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Reliability assessment of multi-state phased mission system with non-repairable multi-state components
Xiang-Yu Li, Hong-Zhong Huang, Yan-Feng Li, Enrico Zio
To cite this version:
Xiang-Yu Li, Hong-Zhong Huang, Yan-Feng Li, Enrico Zio. Reliability assessment of multi-state
phased mission system with non-repairable multi-state components. Applied Mathematical Modelling,
Elsevier, 2018, 61, pp.181-199. �10.1016/j.apm.2018.04.008�. �hal-01988969�
ContentslistsavailableatScienceDirect
Applied Mathematical Modelling
journalhomepage:www.elsevier.com/locate/apm
Reliability assessment of multi-state phased mission system with non-repairable multi-state components
Xiang-YuLia,b, Hong-Zhong Huanga,∗,Yan-Feng Lia, Enrico Ziob,c
aCenter for System Reliability and Safety, School of Mechanical and Electrical Engineering, University of Electronic Science and Technology of China, 611731 Chengdu, China
bChair System Science and the Energy Challenge, Fondation Électricité de France (EDF), CentraleSupélec, Université Paris Saclay, 91192 Gif-sur-Yvette Cedex, France
cEnergy Department, Politecnico di Milano, Milano, Italy
a r t i c l e i n f o
Article history:
Received 17 December 2017 Revised 12 April 2018 Accepted 18 April 2018 Available online 25 April 2018 Keywords:
Reliability assessment
Multi-state multivalued decision diagram (MMDD)
Phased mission system (PMS) Multi-state system Semi-Markov process Monte Carlo simulation Attitude and Orbit Control System
a b s t r a c t
Phasedmissionsystems(PMSs)likesatellitesandspacecraftperformtheirfunctionsover non-overlappingmissionperiods,called phases. Oneofthe challengesinassessing reli- ability ofPMSscomesfrom consideringthe s-dependence among phases, and thecon- siderationonthemulti-statebehavior ofcomponentsand systemsmakesthe reliability analysiseven more difficult. Toeffectively addressthisproblem, amulti-state multival- ueddecisiondiagramalgorithmforPMSandamulti-statemulti-valueddecisiondiagram modelfor phased missionsystem(PMS-MMDD)methodis developedfor the reliability modellingofnon-repairablemulti-statecomponents.BasedontheSemi-Markovprocess,a Markovrenewalequation-basedmethodisdevelopedtodealwithnon-exponentialmulti- statecomponentsandanumericalmethod,thetrapezoidalintegrationmethod,isusedto computethecomplexintegralsinthepathprobabilityevaluation.Acasestudyofamulti- stateattitudeandorbitcontrolsysteminaspacecraftisanalyzedtoillustratetheproposed PMS-MMDDmodelandtheMarkovrenewalequation-basedevaluationmethod.Theaccu- racyandcomputationefficiencyoftheproposedmethodareverifiedbytheMonteCarlo simulationmethod.
© 2018ElsevierInc.Allrightsreserved.
ACRONYM
AOCS Attitudeandorbitcontrolsystem BDD Binarydecisiondiagram
CTMC ContinuoustimeMarkovchain
FT Faulttree
MMDD Multi-statemultivalueddecisiondiagram PDO Phasedependentoperation
PMS Phasedmissionsystem
PMS-BDD Binarydecisiondiagrammodelforphasedmissionsystem
PMS-MMDD Multi-statemulti-valueddecisiondiagrammodelforphasedmissionsystem SMP Semi-MarkovProcess
∗ Corresponding author.
E-mail address: hzhuang@uestc.edu.cn (H.-Z. Huang).
https://doi.org/10.1016/j.apm.2018.04.008 0307-904X/© 2018 Elsevier Inc. All rights reserved.
