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Hindawi Publishing CorporationJournal of Quality and Reliability EngineeringVolume 2013 Article ID 532350 14 pageshttpdxdoiorg1011552013532350

Research ArticleAHeuristic Methodology for Efficient Reduction of LargeMultistate Event Trees

Eftychia C Marcoulaki

ste Reliailit an nstrial aet aratr atinal entre r cientic Research ekrits P 0Agia Paraskevi 15310 Athens Greece

Correspondence should be addressed to Eychia C Marcoulaki emarcoulakiiptademokritosgr

Received 31 May 2012 Revised 22 August 2012 Accepted 26 September 2012

Academic Editor Nikolaos E Limnios

Copyright copy 2013 Eychia C Marcoulaki is is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited

is work proposes a new methodology for the management of event tree information used in the quantitative risk assessment ofcomplex systems e size of event trees increases exponentially with the number of system components and the number of statesthat each component can be found in eir reduction to a manageable set of events can facilitate risk quantication and safetyoptimization tasks e proposed method launches a deductive exploitation of the event space to generate reduced event trees forlarge multistate systems e approach consists in the simultaneous treatment of large subsets of the tree rather than focusing onthe given single components of the system and getting trapped into guesses on their structural arrangement

1 Introduction

For a given system the scope of quantitative risk assessment isto investigate the circumstances giving rise to differentmodesof system operation and to quantify the risk for each opera-tionmode A system can be comprised of hardware sowarehumans or organizational components [1] Each componentcan be found in various states of operation leading to mul-tiple modes of failure and normal operation for the overallsystem Once this mapping of component states to systemoutcomes is known it is theoretically possible to quantifythe risks for different operation modes to occur given theoccurrence probabilities for all the component states [2 3]

e computational effort and the memory requirementsfor risk evaluations increase exponentially as the system com-ponents and the number of component states increases Exactcalculations for binary systems are achieved faster by employ-ing binary decision diagrams to effectively organize the evalu-ation procedure [4] Since the logic behindmultistate systemsis not Boolean multistate behavior cannot be representedby binary models without introducing additional variablesand constraints [5] Rocco and Muselli [6] developed amethodology based onmachine-learning and hamming clus-tering to addressmultistate systems and any success criterion

e required computational resources can be reduced usingapproximate risk estimations [7] or criticality analysis [8]Event trees represent the combination of component statesleading to each mode of system operation Quantication ofevent trees enables faster exact risk evaluation [2] and is notlimited to binary or two-terminal systems it is however verycomputationally intensive Clearly implementation of eventtree quantication to systems with many components inmultiple states needs to be preceded by substantial reductionof the number of tree branches

is work develops an algorithm that can efficientlyexploit a large event space and generate a reduced event treeIt is assumed that every real system has an intrinsic logicbehind the assignment of system outcomes to the systemevents A simple and general methodology is suggested torobustly extract this knowledge by exploiting the system out-come space information as this is stored in a table listing allthe possible event combinations and their associated nal sys-tem outcomes e proposed algorithm is not biased by anyprior information on the functionalities of the system instructural or algebraic form and seeks to acquire knowledgeon the system logic and encapsulate it in the reduced treeealgorithm can be generally applied to any given system andis not affected by the way that the supplied data may be

2 Journal of Quality and Reliability Engineering

organized or sorted e procedure is deductive startingfrom the entire event table and systematically organizing thesystem events in a set of clusters e nal set of clusters canbe translated back to an event tree that is signicantly reducedcompared to original outcome space information supplied tothe algorithm

e paper is organized as follows Section 2 presents somevery basic system denitions used in the system represen-tation and operations presented in Section 3 In Section 3 asuitable system representation is dened using sets of eventsand based on the concepts of Cartesian products Section 4describes a set of clustering and declustering operationsto be applied on the event sets Section 5 discusses the imple-mentation of the proposed developments into an algorithmicprocedure Section 6 presents an illustration example and alarge case study to demonstrate the proposed methodologySection 7 concludes the work

asi Systemenitions

e system considered here is comprised of blocks Eachblock can be found in various states and the system response(or output) at every given instance of time 119905119905 depends on thestate that the blocks occupy at 119905119905 Each block relates to a com-ponent a set of components or a part of a component of thesystem regardless of physical conventions and according tothe choices in the system modeling e basic denitions aretaken from Papazoglou [9 10] where the blocks were inter-connected to form functional block diagrams In the presentwork the blocks are stripped of their networking functional-ities and the denitions are simplied accordingly

21 System Blocks and eir States Consider a system of 119870119870-independent blocks Each block 119887119887119896119896 119896119896 119896 119896119896119896 119896119896119896 119896119870119870119896 can befound in various internal states during the system operationperiod

Let the state set of block 119887119887119896119896 denoted as 119826119826119896119896 = 119896119904119904119896119896119896119896 119904119904119896119896119896119896119896 119896

119904119904119870119870119878119878119896119896119896119896 119896 be a partition over the possible instances of 119887119887119896119896 where 119904119904

119894119894119896119896119896119896

denotes the 119894119894119896119896th state of block 119887119887119896119896 and119870119870119878119878119896119896 denotes the numberof elements in 119826119826119896119896 thus119870119870119878119878119896119896 is the cardinality of 119878119878119896119896119896 |119878119878119896119896|

It is assumed that at every given time instance within theperiod of system operation 119887119887119896119896 is found in exactly one state(eg at 94 of maximum production level) and this staterelates to exactly one member of 119826119826119896119896 (eg 119904119904

119896119896119896 = ldquoat least 90

of maximum production levelrdquo)

22 Event enitions A basic event is an instance of a singlesystem block according to the state set partition of this blockA basic event for block 119887119887119896119896 is denoted as 119896119904119904119894119894119896119896119896119896 119896119896 119904119904

119894119894119896119896119896119896 119896 119826119826119896119896

A joint event 119896119904119904119894119894119896119896119896119896119896119896 119896 119904119904119894119894119896119896119896119896119896119896119896119896 119896 119904119904

119894119894119896119896119898119898119896119896119898119898119896 is the combination ofmore

than one basic events taking place in119898119898 different blocks of thesystem 119887119887119896119896119896 119896 119887119887119896119896119896 119896119896 119896 119887119887119896119896119898119898 Note that the set 119896119904119904

119894119894119896119896119896119896119896119896119896 119904119904119894119894119896119896119896119896119896119896119896119896 119896 119904119904

119894119894119896119896119898119898119896119896119898119898119896 is

the result of the Cartesian product 119896119904119904119894119894119896119896119896119896119896119896 119896 times 119896119904119904119894119894119896119896119896119896119896119896119896 times ⋯ times 119896119904119904

119894119894119896119896119898119898119896119896119898119898119896

A complete joint event 119890119890 is dened here as a joint eventover all the system blocks 119896119904119904119894119894119896119896 119896 119904119904

119894119894119896119896 119896119896 119896 119904119904119894119894119870119870119870119870 119896 For simplicity the

term event is used here instead of complete joint event

Note that the above event denitions are simplied com-pared to Papazoglou [10] where blocks had functionalitiesnot considered here

23 Event Space Subspaces and Event Partitions Let thesystem event space 119812119812 be the set of all the possible completejoint events 119890119890 en 119812119812 = 1198961006594100659411989011989011989611989610065941006594119890119890119896119896119896 11989610065971006597119890119890119870119870119864119864

119896 where 119870119870119864119864 denotesthe number of elements 119812119812 thus 119870119870119864119864 = |119864119864| Note that 119812119812 is theresult of the Cartesian product 119826119826119896 times 119826119826119896 times ⋯ times 119826119826119870119870 therefore|119812119812| = 119812119870119870

119896119896=119896|119826119826119896119896|Consider a nonempty subspace 119808119808 of the system event

space 119808119808 119808 119812119812 119808 119808119808119808119808 Let 119824119824119824119808119808119824 denote a partition appliedover 119808119808 composed of 119870119870119824119824119824119808119808119824 disjoint subspaces of 119808119808 denotedby 119850119850119876119876119894119894 119824119808119808119824119896 119894119894 119896 119896119896119896 119896119896119896 119896119870119870119824119824119824119808119808119824119896

24 Event Table Let 119825119825 = 119896119825119825119896119896 119825119825119896119896119896 119896 119825119825119870119870119825119825119896 denote the set of

the all possible system outcomes and 119870119870119825119825 is their numbererefore119870119870119825119825 is the cardinality |119825119825| of 119825119825

Each event yields a unique system outcome while differ-ent events may yield the same outcome For instance theremight be more than one event that leads to system failure Inevent trees a complete joint event and its associated outcomeare equivalent to a path [9 10]

Let 119879119879 119808 119890119890 119890 119825119825 = 119879119879119824 119890119890119824 where 119890119890 119896 119812119812 and 119825119825 119896 119825119825 denotethemany-to-onemapping from the event space119812119812 to the set ofsystem outcomes119825119825 e 119879119879mapping is recorded in the systemevent table as a complete list of all the system events and theiroutcome

e 119879119879 denes a partition on 119812119812 which is herein calledthe outcome-based partition and denoted by 119824119824119879119879119824119812119812119824 emembers of119824119824119879119879119824119812119812119824 are denoted by 119850119850119824119824

119879119879

119825119825119894119894 119824119812119812119824119896 119825119825119894119894 119896 119825119825Table 1 gives the event table of an example taken from

Papazoglou [10] In this case there are 32 events and 4possible outcomes and119824119824119879119879119824119812119812119824 is comprised of the 4 sets

(1) 119850119850119824119824119879119879

119825119825119896 119824119812119812119824 = 119896100659410065941198901198901198961198961006594100659411989011989031198961006594100659411989011989041198961006596100659611989011989011989611989611989610065961006596119890119890119896119896119896100659610065961198901198901198967119896100659610065961198901198901198968119896100659610065961198901198901198969119896100659610065961198901198901198960119896100659610065961198901198901198965119896100659610065961198901198901198966119896100659610065961198901198901198967119896100659610065961198901198901198968119896

(2) 119850119850119824119824119879119879

119825119825119896 119824119812119812119824 = 11989610065941006594119890119890711989610065961006596119890119890119896511989610065961006596119890119890119896611989610065961006596119890119890119896119896119896100659610065961198901198901198961198961198961006596100659611989011989011989631198961006596100659611989011989011989641198961006596100659611989011989011989691198961006596100659611989011989030119896100659610065961198901198903119896119896100659610065961198901198903119896119896

(3) 119850119850119824119824119879119879

1198251198253 119824119812119812119824 = 119896100659410065941198901198905119896100659410065941198901198906119896100659410065941198901198908119896100659610065961198901198901198963119896100659610065961198901198901198964119896

(4) 119850119850119824119824119879119879

1198251198254 119824119812119812119824 = 11989610065941006594119890119890119896119896100659410065941198901198909119896100659610065961198901198901198960119896

Note that the 119879119879 mapping can be derived from the structuraldependencies among the system blocks as dictated by therational and physical interconnections of components withinthe system and depicted in the form of a fault tree or afunctional block diagram However such information maybe unavailable or too difficult to attain or process is workassumes that the only information available is the systemevent table

3 System Representation

e system representation presented here aims to organizethe information contained in the event table e tabledata are partitioned and organized into vectors summarizing

Journal of Quality and Reliability Engineering 3

T 1 Event table for the motor-operated valve [10]

Complete joint events Block states System outcomes1198871198871 1198871198872 1198871198873

100659410065941198901198901 11990411990411 11990411990412 11990411990413 1199031199031100659410065941198901198902 11990411990411 11990411990412 11990411990423 1199031199034100659410065941198901198903 11990411990411 11990411990412 11990411990433 1199031199031100659410065941198901198904 11990411990411 11990411990412 11990411990443 1199031199031100659410065941198901198905 11990411990411 11990411990422 11990411990413 1199031199033100659410065941198901198906 11990411990411 11990411990422 11990411990423 1199031199033100659410065941198901198907 11990411990411 11990411990422 11990411990433 1199031199032100659410065941198901198908 11990411990411 11990411990422 11990411990443 1199031199033100659410065941198901198909 11990411990411 11990411990432 11990411990413 11990311990341006596100659611989011989010 11990411990411 11990411990432 11990411990423 11990311990341006596100659611989011989011 11990411990411 11990411990432 11990411990433 11990311990311006596100659611989011989012 11990411990411 11990411990432 11990411990443 11990311990311006596100659611989011989013 11990411990411 11990411990442 11990411990413 11990311990331006596100659611989011989014 11990411990411 11990411990442 11990411990423 11990311990331006596100659611989011989015 11990411990411 11990411990442 11990411990433 11990311990321006596100659611989011989016 11990411990411 11990411990442 11990411990443 11990311990321006596100659611989011989017 11990411990421 11990411990412 11990411990413 11990311990311006596100659611989011989018 11990411990421 11990411990412 11990411990423 11990311990311006596100659611989011989019 11990411990421 11990411990412 11990411990433 11990311990311006596100659611989011989020 11990411990421 11990411990412 11990411990443 11990311990311006596100659611989011989021 11990411990421 11990411990422 11990411990413 11990311990321006596100659611989011989022 11990411990421 11990411990422 11990411990423 11990311990321006596100659611989011989023 11990411990421 11990411990422 11990411990433 11990311990321006596100659611989011989024 11990411990421 11990411990422 11990411990443 11990311990321006596100659611989011989025 11990411990421 11990411990432 11990411990413 11990311990311006596100659611989011989026 11990411990421 11990411990432 11990411990423 11990311990311006596100659611989011989027 11990411990421 11990411990432 11990411990433 11990311990311006596100659611989011989028 11990411990421 11990411990432 11990411990443 11990311990311006596100659611989011989029 11990411990421 11990411990442 11990411990413 11990311990321006596100659611989011989030 11990411990421 11990411990442 11990411990423 11990311990321006596100659611989011989031 11990411990421 11990411990442 11990411990433 11990311990321006596100659611989011989032 11990411990421 11990411990442 11990411990443 1199031199032

the contribution of block states in each subspace of thepartition ese vectors will provide the framework to applythe manipulations described in Section 4

31 Cartesian Subspaces and Partitions Let 119826119826119808119808119896119896 sube 119826119826119896119896 119808119808 sube119812119812119812119808119808119812119812 denote the set of 119887119887119896119896 block states 119896119896 119896 1198961 2119896 119896119896119896 inthe events comprising119808119808 Note that since119808119808119812119812 then 119826119826119808119808119896119896 119812119812for all 119896119896

Let 119810119810119810119808119808119810 119808119808 sube 119812119812 119812 119808119808119812119812 denote the Cartesian product1198261198261198081198081 times119826119826

1198081198082 times⋯times119826119826119808119808119896119896 In general119810119810119810119808119808119810 is a superset of119808119808 since it

always contains all the elements of119808119808 and may contain eventsnot included in 119808119808

Consider the system of Table 1 For the subspace 119809119809 1198091198961006594100659411989011989011006594100659411989011989031006594100659411989011989041006596100659611989011989017100659610065961198901198901810065961006596119890119890191006596100659611989011989020119896 we get 1198261198261198091198091 119809 11989611990411990411 119904119904

21119896 119826119826

1198091198092 119809 11989611990411990412119896

1198261198261198091198093 119809 11989611990411990413 11990411990423 119904119904

33 119904119904

43119896 and 119810119810119810119809119809119810 119809 1198961006594100659411989011989011006594100659411989011989021006594100659411989011989031006594100659411989011989041006596100659611989011989017100659610065961198901198901810065961006596119890119890191006596100659611989011989020119896

As expected 119810119810119810119809119809119810 is a superset of 119809119809A Cartesian subspace denoted as119808119808

10066101006610 is herein dened as a

nonempty subspace of 119812119812 such that 11980811980810066101006610119809 119810119810119810119808119808

10066101006610119810 Note that all

the singleton subspaces are CartesianA Cartesian partition over 119808119808 119808119808 sube 119812119812 119812 119808119808119812119812 denoted

as 11982411982410066101006610119810119808119808119810 is herein dened as a partition comprised only of

Cartesian subspaces of 119808119808 e elements of11982411982410066101006610119810119808119808119810 are denoted

as 11985011985010066091006609

11982411982410066091006609119894119894 119810119808119808119810 119894119894 119896 1198961 2119896 119896119896119824119824

10066091006609119810119808119808119810119896 Note that every subspace of

119812119812 has at least one Cartesian partition comprised of singletonsubspaces

For instance the subspace 119809119809 dened above can be parti-tioned into the four subspaces 119850119850

10066091006609119824119824100660910066091 119810119809119809119810 119809 119896100659410065941198901198901119896 119850119850

10066091006609119824119824100660910066092 119810119809119809119810 119809 119896100659410065941198901198903119896

11985011985010066091006609119824119824100660910066093 119810119809119809119810 119809 119896100659410065941198901198904119896 and 119850119850

10066091006609119824119824100660910066094 119810119809119809119810 119809 1198961006596100659611989011989017100659610065961198901198901810065961006596119890119890191006596100659611989011989020119896 ese are all

Cartesian since 11985011985010066091006609119824119824100660910066091 119810119809119809119810 11985011985010066091006609

119824119824100660910066092 119810119809119809119810 11985011985010066091006609

119824119824100660910066093 119810119809119809119810 are singleton and

11985011985010066091006609119824119824100660910066094 119810119809119809119810 119809 11990411990421 times 119904119904

12 times 1198261198263

32 Implicit Subspaces and Partitions e state set 119826119826119808119808119896119896 iscalled a complete state set when it contains all the possiblestates of the 119887119887119896119896 block that is 119826119826

119808119808119896119896 119809 119826119826119896119896

An implicit subspace denoted as 119808119808asymp is herein dened

as a Cartesian subspace when all of its constituent blockstate subsets 119826119826

119808119808asymp119896119896 are either complete or singleton sets Note

that an implicit subspace corresponds to the implicant [11]containing only the block states in the singleton sets esubspace 119850119850

10066091006609119824119824100660910066094 119810119809119809119810 dened above is implicit and corresponds to

the implicant 11989611990411990421 11990411990412119896

For example the sets1198091198091 119809 11990411990421 times 11990411990422 times1198261198263 1198091198092 119809 11990411990421 times 119896119904119904

12 119904119904

22119896 times

1198261198263 and 1198091198093 119809 11990411990421 times 1198261198262 times 1198261198263 are all Cartesian (since they areCartesian products) but 1198091198092 is not implicit

An implicit partition over 119808119808 denoted as 119824119824asymp119810119808119808119810 is herein

dened as a partition comprised only of implicit subspacese elements of 119824119824

asymp119810119808119808119810 are denoted as 119850119850

asymp

11982411982410066091006609119894119894119810119808119808119810 Every subspace

119808119808 has at least oneimplicit partition the 119824119824asymp119810119808119808119810 119809 119808119808 and the

cardinalities of all possible partitions over 119808119808 lie between one(when 119808119808 is an implicit subspace) and |119808119808|

For example the partition over subspace 119809119809 into11985011985010066091006609

11982411982410066091006609119894119894 119810119809119809119810 119894119894 119896 1198961 2 3 4119896 is implicit Also the partition of

1198091198092 119809 11990411990421 times 11989611990411990412 11990411990422119896 times 1198261198263 into 119904119904

21 times 11990411990412 times 1198261198263 and 11990411990421 times 11990411990422 times 1198261198263 is

implicitAn implicit partition conveys the same information as

the set of its corresponding implicants erefore 1198091198092 can bederived from the prime implicants 11989611990411990421 119904119904

12119896 and 119896119904119904

21 119904119904

22119896

33 ContributionVectors e contribution vector of subspace119808119808 denoted by119821119821119810119808119808119810 is dened here as a vector of nonnegativeinteger entries 119899119899119894119894119896119896119896119896 119810119808119808119810 119896 119808+ reporting the sum of the contrib-utions of each state 119904119904119894119894119896119896119896119896 of each block 119887119887119896119896 in the events com-prising 119808119808 namely the number of times that each block statecontributes in the development of 119808119808


For the example of subspace 119809119809 the contribution vectoris 119821119821119821119809119809119821 119821 119821119821119821 119821119821 119821119821 119821119821 119821119821 119821119821 119821119821 119821119821 119821119821 119821119821 where the semicolons areused as separators between the three block compartments

Properties of contribution vectors include the following

(i) e vector length is 120582120582 119821 120582119870119870119896119896119821119821 |119826119826119896119896| therefore 119821119821119821119821119821119821 119821

ℕ120582120582+ for all 119821119821 119808 119808119808

(ii) For each block 119887119887119896119896 the sum of all the block entries in119821119821119821119821119821119821 equals |119821119821|

In general vector 119821119821119821119821119821119821 is an abstract (inductive) represen-tation of the information stored in 119821119821 As the size of 119821119821increases retrieving it from 119821119821119821119821119821119821 is not trivial it might evenbe impossible

34 Bicontribution Vectors Let the bicontribution vector119819119819119821119821119821119821 of subspace 119821119821 be dened as a vector of Boolean entries119897119897119894119894119896119896119896119896 119821119821119821119821 119821 119808O119821 I reporting the contribution or not of each state119904119904119894119894119896119896119896119896 of each block 119887119887119896119896 to the events of 119821119821 namely whether acertain block state contributes (ldquotruerdquo or I) or not (ldquofalserdquo orO) in the development of set 119821119821

Properties of bicontribution vectors include the follow-ing

(i) e vector length is 120582120582 119821 120582119870119870119896119896119821119821 |119826119826119896119896| so 119819119819119821119821119821119821 119821 119808O119821 I

120582120582for all 119821119821 119808 119808119808

(ii) e sum of all the entries corresponding to block 119887119887119896119896equals |119826119826119821119821119896119896 | for all 119896119896 119821 119808119821119821 119821119821119896 119821119870119870

For instance the associated bicontribution vector of 119821119821119821119809119809119821 is119819119819119821119809119809119821 119821 119821I119821 I119821 I119821O119821O119821 I119821 I119821 I119821 I119821 I119821 e sum of entries is 8 andequals the number of block state instances present in set 119809119809

