UNIVERSITAT POLITÈCNICA DE CATALUNYA
P og ama de Doc o a :
AUTOMÀTICA, ROBÒTICA I VISIÓ
Tesi Doc o al
CONTRIBUTION TO RELIABLE CONTROL OF DYNAMIC SYSTEMS
Jean Ca lo Salaza Co es
Di ec o s: D . Fa iha Nejja i Akhi-Ela ab i D . Ramon Sa a e Es uch
Ma ç 2018
Abs ac
This hesis p esen s some con ibu ions o he ield o Heal h-Awa e Con ol (HAC) o
dynamic sys ems.
In he i s pa o his hesis, a e iew o he concep s and me hodologies ela ed o eli-
abili y e sus deg ada ion and aul ole an con ol e sus heal h-awa e con ol is p e-
sen ed. Fi s ly, in an a emp o uni y concep s, an o e iew o HAC, deg ada ion, and
eliabili y modeling including some o he mos ele an heo e ical and applied con i-
bu ions is gi en.
Mo eo e , eliabili y modeling is o malized and exempli ied using he s uc u e unc-
ion, Bayesian ne wo ks (BNs) and Dynamic Bayesian ne wo ks (DBNs) as modeling
ools in eliabili y analysis. In addi ion, some Reliabili y Impo ance Measu es (RIMs)
a e p esen ed.
In pa icula , his hesis de elops BNs models o o e all sys em eliabili y analysis by
using Bayesian in e ence echniques. Bayesian ne wo ks a e powe ul ools in sys em e-
liabili y assessmen due o hei lexibili y in modeling he eliabili y s uc u e o complex
sys ems.
Fo he HAC scheme implemen a ion, his hesis p esen s and discusses he in eg a ion
o ac ua o s heal h in o ma ion by means o RIMs and deg ada ion in Model P edic i e
Con ol (MPC) and Linea Quad a ic Regula o algo i hms.
In he p oposed s a egies, he cos unc ion pa ame e s a e uned using RIMs. The
me hodology is able o a oid he occu ence o ca as ophic and incipien aul s by mon-
i o ing he o e all sys em eliabili y.
The p oposed HAC s a egies a e applied o a D inking Wa e Ne wo k (DWN) and a
mul i o o UAV sys em. Mo eo e , a hi d app oach, which uses MPC and es ic s he
deg ada ion o he sys em componen s is applied o a win o o sys em.
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Finally, his hesis p esen s and discusses wo eliabili y in e p e a ions. These in e p e a-
ions, namely ins an aneous and expec ed, di e in he manne how eliabili y is e alua ed
and how i s e olu ion along ime is conside ed. This compa ison is made wi hin a HAC
amewo k and s udies he sys em eliabili y unde bo h app oaches.
Keywo ds: p ognos ics and heal h-managemen , heal h-awa e con ol, eliabili y anal-
ysis, eliabili y impo ance measu es, Bayesian ne wo ks, Dynamic Bayesian Ne wo ks,
model p edic i e con ol, linea quad a ic egula o , d inking wa e ne wo ks, oc o o o
Resum
Aques a esi p esen a algunes con ibucions al camp del con ol basa en la salu dels
componen s "Heal h-Awa e Con ol"(HAC) de sis emes dinàmics.
A la p ime a pa d’aques a esi, es p esen a una e isió dels concep es i me o-dologies e-
laciona s amb la iabili a e sus deg adació, el con ol ole an a allades e sus el HAC.
En p ime lloc, i pe uni ica els concep es, s’in odueixen els concep es de deg adació i
iabili a , models de iabili a i de HAC incloen algunes de les con ibucions eò iques i
aplicades més elle an s.
La esi, a més, el modela ge de la iabili a es o mali za i exempli ica u ili zan la unció
d’es uc u a del sis ema, xa xes bayesianes (BN) i xa xes bayesianes dinàmiques (DBN)
com a eines de modela i anàlisi de la iabili a com ambé p esen a algunes mesu es
d’impo ància de la iabili a (RIMs).
En pa icula , aques a esi desen olupa models de BNs pe a l’anàlisi de la iabili a del
sis ema a a és de l’ús de ècniques d’in e ència bayesiana. Les xa xes bayesianes són
eines pode oses en l’a aluació de la iabili a del sis ema g àcies a la se a lexibili a en el
modela de la iabili a de sis emes complexos.
Pe a la implemen ació de l’esquema de HAC, aques a esi p esen a i discu eix la in e-
g ació de la in o mació sob e la salu i deg adació dels ac uado s mi jançan les RIMs en
algo i mes de con ol p edic iu basa en models (MPC) i con ol lineal quad à ic (LQR).
En les es a ègies p oposades, els pa àme es de la unció de cos s’ajus en u i-li zan
els RIMs. Aques es écniques de con ol iable pe me an millo a la disponibili a i la
segu e a dels sis emes e i an l’apa ició de allades a a és de la inco po ació d’aques a
in o mació de la salu dels componen s en l’algo i me de con ol.
Les es a ègies de HAC p oposades s’apliquen a una xa xa d’aigua po able (DWN) i a
un sis ema UAV mul i o o . A més, un e ce en ocamen en se i la deg adació dels
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ac uado s com a es icció dins l’algo i me de con ol MPC s’aplica a un sis ema ae i a
dos g aus de llibe a (TRMS).
Finalmen , aques a esi ambé p esen a i discu eix dues in e p e acions de la iabili a .
Aques es in e p e acions, nomenades ins an ània iespe ada, di e eixen en la o ma en què
s’a alua la iabili a i com es conside a la se a e olució al lla g del emps. Aques a com-
pa ació es eali za en el ma c del con ol HAC i es udia la iabili a del sis ema en o s dos
en ocamen s.

Resumen
Es a esis p esen a algunas con ibuciones en el campo del con ol basado en la salud de
los componen es “Heal h-Awa e Con ol” (HAC) de sis emas dinámicos.
En la p ime a pa e de es a esis, se p esen a una e isión de los concep os y me odolo-
gías elacionados con la iabilidad e sus deg adación, el con ol ole an e a allos e sus
el HAC. En p ime luga , y pa a uni ica los concep os, se in oducen los concep os de
deg adación y iabilidad, modelos de iabilidad y de HAC incluyendo algunas de las
con ibuciones eó icas y aplicadas más ele an es.
La esis, demás o maliza y ejempli ica el modelado de iabilidad u ilizando la unción
de es uc u a del sis ema, edes bayesianas (BN) y edes bayesianas diná-micas (DBN)
como he amien as de modelado y análisis de iabilidad como ambién p esen a algunas
medidas de impo ancia de la iabilidad (RIMs).
En pa icula , es a esis desa olla modelos de BNs pa a el análisis de la iabilidad del
sis ema a a és del uso de écnicas de in e encia bayesiana. Las edes bayesianas son
he amien as pode osas en la e aluación de la iabilidad del sis ema g acias a su lexibi-
lidad en el modelado de la iabilidad de sis emas complejos.
Pa a la implemen ación del esquema de HAC, es a esis p esen a y discu e la in eg ación
de la in o mación sob e la salud y deg adación de los ac uado es median e las RIMs en
algo i mos de con ol p edic i o basado en modelos (MPC) y del con ol cuad á ico lineal
(LQR).
En las es a egias p opues as, los pa áme os de la unción de cos e se ajus an u ilizando
las RIMs. Es as écnicas de con ol iable pe mi i án mejo a la disponibilidad y la segu-
idad de los sis emas e i ando la apa ición de allos a a és de la inco po ación de la
in o mación de la salud de los componen es en el algo i mo de con ol.
Las es a egias de HAC p opues as se aplican a una ed de agua po able (DWN) y a
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un sis ema UAV mul i o o . Además, un e ce en oque que usa la deg adación de los
ac uado es como es icción en el algo i mo de con ol MPC se aplica a un sis ema aé eo
con dos g ados de libe ad (TRMS).
Finalmen e, es a esis ambién p esen a y discu e dos in e p e aciones de la iabilidad. Es-
as in e p e aciones, llamadas ins an ánea yespe ada, di ie en en la o ma en que se e alúa
la iabilidad y cómo se conside a su e olución a lo la go del iempo. Es a compa ación
se ealiza en el ma co del con ol HAC y es udia la iabilidad del sis ema en ambos en o-
ques.
Acknowledgemen s
This hesis was ca ied ou a he Resea ch Cen e o Supe ision, Sa e y and Au oma ic Con ol
(CS2AC) wi h he inancial suppo o Minis e io de Economía y Compe i i idad om Spanish
Go e nmen h ough he g an BES-2012-059571.
Fi s ly, I wan o exp ess my since e g a i ude o my supe iso s, P o . Fa iha Nejja i and
P o . Ramon Sa a e, I ha e lea ned a lo om hem, and also I had an excellen en i-
onmen o wo king, lea ning and communica ing my ideas. I o e hem my absolu e
hank ulness o hei suppo and he con idence hey ha e pu on me, o hei ideas,
sugges ions, and wo ds o encou agemen whene e I needed hem.
I also wan o hank P o . Vicenç Puig and P o . Joseba Que edo o hei suppo and
o o e ing me he oppo uni y o ca y ou his hesis.
I would like o hank P o . Louise T a é and P o . Ch is ophe Simon o he aluable ime
spen eading and e iewing his disse a ion. I would also like o hank hem, oge he
wi h P o . Te esa Escobe o accep ing o be pa o he hesis examina ion panel.
Du ing he hesis, I ha e had he p i ilege o wo king a he Cen e de Reche che en
Au oma ique de Nancy, whe e I spen some mon hs ca ying ou wo esea ch isi s.
My since e hanks o P o . Didie Theilliol and P o . Philippe Webe o all he ui ul
scien i ic discussions, hei ad ice, and hei cons an good mood.
A Big Thanks o my amily, who despi e he dis ance, has gi en me hei suppo and
wo ds o encou agemen . A special hanks o my mo he Julia, my wi e So ía o hei
con inuous encou agemen and ca e, and my sis e Ma isol o gi en me he suppo
when I needed i he mos .
I would also like o hank my ellows om CS2AC, who ha e con ibu ed o making hese
yea s enjoyable. I especially es ima e he b eak momen s ha ing a co ee o chocola e a
he ba , he Xocomà iques, and he excu sions o he moun ain. A special ecogni ion o
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he adminis a i e s a o hei suppo , hei p o essionalism helped me o make my
de elopmen as a Ph.D. s uden much easie . Mol es g àcies!
Lis o ac onyms and no a ion
Ac onyms
AFTC Ac i e Faul -Tole an Con ol
AFTM Accele a ed Failu e Time Model
AHM Addi i e Haza d model
ANN A i icial Neu al Ne wo k
CBM Condi ion-Based Main enance
cd Cumula i e dis ibu ion unc ion
CIF C i ical Impo ance Fac o
CPT Condi ional P obabili y Table
DBN Dynamic Bayesian Ne wo k
DFR Dec easing-Failu e-Ra e
DIF Diagnos ic Impo ance Fac o
DWN D inking Wa e Ne wo k
FDI Faul De ec ion and Isola ion
FEM Fini e Elemen Me hod
FFNN Feed o wa d Neu al Ne wo k
FTC Faul -Tole an Con ol
FV Fussell-Vesely impo ance measu e
HAC Heal h-Awa e Con ol
HMM Hidden Ma ko Models
IFR Inc easing-Failu e-Ra e
ISE In eg a ed Squa e E o
JCAR Join Cumula i e Ac ua o Reliabili ies index
LMI Linea Ma ix Inequali ies
LP Linea P og aming
LRM Logis ic Reg ession Model
x

Lis o ac onyms and no a ion x i
MC Ma ko chain
MIF Ma ginal Impo ance Fac o
MPC Model P edic i e Con ol
MTBF Mean Time Be ween Failu es
MTTF Mean Time To Failu e
MTTR Mean Time To Repai
NLMPC Nonlinea Model P edic i e Con ol
pd P obabili y densi y unc ion
PFTC Passi e Faul -Tole an Con ol
PHM P ognos ics and Heal h Managemen
P HM P opo ional Haza d model
PIM P opo ional In ensi y Model
QP Quad a ic P og aming
RAW Reliabili y Achie emen Wo h
RIM Reliabili y Impo ance Measu e
RAW Reliabili y Reduc ion Wo h
RScum Cumula i e sys em eliabili y o he ins an aneous eliabili y in e p e a ion
Rk
Scum Cumula i e sys em eliabili y o he expec ed eliabili y in e p e a ion
RUL Remaining Use ul Li e
SIM S uc u al Impo ance Measu e
SOM Sel -O ganizing Map
SSI S ess-S eng h In e e ence model
SVM Suppo Vec o Machine
TRMS Twin-Ro o MIMO Sys em
UAV Unmanned Ae ial Vehicles
Ucum Cumula i e con ol e o
WPHM Weibull P opo ional Haza d Model
No a ion
IβIden i y ma ix o size β×β
[Λ]βBlock column ma ix composed by β×1blocks o ma ices Λ
Lis o ac onyms and no a ion x ii
TΛ
βBlock lowe iangula ma ix composed by β×βblocks o ma ices Λ
βShape pa ame e (Weibull and Gamma dis ibu ion)
h0Baseline ailu e a e
Bm Viscous ic ion coe icien o he main p opelle
uiLowe bound o he con ol inpu o he iac ua o
uiUppe bound o he con ol inpu o he iac ua o
B Viscous ic ion coe icien o he ail p opelle
Dn Componen /sys em s a e is down o ailed
εWeigh ing pa ame e o cos unc ion objec i es
ηScale pa ame e (Weibull dis ibu ions)
k n Ae odynamic o ce coe icien o he main o o o nega i e ω
ΓComple e Gamma unc ion Γ(β)=(β−1)! (Gamma dis ibu ion)
γColumn ec o o eg ession coe icien s
HcCon ol ho izon
HpP edic ion ho izon
[Λ]T
βBlock ow ma ix composed by 1×βblocks o ma ices Λ
αDiagonal weigh ing ma ix o elemen s αi
δDiagonal weigh ing ma ix o elemen s δi
ρDiagonal weigh ing ma ix o elemen s ρi
˜αDiagonal weigh ma ix o elemen s α
˜
δDiagonal weigh ma ix o elemen s δ
˜ρDiagonal weigh ma ix o elemen s ρ
Y e Ou pu se poin
IBVa iable ep esen ing he Bi nbaum measu e
IβIden i y ma ix o size β×β
TΛ
βBlock lowe iangula ma ix composed by β×βblocks o ma ices Λ
ICIF Va iable ep esen ing C i ical Impo ance Fac o
IDIF Va iable ep esen ing Diagnos ic Impo ance Fac o
Ic
DIF Va iable ep esen ing DIF using minimal cu se s
Ip
DIF Va iable ep esen ing DIF using minimal pa h se s
IRAW Va iable ep esen ing Reliabili y Achie emen Wo h
IRRW Va iable ep esen ing Reliabili y Reduc ion Wo h
Lis o ac onyms and no a ion x iii
Jm Momen o ine ia main DC mo o
J Momen o ine ia in ail mo o
k1Inpu cons an o he ail mo o
k2Inpu cons an o he main mo o
kah/ ϕh/ Physical cons an
kchn Cable o ce coe icien o nega i e θh
kchp Cable o ce coe icien o posi i e θ
k hn Ae odynamic o ce coe icien o he ail o o o nega i e ωh
k hp Ae odynamic o ce coe icien o he ail o o o posi i e ωh
k p Ae odynamic o ce coe icien o he main o o o posi i e ω
kgGy oscopic cons an
kmPosi i e cons an
koh Ho izon al ic ion coe icien o he beam subsys em
ko Ve ical ic ion coe icien o he beam subsys em
k Posi i e cons an
k h D ag ic ion coe icien o he ail p opelle
k D ag ic ion coe icien
λiFailu e a e o he i h componen
Lah/a A ma u e induc ance o ail / main mo o
λExponen ial and Gamma dis ibu ion pa ame e , ailu e a e and in e se scale
espec i ely
[Λ]βBlock column ma ix composed by β×1blocks o ma ices Λ
[Λ]α×βBlock ma ix composed by α×βblocks o ma ices Λ
lbLeng h o coun e -weigh beam
lcb Dis ance be ween he coun e weigh and he join
lmLeng h o main pa o he beam
l Leng h o ail pa o he beam
(·)−1In e se o ma ix
(·)TT anspose o ma ix
mbMass o he coun e -weigh beam
mcb Mass o he coun e -weigh
Ckk h minimal cu se
Lis o ac onyms and no a ion xix
TMMission ime
mmMass o main pa o he beam
mms Mass o he main shield
Pss h minimal pa h se
mm Mass o he main DC mo o
m Mass o he ail pa o he beam
m s Mass o he ail shield
µLoca ion pa ame e (Logno mal dis ibu ion)
ΩhAngula eloci y o he TRMS a ound he e ical axis
ωhRo a ional eloci y o he ail o o
Ω Angula eloci y o he TRMS a ound he ho izon al axis
ω Ro a ional eloci y o he main o o
Φcd o he s anda d no mal dis ibu ion (Logno mal dis ibu ion)
piP obabili y o being up
qiP obabili y o being down
P (A) P obabili y o e en A
ψ(·)Co a ia e unc ion
Rah/a A ma u e esis ance o ail / main mo o
RiReliabili y o he i h componen
ms Radius o he main shield
θhRe olu ions pe minu e
Rs(TM)Sys em eliabili y a he end o he mission
RsReliabili y o he sys em
s Radius o he ail shield
TsSampling ime
σScale pa ame e (Logno mal dis ibu ion)
Φ(·)S uc u e unc ion
SRandom a iable ep esen ing he s a e o he sys em
TLi e ime o a de ice
θ0
Equilib ium pi ch angle co esponding o u = 0.2753V
θhYaw angle o he beam
θ Pi ch angle o he beam
Lis o ac onyms and no a ion xx
Time
m Mass o he ail DC mo o
uhInpu ol age o he ail mo o
Up Componen /sys em s a e is up o unc ioning
u Inpu ol age o he main mo o
zRow ec o o co a ia es
Z( )Deg ada ion p ocess
Z h Deg ada ion ailu e h eshold

