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Relevance of control theory to design and maintenance problems in time-variant reliability: The case of

stochastic viability

C. Rougé, Jean-Denis Mathias, G. Deffuant

To cite this version:

C. Rougé, Jean-Denis Mathias, G. Deffuant. Relevance of control theory to design and maintenance

problems in time-variant reliability: The case of stochastic viability. Reliability Engineering and

System Safety, Elsevier, 2014, 132, pp.250-260. �10.1016/j.ress.2014.07.025�. �hal-01118695�

(2)

Relevance of control theory to design and maintenance

problems in time-variant reliability: the case of stochastic

viability

CharlesRougé

1,2

, Jean-Denis Mathias

1

and GuillaumeDeuant

1

July 22,2014

Abstract

Thegoal ofthispaperistwofold: 1)toshowthat time-variantreliabilityandabranchofcontrol

theorycalledstochasticviabilityaddresssimilarproblemswithdierentpointsofview,and2)to

demonstratetherelevanceofconceptsandmethodsfromstochasticviabilityinreliabilityproblems.

Onthe onehand, reliability aims at evaluatingthe probability of failureof a system subjected

to uncertainty and stochasticity. Onthe otherhand, viability aimsat maintaining a controlled

dynamicalsystemwithinasurvivalset. Whenthedynamicalsystemisstochastic,thisworkshows

thata viabilityproblembelongstoaspecicclassofdesignandmaintenanceproblemsintime-

variantreliability. Dynamicprogramming,whichisusedforsolvingMarkovianstochasticviability

problems,thenyieldsthesetofdesignstatesforwhichthereexistsamaintenancestrategywhich

guaranteesreliability withacondencelevel

β

^for ^a^given^period^of^time

T

^. ^Besides,^it ^leads^to

astraightforwardcomputationofthedateoftherstoutcrossing,informingonwhenthesystem

ismostlikelytofail. Weillustratethisapproachwithasimpleexampleofpopulationdynamics,

includingacasewhereload increaseswithtime.

Keywords: Viabilitytheory,Time-variantreliability,Dynamicalsystems,Dynamicprogramming,

Reliabilitykernel,Designandmaintenanceproblems

1

Irstea,URLISCLaboratoired'ingénieriedessystèmescomplexes,24avenuedesLandais-CS20085,63178Aubière

Cedex,F-63172,France

2

Université Laval, Département de génie civil et de génie des eaux, Pavillon Adrien-Pouliot, 1065 avenue dela

Médecine,QuébecG1V0A6,QC,Canada

Author-produced version of the paper published in Reliability Engineering & System Safety, volume 132, December 2014,

pages 250–260. Original paper available from http://www.sciencedirect.com. DOI : 10.1016/j.ress.2014.07.025

(3)

Thispaperconnectstwolines ofresearch, viabilityandreliability,thathaveignoredeach otherupto 2

nowdespitestrongsimilarities. Bothframeworksstudythepotentialforasystemto retaindesirable 3

properties. Theyweredevelopedindierentcontextsandsometimestackledierentspecictechnical 4

or conceptual issues in relation with the same type of problems, which makes their confrontation 5

promising. Inparticular, this work focuses on showing how conceptsand methods comingfrom the 6

so-called stochastic viability framework (Doyen and De Lara, 2010) are applicable to time-variant 7

reliability. Indeed,theyfostertheresolutionofaparticularclassofdesignandmaintenanceproblems, 8

thatthispaperistodescribewithaccuracy.

9

Reliabilitytheoryinitiallycomesfromtheeldofmechanicalandstructuralengineering(Rackwitz, 10

2001)and hasawide rangeof applications, from material science(Mathias and Lemaire,2012)and 11

industrialmaintenance(Rausand,1998)toecology(Naeem,1998),environmentalmanagement(Aliev 12

andKartvelishvili, 1993)and hydrology(Melching, 1992). In these applications, dierent numerical 13

methods enable the estimation of the response surface and the associated probability of a system 14

to be in theso-calledfailure set. Reliability methods provideever-improvingapproximationsofthis 15

probability of failure in cases of growing complexity, and have been perfected and tailored to an 16

increasingnumberofapplications(DitlevsenandMadsen,1996;Rackwitz,2001;Lemaire,2009). Let 17

uscite for instanceMonte Carlomethods, First and SecondOrderReliabilityMethods (FORM and 18

SORM), or response surface approximations. A central concern is often with understanding and 19

modelingthecorrelationsbetweenthedierentvariables.

20

However,manyofthesedevelopmentsdealwithtime-invariantsystems,sincetheyarecarriedout 21

under a single denite period of time. When the system under consideration evolves in time, the 22

reliability problem is referred to astime-variant. The central issue of representing the correlations 23

betweenvariables is then extendedto account for the time correlations of the processesof interest.

24

Theprobabilityofreachingthefailure setduringtheevolutioniscalledthecumulativeprobabilityof 25

failure. Rice'sformula(Rice, 1944),whichcountstheaveragenumberof timesanergodicstationary 26

processcrossesagivenxedlevel,servesasabasisforcomputingthecumulativeprobabilityoffailure 27

intheoutcrossingapproach. This approachis basedonthecomputationand timeintegrationof the 28

outcrossingrate, i.e. the rateat whichthe statereachesthefailure set, e.g. Liand DerKiureghian 29

(1995). It hasbeenappliedto simplecaseswhereanalyticalderivationsaretractable(Guedes Soares 30

and Garbatov, 1998; Sudret, 2008a) or alongside approaches from time-invariant reliability such as 31

FORM(KuschelandRackwitz,2000),ornite elementsmethods(Sudret,2008b).

32

Thus, bridgesexist between the time-variant and -invariant cases. In fact, someoutcrossing al- 33

gorithmsdecompose thetime-variantprobleminto aseriesoftime-invariantones (HagenandTvedt, 34

1991;Andrieu-Renaudet al.,2004;Sudret, 2008a),andconversely,theoutcrossingratehasbeende- 35

ned on variables other than time (Sudret, 2008b). Some cases can even be solved both with the 36

outcrossingrateapproach, andbyhavingtimeasaparameter(Burgazzi, 2008). Otherstudies treat 37

atime-variantproblem likeatime-invariantone,byconsidering timeasaparameter(Petryna et al., 38

2002;Schotanusetal.,2004)orasyetanotherspacevariable(WangandWang,2013),orbytreating 39

anitenumberofdateslikeaseriessystem(SavageandSon,2011).

40

Mostoftheworkscitedinthetwoparagraphsaboveassumeamonotonicdecreaseofperformance 41

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old, but recenttime-variant reliability studies havequestioned its systematicuse, and suggestusing 43

methodsthatdonotrequirethishypothesis(QuigleyandWalls,2011;Targoutzidis,2012;Wangand 44

Wang,2013). Asecondlimitationoftheexistingliterature,linkedwiththeassumptionofamonotonic 45

decreaseinperformancethroughtime,istheideathat maintenanceisthefactofchoosingbetweena 46

limitedset ofoptionswhichessentiallyareequivalentto rejuvenatingthesystem,e.g. GuedesSoares 47

andGarbatov (1998),Kuscheland Rackwitz (2000),Val andStewart (2003). Without amonotonic 48

decreaseofperformance,othertypesofmaintenanceneedtobetakenintoaccount. Besides,thereisno 49

frameworkwithinthetime-variantreliabilityliteraturethatformallyconsidersdesignandmaintenance 50

together. Nevertheless,designandmaintenancearecloselyrelated,sinceasystemshould bedesigned 51

inawaythatallowsforanappropriatemaintenancethroughoutitslifetime.