1. Introduction
Phasedmission systems(PMSs) aretypical oftheaerospaceindustry.PMSsperforminmultiple,consecutiveandnon- overlapping mission durations, known as phases [1]. In the differentphases, system is required to accomplish different tasks by differentcomponents and system structures in different environments [2]. The degradation process of the sys- tems/componentsofthePMSscanbedescribedintermsoftransitionsamongmulti-states(fromperfectlyworkingtototally failure)based ondiscrete function performancelevels [3,4].Then,the reliabilityofmulti-state phased-mission systemsis definedastheprobabilitythatthesystemresidesinstatesabovethepredefinedfailure thresholdinallphases.Challenges inanalyzing thePMS (evenbinary) aredueto twoaspects [2]: (1)Dependenceamongphases: thecomponentsfailedin one phase will remain in the failure state in the followingphases, in non-repairable PMSs. And the componentstate at beginning of one phase should be equalto its state at theend of last phase; (2) Dynamic behavior among phases:the workingcomponents,thesystemstructures andthefailure criteriaaredifferentduetodifferentmissionrequirements.In multi-state PMS,themulti-state behaviorsrendersmore complicatedthephase dependenceandthedynamicsleadingto thestateexplosionproblem[3].
Mostofthe worksonthe PMSreliability focuson binarystate PMSs,andtheir contributions are mainlyintwo main aspects:(1)improvingcomputationalefficiency[5,6];(2)modellingthePMSwithspecialfeatures,suchasexternal/internal commoncausefailures[7],imperfectcoverage[8]orgeneralizedPMS[9].ThePMSreliabilitymodellingmethodsemployed canbedividedintotwomajorcategories:simulationapproaches[10,11]andanalyticalapproaches[12–16].Comparedtoan- alyticalmethods,thesimulationapproachescandealwithmoregeneralsystemsbutsufferfromhighcomputationalcosts.
Theanalyticalapproachescanbefurthercategorizedintothreetypes:(1)combinatorialmethods,suchasthebinarydeci- siondiagram[13,14](BDD),themultivalueddecisiondiagram (MDD)[6]andthemulti-statemultivalueddecisiondiagram (MMDD)[15]methods.ThesemethodscananalyzethePMSsveryefficiently,especiallyforlargescalesystems.However,the combinatorialmethodscanonlyhandlesystemswiths-independentcomponents.(2)State-spacebasedapproaches,suchas thecontinuoustime Markovchain (CTMC)[17,18],Semi-MarkovProcess (SMP)[19]andPetri-nets[20].State-spacebased approachesareflexible andpowerfulinmodellingthePMSswithcomplexdynamicbehaviors(e.g.,cold standby),such as cold standby,butsufferfromthe state explosionproblem. (3) Modularmethods [21],whichcombined thecombinatorial methodswithstate-spacemodel,mergingtheadvantagesofbothbutlimitedtosomespecialcases.
Inrecentyears,multi-statesystemshavebeenwidelystudied[22–31]todescribemultiplelevelsofworkingperformance orthedegradationprocess ofcomponentsandsystems.The worksonmulti-state systemscanbe categorizedby theirre- search objectives: (1) systemstate probabilities orreliability assessmentmethods, e.g., by the decisiondiagram methods [23],thestochasticprocessmethods[24–26]andtheuniversalgenerationfunctionmethod[27];(2)redundancyoptimiza- tion[28];(3)parametersestimation[29];(4)maintenanceandwarrantyresearch [30],etc.However,alloftheseworksdo notconsiderthemulti-phasedcharacteristicsofPMSs.
Theworkson theanalysisofmulti-statePMSare veryfew.Recently,the MMDDmethodhasbeenapplied toevaluate thereliabilityofmulti-statePMSs[5].TheMMDDmodelsaregeneratedfromthemulti-statefaulttreemodelofeachphase, thenthesystemMMDD modelanddisjointpathsforthesystemaregeneratedbythe generalMMDD manipulationrules.
At last, the CTMC is usedto evaluate the probabilities of thedisjoint paths fromthe system MMDD model.In [5],only repairable MS-PMSsare analyzed, butin practical, manysystems cannot be repaired, such assatellites orthe spacecraft workingintheouterspace.PartofthepathsfromtheMMDDmodelsgeneratedbythegeneralMMDDmanipulationrules may have self-conflict problemsin non-repairable systems. Forexample, one component may be in higher performance statesinlater phasesinthesame pathbecausethey can berepaired, butthesepathsare notavailable innon-repairable PMS.Toaddressthisproblem,thispaperproposesaMMDDalgorithmformulti-statePMSandPMS-MMDDmodelinwhich thedependencyamongphasescanbe fullyconsideredandaddressedinthemodelgeneration processwithoutadditional steps.Furthermore,themoregeneralnon-exponentialdistributionsofthestatetransitiontimes[31,32]areconsidered,and basedontheSemi-MarkovProcess(SMP)[33,34],aMarkovrenewalequation-basedmethodisalsoproposed.Anumerical method,thetrapezoidalintegrationmethod,isusedtoevaluatethepathprobabilities.