Chap e 1
In oduc ion
1.1 Backg ound and mo i a ion
1.1.1 Deg ada ion s. Reliabili y
The deg ada ion o physical componen s in enginee ing sys ems is gene ally ine i able,
in iew o ac o s such as wea due o usage, aging o he ma e ials and hos ile en i on-
men al condi ions. In pa icula , he deg ada ion o ac ua o s in a closed-loop con ol
sys em can lead o poo pe o mance and some imes o a loss o con ollabili y when he
le el o deg ada ion inc eases and he ac ua o educes i s capabili ies such as speed e-
sponse, o ce, p essu e, s eng h, e c; o becomes p one o aul s when i s deg ada ion
le el eaches o goes beyond a ce ain sa e le el, known also as ailu e h eshold o c i i-
cal alue [185].
The e exis wo ypes o deg ada ion: na u al and o ced deg ada ion. On he one hand,
na u al deg ada ion is an age- o ime-dependen in e nal p ocess whe e componen s
g adually deg ade, leading o hei ailu e o b eakdown. On he o he hand, he o ced
deg ada ion is a i icially induced by an ex e nal agen , whe e componen loading g ad-
ually inc eases in esponse o an inc eased demand [16, p. 32, 37, 52]. Such deg ada ion
can be cha ac e ized in o h ee ca ego ies: bina y, deg ada ion wi h a ini e numbe o
le els, and deg ada ion wi h an in ini e numbe o le els [105].
The heal h o sys em componen s is o p imo dial impo ance o he sa e y and eliabili y
o he con olled sys em. Reliabili y p edic ion based on deg ada ion modeling can be an
e icien me hod o es ima e he heal h o some highly eliable componen s o sys ems
whe e obse a ions o ailu es a e uncommon. Thus, o a oid ailu es i is impo an
o enhance sys em sa e y by aking in o conside a ion he deg ada ion o componen s
heal h in he con olle design [73, 124].
As aul and ailu e concep s a e used in di e en ields such as eliabili y, sa e y, and
1
Chap e 1. In oduc ion 2
aul - ole an sys ems, in di e en echnological a eas, hei e minological use is no uni-
o m. In his hesis, he de ini ion gi en by [60] is used.
De ini ion 1.1. Faul : A aul is an impe missible de ia ion o a leas one cha ac e is ic
p ope y ( ea u e) o he sys em om he accep able, usual, s anda d condi ion.
De ini ion 1.2. Failu e: A ailu e is a pe manen in e up ion o a sys em abili y o pe -
o m a equi ed unc ion unde speci ied ope a ing condi ions.
Recen ly, he in e es o he esea ch on pe o mance deg ada ion in con ol sys em de-
sign has inc eased [22, 41, 126, 189]. I he design objec i e is s ill o main ain he o iginal
sys em pe o mance, his may o ce he emaining componen s o wo k beyond hei no -
mal se ice le el o compensa e o he handicaps caused by he deg aded ones. The e-
o e, he ade-o be ween achie able pe o mance and a ailable ac ua o capabili y and
hei impo ance o he eliabili y o he sys em should be ca e ully conside ed in all
con ol designs [146].
Sys em componen s can expe ience physical deg ada ion du ing hei ope a ion, and he
se e i y o such deg ada ion is ela ed o he o al ope a ing li e o he componen . In his
con ex , i may be in e es ing o design con olle s ha can exploi he a ailable knowl-
edge o he deg ada ion dynamics o main ain adequa e pe o mance and ex end he
use ul li e o he componen s.
Some assump ions a e usually aken in o accoun in o de o model he deg ada ion o
a componen . Fo ins ance, in [56, 74, 75, 143–146] i is assumed ha he deg ada ion
is p opo ional o he con ol e o o ac ua o s and i modi ies he ailu e a e o each
ac ua o . Mo e accu a e assump ions can also be aken, o ins ance in [139, 141] he
deg ada ion is assumed o be dependen no only on he load bu also on he ime and
en i onmen al condi ions. Cons ain s could be imposed o ensu e ha he cumula i e
deg ada ion will be accep able a he end o he main enance ho izon [124].
Highe le els o a ailabili y and eliabili y a e impo an objec i es o he design o mos
mode n enginee ing sys ems and ecen ly, a g owing in e es o model he eliabili y o
complex indus ial sys ems by means o Bayesian Ne wo ks (BN) has appea ed [174] and
some wo ks on BN and sys em sa e y ha e been de eloped [178, 180, 181].
This can be pe o med by edis ibu ing he con ol e o among he a ailable compo-
nen s o ac ua o s o alle ia e he wo k load and he s ess ac o s on he equipmen wi h
wo s condi ions o a oid hei b eak down. Fo his pu pose, an app op ia e policy
should be de eloped o edis ibu e his e o un il main enance ac ions can be aken.
Such policy could be de ined in e ms o emaining use ul li e, eliabili y, deg ada ion,
s uc u e impo ance, aging, among o he s [10, 74].
The applica ion o BNs o eliabili y s a ed a he end o 90’s. In [163] he au ho s p esen
Chap e 1. In oduc ion 3
he ad an ages o BNs in compa ison wi h Reliabili y Block Diag ams (RBD). In [15] he
au ho s p opose o model a aul ee using a BN.
Reliabili y is he abili y o a sys em o ope a e success ully long enough o comple e i s
assigned mission unde s a ed condi ions. I can be modeled as an exponen ial unc ion
[43, 182], a Weibull unc ion [9, 67] o a Gamma unc ion [84, 95, 112], among o he s.
Reliabili y can also be exp essed as a s ochas ic p ocess [117]. Fo example, i is common
o use Ma ko Chains (MC) o model he eliabili y o componen s [118]. Un o una ely,
in p ac ice he complexi y o he sys em leads o a combina o ial explosion o s a es e-
sul ing in a MC wi h a e y la ge size. The eliabili y in o ma ion ob ained wi h he MC
is p opaga ed o he sys em using a Dynamic Bayesian Ne wo k (DBN) which includes
empo al in o ma ion o calcula e he impac o he componen eliabili y on he sys em
eliabili y [10].
The esea ch in his ield was mainly ocused on imp o ing he main enance me hod-
ologies. Howe e , he g owing impo ance o main enance has gene a ed an inc easing
in e es in he de elopmen and implemen a ion o op imal main enance s a egies o
imp o ing sys em eliabili y, p e en ing he occu ence o sys em ailu es, and educing
main enance cos s o de e io a ed sys ems [171].
The main enance has been done adi ionally based on one o wo concep ions; p e en-
i e main enance o co ec i e main enance. P e en a i e main enance pe o ms egu-
la ly scheduled main enance ac ions o main ain sys em in good condi ions and a oid
ailu es du ing se ice. Co ec i e main enance lea es he sys em in ope a ion un il i
ails and hen akes es o a i e main enance ac ions. In con as , bo h o hem ha e d aw-
backs. On one hand, p e en i e main enance is expensi e and he li e cycle o sys em and
componen s is no maximized. On he o he hand, he co ec i e main enance maximizes
he li e cycle o componen s bu i has isks o damage o o he componen s when ailu es
occu . Whiche e o he app oach ha is aken, unan icipa ed ailu es esul in down ime
o he sys em, and he e o e he e will always be a eac i e main enance needed.
As a esul , he sys em down ime will be as long as he necessa y spa e pa s and pe -
sonnel be a ailable and he ime necessa y o ca y ou he main enance ask. Condi ion-
Based Main enance (CBM) has appea ed as a new main enance me hodology which in-
ol es he eal- ime analysis o sys em condi ion o sys em heal h s a e and based on his
in o ma ion he main enance asks a e pe o med.
By con as o p e en i e and co ec i e main enance app oaches, CBM has he po en ial
Chap e 1. In oduc ion 4
o minimizing sys em ailu es inciden s, educing scheduled main enance asks, maxi-
miza ion o he li e cycle o componen s, and inc emen o sys em a ailabili y. The ech-
nical capabili ies o in e sys em and componen s condi ion in eal- ime om measu e-
men s o he p ocess a e c i ical o he success o a CBM implemen a ion.
The use o new echnologies in he p oduc ion sys ems allows o imp o e p oduc s qual-
i y, educe cos s and inc ease p oduc i i y. Howe e , new echnologies usually in ol e
a high le el o complexi y and his complexi y may esul in mo e ailu e-p one sys ems.
Faul - ole an con ol (FTC) has eme ged as a esponse o his p oblem [23]. The e o e, i
is also possible o implemen aul ole an con ol (FTC) echniques whose objec i e is o
allow sys em unc ioning in spi e o ha ing aul y componen s such as ac ua o s o sen-
so s [190]. Ne e heless, i would be in e es ing no o wai un il a ailu e occu s bu o
an icipa e and p e en hem om happening, especially o a oid incipien aul s which
a e di icul o de ec . The de ec ion o incipien aul s is an open ield o esea ch due o
he ac ha some de ec ion me hods such as obse e s o pa i y equa ions end o ack
he sys em e en when he e is a aul [39]. To achie e such an objec i e a P ognos ics and
Heal h Managemen (PHM) s a egy is commonly used.
The p oblema ic o aul ole an in sys ems has been widely ea ed by se e al au ho s.
Now days i is possible o alk abou Faul De ec ion and Isola ion (FDI) as a ield o
esea ch add essed o he p oblem o de ec ion and localiza ion o aul s. In FTC and FDI
esea ch, wo ks such as [13, 23, 44, 60–62] a e obliga o y e e ence due o he p og ess
hey ha e made.
In ecen yea s, HAC has been eme ging as a new echnique o handle his p oblem. I
consis s in aking p edic i e ac ions o p e en a ailu e occu ence, ins ead o FTC ha
akes ac ions a e aul occu ence.
Hence, his p oblem can be add essed aking in o accoun he sys em o imp o e he
sys em sa e y. HAC can ex end he li e ime o he en i e sys em o componen s and
a oid ailu es un il he nex main enance ask.
1.1.2 P ognos ics and Heal h Managemen
P ognos ics and Heal h-Managemen (PHM) in ol es he applica ion o h ee concep s:
diagnos ics, p ognos ics and heal h managemen . The i s one iden i ies he s a e o
he sys em du ing i s unc ioning, p o iding an accu a e aul de ec ion and isola ion
capabili y wi h low alse ala m a e [122].
PHM p o ides sys em heal h in o ma ion based on he e alua ion o i s eliabili y and/o
i s emaining use ul li e (RUL) which allows o make p edic ion o he ad en o u u e
Chap e 1. In oduc ion 11
Sa e y 167 (2017). Special Sec ion: Applica ions o P obabilis ic G aphical Models in
Dependabili y, Diagnosis and P ognosis, pages 663 –672.
J. C. Salaza , A. Sanjuan, F. Nejja i, and R. Sa a e. “Heal h-Awa e and Faul -Tol-
e an con ol o an oc o o o UAV sys em based on ac ua o eliabili y”. In: To be
submi ed o In e na ional Jou nal o Applied Ma hema ics and Compu e Science (2018).
J. C. Salaza , R. Sa a e, and F. Nejja i. “Heal h-Awa e Con ol: A selec i e e iew
and su ey o cu en de elopmen ”. In: To be submi ed o Annual Re iews in Con ol
(2018).
• Book chap e
J. C. Salaza , P. Webe , F. Nejja i, D. Theilliol, and R. Sa a e. “MPC F amewo k
o Sys em Reliabili y Op imiza ion”. In: Ad anced and In elligen Compu a ions in
Diagnosis and Con ol. Ed. by Z. Kowalczuk. Ad ances in In elligen Sys ems and
Compu ing 386. Sp inge In e na ional Publishing, 2015, pages 161–177.
• In e na ional con e ences
J. C. Salaza , R. Sa a e, and F. Nejja i. “Uppe bound sys em eliabili y assessmen
o heal h-awa e con ol o complex sys ems”. In: P oceedings o he 4 h Eu opean
Con e ence o he P ognos ics and Heal h Managemen Socie y (PHME 2018). U ech ,
The Ne he lands, 2018.
J. C. Salaza , R. Sa a e, F. Nejja i, P. Webe , and D. Theilliol. “Reliabili y compu-
a ion wi hin an MPC heal h-awa e amewo k”. In: P oceedings o he 20 h Wo ld
Cong ess o he In e na ional Fede a ion o Au oma ic Con ol (IFAC 2017). Toulouse,
F ance, 2017, pages 12230–12235.
J. C. Salaza , A. Sanjuan, F. Nejja i, and R. Sa a e. “Heal h-Awa e Con ol o
an oc o o o UAV sys em based on ac ua o eliabili y”. In: P oceedings o he 4 h
In e na ional Con e ence on Con ol, Decision and In o ma ion Technologies (CoDIT’17).
Ba celona, Spain, 2017.
J. C. Salaza , F. Nejja i, R. Sa a e, P. Webe , and D. Theilliol. “Reliabili y impo ance
measu es o a heal h-awa e con ol o d inking wa e ne wo ks”. In: P oceedings o
he 3 d Con e ence on Con ol and Faul -Tole an Sys ems (SysTol’16). Ba celona, Spain,
2016, pages 572–578.
J. C. Salaza , P. Webe , R. Sa a e, D. Theilliol, and F. Nejja i. “MPC design based
on a DBN eliabili y model: Applica ion o d inking wa e ne wo ks”. In: P oceed-
ings o he 9 h IFAC Symposium on Faul De ec ion, Supe ision and Sa e y o Technical
P ocesses (SAFEPROCESS 2015). Vol. 48. 21. Pa is, F ance: IFAC, 2015, pages 688–

Chap e 1. In oduc ion 12
693.
J. C. Salaza , P. Webe , F. Nejja i, D. Theilliol, and R. Sa a e. “MPC F amewo k o
Sys em Reliabili y Op imiza ion”. In: P oceedings o he 12 h Diagnosis o P ocesses
and Sys ems (DPS 2015). Us ka, Poland, 2015, pages 386.
J. C. Salaza , F. Nejja i, and R. Sa a e. “Reliable con ol o a win o o MIMO
sys em using ac ua o heal h moni o ing”. In: P oceedings o he 22nd Medi e anean
Con e ence o Con ol and Au oma ion (MED’14). Pale mo, I aly, 2014, pages 481–486.
• Collabo a ion
A. Solde ila, J. Caye o, J. C. Salaza , D. Ro ondo, and V. Puig. “Con ol o a quad u-
ple ank p ocess using a mixed economic and s anda d MPC”. in: Ac as de las XXXV
Jo nadas de Au omá ica. Valencia, Spain, 2014.Fi s place on he CEA con es 2014.
1.5.1 Resea ch s ays
Du ing he de elopmen o his hesis, wo esea ch s ays we e made in he Cen e de
Reche che en Au oma ique de Nancy a he Uni e si é de Lo aine wi h a du a ion o 4
mon hs each one. The i s one, om May o July 2014 whe e he wo k was conce ned o
he opic o eliable con ol o complex sys ems, he s udy o he ma hema ical modeling
o he deg ada ion/ eliabili y using Bayesian app oaches, and he in es iga ion o he
in eg a ion o his modeling in he con ol algo i hm.
And he second one, om May o July 2015 whe e he wo k was conce ned wi h he opic
o eliable con ol o complex sys ems, he s udy o eliabili y impo ance measu es, he
s udy o he in eg a ion o deg ada ion/ eliabili y model in he con ol algo i hm, and
also some ideas on he in eg a ion o eliabili y wi h linea ma ix inequali ies o compu e
he con olle eedback gain we e de ined.
13
Pa I
Fundamen als
Chap e 2
P ognos ics and Heal h Managemen
This chap e p esen s he concep o P ognos ics and Heal h Managemen
and o e s a e iew o PHM me hodologies based on he con ol echniques
used o implemen an HAC app oach.
2.1 In oduc ion
In indus ial p ocesses o dynamical sys ems heal h s a us o i s componen s such as
ac ua o s o senso s is o p ima y conce n as i s ailu e will lead o immedia e sys em
shu down o loss o pe o mance in e ms o economical cos o p oduc i i y. A well-
managed sys em ha minimizes he isk o ailu e is he e o e desi able in many appli-
ca ions. The capabili y o accu a ely p edic he heal h o sys em componen s (and con-
sequen ly he sys em heal h i sel ) is he key o ensu ing hei dependabili y, a ailabili y,
eliabili y, sa e y, and secu i y.
A sys em is said o be dependable when i is us wo hy enough o ha e con idence on
he se ice ha i gi es. Fo a sys em o be dependable, i mus be a ailable and eady o
use when is needed; eliable, when i is able o p o ide con inui y o se ice while i is
in use; sa e, when i does no ha e a ca as ophic consequence on he en i onmen ; and
secu e, when i is able o p ese e con iden iali y [147].
Thus, he conce n abou p ese ing he heal h o complex sys em has led o he de elop-
men o di e en echniques such as Faul -Tole an Con ol (FTC) and P ognos ics and
Heal h Managemen (PHM).
B ie ly, hose con ol echniques which ha e he capaci y o main ain he o e all sys em
s abili y and a sa is ac o y pe o mance in p esence o aul s a e called FTC. This means
ha a closed-loop sys em is aul ole an i i is able o ole a e componen mal unc ions
while main aining a desi able pe o mance and s abili y p ope ies.
FTC echniques can be classi ied in o h ee g oups:
14
Chap e 2. P ognos ics and Heal h Managemen 15
• Ha dwa e edundancies echniques:
The ha dwa e edundancy echniques y o achie e aul ole ance by aking ad-
an age o he ha dwa e edundancy in he sys em. Thei main ad an age is i s
simplici y bu i implies a cos o edundan ha dwa e and main enance.
• Analy ical edundancy echniques: Passi e Faul Tole an Con ol (PFTC):
The passi e FTC echniques a e con ol laws ha ake in o accoun he aul appea -
ance as a sys em pe u ba ion. The e o e, wi hin de ined ma gins, he con ol law
has buil -in aul - ole an capabili ies, allowing he sys em o ace aul occu ence,
hanks o i s obus ness agains a class o aul s.
The ad an age o his app oach is ha i needs nei he aul diagnosis no con olle
econ igu a ion, bu as i needs o ake in conside a ion all possible aul s o a sys-
em du ing he design s age i p o ides limi ed aul ole ance capabili ies, hus i
canno be gua an eed ha unconside ed aul s can be handled. Mo eo e , i en ails
a loss o pe o mance wi h espec o he nominal case.
• Analy ical edundancy echniques: Ac i e Faul Tole an Con ol (AFTC):
The ac i e FTC echniques eadjus he con ol law based on he in o ma ion p o-
ided by he aul s diagnosis module. Wi h his in o ma ion, some au oma ic ad-
jus men s in he con ol law a e done a e he aul occu ence a emp ing o mee
he con ol objec i es wi h he minimum pe o mance deg ada ion.
Discussions on FTC a e beyond he scope o his hesis and in e es ed eade s a e e e ed
o [13, 190] and he e e ences he ein which e iew he de elopmen s made in his a ea.
PHM is a me hodology aimed a handling he “Sys em Heal h” unde s ood as sys em
eliabili y, emaining use ul li e (RUL), sys em deg ada ion, e c, and based on a eal- ime
moni o ing and incipien aul de ec ion.
The main di e ence be ween bo h me hodologies is ha while FTC is applied when he
aul has occu ed, PMH is applied du ing he whole sys em unc ioning and i s aim is o
a oid o a leas o delay he aul occu ence.
In o he wo ds, FTC echniques do no p o ide an ac i e econ igu a ion o he con ol
law gi en he heal h componen s a e. In so a , as he applica ion o PHM and he de el-
opmen o on-line p ognos ics echniques ha e e ol ed, a new ype o FTC called P oac-
i e FTC, which has wo p ima y objec i es: damage a oidance while ensu ing p ima y
mission success [160].
Chap e 2. P ognos ics and Heal h Managemen 16
P oac i e Faul -Tole an Con ol is also called Heal h-Awa e Con ol (HAC), and i s unc-
ioning is as ollows: gi en p ope on-line p ognos ic in o ma ion o he sys em, he HAC
modi ies he con olle ac ions o eschedules he mission p o ile in o de o p ese e a
high le el o sys em heal h.
HAC echnique e alua es he heal h sys em while pe o ming con ol o e he sys em
in a non- aul y si ua ion. Mo eo e , i a oids aul y scena ios by mi iga ing he heal h
deg ada ion ia app op ia e con ol ac ions conside ing heal h indica o s in he con ol
objec i es. This is done by cons an ly e alua ing he sys em heal h indica o s and making
co ec ions h ough he con ol ac ions based on hose indica o s.
A e iew on me hodologies which combine he use o eliabili y and con ol heo ies,
hei o igins, and hei applica ions, based on a bibliog aphy sea ch ela ed o he opics
in ol ed in HAC will be p esen ed.
Rema k in Figu e 2.1 he pe sis en inc ease in he wo ks ela ed o he opics o eliabili y
and con ol, including aul - ole an and heal h-awa e echniques om 2000 o 2016.
FIGURE 2.1: Amoun o Publica ions E olu ion.
The e o e, i is clea ha he in e es on his ield has been inc easing. Those wo ks a e
classi ied in he ollowing ields: Enginee ing, Compu e Sciences and Ma hema ics, Ma-
e ial Sciences, Chemical Enginee ing and Mul idisciplina y as i is shown in Figu e 2.2.
Fu he mo e, i is wo h o highligh he wo k ha some esea ch communi ies a e doing
in his ield encou aging he de elopmen o new wo k and p omo ing di e en con e -
ences and jou nals. To men ion jus a ew, he e is he P ognos ics and Heal h Manage-
men Socie y, which suppo s a con e ence each yea and main ains a jou nal. Also, he
IEEE Reliabili y Socie y, which suppo s a con e ence and a jou nal on hese opics.