LetΛ119821119821119821119821119821119821119821119821 119821 119808119821119821 119821119821119896 119821 |119821119821| 119821 119808I119821O denote the opera-tion applied on the contribution vector 119821119821119821119821119821119821 to deriveits associated bicontribution vector 119819119819119821119821119821119821 Based on thedenitions of contribution and bicontribution vectors

119819119819119821119821119821119821 119821 Λ1007714100771411982111982111982111982111982111982110077301007730⟺ 119897119897119894119894119896119896119896119896 119821119821119821119821 1198211007618100761810076341007634100762610076261007634100763410076421007642

O119821 119899119899119894119894119896119896119896119896 119821119821119821119821 119821 119821119821

I119821 119899119899119894119894119896119896119896119896 119821119821119821119821 gt 119821119821forall119896119896119821 119894119894119896119896

(1)

e reverse operation Λminus119821119821119821119821119821119821119821119821119821 yields 119810119810119821119821119821119821 as follows

Λminus1198211007714100771411982111982111982111982111982111982110077301007730

119821 10076821007682 119890119890 119821 10076821007682119904119904119894119894119821119821 119821 119904119904119894119894119821119821 119821119896 119821 119904119904119894119894119870119870119870119870 10076981007698 119821 119808119808 119821 119897119897119894119894119896119896119896119896 119821119821119821119821 119821 I119821 forall119896119896119821 11989411989411989611989610076981007698

(2)

erefore from the denition of 119810119810119821119821119821119821119821 Λminus119821119821119819119819119821119821119821119821119821 119821 119810119810119821119821119821119821Clearly 119819119819119821119821119821119821 always carries less information than its associ-ated vector 119821119821119821119821119821119821 unless 119821119821 is Cartesian

35 Cartesian Contribution Vectors e Cartesian contribu-tion vector of subspace119821119821 denoted as119821119821

10066101006610119821119821119821119821 is dened here as

the contribution vector of 119810119810119821119821119821119821 thus 11982111982110066101006610119821119821119821119821 119808 119821119821119821119810119810119821119821119821119821119821 Let

11989911989910066091006609119894119894119896119896119896119896 119821119821119821119821 denote the entries of 11982111982110066101006610

119821119821119821119821

For instance starting from subspace 119809119809 we get 11982111982110066101006610119821119809119809119821 119821

119821119821119821 119821119821 8119821 119821119821 119821119821 119821119821 119821119821 119821119821 119821119821 119821119821Let Ξ119821119819119819119821119821119821119821119821 denote the operation applied on the bicon-

tribution vector 119819119819119821119821119821119821 to derive the Cartesian contributionvector 119821119821

10066101006610119821119821119821119821 Based on the above vector denitions

11982111982110066101006610119821119821119821119821 119821 Ξ1007714100771411981911981911982111982111982111982110077301007730⟺ 119899119899

10066091006609119894119894119896119896119896119896119821119821119821119821 119821

1007618100761810076341007634100762610076261007634100763410076421007642

119821119821 119897119897119894119894119896119896119896119896 119821119821119821119821 119821 O119821|119810119810119821119821119821119821|10092501009250119826119826119821119821119896119896 10092501009250

119821 119897119897119894119894119896119896119896119896 119821119821119821119821 119821 I119821

forall119896119896119821 119894119894119896119896(3)

For every subspace 119821119821 each entry of 119810119810119821119821119821119821 119899119899119894119894119896119896119896119896 119821119821119821119821 laysbetween zero and 119899119899119894119894119896119896sim119896119896119821119821119821119821 Nonzero entries for which 119899119899

119894119894119896119896119896119896 119821119821119821119821 119821

119899119899119894119894119896119896sim119896119896119821119821119821119821 are herein called Cartesian entriese vector 119821119821119821119821119821119821 satises the Cartesian property if and

only if is 119821119821 a Cartesian subset In this case 119821119821119821119821119821119821 119821 11982111982110066101006610119821119821119821119821

Going back to set 119809119809 and comparing 119821119821119821119809119809119821 to 11982111982110066101006610119821119809119809119821

(i) the entries corresponding to the block states 119904119904119821119821 119904119904119821119821 119904119904

119821119821

and 119904119904119821119821 are Cartesian

(ii) the 119821119821119821119809119809119821 vector does not satisfy the Cartesian prop-erty since 119899119899119894119894119896119896119896119896 119821119809119809119821 119809 119899119899

119894119894119896119896sim119896119896119821119809119809119821 at 119808119896119896119821 119894119894119896119896 119821 119808119821119821 119821 or 119808119821119821 119821

e contribution vectors of implicit subspaces are hereincalled implicit contribution vectors Since implicit subspacesare always Cartesian implicit contribution vectors alwayssatisfy the Cartesian property

36 Vector Partitions Considering a partition119824119824119821119821119821119821 the con-tribution vector partition of 119824119824119821119821119821119821 denoted as 119821119821119821119824119824119821119821119821119821119821 isdened as the set of the contribution vectors 119821119821119821119821119821119824119824119894119894 119821119821119821119821119821

Since 120582119894119894 119899119899119894119894119896119896119896119896 119821119821119821

119876119876119894119894 119821119821119821119821119821 119821 119899119899119894119894119896119896119896119896 119821119821119821119821 for all 119896119896119821 119894119894119896119896 the vector

119821119821119821119821119821119821 is specied as the composite contribution vector of119821119821119821119824119824119821119821119821119821119821 In general composite contribution vectors carryless information than their vector partitions

Likewise to subspace partitions vector partitions can beclassied as Cartesian (when all their associated vectors areCartesian) or implicit (when all their associated vectors areimplicit) Cartesian and implicit partitions are denoted as11982111982110066101006610(bull) and 119821119821

asymp(bull) respectively

Considering subspace 119809119809 and its partition 1198211198211006609100660911982411982410066091006609119821 119821119809119809119821 119821

11980810065941006594119890119890119821 1198211198211006609100660911982411982410066091006609119821 119821119809119809119821 119821 11980810065941006594119890119890119821 119821119821

1006609100660911982411982410066091006609119821 119821119809119809119821 119821 11980810065941006594119890119890119821 and 119821119821

1006609100660911982411982410066091006609119821 119821119809119809119821 119821 11980810065961006596119890119890119821119821119821100659610065961198901198901198218119821100659610065961198901198901198219119821

10065961006596119890119890119821119821 the respective contribution vector partition is

119821119821119821119824119824119821119809119809119821119821 119821

100761810076181007634100763410076341007634100763410076341007634100763410076341007634100763410076341007626100762610076341007634100763410076341007634100763410076341007634100763410076341007634100763410076421007642

119821119821100765110076511198211198211006609100660911982411982410066091006609119821 11982111980911980911982110076671007667

119821119821100765110076511198211198211006609100660911982411982410066091006609119821 11982111980911980911982110076671007667

119821119821100765110076511198211198211006609100660911982411982410066091006609119821 11982111980911980911982110076671007667

119821119821100765110076511198211198211006609100660911982411982410066091006609119821 11982111980911980911982110076671007667

100761910076191007635100763510076351007635100763510076351007635100763510076351007635100763510076351007627100762710076351007635100763510076351007635100763510076351007635100763510076351007635100763510076431007643

11982110076181007618100763410076341007634100763410076261007626100763410076341007634100763410076421007642

119821119821119821 119821119821 119821119821 119821119821 119821119821 119821119821 119821119821 119821119821 119821119821 119821119821119821119821119821 119821119821 119821119821 119821119821 119821119821 119821119821 119821119821 119821119821 119821119821 119821119821119821119821119821 119821119821 119821119821 119821119821 119821119821 119821119821 119821119821 119821119821 119821119821 119821119821119821119821119821 119821119821 119821119821 119821119821 119821119821 119821119821 119821119821 119821119821 119821119821 119821119821

10076191007619100763510076351007635100763510076271007627100763510076351007635100763510076431007643

(4)


e 11985011985010066091006609

11982411982410066091006609119894119894 (119809119809119809 subsets are implicit so 119821119821(119824119824(119809119809119809119809 is an implicit

contribution vector partitionIt should be highlighted that a Cartesian contribution

vector partition is a disjoint-set data structure as it containsall the information necessary to (a) nd which subspaceincludes a certain event and (b) reconstruct the event unionset using the Λminus1[119819119819(bull)] operations

4 Decomposition and RecompositionOperations

Stating from a given outcome-based partition 119824119824119879119879(119812119812119809 thescope here is to derive an implicit partition featuringminimalcardinality for each of the 119850119850119824119824

119879119879

119903119903119894119894 (119812119812119809 119903119903119894119894 isin 119825119825 subspaces iswould be equivalent to a set of prime implicants describingthe event tablee scope is accomplished through the appli-cation of specic operations on the system vectors denedabove ese operations extract knowledge from the infor-mation carried in the composition vectors and store thisknowledge in the minimal possible schemes e naming ofthese operations is aer Shannonrsquos decomposition [11]

41 Decomposition of Contribution Vectors is sectionpresents an operation for the systematic manipulation of acontribution vector to yield a contribution vector partitione operation is called decomposition since it is a case ofmultistate Shannonrsquos decomposition applied on the contribu-tion vectors is operation decreases the abstraction of theinformation stored in the vector since contribution vectorpartitions carry more information than their compositevectors Starting from a contribution vector subsequentdecomposition actions generate a low cardinality implicitpartition of this vector

Consider a subspace 119808119808119808119808 119808 119808119808119808 119808 119808119808119808119808119808119808119808 119808 ge 2 and a non-empty subset 120590120590119808119808 sub 119808119808119808119808119808119808 e decomposition operation of 119808119808according to 120590120590119808119808 is a partition rule applied on 119808119808 to generatethe sets 119808119808120590120590119808119808 119808119808 minus 119808119808120590120590119808119808 such that

119808119808120590120590119808119808 = 10076821007682 119890119890 isin 119808119808 119890 119890120590120590119808119808119808119808 10076981007698

119808119808 minus 119808119808120590120590119808119808 = 10076821007682 119890119890 isin 119808119808 119890 10076501007650119810119810(119808119808119809 minus 119890120590120590119808119808119808119808 1007666100766610076981007698

where 119890120590120590119808119808119808119808 = 100768210076821198081198081198081198081 times 1198081198081198081198082 times ⋯ times 120590120590119808119808 times⋯ times 11980811980811980811980811987011987010076981007698

(5)

Since 120590120590119808119808 119808119808 then 119808119808120590120590119808119808 119808119808 minus 119808119808120590120590119808119808 119808119808e simplest way of applying the decomposition opera-

tion is to go through each one of the events in119808119808 as describedin the operation denition is procedure requires com-putational effort to decide whether a certain event belongsin 119808119808120590120590119808119808 or 119808119808 minus 119808119808120590120590119808119808 If on the other hand 119821119821(119808119808119809 containsCartesian entries the operation can be applied directly on119821119821(119808119808119809 Section 6 shows that decomposing 119821119821(119808119808119809 rather than119808119808 reduces the computational effort by several orders ofmagnitude

Consider a contribution vector 119821119821(119808119808119809 119808 119808119808119808 119808 119808119808119808119808119808119808119808 119808 ge2 and 119808119808119808119808119808119808119808 119808 119808 0 and a nonempty set 120590120590119808119808 sube 119808119808119808119808119808119808 e vectordecomposition operation according to 120590120590119808119808 is a partition rule

applied on 119821119821(119808119808119809 to yield the contribution vector partition119821119821(119808119808120590120590119808119808119809 119821119821(119808119808120590120590119808119808119809 such that

1198211198211007649100764911980811980812059012059011980811980810076651007665 = 119821119821100765010076501198081198081198211198211 times 1198081198081198211198212 times ⋯ times 120590120590119808119808 times⋯ times 11980811980811982111982111987011987010076661007666

1198211198211007649100764911980811980812059012059011980811980810076651007665 = 119821119821(119808119808119809 minus 1198211198211007649100764911980811980812059012059011980811980810076651007665 (6)

e development of 119821119821(119808119808120590120590119808119808119809 ensures that this vector has theCartesian property Instead of theCartesian product1198081198081198211198211 times119808119808

1198211198212 times

⋯ times 120590120590119808119808 times⋯ times 119808119808119821119821119870119870 we can use the bicontribution vector

1198191198191007649100764911980811980812059012059011980811980810076651007665 119808 119897119897119894119894119898119898119898119898 10092501009250119819119819(119808119808120590120590119808119808 119809 =

1007618100761810076341007634100763410076341007634100763410076341007634100762610076261007634100763410076341007634100763410076341007634100763410076421007642

119897119897119894119894119898119898119898119898 10092501009250119819119819(119808119808119809 119808119808 119808119898119898

I 119808119808 = 119898119898 and 119896119896119894119894119898119898119898119898 isin 120590120590119808119808

O 119808119808 = 119898119898 and 119896119896119894119894119898119898119898119898119898119898 notin 120590120590119808119808(7)

e decomposition operation can be applied iteratively Tosimplify the notation let1198211198210 be an initial contribution vectordecomposed into 119821119821

12059012059001 1198211198211 where 119821119821

12059012059001 has the Cartesian

property en 1198211198211 is decomposed into 11982111982112059012059012 1198211198212 where 119821119821

12059012059012

has the Cartesian property and so forth Application of thedecomposition operation over 119894119894 iterations replaces 1198211198210 withthe contribution vector partition 119821119821119894119894 = 119821119821

12059012059001 119821119821

12059012059012 119821119821

12059012059023 hellip

119821119821120590120590119894119894minus1119894119894 119821119821119894119894 e iterations terminate when119821119821119894119894 has the Cartesian

property therefore 119821119821119894119894 is a Cartesian partition Note that the119821119821120590120590119894119894minus1119894119894 vectors can be derived as 119821119821120590120590119894119894minus1

119894119894 = Ξ[119819119819120590120590119894119894minus1119894119894 ] using (7) and

119819119819119894119894minus1 = Λ[119821119821119894119894minus1]

411 Decomposition Example To illustrate the decompo-sition operation consider the set 1198501198501198791198791199031199031 (119812119812119809 of Table 1 and itscontribution vector 1198211198210 = [5 8 7 0 6 0 3 2 4 4]

Starting from 1198211198210 the bicontribution vector is 1198191198190 =[I I IO IO I I I I] and the Cartesian contribution vectoris 119821119821sim0 = [8 8 8 0 8 0 4 4 4 4] Clearly 119821119821sim0 has threeCartesian entries at 11989611989621 119896119896

33 and 11989611989643 Let 1205901205900 = 11989611989633

(i) 11981911981912059012059001 = [I I IO IOOO 119816119816O]

rArr 11982111982112059012059001 = [2 2 2 0 2 0 0 0 120786120786 0]

(ii) 1198211198211 = 1198211198210 minus 11982111982112059012059001 = [3 6 5 0 4 0 3 2 120782120782 4]

1198191198191 = [I I IO IO I I 119822119822 I]

Alternatively the decomposition could be applied simultane-ously at 120590120590lowast0 = 11989611989633 119896119896

43 as follows

(i) 119819119819120590120590lowast01 = [I I IO IOOO 119816119816 119816119816]

rArr 119821119821120590120590lowast01 = [4 4 4 0 4 0 0 0 120786120786 120786120786]

(ii) 119821119821lowast1 = 1198211198210 minus 119821119821

120590120590lowast01 = [1 4 3 0 2 0 3 2 120782120782 120782120782]

119819119819lowast1 = [I I IO IO I I 119822119822119822119822]


In both cases the vectors 1198211198211 and 119821119821lowast1 do not satisfy the

Cartesian property but they contain Cartesian entries Forinstance 119821119821lowast

1 can be decomposed at 120590120590lowast1 = 11990411990421 as follows

(i) 119819119819120590120590lowast12 = [0 120783120783120783 1 0 1 0120783 1 1 0 0120783

rArr 119821119821120590120590lowast12 = [0 120786120786120783 2 0 2 0120783 2 2 0 0120783

(ii) 119821119821lowast2 = 119821119821

lowast1 minus 119821119821

120590120590lowast12 = [1 120782120782120783 1 0 0 0120783 1 0 0 0120783

Both 119821119821lowast2 and 119821119821

120590120590lowast12 satisfy the Cartesian property and no

further decomposition is necessary So the decompositionof 1198211198210 yields the Cartesian contribution vector partition119821119821

12059012059001 119821119821

lowast2 119821119821

120590120590lowast12

e set vectors may contain complete blocks like therst block in 119821119821

12059012059001 e vectors 1198211198211205901205900

1 and 11982111982112059012059012 can be further

decomposed until they give implicit contribution vectorpartitions

(i)

11982111982112059012059001 = [4 4120783 4 0 4 0120783 0 0 4 4120783

sim 10076851007685[2 2120783 4 0 0 0120783 0 0 2 2120783[2 2120783 0 0 4 0120783 0 0 2 2120783

10077011007701

sim10076181007618100763410076341007634100763410076261007626100763410076341007634100763410076421007642

[1 1120783 2 0 0 0120783 0 0 2 0120783[1 1120783 2 0 0 0120783 0 0 0 2120783[1 1120783 0 0 2 0120783 0 0 2 0120783[1 1120783 0 0 2 0120783 0 0 0 2120783

10076191007619100763510076351007635100763510076271007627100763510076351007635100763510076431007643

(8)

(ii)

11982111982112059012059012 = [0 4120783 2 0 2 0120783 2 2 0 0120783

sim 10076851007685[0 2120783 0 0 2 0120783 1 1 0 0120783[0 2120783 2 0 0 0120783 1 1 0 0120783

10077011007701

sim10076181007618100763410076341007634100763410076261007626100763410076341007634100763410076421007642

[0 1120783 0 0 1 0120783 0 1 0 0120783[0 1120783 0 0 1 0120783 1 0 0 0120783[0 1120783 1 0 0 0120783 0 1 0 0120783[0 1120783 1 0 0 0120783 1 0 0 0120783

10076191007619100763510076351007635100763510076271007627100763510076351007635100763510076431007643

(9)

In the previous example the cardinality of 1198501198501198791198791199031199031(119812119812119812 was 13while the cardinality of the implicit partition is equal to 4+4+1 = 9 e latter cardinality could be even smaller if a moreintelligent decomposition strategy were applied For instance1198211198210 has two complete blocks (rst and third block) and threeCartesian entries (one in the rst block and two in the third)e choice of decomposition order that is the sequence of120590120590119894119894rsquos is crucial as discussed in the algorithm implementationsection

42 Recomposition of Contribution Vectors e outcome ofthe decomposition operation is an implicit contribution vec-tor partition Given this we can seek merging opportunitiesto create unions that have complete blocks thus reduce thepartition cardinality Since the contribution vectors are all

implicit the recomposition operation is hereby discussed interms of bicontribution vectors

Consider a block 119887119887119898119898 and its set of states 119826119826119898119898 Let 119819119819 be aset of |119826119826119898119898| bicontribution vectors 119819119819119894119894 = [119897119897119894119894119896119896119894119894119898119898119896119896120783 119894119894119898119898 isin 1 2hellip |119826119826119898119898| 119894119894119896119896 isin 1 2hellip |119826119826119896119896| 119896119896 isin 1 2hellip 119896119896 such that 119897119897119894119894119896119896119894119894119898119898119896119896 =

120582120582119894119894119896119896119896119896 isin O I for all 119896119896 119896119898119898 119894119894119898119898 119894119894119896119896 en the set 119819119819 can bereplaced by the single bicontribution vector 119819119819 = [120582120582119894119894119896119896119896119896 120783 where120582120582119894119894119898119898119898119898 = 119868119868 for all 119894119894119898119898 Note that 119819119819 = 119819119819(1198261198261198191198191

1 times 11982611982611981911981912 times ⋯ times 119826119826119898119898 times

⋯ times 1198261198261198191198191119896119896 119812

421 Recomposition Example Consider the set 119819119819 =10076821007682 [I O120783 I O O O120783 I O O O120783[O I120783 I O O O120783 I O O O120783 10076981007698

From the decomposition example discussed earlier etwo bicomposition vectors have exactly the same entriesin the second and third block but different entries in therst block In addition the rst block has exactly two stateinstances e two vectors can be recomposed into a singleequivalent vector [I I120783 IOOO120783 IOOO120783

5 Algorithm Implementation

Given the table associated with an event tree the proposedalgorithm launches an iterative process of decompositionand recomposition operations until we obtain an implicitpartition of minimal cardinality for each one of the tree out-comes orking with the vectors dened above rather thansets of events decreases signicantly the amount of informa-tion being stored and the computational effort for manipu-lating this information

e algorithm applies the vector decomposition andrecomposition operations using a set of heuristic rulesese rules help in the identication of more promisingentries to apply the operations Sections 51 and 52 describethe heuristic rules and Section 53 discusses how theseprocedures work together within the proposed algorithm

51 Heuristic Rules for Decomposition Order e decompo-sition operations proceed iteratively replacing the original|119825119825| vectors of the outcome partition with |119825119825| contributionvector sets Decompositions are applied locally based onthe features (eg the Cartesian entries) of each contributionvector

Given a composition vector the choice of decompositionorder is crucial in preserving the maximum possible of theinitial complete blocks e set 1198501198501198791198791199031199031 (119812119812119812 of Table 1 has a com-plete block of 2 states and a complete block of 4 states If thedecomposition order is different the nal implicit partitioncould be different In effect applying the decomposition onentries 11990411990433 119904119904

43 of1198211198210 results in a reduction by 4 = 13minus9 on the

cardinality of the nal implicit partition of 1198501198501198791198791199031199031 (119812119812119812 Applyingthe decomposition on 11990411990421mdashfollowed by 11990411990433 119904119904

43 119904119904

13 119904119904

34 119904119904

44


and so forth in any ordermdashpreserves the third block and thenal reduction is by 13 minus 7 = 6

1198211198210 = [5 8 7 0 6 0 3 2 4 4] sim 10076851007685[0 8 4 0 4 0 2 2 2 2][5 0 3 0 2 0 1 0 2 2]

10077011007701

sim10076181007618100763410076341007634100763410076261007626100763410076341007634100763410076421007642

[0 4 4 0 0 0 1 1 1 1][0 4 0 0 4 0 1 1 1 1][4 0 2 0 2 0 0 0 2 2][1 0 1 0 0 0 1 0 0 0]