52

To address these current limits of time-variant reliability, this work uses a stochastic controlled 53

dynamicalsystemformulation. Non-controlleddynamics canbefound in thetime-variantreliability 54

literature(Sørensen et al., 2005; Biondini and Frangopol,2009; Targoutzidis, 2012), and theuse of 55

controlsleadstoageneralformulationfordesignandmaintenanceproblemsbylinkingtheacceptability 56

of a design to the existence of a maintenance strategy such that reliability is guaranteed with a 57

condence

β

^,^i.e.,^such^that^the^cumulativeprobabilityoffailureissmallerthan

1 − β

^.

58

Thelink betweentheinitial congurationof asystemandtheexistenceofstrategiesthat keepit 59

outofafailurestatearecentraltoviabilitytheory(Aubin,1991;Aubinetal.,2011). Thisisacontrol 60

theorythat dealswith controlleddynamic systemsunder stateconstraints,and whose originalfocus 61

ison controlled deterministicsystems. An emphasis is put onnding theviabilitykernel,theset of 62

allinitial stateswhich canbe controlled sothat theirtrajectory ismaintained within theconstraint 63

set at all times. Viability algorithms generally yield both the viability kernel and the associated 64

viablecontrols at once, e.g. Saint-Pierre (1994), Bonneuil (2006), Deuant et al. (2007). Viability 65

tools have been successfully applied to a variety of elds such as nance, robotics, orecology, e.g.

66

DeuantandGilbert(2011). Recentwork hasextendedtheframeworkofviabilitytheoryin discrete 67

timebyconsideringuncertaintiesin thedynamics,leadingto thedenitionofthestochasticviability 68

kernel (De Lara and Doyen, 2008), a set of states for which the respect of the constraints can be 69

guaranteedwithadesiredminimalprobabilityandforadesiredtimeframe. Dynamicprogrammingcan 70

computestochasticviabilitykernelsanddeterminethecontrolstrategythatmaximizestheprobability 71

tomaintainthesystemintheconstraintsetduringthatperiod(DoyenandDeLara,2010). Thisisthe 72

specicdevelopmentwhichapplicabilitytoreliabilityproblemsweproposetodemonstratethroughout 73

thiswork.

74

Thepaperisorganizedasfollows. Section2introducesthenotionof reliabilitykernelto describe 75

atime-variant design problem. Then Section 3 extends this notion to a coupled problem of design 76

andmaintenancethroughacontrolleddynamicalsystemformulation. Afterthat,Section4showshow 77

the framework of viability theory applies to aspecic case of this coupled design and maintenance 78

problem,andsolvesitintheMarkoviancase. Section5proposesanapplicationin ordertoillustrate 79

howdynamicprogrammingcanbeappliedtoareliabilityproblem. ThediscussionofSection6further 80

arguesaboutthe potential of confronting reliability with control theories such asviability. Finally, 81

Section7summarizesthendings.

82

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Thissectionproposesageneralformulationfordesignproblemsintime-variantreliability,whichcomes 84

fromasimilarproblem intime-invariantreliability.

85

2.1 Time-invariant reliability 86

Letus consider a systemand avectorof

n

^random^variables ^X ^which ^represents ^the^system's ^state

87

variablesandtheiruncertainty. Reliabilityisconcernedwiththeperformancefunction

g(

^X

)

^,^and^with

88

theso-calledlimit-state(orfailure)surfacedenedby(DitlevsenandMadsen,1996;Lemaire,2009):

89

g(

^X

) = 0

⁽¹⁾

Thelimit-state surfaceseparatesthefailuredomain

F

^(where

g(

^X

) < 0

⁾^from^the^survival^domain

S

90

(where

g(

^X

) ≥ 0

^). ^The^object^ofreliabilityistodeterminetheprobabilityoffailure

p f

^of^the^system^:

91

p f = P (

^X

∈ F) = P (g(

^X

) < 0)).

⁽²⁾

A diversityof methods havebeendeveloped to computeor approximate the limit-state surfaceand 92

theprobabilityoffailureinthetime-invariantcase(DitlevsenandMadsen,1996;Lemaire,2009).

93

ChoicesregardingthedesignofthesystemmayinuencetherandomvectorXortheperformance 94

function. Without lossof generality, theproblem can be formulatedso these choices only aect the 95

former. Let us representchoices by a xed vector

π

^chosen ⁱⁿ ^a ^space

Π ⊂ R ^m

and

m ∈ N

. Let 96

uscall designthis vector: each designleadsto adistinct random vectorX

(π)

^. ^Then ^the^associated

97

probabilityoffailure

p f (π)

^is:

98

p f (π) = P (g(

^X

(π)) < 0)).

⁽³⁾

Thisworkfocusesonndingvaluesof

π

^such^that^the^system^is^reliable^with^a^condence^level

β

^(i.e.,

99

asignicancelevel

α = 1 − β

^). În ôther^words,^weâre înterestedⁱⁿ ^ndingêlements ^from^the ^setôf

100

designchoices such that reliabilityis achieved with acondence

β

^. ^Letûs întroduce^this ^set âs^the

101

reliabilitykernel,notedRel

π (β)

^and^formally^written ^as^follows:

102

Rel

π (β) = {π ∈ Π|p _f (π) ≤ 1 − β}

⁽⁴⁾

Forinstance,Rel

π (0.99)

îs^the^setôfâvailable^designs^such^that^the^system^hasâ

99%

^chance^of^being

103

inthesurvivalset

S

^. ^Let^us ^now^extend^this^design^problem^to ^thetime-variantcase.

104

2.2 Time-variant reliability 105

Wenowplaceourselvesbetweenaninitialdate

t 0 = 0

^and^nal^date

T

^,^so^that^the^problem^is^studied

106

withinatimeinterval

[0, T ]

^called^the^planning^period. ^Theuncertaintyandstochasticityofthesystem 107

are representedat all dates by the vectorX

(t, π)

^. ^There ^is ^a ^consensus ⁱⁿ ^the reliability literature 108

thatX

(t, π)

âggregatesâ^vectorôf^random^variables^likeⁱⁿ time-invariantviability,aswellasavector 109

of one-dimensional random processes that may be correlated with one another as well as with the 110

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Theseprocessesmayalsobeautocorrelatedin time.

112

Theperformanceofthesystemmayalsoevolvewithtime,andisnownoted

g(t,

^X

(t, π))

^. ^Likewise,

113

thelimit-statesurface

g(t,

^X

(t, π)) = 0

^may^be^dependent^on^time,^and^so^may^the^failure^domain

F(t)

114

(where

g(t,

^X

(t, π)) ≤ 0

⁾^and^the^survival^domain

S(t)

^(where

g(t,

^X

(t, π)) ≥ 0

^).

115

Time-variantreliabilityisconcernedwiththecumulativeprobabilityoffailure

p f (t, π)

^, ^the^proba-

116

bilityof reachingthefailuresetover

[0, t]

^:

117

p f (t, π) = P (∃τ ∈ [0, t],

^X

(τ, π) ∈ F (τ))

⁽⁵⁾

Fromnowon,weshallsimplycallprobabilityoffailure thecumulativeprobabilityoffailure

p f (t, π)

^,

118

because itis the time-variantequivalentof the probability of failure

p f (t)

^from ^equation ^(5). ^Much

119

likeforthetime-invariantcase,thisworkfocusesontheproblemofndingvaluesofthedesignvector 120

π

^such ^that ^the^system ^is ^reliable^over^the ^planning^period

[0, T ]

^with ^a^condence ^level

β

^. ^A ^very

121

similarproblemconsistsinndingelementsfromthereliabilitykernelRel

π (β, T )

^,^dened^as:

122

Rel

π (β, T ) = {π ∈ Π|p _f (T, π) ≤ 1 − β}

⁽⁶⁾

Forinstance,Rel

π (0.99, 100)

îs^the^setôf âvailable^designs^such^that ^the^system^hasâ

99%

^chance^of

123

stayingin thesurvivalset

S(t)

^until^at^least

T = 100

^.