Chap e 2. P ognos ics and Heal h Managemen 17
FIGURE 2.2: Rela ed wo ks by ields.
In addi ion, hese opics a e being add essed in epu ed con e ences, such as he Medi e -
anean Con ol Con e ence (MED), he IFAC Symposium on Faul De ec ion, Supe ision
and Sa e y on Technical P ocess (SAFEPROCESS), he In e na ional Con e ence on Con-
ol and Faul -Tole an Sys ems (Sys ol), he In e na ional Science Con e ence Diagnos ics
o P ocesses and Sys ems (DPS), he In e na ional Con e ence on Con ol, Decision and
In o ma ion echnologies (CoDIT), and he Wo ld Cong ess o he In e na ional Fede a-
ion o Au oma ic Con ol among o he s.
2.2 P ognos ics and Heal h Managemen
The p ognos ics in o ma ion is use ul because i supplies he decision make wi h ade-
qua e in o ma ion abou he expec ed ime o sys em o componen s ailu e and allows o
ake he sui able ac ions o deal wi h hem. Assessing he heal h o a sys em p o ides in-
o ma ion ha can be used o mee se e al c i ical goals: (1) p o iding ad ance wa ning
o ailu es; (2) minimizing unscheduled main enance, ex ending main enance cycles, and
main aining e ec i eness h ough imely epai ac ions; (3) educing he li e-cycle cos o
equipmen by dec easing inspec ion cos s, down ime, and in en o y; and (4) imp o ing
quali ica ion and assis ing in he design and logis ical suppo o ielded and u u e sys-
ems [121]. In his sense, PHM is an eme ging enginee ing discipline ha links s udies o
sys em ailu e o sys em li e cycle managemen [36].
The main aim o PHM is o imp o e sa e y and educe main enance cos . To achie e his
Chap e 2. P ognos ics and Heal h Managemen 18
objec i e some asks, such as sys em moni o ing, ailu e p ognos ics, and RUL compu-
a ion, can be in ol ed. Based on ha , some ac ions like logis ics equi emen s, main e-
nance pe o mance, componen s eplacemen o con olle econ igu a ion, among o h-
e s, should be aken in o de o manage he heal h o he sys em.
The s eps in ol ed in a PHM s a egy a e obse a ion, analysis, and decision making
(see Figu e 2.3). Obse a ion is he s ep whe e he da a is acqui ed and p ocessed. The
second s ep consis s in analyzing he da a and ex ac ing in o ma ion o i , such as heal h
s a e, RUL, o pe o m diagnosis and p ognos ics. And, he hi d s ep consis s in aking
he con enien ac ion based on he analyzed da a o ex end he use ul li e o he sys em
o componen s.
PHM
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























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
Obse a ion(Da a acquisi ion
Da a p ocessing
Analysis
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
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
Heal h assessmen
Diagnosis
P ognos ics
Decision making
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

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
Condi ion-Based Main enance
Logis ic ac ions
Heal h-Awa e Con ol
FIGURE 2.3: S eps in ol ed in a PHM s a egy.
PHM concep has i s o igins in sys em enginee ing and conside s aspec s such as quali y,
eliabili y, and main enance which a e used o p o ide indica ions o anomalies and make
p edic ions o u u e ailu es [40].
Resea ch on PHM me hodologies mo i a ed by he bene i s i b ings has been inc eased
conside ably in he las decade. Fo ins ance, a li e a u e e iew on p ognos ics can be
ound in [122], and a pa icula e iew o da a-d i en me hods o PHM can be ound
in [165], a PHM e iew in manu ac u ing p ocess can be ound in [170]. A e iew on
machine y PHM implemen ing Condi ion-Based Main enance (CBM) which summa izes
he ecen esea ch wi h emphasis on models, algo i hms and echnologies o da a p o-
cessing and main enance decision-making is p esen ed in [64].
In [48] a diagnosis and p ognos ic app oach o powe elec onic d i es and elec ic ma-
chines (AC/DC, DC/DC and DC/AC sys ems) is p esen ed. This app oach inco po a es
a low cos moni o ing o he powe elec onics such as powe MOSFETs and IGBTs. The
p oposed HAC s a egy consis s in educing he pe o mance o he con ol accomplish-
ing he mission wi h educed pe o mance.
P ognos ics s a egies can be implemen ed using di e en echniques. A classi ica ion
Chap e 2. P ognos ics and Heal h Managemen 19
o hem can be ound in [4, 36]. The one gi en in [4] is p esen ed in Figu e 2.4, which
conside s ou ca ego ies: physical-based, da a-d i en, hyb id and expe imen al-based
echniques. These ca ego ies a e explained in de ail below p ima ily based in [4, 36, 64].
FIGURE 2.4: P ognos ics app oaches
Physical-based me hodologies
In he physical-based app oach, he i s p inciples a e used o ob ain accu a e heo e ical
models speci ic o a pa icula ype o componen . In his ca ego y, he e a e he ol-
lowing app oaches: physical model, cumula i e damage, haza d a e and p opo ional
haza d a e, nonlinea dynamics.
The physical models a e used o desc ibe he physics o he sys em and ailu e modes,
such as c ack p opaga ion, wea co osion among o he s. These models combine sys em-
speci ic mechanical knowledge, de ec g ow h equa ions, and condi ion moni o ing o
p o ide be e p ognos ics ou pu . These me hodologies a e mo e accu a e han da a-
d i en ones due ha hey con ain a unc ional mapping o he sys em pa ame e s.
Using an explici model o he sys em and esidual gene a ion me hods such as Kalman
il e , pa ame e es ima ion (o sys em iden i ica ion) and pa i y ela ions, i is possible o
ob ain signals, called esiduals, which a e indica i e o aul p esence in he sys em. The
esiduals a e used o implemen aul de ec ion, isola ion and iden i ica ion echniques.
Chap e 2. P ognos ics and Heal h Managemen 20
Model-based app oaches can be mo e e ec i e han o he model- ee app oaches. How-
e e , a co ec and accu a e model is needed, and an explici ma hema ical model may
no be easible o complex sys ems [64].
The e a e some applica ions o physical models, o ins ance in [90], he au ho s used
he Pa is’ law o model spu gea c ack g ow h om an analysis o he s ess and s ain
ields based on gea oo h load, geome y and ma e ial p ope ies. In [116], he au ho s
p esen ed RUL compu a ion app oach based on a c ack g ow h model and implemen ed
using obse e s and in ensi y s ess measu es. In [87], RUL es ima ion app oach o gea s
based on a igue ee h c ack, gea dynamics, and ac u e models was p oposed.
Physics-based p ognos ics ha e been applied o sys ems in which hei deg ada ion phe-
nomenon can be ma hema ically modeled such as in a gea box p ognos ic module [17].
Physical modeling and pa ame ic iden i ica ion echniques ha e been applied wi h aul
de ec ion an ailu e p edic ion algo i hms in o de o p edic he sys em ime- o- ailu e.
The aul s and ailu e modes a e aced back o physically meaning ul sys em pa ame-
e s, p o iding aluable diagnos ic and p ognos ic in o ma ion.
In [102], a esidual-based ailu e p ognos ic echnique applied o a hyd aulic sys em was
p oposed. The emaining sys em use ul li e is es ima ed based on esidual signals, a
bond g aph model o he sys em dynamics, and he deg ada ion model, which allows
he p edic ion o he u u e heal h s a e o he sys em.
Physical-based p ognos ics app oaches a e e y e ec i e and desc ip i e because sys em
deg ada ion modeling is based on laws o na u e. Rema k ha he accu acy and p ecision
depend on model ideli y [166]. The e a e some disad an ages and limi a ions o his
app oach such as: de eloping a high ideli y model o RUL es ima ion is e y cos ly,
ime-consuming, and compu a ionally cos ly and some imes i canno be ob ained. Also,
i will be componen /sys em speci ic which limi s i s use o o he simila cases. Hence,
some imes he da a-d i en app oach is p e e ably used [36].
Da a-d i en me hodologies
Da a-d i en me hods a e based on he ac ha condi ion moni o ing da a and he ex-
ac ed ea u es a y wi h he de elopmen o ei he he ini ia ion and p opaga ion p o-
cess o he deg ada ion p ocess. These me hods a e use ul when a la ge quan i y o noisy
da a needs o be ans o med in o a logical in o ma ion o es ima e he RUL whose accu-
acy depends on he quan i y and quali y o he da a.
The da a-d i en p ognos ics me hodology is based on s a is ical and lea ning echniques,
mos o which come om he heo y o pa e n ecogni ion. Such app oaches inco po a e
Chap e 2. P ognos ics and Heal h Managemen 27
Robus con ol app oaches a e used o ins ance in [58], whe e he objec i e is o main ain
s abili y and pe o mance o he sys em nea o he desi ed pe o mance in he p esence
o sys em componen aul s, and in ce ain ci cums ances educe he pe o mance e-
qui emen s o achie e he objec i e. In [75] and [74], he au ho s p opose he in eg a ion
o eliabili y and econ igu abili y analysis in an FTC sys em o a ank sys em and ai -
c a model, espec i ely. These wo ks ha e in common he use o componen s eliabili y
as indexes o pe o m he con ol econ igu a ion a e he occu ence o ailu es.
In [73], he au ho s p opose an FTC sys em based on a eedback con olle which gua -
an ees he highes sys em eliabili y. This con olle is syn hesized using linea ma ix
inequali ies (LMIs) and inco po a ing a eliabili y indica o , his eliabili y indica o is
he well known Bi nbaum measu e which indica es hose sys em componen s whose e-
liabili y a e c i ical o he eliabili y o he sys em. No e ha in he men ioned wo ks, he
asse eliabili ies a e modeled using he exponen ial dis ibu ion unc ion.
In [72], he au ho s p opose a econ igu able con ol alloca ion p oblem applied o an
o e -ac ua ed sys em in which he edis ibu ion ac o is de ined in e ms o he ac u-
a o eliabili ies modeled by using he Weibull dis ibu ion unc ion. Then, he con ol
alloca ion p oblem consis s in assigning mo e con ol e o o hose ac ua o s whose e-
liabili ies a e highe and o elie e hose ac ua o s whose eliabili ies a e lowe .
In [10] and [9], a con ol alloca ion p oblem is sol ed by inco po a ing eliabili y im-
po ance measu es o edis ibu e he con ol e o among he a ailable ac ua o s o a
hyd aulic sys em. The eliabili y is modeled using a Weibull dis ibu ion unc ion and i
is compu ed by a Dynamic Bayesian Ne wo k (DBN).
Table 2.2 p esen a lis o applica ion made in his opic. Fo ins ance, in [124] a PHM
scheme using MPC is p esen ed. An applica ion using a ank le el o e -ac ua ed sys em
is p esen ed. The main idea is o manage he ac ua o s deg ada ion o e he main enance
ho izon o achie e he con ol goals using MPC.
In [113] an app oach o design a eliable admissible model ma ching (AMM) aul ole an
con ol (FTC) o LPV sys ems is p oposed. The main idea is o econ igu e he con olle
on-line aking in o accoun changes due o he aul s, main aining a ce ain ac ua o s
eliabili y le el in spi e o he aul s.
In [91] he au ho s p esen a eliable obus acking con olle agains ac ua o aul s
and con ol su ace impai men o ai c a bases in on a mixed linea -quad a ic (LQ)/
H∞pe o mance indexes and mul iobjec i e op imiza ion using linea ma ix inequali-
ies (LMIs). In such wo k, he concep eliable is used o mean us wo hy due o he ac
ha au ho s p oposed a con ol app oach wi h aul ole ance capabili ies unde ac ua-
o s ou ages.

Chap e 2. P ognos ics and Heal h Managemen 28
TABLE 2.2: Applica ion examples o HAC
Applica ions Re e ences
Ai c a Chamseddine, Theilliol, e al. [20], Khelassi, Jiang,
e al. [72], Khelassi, Theilliol, e al. [73, 74], LI,
Zhao, e al. [88], Oca and Puig [113], Salaza , Ne-
jja i, e al. [137], Theilliol, Webe , e al. [162], and
Webe , Boussaid, e al. [175]
Tank le el sys ems Abdel-Geliel, Bad eddin, e al. [1], Bicking, We-
be , e al. [9], Da dinie -Ma on, Hamelin, e al.
[31], G osso, Ocampo, e al. [54], Guenab, Webe ,
e al. [58], Khelassi, Theilliol, e al. [75], Nguyen,
Dieulle, e al. [109–111], Pe ei a, Gal ao, e al. [124,
125], Robles, Puig, e al. [133], Salaza , Nejja i, e
al. [136, 137], Salaza , Webe , e al. [143, 144], and
Webe , Simon, e al. [178]
Elec omechanics sys ems Escobe , Puig, e al. [38], Gina , Ba las, e al. [48],
Gokde e, Chiu, e al. [51], Lee, Kim, e al. [85], and
Tang, Kacp zynski, e al. [159]
In [75] he au ho s p opose he in eg a ion o eliabili y e alua ion in a aul - ole an con-
ol sys em and illus a e wi h a ligh con ol applica ion. The con olle is analyzed wi h
espec o eliabili y equi emen s and i s con ollabili y de ined h ough i s G amian.
The admissible solu ion is p oposed acco ding o eliabili y e alua ion based on ene gy
consump ion unde deg aded unc ional condi ions.
In [132] he au ho s p esen an o e iew o modeling and con ol s a egies including
aul - ole an capabili ies o wind u bines and wa e ene gy de ices. In hese sys ems,
he eliabili y imp o emen is achie ed by a signi ican educ ion o pe iods o null o
e y low powe p oduc ion.
2.5 Conclusions
In his chap e , a li e a u e e iew on P ognos ics and Heal h Managemen , pa icula ly
in he his o ical de elopmen o Heal h-Awa e Con ol me hodologies has been p e-
sen ed. Besides he a emp s o add ess he p oblem o HAC, a lis o applica ions and
con ol echniques used we e gi en.
App oaches o include he sys em heal h in o ma ion in he con olle design has been
p oposed, bu i is s ill an open esea ch ield o explo e. The p oblem o how o include
Chap e 2. P ognos ics and Heal h Managemen 29
he heal h in o ma ion in a sys ema ic way in he con olle design should be u he
in es iga ed.
The in e ac ion be ween p ognos ics and con ol es ablishes a new eedback loop. To un-
de s and he e ec s o his loop in he whole pe o mance o he sys em, a ma hema ical
o mula ion o he p oblem should be conside ed. Due o he combined disc e e-e en /
con inuous na u e o he econ igu a ion/accommoda ion ac ions and he con ol loop,
espec i ely, echniques coming om hyb id sys em heo y could be applied.
The app op ia e heal h indica o o be used o econ igu ing/accommoda ing he con-
olle is also a key issue. Some me hodologies o he designe should be gi en in o de
o acili a e he design o he HAC econ igu a ion/accommoda ion s a egy.
Fu he mo e, model-based p ognos ic app oaches accu acy depends on he a ailabili y
o an accu a e model and da a. Da a-d i en me hods a e p e e ed in he case o quick
es ima ions wi h lowe accu acy. Ne e heless, physics-based app oaches seem o be he
mos adequa e when p ognos ics accu acy is needed and he da a is limi ed.
Chap e 3
Backg ound on eliabili y
In his chap e , a e iew o deg ada ion and eliabili y concep s and hei
modeling app oaches is p esen ed. The esea ch in deg ada ion and eliabil-
i y has been a opic o ele an in e es since hey p o ide an es ima ion o
span li e o sys ems and componen s. The e o e, a e iew o he mos ele-
an o bo h, heo e ical and applied con ibu ions ound in he li e a u e is
gi en.
In Sec ion 3.1, an in oduc ion o eliabili y and deg ada ion is gi en, and he
mos used p obabili y dis ibu ions a e desc ibed. Then, in Sec ion 3.2, an
explana ion abou co a ia es and how hey can be modeled and in eg a ed
in o he p obabili y dis ibu ions is p esen ed (Sec ion 3.2.1). A e , in Sec-
ion 3.2.2, some models o i deg ada ion da a, such as i ing da a o p oba-
bili y dis ibu ion o using eg esso s, a e p esen ed.
3.1 Reliabili y and deg ada ion
Reliabili y analysis o dynamical sys ems helps o p e en ailu es which could be cos ly
and some imes disas ous. In complex sys ems o in sa e y-c i ical sys ems, i is impe -
a i e o iden i y he key componen o he sys em and p e en he sys em ailu e. Gen-
e ally, his is done by implemen ing Condi ioned-Based Main enance (CBM) me hods,
whe e decisions a e suppo ed by he eliabili y analysis in o ma ion.
In he li e a u e, au ho s e e indis inc ly o he concep s o eliabili y, deg ada ion, de e-
io a ion, e c. In gene al e ms, eliabili y leads o he concep o dependabili y, success ul
ope a ion o pe o mance, and he absence o ailu es, whe eas un eliabili y (lack o eli-
abili y) leads o he opposi e [14]. Thus, i is con enien now o gi e a clea de ini ion o
hese concep s.
De ini ion 3.1. Reliabili y is he p obabili y ha an asse will pe o m i s unc ioning co -
ec ly o a speci ied pe iod o ime and unde speci ied ope a ing condi ions [14].
30
Chap e 3. Backg ound on eliabili y 31
Whe eas, deg ada ion can be de ined as:
De ini ion 3.2. Deg ada ion is he he educ ion in pe o mance, eliabili y and li espan o
asse s [52].
Deg ada ion can be iewed as a damage ha he sys em accumula es o e ime and e en-
ually leads o a ailu e when he accumula ed damage eaches a ailu e h eshold.
The condi ion o a componen can be cha ac e ized acco ding o he deg ee o de ail gi en
o he deg ada ion p ocess [105]. I could be cha ac e ized as bina y condi ion (see Fig-
u e 3.1(a)), being equal o 1 i he componen is in i s wo king s a e, i.e., he componen
pe o mance is sa is ac o y o accep able, and 0 o he opposi e si ua ion, whe e he com-
ponen is in he aul y s a e. In his cha ac e iza ion, he componen s a s in he wo king
s a e and changes o he ailed one a e a pe iod o ime ( ailu e ime). This is ep e-
sen ed as a andom a iable because he ime ins an o change om wo king o ailed is
unce ain. An example o cha ac e iza ion is an elec ic bulb whe e i s s a e changes om
wo king o ailed in a e y sho ime which can be assumed o be ins an aneous.
(a) (b)
(c)
FIGURE 3.1: Componen condi ion cha ac e iza ion: (a) bina y s a es, (b)
mul i-s a e wi h ini e s a es, (c) mul i-s a e wi h in ini e s a es.
Chap e 3. Backg ound on eliabili y 32
Also, i could be cha ac e ized wi h a ini e numbe o s a es (see Figu e 3.1(b)) whe e he
condi ion o he componen can assume any alue om he se {1,2, . . . , K}wi h:
• 1 co esponding o componen pe o mance being ully accep able, i.e., he compo-
nen is in he good wo king s a e.
•i,1< i < K co esponding o componen pe o mance being pa ially accep able,
i.e., componen is in a wo king s a e wi h a highe alue o iimplying a highe le el
o deg ada ion and,
•Kco esponding o componen pe o mance being unaccep able, i.e, he compo-
nen is in a aul y s a e.
The ime o ailu e o he componen is gi en by F= in { :Z( ) = K}. An example
o his cha ac e iza ion conside he wea in a i e, whe e no wea co esponds o s a e 1
and comple e wea co esponds o s a e K.
And inally, i could be cha ac e ized wi h an in ini e numbe o le els (see Figu e 3.1(c))
being an ex ension o he abo e case wi h K=∞. He e a highe alue implies a highe
deg ada ion, and he componen ailu e ime is gi en by F= in { :Z( ) = Z h}.
Rema k ha , when he le el o deg ada ion eaches he h eshold Z h he asse ailu e
occu ence inc eases. As a consequence, a ailu e is o en a esul o he e ec o deg a-
da ion.
F om hese wo de ini ions, ema k ha eliabili y declines when asse s deg ade o de-
e io a e. The ailu e h eshold p o ides a link be ween deg ada ion and asse s ailu e,
he e o e, i is possible o use he deg ada ion signals o es ima e he ailu e ime dis ibu-
ion, he RUL, e c. The deg ada ion signals a e ob ained by a p ope deg ada ion model,
which consis s in de eloping a good p obabili y model ha is capable o desc ibing he
deg ada ion p ocess.
Mo e speci ically, eliabili y is he p obabili y ha a sys em o componen will ope a e
p ope ly o a speci ic pe iod o ime unde design ope a ing condi ions (such as empe -
a u e, ol s, e c.) wi hou ailu e. In o he wo ds, eliabili y can be used as a measu e o
he success o he sys em o p o ide i s unc ion adequa ely.
Ma hema ically, eliabili y R( )is he p obabili y ha a sys em will be success ul in he
in e al om ime 0 o ime :
R( ) = P (T > ) ≥0,(3.1)