Thepaperisorganizedasfollows.Section2introduces asimpleMS-PMSexampleandthe basicconcepts ofPMS-BDD methodandgeneralMMDDmethod.InSection3,phasealgebraforthemulti-statePMSandthePMS-MMDDmodelarepro- posedanddescribedindetail.TheMMDDmodelanddisjointpathsgenerationprocessarealsoshownbythePMSexample inSection2.Section4proposestheMarkovrenewalequation-basedmethodforevaluationofpathprobabilities.InSection 5,apracticalexample,anattitudeandorbitcontrol system(AOCS) ofaspacecraftisusedtoillustratetheproposedPMS- MMDD method.Then,theproposed methodiscomparedto theexistingpathevaluationmethod.Moreover,the proposed methodiscomparedtotheMonteCarlo simulationmethodunderdifferentnumberofsimulationrealizationstoshow its accuracyandcomputationefficiency.Section6givestheconclusionsandfutureworks.
2. MMDDmethodfornon-repairablePMSs 2.1. DescriptionoftheexamplePMS
A simple phased-mission system consistingof three components andthree phasesis used to show the PMS-MMDD methodproposedinthispaper.ThesystemconsistsofthreecomponentsA,BandCandthestatetransitiongraphsforthe
Fig. 1. The state transition graphs for components in the example system.
Fig. 2. The multi-state FT model for the example system.
threecomponentsareshowninFig.1.Fi,j(t)representsthecumulativedistributionfunctionofthestatetransitiontimefrom stateitostatejattimet.
The example PMS consists of three consecutive missions: τ1, τ2 andτ3. Mission τ1 needs componentA in state 2 orcomponent Cin state 1; Mission τ2 needs component A above state 1 orcomponent Bin state 2; Mission τ3 needs componentBabovestate1orcomponentCinstate1.Theentiremissionissuccessfulifallthethreeconsecutivemissions arecompletedsuccessfully.ThesystemstructurefunctionforthisexamplePMSisdescribedas,
=1·2·3=
A1,(2)+C1,(1)
A2,(1)+A2,(2)+B2,(2)
B3,(1)+B3,(2)+C3,(1)
(1) InEq.(1), each element representsthat one componentinone specific phaseandstate, forexample,A1,(2) represents thatcomponent Ais instate 2inphase 1.Accordingto Eq.(1), thesystemmulti-state faulttree (FT)modelisshownin Fig.2.
2.2.BasicconceptsofBDDandMMDD
Inthis section, thebasics of the BDD modeland its extended MMDD model,aswell asthe PMS-BDD model forthe binarynon-repairablePMS,areintroducedindetails.
2.2.1. BDDmodelandPMS-BDDmethod
ABDDmodelisadirectedacyclicgraphbasedontheShannondecomposition.LetfbeaBooleanexpressionforabinary componentA, theBoolean function can be expressedas: f=AfA=1+A¯fA=0.The if-then-else (ite) format isused to expressthedecomposition andtheite format of componentAis: f= ite(A, f1, f0). The entireBDD modelis rooted with two sink nodes,0 and 1, representing the system’sfailure andsuccess, respectively. Each non-sink node is also labeled withtwoedgescalledthe0-edgeand1-edgetorepresenttheBooleanexpressionsofthecomponent’sfailureandsuccess, respectively.
TheBDDmodelisgeneratedrecursivelybytheBDDmanipulationwithallthecomponents’Booleanexpressions, gh=ite(x,G1,G2)ite(y,H1,H2)
=
⎧⎨
⎩
ite(x,G1H1,G1H1) index(x)=index(y) ite(x,G1h,G2h) index(x)<index(y) ite(y,gH1,gH2) index(x)>index(y)
(2)
gandhareBooleanexpressions forcomponentsxandy,respectively.♦representsthelogicexpressions(OR,AND)andthe indexdenotesthepredefinedorderofthevariables[8].