124

Existing time-variant reliability methods aim at nding the value of the probability of failure 125

p f (T, π)

^given ^the ^value ^of

π

^. În ôther ^words, ^they ônly âim ât iteratively testing which values of 126

π

^may ^be acceptable. This is the case for instance for outcrossing(or outcrossing-based) methods 127

(KuschelandRackwitz,2000;Rackwitz,2001;Andrieu-Renaudetal.,2004;Sudret,2008a). Theyare 128

basedonthe outcrossingrate

ν ⁺ (t)

^dened^as^theinstantaneousrateatwhich thesystemleavesthe 129

survivalset (Andrieu-Renaudet al.,2004):

130

ν ⁺ (t) = lim

∆t →0 ^>

P ({g(t,

^X

(t, π)) ≥ 0} ∩ {g(t + ∆t,

^X

(t + ∆t, π)) < 0})

∆t

⁽⁷⁾

Timeintegrationof

ν ⁺ (t)

^throughout^the^planning^period^provides^an^upper^bound^for^theprobability 131

offailure:

132

p f (T, π) ≤ Z T

0 ν ⁺ (t)dt

⁽⁸⁾

andthisinequalitybecomesanequalityundertheassumptionthatfailureoccursonlyonce. Theaim 133

isthentodeterminewhenthetime-dependentsystemcrossesthelimit-state surface.

134

3 A general design and maintenance problem

135

Letusnowexploretheconsequencesofchangingthetime-variantdesignproblemofSection2.2soas 136

toincorporatethe maintenanceof thesystem. Since maintenanceis doneat discretedates, itis the 137

valuesofX

(t, π)

ât^these^dates^that^matter. ^Thus,^we^rstîntroduceâdiscrete-timedynamicalsystem 138

formulation,thenweaddto thisformulationthepossibilitytocontrolthesystem.

139

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Insteadofarepresentation ofthetime evolutionof asystemthroughitscorrelationstructure, usual 141

withinreliabilitytheory,inthisworktimedependenceisexpressedexplicitlyusingadynamicalsystem 142

formulation. Dynamicalsystemsarepresentinthetime-variantreliabilityliterature(Sørensenet al., 143

2005; Targoutzidis, 2012), and including under discrete time formulations(Biondini and Frangopol, 144

2009). In this Section, let us assume that reliability assessmentsfocus on X

(t, π)

^at ^discrete ^dates

145

t = 0, 1, . . . , T − 1, T

^, ^where ^the ^time ^interval ^between^two consecutive dates may not be constant.

146

Then,thetime correlationstructure ofthestochastic processesofinterestmaynotbeentirelygiven 147

bythe discretesequence of randomvectors

(

^X

(0, π),

^X

(1, π), . . . ,

^X

(T − 1, π),

^X

(T, π))

^. ^This ^is ^why

148

weintroduceanotherrandom vectorW

(t)

^which^represents^howuncertaintyandstochasticityaect 149

thesystembetweentwoconsecutivedates 1

: 150

X

(t + 1, π) = f (t, π,

^X

(t, π),

^W

(t))

⁽⁹⁾

where

f

îs^the^dynamic²^. ^Fôllowingâ^convention^from^De^Laraând^Doyen^(2008),ârealizationofthe 151

sequenceW

= (

^W

(0),

^W

(1), . . . ,

^W

(T − 1))

îs^called â^scenario,ândît îsânêlement^from ^the^setôf

152

allscenarios

S

. ThecorrelationstructureofWreectsthetimecorrelationstructureofthestochastic 153

processesof interest. Thus,thevectorsW

(t 1 )

^and^W

(t 2 )

^can^be^correlated^for^any^two^dates

t 1

^and

154

t 2

^. ^Onlyⁱⁿ^the^Markovian^case^are^W

(t 1 )

^and^W

(t 2 )

statisticallyindependentforall

t 1

^and

t 2

^within

155

[0, T ]

^.

156

Throughtheiterativeapplicationequation(9)betweentheinitialdate

0

^and^a^later^date

t

^,^X

(t, π)

157

onlydependsonthedesign

π

^, ^the^initial^value^X

(0, π)

^of^the^random^vector^X

(t, π)

^, ^and^the^scenario

158

W. Furtherassuming,similartothetime-invariantcase(Section2.1),thatX

(0, π)

îsônlyâ^function

159

ofthedesign

π

^,^X

(t, π)

^can^be^writtenâsâ^functionôf^the^following:

160

X

(t, π) = f (t, π,

^W

)

⁽¹⁰⁾

andwecalltrajectorythesequence

(

^X

(0, π),

^X

(1, π), . . . ,

^X

(T − 1, π),

^X

(T, π))

^.

161

Then,theprobabilityoffailureiscomputedusingtheprobabilitydistributionofthescenarios. The 162

denitionsof

p f (t, π)

^from^equation⁽⁵⁾ ^and^Rel

π (β, T )

^from ⁽⁶⁾^apply^to ^the^discrete^time^dynamics

163

fromequation(9),theonlychangebeingthatonlyasetofdiscretedateswithin

[0, T ]

îs^nowôfînterest.

164

3.2 Design and maintenance in time-variant reliability 165

Stillonaplanningperiod

[0, T ]

^,^let^us^now^consider^thepossibilityofactingonthesystemateachof 166

thediscretedatesintroducedinSection3.1. Maintenanceactionsarerepresentedbyavector

u

^called

167

acontrol orcontrolvector. Similartothedesign

π

^,â^control^vectorîsâ^choice^xed^by^theêntity^that

168

maintainsthesystem. Theset ofavailable controlsat agivendiscretedate

t

^is^noted

U (t, π)

^, ^and^it

169

1

Sincethedependenceofthestateontheparameter

π

îsâlready^clearⁱⁿêquation^(9),^W

(t)

^can^be^madeindependent fromtheparameter

π

^,^which^is^why^we^do^not^use^the^notation^W

(t, π)

^.

2

Thedierentdynamicsinthetextwillalwaysbenotedas

f

^,^even^though^they^are^not^the^same ^.

(8)

isasubsetof

R ^q

. Therepresentationofmaintenanceactionsleadsto updatingequation(9)into:

170

X

(t + 1, π) = f (t, π,

^X

(t, π), u(t),

^W

(t))

⁽¹¹⁾

Incontrol theoryterms,thisisastochasticcontrolleddiscrete-timedynamicalsystem.

171

Inthis work,controls areapplied sothat thesystemdoesnotreachthe failureset. Thisis what 172

thereliabilityliteraturecalls preventivemaintenance(Rausand,1998). Toassesshowcontrolsaects 173

reliability, the whole sequence of controls applied at thedates

0, 1, . . . , T − 1

îs ôf înterest. ^We ^call

174

strategythesequence

u(.) = (u(0), u(1), . . . , u(T −1))

^. ^The^set^of^all^strategies^that^can^beimplemented 175

duringthetimeframe

[0, T ]

^is^noted

U(T )

^. ^Notations^involvedⁱⁿ^thisformulationaresummarizedin 176

Table1. Thenotionoftrajectoryintroducedbyequation(10)canbeextendedtoaccountfor

u(.)

^,^so

177

thattheiterativeapplicationofequation(11)betweentheinitialdateandadate

t

^leads^to:

178

X

(t, π) = f (t, π, u(.),

^W

)

⁽¹²⁾

Given the design

π

^, ^the ^strategy

u(.)

ând ^the ^scenario ^W, ^there îs ônly ône ^sequence ôf ^random

179

variables,

(

^X

(0, π),

^X

(1, π), . . . ,

^X

(T − 1, π),

^X

(T, π))

^.

180

Throughthelatterequation(12),theprobabilityoffailurebecomesafunctionof

t

^,

π

^and

u(.)