10076191007619100763510076351007635100763510076271007627100763510076351007635100763510076431007643

sim10076181007618100763410076341007634100763410076261007626100763410076341007634100763410076421007642

[0 4 4 0 0 0 1 1 1 1][0 4 0 0 4 0 1 1 1 1][1 0 1 0 0 0 1 0 0 0]

[1 0 1 0 0 0 0 0 1 0][1 0 1 0 0 0 0 0 0 1][1 0 0 0 1 0 0 0 1 0][1 0 0 0 1 0 0 0 0 1]

10076191007619100763510076351007635100763510076271007627100763510076351007635100763510076431007643

(10)

e following decomposition rules support the generationof the smallest possible implicit partitions at the minimumpossible execution time and they are applied on each contri-bution vector that is not Cartesian

(a) If there are Cartesian entries in the current contribu-tion vector

(i) it is prefered to decompose at Cartesian entriesin incomplete blocks to avoid breaking thecomplete blocks Note that in this case thedecomposition order makes no difference to thenal partitions

(ii) if the only Cartesian entries are within completeblocks start decomposing the complete blockswith the largest span between their contributionvalues relatively to their number of statesthat is sort blocks according to (max119896119896119899119899

119894119894119896119896119896119896 minus

min119896119896119899119899119894119894119896119896119896119896 )|119826119826119896119896|

(b) If there are no Cartesian entries in the current con-tribution vector then

(i) if no complete blocks are present prefer incom-plete blocks with more states being present anddecompose them into as many vectors as thenumber of states present

(ii) if only complete blocks are present decomposethe block featuring the entry with the maxi-mum departure from its Cartesian value that ismax11989611989611989911989910066091006609

119894119894119896119896119896119896 minus 119899119899

119894119894119896119896119896119896

Before the application of these rules it is essential to recognizewhich of the complete blocks make sense according to thevalues of the contribution vector entries For instance thevector [5 8 7 0 6 0 3 2 4 4] could lead to a partition thatincludes a bicontribution vector of 2 complete blocks butthis not possible for the vector [3 8 5 0 6 0 2 2 4 3] sincethe rst entry in the compartment of the rst block is lowerthan the number of states in the third block Similarly in thevector [0 4 2 0 2 0 1 1 1 1] the third block cannot remaincomplete under any decomposition order since there is no

single entry in the second block compartment greater orequal to |1198261198263| = 4

e nal Cartesian partition includes vectors and mayneed to be further decomposed to derive implicit parti-tions In this case the vectors are decomposed in all theirincomplete block entries (which are all Cartesian) and thedecomposition order makes no difference

52 Heuristic Rules for Recomposition Order Once thedecomposition stage is completed the nal implicit parti-tions may have vectors that can be merged is reduces thepartition cardinalities e choice of recomposition order iscrucial since the application of decomposition operationson large event tables can produce numerous vectors ascandidates for recomposition

Let 1198241198240(119850119850119879119879119903119903119894119894 (119812119812)) 119903119903119894119894 isin 119825119825 denote the implicit partitions

output from the decomposition stage e recompositionalgorithm proceeds according to the following steps appliedon each vector partition 119821119821

100661010066100 = 11982111982110066101006610(1198241198240(119850119850

119879119879119903119903119894119894 (119812119812))) (or the asso-

ciated bicontribution vectors)

(i) Partition11982111982110066101006610into sets having the same complete blocks

Let 11982111982110066101006610119886119886 be such a subset of 119821119821

10066101006610

(ii) e set 11982111982110066101006610119886119886 is a candidate for creating an additional

complete block 119887119887119898119898 if there is at least one subset11982111982110066101006610119898119898119894119894119886119886 sube

11982111982110066101006610119886119886 of cardinality |119826119826119898119898| where 119894119894 isin 119894119894119898119898 is the index ofthe particular subset for 119887119887119898119898 such that (i) its overallcontribution vector features 119887119887119898119898 as a complete blockand (ii) the remaining blocks have the same state sets119826119826119898119898119894119894119896119896 for all 119896119896 119896119898119898

(iii) If11982111982110066101006610119898119898119894119894119886119886 is replaced by its composite contribution vector

the resulting vector partition 119821119821100661010066101 = 119821119821

10066101006610(1198241198241(119850119850

119879119879119903119903119894119894 (119812119812))) is

implicit

(iv) e process is repeated on the new set119821119821100661010066101 to reduce it

to 119821119821100661010066102 and so forth and terminates when there are no

more recomposition candidates le

e set 11982111982110066101006610119886119886 may have several subsets 119821119821

10066101006610119898119898119894119894119886119886 that relate to the

creation of different complete blocks andor to different waysof creating a particular complete block e recompositionprocedure is supported by intelligent selection biases toensure that the merging opportunities are properly exploitedDuring the iterative recomposition process the followingrecomposition rules are applied on every set 119821119821

10066101006610119886119886 of Cartesiancontribution vectors featuring the same complete blocks

(i) Find all the candidate sets 11982111982110066101006610119898119898119894119894119886119886 sube 119821119821

10066101006610119886119886 119894119894 isin 119894119894119898119898 forcreating each new complete block at 119887119887119898119898 for all 119898119898 isin1 2hellip 119870119870

(ii) Associate each candidate to a score 119908119908119898119898119894119894119886119886 proportional

to |119826119826119898119898| and 119894119894119898119898 and incorporate the potential to createtwo complete blocks in one go

(iii) Select among conicting sets according to their 119908119908119898119898119894119894119886119886


T 2 Recomposition example initial set of bicontribution vectors (Step 1)

119819119819119894119894 1198871198871 1198871198872 1198871198873 1198871198874 1198871198875 1198871198876 1198871198877 1198871198878 1198871198879 1198871198871024 O O O I I O O I O O I O I I O I O I O I O I O26 O O O I I O O I O O I O I I O I O I O O I I O28 O O O I I O O I O O I O I O I I O I O I I I O21 I O O O I O I I I O I O I I O I O I O I O I O25 I O O O I O I I I O I O I I O I O I O O I I O11 O O O I I O O I O I O I O I I I O I O I I I O12 O O O I I O O I O O I I O I I I O I O I I I O18 O O O I I O O I O I O O I I I I O I O I I I O22 O O O I I O I O O O I O I I O I I I I I O I O23 O O O I I O O O I O I O I I O I I I I I O I O27 I O O O I O I I I O I O I O I I O I O I I I O5 I O O O I O I I I I O I O I I I O I O I I I O6 I O O O I O I I I O I I O I I I O I O I I I O14 I O O O I O I I I I O O I I I I O I O I I I O15 O O I O I O I I I I O O I I O I I I I I O I O19 O I O O I O I I I O I O I I O I I I I I O I O20 O O I O I O I I I O I O I I O I I I I I O I O7 O O O I I O I O O I O I O I I I I I I I I I O8 O O O I I O I O O O I I O I I I I I I I I I O9 O O O I I O O O I I O I O I I I I I I I I I O10 O O O I I O O O I O I I O I I I I I I I I I O16 O O O I I O I O O I O O I I I I I I I I I I O17 O O O I I O O O I I O O I I I I I I I I I I O29 O O O I I O I O O I O I O I I I I I I I I O I30 O O O I I O I O O I O O I I I I I I I I I O I1 O I O O I O I I I I O I O I I I I I I I I I O2 O I O O I O I I I O I I O I I I I I I I I I O3 O O I O I O I I I I O I O I I I I I I I I I O4 O O I O I O I I I O I I O I I I I I I I I I O13 O I O O I O I I I I O O I I I I I I I I I I O

(iv) Apply the recomposition operation on the selectedsets

Note that in the new partition the composite vectors shouldbe removed form the subset119821119821

10066101006610119886119886 and possibly included in othersubsets including the complete blocks of 119821119821

10066101006610119886119886 and completeblocks created during the recompositione reason why thevalue of 119868119868119898119898 is taken into account is that larger 119868119868119898119898rsquos increasethe probability of nding recomposable sets in the nextiteration of the recomposition process

Consider for example the vectors of Table 2 representingthe implicit partition for the h outcome of the R example solved in Section 61 As explained above recompositionoperations are applied on bicontribution vectors e vectorsare sorted and divided into different sets according to theircomplete blocks e procedure starts from the subset withthe fewer complete blocks is includes the vectors 11981911981924and 11981911981926 which are merged to give 11981911981924+26 Table 3 showsthe Cartesian set updated with 11981911981924+26 which can now be

merged with 11981911981928 and so forthe recomposition choices arenot always so few and they can be conicting e subsetof Table 4 appears aer a few iterations A crude analysisindicates the potential of creating new complete blocks at

(i) block 1198871198874 by merging 119819119819i amp 119819119819iii andor 119819119819iv amp 119819119819viii

(ii) block 1198871198875 bymerging119819119819ii amp119819119819iii 119819119819v amp119819119819vii andor119819119819vi amp119819119819viii

(iii) block 11988711988710 by merging 119819119819v amp 119819119819vi andor 119819119819vii amp 119819119819viii

e above seven merging actions involve binary state blocksso their initial scores are equal to 2 is score is multipliedby the number of candidate merges per block (119868119868119898119898) giving ascore of 4 for the 4 merges in blocks 1198871198874 and 11988711988710 and 6 forthe 3 merges in block 1198871198875 e possible merging actions aresorted according to their score and an action can take place ifit does not conict with any of the higher score actions In thissense the proposed actions are 119819119819ii amp 119819119819iii 119819119819v amp 119819119819vii and 119819119819vi amp119819119819viii


is involvement of 119868119868119898119898 in the score calculation stemsfrom the observation that the resulting vectors have manycommon entries so the possibility of these vectors beingtreated as candidates for subsequent merges is very highIn effect amongst the merged vectors of Table 5 vectors119819119819v+vii and 119819119819vi+viii are candidates to be merged Howeverthis potential should be examined along with other vectorsfeaturing the same complete blocks

53 Algorithm Implementation e decomposition andrecomposition operations discussed above are each imple-mented into an iterative procedure Following is a step-by-step description of the algorithm using the motor-operatedvalve (MOV) example of Table 1 to illustrate the differentprocedures

Step 1 Acquire event outcome datais step returns a table of119870119870 columns for the119870119870 component blocks and 1 column for theoutcomes Following is the data array for the MOV example

1 1 1

1

1 1 2 4

1 1 3 1

1 1 4 1

1 2 1 3

1 2 2 3

1

2 3 2

1 2 4 3

1 3 1 4

1 3 2 4

1 3 3 1

1 3 4

1

1 4 1 3

1 4 2 3

1 4 3 2

1 4 4 2

2 1 1 1

2

1 2 1

2 1 3 1

2 1 4 1

2 2 1 2

2 2 2 2

2 2 3

2

2 2 4 2

2 3 1 1

2 3 2 1

2 3 3 1

2 3 4 1

2

4 1 2

2 4 2 2

2 4 3 2

2 4 4

2

Step 2 Get the contribution vectors referring to the outcomepartition e 32 events in columns 1 to 3 of the above arraycan be divided into 13 11 5 and 3 events according to the

outcome they yield e following arrays can be derived forthe 4 outcomes

1 1 1 1

1 1 3 1

1 1 4 1

1 3 3 1

1 3 4 1

2 1 1 1

2 1 2 1

2 1 3 1

2 1 4 1

2 3 1 1

2 3 2 1

2 3 3 1

2 3 4 1

1 2 3 2

1 4 3 2

1 4 4 2

2 2 1 2

2 2 2 2

2 2 3 2

2 2 4 2

2 4 1 2

2 4 2 2

2 4 3 2

2 4 4 2

1 2 1 3

1 2 2 3

1 2 4 3

1 4 1 3

1 4 2 3

1 1 2 4

1 3 1 4

1 3 2 4

Using composition vectors each of these arrays gives a row ofthe following array

119873119873119877119877 = 100762010076201007636100763610076361007636

10076441007644

5 8 7 0 6 0 3 2 4 4 13 8 0 5 0 6 2 2 4 3 25 0 0 3 0 2 2 2 0 1 33 0 1 0 2 0 1 2 0 0 4

100762110076211007637100763710076371007637

10076451007645

(11)

e 119873119873119877119877array now stores the contribution vector partitionaccording to the system outcome e superscript 119877119877 is used

to indicate that the last column of119873119873119877119877 also stores the numberof the outcome associated with each contribution vector

Step 3 Check data consistency is step exposes any irreg-ularities present in the original data by checking that allthe possible complete events are present in the event tableonly these events are present and the table has no duplicateentries

st check involves the columns of the contributionvector array Based on the number of block states we estimatethe exact number of occurrences for each one of them andsee if the sums of entries in the array satisfy this constrainte MOV example has 3 blocks of 2 4 and 4 states eacherefore each one of the 2 states of block 1 should occur16 times each one of the 4 states of blocks 2 and 3 shouldoccur 8 times each In effect 16 = 5+3+5+3 = 8+8+0+08 = 7 + 0 + 0 + 1 = 0 + 5 + 3 + 0 = ⋯ = 4 + 3 + 1 + 0

e second check involves the rows of the contributionvector array Each entry refers to a specic state of a specicblock In each row the total number of occurrences of theblock 1 states should be equal to the number of occurrencesof blocks 2 and 3 Looking at the rst row of the contributionvector array 5 + 8 = 7 + 0 + 6 + 0 = 3 + 2 + 4 + 4

Step 3 is repeated for the other rows of119873119873119877119877

Step 4 Contribution vector decomposition is stepapplies the decomposition operation (Section 41) andthe decomposition rules (Section 51) to the rst row of

array 119873119873119877119877 Let for instance the contribution vector be[5 8 7 0 6 0 3 2 4 4] e vector exhibits completestate blocks in block 1 (2nd state) and block 3 (3rd


T 3 Recomposition example set of bicontribution vectors

119819119819119894119894 1198871198871 1198871198872 1198871198873 1198871198874 1198871198875 1198871198876 1198871198877 1198871198878 1198871198879 1198871198871024 + 26 O O O I I O O I O O I O I I O I O I O I I I O28 O O O I I O O I O O I O I O I I O I O I I I O21 I O O O I O I I I O I O I I O I O I O I O I O25 I O O O I O I I I O I O I I O I O I O O I I O11 O O O I I O O I O I O I O I I I O I O I I I Ohellip

T 4 Recomposition example subset of bicontribution vectors including only 1198871198876 1198871198877 1198871198878 and 1198871198879 as complete blocks

119819119819119894119894 1198871198871 1198871198872 1198871198873 1198871198874 1198871198875 1198871198876 1198871198877 1198871198878 1198871198879 11988711988710i O O O I I O O O I O I I O I I I I I I I I I Oii O O O I I O O O I I O O I I I I I I I I I I Oiii O O O I I O O O I I O I O I I I I I I I I I Oiv O O O I I O I O O O I I O I I I I I I I I I Ov O O O I I O I O O I O O I I I I I I I I I O Ivi O O O I I O I O O I O O I I I I I I I I I I Ovii O O O I I O I O O I O I O I I I I I I I I O Iviii O O O I I O I O O I O I O I I I I I I I I I O

4th states) e rules indicate that the decomposition isrst applied to block 1 and later to block 3 ccordingto Section 51 the vector [5 8 7 0 6 0 3 2 4 4]is decomposed into the Cartesian contribution vector[0 8 4 0 4 0 2 2 2 2] and the remaining contributionvector [5 0 3 0 2 0 1 0 2 2]

Step 5 Update system arrays ere are 1 + 119877119877 working arrayswhere contribution vectors (and their outcome) are storederst one is119873119873119877119877e others are denoted by119873119873

10066111006611119894119894 and store theCartesian vectors generated during Step 4 for each response

119898119898 starting from 119894119894 119894 1 Each row of 119873119873119877119877 treated during Step4 is replaced by the remaining contribution vector In our

example during the rst iteration119873119873119877119877 and119873119873100661110066111 become

119873119873119877119877 119894 100762010076201007636100763610076361007636

10076441007644

5 0 3 0 2 0 1 0 2 2 13 8 0 5 0 6 2 2 4 3 25 0 0 3 0 2 2 2 0 1 33 0 1 0 2 0 1 2 0 0 4

100762110076211007637100763710076371007637

10076451007645

119873119873100661110066111

119894 [0 8 4 0 4 0 2 2 2 2]

(12)

Note that there is no need to store outcome information inthe arrays119873119873

10066111006611119894119894Steps 4 and 5 are repeated until a decomposition oper-

ation leads to two Cartesian vectors en both vectors areadded to119873119873

10066111006611119894119894 the rst row of119873119873119877119877 is removed and 119894119894 is replacedby 119894119894 + 1

Step 6 Decompose 11987311987310066111006611119894119894 e array 119873119873

10066111006611119894119894 is further decomposedto get an array119873119873

asymp 119894119894of implicit contribution vectors Since the

vectors have only Cartesian entries the operations can be

easily applied on the bicontribution vectors of 11987311987310066111006611119894119894 and 119873119873

asymp 119894119894

denoted by 11987111987110066091006609119894119894and 119871119871

asymp119894119894 respectively e arrays referring to the

contribution vector [5 8 7 0 6 0 3 2 4 4] are

119871119871100660910066091

119894 10076201007620

10076441007644

0 1 1 0 1 0 1 1 1 11 1 0 1 0 1 0 0 0 11 0 1 0 0 0 1 0 0 0

10076211007621

10076451007645(13)

and aer further decomposition

119871119871asymp1

119894

100762010076201007636100763610076361007636100763610076361007636100763610076361007636100763610076361007636100763610076361007636

10076441007644

0 1 0 0 0 1 1 1 1 10 1 0 0 1 0 1 1 1 11 0 1 0 0 0 0 0 1 01 0 0 0 1 0 0 0 1 01 0 1 0 0 0 0 0 0 11 0 0 0 1 0 0 0 0 11 0 1 0 0 0 1 0 0 0

100762110076211007637100763710076371007637100763710076371007637100763710076371007637100763710076371007637100763710076371007637

10076451007645

(14)

Step 6 supports the following recomposition actions

Step 7 Apply recomposition actions is step applies therecomposition operation (Section 42) and the recompositionrules (Section 52) to the parts of 119871119871

asymp119894119894sharing the same

outcome Note that the example considered here is too smalland simple to offer potential for recomposition

Steps 4ndash7 are repeated until the array119873119873119877119877 is empty

Step 8 Algorithm termination e nal output of the pro-cedures described here is the arrays 119871119871

asymp119898119898 which represent

implicit partitions of signicantly reduced cardinality com-pared to the size of the system event table Note that thedecomposition and the recomposition operations developedhere ensure that the consistency of the data is preservedthroughout the vector processing


T 5 Recomposition example subset of bicontribution vectors including only the vectors appearing in Table 4

119819119819119894119894 1198871198871 1198871198872 1198871198873 1198871198874 1198871198875 1198871198876 1198871198877 1198871198878 1198871198879 11988711988710i O O O I I O O O I O I I O I I I I I I I I I Oiv O O O I I O I O O O I I O I I I I I I I I I Ov + vii O O O I I O I O O I O I I I I I I I I I I O Ivi + viii O O O I I O I O O I O I I I I I I I I I I I Oii + iii O O O I I O O O I I O I I I I I I I I I I I O

e Matlab environment is chosen as suitable for thefast manipulation of matrices using the built-in matrixoperations For instance the decomposition when there areno Cartesian entries requires knowledge of the exact eventsubspace that corresponds to the processed contributionvector e algorithm can either keep track of the eventscontributing in each vector or go through the original eventtable to isolate the subspace relating to each vector eformer though it is more sophisticated takes up a lot ofmemory even for relatively small problems Matlab takesadvantage of its built-in matrix operations for sort and ndto reduce signicantly the execution time

6 Case Studies

61 Case Study 1 e rst case study is taken from Papa-zoglou [10] and concerns the development of an event tree fora boiling water nuclear reactor e system involves 10 stateblocks with 2 to 4 block states each e event space consistsof 3072 complete events and the system has 5 outcomesPapazoglou [10] provided a set of Boolean equations anddeveloped functional block diagrams that embedded infor-mation on the dependencies between the blocks He nallypresented a reduced event tree of 41 branches Note thatif the reactor system is treated in BowTieBuilder [12 13]without providing dependency information the resultingevent tree has 110 branches is conrms that the efficiencyof functional block diagram applications in reducing the sizeof event trees depends on the structure of the Booleanmodelthat dictates the dependencies between the blocks

e methodology proposed here takes as input the orig-inal 3072 times 11 event table and produces the results reportedin Table 6 Note that

(i) the states 1 2 3 4 of block I correspond toLMNT and

(ii) the outcomes 1 2 3 4 5 correspond to CI CIICIII CIV Success

of Papazoglou [10] Each row of Table 6 gives a Cartesianvector (or an implicant) corresponding to a branch of theevent tree In this sense the reduced tree described herehas only 38 branches e proposed algorithm identies aninconsistency in the partition of block C of the originaldata Resolving it leads to different results for outcome CIVand this explains the three branches difference between 41and 38 e rest of the branchesimplicants are notably thesame with a single exception involving the choice to expandblock U rather than block Q (see bold cells of Table 6 lines

12ndash15) While both choices yield four branchesimplicantsthis differentiation shows that the procedure proposed hereis not biased by the order of the blocks in the event table data

62 Case Study 2 e proposed methodology is testedagainst a large problem involving 16 blocks including oneblock having four state instances four blocks with three statesand eleven binary e original event table has 663552 times17 cells e system has 5 possible outcomes e initialevent table is constructed via recursive partitions of the eventsubspace

e resulting implicit partition has totally 273 vectors inparticular 86 115 34 28 and 71 for the ve outcomes erecomposition stage requires 108 CPU seconds en thenal implicit partitions have totally 178 vectors in particular31 54 16 20 and 57 for the respective partitions of the veoutcomes

CPU times can give an idea of the relative effort investedin the different activities taking place during a run In thisrelatively large problem the preparatory Steps 1ndash3 of Section53 require 127 CPU seconds e decomposition stepsrequire only 00469CPU seconds for a total of 86 decom-positions using rule (a) and 124CPU seconds for a total of52 decompositions using rule (b) of Section 51 ereforethe application of decompositions on the basis of Cartesianentries reduces the computational effort by almost 4 orders ofmagnitude Clearly an intelligent reduction of the frequencyof visiting the event table would bring signicant benets inthe computational times Note that CPU times refer to anIntel Core Quad 250GHz processor with 195GBRAM