Chap e 3. Backg ound on eliabili y 33
whe e Tis a andom a iable deno ing he ime- o- ailu e o ailu e ime. Which is he
ime un il he sys em i s en e he down ( ailu e) s a e:
T= in { :sys em s a e =down},(3.2)
And he un eliabili y F( ), which is a measu e o ailu e, is de ined as he p obabili y ha
a sys em will ail by ime :
F( ) = P (T≤ )∀ ≥0,(3.3)
in o he wo ds, F( )is he cumula i e dis ibu ion unc ion (cd ), also called ailu e dis-
ibu ion unc ion.
I he ime- o- ailu e andom a iable Thas a densi y unc ion ( ), hen he eliabili y
unc ion R( )is de ined as [127, p. 10]:
R( ) = Z
0
(x)dx, (3.4)
and he unc ion:
h( ) = ( )
R( )
= ( )
1−F( ),
(3.5)
is called he ailu e o haza d a e [47, p. 25]. The e m h( )d is he p obabili y ha a
de ice a he age o will ail in he ime in e al o ( +d ). The impo ance o he
haza d unc ion lies in ha i indica es he change in he ailu e a e o e he li e o a
popula ion o componen s by plo ing hei haza d unc ions on a single axis.
In eliabili y heo y, se e al ypes o p obabili y dis ibu ions a e used; o example, ex-
ponen ial, Weibull, gamma, logno mal, among o he s. A b ie desc ip ion o some o
hese dis ibu ion unc ions will be gi en below.
The exponen ial dis ibu ion is one o he mos widely used in eliabili y enginee ing
because i is ela i ely easy o handle in pe o ming eliabili y analysis, and many en-
ginee ing i ems exhibi cons an haza d a e du ing hei use ul li e [32]. I s p obabili y
densi y unc ion (pd ) is de ined by
( ) = λe−λ ≥0, λ > 0,(3.6)
whe e λis he dis ibu ion pa ame e which is also known as he cons an ailu e a e.
Chap e 3. Backg ound on eliabili y 34
And he eliabili y unc ion gi en by (3.4) and he exponen ial dis ibu ion (3.6), is de-
ined as:
R( ) = Z∞
λe−λ d
=e−λ .
(3.7)
Reliabili y (3.7) e e s o he p obabili y ha a de ice’s li e ime is la ge han , he p ob-
abili y ha he de ice will su i e beyond ime , o he p obabili y ha he de ice ail
a e ime . Rema k ha R(0) = 1 and R(∞) = 0 and ha eliabili y unc ion is a non
inc easing unc ion o .
F om (3.5) and using (3.6) and (3.7) i is e iden ha , o he exponen ial dis ibu ion, he
ailu e a e unc ion is he ailu e a e:
h( ) = λe−λ
e−λ =λ. (3.8)
The exponen ial dis ibu ion has he memo yless p ope y, which means ha he cu en
eliabili y s a us does no depend on he p e ious one. This si ua ion does no ep esen s
he phenomenon o aging which is e y impo an o eliabili y heo y. In ui i ely, ag-
ing ep esen s an inc ease o ailu e isk as a unc ion o ime in use. To in oduce his
dependency, he ollowing de ini ion o eliabili y can be used [47]:
P (T > ) = R( ) = e−R
0λ( )d .(3.9)
In he case o he Weibull dis ibu ion which is used o ep esen se e al physical phe-
nomena, i s p obabili y densi y unc ion is de ined by [179]:
( ) = β β−1
ηβexp −
ηβ
,(3.10)
whe e βand ηa e he shape and scale pa ame e s, espec i ely.
The popula i y o his dis ibu ion s ands on he ac ha , depending on he pa ame e s,
i may desc ibe bo h inc easing and dec easing ailu e a es.
The gamma dis ibu ion is especially use ul o eliabili y modeling o hose asse li e-
imes which deg ada ion can be explained by he shock accumula ion [47].
The logno mal dis ibu ion is a con inuous p obabili y dis ibu ion o a andom a iable
whose loga i hm is no mally dis ibu ed. The logno mal dis ibu ion is applied o he
desc ip ion o he dispe sion o he componen ailu e a e da a.
Chap e 3. Backg ound on eliabili y 35
Table 3.1 p esen s some o he mos widely dis ibu ion unc ions used in eliabili y he-
o y and summa izes hei pd ,cd , eliabili y unc ions, and haza d a es.
TABLE 3.1: Common used con inuous dis ibu ions.
Dis ibu ion Fo mulae Re e ences
Exponen ial ( ) =λe−λ , ≥0
F( ) =1 −e−λ (3.11)
R( ) =e−λ , ≥0
h( ) =λ
Deloux, Cas anie , e al. [33],
Finkels ein [43], Guenab, We-
be , e al. [58], Khelassi, Theil-
liol, e al. [73–75], Oli ei a
and Yoneyama [115], and Wu,
Wang, e al. [182]
Weibull ( ) =β β−1
ηβexp −
ηβ
F( ) =1 −exp −
ηβ(3.12)
R( ) = exp −
ηβ(3.13)
h( ) =β
η
ηβ−1(3.14)
Bicking, Webe , e al. [9],
G osso, Ocampo, e al. [54],
Jiang and Ja dine [67], Khela-
ssi, Jiang, e al. [72], and Tos-
cano and Lyonne [164]
Gamma ( ) = λβ
Γ (β) β−1e−λ (3.15)
F( ) =λβ
ΓZ
0
xβ−1e−λxdx (3.16)
R( ) = λβ
Γ(β)Z∞
xβ−1e−λxdx (3.17)
h( ) = β−1e−λ
R∞
0xβ−1e−λxdx (3.18)
Çinla [19], Lange on, G all, e
al. [82], Lawless and C owde
[84], Lu and Meeke [95], and
Noo wijk [112]
Logno mal ( ) = 1
σ √2πexp "−(ln −µ)2
2σ2#(3.19)
F( ) =Φ ln −µ
σ(3.20)
R( ) =1 −Φ(ln −µ
σ)(3.21)
h( ) = ( )
1−Φ [(ln −µ)/σ](3.22)
Chen and Zheng [24],
Lange on, G all, e al. [82], and
Meeke , Escoba , e al. [103]
F om he eliabili y heo y, ano he use ul concep s like he Mean Time To Failu e (MTTF)
Chap e 3. Backg ound on eliabili y 36
and A ailabili y will be explained. The MTTF is he expec ed li e o he de ice and i can
be e alua ed h ough he ollowing s anda d equa ion [34]:
MTTF =E(T) = Z∞
0
( )d
=Z∞
0
R( )d .
(3.23)
Fo epai able de ices, he MTTF ep esen s he mean ime o he i s ailu e. A e i is
epai ed and pu in o ope a ion again, he a e age ime o he nex ailu e is indica ed by
he mean ime be ween ailu es (MTBF). Unde pe ec epai s, MTBF is equal o MTTF.
Since he e is usually an aging e ec in mos de ices, MTBF dec eases as mo e ailu es
a e expe ienced by he de ice. The ime needed o pe o m a epai is called mean ime
o epai (MTTR). Then, i is possible o de ine he a ailabili y as:
De ini ion 3.3. A ailabili y: The a ailabili y is de ined as he p obabili y ha he de ice
is a ailable when is needed and is o en used as a measu e o i s pe o mance.
I is exp essed as:
A=MTTF
MTTF +MTTR.(3.24)
The ailu e a e o many de ices exhibi s he ba h ub cu e shown in Figu e 3.2 which is
di ided in o h ee sec ions [78]:
FIGURE 3.2: Ba h ub cu e o haza d a e.
• In he in e al (0, 1), which is usually sho , a dec easing- ailu e- a e (DFR) is ob-
se ed. This is o en e e ed o as he ea ly ailu e pe iod and he ailu es ha occu
in his in e al a e called ea ly ailu es, bu n-in ailu es, o in an mo ali y ailu es.
They a e mainly due o manu ac u ing de ec s and can be elimina ed using bu n-in
echniques.
Chap e 3. Backg ound on eliabili y 43
be desc ibed as:
R( ) = P (T≥ ) = P (X( )> D) = exp −Dβ
b e−a !,(3.39)
whe e, aand ba e cons an s, X( )is deg ada ion le el a ime , and Dis he
h eshold le el.
Mix u e model o ha d and so ailu es: his me hod consis s in ob-
aining wo obse a ion samples: ca as ophic ailu es (ha d) and deg ada ion
(so ), and hen builds a mix u e model wi h bo h da a [191]. This can be mod-
eled as:
F( ) = p(z)Fc( ) + [1 −p(z)]Fd( ),(3.40)
whe e, F( )is he cumula i e densi y unc ion (cd ) o ,p(z)is he p opo ion
o componen s ailed ca as ophically wi hin speci ied obse ed deg ada ion
alue z,Fc( )and Fd( )a e espec i ely he ca as ophic ailu e cd o and
deg ada ion ailu e cd o (li e ime).
Time se ies model: i is a echnique used o p edic indi idual sys em
pe o mance eliabili y in eal- ime conside ing mul iple ailu e modes. I in-
cludes an on-line mul i a ia e moni o ing and an icipa ion (using a Kalman
il e ) o selec ed pe o mance measu es and condi ional pe o mance eliabil-
i y es ima es [96].
O he commonly used deg ada ion models: B ownian mo ion (o Wiene
p ocess) and Gamma p ocesses a e con inuous- ime models ha a e app op i-
a e o modeling con inuous deg ada ion p ocesses. Nowadays, hese models
as well as Ma ko models a e widely applied as deg ada ion modeling ech-
niques [14, 71, 89, 98, 112, 128, 153].
ii) Accele a ed deg ada ion models: accele a ed deg ada ion models p o ide in o -
ma ion abou eliabili y a no mal condi ions using deg ada ion da a ob ained a
an accele a ed ime wi h o wi hou s ess condi ions. Deg ada ion p ocess can be
e y slow in an indus ial applica ion a no mal s ess le el and ha e a high MTBF,
which makes di icul o es ima e he ailu e ime dis ibu ion o componen s wi h
high- eliabili y [103, 161, 185].
The e o e, o ob ain da a quickly om a deg ada ion es , an accele a ed li e es is
pe o med. This es consis s in applying an inc easing le el o accele a ion a i-
ables, such as ib a ion ampli ude, empe a u e, co osi e media, load, ol age,
p essu e [150]. Howe e , his kind o es s is a cos ly app oach.

Chap e 3. Backg ound on eliabili y 44
Gene al assump ions o accele a ed deg ada ion models a e:
(a) Deg ada ion is no a e e sible p ocess.
(b) A model applies o a single deg ada ion p ocess, mechanism, o ailu e mode.
(c) Deg ada ion o a es asse ’s pe o mance be o e he es s a s is negligible.
(d) The ailu e p ocesses a highe s ess le els a e he same as a he design o use
s ess le els.
Accele a ed deg ada ion models can be physics-based o s a is ical based. In he
ollowing, a quick e iew on hose ca ego ies is p esen ed.
• Physics-based models: hese models a e used o accele a ed li e es s when
he de e io a ion is caused by he mal and non- he mal pa ame e s, speed,
load, co osi e en i onmen , ib a ion ampli ude, e c., in o de o es ima e
hei se ice li es. Applica ions can include dielec ics, semiconduc o s, ba -
e y cells, lub ican , plas ic, insula ing luids, capaci o s, bea ings, and spin-
dles.
A henius model: his model is used when he damage is caused by em-
pe a u e, especially o : dielec ics [49], semi-conduc o s [103, 123], ba e y
cells, lub ican , and plas ic.. In gene al, i is used o desc ibe many p oduc s
ha ail as a esul o deg ada ion due o chemical eac ions o me al di usion.
The nominal ime τ o ailu e is [108]:
τ=A e
E
(k T),(3.41)
whe e Eis he ac i a ion ene gy o he eac ion in eV, kis he Bol zmann’s con-
s an , 8.3171 ×105eV/K, Tis he absolu e Kel in empe a u e, Ais a cons an
ha depends on p oduc geome y, specimen size and ab ica ion, es me hod
and o he s ac o s.
Ey ing model: his model is used o accele a ed li e es wi h espec o
he he mal and non- he mal a iable. The Ey ing ela ionship o nominal
li e τas a unc ion o absolu e empe a u e Tis:
τ=A
Te
B
(k T),(3.42)
he e Aand Ba e cons an pa ame e s o he p oduc and es me hod, and k
is Bol zmann’s cons an . Fo he small ange o absolu e empe a u e in mos
Chap e 3. Backg ound on eliabili y 45
applica ions, (A/T is essen ially cons an , and (3.42) is close o he A henius
ela ionship (3.41)).
In e se powe model: his model is used o analyze accele a ed li e es
da a o many elec onic and mechanical componen s such as insula ing luids,
capaci o s, bea ings, and spindles in o de o es ima e hei se ice li es when
he accele a ion ope a ing pa ame e s a e non- he mal e.g. speed, load, co -
osi e medium, and ib a ion ampli ude [69, 148]. I is based on he in e se
powe law ela ionship be ween nominal li e τo a p oduc and he accele a -
ing a iable V, and i is exp essed as:
τ(V) = A
Vγ1,(3.43)
he e Aand γ1a e pa ame e s cha ac e is ics o he p oduc , specimen geome-
y and ab ica ion, he es me hod, e c.
• S a is ics-based models: hese models ha e been de eloped o es ima e he
haza d o asse s wi h co a ia es in bo h he eliabili y and biomedical ields.
A e iew on s a is ics-based models can be ound in [53].
In hese me hods, echniques as s anda d eg essions can be applied o mos
aging deg ada ion da a, as such da a a e usually comple e. Howe e , hese
models a e usually nonlinea in he pa ame e s. In such cases, nonlinea e-
g ession me hods mus be used, which in oduces conside able complexi y
[7].
3.3 Conclusions
In his chap e , he link be ween eliabili y and deg ada ion has been explained, ollowed
by a li e a u e e iew on eliabili y modeling and deg ada ion models o eliabili y es-
ima ion. The deg ada ion modeling consis s in i ing deg ada ion da a o a model o a
p obabili y dis ibu ion. The deg ada ion da a can be ob ained a he no mal ope a ing
condi ion o a accele a ed ones.
The eliabili y modeling app oaches a e ela ed o p obabili y dis ibu ions. In his sense,
he choice o he p obabili y dis ibu ion depends on he applica ion, asse and, sys em
na u e.
Chap e 3. Backg ound on eliabili y 46
The haza d model wi h co a ia es is one o he mos common s a is ical models in elia-
bili y and su i al analysis. Some o hem ha e been p esen ed and g ouped in o non-
pa ame ic and semi-pa ame ic models. In such a way, an explana ion abou co a ia es
and how hey explain he ailu e a e and i s in eg a ion in o p obabili y dis ibu ions has
been add essed.
Some models o speci ic physical sys em ha e been e iewed, hose models can be used
o i deg ada ion da a by adjus ing pa ame e s.
In his hesis, he exponen ial dis ibu ion will be used o model he eliabili y o he as-
se s and hei aging phenomenon as desc ibed in (3.9). Mo eo e , he haza d models wi h
co a ia es will be used o explain he aging p ocess, pa icula ly, he P HM p oposed by
[30] because i can explain he deg ada ion p ocess o a la ge a ie y o physical sys ems
in a simpli ied way.
Chap e 4
Reliabili y Assessmen
This chap e add esses he eliabili y modeling and assessmen o dynamic
sys em using s uc u e unc ion, Bayesian and Dynamic Bayesian ne wo ks.
This chap e also p esen s he complexi y o eliabili y modeling and how he
in e ence algo i hms a e able o handle i . Reliabili y Impo ance Measu es
as indica o s o componen s impo ance in o he sys em eliabili y a e also
p esen ed and explained. All hese concep s a e hen illus a ed h ough an
example consis ing o a D inking Wa e Ne wo k sys em.
4.1 Sys em eliabili y
Gene ally, sys ems a e composed by subsys ems o componen s. I he s a e o he sub-
sys ems o componen s can be known, he s a e o he sys em can also be known. Sys em
s uc u e can be explained h ough wo undamen al ela ions: se ies and pa allel [60].
Only bina y componen s will be conside ed, i.e. componen s ha ing only wo s a es:
ope a ional (up) and ailed (down). Le xideno e he s a e o componen i, whe e i=
1, . . . , n and,
xi=(1 i componen iis up
0 i componen iis down .(4.1)
Also, i will be assumed ha he sys em can only ha e wo s a es: up o down. The
dependence o a sys em s a e on he s a es o i s componen s will be de e mined by means
he so-called s uc u e unc ion.
The s uc u e unc ion allows o de e mine he dependency o he s a e o he sys em
ega ding he s a e o i s componen s. This unc ion, deno ed as Φ(x), indica es he s a us
o he sys em (success o ailu e) gi en he s a e o each componen .
Now, le x= (x1, x2, ..., xn)deno es he s a e o he ncomponen s. I can ha e one o 2n
alues which co espond o he possible combina ions o he s a es (wo king o ailed)
47
Chap e 4. Reliabili y Assessmen 48
o he ncomponen s. The e o e, he s a e o he sys em is cha ac e ized by Φ(x)which
is a bina y alue unc ion whe e
Φ(x) = (1 i he sys em is in wo king s a e
0 i he sys em is in ailed s a e .(4.2)
4.1.1 Reliabili y o se ies and pa allel sys ems
A se ial sys em is called o be up i and only i all i s componen s a e up (see Figu e 4.1(a)).
Fo mally, he eliabili y o a se ial sys em is deno ed as:
Φ(x) = x1·x2···xn=
n
Y
i=1
xi.(4.3)
A pa allel sys em is called o be up i and only i one o i s componen s is up (see Fig-
u e 4.1(b)). Fo mally, he eliabili y o a pa allel sys em is deno ed as:
Φ(x)=1−(1 −x1)(1 −x2)···(1 −xn)=1−
n
Y
i=1
(1 −xi).(4.4)
(a) (b)
FIGURE 4.1: Rep esen a ion o se ies (a) and pa allel (b) sys ems.
Fo ins ance, he s uc u e unc ion can be compu ed using ei he he minimal pa h se s
(P1, P2, . . . , Ps) which a e he minimal se s o elemen s o he sys em whose unc ioning
(i.e., being up) ensu es ha he sys em is up,
Φp(x)=1−
s
Y
j=1 
1−Y
i∈Pj
xi
.(4.5)

Chap e 4. Reliabili y Assessmen 49
O he minimal cu se s (C1, C2, . . . , Ck) which a e he minimal se s o elemen s o he
sys em whose ailu e (i.e. being down) causes he ailu e o he sys em,
Φc(x) =
k
Y
j=1 
1−Y
i∈Cj
(1 −xi)
.(4.6)
Now, le us assume ha he s a e o he i h componen is desc ibed by a bina y andom
a iable Xi, de ined by
P (Xi= 1) = pi,P (Xi= 0) = qi= 1 −pi.(4.7)
whe e 1 co esponds o he ope a ional (up) s a e and 0 co esponds o he ailu e (down)
s a e.
I will be assumed ha all componen s a e mu ually independen . This implies a consid-
e able simpli ica ion, due ha o independen componen s, he join dis ibu ion o X1,
X2, . . . , Xn,is comple ely de e mined by componen s eliabili ies p1, p2, . . . , pn.
Le X= (X1, X2, . . . , Xn)be he sys em s a e ec o , which is a andom ec o . Con-
sequen ly, he sys em s uc u e unc ion Φ(X) = Φ(X1, X2, . . . , Xn)becomes a bina y
andom a iable, i.e. Φ(X) = 1 co esponds o he sys em up s a e and Φ(X) = 0 co e-
sponds o he sys em down s a e.
Hence, he sys em eliabili y ( 0) is he p obabili y ha he sys em s uc u e unc ion
equals 1:
0=P (Φ(X) = 1).(4.8)
Since, Φ(·)is a bina y andom a iable, i can be w i en as:
0=E[Φ(X)] .(4.9)
The e o e, in a se ial sys em, i s s uc u e unc ion is gi en by Φ(X) = Qn
i=1 Xi. Thus:
0=E[Φ(X)] =
n
Y
i=1
pi.(4.10)
And a pa allel sys em, i s s uc u e unc ion is gi en by Φ(X)=1−Qn
i=1(1 −Xi). Thus:
0=E[Φ(X)] = 1 −
n
Y
i=1
(1 −pi).(4.11)
Chap e 4. Reliabili y Assessmen 50
A se ies sys em does no ha e any edundancy because he e is only one way o he
sys em o wo k p ope ly; ha is, all componen s ha e o wo k p ope ly. In a pa allel
sys em, he e a e 2n−1di e en ways o he sys em o wo k p ope ly in which each
componen cons i u es a di e en way [78].
4.1.2 Reliabili y o se ies-pa allel sys ems
The e a e mo e complex s uc u es ha can be explained as combina ions o se ies and
pa allel s uc u es. Ne e heless, in he cases whe e such s uc u e educ ion canno be
pe o med, e.g. he case o b idge s uc u e (Figu e 4.2), he pi o al decomposi ion ap-
p oach is used o compu e he sys em eliabili y.
FIGURE 4.2: B idge s uc u e.
Le (αi;p)deno e he ec o pwi h i s i h componen eplaced by αi. Hence, (1i;p) =
(p1, ..., pi−l,1, pi+l, . . . , pn).
The pi o al decomposi ion me hod consis s in compu ing he eliabili y o he sys em
pi o ing a ound a componen by aking i as ully eliable (up) o comple ely un eliable
(down). In his way, he p oblem is educed o compu e he sys em eliabili y o se ial
and pa allel sys ems.
Example 4.1.1. Conside he b idge s uc u e sys em shown in Figu e 4.2. The bes choice
is o pi o a ound elemen 3. Suppose ha componen 3 is up. Then he b idge becomes
a se ies connec ion o wo pa allel subsys ems consis ing o elemen s 1, 2 and 4, 5, espec-
i ely. I s eliabili y is
(13;p) = [1 −(1 −p1)(1 −p2)][1 −(1 −p4)(1 −p5)].(4.12)
Now, conside ha componen 3 is down, hen he b idge becomes a pa allel connec ion
o wo se ies sys ems: one wi h componen s 1, 4 and he second wi h componen s 2, 5.
I s eliabili y is (03;p) = 1 −(1 −p1p4)(1 −p2p5). The e o e, he sys em eliabili y is
o=p3 (13;p) + (1 −p3) (03;p).
Chap e 4. Reliabili y Assessmen 51
The inal esul is:
o=E[Φ(X)] = p1p3p5+p2p3p4+p2p5+p1p4−p1p2p3p5(4.13)
−p1p2p4p5−p1p3p4p5−p1p2p3p4−p2p3p4p5+ 2p1p2p3p4p5.(4.14)
The same esul s can be ob ained by using minimal cu s and pa h se s. Fo example, he
b idge has ou minimal pa h se s: {1,3,5},{2,3,4},{1,4}, and {2,5}. Thus, he andom
s uc u e unc ion is:
Φ(X)=1−(1 −X1X3X5)(1 −X2X3X4)(1 −X2X5)(1 −X1X4).(4.15)
Finally, he e ms in pa en heses a e expanded, he exp ession is simpli ied using he ac
ha Xk
i=Xiand eplacing he eliabili y o each componen .
4.2 Reliabili y assessmen using BNs
4.2.1 Bayesian Ne wo ks
Basically, a Bayesian Ne wo ks (BN) compu es he p obabili y dis ibu ion in a se o a i-
ables acco ding o he p io knowledge o some a iables and he obse a ion o o he s
[65]. The BNs a e also called as Di ec Acyclic G aphs (DAGs).
Le Aand Bbe wo nodes wi h wo possible s a es (S1and S2, see Figu e 4.3). A p ob-
abili y is associa ed o each s a e o he node. This p obabili y is de ined a p io i o oo
nodes and compu ed by in e ence o he o he s. The a p io i p obabili ies o node Aa e
P (A=SA1)and P (A=SA2).
A Condi ional P obabili y Table (CPT) is associa ed o node Band de ines he condi ional
p obabili y o he s a e o Bgi en he s a e o A(P (B|A)). Thus, he BN in e ence com-
pu es he ma ginal dis ibu ion P (B=SB1):
P (B=SB1) =P (B=SB1|A=SA1)P (A=SA1)
+P (B=SB1|A=SA2)P (A=SA2).(4.16)
In he Bayesian ne wo k app oach he p obabilis ic in e ac ions o he componen s o a
sys em a e ep esen ed using a DAG which nodes ep esen he a iables and he a cs
be ween nodes ep esen he causal ela ionships be ween a iables [35]. Basically, BNs
Chap e 4. Reliabili y Assessmen 52
compu e he p obabili y dis ibu ion in a se o a iables acco ding o he p io knowl-
edge o some a iables and he obse a ion o o he s [66].
FIGURE 4.3: Basic Bayesian Ne wo k.
A Condi ional P obabili y Table (CPT) is associa ed wi h node Band de ines he condi-
ional p obabili y o he s a e o Bgi en he s a e o A(P (B|A)) (Table 4.1). The condi-
ional p obabili y is a measu e o he p obabili y o an e en gi en ha ano he e en has
occu ed.
TABLE 4.1: CPT o he BN shown in Figu e 4.3
AB
SB1SB2
SA1P (B=SB1|A=SA1) P (B=SB2|A=SB1)
SA2P (B=SB1|A=SA2) P (B=SB2|A=SB2)
4.2.2 In e ence mechanism
Bayesian ne wo ks a e easy o use hanks o hei g aphical in e p e a ion. Bu , he p ob-
abilis ic in e ence mechanism cons i u es hei eal s eng h. The in e ence o a BN is able
o compu e he ma ginal p obabili y dis ibu ion o any a iable acco ding o:
• Obse a ion o measu emen s o a iables (e idence).
• The likelihood ega ding he s a e o ce ain a iables.
• The condi ional p obabili y dis ibu ion be ween a iables.
In his hesis, he BN and DBN models ha e been p og amed using he Bayes Ne oolbox
[104], which suppo s many di e en in e ence algo i hms, such as:
• Exac in e ence o s a ic BNs:
junc ion ee
a iable elimina ion
b u e o ce enume a ion ( o disc e e ne s)
Chap e 4. Reliabili y Assessmen 59
C100CFE
d100CFE
d10COR
Legend: Componen s
Sou ces Rese oi
Demand
Sec o
Pumping
s a ion
FIGURE 4.9: D inking wa e ne wo k diag am.
This ne wo k consis s o 5 sou ces and 1 sink. I is assumed ha he demand o ecas
a he sink (dm(k)) is known (Figu e 4.10), and ha any single sou ce can sa is y his e-
qui ed wa e demand. I is also assumed ha he olume o he anks should be be ween
a minimum and maximum sa e y le els.
0 50 100 150 200 250 300
0.2
0.3
0.4
0.5
0.6
0.7
0.8
Time [h]
Flow [m3/s]
FIGURE 4.10: D inking wa e demand.
Rega ding he eliabili y o a DWN, in he li e a u e, i is classi ied in o wo main ca e-
go ies. The i s one, named hyd aulic eliabili y, is ela ed o he p obabili y ha a DWN
can supply he consume demands o e a speci ied ime in e al unde speci ied en i on-
men al condi ions, i.e., he anspo o desi ed quan i ies and quali ies o wa e a equi ed

