 (119905 119895) sdot 119870

= 120572 (119905 119895) +

119873119861

01(119905 119895)

119899 (119905 119895) sdot 119870

+ 1 minus 119901119894119895(1)

(B3)


119894119895(1) are the


119902119894119895(1) =

119873119867

01(119905 119895)

119899 (119905 119895) sdot 119870

or 119901119894119895(1) = 1 minus

119873119867

01(119905 119895)

119899 (119905 119895) sdot 119870

(B4)


(119905 119895) =

min 1198730(119905 119895) 119873

1(119905 119895)

119899 (119905 119895) sdot 119870

minus 119902119894119895(1) (B5)

or

(119905 119895) =

min 1198730(119905 119895) 119873

1(119905 119895)

119899 (119905 119895) sdot 119870

+ 119901119894119895(1) minus 1 (B6)

Because of

119901119894119895(1) ge 1 minus

max119896(1003816100381610038161003816d119905119896

1003816100381610038161003816) minusmin

119896(1003816100381610038161003816d119905119896

1003816100381610038161003816)

119899 (119905 119895)

(B7)

or

119902119894119895(1) le

max119896(1003816100381610038161003816d119905119896

1003816100381610038161003816) minusmin

119896(1003816100381610038161003816d119905119896

1003816100381610038161003816)

119899 (119905 119895)

(B8)

we get

(119905 119895)

ge

min 1198730(119905 119895) 119873

1(119905 119895) minus 119870 sdot [max

119895(

10038161003816100381610038161003816d119905119895

10038161003816100381610038161003816) minusmin

119895(

10038161003816100381610038161003816d119905119895

10038161003816100381610038161003816)]

119899 (119905 119895) sdot 119870

(B9)



Acknowledgments


References




























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation












min 1198730(119905 119895) 119873

1(119905 119895)

119899 (119905 119895) sdot 119870

= 120572 (119905 119895) +

119873119861

01(119905 119895)

119899 (119905 119895) sdot 119870

+

119873119867

01(119905 119895)

119899 (119905 119895) sdot 119870

(B1)



01(119905 119895) in (B1) is denoted as


min 1198730(119905 119895) 119873

1(119905 119895)

119899 (119905 119895) sdot 119870

= 120572 (119905 119895) +

119873119861

01(119905 119895)

119899 (119905 119895) sdot 119870

+ 119902119894119895(1)

(B2)


min 1198730(119905 119895) 119873

1(119905 119895)

119899 (119905 119895) sdot 119870

= 120572 (119905 119895) +

119873119861

01(119905 119895)

119899 (119905 119895) sdot 119870

+ 1 minus 119901119894119895(1)

(B3)


119894119895(1) are the


119902119894119895(1) =

119873119867

01(119905 119895)

119899 (119905 119895) sdot 119870

or 119901119894119895(1) = 1 minus

119873119867

01(119905 119895)

119899 (119905 119895) sdot 119870

(B4)


(119905 119895) =

min 1198730(119905 119895) 119873

1(119905 119895)

119899 (119905 119895) sdot 119870

minus 119902119894119895(1) (B5)

or

(119905 119895) =

min 1198730(119905 119895) 119873

1(119905 119895)

119899 (119905 119895) sdot 119870

+ 119901119894119895(1) minus 1 (B6)

Because of

119901119894119895(1) ge 1 minus

max119896(1003816100381610038161003816d119905119896

1003816100381610038161003816) minusmin

119896(1003816100381610038161003816d119905119896

1003816100381610038161003816)

119899 (119905 119895)

(B7)

or

119902119894119895(1) le

max119896(1003816100381610038161003816d119905119896

1003816100381610038161003816) minusmin

119896(1003816100381610038161003816d119905119896

1003816100381610038161003816)

119899 (119905 119895)

(B8)

we get

(119905 119895)

ge

min 1198730(119905 119895) 119873

1(119905 119895) minus 119870 sdot [max

119895(

10038161003816100381610038161003816d119905119895

10038161003816100381610038161003816) minusmin

119895(

10038161003816100381610038161003816d119905119895

10038161003816100381610038161003816)]

119899 (119905 119895) sdot 119870

(B9)



Acknowledgments


References




























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation



























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation









Research Article Distributed Detection in Wireless Sensor...

Documents

Transcript of Research Article Distributed Detection in Wireless Sensor...