Finally the proposed procedure manages to reduce theexpanded event tree to just 00268 of its original size enal partitions are easily translated into a set of implicantsere is no proof that this is a prime set since there is lackof theoretical background on sufficient and necessary mini-mality conditions In any case the proposed methodology isa fast effective and intelligent way to reduce substantially alarge event tree and facilitate the quantication of risk

7 Conclusions

is work presents a new methodology for the reductionof event trees without the use of structural or functionalinformation on the system e work applies a holisticapproach based on the concept of contribution vectors togenerate a minimal set of implicants representative of thesystem behavior e method inherits the advantages andlimitations of event tree representation In this sense themethod is not hindered by component interdependences


T 6 Final reduced event table for case study 1

No of event tree branch (or implicant) Blocks states Outcome 119903119903119894119894I C M Q U X E I V W1 mdash 2 mdash mdash mdash mdash mdash mdash mdash mdash 42 1 1 mdash mdash mdash mdash 1 1 mdash 1 53 1 1 mdash mdash mdash mdash 1 1 mdash 2 24 1 1 mdash mdash mdash mdash 1 2 mdash mdash 35 1 1 mdash mdash mdash mdash 2 mdash mdash mdash 36 2 1 mdash 1 mdash mdash mdash mdash mdash 1 57 2 1 mdash 1 mdash mdash mdash mdash mdash 2 28 2 1 mdash 2 1 mdash mdash mdash mdash 1 59 2 1 mdash 2 1 mdash mdash mdash mdash 2 210 2 1 mdash 2 2 1 mdash mdash 1 1 511 2 1 mdash 2 2 1 mdash mdash 1 2 212 2 1 mdash 2 2 1 mdash mdash 2 mdash 113 2 1 mdash 2 2 2 mdash mdash mdash mdash 114 3 1 mdash mdash 1 mdash mdash mdash mdash 1 515 3 1 mdash mdash 1 mdash mdash mdash mdash 2 216 3 1 mdash mdash 2 1 mdash mdash 1 1 517 3 1 mdash mdash 2 1 mdash mdash 1 2 218 3 1 mdash mdash 2 1 mdash mdash 2 mdash 119 3 1 mdash mdash 2 2 mdash mdash mdash mdash 120 4 1 1 1 mdash mdash mdash mdash mdash mdash 521 4 1 1 2 1 mdash mdash mdash mdash 1 522 4 1 1 2 1 mdash mdash mdash mdash 2 223 4 1 1 2 2 1 mdash mdash 1 1 524 4 1 1 2 2 1 mdash mdash 1 2 225 4 1 1 2 2 1 mdash mdash 2 mdash 126 4 1 1 2 2 2 mdash mdash mdash mdash 127 4 1 2 mdash mdash mdash 1 1 mdash 1 528 4 1 2 mdash mdash mdash 1 1 mdash 2 229 4 1 2 mdash mdash mdash 1 2 mdash mdash 330 4 1 2 mdash mdash mdash 2 mdash mdash mdash 331 4 1 3 mdash 1 mdash mdash mdash mdash 1 532 4 1 3 mdash 1 mdash mdash mdash mdash 2 233 4 1 3 1 2 mdash mdash mdash mdash 1 534 4 1 3 1 2 mdash mdash mdash mdash 2 235 4 1 3 2 2 1 mdash mdash 1 1 536 4 1 3 2 2 1 mdash mdash 1 2 237 4 1 3 2 2 1 mdash mdash 2 mdash 138 4 1 3 2 2 2 mdash mdash mdash mdash 1

and noncoherent behavior in the considered systems eproposed representation framework stems from Cartesianproducts to dene partitions using composition vectorse representation provides the basis for the applicationof decomposition and recomposition operations on sin-gle composition vectors and composition vector partitionsImplementation issues for the efficient use of these operationswithin an iterative algorithmic framework are discussedthoroughly

e proposed method is tested against two case studiesone found in the literature and a ctitious large scale problem

In the former the method provides a set of prime implicantsvery similar to the one reported in the literature e latterillustrates the efficiency of themethod in handling large-scaleproblems and proves the computational advantages from theproposed representation and operations

Future work considers the use of the theoretical back-ground presented here to develop necessary and sufficientconditions for the minimality of the nal set of implicantsese conditions could then be incorporated in the recom-position stage to guide an optimal search algorithm towardsthe set of prime implicants


Nomenclature

119870119870 Number of system blocks119887119887119896119896 System block 119896119896 119896 119896119896119896 119896119896119896 119896119870119870119896119826119826119896119896 Set of internal states of block

119887119887119896119896 119826119826119896119896 = 119896119904119904119896119896119896119896 119904119904119896119896119896119896119896 119896 119904119904

119870119870S119896119896119896119896 119896

119904119904119894119894119896119896119896119896 119894119894119896119896th internal state of block119887119887119896119896 119894119894119896119896 119896 119896119896119896 119896119896119896 119896119870119870119904119904119894119894119896119896119896119896

119896119812119812 System event space119890119890 Complete joint event 119890119890 119896 119812119812119825119825 Set of the all possible system outcomes119903119903119895119895 System outcome 119895119895 119896 119896119896119896 119896119896119896 119896 119895119825119825119895119896119879119879 Event table mapping 119879119879 119879 119890119890 119890 119903119903 = 119879119879119890 119890119890119890

where 119890119890 119896 119812119812 and 119903119903 119896 119825119825119808119808119896119808119808 Nonempty subspaces of 119812119812119895119831119831119895119896 119870119870119831119831 Number of elements (cardinality) of set119831119831119824119824119890119808119808119890 Partition applied over 119808119808119850119850119876119876119894119894 119890119808119808119890 119894119894th subset of 119808119808 according to

119824119824119890119808119808119890 119894119894 119896 119896119896119896 119896119896119896 119896119894119894119824119824119890119808119808119890119896119824119824119879119879119890119808119808119890 Outcome-based partition of 119808119808

(according to mapping 119827119827)119826119826119808119808119896119896 Set of 119887119887119896119896-block states 119896119896 119896 119896119896119896 119896119896119896 119896119870119870119896 in

the events comprising 119808119808119810119810119890119808119808119890 Cartesian product 119826119826119808119808119896 times 119826119826

119808119808119896 times ⋯ times 119826119826119808119808119870119870

11980811980810066101006610 Cartesian subspace119808119808asymp Implicit subspace

11982411982410066101006610119890119808119808119890 Cartesian partition over subspace 119808119808

1198501198501006609100660911982411982410066091006609119894119894 119890119808119808119890 119894119894th subset of119824119824

10066101006610119890119808119808119890 119894119894 119896 119896119896119896 119896119896119896 119896119870119870119824119824

10066091006609119890119808119808119890119896

119824119824asymp119890119808119808119890 Implicit partition over 119808119808

119850119850asymp

11982411982410066091006609119894119894119890119808119808119890 119894119894th subset of119824119824

asymp119890119808119808119890 119894119894 119896 119896119896119896 119896119896119896 119896119870119870119824119824

asymp119890119808119808119890119896

119821119821119890119808119808119890 Contribution vector of 119808119808119899119899119894119894119896119896119896119896 119890119808119808119890 Entry of 119821119821119890119808119808119890 119899119899119894119894119896119896119896119896 119890119808119808119890 119896 119808+ and

119896119896 119896 119896119896119896 119896119896119896 119896119870119870119896 119894119894119896119896 119896 119896119896119896 119896119896119896 119896119870119870119904119904119894119894119896119896119896119896119896

119819119819119890119808119808119890 Bicontribution vector of subspace 119808119808119897119897119894119894119896119896119896119896 119890119808119808119890 Entry of 119819119819119890119808119808119890 119897119897119894119894119896119896119896119896 119890119808119808119890 119896 119896O119896 I119896 and

119896119896 119896 119896119896119896 119896119896119896 119896119870119870119896 119894119894119896119896 119896 119896119896119896 119896119896119896 119896119870119870119904119904119894119894119896119896119896119896119896

11982111982110066101006610119890119808119808119890 Cartesian contribution vector of 119808119808

11989911989910066091006609119894119894119896119896119896119896 119890119808119808119890 Entry of 119821119821

10066101006610119890119808119808119890 119899119899

asymp119894119894119896119896119896119896119890119808119808119890 119896 119808+ and

119896119896 119896 119896119896119896 119896119896119896 119896119870119870119896 119894119894119896119896 119896 119896119896119896 119896119896119896 119896119870119870119904119904119894119894119896119896119896119896119896

119821119821119890119824119824119890119808119808119890119890 Contribution vector partition of119824119824119890119808119808119890119821119821119890119850119850119824119824119894119894 119890119808119808119890119890 119894119894th member of

119821119821119890119824119824119890119808119808119890119890 119894119894 119896 119896119896119896 119896119896119896 119896119870119870119824119824119890119808119808119890119896120582120582 Vector length119884119884 Vector119884119884 Vector partition (ie set of vectors)11988411988410066101006610 Entity obeying the Cartesian property119884119884asymp Entity obeying the property of

implicitnessΛ[119821119821119890119808119808119890119821 Operation applied on 119821119821119890119808119808119890 to obtain

119819119819119890119808119808119890Ξ[119819119819119890119808119808119890119821 Operation applied on 119819119819119890119808119808119890 to obtain

11982111982110066101006610119890119808119808119890

120590120590119896119896 Subset of 119826119826119896119896119808119808120590120590119896119896 Subset of119808119808 such that 119808119808120590120590119896119896 = 119896 119890119890 119896 119808119808 119890 119890120590120590119896119896119896119890120590120590119896119896119808119808 Event subspace such that

119890120590120590119896119896119808119808 = 119896119826119826119808119808119896 times 119826119826

119808119808119896 times ⋯ times 120590120590119896119896 times⋯ times 119826119826119808119808119870119870119896

Glossary

Complete set of block states Set of all the possible statesof a certain system block

Complete joint event Joint event containing aninstance of each one of thesystem blocks

Cartesian property Event subspaces andcontribution vectors that canbe generated by a Cartesianproduct Also subspacespartitions and partitioncontribution vectors that canbe generated by a set ofCartesian products

Property of implicitness Cartesian entities (iesubspaces vectorspartitions) whose associatedCartesian products containonly complete or singletonsets of block states

Cartesian entries Nonzero entries of acontribution vector equal totheir relative Cartesiancontribution vector entries

References

[1] E Zio ldquoReliability engineering old problems and new chal-lengesrdquo Reliability Engineering and System Safety vol 94 no 2pp 125ndash141 2009

[2] J Andrews ldquoSystem reliability modelling the current capabilityand potential future developmentsrdquo Proceedings of the Institu-tion of Mechanical Engineers C vol 223 no 12 pp 2881ndash28972009

[3] I A Papazoglou and B J M Ale ldquoA logical model forquantication of occupational riskrdquo Reliability Engineering andSystem Safety vol 92 no 6 pp 785ndash803 2007

[4] R Remenyte-Prescott and J D Andrews ldquoAn efficient real-timemethod of analysis for non-coherent fault treesrdquo Quality andReliability Engineering International vol 25 no 2 pp 129ndash1502009

[5] A Lisnianski and G Levitin Multi-State System ReliabilityAssessment Optimization and Applications World Scientic2003

[6] S C M Rocco and M Muselli ldquoApproximate multi-statereliability expressions using a newmachine learning techniquerdquoReliability Engineering and System Safety vol 89 no 3 pp261ndash270 2005

[7] Y Dutuit and A Rauzy ldquoApproximate estimation of systemreliability via fault treesrdquo Reliability Engineering and SystemSafety vol 87 no 2 pp 163ndash172 2005

[8] W S Jung J-E Yang and J Ha ldquoDevelopment of measuresto estimate truncation error in fault tree analysisrdquo ReliabilityEngineering and System Safety vol 90 no 1 pp 30ndash36 2005


[9] I A Papazoglou ldquoMathematical foundations of event treesrdquoReliability Engineering and System Safety vol 61 no 3 pp169ndash183 1998

[10] I A Papazoglou ldquoFunctional block diagrams and automatedconstruction of event treesrdquo Reliability Engineering and SystemSafety vol 61 no 3 pp 185ndash214 1998

[11] A Rauzy and Y Dutuit ldquoExact and truncated computationsof prime implicants of coherent and non-coherent fault treeswithin Araliardquo Reliability Engineering and System Safety vol 58no 2 pp 127ndash144 1997

[12] B J M Ale H Baksteen L J Bellamy et al ldquoQuantifyingoccupational risk the development of an occupational riskmodelrdquo Safety Science no 2 pp 176ndash185 2008

[13] A R Hale B J M Ale L H J Goossens et al ldquoModelingaccidents for prioritizing preventionrdquo Reliability Engineering ampSystem Safety vol 92 no 12 pp 1701ndash1715 2007

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



organized or sorted e procedure is deductive startingfrom the entire event table and systematically organizing thesystem events in a set of clusters e nal set of clusters canbe translated back to an event tree that is signicantly reducedcompared to original outcome space information supplied tothe algorithm

e paper is organized as follows Section 2 presents somevery basic system denitions used in the system represen-tation and operations presented in Section 3 In Section 3 asuitable system representation is dened using sets of eventsand based on the concepts of Cartesian products Section 4describes a set of clustering and declustering operationsto be applied on the event sets Section 5 discusses the imple-mentation of the proposed developments into an algorithmicprocedure Section 6 presents an illustration example and alarge case study to demonstrate the proposed methodologySection 7 concludes the work

asi Systemenitions

e system considered here is comprised of blocks Eachblock can be found in various states and the system response(or output) at every given instance of time 119905119905 depends on thestate that the blocks occupy at 119905119905 Each block relates to a com-ponent a set of components or a part of a component of thesystem regardless of physical conventions and according tothe choices in the system modeling e basic denitions aretaken from Papazoglou [9 10] where the blocks were inter-connected to form functional block diagrams In the presentwork the blocks are stripped of their networking functional-ities and the denitions are simplied accordingly

21 System Blocks and eir States Consider a system of 119870119870-independent blocks Each block 119887119887119896119896 119896119896 119896 119896119896119896 119896119896119896 119896119870119870119896 can befound in various internal states during the system operationperiod

Let the state set of block 119887119887119896119896 denoted as 119826119826119896119896 = 119896119904119904119896119896119896119896 119904119904119896119896119896119896119896 119896

119904119904119870119870119878119878119896119896119896119896 119896 be a partition over the possible instances of 119887119887119896119896 where 119904119904

119894119894119896119896119896119896

denotes the 119894119894119896119896th state of block 119887119887119896119896 and119870119870119878119878119896119896 denotes the numberof elements in 119826119826119896119896 thus119870119870119878119878119896119896 is the cardinality of 119878119878119896119896119896 |119878119878119896119896|

It is assumed that at every given time instance within theperiod of system operation 119887119887119896119896 is found in exactly one state(eg at 94 of maximum production level) and this staterelates to exactly one member of 119826119826119896119896 (eg 119904119904

119896119896119896 = ldquoat least 90

of maximum production levelrdquo)

22 Event enitions A basic event is an instance of a singlesystem block according to the state set partition of this blockA basic event for block 119887119887119896119896 is denoted as 119896119904119904119894119894119896119896119896119896 119896119896 119904119904

119894119894119896119896119896119896 119896 119826119826119896119896

A joint event 119896119904119904119894119894119896119896119896119896119896119896 119896 119904119904119894119894119896119896119896119896119896119896119896119896 119896 119904119904

119894119894119896119896119898119898119896119896119898119898119896 is the combination ofmore

than one basic events taking place in119898119898 different blocks of thesystem 119887119887119896119896119896 119896 119887119887119896119896119896 119896119896 119896 119887119887119896119896119898119898 Note that the set 119896119904119904

119894119894119896119896119896119896119896119896119896 119904119904119894119894119896119896119896119896119896119896119896119896 119896 119904119904

119894119894119896119896119898119898119896119896119898119898119896 is

the result of the Cartesian product 119896119904119904119894119894119896119896119896119896119896119896 119896 times 119896119904119904119894119894119896119896119896119896119896119896119896 times ⋯ times 119896119904119904

119894119894119896119896119898119898119896119896119898119898119896

A complete joint event 119890119890 is dened here as a joint eventover all the system blocks 119896119904119904119894119894119896119896 119896 119904119904

119894119894119896119896 119896119896 119896 119904119904119894119894119870119870119870119870 119896 For simplicity the

term event is used here instead of complete joint event

Note that the above event denitions are simplied com-pared to Papazoglou [10] where blocks had functionalitiesnot considered here

23 Event Space Subspaces and Event Partitions Let thesystem event space 119812119812 be the set of all the possible completejoint events 119890119890 en 119812119812 = 1198961006594100659411989011989011989611989610065941006594119890119890119896119896119896 11989610065971006597119890119890119870119870119864119864

119896 where 119870119870119864119864 denotesthe number of elements 119812119812 thus 119870119870119864119864 = |119864119864| Note that 119812119812 is theresult of the Cartesian product 119826119826119896 times 119826119826119896 times ⋯ times 119826119826119870119870 therefore|119812119812| = 119812119870119870

119896119896=119896|119826119826119896119896|Consider a nonempty subspace 119808119808 of the system event

space 119808119808 119808 119812119812 119808 119808119808119808119808 Let 119824119824119824119808119808119824 denote a partition appliedover 119808119808 composed of 119870119870119824119824119824119808119808119824 disjoint subspaces of 119808119808 denotedby 119850119850119876119876119894119894 119824119808119808119824119896 119894119894 119896 119896119896119896 119896119896119896 119896119870119870119824119824119824119808119808119824119896

24 Event Table Let 119825119825 = 119896119825119825119896119896 119825119825119896119896119896 119896 119825119825119870119870119825119825119896 denote the set of

the all possible system outcomes and 119870119870119825119825 is their numbererefore119870119870119825119825 is the cardinality |119825119825| of 119825119825

Each event yields a unique system outcome while differ-ent events may yield the same outcome For instance theremight be more than one event that leads to system failure Inevent trees a complete joint event and its associated outcomeare equivalent to a path [9 10]

Let 119879119879 119808 119890119890 119890 119825119825 = 119879119879119824 119890119890119824 where 119890119890 119896 119812119812 and 119825119825 119896 119825119825 denotethemany-to-onemapping from the event space119812119812 to the set ofsystem outcomes119825119825 e 119879119879mapping is recorded in the systemevent table as a complete list of all the system events and theiroutcome

e 119879119879 denes a partition on 119812119812 which is herein calledthe outcome-based partition and denoted by 119824119824119879119879119824119812119812119824 emembers of119824119824119879119879119824119812119812119824 are denoted by 119850119850119824119824

119879119879

119825119825119894119894 119824119812119812119824119896 119825119825119894119894 119896 119825119825Table 1 gives the event table of an example taken from

Papazoglou [10] In this case there are 32 events and 4possible outcomes and119824119824119879119879119824119812119812119824 is comprised of the 4 sets

(1) 119850119850119824119824119879119879

119825119825119896 119824119812119812119824 = 119896100659410065941198901198901198961198961006594100659411989011989031198961006594100659411989011989041198961006596100659611989011989011989611989611989610065961006596119890119890119896119896119896100659610065961198901198901198967119896100659610065961198901198901198968119896100659610065961198901198901198969119896100659610065961198901198901198960119896100659610065961198901198901198965119896100659610065961198901198901198966119896100659610065961198901198901198967119896100659610065961198901198901198968119896

(2) 119850119850119824119824119879119879

119825119825119896 119824119812119812119824 = 11989610065941006594119890119890711989610065961006596119890119890119896511989610065961006596119890119890119896611989610065961006596119890119890119896119896119896100659610065961198901198901198961198961198961006596100659611989011989011989631198961006596100659611989011989011989641198961006596100659611989011989011989691198961006596100659611989011989030119896100659610065961198901198903119896119896100659610065961198901198903119896119896

(3) 119850119850119824119824119879119879

1198251198253 119824119812119812119824 = 119896100659410065941198901198905119896100659410065941198901198906119896100659410065941198901198908119896100659610065961198901198901198963119896100659610065961198901198901198964119896

(4) 119850119850119824119824119879119879

1198251198254 119824119812119812119824 = 11989610065941006594119890119890119896119896100659410065941198901198909119896100659610065961198901198901198960119896

Note that the 119879119879 mapping can be derived from the structuraldependencies among the system blocks as dictated by therational and physical interconnections of components withinthe system and depicted in the form of a fault tree or afunctional block diagram However such information maybe unavailable or too difficult to attain or process is workassumes that the only information available is the systemevent table

3 System Representation

e system representation presented here aims to organizethe information contained in the event table e tabledata are partitioned and organized into vectors summarizing




100659410065941198901198901 11990411990411 11990411990412 11990411990413 1199031199031100659410065941198901198902 11990411990411 11990411990412 11990411990423 1199031199034100659410065941198901198903 11990411990411 11990411990412 11990411990433 1199031199031100659410065941198901198904 11990411990411 11990411990412 11990411990443 1199031199031100659410065941198901198905 11990411990411 11990411990422 11990411990413 1199031199033100659410065941198901198906 11990411990411 11990411990422 11990411990423 1199031199033100659410065941198901198907 11990411990411 11990411990422 11990411990433 1199031199032100659410065941198901198908 11990411990411 11990411990422 11990411990443 1199031199033100659410065941198901198909 11990411990411 11990411990432 11990411990413 11990311990341006596100659611989011989010 11990411990411 11990411990432 11990411990423 11990311990341006596100659611989011989011 11990411990411 11990411990432 11990411990433 11990311990311006596100659611989011989012 11990411990411 11990411990432 11990411990443 11990311990311006596100659611989011989013 11990411990411 11990411990442 11990411990413 11990311990331006596100659611989011989014 11990411990411 11990411990442 11990411990423 11990311990331006596100659611989011989015 11990411990411 11990411990442 11990411990433 11990311990321006596100659611989011989016 11990411990411 11990411990442 11990411990443 11990311990321006596100659611989011989017 11990411990421 11990411990412 11990411990413 11990311990311006596100659611989011989018 11990411990421 11990411990412 11990411990423 11990311990311006596100659611989011989019 11990411990421 11990411990412 11990411990433 11990311990311006596100659611989011989020 11990411990421 11990411990412 11990411990443 11990311990311006596100659611989011989021 11990411990421 11990411990422 11990411990413 11990311990321006596100659611989011989022 11990411990421 11990411990422 11990411990423 11990311990321006596100659611989011989023 11990411990421 11990411990422 11990411990433 11990311990321006596100659611989011989024 11990411990421 11990411990422 11990411990443 11990311990321006596100659611989011989025 11990411990421 11990411990432 11990411990413 11990311990311006596100659611989011989026 11990411990421 11990411990432 11990411990423 11990311990311006596100659611989011989027 11990411990421 11990411990432 11990411990433 11990311990311006596100659611989011989028 11990411990421 11990411990432 11990411990443 11990311990311006596100659611989011989029 11990411990421 11990411990442 11990411990413 11990311990321006596100659611989011989030 11990411990421 11990411990442 11990411990423 11990311990321006596100659611989011989031 11990411990421 11990411990442 11990411990433 11990311990321006596100659611989011989032 11990411990421 11990411990442 11990411990443 1199031199032