Chap e 4. Reliabili y Assessmen 60
p essu es o desi ed app op ia e loca ions a desi ed app op ia e imes. The second one,
named opological eliabili y, e e s o he p obabili y ha a gi en ne wo k is physically
connec ed gi en he mechanical eliabili ies o i s componen s [119].
This hesis and he me hodology p oposed he e a e ocused only on he opological e-
liabili y. Mo eo e , he eliabili y modeling illus a ed he e conce ns only o he ac i e
componen s which can be di ec ly commanded.
The DWN eliabili y is modeled using a DBN as ollows: i s , sys em componen s mus
be iden i ied. In his case he e a e 10 pumps, 5 sou ces, 4 anks and se e al pipes.
Secondly, he minimal pa h se s should be de e mined. A minimal pa h se is composed
by hose componen s which allow a low pa h be ween sou ces and sinks, such as pipes,
anks and pumps. A lis o he componen s ha co espond o each minimal pa h se is
p esen ed in Table 4.4. The e a e nine minimal pa h se s in he sys em o Figu e 4.9. Each
minimal pa h se is a ailable depending on he eliabili y o i s componen s.
TABLE 4.4: Componen s and minimal pa h se s ela ionship.
A1A2A3A4A5e1e2e3e4e5e6e7e8e9e10
P1× × ×
P2× × ×
P3× × ×
P4× × × ×
P5× × × ×
P6× × × × ×
P7× × × ×
P8× × × ×
P9× × × × ×
No e ha pipes and anks a e conside ed pe ec ly eliable so hey do no p o ide signi i-
can in o ma ion o he ne wo k. Ne e heless, sou ces a e included in he minimal pa h
se s me ely o illus a ing he p ocedu e.
P o ided he in o ma ion o Table 4.4, he DBN p esen ed in Figu e 4.11 is buil as ol-
lows: nodes eiand Aia e d awn o each componen . No e ha nodes eiha e wo ime
slices in ime kand k+ 1 ollowing he app oach o Sec ion 4.2.4.
Then, hese nodes a e in e connec ed o hei minimal pa h se nodes Piusing a cs. Fi-
nally, each minimal pa h se node is in e connec ed o he sys em eliabili y node S[178].
Ini ially, a ins an k= 0, he pumps and he sys em a e assumed o be ully eliable, i.e.
hei eliabili y is 1. Then, he p obabili y o each node is compu ed using hei CPT.
Chap e 4. Reliabili y Assessmen 61
FIGURE 4.11: Dynamic Bayesian ne wo k model o he DWN.
A each sampling ime, he eliabili y Rio each pump is compu ed acco ding o i s ailu e
a e using a MC (Figu e 4.11). I s beha io ollows an exponen ial dis ibu ion as s a ed
in (3.9). No e ha i is independen o he p e ious s a es o he componen . I only
depends on i s p esen s a e. In he DBN, his co esponds o he CPT shown in Table 4.5.
TABLE 4.5: In e - ime slices CPT o node ei(k+ 1).
ei(k)ei(k+ 1)
Up Dn
Up 1-λ0
i·Tsλ0
i·Ts
Dn 0 1
The CPT o node P1is shown in Table 4.6. This CPT depends on he s a es o he sou ce
1 (A1) and pumps 1 and 5 (e1, e5). I s beha io co esponds o an AND ga e.
I is assumed ha wi h one sou ce i is possible o sa is y he wa e demand. Thus, he
a ailabili y o he sys em can be assu ed as long as a leas one o pa hs Piis a ailable,
which co esponds o he CPT o node Sshown in Table 4.7. I depends on he s a e o
nodes P1 o P9and has he beha io o an OR ga e.
Chap e 4. Reliabili y Assessmen 62
TABLE 4.6: CPTs o nodes P1.
A1e1(k+ 1) e6(k+ 1) P1
Up Dn
Up Up Up 1 0
Up Up Dn 0 1
Up Dn Up 0 1
Up Dn Dn 0 1
Dn Up Up 0 1
Dn Up Dn 0 1
Dn Dn Up 0 1
Dn Dn Dn 0 1
TABLE 4.7: CPT o node S.
P1P2P3. . . P9
S
Up Dn
Up Up Up . . . Up 1 0
Up Up Up . . . Dn 1 0
---. . . Up 1 0
.
.
..
.
..
.
.....
.
..
.
..
.
.
Dn Dn Dn . . . Dn 0 1
4.4 Reliabili y Impo ance Measu es
The e a e indices ha can be used o ake ac ions acco ding o he analysis made o he
sys em eliabili y and a ailabili y. These measu es a e equen ly o signi ican alue in
pe o ming ade-o analysis in sys em design o sugges ing he mos e icien way o
ope a e and main ain a sys em o p io i izing imp o emen e o s.
Reliabili y impo ance measu es we e i s in oduced by [12], and hey a e classi ied in o
wo g oups: Reliabili y Impo ance Measu es (RIMs) and S uc u al Impo ance Mea-
su es (SIMs). The RIMs e alua e he ela i e impo ance o a componen aking in o ac-
coun i s con ibu ion o he o e all sys em eliabili y while he SIMs p o ide he ela i e
impo ance o a componen aking in o accoun i s posi ion in o he sys em s uc u e.
These me ics can be de ined ei he acco ding o hei unc ional aspec , aking in o ac-
coun he minimal pa h se s, o acco ding o hei dys unc ional aspec , conside ing he
minimal cu se s. As bo h a e equi alen , in his hesis only he unc ional aspec is used.
The aim om he sys em eliabili y analysis poin o iew, is o use he RIMs o iden i y
Chap e 4. Reliabili y Assessmen 63
he weakness o s eng hs in he sys em and o quan i y he impac o componen ailu es
o e sys em unc ioning.
4.4.1 Bi nbaum’s Impo ance Measu e
The Bi nbaum impo ance measu e [12] also known as Ma ginal Impo ance Fac o (MIF)
is ela ed o he p obabili y o a componen o be c i ical o he sys em unc ioning. I is
de ined as:
De ini ion 4.1. The B- eliabili y impo ance o componen i o he unc ioning o he sys-
em, deno ed as IB(i;p), o a cohe en sys em wi h independen componen s is de ined
as:
IB(i;p) =P (Φ(X)=1|Xi= 1) −P (Φ(X)=1|Xi= 0)
=∂R(p)
∂pi
=R(1i;p)−R(0i;p).
(4.24)
The no a ion R(1i;p)deno es he eliabili y o he sys em in which he i h componen s is
eplaced by an absolu ely eliable one, while R(0i;p)deno es he eliabili y o he sys em
in which he i h componen is ailed.
The Bi nbaum’s measu e is he p obabili y ha he ailu e o unc ioning o he i h com-
ponen coincides wi h sys em ailu e o unc ioning. This app oach is well known om
classical sensi i i y analysis. Mo eo e , i can be in e p e ed as he maximum los in sys-
em eliabili y when he i h componen changes om he condi ion o pe ec unc ioning
o a ailed condi ion.
No e ha Bi nbaum’s impo ance measu e (IB(i;p)) o he i h componen depends only
on he s uc u e o he sys em and he eliabili ies o he o he componen s. IB(i;p)is
independen o he ac ual eliabili y o he i h componen (pi). This may be ega ded as a
weakness o Bi nbaum’s measu e.
4.4.2 C i ical Reliabili y Impo ance Measu e
The C i ical Reliabili y Impo ance, also known as C i ical Impo ance Fac o (CIF), was
in oduced by Lambe [81] and i is de ined as:
De ini ion 4.2. The c i ical eliabili y impo ance o componen i o sys em unc ion-
ing, deno ed by ICIF (i;p), is de ined as he p obabili y ha i h componen wo ks and is
Chap e 4. Reliabili y Assessmen 64
c i ical o he sys em unc ioning gi en ha he sys em is unc ioning.
ICIF (i;p) =piP (Φ(X)=1|Xi= 1) −P (Φ(X)=1|Xi= 0)
P (Φ(X) = 1)
=pi
R(p)IB(i;p).
(4.25)
Mo eo e , his can be in e p e ed as he p obabili y ha he i h componen has caused a
sys em ailu e when i is known ha he sys em is ailed.
4.4.3 Fussell-Vesely Reliabili y Impo ance Measu e
The Fussell-Veselly (FV) impo ance measu e also known as he Diagnos ic Impo ance
Fac o (DIF) was p oposed ini ially in he con ex o aul ee [45, 168]. I is classi ied
in c- ype and p- ype. The c- ype FV impo ance, akes in o accoun he con ibu ion o
componen o sys em ailu e, and i s de ini ion is based on minimal cu s. The p- ype FV
impo ance, akes in o accoun he con ibu ion o a componen o sys em unc ioning,
and i is de i ed om he minimal pa h se s. I ep esen s he p obabili y ha a leas one
minimal pa h con aining he i h componen wo ks, gi en ha he sys em is unc ioning.
De ini ion 4.3. The Fussell-Veselly, p-FV (c-FV) eliabili y impo ance measu e o com-
ponen i, deno ed by Ip
DIF (i;p)(Ic
DIF (i;p)), is de ined as he p obabili y ha a minimal
pa h (cu ) con aining he i h componen exis s and causes sys em unc ion ( ailu e).
Ip
DIF (i;p) =P {∃P∈Pis. .Xj= 1 ∀j∈P|Φ(X)=1}
=piP {(1i,X) : ∃P∈Pis. . Xj= 1 ∀j∈P}
R(p)
=P (Xi= 1|φ(X) = 1)
(4.26)
Ic
DIF (i;p) =P {∃C∈Cis. .Xj= 0 ∀j∈C|φ(X)=0}
=qiP {(0i,X) : ∃C∈Xj= 0 ∀j∈C}
1−R(p)
=P (0i|φ(X) = 0) = P (Xi= 0|φ(X) = 0)
(4.27)
whe e P∈Pi(C∈Ci) deno es he minimal pa h (cu ) con aining he i h componen .
As bo h a e equi alen , he no a ion IDIF will be used.
IDIF can be in e p e ed as he p obabili y ha he unc ioning o componen icon ibu es
o he unc ioning o he sys em gi en ha he sys em is no ailed.

Chap e 4. Reliabili y Assessmen 65
4.4.4 Reliabili y Achie emen Wo h
The Reliabili y Achie emen Wo h (RAW) desc ibes he inc ease o he sys em eliabili y
i he i h componen is eplaced by a pe ec eliable one. I is de ined as:
De ini ion 4.4. The RAW, deno ed by IRAW (i;p)quali ies he maximum possible pe -
cen age o sys em eliabili y inc ease gene a ed by he i h componen . I is exp essed
as:
IRAW (i;p) =P (Φ(1i,X) = 1)
P (Φ(X) = 1)
=P (Φ(X)=1|Xi= 1)
P (Φ(X) = 1)
=1 + qi
R(p)IB(i;p).(4.28)
4.4.5 Reliabili y Reduc ion Wo h, RRW
The Reliabili y Reduc ion Wo h (RRW) measu e [86] e lec s he educ ion o sys em
eliabili y i he i h componen is ailed. I is de ined as:
De ini ion 4.5. The RRW, deno ed as IRRW (i;p)exp esses he po en ial damage p o-
duced o he sys em by he ailu e o he i h componen .
IRRW (i;p) = P (Φ(X) = 1)
P (Φ(0i,X) = 1)
=P (Φ(X) = 1)
P (Φ(X)=1|Xi= 0)
=1
1−pi
R(p)IB(i;p).(4.29)
4.5 Example
Fo he compu a ion o he RIMs conside he DWN sys em desc ibed in Example 4.3. i
is supposed ha sou ces, anks ans pipelines a e pe ec ly eliable and only ac ua o s a e
a ec ed by a loss o eliabili y acco ding o (3.9).
The aim o compu ing he RIMs in his example is o know, om di e en poin s o iew,
he impo ance o he pumps eliabili y o he o e all sys em eliabili y. In his Sec ion i
is done in a s a ic app oach, since he ailu e a e is cons an . La e , in Sec ion 6.3.5, he
analysis is done dynamically aking in o accoun he e olu ion o he ailu e a e.
Chap e 4. Reliabili y Assessmen 66
Fi s o all, a s a ic RIM analysis is pe o med in o de o ge be e knowledge on hem.
Componen eliabili y is assumed o ollow (3.9) wi h λi(Table 4.8) and he mission ime
is =TM(2000 hou s). The co esponding esul s a e p esen ed in Table 4.9 and Ta-
ble 4.10.
TABLE 4.8: Pump ailu e a es
λ[h−1·10−4]
p1p2p3p4p5p6p7p8p9p10
9.85 10.70 10.50 1.40 0.85 0.80 11.70 0.60 0.74 0.78
TABLE 4.9: Pumps Reliabili y Impo ance Measu es a TM=2000
Pump λ[·10−4]Ri[%] IB[%] IDIF [%] ICIF [%] IRAW [%] IRRW [%]
p19.85 13.94 4.54 14.61 0.77 0.10 0.10
p210.70 11.76 4.43 12.32 0.63 0.10 0.10
p310.50 12.24 4.45 12.83 0.66 0.10 0.10
p41.40 75.57 8.78 77.54 8.05 0.10 0.11
p50.85 84.37 13.72 86.56 14.04 0.10 0.11
p60.80 85.21 87.48 98.58 90.39 0.11 1.04
p711.70 9.63 12.89 10.99 1.50 0.11 0.10
p80.60 88.69 5.81 89.40 6.25 0.10 0.11
p90.74 86.24 12.66 88.06 13.24 0.10 0.11
p10 0.78 85.55 8.11 86.77 8.42 0.10 0.11
In Table 4.10 pumps a e so ed acco ding o hei eliabili y impo ance measu es. Re-
ma k ha pump 6 is he mos c i ical acco ding o all RIMs. I is also in e es ing o high-
ligh ha some RIMs gi e a simila pump c i icali y o de : CIF and RRW a e equi alen ,
and MIF p o ides a close esul .
This a p io i knowledge can be used o decide how o dis ibu e he con ol e o s h ough
he con ol algo i hm. A dynamical RIM analysis will be p esen ed in Sec ion 6.3.5.
4.6 Conclusions
In his chap e , he sys em eliabili y compu a ion om he eliabili y o i s componen s
o subsys ems has been p esen ed. Basically, his can be done by using he sys em con ig-
u a ion s uc u e, i.e. se ial and pa allel sys ems, and i s co esponding exp ession o in
he case o complex sys em con igu a ions by using he pi o al decomposi ion me hod.
Chap e 4. Reliabili y Assessmen 67
TABLE 4.10: A p io i classi ica ion o he pumps
λiRiIBIDIF ICIF IRAW IRRW
p7p8p6p6p6p6p6
p2p9p5p8p5p7p5
p3p10 p7p9p9p1p9
p1p6p9p10 p10 p2p10
p4p5p4p5p4p3p4
p5p4p10 p4p8p4p8
p6p1p8p1p7p5p7
p10 p3p1p3p1p9p1
p9p2p3p2p3p10 p3
p8p7p2p7p2p8p2
I has also add essed he concep s o MC, BN, and DBN. These concep s ha e been ap-
plied o model he eliabili y o a DWN sys em.
In his chap e , a e iew o he a ailable RIMs has been pe o med. These eliabili y index
measu es ha e been e alua ed o a DWN sys em as a ool o iden i y he impo ance o
each pump o he sys em unc ioning.
The RIMs ha e shown o be an objec i e c i e ion ha can be used o edis ibu e he con-
ol e o among he a ailable sys em ac ua o s o a oid, o ins ance, hei deg ada ion.
Chap e 5
Con ol Sys em
This chap e add esses he concep s in ol ed in con ol sys ems and p esen s
he con ol app oaches used in his hesis. Those con ol me hodologies in-
clude he Model P edic i e Con ol (MPC) and he Linea -Quad a ic Regula-
o (LQR), which can be used o implemen a Heal h-Awa e Con ol scheme
as i will be demons a ed la e .
5.1 Model P edic i e Con ol
Model P edic i e Con ol (MPC) was de eloped in he 70’s and has e ol ed conside ably
p o iding a wide ange o con ol me hods ha ha e in common he use o an explici
model o he p ocess o calcula e he u u e con ol inpu by he minimiza ion o an ob-
jec i e unc ion. These con olle design me hods lead o schemes which basically ha e
he ollowing ideas [18]:
• The use o an explici model o p edic he p ocess ou pu a u u e ime ins an s
(p edic ion ho izon).
• The compu a ion o a con ol sequence ha in ol es he minimiza ion o an objec-
i e unc ion.
• A eceding s a egy which slides he p edic ion ho izon owa ds he u u e a each
ime ins an , and he applica ion a each s ep o he i s elemen o he compu ed
con ol sequence.
This con ol me hod has he ollowing ad an ages:
• I in ol es e y in ui i e concep s which make i ela i ely easy o implemen and
une.
• I can be used o con ol a wide a ie y o p ocesses, om hose wi h simple dy-
namics o hose wi h long delay imes o nonminimum phase o uns able.
68
Chap e 5. Con ol Sys em 75
5.2 Linea -Quad a ic Regula o
The Linea -Quad a ic Regula o (LQR) is a well known con ol design echnique ha
p o ides eedback gains in a p ac ical manne [114].
Conside he disc e e- ime, linea ime-in a ian (LTI) sys em gi en in (5.1), and gi en a
cos unc ion de ined in a quad a ic o m as:
JLQR =1
2
∞
X
k=0 xT(k)QLQRx(k) + uT(k)RLQRu(k)(5.34)
whe e QLRQ ∈Rnx×nxand RLRQ ∈Rnu×nua e He mi ian posi i e-de ini e ma ices. I
he sys em is con ollable and obse able, a eedback con ol law can be de ined as:
u(k) = −Kx(k)(5.35)
whe e he op imal eedback gain (K) is he solu ion o he cos unc ion (5.34):
K=RLQR +BTPB−1BTP A. (5.36)
The posi i e-de ini e symme ic ma ix Pis he solu ion o he disc e e- ime algeb aic
Ricca i equa ion:
P=QLQR +ATPA −ATPB(RLQR +BTPB)−1BTPA. (5.37)
The aim o he cos unc ion in his hesis is o use ma ix RLQR as a weigh ing ma ix o
he con ol e o and o edis ibu e he con ol inpu o e he a ailable ac ua o s based
on hei eliabili y in o ma ion, while ma ix QLQR is used o weigh he sys em s a es o
ajec o y acking.
5.3 Conclusions
This chap e p esen ed he con ol me hodologies used in he de elopmen o he hesis.
In he case o MPC me hodology, he op imiza ion p oblem has been o mula ed using
bo h, a common quad a ic and linea cos unc ions which include e ms o minimize
he con ol ac ion and i s a ia ions, and depending he con ol p oblem i can include a
e m o he minimiza ion o he acking e o .
A he i s glance, bo h, MPC and LQR a e op imal con ol echniques, bu hey di e