^:

181

p f (t, π, u(.)) = P (∃τ ∈ [0, t],

^X

(τ, π) ∈ F (τ))

⁽¹³⁾

and we can nowintroduce the reliability kernelof the design problem, in cases where the designer 182

should also be concerned with the system's subsequent maintenance. Then, the goal is to nd a 183

design

π

^such^that ^there ^exists ^a^control ^strategy

u(.)

^that ^guarantees^the^system'sreliabilitywith a 184

condence level

β

^. ^Its reliabilitykernelis thus redened from equation (6) to becomeRel

π,u (β, T)

^,

185

formallydened asfollows:

186

Rel

π,u (β, T ) = {π ∈ Π|∃u(.) ∈ U(T ), p f (T, π, u(.)) ≤ 1 − β}

⁽¹⁴⁾

Equation(6)correspondtothecasewherethereisnomaintenance,whichisequivalenttoconsidering 187

thatonlyonestrategyisavailable,andtherefore,thatitisnecessaryappliedtothesystem. Conversely, 188

therearecaseswherethereisonlymaintenance,andtheproblemistondwhetherthereisanadequate 189

maintenancestrategy: these aremaintenanceproblems. Such casesamounttoconsidering thatthere 190

is only onepossible design (

Π = {π}

^), ^so ^that ^either ^Rel

π,u (β, T ) = ∅

^or^Rel

π,u (β, T ) = Π

^. ^Thus,

191

introducingcontrolsintime-variantreliabilityleadstoacoupleddesignandmaintenanceproblem,and 192

designandmaintenancemustbeconsideredtogether. Twootherimportantremarksmustbemadeat 193

thisstage.

194

First, to call a design

π

^reliable, ît îs ^necessary ^to ^nd ân âdequate ^strategy^, ^but îts ^search îs

195

verychallengingbecauseit is necessaryto choose among multiple optionsat each time step. Thus, 196

searchingastrategyin

U (T )

^means^searching^a^space^of^very^highdimensionality. Existingtime-variant 197

reliabilitymethods ratheraimat computingtheprobabilityof failure

p f (t, π, u(.))

^for ^a^given^design

198

andmaintenancestrategy,thereforetheyarenotsuitedtohandletheproposeddesignandmaintenance 199

problemontheirown. Thegeneralaimofcontroltheoriesistondtheappropriatecontrolsinagiven 200

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X

(t, π) R ⁿ

State(randomvector)atthediscretedate

t

W

(t) R ^p (p ≤ n)

^Eect^of^the^stochastic^processes^between

t

^and

t + 1

W

( R ^p ) ^T

^Scenario,^sequence

(

^W

(0),

^W

(1), . . . ,

^W

(T − 1))

π Π

^Design^(or ^design^choice): ^xed

u U (t, x) ⊂ R ^q

Control(decision): xed

u(.) U (T )

^Strategy^(control^sequence)

Table 1: Vectornotation summary for the general designand maintenance problem in time-variant

reliability: withtherandomvectorsinthetophalfandthedecisionstotake(deterministicvectors)in

thebottomhalf.

situation,whichjustiestheideaofexploringhowaparticularcontroltheory,namelyviabilitytheory, 201

mayhelpndreliabledesigns.

202

Second, thereliability kernel depends heavily on theinformation that is available onthe system 203

ateachdatewhenactionmustbeundertaken. Indeed,informationonX

(t, π)

^may^lead^to^adapt ^the

204

control to that information. Although somereliability studiesuse bayesianupdatingto empirically 205

updatereliabilityestimates,e.g. Hsiaoetal.(2008),QuigleyandWalls(2011),theydonotprovidea 206

formalframeworkto dealwith theissueofinformation. Controltheory didformalizetheimportance 207

ofinformation onthestateof thesystemwhenit comes tocontrollingitso it avoidsfailure(Aubin, 208

1991;Clarkeet al., 1995; Doyenand De Lara,2010; Aubin et al., 2011). Thus, open-loopfeedbacks 209

designatepredetermined controlstrategiesthat areapplied withouttakinginformation collectedat

t

210

onX

(t, π)

^into ^account. Closed-loopfeedbacks indicate, to thecontrary, that the control is chosen 211

basedontheacquisitionofnewknowledgeonX

(t, π)

^. În^fact,^viability^theoryâpplies^to^both^typesôf

212

feedbacks,butthedevelopmentsfromstochasticviabilitythat aretobediscussedinthenextSection 213

useclosed-loopfeedbacks.

214

4 Relevance of stochastic viability in time-variant reliability 215

Therelevanceofstochasticviabilitytosolvedesignandmaintenanceproblemsintime-variantreliabil- 216

ityisnowdemonstrated. Weuseclosed-loopfeedbacks,sothatthecontrol

u

^at^date

t

^also^depends^on

217

informationaboutthesystem. Wewill explorehowthischangesthegeneraldesignandmaintenance 218

problemofSection 3.2, rstin thefullinformationcase inSection 4.1, andthenin themoregeneral 219

partialinformationcaseinSection4.2. Followingthat,Section4.3presentsstochasticviabilityandthe 220

associatedstochasticviabilitykernel,whichitrelatestothereliabilitykerneloftheproblemofSection 221

4.2. This relationship enables the introduction of dynamic programming to solve that problem, as 222

detailedin Section4.4. ThismethodalsoleadstoapproximationsoftheoutcrossingrateasinSection 223

4.5.

224

4.1 Closed-loop feedbacks with full information 225

Fullinformationmeansthatatdate

t

^,^we^have^access^to^complete^knowledge^of^the^state

x(t, π)

^,^which

226

isarealizationoftherandomvariableX

(t, π)

^. ^The^existence^ofclosed-loopfeedbacksmeansthat the 227

choiceof

u

^at^date

t

^depends^on^the^state

x(t, π)

^. ^Within^aclosed-loopformulation,astrategy

u(.)

^(as

228

(10)

introducedin Section3.2)associatesamaintenancedecision

u(t, x(t, π))

^to^each^date

t

^and^state

y

^.

229

SinceweareworkingwithrealizationsratherthanwiththerandomvectorX

(t, π)

^itself,^equation

230

(11)becomes:

231

x(t + 1, π) = f (t, π, x(t, π), u(t, x(t, π)), w(t))

⁽¹⁵⁾

where

w(t)

^represents^therealizationofW

(t)

ⁱⁿêquation^(15). Ît^represents^the^randomnessⁱⁿûpdat-

232

ingthestatefromdate

t

^to

t+1

^,^and^we^also^call^scenario^the^sequence

w(.) = (w(0), w(1), . . . , w(T −1))

^.

233

Let us now introduce

y(t) = (x(t, π), π)

^, â ^vector^which âggregates^the ^stateând ^design ^vectors.

234

Theframeworkofso-calledstochasticviabilitytheory 1

DeLaraandDoyen(2008)andDoyenandDe 235

Lara(2010)focusesonthedynamic of

y(t)

însteadôf^thatôf

x(t, π)

^. ^The^dynamic⁽¹⁵⁾^becomes:

236

y(t + 1) = f (t, y(t), u(t, y(t)), w(t))

⁽¹⁶⁾

Stochasticviabilitythencalls

y(t)

^the^state^vector,^andclosed-loopfeedbacksaredeterminedby

y(t)

^.

237

Yet,ifneither

x

^nor

f

^depend^on^the^design,^setting

y(t) = x(t)

^putsêquation⁽¹⁵⁾ûnder^the^formôf

238

equation(16)(seeSection5).