21119896 119826119826

1198091198092 119809 11989611990411990412119896

1198261198261198091198093 119809 11989611990411990413 11990411990423 119904119904

33 119904119904

43119896 and 119810119810119810119809119809119810 119809 1198961006594100659411989011989011006594100659411989011989021006594100659411989011989031006594100659411989011989041006596100659611989011989017100659610065961198901198901810065961006596119890119890191006596100659611989011989020119896




10066101006610119810 Note that all




as 11985011985010066091006609

11982411982410066091006609119894119894 119810119808119808119810 119894119894 119896 1198961 2119896 119896119896119824119824




10066091006609119824119824100660910066091 119810119809119809119810 119809 119896100659410065941198901198901119896 119850119850

10066091006609119824119824100660910066092 119810119809119809119810 119809 119896100659410065941198901198903119896

11985011985010066091006609119824119824100660910066093 119810119809119809119810 119809 119896100659410065941198901198904119896 and 119850119850

10066091006609119824119824100660910066094 119810119809119809119810 119809 1198961006596100659611989011989017100659610065961198901198901810065961006596119890119890191006596100659611989011989020119896 ese are all

Cartesian since 11985011985010066091006609119824119824100660910066091 119810119809119809119810 11985011985010066091006609

119824119824100660910066092 119810119809119809119810 11985011985010066091006609


11985011985010066091006609119824119824100660910066094 119810119809119809119810 119809 11990411990421 times 119904119904

12 times 1198261198263


119808119808119896119896 119809 119826119826119896119896






the implicant 11989611990411990421 11990411990412119896


12 119904119904

22119896 times





asymp

11982411982410066091006609119894119894119810119808119808119810 Every subspace





1198091198092 119809 11990411990421 times 11989611990411990412 11990411990422119896 times 1198261198263 into 119904119904




12119896 and 119896119904119904

21 119904119904

22119896






ℕ120582120582+ for all 119821119821 119808 119808119808






120582120582for all 119821119821 119808 119808119808




119819119819119821119821119821119821 119821 Λ1007714100771411982111982111982111982111982111982110077301007730⟺ 119897119897119894119894119896119896119896119896 119821119821119821119821 1198211007618100761810076341007634100762610076261007634100763410076421007642

O119821 119899119899119894119894119896119896119896119896 119821119821119821119821 119821 119821119821

I119821 119899119899119894119894119896119896119896119896 119821119821119821119821 gt 119821119821forall119896119896119821 119894119894119896119896

(1)


Λminus1198211007714100771411982111982111982111982111982111982110077301007730

119821 10076821007682 119890119890 119821 10076821007682119904119904119894119894119821119821 119821 119904119904119894119894119821119821 119821119896 119821 119904119904119894119894119870119870119870119870 10076981007698 119821 119808119808 119821 119897119897119894119894119896119896119896119896 119821119821119821119821 119821 I119821 forall119896119896119821 11989411989411989611989610076981007698

(2)



10066101006610119821119821119821119821 is dened here as



119821119821119821119821





11982111982110066101006610119821119821119821119821 119821 Ξ1007714100771411981911981911982111982111982111982110077301007730⟺ 119899119899

10066091006609119894119894119896119896119896119896119821119821119821119821 119821

1007618100761810076341007634100762610076261007634100763410076421007642

119821119821 119897119897119894119894119896119896119896119896 119821119821119821119821 119821 O119821|119810119810119821119821119821119821|10092501009250119826119826119821119821119896119896 10092501009250

119821 119897119897119894119894119896119896119896119896 119821119821119821119821 119821 I119821

forall119896119896119821 119894119894119896119896(3)


119894119894119896119896119896119896 119821119821119821119821 119821





119821119821



119894119894119896119896sim119896119896119821119809119809119821 at 119808119896119896119821 119894119894119896119896 119821 119808119821119821 119821 or 119808119821119821 119821



Since 120582119894119894 119899119899119894119894119896119896119896119896 119821119821119821






11980810065941006594119890119890119821 1198211198211006609100660911982411982410066091006609119821 119821119809119809119821 119821 11980810065941006594119890119890119821 119821119821

1006609100660911982411982410066091006609119821 119821119809119809119821 119821 11980810065941006594119890119890119821 and 119821119821

1006609100660911982411982410066091006609119821 119821119809119809119821 119821 11980810065961006596119890119890119821119821119821100659610065961198901198901198218119821100659610065961198901198901198219119821


119821119821119821119824119824119821119809119809119821119821 119821

100761810076181007634100763410076341007634100763410076341007634100763410076341007634100763410076341007626100762610076341007634100763410076341007634100763410076341007634100763410076341007634100763410076421007642

119821119821100765110076511198211198211006609100660911982411982410066091006609119821 11982111980911980911982110076671007667

119821119821100765110076511198211198211006609100660911982411982410066091006609119821 11982111980911980911982110076671007667

119821119821100765110076511198211198211006609100660911982411982410066091006609119821 11982111980911980911982110076671007667

119821119821100765110076511198211198211006609100660911982411982410066091006609119821 11982111980911980911982110076671007667

100761910076191007635100763510076351007635100763510076351007635100763510076351007635100763510076351007627100762710076351007635100763510076351007635100763510076351007635100763510076351007635100763510076431007643

11982110076181007618100763410076341007634100763410076261007626100763410076341007634100763410076421007642

119821119821119821 119821119821 119821119821 119821119821 119821119821 119821119821 119821119821 119821119821 119821119821 119821119821119821119821119821 119821119821 119821119821 119821119821 119821119821 119821119821 119821119821 119821119821 119821119821 119821119821119821119821119821 119821119821 119821119821 119821119821 119821119821 119821119821 119821119821 119821119821 119821119821 119821119821119821119821119821 119821119821 119821119821 119821119821 119821119821 119821119821 119821119821 119821119821 119821119821 119821119821

10076191007619100763510076351007635100763510076271007627100763510076351007635100763510076431007643

(4)


e 11985011985010066091006609






119879119879




119808119808120590120590119808119808 = 10076821007682 119890119890 isin 119808119808 119890 119890120590120590119808119808119808119808 10076981007698

119808119808 minus 119808119808120590120590119808119808 = 10076821007682 119890119890 isin 119808119808 119890 10076501007650119810119810(119808119808119809 minus 119890120590120590119808119808119808119808 1007666100766610076981007698


(5)






1198211198211007649100764911980811980812059012059011980811980810076651007665 = 119821119821(119808119808119809 minus 1198211198211007649100764911980811980812059012059011980811980810076651007665 (6)


1198211198212 times


1198191198191007649100764911980811980812059012059011980811980810076651007665 119808 119897119897119894119894119898119898119898119898 10092501009250119819119819(119808119808120590120590119808119808 119809 =

1007618100761810076341007634100763410076341007634100763410076341007634100762610076261007634100763410076341007634100763410076341007634100763410076421007642

119897119897119894119894119898119898119898119898 10092501009250119819119819(119808119808119809 119808119808 119808119898119898

I 119808119808 = 119898119898 and 119896119896119894119894119898119898119898119898 isin 120590120590119808119808

O 119808119808 = 119898119898 and 119896119896119894119894119898119898119898119898119898119898 notin 120590120590119808119808(7)


12059012059001 1198211198211 where 119821119821



12059012059012


12059012059001 119821119821

12059012059012 119821119821

12059012059023 hellip



119894119894 = Ξ[119819119819120590120590119894119894minus1119894119894 ] using (7) and

119819119819119894119894minus1 = Λ[119821119821119894119894minus1]



33 and 11989611989643 Let 1205901205900 = 11989611989633

(i) 11981911981912059012059001 = [I I IO IOOO 119816119816O]

rArr 11982111982112059012059001 = [2 2 2 0 2 0 0 0 120786120786 0]

(ii) 1198211198211 = 1198211198210 minus 11982111982112059012059001 = [3 6 5 0 4 0 3 2 120782120782 4]

1198191198191 = [I I IO IO I I 119822119822 I]


43 as follows


rArr 119821119821120590120590lowast01 = [4 4 4 0 4 0 0 0 120786120786 120786120786]

(ii) 119821119821lowast1 = 1198211198210 minus 119821119821

120590120590lowast01 = [1 4 3 0 2 0 3 2 120782120782 120782120782]

119819119819lowast1 = [I I IO IO I I 119822119822119822119822]





(i) 119819119819120590120590lowast12 = [0 120783120783120783 1 0 1 0120783 1 1 0 0120783

rArr 119821119821120590120590lowast12 = [0 120786120786120783 2 0 2 0120783 2 2 0 0120783

(ii) 119821119821lowast2 = 119821119821


120590120590lowast12 = [1 120782120782120783 1 0 0 0120783 1 0 0 0120783

Both 119821119821lowast2 and 119821119821



12059012059001 119821119821

lowast2 119821119821

120590120590lowast12


12059012059001 e vectors 1198211198211205901205900



(i)

11982111982112059012059001 = [4 4120783 4 0 4 0120783 0 0 4 4120783

sim 10076851007685[2 2120783 4 0 0 0120783 0 0 2 2120783[2 2120783 0 0 4 0120783 0 0 2 2120783

10077011007701

sim10076181007618100763410076341007634100763410076261007626100763410076341007634100763410076421007642

[1 1120783 2 0 0 0120783 0 0 2 0120783[1 1120783 2 0 0 0120783 0 0 0 2120783[1 1120783 0 0 2 0120783 0 0 2 0120783[1 1120783 0 0 2 0120783 0 0 0 2120783

10076191007619100763510076351007635100763510076271007627100763510076351007635100763510076431007643

(8)

(ii)

11982111982112059012059012 = [0 4120783 2 0 2 0120783 2 2 0 0120783

sim 10076851007685[0 2120783 0 0 2 0120783 1 1 0 0120783[0 2120783 2 0 0 0120783 1 1 0 0120783

10077011007701

sim10076181007618100763410076341007634100763410076261007626100763410076341007634100763410076421007642

[0 1120783 0 0 1 0120783 0 1 0 0120783[0 1120783 0 0 1 0120783 1 0 0 0120783[0 1120783 1 0 0 0120783 0 1 0 0120783[0 1120783 1 0 0 0120783 1 0 0 0120783

10076191007619100763510076351007635100763510076271007627100763510076351007635100763510076431007643

(9)







⋯ times 1198261198261198191198191119896119896 119812










43 119904119904

13 119904119904

34 119904119904

44



1198211198210 = [5 8 7 0 6 0 3 2 4 4] sim 10076851007685[0 8 4 0 4 0 2 2 2 2][5 0 3 0 2 0 1 0 2 2]

10077011007701

sim10076181007618100763410076341007634100763410076261007626100763410076341007634100763410076421007642

[0 4 4 0 0 0 1 1 1 1][0 4 0 0 4 0 1 1 1 1][4 0 2 0 2 0 0 0 2 2][1 0 1 0 0 0 1 0 0 0]

10076191007619100763510076351007635100763510076271007627100763510076351007635100763510076431007643

sim10076181007618100763410076341007634100763410076261007626100763410076341007634100763410076421007642

[0 4 4 0 0 0 1 1 1 1][0 4 0 0 4 0 1 1 1 1][1 0 1 0 0 0 1 0 0 0]

[1 0 1 0 0 0 0 0 1 0][1 0 1 0 0 0 0 0 0 1][1 0 0 0 1 0 0 0 1 0][1 0 0 0 1 0 0 0 0 1]

10076191007619100763510076351007635100763510076271007627100763510076351007635100763510076431007643

(10)





119894119894119896119896119896119896 minus

min119896119896119899119899119894119894119896119896119896119896 )|119826119826119896119896|




119894119894119896119896119896119896 minus 119899119899

119894119894119896119896119896119896







100661010066100 = 11982111982110066101006610(1198241198240(119850119850

119879119879119903119903119894119894 (119812119812))) (or the asso-




10066101006610






10066101006610(1198241198241(119850119850

119879119879119903119903119894119894 (119812119812))) is

implicit






























1 1 1

1

1 1 2 4

1 1 3 1

1 1 4 1

1 2 1 3

1 2 2 3

1

2 3 2

1 2 4 3

1 3 1 4

1 3 2 4

1 3 3 1

1 3 4

1

1 4 1 3

1 4 2 3

1 4 3 2

1 4 4 2

2 1 1 1

2

1 2 1

2 1 3 1

2 1 4 1

2 2 1 2

2 2 2 2

2 2 3

2

2 2 4 2

2 3 1 1

2 3 2 1

2 3 3 1

2 3 4 1

2

4 1 2

2 4 2 2

2 4 3 2

2 4 4

2



1 1 1 1

1 1 3 1

1 1 4 1

1 3 3 1

1 3 4 1

2 1 1 1

2 1 2 1

2 1 3 1

2 1 4 1

2 3 1 1

2 3 2 1

2 3 3 1

2 3 4 1

1 2 3 2

1 4 3 2

1 4 4 2

2 2 1 2

2 2 2 2

2 2 3 2

2 2 4 2

2 4 1 2

2 4 2 2

2 4 3 2

2 4 4 2

1 2 1 3

1 2 2 3

1 2 4 3

1 4 1 3

1 4 2 3

1 1 2 4

1 3 1 4

1 3 2 4


119873119873119877119877 = 100762010076201007636100763610076361007636

10076441007644

5 8 7 0 6 0 3 2 4 4 13 8 0 5 0 6 2 2 4 3 25 0 0 3 0 2 2 2 0 1 33 0 1 0 2 0 1 2 0 0 4

100762110076211007637100763710076371007637

10076451007645

(11)



















119873119873119877119877 119894 100762010076201007636100763610076361007636

10076441007644

5 0 3 0 2 0 1 0 2 2 13 8 0 5 0 6 2 2 4 3 25 0 0 3 0 2 2 2 0 1 33 0 1 0 2 0 1 2 0 0 4

100762110076211007637100763710076371007637

10076451007645

119873119873100661110066111

119894 [0 8 4 0 4 0 2 2 2 2]

(12)










asymp 119894119894

denoted by 11987111987110066091006609119894119894and 119871119871



119871119871100660910066091

119894 10076201007620

10076441007644

0 1 1 0 1 0 1 1 1 11 1 0 1 0 1 0 0 0 11 0 1 0 0 0 1 0 0 0

10076211007621

10076451007645(13)


119871119871asymp1

119894

100762010076201007636100763610076361007636100763610076361007636100763610076361007636100763610076361007636100763610076361007636

10076441007644

0 1 0 0 0 1 1 1 1 10 1 0 0 1 0 1 1 1 11 0 1 0 0 0 0 0 1 01 0 0 0 1 0 0 0 1 01 0 1 0 0 0 0 0 0 11 0 0 0 1 0 0 0 0 11 0 1 0 0 0 1 0 0 0

100762110076211007637100763710076371007637100763710076371007637100763710076371007637100763710076371007637100763710076371007637

10076451007645

(14)













6 Case Studies











7 Conclusions










Nomenclature


119887119887119896119896 119826119826119896119896 = 119896119904119904119896119896119896119896 119904119904119896119896119896119896119896 119896 119904119904

119870119870S119896119896119896119896 119896







119808119808119896 times ⋯ times 119826119826119808119808119870119870



1198501198501006609100660911982411982410066091006609119894119894 119890119808119808119890 119894119894th subset of119824119824

10066101006610119890119808119808119890 119894119894 119896 119896119896119896 119896119896119896 119896119870119870119824119824

10066091006609119890119808119808119890119896


119850119850asymp

11982411982410066091006609119894119894119890119808119808119890 119894119894th subset of119824119824

asymp119890119808119808119890 119894119894 119896 119896119896119896 119896119896119896 119896119870119870119824119824

asymp119890119808119808119890119896


119896119896 119896 119896119896119896 119896119896119896 119896119870119870119896 119894119894119896119896 119896 119896119896119896 119896119896119896 119896119870119870119904119904119894119894119896119896119896119896119896


119896119896 119896 119896119896119896 119896119896119896 119896119870119870119896 119894119894119896119896 119896 119896119896119896 119896119896119896 119896119870119870119904119904119894119894119896119896119896119896119896


11989911989910066091006609119894119894119896119896119896119896 119890119808119808119890 Entry of 119821119821

10066101006610119890119808119808119890 119899119899

asymp119894119894119896119896119896119896119890119808119808119890 119896 119808+ and

119896119896 119896 119896119896119896 119896119896119896 119896119870119870119896 119894119894119896119896 119896 119896119896119896 119896119896119896 119896119870119870119904119904119894119894119896119896119896119896119896





11982111982110066101006610119890119808119808119890


119890120590120590119896119896119808119808 = 119896119826119826119808119808119896 times 119826119826


Glossary






References

















RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation












100659410065941198901198901 11990411990411 11990411990412 11990411990413 1199031199031100659410065941198901198902 11990411990411 11990411990412 11990411990423 1199031199034100659410065941198901198903 11990411990411 11990411990412 11990411990433 1199031199031100659410065941198901198904 11990411990411 11990411990412 11990411990443 1199031199031100659410065941198901198905 11990411990411 11990411990422 11990411990413 1199031199033100659410065941198901198906 11990411990411 11990411990422 11990411990423 1199031199033100659410065941198901198907 11990411990411 11990411990422 11990411990433 1199031199032100659410065941198901198908 11990411990411 11990411990422 11990411990443 1199031199033100659410065941198901198909 11990411990411 11990411990432 11990411990413 11990311990341006596100659611989011989010 11990411990411 11990411990432 11990411990423 11990311990341006596100659611989011989011 11990411990411 11990411990432 11990411990433 11990311990311006596100659611989011989012 11990411990411 11990411990432 11990411990443 11990311990311006596100659611989011989013 11990411990411 11990411990442 11990411990413 11990311990331006596100659611989011989014 11990411990411 11990411990442 11990411990423 11990311990331006596100659611989011989015 11990411990411 11990411990442 11990411990433 11990311990321006596100659611989011989016 11990411990411 11990411990442 11990411990443 11990311990321006596100659611989011989017 11990411990421 11990411990412 11990411990413 11990311990311006596100659611989011989018 11990411990421 11990411990412 11990411990423 11990311990311006596100659611989011989019 11990411990421 11990411990412 11990411990433 11990311990311006596100659611989011989020 11990411990421 11990411990412 11990411990443 11990311990311006596100659611989011989021 11990411990421 11990411990422 11990411990413 11990311990321006596100659611989011989022 11990411990421 11990411990422 11990411990423 11990311990321006596100659611989011989023 11990411990421 11990411990422 11990411990433 11990311990321006596100659611989011989024 11990411990421 11990411990422 11990411990443 11990311990321006596100659611989011989025 11990411990421 11990411990432 11990411990413 11990311990311006596100659611989011989026 11990411990421 11990411990432 11990411990423 11990311990311006596100659611989011989027 11990411990421 11990411990432 11990411990433 11990311990311006596100659611989011989028 11990411990421 11990411990432 11990411990443 11990311990311006596100659611989011989029 11990411990421 11990411990442 11990411990413 11990311990321006596100659611989011989030 11990411990421 11990411990442 11990411990423 11990311990321006596100659611989011989031 11990411990421 11990411990442 11990411990433 11990311990321006596100659611989011989032 11990411990421 11990411990442 11990411990443 1199031199032







21119896 119826119826

1198091198092 119809 11989611990411990412119896

1198261198261198091198093 119809 11989611990411990413 11990411990423 119904119904

33 119904119904

43119896 and 119810119810119810119809119809119810 119809 1198961006594100659411989011989011006594100659411989011989021006594100659411989011989031006594100659411989011989041006596100659611989011989017100659610065961198901198901810065961006596119890119890191006596100659611989011989020119896




10066101006610119810 Note that all




as 11985011985010066091006609

11982411982410066091006609119894119894 119810119808119808119810 119894119894 119896 1198961 2119896 119896119896119824119824




10066091006609119824119824100660910066091 119810119809119809119810 119809 119896100659410065941198901198901119896 119850119850

10066091006609119824119824100660910066092 119810119809119809119810 119809 119896100659410065941198901198903119896

11985011985010066091006609119824119824100660910066093 119810119809119809119810 119809 119896100659410065941198901198904119896 and 119850119850

10066091006609119824119824100660910066094 119810119809119809119810 119809 1198961006596100659611989011989017100659610065961198901198901810065961006596119890119890191006596100659611989011989020119896 ese are all

Cartesian since 11985011985010066091006609119824119824100660910066091 119810119809119809119810 11985011985010066091006609

119824119824100660910066092 119810119809119809119810 11985011985010066091006609


11985011985010066091006609119824119824100660910066094 119810119809119809119810 119809 11990411990421 times 119904119904

12 times 1198261198263


119808119808119896119896 119809 119826119826119896119896






the implicant 11989611990411990421 11990411990412119896


12 119904119904

22119896 times





asymp

11982411982410066091006609119894119894119810119808119808119810 Every subspace





1198091198092 119809 11990411990421 times 11989611990411990412 11990411990422119896 times 1198261198263 into 119904119904