Chap e 5. Con ol Sys em 76
om he ac ha MPC sol es he op imiza ion p oblem in a ini e ime ho izon and
uses he eceding ho izon app oach, while LQR sol es an op imiza ion p oblem o e
an in ini y ime ho izon. An impo an ea u e o MPC is he possibili y o explici ly
including inpu and s a e cons ain s in he con ol law.
In he MPC echnique, i s weigh s play an impo an ole in sol ing he p oblem. The
une o hese pa ame e s can lead o agg essi e con ol esponses when he a io be-
ween acking e o and con ol e o weigh s is la ge . Also, la ge ho izon p edic ion
p oduces a mo e “op imal” con olle bu inc eases i complexi y.
Ne e heless, he con ol e o weigh s o bo h, MPC and LQR, can be selec ed in o de
o assign mo e o less ela i e impo ance o each ac ua o p oducing lowe o highe
ela i e ac ua o use. This ac can be used o assign weigh s acco ding o ac ua o s eli-
abili ies o RIMs and in his manne achie e be e le els o sys em eliabili y.
An open issue ha equi es u he esea ch is he p ocedu e o ob ain a cos unc ion
ha can include o he e ms o hei minimiza ion bu main ain i s simplici y. Howe e ,
i can lead o a nonlinea o non-con ex op imiza ion p ocess whose solu ion is compu-
a ionally hea y and could lead o non-implemen able con ol schemes.
77
Pa II
Con ibu ions
Chap e 6
Heal h-Awa e Con ol
This chap e p esen s he in eg a ion o eliabili y (as a measu e o he
heal h o he sys em o i s componen s) in he con ol objec i es in o de
o a oid he occu ence o ca as ophic and incipien aul s. Reliabili y
analysis and ailu es conce n he ac ua o s o he sys em. This in e-
g a ion is o mula ed using he eliabili y impo ance measu es in he
pa ame e s o he op imiza ion unc ion. The sensi i i y o he sys em
eliabili y o he deg ada ion o i s ac ua o s due o hei wo king load
p oduced by he con ol ac ion is key o edis ibu e he con ol e o s.
MPC and LQR echniques a e in es iga ed o implemen such HAC ap-
p oach. MPC will be applied o a DWN, and o a mul i o o UAV. Mo e-
o e , an app oach o educe he deg ada ion o sys em componen s is
applied o a Twin Ro o sys em using MPC.
6.1 Reliabili y assessmen o HAC
De ini ion 3.1 and (3.9), he eliabili y o he i h componen o he sys em will be modeled
using he exponen ial unc ion as:
Ri( ) = e−R
0λi( )d ∀i= 1, . . . , p,
whe e λi( )is he ailu e a e depending on ime.
In his hesis, he o e all sys em eliabili y will be compu ed om he eliabili y o i s
componen s using he Bayesian ne wo ks (see Sec ion 4.2). Mo eo e , i is assumed ha
he o e all sys em eliabili y is compu ed by he eliabili y o i s ac ua o s since hey
a e key in achie ing he sys em con ollabili y. Ne e heless, his me hodology could
be ex ended o o he componen s o he sys em. Fo ins ance, in a DWN, he eliabili y
o he pipelines could be modeled as a unc ion o he wa e low and p essu e [5, 80]
78
Chap e 6. Heal h-Awa e Con ol 79
and include i in o he o e all sys em eliabili y compu a ion, o he eliabili y o he
ese oi s could be modeled as a unc ion o hei ope a ional ime and he olume o
wa e hey s o e.
In any case, o achie e he goal o boos ing sys em eliabili y, eliabili y compu a ion
should depend on con olled a iables, such as ol age, cu en , low, p essu e, e c.
6.1.1 Failu e a e
The ailu e a e o he ac ua o a ies wi h ime and he ac ua o usage. The ailu e a e
in he use ul li e pe iod o he ac ua o is assumed o depend on he impac o he load
(usage) and i s age.
In his hesis, and adap a ion o he P HM, p oposed by Cox [30], and de ined as (3.25) is
used:
λi( ) = λ0
i·g(u)∀i= 1, . . . , p, (6.1)
whe e λ0
iis he baseline ailu e a e (nominal ailu e a e) o he i h ac ua o , and g(u)
ep esen s he e ec o he co a ia es depending on he applied load u. In he e, is i
assumed ha he pa ame e λ0
iis equi alen o a cons an h0, and g(u)is equi alen o
ψ(γz), gi en in (3.25).
Di e en de ini ions o unc ion g(u)exis s in he li e a u e. Howe e , he exponen ial
o m is he mos commonly used. In [74, 75] he au ho s p opose a load unc ion based
on he oo -mean-squa e o he applied con ol inpu (ui) up o he end o he mission
(TM), and an ac ua o pa ame e de ined om he uppe and lowe bounds o con ol
inpu .
In Guenab [57] a load unc ion is p oposed as:
gi(u) = eα up om
i∀i= 1, ..., p, (6.2)
whe e up om
iis he a e age le el o he applied con ol.
In [55], he co a ia e unc ion is de ined as:
gi(u) = eβi||ui,o:k||2
2∀i= 1, ..., p, (6.3)
whe e βi= ( M(ui,max −ui,min))−1is a shape pa ame e o he ac ua o ailu e o an
expec ed li e M.
Chap e 6. Heal h-Awa e Con ol 80
In [74], he au ho s p opose:
gi(u) = eβiR M
0u2
i( )d ∀i= 1, ..., p, (6.4)
whe e he load unc ion is de ined acco ding o he oo -mean-squa e o he applied con-
ol inpu un il he end o he mission ( M), and βiis an ac ua o pa ame e de ined as:
βi= ( M( ¯ui−ui))−1, and ¯uiand uia e he uppe and lowe sa u a ion bound o ui,
espec i ely.
Ano he exp ession p oposed in [75] is de ined as:
gi(u) = eα ui
nom ∀i= 1, ..., p, (6.5)
whe e αis a ixed ac o depending on he ac ua o p ope y, ui
nom is he nominal con ol
law deli e ed by he i h ac ua o o achie e he con ol objec i e.
In his hesis, di e en de ini ions o he co a ia e unc ion ha e been s udied. Ini ially, in
[143, 144, 146], he co a ia e unc ion used ep esen s he amoun o load on he ac ua o
co esponding o he no malized ins an aneous ac ua o usage a each sampling ime:
gi(u) = ui(k)−ui
ui−ui∀i= 1, ..., p, (6.6)
whe e ui(k)is he con ol e o a ime k,uiand uia e he minimum and maximum con-
ol e o s allowed o he i h ac ua o . In his case, he highes ac ua o load co esponds
o ui(k) = ui, which leads o he highes ailu e a e λi=λ0
i.
Exp ession (6.6) akes in o accoun he impac o he load a each ime ins an bu no he
p e ious ime. In o de o include he his o ical use, equi alen o he age o he ac ua o ,
he ollowing co a ia e unc ion was p oposed in [139, 141]:
gi(u( )) = 1 + βiZ
0|ui( )|d ∀i= 1, . . . , p, (6.7)
whe e gi(ui( )) is de ined as he cumula i e applied con ol e o o he i h ac ua o om
he beginning o he mission up o ime ins an and βiis a cons an pa ame e .
Using (6.1) in (6.7) i yields,
λi( ) = λ0
i1 + βiZ
0|ui( )|d ∀i= 1, . . . , p (6.8)
his de ini ion implies ha ac ua o s a e unde a cons an eliabili y decay due o he
baseline ailu e a e which is inc eased when he ac ua o s a e used.

Chap e 6. Heal h-Awa e Con ol 81
Exp essions (6.6) o (6.3) only ake in o accoun he asse load as a ac o in he co a i-
a e, whe eas he de ini ion (6.8) ep esen s a mo e ealis ic si ua ion because i akes in o
accoun no only he load o he asse bu also he aging p oduced by he pass o he ime.
6.2 Heal h-Awa e Con ol app oaches
In he p oposed HAC app oach, he con olle is en iched wi h sys em heal h in o ma-
ion p o ided by a moni o ing module and, an accommoda ion p ocess is pe o med
o ackled he ac ua o s deg ada ion by adjus ing he con olle pa ame e s. Figu e 6.1
p esen s a gene ic block diag am o he p oposed app oach, whe e he cos unc ion o
an op imal con ol s a egy is uned up on he basis o heal h in o ma ion p o ided by
he moni o ing module.
FIGURE 6.1: Gene ic block diag am o he p oposed HAC
In he ollowing sec ions, h ee app oaches o implemen he p oposed HAC me hodol-
ogy will be p esen ed:
1. The i s is based on a MPC con ol and includes in o ma ion abou he compo-
nen s and sys em eliabili ies. An MPC con olle is se up based on sys em and
componen eliabili y.
2. An LQR con olle is se up based on sys em and componen eliabili y.
3. An MPC scheme is se up based on ac ua o usage in o ma ion.
Chap e 6. Heal h-Awa e Con ol 82
To e alua e he bene i o hese app oaches, some pe o mance indica o s will nex p o-
posed.
6.2.1 Pe o mance e alua ion
Di e en indexes a e p oposed o e alua e he pe o mance, in con ol and eliabili y
aspec s, o he p oposed HAC me hodologies.
De ini ion 6.1. The Cumula i e Con ol E o (Ucum) index indica es he amoun o en-
e gy spen con olling he sys em, and is gi en by:
Ucum =Ts
TM/Ts
X
k=0
u(k)Tu(k)(6.9)
whe e Tsand TMa e he sampling ime and he mission ime, espec i ely.
De ini ion 6.2. The Join Ac ua o Reliabili y (JAR) index measu es he emaining o e all
ac ua o eliabili y and is de ined as:
JAR =
p
Y
i=1
Ri(TM).(6.10)
And, he In eg al Squa e E o de ined as:
De ini ion 6.3. The ISE measu es how well he con olle ollows he acking e e ence.
ISE =
TM/Ts
X
k=0
q
X
i=1
(ˆyi(k)−y e ,i(k))2.(6.11)
Addi ionally, he sys em eliabili y a he end o he mission ime (Rs(TM)) will also be
used as a pe o mance measu e.
6.3 MPC amewo k o sys em eliabili y op imiza ion
6.3.1 MPC cos unc ion
As p esen ed in Sec ion 5.1, he MPC algo i hm allows including as many objec i es as
needed in he cos unc ion. The impo ance o hese objec i es in he op imiza ion p ob-
lem is handled by he weigh ing pa ame e s, such as αi(k),ρi(k), and δi(k)(5.4).
Chap e 6. Heal h-Awa e Con ol 83
In his scena io, he con ol objec i es s udy is done by selec ing and δi(k) = 1. This
means ha he MPC pe o ms a smoo h con ol bu his e ec is no pa o he s udy,
and ha he e e ence acking is s udy by di ec ly assigning he weigh ε.
In his sec ion, he op imiza ion p oblem in (5.4) is e o mula ed as ollows:
Consequen ly, he cos unc ion used is:
minimize
(ˆu(k|k), ...,
ˆu(k+Hc−1|k))
∆ˆu(k|k), ...,
∆ˆu(k+Hc−1|k))
J(k) = ε

Hp−1
X
j=0
q
X
i=1
(ˆyi(k+j|k)−y e ,i(k))2

+(1 −ε)

Hc−1
X
j=0
p
X
i=1
ρi(k)ˆui(k+j|k)2+
Hc−1
X
j=0
p
X
i=1
∆ˆui(k+j|k)2

subjec o u≤ˆu(k+j|k)≤uj= 0, .., Hc−1
x≤ˆx(k+i|k)≤xi= 1, .., Hp
(6.12)
whe e αi= 1 and δi= 1 o all i.
No e ha a ade-o be ween e e ence acking and ene gy consump ion (con ol e -
o ) a ise. To manage his ade-o , a new weigh is added o he o mula ion p oblem.
The e o e, pa ame e εcan be used o ind he app op ia e equilib ium be ween bo h
op imiza ion objec i es ollowing he me hodology p esen ed la e in Sec ion 6.3.2.
6.3.2 MPC uning me hodology
The MPC uning consis s in inding he app op ia e alues o he weigh ing pa ame e s
in he cos unc ion.
The alues o ρiand εin (6.12) should be selec ed as ollows:
S ep 1:
Se ε= 0 and une ρisuch ha he eliabili y o he sys em a he end o he mission
ime is he highes .
The compa ison and selec ion o he bes app oach will be pe o med based on he
JAR c i e ia (6.10), and he UCum index (6.9).
S ep 2:
Tune εsuch ha he highes sys em eliabili y is achie ed a mission ime Rs(TM)
while ISE index (6.11) is he lowes .
The goal is o s udy he e ec p oduced by he a ia ion o pa ame e εin he sys-
em eliabili y o in o he wo ds, he impac o he acking e o in he eliabili y.
Chap e 6. Heal h-Awa e Con ol 84
Thus, he op imal alue o εwhich co esponds o he highes sys em eliabili y
and he lowes ISE, balancing a he same ime bo h objec i es will be ound.
6.3.3 Con ol e o edis ibu ion
In his hesis, wo app oaches o imp o e he sys em eliabili y a e s udied. On one hand,
a local app oach which ocuses on he ac ua o s eliabili y, and on he o he hand, he
global app oach which ocuses on he o e all sys em eliabili y.
To implemen such app oaches, he weigh ρi(k)in he cos unc ion (6.12) is used as a
way o edis ibu e he con ol e o s among he ac ua o s [144]. The g ea e he alue
gi en o componen ρiis, he highe impo ance he i h ac ua o will ha e and he mo e
penaliza ion i will ha e in he op imiza ion p oblem.
The local app oach a emp s o p ese e he componen eliabili y:
ρi(k)=1−Ri(k)∀i= 1,2, . . . , p. (6.13)
This c i e ia aims a inding he op imal con ol ac ions and dis ibu ing hem among he
a ailable ac ua o s in such a way ha ac ua o s wi h lowe eliabili y le el a e elie ed.
Hence, he use o highly eliable componen s is p io i ized.
The local app oach assumes an equi alen con ibu ion o componen eliabili y o sys-
em eliabili y. Howe e , his is ha dly e e ue because hei con ibu ion depends on
he sys em s uc u e and he in e connec ion o he ac ua o s wi hin such s uc u e.
The aim o he global app oach is o de e mine he ela i e impo ance o he ac ua o s
wi h he objec i e o imp o ing he o e all sys em eliabili y. This s udy is based on
he s udy o he RIMs (see Sec ion 4.4), which p o ide di e en measu es o ac ua o
impo ance. Fo ins ance, by using he Bi nbaum’s measu e as ollows:
ρi(k) = IB,i(k)∀i= 1,2, . . . , p, (6.14)
whe e IB,i(k)deno es he Bi nbaum’s measu e o he i h ac ua o a ins an ime k. In his
case, i is expec ed ha componen s wi h a g ea e con ibu ion o he sys em eliabili y
a e used less han he o he s.
The con ol s a egy scheme is p esen ed in Figu e 6.2. The MPC compu es he con ol
inpu s acco ding o: he cos unc ion, a se o bounds, he cu en sys em s a e and he
weigh s ρi. Then, he con ol inpu is injec ed o he sys em and used o compu e he
componen ailu e a es.
Chap e 6. Heal h-Awa e Con ol 91
FIGURE 6.8: Pumping inpu s [m3/h]: blue line co esponds o ρi= 1 and
ed line co esponds o ρi=ICIF,i ·IRRW,i.
be: ρi(k) = ICIF,i(k)·IRRW,i(k), and ε= 3.162 ×10−11 which esul in a highe sys em
eliabili y and lowe ISE. Figu e 6.11 p esen s he e e ence acking esponse o he con-
ol algo i hm. Al hough he sys em ollows he gi en e e ences o he ou anks, he e
is some ipples especially o olume o ank 1. This could be due o he less weigh as-
signed o his objec i e and o he ac ha he wa e demand sec o dis u bs he olume
o ank 1 o a g ea e ex en han he o he s.

Chap e 6. Heal h-Awa e Con ol 92
FIGURE 6.9: O e all sys em eliabili y e olu ion o he 4 ese oi s in
semi-log scale
FIGURE 6.10: T acking e o o he 4 ese oi s in semi-log scale
6.3.6 Applica ion o a DWN wi h mul iple demands
Now, conside he DWN sys em p esen ed in Figu e 6.12 composed by mul iple sou ces
and demand sec o s o illus a e he me hodology o compu e he o e all sys em eliabil-
i y o a mo e complex example.
Chap e 6. Heal h-Awa e Con ol 93
0 200 400 600 800 1000 1200 1400 1600 1800 2000
0
1
2
3
4
5
x 104
Time [h]
Tanks olume [m3]
ank 1 olume
se poin ank 1
ank 2 olume
se poin ank 2
ank 3 olume
se poin ank 3
ank 4 olume
se poin ank 4
FIGURE 6.11: T acking e e ences o anks
As s a ed in Sec ion 4.3, his hesis and he me hodology p oposed he e a e ocused only
on he opological eliabili y.
F om he sys em s uc u e, i is possible o ob ain a desc ip ion o he sys em which only
akes in o accoun hose componen s which ha e a eliabili y deg ada ion p ocess, in his
hesis hey a e he ac ua o s o he sys em.
Al hough, i is possible o make some educ ions on he s uc u e o complex sys ems
by doing se ies and pa allels equi alences, he e a e some s uc u es ha canno be e-
duced and, in such cases o he me hods o ob ain he o e all sys em eliabili y exp ession
should be used, e.g. he pi o al decomposi ion o he s uc u e unc ion compu a ion
om he minimal pa h/cu se s me hods discussed in Sec ion 4.1.
Mo eo e , his sys em has mo e han one sou ce whe e wa e can be aken and mo e
han one demand sec o whe e wa e mus be supplied. The e o e, he compu a ion o
i s o e all eliabili y is di e en o he sys em wi h one wa e demand sec o . In he
ollowing, a me hodology o compu e i is p esen ed:
S ep 1: Assume a ne wo k wi h l demands. Find he hjminimal pa h se s o each demand
i(mpsj
k, co esponding o he k h minimal pa h se s o he j h demand). This in-
ol es all pumps which connec a demand wi h he sou ces.
Chap e 6. Heal h-Awa e Con ol 94
FIGURE 6.12: DWN wi h 3 sou ces and 4 demands sec o s.
Fo ins ance, he sys em o Figu e 6.12 has 20 minimal pa h se s dis ibu ed as ol-
lows: 9 o demand 1, 4 o demand 2, 5 o demand 3, and 2 o demand 4:
mps1
1={p1, p3, p7}(6.15)
mps1
2={p1, p5, p10}(6.16)
mps1
3={p2, p4, p7}(6.17)
mps1
4={p2, p6, p10}(6.18)
mps1
5={p14, p11, p12}(6.19)
mps1
6={p1, p3, p12, p13}(6.20)
mps1
7={p2, p4, p12, p13}(6.21)
mps1
8={p1, p5, p15, p11, p12}(6.22)
mps1
9={p2, p6, p15, p11, p12}(6.23)
mps2
1={p1, p3, p9}(6.24)
mps2
2={p1, p5, p8}(6.25)
mps2
3={p2, p6, p8}(6.26)
Chap e 6. Heal h-Awa e Con ol 95
mps2
4={p2, p4, p9}(6.27)
mps3
1={p14, p11}(6.28)
mps3
2={p1, p3, p13}(6.29)
mps3
3={p2, p4, p13}(6.30)
mps3
4={p1, p5, p15, p13, p11}(6.31)
mps3
5={p2, p6, p15, p13}(6.32)
mps4
1={p1, p5, p16}(6.33)
mps4
2={p2, p6, p16}(6.34)
S ep 2: Nex , ob ain he s uc u e unc ion o he sys em as:
Φ(X) =
l
Y
j=1 