239

4.2 Closed-loop feedback with partial information 240

In many cases, there is no direct access to the realization

x(t, π)

^. ^In ^this ^paper, ^we ^assume ^that

241

thispartial information is arealization

z(t, π)

^of ^a^known ^random^variable ^Z

(x(t, π), π)

^. ^Under ^this

242

assumption,thedynamicsofthis realization

z(t, π)

^can^be^deduced^from^that ^of^X

(t, π)

^:

243

z(t + 1, π) = f (t, π, z(t, π), u(t, z(t, π)), w(t))

⁽¹⁷⁾

where

u

^depends ^on^the^vector

z(t, π)

^because^we^still^are ⁱⁿ ^theclosed-loop feedbackcase. Thefull 244

information caseof Section 4.1 correspondsto thecase

P [

^Z

(x(t, π), π) = x(t, π)] = 1

^, ^Then, ^setting

245

y(t) = (z(t, π), π)

^yields^the^same^equation^as^(16):

246

y(t + 1) = f (t, y(t), u(t, y(t)), w(t))

⁽¹⁸⁾

where

y(t)

îsâgain^called^the^stateôf ^the^system^fromâ^stochastic^viabilityperspective. Yet again,if 247

z

^and

f

^do^not^explicitly^depend^on

π

^,^then^using

y(t) = z(t)

^turns^equation⁽¹⁷⁾^into ^(16).

248

Workingwiththedynamicsofequation(16)onlymakessenseiftheknowledgeof

y(t) = (z(t, π), π)

249

helpsinassessingthereliabilityofthesystem. Therefore,intheremainderofthisarticle,wealsoas- 250

sumethatitispossibletocomputetheconditionalprobability

P (

^X

(t, π) ∈ S(t)|y(t))

^. ^This^assumption

251

holdsin thefullinformationcasebecausethen,

P (

^X

(t, π) ∈ S(t)|y(t)) = 1

^if

y(t) = (x(t, π), π)

^where

252

x(t, π)

^is^therealizationofX

(t, π)

^,^and

0

^otherwise.

253

Since

π

îs^xed^beforehandând^does^not^dependôn^time, îtîs^given^by^theînitial^state

y 0 = y(0)

^.

254

Asforequation(12),foragiveninitialstate

y 0

^,^a^given^strategy

u(.)

^and^a^given^scenario

w(.)

^,^there^is

255

auniquetrajectory

y(y 0 , u(.), w(.)) = (y(0), y(1), . . . , y(T − 1), y(T ))

^. ^With^these^notations,^and^since

256

probabilitiesthentakeinto accountallthescenarios,wecanthenwritetheprobabilityoffailureasa 257

1

Asamatteroffact,probabilistic maybeamoreaccuratewordthanstochastic.

(11)

functionof

t

^,

y 0

^and

u(.)

^:

258

p f (t, y 0 , u(.)) = P [∃τ ∈ [0, t],

^X

(t, π) ∈ F (τ)|y 0 , u(.)]

⁽¹⁹⁾

Thesearchof reliabledesignscanbeturned into that ofinitial states

y 0

^such^that ^there ^is^a^control

259

strategy

u(.)

^which^guarantees^that^the^system^is^reliable^with^a^condence^level

β

^during^the^planning

260

period. The corresponding reliability kernel, derived from equation (14), is simply noted Rel

(β, T )

261

(insteadofRel

π,u (β, T )

^). ^It ^is^the^following^set:

262

Rel

(β, T ) = {y 0 ∈ Y |∃u(.) ∈ U(T ), p f (T, y 0 , u(.)) ≤ 1 − β}

⁽²⁰⁾

wheretheset

Y

isthestatespace. Inordertocomputethiskernel,letusnowrelateittoamathematical 263

objectfromstochasticviabilitytheory,thestochasticviabilitykernel.

264

4.3 The stochastic viabilitykernel 265

Similartoreliability,stochasticviabilitytheory(DeLaraandDoyen,2008;DoyenandDeLara,2010) 266

focusesontheprobabilityforasystemtostayinthesurvivalsetduringagiventimeframe. Indiscrete 267

time,itfocusesonthetimeevolutionofastatevector,throughagoverningequationthatisnoneother 268

thanequation(16). Stochasticviabilityassumesthatthestatevector

y(t)

îs^knownâtêach^time^step,

269

andthatgiven

y(t)

^,^theprobabilityofbeingin thesurvivalset at

t

^is^also^known.

270

Thus,stochastic viabilityfocusesonaverysimilarproblem tothat describedin Section4.2. One 271

of its central concepts is the so-called stochastic viability kernel, which importance comes from the 272

originaldeterministiccontrolframeworkofviabilitytheory(foraquickoverviewofviabilitytheoryand 273

theviabilitykernel,seeAppendix A). Itisdenedasthesetofallstatesforwhichthereisastrategy 274

suchthat thesystemhasaprobability

β

^or^higher^of^stayingⁱⁿ ^the^survival^set

S(t)

^for^a^given^time

275

horizon

T

^. Ît ^can^be^formally^dened ^by^the^followingêquationⁱⁿ ^which îtîs^noted^Viab

(β, T )

^:

276

Viab

(β, T ) = {y 0 ∈ Y |∃u(.) ∈ U (T ), P (∀t ∈ [0, T ],

^X

(t, π) ∈ S(t)|y 0 , u(.)) ≥ β }

⁽²¹⁾

Stochastic viability is related to the closed-loop reliability problem of Section 4.2 through the 277

remarkthat:

278

p f (T, y 0 , u(.)) = 1 − P (∀t ∈ [0, T ],

^X

(t, π) ∈ S(t)|y 0 , u(.))

⁽²²⁾

Throughequations (20) and (21), the stochastic viability kernel is the reliability kernelof equation 279

(20):

280

Rel

(β, T ) =

^Viab

(β, T )

⁽²³⁾

Yet,byitself,equation(23)doesnotallowforthecomputation ofthereliabilitykernelRel

(β, T)

^.

281

Itsinterestcomes from thefact that there exists adynamicprogrammingalgorithm tocompute the 282

stochasticviabilitykernel.

283

(12)

InthisSection,weareintheMarkoviancase,meaningthatallthe

w(t)

ôfêquation¹⁶ârestatistically 285

independentfromeachother. Then,DoyenandDeLara(2010)establishthattheproblemofndingthe 286

stochasticviabilitykernelcanbesolvedbydynamicprogramming,awidespreadcategoryofrecursive 287

algorithmsdesignedto solvetheproblem backwardsfrom date

T

^to ^the^initial ^date. ^Thus, ^dynamic

288

programmingalsoallowsforsolvingthereliability-viabilityproblemin theMarkoviancase.

289

Letusassumethatthecontinuousboundedsets thatformthestateandcontrolspaceshavebeen 290

discretized,as isthecasein practice. Thestateisthen representedbyanite setof points

y i

^of ^the

291

discretespace

A d

^. ^Likewise,^the^discrete^control^space

U d

^isrepresentedbypointsnoted

u j

^,^and^each

292

controlspace

U _d (t, y i )

îsâ^subsetôf

U d

^. ^Then^the^transition^equation⁽¹⁶⁾^is^given^by^theprobabilities 293

P (f (t, y i , u j , w(t)) = y k )

^,^which ^we^assume^to^be^handilycomputable. AsspeciedinSection 4.2,we 294

alsoneed toassume that atdate

t

^,^we^know ^enough^about^theprobabilitydistribution of X

(t, π)

^to

295

compute

P (

^X

(t, π) ∈ S(t)|y i )

^,^and ^that^we^know^the^value^of

y i

^.

296

Then, DoyenandDeLara(2010)link Viab

(β, T )

^to^a^value^function

V (t, y i )

^that^is^dened ^both

297

byaninitialequationatdate

T

ând^byâ^recursiveêquation. ^Theînitialêquationîsâs^follows:

298

V (T, y i ) = P (

^X

(T, π) ∈ S(T )|y i )

⁽²⁴⁾

whilethelatterreadsforall

t

^between^dates

0

^and

T − 1

^:

299

V (t, y i ) =



 max

u j ∈U _d (t,y i )

X

y k ∈ A d

V (t + 1, y k ). P (f (t, y i , u j , w(t)) = y k )



 . P (

^X

(t, π) ∈ S(t)|y i )

⁽²⁵⁾

DoyenandDe Larademonstratethat Viab

(β, T )

îs^the^setôfâll^states^such^that

V (0, y i ) ≥ β

^. ^Thus,

300

thestates for which there exists areliable strategy

u(.)