12119896 and 119896119904119904

21 119904119904

22119896






ℕ120582120582+ for all 119821119821 119808 119808119808






120582120582for all 119821119821 119808 119808119808




119819119819119821119821119821119821 119821 Λ1007714100771411982111982111982111982111982111982110077301007730⟺ 119897119897119894119894119896119896119896119896 119821119821119821119821 1198211007618100761810076341007634100762610076261007634100763410076421007642

O119821 119899119899119894119894119896119896119896119896 119821119821119821119821 119821 119821119821

I119821 119899119899119894119894119896119896119896119896 119821119821119821119821 gt 119821119821forall119896119896119821 119894119894119896119896

(1)


Λminus1198211007714100771411982111982111982111982111982111982110077301007730

119821 10076821007682 119890119890 119821 10076821007682119904119904119894119894119821119821 119821 119904119904119894119894119821119821 119821119896 119821 119904119904119894119894119870119870119870119870 10076981007698 119821 119808119808 119821 119897119897119894119894119896119896119896119896 119821119821119821119821 119821 I119821 forall119896119896119821 11989411989411989611989610076981007698

(2)



10066101006610119821119821119821119821 is dened here as



119821119821119821119821





11982111982110066101006610119821119821119821119821 119821 Ξ1007714100771411981911981911982111982111982111982110077301007730⟺ 119899119899

10066091006609119894119894119896119896119896119896119821119821119821119821 119821

1007618100761810076341007634100762610076261007634100763410076421007642

119821119821 119897119897119894119894119896119896119896119896 119821119821119821119821 119821 O119821|119810119810119821119821119821119821|10092501009250119826119826119821119821119896119896 10092501009250

119821 119897119897119894119894119896119896119896119896 119821119821119821119821 119821 I119821

forall119896119896119821 119894119894119896119896(3)


119894119894119896119896119896119896 119821119821119821119821 119821





119821119821



119894119894119896119896sim119896119896119821119809119809119821 at 119808119896119896119821 119894119894119896119896 119821 119808119821119821 119821 or 119808119821119821 119821



Since 120582119894119894 119899119899119894119894119896119896119896119896 119821119821119821






11980810065941006594119890119890119821 1198211198211006609100660911982411982410066091006609119821 119821119809119809119821 119821 11980810065941006594119890119890119821 119821119821

1006609100660911982411982410066091006609119821 119821119809119809119821 119821 11980810065941006594119890119890119821 and 119821119821

1006609100660911982411982410066091006609119821 119821119809119809119821 119821 11980810065961006596119890119890119821119821119821100659610065961198901198901198218119821100659610065961198901198901198219119821


119821119821119821119824119824119821119809119809119821119821 119821

100761810076181007634100763410076341007634100763410076341007634100763410076341007634100763410076341007626100762610076341007634100763410076341007634100763410076341007634100763410076341007634100763410076421007642

119821119821100765110076511198211198211006609100660911982411982410066091006609119821 11982111980911980911982110076671007667

119821119821100765110076511198211198211006609100660911982411982410066091006609119821 11982111980911980911982110076671007667

119821119821100765110076511198211198211006609100660911982411982410066091006609119821 11982111980911980911982110076671007667

119821119821100765110076511198211198211006609100660911982411982410066091006609119821 11982111980911980911982110076671007667

100761910076191007635100763510076351007635100763510076351007635100763510076351007635100763510076351007627100762710076351007635100763510076351007635100763510076351007635100763510076351007635100763510076431007643

11982110076181007618100763410076341007634100763410076261007626100763410076341007634100763410076421007642

119821119821119821 119821119821 119821119821 119821119821 119821119821 119821119821 119821119821 119821119821 119821119821 119821119821119821119821119821 119821119821 119821119821 119821119821 119821119821 119821119821 119821119821 119821119821 119821119821 119821119821119821119821119821 119821119821 119821119821 119821119821 119821119821 119821119821 119821119821 119821119821 119821119821 119821119821119821119821119821 119821119821 119821119821 119821119821 119821119821 119821119821 119821119821 119821119821 119821119821 119821119821

10076191007619100763510076351007635100763510076271007627100763510076351007635100763510076431007643

(4)


e 11985011985010066091006609






119879119879




119808119808120590120590119808119808 = 10076821007682 119890119890 isin 119808119808 119890 119890120590120590119808119808119808119808 10076981007698

119808119808 minus 119808119808120590120590119808119808 = 10076821007682 119890119890 isin 119808119808 119890 10076501007650119810119810(119808119808119809 minus 119890120590120590119808119808119808119808 1007666100766610076981007698


(5)






1198211198211007649100764911980811980812059012059011980811980810076651007665 = 119821119821(119808119808119809 minus 1198211198211007649100764911980811980812059012059011980811980810076651007665 (6)


1198211198212 times


1198191198191007649100764911980811980812059012059011980811980810076651007665 119808 119897119897119894119894119898119898119898119898 10092501009250119819119819(119808119808120590120590119808119808 119809 =

1007618100761810076341007634100763410076341007634100763410076341007634100762610076261007634100763410076341007634100763410076341007634100763410076421007642

119897119897119894119894119898119898119898119898 10092501009250119819119819(119808119808119809 119808119808 119808119898119898

I 119808119808 = 119898119898 and 119896119896119894119894119898119898119898119898 isin 120590120590119808119808

O 119808119808 = 119898119898 and 119896119896119894119894119898119898119898119898119898119898 notin 120590120590119808119808(7)


12059012059001 1198211198211 where 119821119821



12059012059012


12059012059001 119821119821

12059012059012 119821119821

12059012059023 hellip



119894119894 = Ξ[119819119819120590120590119894119894minus1119894119894 ] using (7) and

119819119819119894119894minus1 = Λ[119821119821119894119894minus1]



33 and 11989611989643 Let 1205901205900 = 11989611989633

(i) 11981911981912059012059001 = [I I IO IOOO 119816119816O]

rArr 11982111982112059012059001 = [2 2 2 0 2 0 0 0 120786120786 0]

(ii) 1198211198211 = 1198211198210 minus 11982111982112059012059001 = [3 6 5 0 4 0 3 2 120782120782 4]

1198191198191 = [I I IO IO I I 119822119822 I]


43 as follows


rArr 119821119821120590120590lowast01 = [4 4 4 0 4 0 0 0 120786120786 120786120786]

(ii) 119821119821lowast1 = 1198211198210 minus 119821119821

120590120590lowast01 = [1 4 3 0 2 0 3 2 120782120782 120782120782]

119819119819lowast1 = [I I IO IO I I 119822119822119822119822]





(i) 119819119819120590120590lowast12 = [0 120783120783120783 1 0 1 0120783 1 1 0 0120783

rArr 119821119821120590120590lowast12 = [0 120786120786120783 2 0 2 0120783 2 2 0 0120783

(ii) 119821119821lowast2 = 119821119821


120590120590lowast12 = [1 120782120782120783 1 0 0 0120783 1 0 0 0120783

Both 119821119821lowast2 and 119821119821



12059012059001 119821119821

lowast2 119821119821

120590120590lowast12


12059012059001 e vectors 1198211198211205901205900



(i)

11982111982112059012059001 = [4 4120783 4 0 4 0120783 0 0 4 4120783

sim 10076851007685[2 2120783 4 0 0 0120783 0 0 2 2120783[2 2120783 0 0 4 0120783 0 0 2 2120783

10077011007701

sim10076181007618100763410076341007634100763410076261007626100763410076341007634100763410076421007642

[1 1120783 2 0 0 0120783 0 0 2 0120783[1 1120783 2 0 0 0120783 0 0 0 2120783[1 1120783 0 0 2 0120783 0 0 2 0120783[1 1120783 0 0 2 0120783 0 0 0 2120783

10076191007619100763510076351007635100763510076271007627100763510076351007635100763510076431007643

(8)

(ii)

11982111982112059012059012 = [0 4120783 2 0 2 0120783 2 2 0 0120783

sim 10076851007685[0 2120783 0 0 2 0120783 1 1 0 0120783[0 2120783 2 0 0 0120783 1 1 0 0120783

10077011007701

sim10076181007618100763410076341007634100763410076261007626100763410076341007634100763410076421007642

[0 1120783 0 0 1 0120783 0 1 0 0120783[0 1120783 0 0 1 0120783 1 0 0 0120783[0 1120783 1 0 0 0120783 0 1 0 0120783[0 1120783 1 0 0 0120783 1 0 0 0120783

10076191007619100763510076351007635100763510076271007627100763510076351007635100763510076431007643

(9)







⋯ times 1198261198261198191198191119896119896 119812










43 119904119904

13 119904119904

34 119904119904

44



1198211198210 = [5 8 7 0 6 0 3 2 4 4] sim 10076851007685[0 8 4 0 4 0 2 2 2 2][5 0 3 0 2 0 1 0 2 2]

10077011007701

sim10076181007618100763410076341007634100763410076261007626100763410076341007634100763410076421007642

[0 4 4 0 0 0 1 1 1 1][0 4 0 0 4 0 1 1 1 1][4 0 2 0 2 0 0 0 2 2][1 0 1 0 0 0 1 0 0 0]

10076191007619100763510076351007635100763510076271007627100763510076351007635100763510076431007643

sim10076181007618100763410076341007634100763410076261007626100763410076341007634100763410076421007642

[0 4 4 0 0 0 1 1 1 1][0 4 0 0 4 0 1 1 1 1][1 0 1 0 0 0 1 0 0 0]

[1 0 1 0 0 0 0 0 1 0][1 0 1 0 0 0 0 0 0 1][1 0 0 0 1 0 0 0 1 0][1 0 0 0 1 0 0 0 0 1]

10076191007619100763510076351007635100763510076271007627100763510076351007635100763510076431007643

(10)





119894119894119896119896119896119896 minus

min119896119896119899119899119894119894119896119896119896119896 )|119826119826119896119896|




119894119894119896119896119896119896 minus 119899119899

119894119894119896119896119896119896







100661010066100 = 11982111982110066101006610(1198241198240(119850119850

119879119879119903119903119894119894 (119812119812))) (or the asso-




10066101006610






10066101006610(1198241198241(119850119850

119879119879119903119903119894119894 (119812119812))) is

implicit






























1 1 1

1

1 1 2 4

1 1 3 1

1 1 4 1

1 2 1 3

1 2 2 3

1

2 3 2

1 2 4 3

1 3 1 4

1 3 2 4

1 3 3 1

1 3 4

1

1 4 1 3

1 4 2 3

1 4 3 2

1 4 4 2

2 1 1 1

2

1 2 1

2 1 3 1

2 1 4 1

2 2 1 2

2 2 2 2

2 2 3

2

2 2 4 2

2 3 1 1

2 3 2 1

2 3 3 1

2 3 4 1

2

4 1 2

2 4 2 2

2 4 3 2

2 4 4

2



1 1 1 1

1 1 3 1

1 1 4 1

1 3 3 1

1 3 4 1

2 1 1 1

2 1 2 1

2 1 3 1

2 1 4 1

2 3 1 1

2 3 2 1

2 3 3 1

2 3 4 1

1 2 3 2

1 4 3 2

1 4 4 2

2 2 1 2

2 2 2 2

2 2 3 2

2 2 4 2

2 4 1 2

2 4 2 2

2 4 3 2

2 4 4 2

1 2 1 3

1 2 2 3

1 2 4 3

1 4 1 3

1 4 2 3

1 1 2 4

1 3 1 4

1 3 2 4


119873119873119877119877 = 100762010076201007636100763610076361007636

10076441007644

5 8 7 0 6 0 3 2 4 4 13 8 0 5 0 6 2 2 4 3 25 0 0 3 0 2 2 2 0 1 33 0 1 0 2 0 1 2 0 0 4

100762110076211007637100763710076371007637

10076451007645

(11)



















119873119873119877119877 119894 100762010076201007636100763610076361007636

10076441007644

5 0 3 0 2 0 1 0 2 2 13 8 0 5 0 6 2 2 4 3 25 0 0 3 0 2 2 2 0 1 33 0 1 0 2 0 1 2 0 0 4

100762110076211007637100763710076371007637

10076451007645

119873119873100661110066111

119894 [0 8 4 0 4 0 2 2 2 2]

(12)










asymp 119894119894

denoted by 11987111987110066091006609119894119894and 119871119871



119871119871100660910066091

119894 10076201007620

10076441007644

0 1 1 0 1 0 1 1 1 11 1 0 1 0 1 0 0 0 11 0 1 0 0 0 1 0 0 0

10076211007621

10076451007645(13)


119871119871asymp1

119894

100762010076201007636100763610076361007636100763610076361007636100763610076361007636100763610076361007636100763610076361007636

10076441007644

0 1 0 0 0 1 1 1 1 10 1 0 0 1 0 1 1 1 11 0 1 0 0 0 0 0 1 01 0 0 0 1 0 0 0 1 01 0 1 0 0 0 0 0 0 11 0 0 0 1 0 0 0 0 11 0 1 0 0 0 1 0 0 0

100762110076211007637100763710076371007637100763710076371007637100763710076371007637100763710076371007637100763710076371007637

10076451007645

(14)













6 Case Studies











7 Conclusions










Nomenclature


119887119887119896119896 119826119826119896119896 = 119896119904119904119896119896119896119896 119904119904119896119896119896119896119896 119896 119904119904

119870119870S119896119896119896119896 119896







119808119808119896 times ⋯ times 119826119826119808119808119870119870



1198501198501006609100660911982411982410066091006609119894119894 119890119808119808119890 119894119894th subset of119824119824

10066101006610119890119808119808119890 119894119894 119896 119896119896119896 119896119896119896 119896119870119870119824119824

10066091006609119890119808119808119890119896


119850119850asymp

11982411982410066091006609119894119894119890119808119808119890 119894119894th subset of119824119824

asymp119890119808119808119890 119894119894 119896 119896119896119896 119896119896119896 119896119870119870119824119824

asymp119890119808119808119890119896


119896119896 119896 119896119896119896 119896119896119896 119896119870119870119896 119894119894119896119896 119896 119896119896119896 119896119896119896 119896119870119870119904119904119894119894119896119896119896119896119896


119896119896 119896 119896119896119896 119896119896119896 119896119870119870119896 119894119894119896119896 119896 119896119896119896 119896119896119896 119896119870119870119904119904119894119894119896119896119896119896119896


11989911989910066091006609119894119894119896119896119896119896 119890119808119808119890 Entry of 119821119821

10066101006610119890119808119808119890 119899119899

asymp119894119894119896119896119896119896119890119808119808119890 119896 119808+ and

119896119896 119896 119896119896119896 119896119896119896 119896119870119870119896 119894119894119896119896 119896 119896119896119896 119896119896119896 119896119870119870119904119904119894119894119896119896119896119896119896





11982111982110066101006610119890119808119808119890


119890120590120590119896119896119808119808 = 119896119826119826119808119808119896 times 119826119826


Glossary






References

















RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation













ℕ120582120582+ for all 119821119821 119808 119808119808






120582120582for all 119821119821 119808 119808119808




119819119819119821119821119821119821 119821 Λ1007714100771411982111982111982111982111982111982110077301007730⟺ 119897119897119894119894119896119896119896119896 119821119821119821119821 1198211007618100761810076341007634100762610076261007634100763410076421007642

O119821 119899119899119894119894119896119896119896119896 119821119821119821119821 119821 119821119821

I119821 119899119899119894119894119896119896119896119896 119821119821119821119821 gt 119821119821forall119896119896119821 119894119894119896119896

(1)


Λminus1198211007714100771411982111982111982111982111982111982110077301007730

119821 10076821007682 119890119890 119821 10076821007682119904119904119894119894119821119821 119821 119904119904119894119894119821119821 119821119896 119821 119904119904119894119894119870119870119870119870 10076981007698 119821 119808119808 119821 119897119897119894119894119896119896119896119896 119821119821119821119821 119821 I119821 forall119896119896119821 11989411989411989611989610076981007698

(2)



10066101006610119821119821119821119821 is dened here as



119821119821119821119821





11982111982110066101006610119821119821119821119821 119821 Ξ1007714100771411981911981911982111982111982111982110077301007730⟺ 119899119899

10066091006609119894119894119896119896119896119896119821119821119821119821 119821

1007618100761810076341007634100762610076261007634100763410076421007642

119821119821 119897119897119894119894119896119896119896119896 119821119821119821119821 119821 O119821|119810119810119821119821119821119821|10092501009250119826119826119821119821119896119896 10092501009250

119821 119897119897119894119894119896119896119896119896 119821119821119821119821 119821 I119821

forall119896119896119821 119894119894119896119896(3)


119894119894119896119896119896119896 119821119821119821119821 119821





119821119821



119894119894119896119896sim119896119896119821119809119809119821 at 119808119896119896119821 119894119894119896119896 119821 119808119821119821 119821 or 119808119821119821 119821



Since 120582119894119894 119899119899119894119894119896119896119896119896 119821119821119821






11980810065941006594119890119890119821 1198211198211006609100660911982411982410066091006609119821 119821119809119809119821 119821 11980810065941006594119890119890119821 119821119821

1006609100660911982411982410066091006609119821 119821119809119809119821 119821 11980810065941006594119890119890119821 and 119821119821

1006609100660911982411982410066091006609119821 119821119809119809119821 119821 11980810065961006596119890119890119821119821119821100659610065961198901198901198218119821100659610065961198901198901198219119821


119821119821119821119824119824119821119809119809119821119821 119821

100761810076181007634100763410076341007634100763410076341007634100763410076341007634100763410076341007626100762610076341007634100763410076341007634100763410076341007634100763410076341007634100763410076421007642

119821119821100765110076511198211198211006609100660911982411982410066091006609119821 11982111980911980911982110076671007667

119821119821100765110076511198211198211006609100660911982411982410066091006609119821 11982111980911980911982110076671007667

119821119821100765110076511198211198211006609100660911982411982410066091006609119821 11982111980911980911982110076671007667

119821119821100765110076511198211198211006609100660911982411982410066091006609119821 11982111980911980911982110076671007667

100761910076191007635100763510076351007635100763510076351007635100763510076351007635100763510076351007627100762710076351007635100763510076351007635100763510076351007635100763510076351007635100763510076431007643

11982110076181007618100763410076341007634100763410076261007626100763410076341007634100763410076421007642

119821119821119821 119821119821 119821119821 119821119821 119821119821 119821119821 119821119821 119821119821 119821119821 119821119821119821119821119821 119821119821 119821119821 119821119821 119821119821 119821119821 119821119821 119821119821 119821119821 119821119821119821119821119821 119821119821 119821119821 119821119821 119821119821 119821119821 119821119821 119821119821 119821119821 119821119821119821119821119821 119821119821 119821119821 119821119821 119821119821 119821119821 119821119821 119821119821 119821119821 119821119821

10076191007619100763510076351007635100763510076271007627100763510076351007635100763510076431007643

(4)


e 11985011985010066091006609






119879119879




119808119808120590120590119808119808 = 10076821007682 119890119890 isin 119808119808 119890 119890120590120590119808119808119808119808 10076981007698

119808119808 minus 119808119808120590120590119808119808 = 10076821007682 119890119890 isin 119808119808 119890 10076501007650119810119810(119808119808119809 minus 119890120590120590119808119808119808119808 1007666100766610076981007698


(5)






1198211198211007649100764911980811980812059012059011980811980810076651007665 = 119821119821(119808119808119809 minus 1198211198211007649100764911980811980812059012059011980811980810076651007665 (6)


1198211198212 times


1198191198191007649100764911980811980812059012059011980811980810076651007665 119808 119897119897119894119894119898119898119898119898 10092501009250119819119819(119808119808120590120590119808119808 119809 =

1007618100761810076341007634100763410076341007634100763410076341007634100762610076261007634100763410076341007634100763410076341007634100763410076421007642

119897119897119894119894119898119898119898119898 10092501009250119819119819(119808119808119809 119808119808 119808119898119898

I 119808119808 = 119898119898 and 119896119896119894119894119898119898119898119898 isin 120590120590119808119808

O 119808119808 = 119898119898 and 119896119896119894119894119898119898119898119898119898119898 notin 120590120590119808119808(7)


12059012059001 1198211198211 where 119821119821



12059012059012


12059012059001 119821119821

12059012059012 119821119821

12059012059023 hellip



119894119894 = Ξ[119819119819120590120590119894119894minus1119894119894 ] using (7) and

119819119819119894119894minus1 = Λ[119821119821119894119894minus1]



33 and 11989611989643 Let 1205901205900 = 11989611989633

(i) 11981911981912059012059001 = [I I IO IOOO 119816119816O]

rArr 11982111982112059012059001 = [2 2 2 0 2 0 0 0 120786120786 0]

(ii) 1198211198211 = 1198211198210 minus 11982111982112059012059001 = [3 6 5 0 4 0 3 2 120782120782 4]

1198191198191 = [I I IO IO I I 119822119822 I]


43 as follows


rArr 119821119821120590120590lowast01 = [4 4 4 0 4 0 0 0 120786120786 120786120786]

(ii) 119821119821lowast1 = 1198211198210 minus 119821119821

120590120590lowast01 = [1 4 3 0 2 0 3 2 120782120782 120782120782]

119819119819lowast1 = [I I IO IO I I 119822119822119822119822]





(i) 119819119819120590120590lowast12 = [0 120783120783120783 1 0 1 0120783 1 1 0 0120783

rArr 119821119821120590120590lowast12 = [0 120786120786120783 2 0 2 0120783 2 2 0 0120783

(ii) 119821119821lowast2 = 119821119821


120590120590lowast12 = [1 120782120782120783 1 0 0 0120783 1 0 0 0120783

Both 119821119821lowast2 and 119821119821



12059012059001 119821119821

lowast2 119821119821

120590120590lowast12


12059012059001 e vectors 1198211198211205901205900



(i)

11982111982112059012059001 = [4 4120783 4 0 4 0120783 0 0 4 4120783

sim 10076851007685[2 2120783 4 0 0 0120783 0 0 2 2120783[2 2120783 0 0 4 0120783 0 0 2 2120783

10077011007701

sim10076181007618100763410076341007634100763410076261007626100763410076341007634100763410076421007642