1−
hj
Y
k=1


1−Y
pi∈mpsj
k
Xi



(6.35)
whe e Xiis a bina y andom a iable ep esen ing he s a e o he i h componen
(as de ined in (4.7)).
S ep 3: Now, he o e all sys em eliabili y should be ob ained by expanding he s uc u e
unc ion exp ession (6.35) and hen simpli y i using he ac ha Xα
i=Xiand aking
he expec a ion (and eplacing P(Xi= 1) by Ri).
Rs=E[Φ(X)] (6.36)
Fo he sys em o his example, he ime spen o compu e he o e all sys em eliabili y
was 321.5709 seconds and he eliabili y exp ession has 610 e ms, which could inc ease
exponen ially as he numbe o minimal pa h inc eases.
Rega ding he RIMs needed o apply he me hodology p oposed in Sec ion 6.2, i is pos-
sible o compu e hem om he s uc u e unc ion as p esen ed in Sec ion 4.4.
6.4 Heal h-Awa e LQR amewo k
6.4.1 LQR amewo k o HAC implemen a ion
As p esen ed in Sec ion 5.2, he LQR algo i hm minimizes he cos unc ion (5.34), ha
includes wo weigh ing ma ices, RLQR and QLQR. The RLQR ma ix weigh s he con ol
e o magni ude and he QLQR one weigh s he sys em s a es.
Chap e 6. Heal h-Awa e Con ol 96
Thus, ma ix RLQR can be used o edis ibu e he con ol e o be ween he ac ua o s
based on eliabili y in o ma ion. The ma ix could be assigned o a combina ion o RIM
indexes, ollowing a simila s udy o ha in he p e ious sec ion. Howe e , jus he
ollowing assignmen will be analyzed:
RLQR(k) = diag(IB(k)),(6.37)
wi h IB= [IB,1, IB,2, . . . , IB,p].
This me hodology is illus a ed h ough an example o e an oc o o o sys em. Fo com-
pa ison pu poses, an addi ional scena io will be conside ed, whe e no ac ua o eliabili y
is aken in o accoun in he LQR cos unc ion. In his second scena io RLQR will be se
as RLQR(k) = I(Iis he iden i y ma ix o p ope dimensions).
6.4.2 Oc o o o UAV model
To desc ibe he dynamics o a mul i o o , i is necessa y o de ine he wo ames in which
i will ope a e: Ine ial ame and Body ame, which a e ela ed by he o a ion ma ix.
The ine ial ame {I}is s a ic and ep esen s he e e ence o he mul i o o while he
body ame {B}is de ined by he o ien a ion o he mul i o o and is si ua ed in i s cen e
o mass (see Figu e 6.13).
FIGURE 6.13: Oc o o o PPNNPPNN s uc u e.
The dynamics o a mul i o o is gi en by he ollowing equa ions [99]:
˙xI= x(6.38)

Chap e 6. Heal h-Awa e Con ol 97
˙yI= y(6.39)
˙zI= z(6.40)
˙ x=1
m[cos(φI) sin(θI) cos(ψI) + sin(φI) sin(ψI)]T(6.41)
˙ y=1
m[cos(φI) sin(θI) sin(ψI)−sin(φI) cos(ψI)]T(6.42)
˙ z=1
m[cos(φI) cos(θI)]T−g(6.43)
˙
φI=p+ sin(φ) an(θ)q+ cos(φ) an(θ) (6.44)
˙
θI= cos(φ)q−sin(φ) (6.45)
˙
ψI=sin(φ)
cos(θ)q+cos(φ)
cos(θ) (6.46)
˙p=1
Jxx
[−(Jzz −Jyy)q −JpqΩp+τx](6.47)
˙q=1
Jyy
[(Jzz −Jxx)p +JppΩp+τy](6.48)
˙ =1
Jzz
[−(Jyy −Jxx)pq +τz](6.49)
whe e Jpis he ine ia momen o he mo o ( o a ing pa s) and he p opelle a ound z
axis, Tis he li o ce, and J=diag(Jxx, Jyy, Jzz)is he ine ia enso o he oc o o o
body.
Then, o he oc o o o wi h he s uc u e PPNNPPNN as he one p esen ed in Fig-
u e 6.13, whe e P and N de ine a posi i e and nega i e eac i e mo o o que espec i ely
( ep esen ed as a ows), Ωpis:
Ωp=−|Ω1|−|Ω2|+|Ω3|+|Ω4|−|Ω5|−|Ω6|+|Ω7|+|Ω8|(6.50)
whe e Ωiis he angula eloci y o he i h mo o .
Fou p opelle s can o a e in a clockwise di ec ion, while he emaining can o a e an i-
clockwise. The oc o o o is mo ed by changing he o o speeds. Fo example, inc easing
o dec easing oge he he eigh p opelle s speeds, e ical mo ion is achie ed. Changing
only he speeds o he p opelle s si ua ed opposi ely p oduces ei he oll o pi ch o a-
ion, coupled wi h he co esponding la e al mo ion. Finally, yaw o a ion esul s om
he di e ence in he coun e - o que be ween each pai o p opelle s. Mo eo e , he oc-
o o o has ac ua o edundancy and can unc ion wi h a leas ou p opelle s o ming a
quad o o s uc u e.
No ice ha he a ows in Figu e 6.13 ep esen he eac i e o que o he mo o s. Fu he -
mo e, he sys em inpu s Ωip oduce a li o ce Tand o ques τin x,y, and zaxis gi en
Chap e 6. Heal h-Awa e Con ol 98
by:
u =Bs uΩ(6.51)
which in i s comple e o m is:







T
τx
τy
τz







=






kbkbkbkbkbkbkbkb
0−kbls(45) −kbl−kbls(45) 0 kbls(45) kbl kbls(45)
−kbl−kblc(45) 0 +kblc(45) kbl kblc(45) 0 −kblc(45)
+kd+kd−kd−kd+kd+kd−kd−kd

























Ω2
1
Ω2
2
Ω2
3
Ω2
4
Ω2
5
Ω2
6
Ω2
7
Ω2
8


















,
(6.52)
whe e kband kda e coe icien s o he mo o and lis he dis ance be ween he cen e o
mass and he cen e o he o o . The pa ame e s alue which de ine he oc o o o model
a e p esen ed in Table 6.4.
TABLE 6.4: Pa ame e s alue
Pa ame e Symbol Value
Body ine ia Jxx =Jyy 25 ·10−3[kgm2]
Body ine ia Jzz 42 ·10−3[kgm2]
P opelle ine ia Jp104 ·10−6[kgm2]
Mass m1.86 [kg]
A m leng h l0.4[m]
Th us ac o kb54.2·10−6[Ns2]
D ag ac o kd1.1·10−6[Nms2]
Equa ions (6.38)-(6.49) de ine he nonlinea s a e space model o an oc o o o ha should
be linea ized o apply a heal h awa e linea -quad a ic con olle (LQR) o he UAV sys-
em.
The s a e and inpu s ec o s conside ed a e x= [x y z φ θ ψ x y zp q ]Tand
u= [Ω2
1Ω2
2Ω2
3Ω2
4Ω2
5Ω2
6Ω2
7Ω2
8]T, espec i ely, he Taylo se ies app oxima ion a he
ho e posi ion is applied.
The ho e posi ion co esponds o he si ua ion whe e he planes xy o bo h ames ({I}
and {B}) a e in pa allel and he mo o s a e gene a ing a li ing o ce equal o he weigh
o he oc o o o .
Chap e 6. Heal h-Awa e Con ol 99
6.4.3 Oc o o o LQR con olle
In his example, he con ol o he UAV consis s in a cascade s uc u e (see Figu e 6.14),
whe e he ou e loop con ols he pose o he UAV in axis xand yand he inne loop
con ols he pose in axis zand he o ien a ion in axis x,yand z. No e ha he inne loop
equency is as e han he ou e loop one.
FIGURE 6.14: Con ol scheme.
The e o e, he linea model will be di ided in o wo subsys ems. On one hand, he inne
con ol loop, whe e eIis he inne s a e ec o deno ed as:
eI=x e i−xi
= [ezeφeθeψe zepeqe ]T,(6.53)
and ∆uIis he inne inpu ec o deno ed as:
∆uI=h∆Ω2
1∆Ω2
2∆Ω2
3∆Ω2
4∆Ω2
5∆Ω2
6∆Ω2
7∆Ω2
8iT,(6.54)
wi h ∆Ω2
i= Ω2
i−u = Ω2
i−mg/(8kb), whe e u s ands o he inpu p o iding he
equilib ium poin .
The e o e, he inne loop model is gi en by:
˙eI( ) = AIeI( ) + BIBs ∆uI( )(6.55)
whe e
AI="04×4I4×4
04×404×4#,
and
BI="04×4
βI#,
wi h βI=diag(1/m, 1/Jxx,1/Jyy,1/Jzz)is a diagonal ma ix, I4×4is he iden i y ma ix
and Bs is he s uc u al ma ix (6.52).
Chap e 6. Heal h-Awa e Con ol 100
On he o he hand, he ou e con ol loop, whe e eois he ou e s a e ec o deno ed as:
eo=x e o−xo
=exeye xe yT,(6.56)
and ∆uois he ou e inpu ec o deno ed as:
∆uo= [∆φ∆θ]T
= [φ e θ e ]T.(6.57)
The e o e, he ou e loop model is:
˙eo( ) = Aoeo( ) + Bo∆uo( ),(6.58)
whe e
Ao="02×2I2×2
02×202×2#,
and
Bo=






0 0
0 0
0g
−g0







,
wi h gis he g a i a ional accele a ion equal o 9.81 [m/s2].
Rega ding he HAC LQR-based, Figu e 6.15 p esen s i s scheme, whe e a module o com-
pu e he ac ua o s and sys em eliabili ies is a ached o he inne con ol loop and is used
o adap he pa ame e s o he con olle , i.e. he RLQR ma ix o he inne loop.
6.4.4 Oc o o o eliabili y model
In his applica ion, he co a ia e is exp essed as a unc ion o he load and he age o he
ac ua o . The e o e, he co a ia e unc ion and he ailu e a e used a e (6.7) and (6.8),
espec i ely.
Rega ding he o e all sys em eliabili y, i is compu ed using he sys em s uc u e unc-
ion as p esen ed in Sec ion 4.1. I is assumed ha he o e all sys em eliabili y is de e -
mined by he eliabili y o i s ac ua o s and he sys em con ollabili y.
Al hough he oc o o o sys em has 8 ac ua o s ( i∀i∈[1,8]) in e ms o con ollabili y,
i can ligh wi hou any p oblem wi h a leas 4 o hem. In such scena io he sys em
Chap e 6. Heal h-Awa e Con ol 107
FIGURE 6.21: Di e ence be ween sys em eliabili ies RIB
Sand RI
S.
Assuming ha he ac ua o wea is p opo ional o he exe ed con ol e o u, a model
o he cumula i e deg ada ion z(k)∈Rpcan be w i en as:
z(k+ 1) = z(k) + Γ|u(k)|,(6.67)
whe e Γ=diag(γ1, γ2, . . . , γp)is a diagonal ma ix o deg ada ion coe icien s associa ed
wi h he pelemen s o z(k). These coe icien s a e assumed o be cons an and calib a ed
h ough expe imen al da a and hey may be ela ed o measu able a iables such as i-
b a ion, in e nal elec ical esis ance, o empe a u e, which a e assumed o be known.
Assume ha a main enance se ice is scheduled a sample k=kM. The goal consis s in
gua an eeing ha he cumula i e deg ada ion emains below a sa e h eshold z h du ing
he main enance ho izon,
z(k)≤z h,∀k∈[0, kM].(6.68)
6.5.2 MPC o mula ion unde ac ua o usage cons ain s
A cumula i e deg ada ion model which assumes ha he deg ada ion o he ac ua o s
is p opo ional o he exe ed con ol e o uand i s a ia ions ∆uis p oposed in [124,
125]. In such an app oach, Pe ei a, Gal ao, e al. [124] p opose o manage he ac ua o s
deg ada ion h ough he con ol ac ion, imposing a h eshold in he cumula i e deg a-
da ion o he ac ua o s, and adap ing he used ac ua o o his h eshold, by in oducing
he deg ada ion as a cons ain in he cos unc ion. In his case, he sys em pe o mance
is no a ec ed because o he edundancy o he ac ua o s in he sys em. I an ac ua o
eaches i s maxim deg ada ion, i s usage is educed and compensa ed by he edundan
ac ua o .

Chap e 6. Heal h-Awa e Con ol 108
The ac ua o s deg ada ion model will be in eg a ed in he cos unc ion (5.12) which min-
imizes he acking e o , he con ol inc emen s, and he magni ude o he con ol inpu .
Cons ain (6.68) should be included in he op imiza ion p oblem. Howe e , his would
equi e ex ending he p edic ion ho izon o e he main enance ho izon, becoming an
in ac able p oblem. Ins ead he ollowing cons ain will be conside ed:
z(k+Hp)≤zmax(k)(6.69)
To de e mine zmax he uni o m " a ioning" heu is ic p oposed in [124] will be ollowed.
This heu is ic s a es ha he deg ada ion o e a p edic ion ho izon o leng h Hp< kMis
allowed o inc ease by
Hp
z h −z(k)
kM+Hp−k(6.70)
The e o e, he maximum deg ada ion a he end o p edic ion ho izon mus no exceed:
zmax(k) = z(k) + Hp
z h −z(k)
kM+Hp−k(6.71)
F om (6.67), a p edic ion equa ion o he deg ada ion index can be w i en as
ˆ
z(k+Hp|k) = z(k) + Ξ|ˆ
u(k)|(6.72)
whe e
Ξ=













Γ 0p×p··· 0p×p
0p×pΓ··· 0p×p
.
.
..
.
.....
.
.
0p×p0p×p··· Γ
.
.
..
.
.....
.
.
0p×p0p×p··· (Hp−Hc+ 1)Γ













(6.73)
Thus, using equa ions (6.72) and (5.18), he cons ain (6.69) can be eplaced wi h
z(k) + ΞΘ≤zmax (6.74)
Chap e 6. Heal h-Awa e Con ol 109
Finally, (6.74) cons ain should be added o he LP p oblem as ollows:
minimize
a(k)cTa(k)
s. . " 1
2#a(k)≤"b1(k)
b2(k)#(6.75)
whe e
2=h0p×pHc0p×qHpΞ 0p×pHc0pi(6.76)
and
b2(k) = hzmax(k)−z(k)i(6.77)
The idea o include ac ua o s usage cons ain s in he op imiza ion p oblem was al eady
p esen ed in [124]. The con ibu ion in his hesis consis s in analyzing he ole o he cos
unc ion weigh s in imp o ing he sys em sa e y. Nex , his me hodology will be applied
o a Twin Ro o MIMO sys em.
6.5.3 Twin Ro o MIMO sys em
The Twin-Ro o MIMO Sys em (TRMS) is a labo a o y se up (Figu e 6.22) de eloped by
Feedback Ins umen s Limi ed o con ol expe imen s. The sys em is pe cei ed as a
challenging enginee ing p oblem due o i s high non-linea i y, c oss-coupling be ween
i s wo axes, and inaccessibili y o some o i s s a es measu es.
FIGURE 6.22: Componen s o he Twin Ro o MIMO Sys em
The TRMS mechanical uni has wo o o s ( he main and he ail, bo h d i en by DC
Chap e 6. Heal h-Awa e Con ol 110
mo o s) placed on a beam oge he wi h a coun e balance whose a m wi h a weigh a i s
end is ixed o he beam a he pi o and i de e mines a s able equilib ium posi ion. The
beam can o a e eely bo h in he ho izon al and e ical planes.
This applica ion aims a showing he e ec o he deg ada ion on he sys em pe o mance
h ough he con olle . This is pa o he i s esul s ob ained in he esea ch on his
hesis.
Accu a e models o TRMS a e p oposed by [27, 46, 130] and [135]. Each o hese models
lead o a se o nonlinea di e en ial equa ions whe e he TRMS is spli in o simple
subsys ems: he DC-Mo o s, he p opelle s and he beam. The i s wo ha e independen
dynamics, ha is, he main mo o does no a ec he beha io o he ail mo o , and
ice e sa. The same is ue o he p opelle s. On he o he hand, he dynamics o
he beam a e s ongly nonlinea wi h he p esence o in e ac ion phenomenon among
he ho izon al and he e ical dynamics. The s a e o he beam is desc ibed by ou
p ocess a iables: ho izon al and e ical angles measu ed by posi ion senso s i ed a
he pi o , and hei wo co esponding angula eloci ies. The acho-gene a o s a e used
o measu e he angula eloci ies o he o o s.
The cons an s o he nonlinea model a e p esen ed in Table 6.7.
Pa ame e Symbol Value
Ae odynamic o ce coe . o he ail o o o posi i e
ωh
k hp 1.84 ·10−6[N/ pm2]
Ae odynamic o ce coe . o he ail o o o nega i e
ωh
k hn 2.2·10−6[N/ pm2]
Ae odynamic o ce coe . o he main o o o posi-
i e ω
k p 1.62 ·10−5[N/ pm2]
Ae odynamic o ce coe . o he main o o o nega-
i e ω
k n 1.08 ·10−5[N/ pm2]
Angula eloci y o he TRMS a ound he e ical
axis
Ωh[ pm]
Angula eloci y o he TRMS a ound he ho izon al
axis
Ω [ pm]
A ma u e induc ance o ail / main mo o Lah/a 0.86 ×10−3[H]
A ma u e esis ance o ail / main mo o Rah/a 8[Ω]
Cable o ce coe icien o nega i e θhkchn kchp ∗0.9
Cable o ce coe icien o posi i e θ kchp 8.54 ·10−3
Ho izon al ic ion coe icien o he beam subsys em koh 4.7·10−3
Ve ical ic ion coe icien o he beam subsys em ko 1.31 ·10−3
Dis ance be ween he coun e weigh and he join lcb 0.276[m]
D ag ic ion coe icien o he ail p opelle k h 5·10−8
D ag ic ion coe icien k 5.6·10−7
Chap e 6. Heal h-Awa e Con ol 111
Equilib ium pi ch angle (u = 0.2753V)θ0
0[◦]
Gy oscopic cons an kg0.2
Inpu cons an o he main mo o k28.5
Inpu ol age o he ail mo o uh[V]
Inpu ol age o he main mo o u [V]
Inpu cons an o he ail mo o k16.5
Leng h o ail pa o he beam l 0.282[m]
Leng h o main pa o he beam lm0.246[m]
Leng h o coun e -weigh beam lb0.290[m]
Mass o he coun e -weigh mcb 0.068[kg]
Mass o he coun e -weigh beam mb0.022[kg]
Mass o main pa o he beam mm0.014[kg]
Mass o he ail shield m s 0.119[kg]
Mass o he main DC mo o mm 0.236[kg]
Mass o he ail DC mo o m 0.221[kg]
Momen o ine ia main DC mo o Jm 2.16 ·10−4[kgm2]
Momen o ine ia in ail mo o J 3.14 ·10−5[kgm2]
Posi i e cons an k 2.6·10−5
Mass o he main shield mms 0.219[kg]
Mass o he ail pa o he beam m 0.015[kg]
Pi ch angle o he beam θ [◦]
Physical cons an kah/ ϕh/ 0.0202[Nm/A]
Posi i e cons an km2·10−4
Radius o he ail shield s 0.1[m]
Radius o he main shield ms 0.155[m]
Ro a ional eloci y o he ail o o ωh[ pm]
Ro a ional eloci y o he main o o ω [ pm]
Viscous ic ion coe icien o he ail p opelle B 2.3·10−5[kg.m2/s]
Viscous ic ion coe icien o he main p opelle Bm 4.5·10−5[kg.m2/s]
Yaw angle o he beam θh[◦]
The ma hema ical model o he TRMS is ep esen ed by he ollowing se o nonlinea
di e en ial equa ions [107]:
diah/
d =−Rah/
Lah/
iah/ −kah/ ϕh/
Lah/
ωh/ +k1/2
Lah/
uh/ (6.78)
dωh/
d =kah/ ϕh/
J /m
iah/ −B /m
J /m
ωh/ − 1/4(ωh/ )
J /m
(6.79)
dΩh
d =l 2(ωh) cos θ −kohΩh− 3(θh)
Dcos2θ +Esin2θ +F
+kmω sin θ Ω Dcos2θ −Esin2θ −F−2Ecos2θ 
Dcos2θ +Esin2θ +F2
Chap e 6. Heal h-Awa e Con ol 112
+kmcos θ (ka ϕ ia −Bm ω − 4(ω ))
Dcos2θ +Esin2θ +FJm
(6.80)
dθh
d =Ωh(6.81)
dΩ
d =lm 5(ω ) + kgΩh 5(ω ) cos θ −ko Ω
J
+g((A−B) cos θ −Csin θ )−0.5Ωh2Hsin 2θ
J
+k (kahϕhiah −B ωh− 1(ωh))
J J
(6.82)
dθ
d =Ω (6.83)
whe e uh/ is he inpu ol age o he ail/main mo o , Ωh/ is he angula eloci y a ound
he e ical/ho izon al axis, θh/ is he azimu h/pi ch beam angle (ho izon al/ e ical
plane), ωh/ is he o a ional eloci y o he ail/main o o , J /m is he momen o in-
e ia in DC-mo o ail/main p opelle subsys em, kah/ ϕh/ is he o que cons an o he
ail/main mo o , and J is he momen o ine ia abou he ho izon al axis. Func ions i
a e de ined as:
1(ωh) = sign(ωh)k hω2
h(6.84)
2(ωh) = (k hpω2
hi ωh≥0
−k hnω2
hi ωh<0(6.85)
3(θh) = (kchpθhi θh≥0
kchnθhi θh<0(6.86)
4(ω ) = sign(ω )k ω2
(6.87)
5(ω ) = (k pω2
i ω ≥0
−k nω2
i ω <0(6.88)
Finally, he cons an s o he nonlinea model (6.78)-(6.83) a e de ined as:
A=m
2+m +m sl B=mm
2+mm +mmslm
C=mb
2lb+mcblcb H=Al +Blm+mb
2l2
b+mcbl2
cb
D=mb
3l2
b+mcbl2
cb F=mms 2
ms +m s
2 2
s
E=mm
3+mm +mmsl2
m+m
3+m +m sl2
whe e mms and m s a e he masses o he main and ail shields, mmand m a e he masses
o he main and he ail pa s o he beam, mm and m a e he masses o he main and
he ail DC-mo o wi h main and ail o o , mband lba e he mass and he leng h o