^with ^a^condence ^level

β

^are ^also ^given ^by

301

V (0, y i ) ≥ β

^. ^Besides,ânother^result^from ^Doyenând^De ^Lara⁽²⁰¹⁰⁾îs^that ^thecomputationof the 302

valuefunctionyieldsthecontrolstrategy

u ^∗ (.)

^that^minimizes^theprobabilityoffailureforatrajectory 303

startingat astate

y 0

^,^so^that:

304

V (0, y 0 ) = 1 − p f (T, y 0 , u ^∗ (.))

⁽²⁶⁾

Amajoradvantageofthestochasticviabilityapproachisthatthemaintenancecontrolsarenotxed 305

beforehand. Itdynamicallyandsimultaneouslycomputesboththeoptimalmaintenancestrategyand 306

theassociatedprobabilityofviability(or reliability)associatedtothatstrategy.

307

4.5 Approximating the date of rst outcrossing 308

Whereas theoutcrossing rate(equation (7)) is usually integrated overa period of time to yield the 309

probability offailure, thedynamic programmingalgorithm presentedabovedoes notcomputeit di- 310

rectly. Nevertheless,givenaninitial state

y 0

^and^a^strategy

u(.) ∈ U (T )

^, îtîs ûseful^to^knowâround

311

whichdatesthesystemismostlikelytoleavethesurvivalset. Noting

t out

^the^date^at^which^the^system

312

rstleavesthesurvivalset forthersttime, the probability

P (t out = t)

^is ^related^to ^theprobability 313

(13)

offailure

p f (t, y 0 , u(.))

^denedⁱⁿ^equation^(19):

314

p f (t, y 0 , u(.)) =

t

X

τ=0

P (t out = τ )

⁽²⁷⁾

Applyingthelatterequationatdates

t

^and

t − 1

^directly^leads^to:

315

P (t out = t) = p f (t, y 0 , u(.)) − p f (t − 1, y 0 , u(.))

⁽²⁸⁾

Ifwenote

∆(t)

^the^amount^of^time^between^dates

t

^and

t+1

^,^then

P (t out = t)/∆(t)

^is^anapproximation 316

oftherateof therst outcrossing. Computation ofthe probability offailure

p f (t, y 0 , u(.))

^for

t < T

317

andaxedmaintenancestrategy

u(.)

^can^be^achieved^both^by^backward^and ^forwardprogramming, 318

andbothmethodsarepresentedinAppendixB.

319

5 Application

320

InthisSectionweapplythedynamicprogrammingtechniquesfromSection4.4toasimpledynamical 321

model of controlled population growth. The aim throughthis is to demonstrate that

1)

^despite ^the

322

apparentsimplicityoftheequations,complexcontrolstrategiescanandmayhavetobedevised,and 323

2)

^that^dynamicprogrammingisadequatetodoso. Thereadershouldkeepinmindthatthisapplica- 324

tiondoesnotintendto showcasethat dynamicprogrammingismoreprecise orlesscomputationally 325

demanding than other techniques in time-variant reliability; yet,these techniques are not meant to 326

ndappropriatemaintenancestrategies. ThisSectionalsointendstoshowthatdynamicprogramming 327

alsoappliesto thecase, classicalin time-variantreliability, ofaperformancefunction thatdecreases 328

withtime,anditusestheresultsfromSection4.5to computetheoutcrossingrate.

329

5.1 A simple population model 330

WeconsideramodiedversionofasimplemodelofpopulationgrowthintroducedbyAubinandSaint- 331

Pierre (2002). It is discretized and uncertainty is integrated as an additive term to the population 332

variableat each time step. All quantities beingnon dimensional for simplicity, the evolution of the 333

state

x = (a, b)

^reads:

334

( a(t + 1) = a(t) + (a(t)b(t) + w(t))∆t b(t + 1) = b(t) + u(t, a(t), b(t))

(29)

Thisisthefullinformationcasefrom Section4.1. Theinitialstate

x 0

^represents^the^system's^design,

335

sowecanwrite

π = x 0 = (a 0 , b 0 )

^. ^Since^neither^the^dynamic^nor^the^state^vector^depend^explicitly^on

336

x 0

^,êquation⁽²⁹⁾îsûnder^the^form

x(t +1) = f (t, x(t), u(t, x(t)), w(t))

^likeⁱⁿ^equation^(16). ^Therefore,

337

dynamicprogrammingandothercomputationscanbecarriedoutusing

y(t) = x(t)

^,^and^we^shall^keep

338

thenotation

x(t)

^instead

y(t)

^throughout^this^section.

339

Thestatevariablesarethepopulation

a(t)

^and^its^growth^coecient

b(t)

^. ^The^time^interval^between

340

twoconsecutivedatesis constantat

∆t = 1

^. ^The^state^variable

b(t)

îs^controlled ^byâûnique^control

341

variable

u(t, x(t))

^. ^The ^feedback ^rule, ^that îs, ^the ^control ^to ^be âssociated ^to êach ^state, îs ^to ^be

342

(14)

determinedin order to maximizethesystem's reliability. Thecontrolspace is

U = [U min , U max ]

^and

343

representstheinertia in theevolutionofthepopulation. These boundsare takentobe

U min = −0.5

344

and

U max = 0.5

^.

345

Theuncertainty

w(t)

^is^arealizationofaGaussianrandomvariableW

(t)

^of^mean

0

^and^standard

346

deviation

0.25

^. În^the^same^wayâsⁱⁿ^Section^4.4,^weâreⁱⁿ^the^Markovian^case,^so ^that^the^Gaussian

347

randomvariablesW

(t 1 )

^and^W

(t 2 )

^at ^two^dierent^dates

t 1

^and

t 2

^arestatistically independent. In 348

fact,theterm

w(t).∆t

^from ^equation ⁽²⁹⁾^reectsⁱⁿ ^discrete ^time^the^hypothesis ^that ⁱⁿ ^continuous

349

time,thestatevariable

a(t)

^is^disturbed^by^a^white^noise^process.

350

Thesize of thepopulationis constrained,sothat the survivalset is representedby thefollowing 351

performancefunction:

352

g(t, x(t)) = g(t, a(t), b(t)) = (a(t) − 0.2)(c(t) − a(t))

⁽³⁰⁾

sothatthesurvivalsetisdenedby

a(t) ∈ [0.2, c(t)]

^where

c(t)

^is^the^carrying^capacity^of^the^system

353

(

c(t) ≥ 0.2

^). În êcology, ^the ^carrying ^capacity îs ^the ^maximal ^size ôf ^the ^population ^that ^can ^be

354

sustained by the environment it lives in. For a given expression of

g(t, x(t))

ⁱⁿ ^equation ^(30), ^the

355

designproblemisrelatedtoassessingreliabilityatatimehorizon

T

^for^a^given^initial^state

x 0

^. ^Since

356

reliabilityalsodependsonthewaythesystemissubsequentlycontrolledormaintainedthisproblem 357

isadesignandmaintenanceproblemasdescribedin Section3.2.

358

In this study, the state space has been discretized, with resolutions

∆a = 0.01

^and

∆b = 0.05

^,

359

andthe control spaceis likewisediscretized witharesolution

∆u = 0.05

^. ^In ^this^discrete ^space, ^the

360

transitionfunctionbetweentwotimestepswasobtainedbyinterpolatingfromequation(29). Inwhat 361

follows,therelevantrangefor

b

^was^found^to^be

[−1.5, 2.5]

^.

362

5.2 Constant carrying capacity 363

Letusassumethat thecarryingcapacity isconstantat

c = 3

^. ^Then, ^theperformancefunction from 364

equation(30)isunder thefollowingform

g 1

^:

365

g 1 (t, x(t)) = (a(t) − 0.2)(3 − a(t))

⁽³¹⁾

Dynamicprogrammingleadstothestrategy

u(.)