[1 1120783 2 0 0 0120783 0 0 2 0120783[1 1120783 2 0 0 0120783 0 0 0 2120783[1 1120783 0 0 2 0120783 0 0 2 0120783[1 1120783 0 0 2 0120783 0 0 0 2120783

10076191007619100763510076351007635100763510076271007627100763510076351007635100763510076431007643

(8)

(ii)

11982111982112059012059012 = [0 4120783 2 0 2 0120783 2 2 0 0120783

sim 10076851007685[0 2120783 0 0 2 0120783 1 1 0 0120783[0 2120783 2 0 0 0120783 1 1 0 0120783

10077011007701

sim10076181007618100763410076341007634100763410076261007626100763410076341007634100763410076421007642

[0 1120783 0 0 1 0120783 0 1 0 0120783[0 1120783 0 0 1 0120783 1 0 0 0120783[0 1120783 1 0 0 0120783 0 1 0 0120783[0 1120783 1 0 0 0120783 1 0 0 0120783

10076191007619100763510076351007635100763510076271007627100763510076351007635100763510076431007643

(9)







⋯ times 1198261198261198191198191119896119896 119812










43 119904119904

13 119904119904

34 119904119904

44



1198211198210 = [5 8 7 0 6 0 3 2 4 4] sim 10076851007685[0 8 4 0 4 0 2 2 2 2][5 0 3 0 2 0 1 0 2 2]

10077011007701

sim10076181007618100763410076341007634100763410076261007626100763410076341007634100763410076421007642

[0 4 4 0 0 0 1 1 1 1][0 4 0 0 4 0 1 1 1 1][4 0 2 0 2 0 0 0 2 2][1 0 1 0 0 0 1 0 0 0]

10076191007619100763510076351007635100763510076271007627100763510076351007635100763510076431007643

sim10076181007618100763410076341007634100763410076261007626100763410076341007634100763410076421007642

[0 4 4 0 0 0 1 1 1 1][0 4 0 0 4 0 1 1 1 1][1 0 1 0 0 0 1 0 0 0]

[1 0 1 0 0 0 0 0 1 0][1 0 1 0 0 0 0 0 0 1][1 0 0 0 1 0 0 0 1 0][1 0 0 0 1 0 0 0 0 1]

10076191007619100763510076351007635100763510076271007627100763510076351007635100763510076431007643

(10)





119894119894119896119896119896119896 minus

min119896119896119899119899119894119894119896119896119896119896 )|119826119826119896119896|




119894119894119896119896119896119896 minus 119899119899

119894119894119896119896119896119896







100661010066100 = 11982111982110066101006610(1198241198240(119850119850

119879119879119903119903119894119894 (119812119812))) (or the asso-




10066101006610






10066101006610(1198241198241(119850119850

119879119879119903119903119894119894 (119812119812))) is

implicit






























1 1 1

1

1 1 2 4

1 1 3 1

1 1 4 1

1 2 1 3

1 2 2 3

1

2 3 2

1 2 4 3

1 3 1 4

1 3 2 4

1 3 3 1

1 3 4

1

1 4 1 3

1 4 2 3

1 4 3 2

1 4 4 2

2 1 1 1

2

1 2 1

2 1 3 1

2 1 4 1

2 2 1 2

2 2 2 2

2 2 3

2

2 2 4 2

2 3 1 1

2 3 2 1

2 3 3 1

2 3 4 1

2

4 1 2

2 4 2 2

2 4 3 2

2 4 4

2



1 1 1 1

1 1 3 1

1 1 4 1

1 3 3 1

1 3 4 1

2 1 1 1

2 1 2 1

2 1 3 1

2 1 4 1

2 3 1 1

2 3 2 1

2 3 3 1

2 3 4 1

1 2 3 2

1 4 3 2

1 4 4 2

2 2 1 2

2 2 2 2

2 2 3 2

2 2 4 2

2 4 1 2

2 4 2 2

2 4 3 2

2 4 4 2

1 2 1 3

1 2 2 3

1 2 4 3

1 4 1 3

1 4 2 3

1 1 2 4

1 3 1 4

1 3 2 4


119873119873119877119877 = 100762010076201007636100763610076361007636

10076441007644

5 8 7 0 6 0 3 2 4 4 13 8 0 5 0 6 2 2 4 3 25 0 0 3 0 2 2 2 0 1 33 0 1 0 2 0 1 2 0 0 4

100762110076211007637100763710076371007637

10076451007645

(11)



















119873119873119877119877 119894 100762010076201007636100763610076361007636

10076441007644

5 0 3 0 2 0 1 0 2 2 13 8 0 5 0 6 2 2 4 3 25 0 0 3 0 2 2 2 0 1 33 0 1 0 2 0 1 2 0 0 4

100762110076211007637100763710076371007637

10076451007645

119873119873100661110066111

119894 [0 8 4 0 4 0 2 2 2 2]

(12)










asymp 119894119894

denoted by 11987111987110066091006609119894119894and 119871119871



119871119871100660910066091

119894 10076201007620

10076441007644

0 1 1 0 1 0 1 1 1 11 1 0 1 0 1 0 0 0 11 0 1 0 0 0 1 0 0 0

10076211007621

10076451007645(13)


119871119871asymp1

119894

100762010076201007636100763610076361007636100763610076361007636100763610076361007636100763610076361007636100763610076361007636

10076441007644

0 1 0 0 0 1 1 1 1 10 1 0 0 1 0 1 1 1 11 0 1 0 0 0 0 0 1 01 0 0 0 1 0 0 0 1 01 0 1 0 0 0 0 0 0 11 0 0 0 1 0 0 0 0 11 0 1 0 0 0 1 0 0 0

100762110076211007637100763710076371007637100763710076371007637100763710076371007637100763710076371007637100763710076371007637

10076451007645

(14)













6 Case Studies











7 Conclusions










Nomenclature


119887119887119896119896 119826119826119896119896 = 119896119904119904119896119896119896119896 119904119904119896119896119896119896119896 119896 119904119904

119870119870S119896119896119896119896 119896







119808119808119896 times ⋯ times 119826119826119808119808119870119870



1198501198501006609100660911982411982410066091006609119894119894 119890119808119808119890 119894119894th subset of119824119824

10066101006610119890119808119808119890 119894119894 119896 119896119896119896 119896119896119896 119896119870119870119824119824

10066091006609119890119808119808119890119896


119850119850asymp

11982411982410066091006609119894119894119890119808119808119890 119894119894th subset of119824119824

asymp119890119808119808119890 119894119894 119896 119896119896119896 119896119896119896 119896119870119870119824119824

asymp119890119808119808119890119896


119896119896 119896 119896119896119896 119896119896119896 119896119870119870119896 119894119894119896119896 119896 119896119896119896 119896119896119896 119896119870119870119904119904119894119894119896119896119896119896119896


119896119896 119896 119896119896119896 119896119896119896 119896119870119870119896 119894119894119896119896 119896 119896119896119896 119896119896119896 119896119870119870119904119904119894119894119896119896119896119896119896


11989911989910066091006609119894119894119896119896119896119896 119890119808119808119890 Entry of 119821119821

10066101006610119890119808119808119890 119899119899

asymp119894119894119896119896119896119896119890119808119808119890 119896 119808+ and

119896119896 119896 119896119896119896 119896119896119896 119896119870119870119896 119894119894119896119896 119896 119896119896119896 119896119896119896 119896119870119870119904119904119894119894119896119896119896119896119896





11982111982110066101006610119890119808119808119890


119890120590120590119896119896119808119808 = 119896119826119826119808119808119896 times 119826119826


Glossary






References

















RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










e 11985011985010066091006609






119879119879




119808119808120590120590119808119808 = 10076821007682 119890119890 isin 119808119808 119890 119890120590120590119808119808119808119808 10076981007698

119808119808 minus 119808119808120590120590119808119808 = 10076821007682 119890119890 isin 119808119808 119890 10076501007650119810119810(119808119808119809 minus 119890120590120590119808119808119808119808 1007666100766610076981007698


(5)






1198211198211007649100764911980811980812059012059011980811980810076651007665 = 119821119821(119808119808119809 minus 1198211198211007649100764911980811980812059012059011980811980810076651007665 (6)


1198211198212 times


1198191198191007649100764911980811980812059012059011980811980810076651007665 119808 119897119897119894119894119898119898119898119898 10092501009250119819119819(119808119808120590120590119808119808 119809 =

1007618100761810076341007634100763410076341007634100763410076341007634100762610076261007634100763410076341007634100763410076341007634100763410076421007642

119897119897119894119894119898119898119898119898 10092501009250119819119819(119808119808119809 119808119808 119808119898119898

I 119808119808 = 119898119898 and 119896119896119894119894119898119898119898119898 isin 120590120590119808119808

O 119808119808 = 119898119898 and 119896119896119894119894119898119898119898119898119898119898 notin 120590120590119808119808(7)


12059012059001 1198211198211 where 119821119821



12059012059012


12059012059001 119821119821

12059012059012 119821119821

12059012059023 hellip



119894119894 = Ξ[119819119819120590120590119894119894minus1119894119894 ] using (7) and

119819119819119894119894minus1 = Λ[119821119821119894119894minus1]



33 and 11989611989643 Let 1205901205900 = 11989611989633

(i) 11981911981912059012059001 = [I I IO IOOO 119816119816O]

rArr 11982111982112059012059001 = [2 2 2 0 2 0 0 0 120786120786 0]

(ii) 1198211198211 = 1198211198210 minus 11982111982112059012059001 = [3 6 5 0 4 0 3 2 120782120782 4]

1198191198191 = [I I IO IO I I 119822119822 I]


43 as follows


rArr 119821119821120590120590lowast01 = [4 4 4 0 4 0 0 0 120786120786 120786120786]

(ii) 119821119821lowast1 = 1198211198210 minus 119821119821

120590120590lowast01 = [1 4 3 0 2 0 3 2 120782120782 120782120782]

119819119819lowast1 = [I I IO IO I I 119822119822119822119822]





(i) 119819119819120590120590lowast12 = [0 120783120783120783 1 0 1 0120783 1 1 0 0120783

rArr 119821119821120590120590lowast12 = [0 120786120786120783 2 0 2 0120783 2 2 0 0120783

(ii) 119821119821lowast2 = 119821119821


120590120590lowast12 = [1 120782120782120783 1 0 0 0120783 1 0 0 0120783

Both 119821119821lowast2 and 119821119821



12059012059001 119821119821

lowast2 119821119821

120590120590lowast12


12059012059001 e vectors 1198211198211205901205900



(i)

11982111982112059012059001 = [4 4120783 4 0 4 0120783 0 0 4 4120783

sim 10076851007685[2 2120783 4 0 0 0120783 0 0 2 2120783[2 2120783 0 0 4 0120783 0 0 2 2120783

10077011007701

sim10076181007618100763410076341007634100763410076261007626100763410076341007634100763410076421007642

[1 1120783 2 0 0 0120783 0 0 2 0120783[1 1120783 2 0 0 0120783 0 0 0 2120783[1 1120783 0 0 2 0120783 0 0 2 0120783[1 1120783 0 0 2 0120783 0 0 0 2120783

10076191007619100763510076351007635100763510076271007627100763510076351007635100763510076431007643

(8)

(ii)

11982111982112059012059012 = [0 4120783 2 0 2 0120783 2 2 0 0120783

sim 10076851007685[0 2120783 0 0 2 0120783 1 1 0 0120783[0 2120783 2 0 0 0120783 1 1 0 0120783

10077011007701

sim10076181007618100763410076341007634100763410076261007626100763410076341007634100763410076421007642

[0 1120783 0 0 1 0120783 0 1 0 0120783[0 1120783 0 0 1 0120783 1 0 0 0120783[0 1120783 1 0 0 0120783 0 1 0 0120783[0 1120783 1 0 0 0120783 1 0 0 0120783

10076191007619100763510076351007635100763510076271007627100763510076351007635100763510076431007643

(9)







⋯ times 1198261198261198191198191119896119896 119812










43 119904119904

13 119904119904

34 119904119904

44



1198211198210 = [5 8 7 0 6 0 3 2 4 4] sim 10076851007685[0 8 4 0 4 0 2 2 2 2][5 0 3 0 2 0 1 0 2 2]

10077011007701

sim10076181007618100763410076341007634100763410076261007626100763410076341007634100763410076421007642

[0 4 4 0 0 0 1 1 1 1][0 4 0 0 4 0 1 1 1 1][4 0 2 0 2 0 0 0 2 2][1 0 1 0 0 0 1 0 0 0]

10076191007619100763510076351007635100763510076271007627100763510076351007635100763510076431007643

sim10076181007618100763410076341007634100763410076261007626100763410076341007634100763410076421007642

[0 4 4 0 0 0 1 1 1 1][0 4 0 0 4 0 1 1 1 1][1 0 1 0 0 0 1 0 0 0]

[1 0 1 0 0 0 0 0 1 0][1 0 1 0 0 0 0 0 0 1][1 0 0 0 1 0 0 0 1 0][1 0 0 0 1 0 0 0 0 1]

10076191007619100763510076351007635100763510076271007627100763510076351007635100763510076431007643

(10)





119894119894119896119896119896119896 minus

min119896119896119899119899119894119894119896119896119896119896 )|119826119826119896119896|




119894119894119896119896119896119896 minus 119899119899

119894119894119896119896119896119896







100661010066100 = 11982111982110066101006610(1198241198240(119850119850

119879119879119903119903119894119894 (119812119812))) (or the asso-




10066101006610






10066101006610(1198241198241(119850119850

119879119879119903119903119894119894 (119812119812))) is

implicit






























1 1 1

1

1 1 2 4

1 1 3 1

1 1 4 1

1 2 1 3

1 2 2 3

1

2 3 2

1 2 4 3

1 3 1 4

1 3 2 4

1 3 3 1

1 3 4

1

1 4 1 3

1 4 2 3

1 4 3 2

1 4 4 2

2 1 1 1

2

1 2 1

2 1 3 1

2 1 4 1

2 2 1 2

2 2 2 2

2 2 3

2

2 2 4 2

2 3 1 1

2 3 2 1

2 3 3 1

2 3 4 1

2

4 1 2

2 4 2 2

2 4 3 2

2 4 4

2



1 1 1 1

1 1 3 1

1 1 4 1

1 3 3 1

1 3 4 1

2 1 1 1

2 1 2 1

2 1 3 1

2 1 4 1

2 3 1 1

2 3 2 1

2 3 3 1

2 3 4 1

1 2 3 2

1 4 3 2

1 4 4 2

2 2 1 2

2 2 2 2

2 2 3 2

2 2 4 2

2 4 1 2

2 4 2 2

2 4 3 2

2 4 4 2

1 2 1 3

1 2 2 3

1 2 4 3

1 4 1 3

1 4 2 3

1 1 2 4

1 3 1 4

1 3 2 4


119873119873119877119877 = 100762010076201007636100763610076361007636

10076441007644

5 8 7 0 6 0 3 2 4 4 13 8 0 5 0 6 2 2 4 3 25 0 0 3 0 2 2 2 0 1 33 0 1 0 2 0 1 2 0 0 4

100762110076211007637100763710076371007637

10076451007645

(11)



















119873119873119877119877 119894 100762010076201007636100763610076361007636

10076441007644

5 0 3 0 2 0 1 0 2 2 13 8 0 5 0 6 2 2 4 3 25 0 0 3 0 2 2 2 0 1 33 0 1 0 2 0 1 2 0 0 4

100762110076211007637100763710076371007637

10076451007645

119873119873100661110066111

119894 [0 8 4 0 4 0 2 2 2 2]

(12)










asymp 119894119894

denoted by 11987111987110066091006609119894119894and 119871119871



119871119871100660910066091

119894 10076201007620

10076441007644

0 1 1 0 1 0 1 1 1 11 1 0 1 0 1 0 0 0 11 0 1 0 0 0 1 0 0 0

10076211007621

10076451007645(13)


119871119871asymp1

119894

100762010076201007636100763610076361007636100763610076361007636100763610076361007636100763610076361007636100763610076361007636

10076441007644

0 1 0 0 0 1 1 1 1 10 1 0 0 1 0 1 1 1 11 0 1 0 0 0 0 0 1 01 0 0 0 1 0 0 0 1 01 0 1 0 0 0 0 0 0 11 0 0 0 1 0 0 0 0 11 0 1 0 0 0 1 0 0 0

100762110076211007637100763710076371007637100763710076371007637100763710076371007637100763710076371007637100763710076371007637

10076451007645

(14)













6 Case Studies











7 Conclusions










Nomenclature


119887119887119896119896 119826119826119896119896 = 119896119904119904119896119896119896119896 119904119904119896119896119896119896119896 119896 119904119904

119870119870S119896119896119896119896 119896







119808119808119896 times ⋯ times 119826119826119808119808119870119870



1198501198501006609100660911982411982410066091006609119894119894 119890119808119808119890 119894119894th subset of119824119824

10066101006610119890119808119808119890 119894119894 119896 119896119896119896 119896119896119896 119896119870119870119824119824

10066091006609119890119808119808119890119896


119850119850asymp

11982411982410066091006609119894119894119890119808119808119890 119894119894th subset of119824119824

asymp119890119808119808119890 119894119894 119896 119896119896119896 119896119896119896 119896119870119870119824119824

asymp119890119808119808119890119896


119896119896 119896 119896119896119896 119896119896119896 119896119870119870119896 119894119894119896119896 119896 119896119896119896 119896119896119896 119896119870119870119904119904119894119894119896119896119896119896119896


119896119896 119896 119896119896119896 119896119896119896 119896119870119870119896 119894119894119896119896 119896 119896119896119896 119896119896119896 119896119870119870119904119904119894119894119896119896119896119896119896


11989911989910066091006609119894119894119896119896119896119896 119890119808119808119890 Entry of 119821119821

10066101006610119890119808119808119890 119899119899

asymp119894119894119896119896119896119896119890119808119808119890 119896 119808+ and

119896119896 119896 119896119896119896 119896119896119896 119896119870119870119896 119894119894119896119896 119896 119896119896119896 119896119896119896 119896119870119870119904119904119894119894119896119896119896119896119896





11982111982110066101006610119890119808119808119890


119890120590120590119896119896119808119808 = 119896119826119826119808119808119896 times 119826119826


Glossary






References

















RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation













(i) 119819119819120590120590lowast12 = [0 120783120783120783 1 0 1 0120783 1 1 0 0120783

rArr 119821119821120590120590lowast12 = [0 120786120786120783 2 0 2 0120783 2 2 0 0120783

(ii) 119821119821lowast2 = 119821119821


120590120590lowast12 = [1 120782120782120783 1 0 0 0120783 1 0 0 0120783

Both 119821119821lowast2 and 119821119821



12059012059001 119821119821

lowast2 119821119821

120590120590lowast12


12059012059001 e vectors 1198211198211205901205900



(i)

11982111982112059012059001 = [4 4120783 4 0 4 0120783 0 0 4 4120783

sim 10076851007685[2 2120783 4 0 0 0120783 0 0 2 2120783[2 2120783 0 0 4 0120783 0 0 2 2120783

10077011007701

sim10076181007618100763410076341007634100763410076261007626100763410076341007634100763410076421007642

[1 1120783 2 0 0 0120783 0 0 2 0120783[1 1120783 2 0 0 0120783 0 0 0 2120783[1 1120783 0 0 2 0120783 0 0 2 0120783[1 1120783 0 0 2 0120783 0 0 0 2120783

10076191007619100763510076351007635100763510076271007627100763510076351007635100763510076431007643

(8)

(ii)

11982111982112059012059012 = [0 4120783 2 0 2 0120783 2 2 0 0120783

sim 10076851007685[0 2120783 0 0 2 0120783 1 1 0 0120783[0 2120783 2 0 0 0120783 1 1 0 0120783

10077011007701

sim10076181007618100763410076341007634100763410076261007626100763410076341007634100763410076421007642

[0 1120783 0 0 1 0120783 0 1 0 0120783[0 1120783 0 0 1 0120783 1 0 0 0120783[0 1120783 1 0 0 0120783 0 1 0 0120783[0 1120783 1 0 0 0120783 1 0 0 0120783

10076191007619100763510076351007635100763510076271007627100763510076351007635100763510076431007643

(9)







⋯ times 1198261198261198191198191119896119896 119812










43 119904119904

13 119904119904

34 119904119904

44



1198211198210 = [5 8 7 0 6 0 3 2 4 4] sim 10076851007685[0 8 4 0 4 0 2 2 2 2][5 0 3 0 2 0 1 0 2 2]

10077011007701

sim10076181007618100763410076341007634100763410076261007626100763410076341007634100763410076421007642

[0 4 4 0 0 0 1 1 1 1][0 4 0 0 4 0 1 1 1 1][4 0 2 0 2 0 0 0 2 2][1 0 1 0 0 0 1 0 0 0]

10076191007619100763510076351007635100763510076271007627100763510076351007635100763510076431007643

sim10076181007618100763410076341007634100763410076261007626100763410076341007634100763410076421007642

[0 4 4 0 0 0 1 1 1 1][0 4 0 0 4 0 1 1 1 1][1 0 1 0 0 0 1 0 0 0]

[1 0 1 0 0 0 0 0 1 0][1 0 1 0 0 0 0 0 0 1][1 0 0 0 1 0 0 0 1 0][1 0 0 0 1 0 0 0 0 1]

10076191007619100763510076351007635100763510076271007627100763510076351007635100763510076431007643

(10)





119894119894119896119896119896119896 minus

min119896119896119899119899119894119894119896119896119896119896 )|119826119826119896119896|