Chap e 6. Heal h-Awa e Con ol 113
he coun e -weigh beam, mcb and lcb ep esen he mass o he coun e -weigh and he
dis ance be ween he coun e -weigh and he join , and ms and s a e he adius o he
main and ail shield.
6.5.4 MPC o TRMS sys em
Using he nonlinea model o he TRMS (6.78)-(6.83) a linea model has been ob ained by
linea izing a ound an equilib ium poin (¯uh= 0.9865,¯u = 0.2753,¯
θh= 1.5700,¯
θ = 0)
and disc e izing wi h a sample ime Ts= 0.015 s:
A=


















1.0 0.015 4.1e−7 2.8e−8 2.3e−5−6.5e−6−1.9e−7 9.1e−9
−2.8e−3 1.0 5.4e−5 3.7e−6 3.1e−3−8.5e−4−2.5e−5 4.9e−7
0 0 0.96 0.066 0 0 0 0
0 0 −2.4e−3−1.7e−4 0 0 0 0
−1.2e−8 1.3e−5−7.6e−8 5.1e−9 1.0 0.015 8.2e−7 8.1e−9
−2.5e−6 1.8e−3−1.0e−5 5.4e−9−0.075 1.0 1.1e−4 1.1e−6
0 0 0 0 0 0 0.99 0.01
0 0 0 0 0 0 −2.5e−3−2.5e−5


















,
(6.89)
B=


















1.0e−6 7.2e−7
2.1e−4 9.0e−5
7.6 0
0.79 0
3.9e−7 4.0e−7
3.9e−5 8.0e−5
0 1.5
0 1.1


















,
(6.90)
and C="10000000
00001000#
(6.91)
The sys em inpu ec o is u= [uhu ]Tand he sys em s a es a e x= [θhΩhωhiah θ Ω
ω ia ]T. The con ol objec i e consis s in main aining he pi ch angle θ and he azimu h
angle θhin he desi ed angula posi ions unde decoupling and ac ua o deg ada ion
e ec s. Fo he azimu h angle, a squa e e e ence signal has been de ined, whe eas he
pi ch angle e e ence signal has been se o 0◦. As no all s a e a iables a e measu ed, a
educed-o de s a e obse e is used o es ima e nonmeasu ed a iables, which a e he
angula momen ums o he beams and he a ma u e cu en s o he o o s (Ωh,Ω ,ia ,
iah).
Rega ding he HAC o he TRMS, Figu e 6.23 p esen s he con ol scheme o he ap-
p oach. He e, an MPC cos unc ion is uned based on he deg ada ion coe icien o he
Chap e 6. Heal h-Awa e Con ol 114
ac ua o s. Addi ionally, he da a p o ided by a moni o ing module is used o changes
he cons ains in he op imiza ion p oblem. This module moni o s sys em con ol inpu s
and compu e he deg ada ion o he ac ua o s acco ding o hei use.
FIGURE 6.23: HAC MPC-based o TRMS.
Since ac ua o deg ada ion depends on he ime e olu ion o u, he design o a heal h-
awa e MPC law can be aken in o accoun h ough wo app oaches: he inclusion o
he deg ada ion cons ain (6.69) and he p ope uning o he weigh ing ac o s ρand
α. To s udy his dependency, 4 case s udies a e simula ed using he nonlinea model o
he TRMS. The gene al MPC pa ame e s a e summa ized in Table 6.8, and he speci ic
pa ame e s co esponding o each case a e shown in Table 6.9. In bo h ables, subindex
1 (2) co esponds o inpu uh(u ), excep o pa ame e µwhich is ela ed o θh(θ ).
HMis he main enance in e al, and is ela ed o he main enance ho izon as ollows:
HM=kMTs.
Simula ion esul s a e shown in Figu es 6.24-6.27, o azimu h angle θhand pi ch angle
θ . Cumula i e deg ada ion zis shown in Figu e 6.28(a) o ail and Figu e 6.28(b) o
main mo o .
In Table 6.10 some pe o mance indexes ha e been e alua ed o a simula ion in e al
o 1 hou (Tsim). Rows 2 and 3 indica e he in eg al o he squa e o he acking e o
(ISE) co esponding o bo h ou pu angles. The smalle he ISE indexes a e, he be e he
con ol pe o mance is. Rows 4 and 5 co espond o he cumula i e deg ada ion o bo h
ac ua o s ( ail/main) a he end o simula ion.
Chap e 6. Heal h-Awa e Con ol 115
TABLE 6.8: Lis o MPC pa ame e s
Pa ame e Symbol Value
P edic ion ho izon Hp80 [s]
Con ol ho izon Hc4 [s]
Main enance ho izon HM2 [yea s]
Sampling ime Ts0.015 [s]
Simula ion ime Tsim 1 [h]
T acking weigh s µ1/µ21
Con ol e o weigh s α1/α2
Con ol e o a ia ion weigh s ρ1/ρ2
Uppe con ol e o s u1max/u2max 12 [V]
Lowe con ol e o s u1mim/u2mim -12 [V]
Deg ada ion h eshold z h11
Deg ada ion h eshold z h21
Deg ada ion a e γ11.1·10−11
Deg ada ion a e γ22.7·10−10
TABLE 6.9: Case s udy pa ame e s
Case ρ1ρ2α1α2Deg. cons ain
1 0 0 0 0 No
2 0 0 0 0 Yes
3 1/120 1/120 1/120 1/120 Yes
4 1/1000 1/120 1/1000 1/120 Yes
TABLE 6.10: Pe o mance e alua ion
Case 1 2 3 4
ISE Θh20.2395 20.2317 21.9480 21.5749
ISE Θ 1.4016 1.4246 1.1740 1.1289
Zhcum. (·10−4)0.0305 0.0307 0.0261 0.0266
Z cum. (·10−4)0.4434 0.4390 0.3109 0.3114
Case 1 co esponds o a neu al MPC con ol, because i does no ake in o accoun he
deg ada ion cons ain , and only acking e o s a e penalized in he cos unc ion. As a
esul , con ol pe o mance is good, bu he main mo o deg ada ion a Tsim is highe in
compa ison o he o he cases.
Chap e 6. Heal h-Awa e Con ol 116
3000 3010 3020 3030 3040 3050 3060 3070 3080 3090 3100
0
20
40
60
80
100
Time [s]
yaw Θ [°]
Θh
Re e ence
3000 3010 3020 3030 3040 3050 3060 3070 3080 3090 3100
−10
0
10
20
Time [s]
pi ch angle Θ [°]
Θ
Re e ence
FIGURE 6.24: Response o azimu h and pi ch angles o Case 1.
3000 3010 3020 3030 3040 3050 3060 3070 3080 3090 3100
0
20
40
60
80
100
Time [s]
yaw Θ [°]
Θh
Re e ence
3000 3010 3020 3030 3040 3050 3060 3070 3080 3090 3100
−10
0
10
20
Time [s]
pi ch angle Θ [°]
Θ
Re e ence
FIGURE 6.25: Response o azimu h and pi ch angles o Case 2.
In Case 2, he deg ada ion cons ain is aken in o accoun and he main mo o deg ada-
ion is no allowed o exceed he maximum h eshold be o e he main enance ho izon. As
a consequence he pi ch con ol pe o mance is deg aded (i.e., ISE inc eases), bu deg a-
da ion is educed (i.e., cumula i e zdec eases).
Chap e 7. Reliabili y In e p e a ions 123
FIGURE 7.2: Expec ed eliabili y o he asse Rτ
i(TM).
in e al [τ, TM]. Hence, he co esponding eliabili y o he asse becomes:
Rτ
i(TM) = e−λτ
i×(TM−τ)∀i= 1, . . . , p (7.4)
whe e λi(τ)is he i h asse ailu e a e compu ed a ime τ.
Rega ding sys em eliabili y, unde he i s in e p e a ion i will be called he ins an a-
neous sys em eliabili y, deno ed as RS( ). Unde he second in e p e a ion i will be called
he expec ed sys em eliabili y, deno ed as Rτ
S(TM).
The Reliabili y Impo ance Measu e (RIM) should be exp essed acco dingly wi h he ex-
pec ed eliabili y in e p e a ion as i is p esen ed below.
7.1.1 Impo ance eliabili y measu es
As p esen ed in Sec ion 4.4, o measu e and quan i y he impac o asse ailu es o e he
unc ioning o he sys em, se e al indica o s conce ning eliabili y impo ance ha e been
p oposed, each o hem wi h a pa icula pu pose [79].
Fo ins ance, he Bi nbaum’s impo ance measu e IBide ined in (4.24) quan i ies he max-
imum dec ease o sys em eliabili y due o eliabili y changes o he i h ac ua o .
Acco ding o bo h eliabili y in e p e a ions, wo Bi nbaum’s measu es a e p oposed. On
he one hand, unde he ins an aneous eliabili y in e p e a ion, he asse Bi nbaum’s
impo ance measu e will be compu ed as in (4.24), whe eas unde he expec ed eliabili y
in e p e a ion he Bi nbaum’s impo ance measu e will be de e mined as ollows:
Iτ
Bi(TM) =∂Rτ
S(TM)
∂Rτ
i(TM)=Rτ
S(1i, TM)−Rτ
S(0i, TM)(7.5)

Chap e 7. Reliabili y In e p e a ions 124
which deno es he asse Bi nbaum’s impo ance measu e a mission ime ins an TMcom-
pu ed a cu en ime τ.
7.1.2 Redis ibu ion policy
The edis ibu ion policy discussed in Sec ion 6.3.3 which is based on he ac ua o s RIMs
is now used o compa e bo h eliabili y in e p e a ions (i.e., ins an aneous and expec ed).
The MPC echnique acili a es he implemen a ion o a Heal h-Awa e Con ol and he
explo a ion o di e en edis ibu ion policies wi hou signi ican changes in he con ol
algo i hm.
Fi e scena ios a e p oposed se ing a di e en weigh (ρi) unde he wo eliabili y in e -
p e a ions.
In he i s scena io ac ua o eliabili y is a ge ed. Acco dingly, ρiis se unde he ins an-
aneous eliabili y in e p e a ion as ollows:
ρi(k)=1−Ri(k).(7.6)
In he i s scena io, he ins an aneous eliabili y in e p e a ion is assumed.
In he second scena io he o e all sys em eliabili y is a ge ed using he Bi nbaum’s im-
po ance measu e. Acco dingly, ρiis se unde he ins an aneous eliabili y in e p e a ion
as ollows:
ρi(k) = IBi(k).(7.7)
Simila ly, unde he expec ed eliabili y in e p e a ion, he hi d scena io co esponds o:
ρi(k)=1−Rk
i(k )(7.8)
and he ou h scena io o:
ρi(k) = Ik
Bi(k )(7.9)
whe e k is he end o mission sample, co esponding o TM.
In he i h weigh assignmen , which is common o bo h eliabili y in e p e a ions, no
eliabili y eedback is aken in o accoun , i.e., ρi(k)=1.
Chap e 7. Reliabili y In e p e a ions 125
7.1.3 Pe o mance e alua ion
In addi ion o he cumula i e con ol e o index (Ucum) new indices a e de ined nex ,
which will be used in assessing he heal h-awa e con ol pe o mance unde bo h elia-
bili y in e p e a ions.
De ini ion 7.1. The Cumula i e Sys em Reliabili y indices indica e he agg ega ed sys-
em eliabili y o e he mission ime.
Unde he ins an aneous eliabili y in e p e a ion, i is deno ed in disc e e ime as:
RScum =Ts
TM/Ts
X
k=0
Rs(k).(7.10)
And unde he expec ed eliabili y in e p e a ion i is deno ed as:
Rk
Scum =Ts
TM/Ts
X
k=0
Rk
s(k ).(7.11)
A highe alue indica es a be e managemen o he asse s eliabili ies wi h he objec i e
o imp o ing he o e all sys em eliabili y.
7.2 D inking Wa e Ne wo k example
Bo h eliabili y in e p e a ions will be compa ed on an applica ion o he DWN p esen ed
in Sec ion 6.3.4. The objec i e is o apply he MPC HAC me hodology o main ain he
pumps and anks wi hin hei bounds and ex end he eliabili y o he sys em. The cos
unc ion used in his example is (6.12) wi h ε= 0, and he ailu e a e o he pumps is
compu ed using (6.8) in acco dance wi h he eliabili y in e p e a ion.
The pa ame e s o he simula ion a e p esen ed in Table 7.1.
Figu e 7.3 shows he ins an aneous sys em eliabili y in he scena io whe e no eliabil-
i y eedback me hod is applied (ρi= 1) whe e he o e all sys em eliabili y p esen s a
g adual dec easing beha io , which is cha ac e is ic o he ins an aneous eliabili y in e -
p e a ion. This is he nominal scena io and will be compa ed wi h he esul s o he o he
ρise ings.
To illus a e how eliabili y beha es unde he expec ed eliabili y in e p e a ion, he e-
liabili y compu ed a each sampling ime we e p ojec ed τ+TMhou s in o he u u e.
Chap e 7. Reliabili y In e p e a ions 126
TABLE 7.1: Simula ion pa ame e s
Pa ame e Symbol Value
P edic ion ho izon Hp24 [h]
Con ol ho izon Hc8 [h]
Sampling ime Ts1 [h]
Componen pa ame e βi10−2∀i∈[1,10]
Uppe con ol bound ui
0.75 0.75 0.75 1.20 0.85 [m3/s]
1.60 1.70 0.85 1.70 1.60
Lowe con ol bound ui0∀i∈[1,10] [m3/s]
Baseline ailu e a e λ0
i
9.85 10.70 10.50 1.40 0.85 [h−1·10−4]
0.80 11.70 0.60 0.74 0.78
Uppe s a e bound xi65200 3100 14450 11745 [m3]
Lowe s a e bound xi25000 2200 5200 3500 [m3]
Ini ial s a es xi(0) 45100 2650 9825 7622 [m3]
0 200 400 600 800 1000 1200 1400 1600 1800 2000
0.4
0.5
0.6
0.7
0.8
0.9
1
FIGURE 7.3: Ins an aneous o e all sys em eliabili y p o ile e olu ion wi h
ρi= 1.
Some o hose alues a e p esen ed in Figu e 7.4, whe e he o e all sys em eliabili y
compu ed a τ= 1,500,1000,1500, and 2000 hou s is p ojec ed in o he u u e.
Nex , Figu e 7.5 p esen s he expec ed o e all sys em eliabili y e olu ion a ime ins an
TM, which consis s o he alues o each sys em eliabili y p ojec ion a TMcompu ed a
each sampling ime. In his case, he eliabili y s a s om a alue be ween 0 and less
han 1 and mo es owa ds 1.
Chap e 7. Reliabili y In e p e a ions 127
0 500 1000 1500 2000 2500 3000 3500 4000
0.4
0.5
0.6
0.7
0.8
0.9
1
FIGURE 7.4: Expec ed o e all sys em eliabili y p o ile e olu ion a di e -
en ime ins an s wi h ρi= 1.
0 200 400 600 800 1000 1200 1400 1600 1800 2000
0.65
0.7
0.75
0.8
0.85
0.9
0.95
1
FIGURE 7.5: Expec ed sys em eliabili y a mission ime TM= 2000he o-
lu ion wi h ρi= 1.
7.2.1 Reliabili ies compa ison
The i e scena ios p oposed in Sec ion 7.1.2 ha e been conside ed in he MPC con ol o
he DWN. All cases will be assessed unde bo h cumula i e eliabili y indices (7.10) and
(7.11).
Figu e 7.6 shows he ins an aneous sys em eliabili y e olu ion o he i e scena ios.
Rema k ha he mos sui able policies ha imp o e sys em eliabili y a e hose based on
he Bi nbaum’s measu e.
Chap e 7. Reliabili y In e p e a ions 128
0 200 400 600 800 1000 1200 1400 1600 1800 2000
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
FIGURE 7.6: Ins an aneous sys em eliabili y.
Figu e 7.7 p o ides he expec ed sys em eliabili y a he end o he mission ime TM o
he i e scena ios. Again he bes esul s co espond o hose based on he Bi nbaum’s
measu e.
0 200 400 600 800 1000 1200 1400 1600 1800 2000
0.6
0.65
0.7
0.75
0.8
0.85
0.9
0.95
1
FIGURE 7.7: Expec ed sys em eliabili y a mission ime .
The eliabili y pe o mance indices we e also compu ed o each scena io and a e p e-
sen ed in Table 7.2.
The pe o mance indices con i m ha he bes eliabili y pe o mance is a ained when
using a edis ibu ion policy based on he Bi nbaum’s impo ance measu e, i.e., (7.7) and

Chap e 7. Reliabili y In e p e a ions 129
TABLE 7.2: Reliabili y pe o mance indexes.
ρi(k)RScum [·106]Rk
Scum [·106]Ucum [·106]
1 5.6131 5.5583 1.5370
1−Ri(k) 5.4046 5.4054 1.9687
1−Rk
i(k ) 5.4525 5.4340 1.9002
IBi(k) 6.1006 6.1653 3.2158
Ik
Bi(k ) 6.0915 6.1447 3.5040
(7.9), wi h a small imp o emen when he ins an aneous eliabili y in e p e a ion is ol-
lowed, i.e., (7.7). Focusing on ac ua o eliabili y (i.e., (7.6) and (7.8)) does no op imize
sys em eliabili y. Howe e , a ge ing sys em eliabili y leads o a g ea e ac ua o ene gy
expendi u e.
To illus a e he pe o mance o he con ol algo i hm, ank olumes o he bes edis i-
bu ion policy, co esponding o (7.7), a e p esen ed in Figu e 7.8. No e ha , he DWN is
able o supply he equi ed wa e demand main aining ank olumes wi hin he speci ied
bounds.
0 200 400 600 800 1000 1200 1400 1600 1800 2000
0
0.5
1
1.5
2
2.5
3
3.5 10 4
FIGURE 7.8: Tank olumes co esponding o ρi(k) = IBi(k).
And he pump con ol ac ions co esponding o he policy gi en in (7.7) a e p esen ed in
Figu e 7.9. No e ha he con ol ac ions e ol e in ime acco ding o he impo ance o
each ac ua o , which change dynamically as hei eliabili ies change.
Rema k ha in his scena io, he cos unc ion which compu es he con ol ac ions does
no ake in o accoun any acking objec i e. The e o e, ank olumes eely e ol e wi hin
Chap e 7. Reliabili y In e p e a ions 130
FIGURE 7.9: Pump commands co esponding o ρi(k) = IBi(k).
hei bounds.
7.3 Conclusions
In his chap e , wo eliabili y in e p e a ions ha e been p esen ed and illus a ed using
a DWN sys em. Bo h app oaches ha e been applied o a Heal h-Awa e Con ol scheme
based on an MPC algo i hm wi h he objec i e o imp o ing sys em eliabili y.
Chap e 7. Reliabili y In e p e a ions 131
The s udy o wo di e en in e p e a ions o eliabili y we e p esen ed. This s udy was
done due o he need o cla i y wha is mean by eliabili y in bo h cases and wha i is
ob ained by using each o hose in e p e a ions.
The esul s o he edis ibu ion policy p o ide simila esul s in e ms o eliabili y en-
hancemen independen ly o he eliabili y in e p e a ion. Thus, bo h in e p e a ions a e
i ually equi alen .
This Chap e aims o illus a e ano he in e p e a ion o he eliabili y. I is no in ended
o do he same expe imen a ions done in he p e ious chap e whe e di e en weigh ing
c i e ia we e compa ed, and only one RIM was used in he expe imen . Mo eo e , an-
o he RIMs o a mix o hem can gi e be e esul s in e ms o eliabili y imp o emen ,
as i was demons a ed in he p e ious chap e .
Mo eo e , by applying bo h o hem esul s coincide in de e mining ha he Bi nbaum’s
measu e is he bes app oach o in eg a e he asse s eliabili y in o he con ol law.
132
Pa III
Conclusions and pe spec i es