Table I
Phase algebra for the BDD model (1 ≤i < j ≤n ).
A i,(1)•A j,(1)→ A j,(1) A i,(1)+ A j,(1)→ A i,(1)
A ¯i,(1)•A ¯j,(1)→ A ¯i,(1) A ¯i,(1)+ A ¯j,(1)→ A ¯j,(1) A ¯i,(1)•A j,(1)→ 0 A ¯i,(1)+ A j,(1)→ 1
Fig. 3. MMDD models.
ThePMS-BDD methodisproposedin[8]todealwiththedependenceamongphasesinthePMS withtheBDD model.
Assume that component A isa binary components workingin nphases. In the non-repairable binaryPMS, accordingto therelationshipofthesamecomponentindifferentphases, phasealgebraforbinaryPMS isproposedin[8],asshownin TableI.
InTableI,componentAonlyhastwostates:Ai,(1)=1meanscomponentAworksnormalinphaseiandA¯i,(1)=1means thatcomponentAisinfailurestate inphasei.InThephysicalmeaningoftheelementsintheleft columnare:(1)Ai,(1)•
Aj,(1)→Aj,(1),ComponentAisworkinginphaseiandphasejimpliescomponentAworksinphasej.(2)A¯i,(1)A¯j,(1)→A¯i,(1), componentAisfailedinphaseiandphasejimpliescomponentAisfailedinphasei.(3)A¯i,(1)Aj,(1)→0,componentAis failedinphaseiandworksinphasejisnotpossible.
With this phase algebra and the backward phase dependent operation (PDO), two Boolean expressions Ei= ite(A¯i,(1),G1,G2)andEj=ite(A¯j,(1),H1,H2),representingthebehaviorsofcomponentAinphaseiandphasejrespectively, theBDDmanipulationforthenon-repairablePMScanbechangedas,
ite¯
Ai,(1),G1,G2
ite¯
Aj,(1),H1,H2
=ite¯
Aj,(1),EiH1,G2H2
(3)
Withthebackward PDOandthePMS-BDD method,thecancellationofthecommoncomponentscanbe automatically doneinthemodelgenerationprocess,whichcanmakethesystemBDDmodelsmaller.Moreinformationanddetailsabout theBDDmodelandPMS-BDDmethodcanbefoundin[15].
2.2.2. MMDDmodel
MMDD isadirected acyclicgraphextended fromBDD andbasedonShannon’s decompositionforthesymbolicrepre- sentationandmanipulationofmulti-valuedlogicfunctions[1].LiketheiteformatintheBDDoperation,thecaseformatof logicexpressionFforamulti-statecomponentAwithmstatesisdefinedas[15],
F =A1FxA=1+A2FxA=2+· · · +AmFxA=m
=case
A,FxA=1,FxA=2,...,FxA=m
=case(A,F1,F2,...,Fm) (4)
TheMMDDformatofEq.(4)andthebasiceventAi,whichmeansthecomponentisinstatei(1≤i≤m),arerespectively showninFig.3(a)and(b).
WiththebasicMMDDeventforeachstateofeachcomponent,thesystemMMDDmodelcanbegeneratedbytheMMDD manipulationrules,whichisalsoextendedfromtheBDDmanipulationrules,shownas[15],
gh=case(x,G1,. . .Gm)casey(y,H1,. . .Hm)
=
case(x,G
1H1,. . .,GmHm) index(x)=index(y) case(x,G1h,...,Gmh) index(x)<index(y)
case(y,gH1,...,gHm) index(x)>index(y) (5)
wheregandhareBooleanexpressionswithmulti-valuedinputsand♦representsthelogicoperations(OR,AND).Theindex meansthepredefinedorderofthevariables.
InthesystemMMDD model,therearetwo sinknodes‘1and‘0,representingthecomponentbeing/notbeinginone specificstate.Eachpathfromtherootnodetothesinknode‘1representsthecombinationsofcomponents’statesthatthe systemisinthisspecificstate.