^that^optimizesreliabilityatanyhorizon,andtheonly 366

approximationisthat ofthediscretization. This is showcased for

T = 100

^by ^Figure^1,^which ^shows

367

that only initial states grouped around

b 0 = 0

^have ^a^good reliability. There is, however, asizable 368

reliabilitykernelRel

(0.95, 100)

^, ^which ^shows^which ^initial^states^or^which ^system^designs^lead^to

369

thelowest probabilityoffailure. Suchreliabilitykernelscanalsobecomputedatanyhorizon,which 370

allowsfor observingtheevolutionof Rel

(0.95, T )

^as

T

încreases. Îts^size ^decreases ^very^littleûntilît

371

abruptlyceasestoexistwhen

T

^tops

254

^(Figure^2).

372

This stability of thereliabilitykernelRel

(0.95, T )

âs^the ^horizon încreasesîs ^matched ^by^that ôf

373

the optimal strategy. Whateverthe horizon, the backward sequence of feedback maps

x 7→ u(t, x)

374

fromthenaldate

T

^to^theînitial^dateîs^the^same,ând^whatîs^more,^the^map^becomes^constant^for

375

t ≤ T −10

^. ^It^is^noted

u ^∗

^andrepresentedonFigure3. Foragivenvalueof

a

^,^the^value^of

u ^∗

^increases

376

as

b

^increases. ^Yet^therelationshipbetween

u ^∗

^and

b

îs^dierent^forêach^single^valueôf

a

^,^so ^that^the

377

(15)

Population a

Growth coefficient b

0.5 1 1.5 2 2.5 3

−1.5

−1

−0.5 0 0.5 1 1.5 2 2.5

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 1−p 0.9

f (100,x 0 ,u*(.)) Limit of Rel(0.95,100)

Figure1: Reliabilitywiththeperformancefunction

g 1

ândâ^time^horizonôfâ^hundred^time^steps.

(16)

0.5 1 1.5 2 2.5 3

−1 −1.5 0 −0.5

1 0.5 2 1.5

2.5 0

50 100 150 200 250 300

Growth coefficient b Population a

Time horizon T

Figure 2: Initial states

x 0

^belonging ^to ^Rel

(0.95, T )

^, ^for ^dierent ^values ^of ^the ^horizon

T

^and ^the

performancefunction

g 1

^.

(17)

isimportantto recallthat usualtime-variantreliabilitykernelare notdevisedtoyield such complex 379

maintenancestrategies.

380

Population a

Growth coefficient b

0.5 1 1.5 2 2.5 3

−1.5

−1

−0.5 0 0.5 1 1.5 2 2.5

−0.5

−0.4

−0.3

−0.2

−0.1 0 0.1 0.2 0.3 0.4 0.5

Figure3: Mapof theoptimalcontrols

u ^∗ (x)

^for ^theperformance function

g 1

^,

10

^time ^steps^or^more

fromthehorizon.

Outcrossingratesasapproximatedby

P (t out = t)/∆t

^can^be^estimated ^using ^this^feedback^map.

381

Since

∆t = 1

^,^using^equation⁽²⁸⁾^theoutcrossingratetakestheapproximatevalueof

(λ(1−p f (t, x 0 , u ^∗ (.)))

382

afterlessthan

10

^time^steps,^where

λ ≈ 2×10 ⁻ ⁴

^is^theprobabilityofleavingat

t

conditionalonstaying 383

inthesurvivalsetupto

t − 1

^.

λ

^isindependentontheinitialstate,sothatthedierencesinreliability 384

displayedinFigure1accountfortheprobabilityofleavingthesurvivalset withinthesersttentime 385

steps. After

t = 10

^,^theprobabilityof failureincreasesveryslowly.

386

5.3 Decreasing carrying capacity 387

Letus nowsupposethat the systemperformance decreases overtime. Wechooseasimplemodelof 388

lineardecreasefromSavageandSon(2011)withadiminishingcarryingcapacity

c(t) = 3 − 0.01t

^. ^Let

389

(18)

usnotethisfunction

g 2

^, ^equation⁽³⁰⁾^becomes:

390

g 2 (t, x(t)) = (a(t) − 0.2)(3 − 0.01t − a(t))

⁽³²⁾

Asexpected,thislineardecreaseinperformanceaectsreliability,sothatRel

(0.95, T )

^,^even^though

391

itassumes asimilar shape asfor

g 1

^, ^vanishes^for

T > 54

^(Figure ^4). ^Besides, ^unlike ^for^the ^case^of

392

aconstantcarrying capacity, theoptimal control maps change at each time step. This would make 393

themverydicultto ndifitwerenotfordynamicprogramming.

394

0.5 1 1.5 2 2.5 3

−1.5

−1

−0.5 0.5 0

1 1.5 2

2.5 0

10 20 30 40 50 60

Growth coefficient b Population a

Time horizon T

Figure 4: Initial states

x 0

^belonging ^to ^Rel

(0.95, T )

^, ^for ^dierent ^values ^of ^the ^horizon

T

^and ^the

performancefunction

g 2

^.

Yet,for

t ≤ T −10

^it^seems^that^the^map

u ^∗ (t, .)

^does^not^depend^on^the^horizon

T

^. ^Thus,^the^maps

395

for

T = 100

^and

T = 200

âre îdentical ûntil ^the^date

t = 92

^, ^while ^those ^for

T = 150

^and

T = 200

396

are identical until

t = 145

^. ^This ^makes ^the computation of the outcrossing rates valuescomputed 397

withtheoptimalstrategyfor

T = 200

^applicable^to ^lower^time^horizons. ^No^matter^the^initial ^state,

398

theoutcrossingrateis low after therst tentime stepsthen gradually increasesto peakat

t = 124

399

(Figure 5). Then, it decreases because thedecreasing quantity

(1 − p f (t, x 0 , u ^∗ (.))

ⁱⁿ ^equation ⁽²⁸⁾

400

compensates thegrowth of the probability of leaving the survival set at

t

conditional on staying in 401

ituntil

t − 1

^. ^Like ^for^the^previous^case, ^the^amplitude^of ^theoutcrossingrateafter

t = 10

^depends

402

ontheoddsof leavingthesurvivalsetwithin therst fewtimesteps. The cumulativeprobabilityof 403

failurethroughtimecanbecomputedalongsidetheoutcrossingrate(Figure6).

404

(19)

0 20 40 60 80 100 120 140 160 180 200 10 ⁻⁵

10 ⁻⁴ 10 ⁻³ 10 ⁻² 10 ⁻¹ 10 ⁰

Time

Out−crossing rate

x ₀ =(0.3,0) x 0 =(0.5,0) x 0 =(1.5,0)

Figure5: Evolutionoftheoutcrossingrateundertheperformancefunction

g 2

^.

(20)

0 20 40 60 80 100 120 140 160 180 200 0

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Time Failure probability p f (t,x 0 ,u*(.))

x 0 =(0.3,0) x ₀ =(0.5,0) x 0 =(1.5,0)

Figure6: Evolutionoftheprobabilityoffailureundertheperformancefunction

g 2

^.

(21)

Thelimitations of dynamic programming algorithmsshould be keptin mind. In practice, theycan 406

onlysolvesystemswhichstatespacehasalowdimension. Toohighadimensionleadstotheso-called 407

curseofdimensionality whichdesignatestheexponentialincreaseoftheneededcomputationaltime 408

andmemory. There exist approximationand decomposition algorithms that havebeenused to deal 409

withthe dimensionproblem in dynamic programming,such astheBenders decomposition (Perreira 410

andPinto,1985)or dualapproximations,e.g. Shapiro(2011),buttheirapplicabilitylieswelloutside 411

thescopeofthiswork.