119894119894119896119896119896119896 minus 119899119899

119894119894119896119896119896119896







100661010066100 = 11982111982110066101006610(1198241198240(119850119850

119879119879119903119903119894119894 (119812119812))) (or the asso-




10066101006610






10066101006610(1198241198241(119850119850

119879119879119903119903119894119894 (119812119812))) is

implicit






























1 1 1

1

1 1 2 4

1 1 3 1

1 1 4 1

1 2 1 3

1 2 2 3

1

2 3 2

1 2 4 3

1 3 1 4

1 3 2 4

1 3 3 1

1 3 4

1

1 4 1 3

1 4 2 3

1 4 3 2

1 4 4 2

2 1 1 1

2

1 2 1

2 1 3 1

2 1 4 1

2 2 1 2

2 2 2 2

2 2 3

2

2 2 4 2

2 3 1 1

2 3 2 1

2 3 3 1

2 3 4 1

2

4 1 2

2 4 2 2

2 4 3 2

2 4 4

2



1 1 1 1

1 1 3 1

1 1 4 1

1 3 3 1

1 3 4 1

2 1 1 1

2 1 2 1

2 1 3 1

2 1 4 1

2 3 1 1

2 3 2 1

2 3 3 1

2 3 4 1

1 2 3 2

1 4 3 2

1 4 4 2

2 2 1 2

2 2 2 2

2 2 3 2

2 2 4 2

2 4 1 2

2 4 2 2

2 4 3 2

2 4 4 2

1 2 1 3

1 2 2 3

1 2 4 3

1 4 1 3

1 4 2 3

1 1 2 4

1 3 1 4

1 3 2 4


119873119873119877119877 = 100762010076201007636100763610076361007636

10076441007644

5 8 7 0 6 0 3 2 4 4 13 8 0 5 0 6 2 2 4 3 25 0 0 3 0 2 2 2 0 1 33 0 1 0 2 0 1 2 0 0 4

100762110076211007637100763710076371007637

10076451007645

(11)



















119873119873119877119877 119894 100762010076201007636100763610076361007636

10076441007644

5 0 3 0 2 0 1 0 2 2 13 8 0 5 0 6 2 2 4 3 25 0 0 3 0 2 2 2 0 1 33 0 1 0 2 0 1 2 0 0 4

100762110076211007637100763710076371007637

10076451007645

119873119873100661110066111

119894 [0 8 4 0 4 0 2 2 2 2]

(12)










asymp 119894119894

denoted by 11987111987110066091006609119894119894and 119871119871



119871119871100660910066091

119894 10076201007620

10076441007644

0 1 1 0 1 0 1 1 1 11 1 0 1 0 1 0 0 0 11 0 1 0 0 0 1 0 0 0

10076211007621

10076451007645(13)


119871119871asymp1

119894

100762010076201007636100763610076361007636100763610076361007636100763610076361007636100763610076361007636100763610076361007636

10076441007644

0 1 0 0 0 1 1 1 1 10 1 0 0 1 0 1 1 1 11 0 1 0 0 0 0 0 1 01 0 0 0 1 0 0 0 1 01 0 1 0 0 0 0 0 0 11 0 0 0 1 0 0 0 0 11 0 1 0 0 0 1 0 0 0

100762110076211007637100763710076371007637100763710076371007637100763710076371007637100763710076371007637100763710076371007637

10076451007645

(14)













6 Case Studies











7 Conclusions










Nomenclature


119887119887119896119896 119826119826119896119896 = 119896119904119904119896119896119896119896 119904119904119896119896119896119896119896 119896 119904119904

119870119870S119896119896119896119896 119896







119808119808119896 times ⋯ times 119826119826119808119808119870119870



1198501198501006609100660911982411982410066091006609119894119894 119890119808119808119890 119894119894th subset of119824119824

10066101006610119890119808119808119890 119894119894 119896 119896119896119896 119896119896119896 119896119870119870119824119824

10066091006609119890119808119808119890119896


119850119850asymp

11982411982410066091006609119894119894119890119808119808119890 119894119894th subset of119824119824

asymp119890119808119808119890 119894119894 119896 119896119896119896 119896119896119896 119896119870119870119824119824

asymp119890119808119808119890119896


119896119896 119896 119896119896119896 119896119896119896 119896119870119870119896 119894119894119896119896 119896 119896119896119896 119896119896119896 119896119870119870119904119904119894119894119896119896119896119896119896


119896119896 119896 119896119896119896 119896119896119896 119896119870119870119896 119894119894119896119896 119896 119896119896119896 119896119896119896 119896119870119870119904119904119894119894119896119896119896119896119896


11989911989910066091006609119894119894119896119896119896119896 119890119808119808119890 Entry of 119821119821

10066101006610119890119808119808119890 119899119899

asymp119894119894119896119896119896119896119890119808119808119890 119896 119808+ and

119896119896 119896 119896119896119896 119896119896119896 119896119870119870119896 119894119894119896119896 119896 119896119896119896 119896119896119896 119896119870119870119904119904119894119894119896119896119896119896119896





11982111982110066101006610119890119808119808119890


119890120590120590119896119896119808119808 = 119896119826119826119808119808119896 times 119826119826


Glossary






References

















RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











1198211198210 = [5 8 7 0 6 0 3 2 4 4] sim 10076851007685[0 8 4 0 4 0 2 2 2 2][5 0 3 0 2 0 1 0 2 2]

10077011007701

sim10076181007618100763410076341007634100763410076261007626100763410076341007634100763410076421007642

[0 4 4 0 0 0 1 1 1 1][0 4 0 0 4 0 1 1 1 1][4 0 2 0 2 0 0 0 2 2][1 0 1 0 0 0 1 0 0 0]

10076191007619100763510076351007635100763510076271007627100763510076351007635100763510076431007643

sim10076181007618100763410076341007634100763410076261007626100763410076341007634100763410076421007642

[0 4 4 0 0 0 1 1 1 1][0 4 0 0 4 0 1 1 1 1][1 0 1 0 0 0 1 0 0 0]

[1 0 1 0 0 0 0 0 1 0][1 0 1 0 0 0 0 0 0 1][1 0 0 0 1 0 0 0 1 0][1 0 0 0 1 0 0 0 0 1]

10076191007619100763510076351007635100763510076271007627100763510076351007635100763510076431007643

(10)





119894119894119896119896119896119896 minus

min119896119896119899119899119894119894119896119896119896119896 )|119826119826119896119896|




119894119894119896119896119896119896 minus 119899119899

119894119894119896119896119896119896







100661010066100 = 11982111982110066101006610(1198241198240(119850119850

119879119879119903119903119894119894 (119812119812))) (or the asso-




10066101006610






10066101006610(1198241198241(119850119850

119879119879119903119903119894119894 (119812119812))) is

implicit






























1 1 1

1

1 1 2 4

1 1 3 1

1 1 4 1

1 2 1 3

1 2 2 3

1

2 3 2

1 2 4 3

1 3 1 4

1 3 2 4

1 3 3 1

1 3 4

1

1 4 1 3

1 4 2 3

1 4 3 2

1 4 4 2

2 1 1 1

2

1 2 1

2 1 3 1

2 1 4 1

2 2 1 2

2 2 2 2

2 2 3

2

2 2 4 2

2 3 1 1

2 3 2 1

2 3 3 1

2 3 4 1

2

4 1 2

2 4 2 2

2 4 3 2

2 4 4

2



1 1 1 1

1 1 3 1

1 1 4 1

1 3 3 1

1 3 4 1

2 1 1 1

2 1 2 1

2 1 3 1

2 1 4 1

2 3 1 1

2 3 2 1

2 3 3 1

2 3 4 1

1 2 3 2

1 4 3 2

1 4 4 2

2 2 1 2

2 2 2 2

2 2 3 2

2 2 4 2

2 4 1 2

2 4 2 2

2 4 3 2

2 4 4 2

1 2 1 3

1 2 2 3

1 2 4 3

1 4 1 3

1 4 2 3

1 1 2 4

1 3 1 4

1 3 2 4


119873119873119877119877 = 100762010076201007636100763610076361007636

10076441007644

5 8 7 0 6 0 3 2 4 4 13 8 0 5 0 6 2 2 4 3 25 0 0 3 0 2 2 2 0 1 33 0 1 0 2 0 1 2 0 0 4

100762110076211007637100763710076371007637

10076451007645

(11)



















119873119873119877119877 119894 100762010076201007636100763610076361007636

10076441007644

5 0 3 0 2 0 1 0 2 2 13 8 0 5 0 6 2 2 4 3 25 0 0 3 0 2 2 2 0 1 33 0 1 0 2 0 1 2 0 0 4

100762110076211007637100763710076371007637

10076451007645

119873119873100661110066111

119894 [0 8 4 0 4 0 2 2 2 2]

(12)










asymp 119894119894

denoted by 11987111987110066091006609119894119894and 119871119871



119871119871100660910066091

119894 10076201007620

10076441007644

0 1 1 0 1 0 1 1 1 11 1 0 1 0 1 0 0 0 11 0 1 0 0 0 1 0 0 0

10076211007621

10076451007645(13)


119871119871asymp1

119894

100762010076201007636100763610076361007636100763610076361007636100763610076361007636100763610076361007636100763610076361007636

10076441007644

0 1 0 0 0 1 1 1 1 10 1 0 0 1 0 1 1 1 11 0 1 0 0 0 0 0 1 01 0 0 0 1 0 0 0 1 01 0 1 0 0 0 0 0 0 11 0 0 0 1 0 0 0 0 11 0 1 0 0 0 1 0 0 0

100762110076211007637100763710076371007637100763710076371007637100763710076371007637100763710076371007637100763710076371007637

10076451007645

(14)













6 Case Studies











7 Conclusions










Nomenclature


119887119887119896119896 119826119826119896119896 = 119896119904119904119896119896119896119896 119904119904119896119896119896119896119896 119896 119904119904

119870119870S119896119896119896119896 119896







119808119808119896 times ⋯ times 119826119826119808119808119870119870



1198501198501006609100660911982411982410066091006609119894119894 119890119808119808119890 119894119894th subset of119824119824

10066101006610119890119808119808119890 119894119894 119896 119896119896119896 119896119896119896 119896119870119870119824119824

10066091006609119890119808119808119890119896


119850119850asymp

11982411982410066091006609119894119894119890119808119808119890 119894119894th subset of119824119824

asymp119890119808119808119890 119894119894 119896 119896119896119896 119896119896119896 119896119870119870119824119824

asymp119890119808119808119890119896


119896119896 119896 119896119896119896 119896119896119896 119896119870119870119896 119894119894119896119896 119896 119896119896119896 119896119896119896 119896119870119870119904119904119894119894119896119896119896119896119896


119896119896 119896 119896119896119896 119896119896119896 119896119870119870119896 119894119894119896119896 119896 119896119896119896 119896119896119896 119896119870119870119904119904119894119894119896119896119896119896119896


11989911989910066091006609119894119894119896119896119896119896 119890119808119808119890 Entry of 119821119821

10066101006610119890119808119808119890 119899119899

asymp119894119894119896119896119896119896119890119808119808119890 119896 119808+ and

119896119896 119896 119896119896119896 119896119896119896 119896119870119870119896 119894119894119896119896 119896 119896119896119896 119896119896119896 119896119870119870119904119904119894119894119896119896119896119896119896





11982111982110066101006610119890119808119808119890


119890120590120590119896119896119808119808 = 119896119826119826119808119808119896 times 119826119826


Glossary






References

















RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation


























1 1 1

1

1 1 2 4

1 1 3 1

1 1 4 1

1 2 1 3

1 2 2 3

1

2 3 2

1 2 4 3

1 3 1 4

1 3 2 4

1 3 3 1

1 3 4

1

1 4 1 3

1 4 2 3

1 4 3 2

1 4 4 2

2 1 1 1

2

1 2 1

2 1 3 1

2 1 4 1

2 2 1 2

2 2 2 2

2 2 3

2

2 2 4 2

2 3 1 1

2 3 2 1

2 3 3 1

2 3 4 1

2

4 1 2

2 4 2 2

2 4 3 2

2 4 4

2



1 1 1 1

1 1 3 1

1 1 4 1

1 3 3 1

1 3 4 1

2 1 1 1

2 1 2 1

2 1 3 1

2 1 4 1

2 3 1 1

2 3 2 1

2 3 3 1

2 3 4 1

1 2 3 2

1 4 3 2

1 4 4 2

2 2 1 2

2 2 2 2

2 2 3 2

2 2 4 2

2 4 1 2

2 4 2 2

2 4 3 2

2 4 4 2

1 2 1 3

1 2 2 3

1 2 4 3

1 4 1 3

1 4 2 3

1 1 2 4

1 3 1 4

1 3 2 4


119873119873119877119877 = 100762010076201007636100763610076361007636

10076441007644

5 8 7 0 6 0 3 2 4 4 13 8 0 5 0 6 2 2 4 3 25 0 0 3 0 2 2 2 0 1 33 0 1 0 2 0 1 2 0 0 4

100762110076211007637100763710076371007637

10076451007645

(11)



















119873119873119877119877 119894 100762010076201007636100763610076361007636

10076441007644

5 0 3 0 2 0 1 0 2 2 13 8 0 5 0 6 2 2 4 3 25 0 0 3 0 2 2 2 0 1 33 0 1 0 2 0 1 2 0 0 4

100762110076211007637100763710076371007637

10076451007645

119873119873100661110066111

119894 [0 8 4 0 4 0 2 2 2 2]

(12)










asymp 119894119894

denoted by 11987111987110066091006609119894119894and 119871119871



119871119871100660910066091

119894 10076201007620

10076441007644

0 1 1 0 1 0 1 1 1 11 1 0 1 0 1 0 0 0 11 0 1 0 0 0 1 0 0 0

10076211007621

10076451007645(13)


119871119871asymp1

119894

100762010076201007636100763610076361007636100763610076361007636100763610076361007636100763610076361007636100763610076361007636

10076441007644

0 1 0 0 0 1 1 1 1 10 1 0 0 1 0 1 1 1 11 0 1 0 0 0 0 0 1 01 0 0 0 1 0 0 0 1 01 0 1 0 0 0 0 0 0 11 0 0 0 1 0 0 0 0 11 0 1 0 0 0 1 0 0 0

100762110076211007637100763710076371007637100763710076371007637100763710076371007637100763710076371007637100763710076371007637

10076451007645

(14)













6 Case Studies











7 Conclusions










Nomenclature


119887119887119896119896 119826119826119896119896 = 119896119904119904119896119896119896119896 119904119904119896119896119896119896119896 119896 119904119904

119870119870S119896119896119896119896 119896







119808119808119896 times ⋯ times 119826119826119808119808119870119870



1198501198501006609100660911982411982410066091006609119894119894 119890119808119808119890 119894119894th subset of119824119824

10066101006610119890119808119808119890 119894119894 119896 119896119896119896 119896119896119896 119896119870119870119824119824

10066091006609119890119808119808119890119896


119850119850asymp

11982411982410066091006609119894119894119890119808119808119890 119894119894th subset of119824119824

asymp119890119808119808119890 119894119894 119896 119896119896119896 119896119896119896 119896119870119870119824119824

asymp119890119808119808119890119896


119896119896 119896 119896119896119896 119896119896119896 119896119870119870119896 119894119894119896119896 119896 119896119896119896 119896119896119896 119896119870119870119904119904119894119894119896119896119896119896119896


119896119896 119896 119896119896119896 119896119896119896 119896119870119870119896 119894119894119896119896 119896 119896119896119896 119896119896119896 119896119870119870119904119904119894119894119896119896119896119896119896


11989911989910066091006609119894119894119896119896119896119896 119890119808119808119890 Entry of 119821119821

10066101006610119890119808119808119890 119899119899

asymp119894119894119896119896119896119896119890119808119808119890 119896 119808+ and

119896119896 119896 119896119896119896 119896119896119896 119896119870119870119896 119894119894119896119896 119896 119896119896119896 119896119896119896 119896119870119870119904119904119894119894119896119896119896119896119896





11982111982110066101006610119890119808119808119890


119890120590120590119896119896119808119808 = 119896119826119826119808119808119896 times 119826119826


Glossary






References

















RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation



















119873119873119877119877 119894 100762010076201007636100763610076361007636

10076441007644

5 0 3 0 2 0 1 0 2 2 13 8 0 5 0 6 2 2 4 3 25 0 0 3 0 2 2 2 0 1 33 0 1 0 2 0 1 2 0 0 4

100762110076211007637100763710076371007637

10076451007645

119873119873100661110066111

119894 [0 8 4 0 4 0 2 2 2 2]

(12)










asymp 119894119894

denoted by 11987111987110066091006609119894119894and 119871119871



119871119871100660910066091

119894 10076201007620

10076441007644

0 1 1 0 1 0 1 1 1 11 1 0 1 0 1 0 0 0 11 0 1 0 0 0 1 0 0 0

10076211007621

10076451007645(13)


119871119871asymp1

119894

100762010076201007636100763610076361007636100763610076361007636100763610076361007636100763610076361007636100763610076361007636

10076441007644

0 1 0 0 0 1 1 1 1 10 1 0 0 1 0 1 1 1 11 0 1 0 0 0 0 0 1 01 0 0 0 1 0 0 0 1 01 0 1 0 0 0 0 0 0 11 0 0 0 1 0 0 0 0 11 0 1 0 0 0 1 0 0 0

100762110076211007637100763710076371007637100763710076371007637100763710076371007637100763710076371007637100763710076371007637

10076451007645

(14)













6 Case Studies











7 Conclusions










Nomenclature


119887119887119896119896 119826119826119896119896 = 119896119904119904119896119896119896119896 119904119904119896119896119896119896119896 119896 119904119904

119870119870S119896119896119896119896 119896







119808119808119896 times ⋯ times 119826119826119808119808119870119870



1198501198501006609100660911982411982410066091006609119894119894 119890119808119808119890 119894119894th subset of119824119824

10066101006610119890119808119808119890 119894119894 119896 119896119896119896 119896119896119896 119896119870119870119824119824

10066091006609119890119808119808119890119896


119850119850asymp

11982411982410066091006609119894119894119890119808119808119890 119894119894th subset of119824119824

asymp119890119808119808119890 119894119894 119896 119896119896119896 119896119896119896 119896119870119870119824119824

asymp119890119808119808119890119896


119896119896 119896 119896119896119896 119896119896119896 119896119870119870119896 119894119894119896119896 119896 119896119896119896 119896119896119896 119896119870119870119904119904119894119894119896119896119896119896119896


119896119896 119896 119896119896119896 119896119896119896 119896119870119870119896 119894119894119896119896 119896 119896119896119896 119896119896119896 119896119870119870119904119904119894119894119896119896119896119896119896


11989911989910066091006609119894119894119896119896119896119896 119890119808119808119890 Entry of 119821119821

10066101006610119890119808119808119890 119899119899

asymp119894119894119896119896119896119896119890119808119808119890 119896 119808+ and

119896119896 119896 119896119896119896 119896119896119896 119896119870119870119896 119894119894119896119896 119896 119896119896119896 119896119896119896 119896119870119870119904119904119894119894119896119896119896119896119896





11982111982110066101006610119890119808119808119890


119890120590120590119896119896119808119808 = 119896119826119826119808119808119896 times 119826119826


Glossary






References

















RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation













6 Case Studies











7 Conclusions










Nomenclature


119887119887119896119896 119826119826119896119896 = 119896119904119904119896119896119896119896 119904119904119896119896119896119896119896 119896 119904119904

119870119870S119896119896119896119896 119896







119808119808119896 times ⋯ times 119826119826119808119808119870119870



1198501198501006609100660911982411982410066091006609119894119894 119890119808119808119890 119894119894th subset of119824119824

10066101006610119890119808119808119890 119894119894 119896 119896119896119896 119896119896119896 119896119870119870119824119824

10066091006609119890119808119808119890119896


119850119850asymp

11982411982410066091006609119894119894119890119808119808119890 119894119894th subset of119824119824

asymp119890119808119808119890 119894119894 119896 119896119896119896 119896119896119896 119896119870119870119824119824

asymp119890119808119808119890119896


119896119896 119896 119896119896119896 119896119896119896 119896119870119870119896 119894119894119896119896 119896 119896119896119896 119896119896119896 119896119870119870119904119904119894119894119896119896119896119896119896


119896119896 119896 119896119896119896 119896119896119896 119896119870119870119896 119894119894119896119896 119896 119896119896119896 119896119896119896 119896119870119870119904119904119894119894119896119896119896119896119896


11989911989910066091006609119894119894119896119896119896119896 119890119808119808119890 Entry of 119821119821

10066101006610119890119808119808119890 119899119899

asymp119894119894119896119896119896119896119890119808119808119890 119896 119808+ and

119896119896 119896 119896119896119896 119896119896119896 119896119870119870119896 119894119894119896119896 119896 119896119896119896 119896119896119896 119896119870119870119904119904119894119894119896119896119896119896119896





11982111982110066101006610119890119808119808119890


119890120590120590119896119896119808119808 = 119896119826119826119808119808119896 times 119826119826


Glossary






References

















RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation

















Nomenclature


119887119887119896119896 119826119826119896119896 = 119896119904119904119896119896119896119896 119904119904119896119896119896119896119896 119896 119904119904

119870119870S119896119896119896119896 119896







119808119808119896 times ⋯ times 119826119826119808119808119870119870



1198501198501006609100660911982411982410066091006609119894119894 119890119808119808119890 119894119894th subset of119824119824

10066101006610119890119808119808119890 119894119894 119896 119896119896119896 119896119896119896 119896119870119870119824119824

10066091006609119890119808119808119890119896


119850119850asymp

11982411982410066091006609119894119894119890119808119808119890 119894119894th subset of119824119824

asymp119890119808119808119890 119894119894 119896 119896119896119896 119896119896119896 119896119870119870119824119824

asymp119890119808119808119890119896


119896119896 119896 119896119896119896 119896119896119896 119896119870119870119896 119894119894119896119896 119896 119896119896119896 119896119896119896 119896119870119870119904119904119894119894119896119896119896119896119896


119896119896 119896 119896119896119896 119896119896119896 119896119870119870119896 119894119894119896119896 119896 119896119896119896 119896119896119896 119896119870119870119904119904119894119894119896119896119896119896119896


11989911989910066091006609119894119894119896119896119896119896 119890119808119808119890 Entry of 119821119821

10066101006610119890119808119808119890 119899119899

asymp119894119894119896119896119896119896119890119808119808119890 119896 119808+ and

119896119896 119896 119896119896119896 119896119896119896 119896119870119870119896 119894119894119896119896 119896 119896119896119896 119896119896119896 119896119870119870119904119904119894119894119896119896119896119896119896





11982111982110066101006610119890119808119808119890


119890120590120590119896119896119808119808 = 119896119826119826119808119808119896 times 119826119826


Glossary






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









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