412

Yet,theconfrontationofreliabilitywithcontrol theorylooksverypromising. Inthispaper,itled 413

to the formaldenition of coupled designand maintenance problems without restrictivehypotheses 414

onthe aging of the systemwith time, nor on the eects of maintenance. While stochastic viability 415

wasused,otherbranchesofcontroltheorymayberelevanttosuchdesignandmaintenanceproblems, 416

bothintheclosed-andopen-loopcases. Infactsomeworksareveryclosetoviabilitytheory,suchas 417

theinvarianceframework(Clarkeetal.,1995)orreach-avoidproblemsinprobabilistichybridsystems, 418

e.g.Abateetal.(2008),SummersandLygeros(2010). Theymayhelpintheformulationand/orthe 419

resolutionof time-variantreliabilityproblems. Conversely, reliability maybeinstrumental in solving 420

controlproblems. Forinstance,itisnecessaryto know

P (

^X

(t, y(t)) ∈ S(t))

ⁱⁿ^viability^problems,^but

421

thisknowledgemaynotbeeasytoget. Time-invariantreliabilitymethodsmaythenprovideecient 422

approximationsoftheseprobabilities.

423

Theinterestoftheconfrontationofreliabilityandviabilitydoesnotstopwiththepotentialmethod- 424

ologicaldevelopments. Boththeories,sincetheytackleverysimilarperformanceproblems,havebeen 425

usedin elds concernedwith environmental andresourcesmanagement. Thus, reliabilitytheory has 426

beenusedfor morethanthree decadesforwaterresourcessystems,followingthepioneering work by 427

Hashimotoetal.(1982)latercompletedbyMoyetal.(1986)andKundzewiczandLaski(1995). Ithas 428

alsobeenusedingroundwatermanagement,beitforwaterquantity(Oviedo-Salcedo,2012)orquality 429

(SkaggsandBarry, 1997)issues. Inecology, thedenition of ecosystemfailure byNaeem (1998)has 430

fostereddiscussionsonthelink betweenspeciesredundancyandecosystemreliability(NaeemandLi, 431

1997;Rastetteretal.,1999). Thegoalistoassesswhethertheperformanceofthesystemisconsistent 432

orsatisfactoryoveragiventimeframe,whichcanbeexpressedintermsofwhetherandhowreliability 433

mayreachorbest athresholdvalue. Thestochasticviabilityframework,which usesthewordofvia- 434

bilityinsteadofreliability,hastackledthesametypeofperformanceproblemforsherymanagement 435

(Doyen et al., 2007; De Lara and Martinet, 2009; Doyen et al., 2012), with an emphasis on nding 436

managementoptionsthat maintainboththeshstocksandeconomicbenetsfromshing. Viability 437

hasalso beusedfortheaptlynamed populationviabilityanalysis(De LaraandDoyen, 2008)which 438

triesand assessestheriskthat aspeciesmaygoextinct. Thesimpleexampleofpopulationviability 439

analysis from Section 5showcases how bothstochastic viability and time-variantreliability may be 440

relevanttosimilarproblems.

441

Finally, both viability and reliability have been used to explore other concepts related to the 442

performance of a system. On the one hand, resilience has been dened as the possibility for the 443

systemto recoverandget toasetof statesrobustto uncertaintyafter amajoreventdraggeditinto 444

thefailureset,this robustset beingthestochastic viability kernel(Rougéet al., 2013). Inthesame 445

(22)

of reaching a given stochastic viability kernel within a given time frame after an event. On the 447

other hand, resilience but also vulnerability have been dened alongside reliability as performance 448

indicatorsforwaterresourcessystems(Hashimoto et al.,1982)and further, amethodcomputing all 449

three conceptsusing FORM also exists (Maieret al., 2001). Thus, bringing viabilityand reliability 450

methods together may improve the denition and computation of other related concepts such as 451

resilienceandvulnerability. 452

7 Conclusion

453

Stochasticviabilityandreliabilityhavethesamebroadgoalofcomputingtheprobabilityforasystem 454

tonotviolateitsconstraints. Thisworkusedstochasticviabilitytoproposenewconceptsandmethods 455

intime-variantreliability.

456

Conceptually, similarities between stochastic viability and reliability led to the denition of the 457

reliability kernel to deal with design problems: this is an analog of the stochastic viability kernel.

458

Viabilitybeingabranch ofcontrol theory,thenotionofreliabilitykernelhasbeenextendedto cases 459

whereboththedesignandmaintenanceofthesystemhavetobeconsideredtogether. Thiskernelthen 460

regroupsthepossibledesignssuchthat thesystemcanbemaintainedinthe survivalsetwith ahigh 461

probabilityand forasucient amountof time. Besides, its denitionrelies on aformulationthatis 462

independentfromtheassumptionofamonotonicdecreaseinperformanceovertime,eventhoughthe 463

applicationshowsthattheframeworkiswell-suitedtotacklethesecasesaswell.

464

As for the method, dynamic programmingis applicableto time-variantreliabilityin thespecic 465

casesin whichadesignandmaintenanceproblemisalsoastochasticviabilityproblem, namelywhen 466

theuncertaintyandstochasticityofthesystemcanbesummarizedatdiscretedatesbyavectorcalled 467

the state vector, and when the search of the design is related to that of the initial state. If the 468

uncertainty can be expressed by independent random variables, then dynamic programmingyields 469

boththereliabledesignsandtheadequatemaintenancestrategyforthese designs.

470

Acknowledgements 471

ThisworkhasbeensupportedbyaRégionAuvergne'sscholarship. Theauthorsalsowanttothankan 472

anonymousreviewer, whoseconstructiveremarks onearlier versions of this manuscriptconsiderably 473

improveditscontent.

474

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Relevance of control theory to design and maintenance problems in time-variant reliability: The case of

stochastic viability

C. Rougé, Jean-Denis Mathias, G. Deffuant

To cite this version:

C. Rougé, Jean-Denis Mathias, G. Deffuant. Relevance of control theory to design and maintenance

problems in time-variant reliability: The case of stochastic viability. Reliability Engineering and

System Safety, Elsevier, 2014, 132, pp.250-260. �10.1016/j.ress.2014.07.025�. �hal-01118695�

1,2

1

1

β

T

Author-produced version of the paper published in Reliability Engineering & System Safety, volume 132, December 2014,

pages 250–260. Original paper available from http://www.sciencedirect.com. DOI : 10.1016/j.ress.2014.07.025

β

1 − β

n

g(

)

g(

) = 0

F

g(

) < 0

S

g(

) ≥ 0

p f

p f = P (

∈ F) = P (g(

) < 0)).

π

Π ⊂ R m

m ∈ N

(π)

p f (π)

p f (π) = P (g(

(π)) < 0)).

π

β

α = 1 − β

β

π (β)

π (β) = {π ∈ Π|p f (π) ≤ 1 − β}

π (0.99)

99%

S

t 0 = 0

T

[0, T ]

(t, π)

(t, π)

g(t,

(t, π))

g(t,

(t, π)) = 0

F(t)

g(t,

(t, π)) ≤ 0

S(t)

g(t,

(t, π)) ≥ 0

p f (t, π)

[0, t]

p f (t, π) = P (∃τ ∈ [0, t],

(τ, π) ∈ F (τ))

p f (t, π)

p f (t)

π

[0, T ]

β

π (β, T )

π (β, T ) = {π ∈ Π|p f (T, π) ≤ 1 − β}

π (0.99, 100)

Π ⊂ R ^m

π (β) = {π ∈ Π|p _f (π) ≤ 1 − β}

π (β, T ) = {π ∈ Π|p _f (T, π) ≤ 1 − β}

ν ⁺ (t)

ν ⁺ (t) = lim

∆t →0 ^>

ν ⁺ (t)

ν ⁺ (t)dt