Robust fault detection of singular LPV systems with multiple time-varying delays

In . J. Appl. Ma h. Compu . Sci., 2016, Vol. 26, No. 1, 45–61
DOI: 10.1515/amcs-2016-0004
ROBUST FAULT DETECTION OF SINGULAR LPV SYSTEMS WITH MULTIPLE
TIME–VARYING DELAYS
AMIR HOSSEIN HASSANABADIa,MASOUD SHAFIEEa,b, ∗,VICENÇ PUIGc
aDepa men o Elec ical Enginee ing
Ami kabi Uni e si y o Technology (Teh an Poly echnic), 424, Ha ez A ., Teh an, I an
e-mail: {a.hassanabadi,msha iee}@au .ac.i
bSys em, In o ma ion and Con ol Resea ch Cen e
Ami kabi Uni e si y o Technology (Teh an Poly echnic), 424, Ha ez A e., Teh an, I an
cAd anced Con ol Sys ems G oup (SAC)
Uni e si a Poli ècnica de Ca alunya (UPC), Pau Ga gallo, 5, 08028 Ba celona, Spain
e-mail: [email p o ec ed]
In his pape , he obus aul de ec ion p oblem o LPV singula delayed sys ems in he p esence o dis u bances and
ac ua o aul s is conside ed. Fo bo h dis u bance decoupling and ac ua o aul de ec ion, an unknown inpu obse e
(UIO) is p oposed. The aim is o compu e a esidual signal which has minimum sensi i i y o dis u bances while ha ing
maximum sensi i i y o aul s. Robus ness o unknown inpu s is o mula ed in he sense o he H∞-no m by means o he
bounded eal lemma (BRL) o LPV delayed sys ems. In o de o o mula e aul sensi i i y condi ions, a e e ence model
which cha ac e izes he ideal esidual beha io in a aul y si ua ion is conside ed. The esidual e o wi h espec o his
e e ence model is compu ed. Then, he maximiza ion o he esidual aul e ec is con e ed o minimiza ion o i s e ec
on he esidual e o and is add essed by using he BRL. The comp omise be ween he unknown inpu e ec and he aul
e ec on he esidual is ansla ed in o a mul i-objec i e op imiza ion p oblem wi h some LMI cons ain s. In o de o show
he e iciency and applicabili y o he p oposed me hod, a pa o he Ba celona sewe sys em is conside ed.
Keywo ds: singula delayed LPV sys ems, aul de ec ion, unknown inpu obse e (UIO), obus ness, aul sensi i i y.
1. In oduc ion
Mode n sys ems ha e become mo e complex and he need
o highe pe o mance and eliabili y has made aul
diagnosis a c ucial elemen . The basis o model-based
aul diagnosis me hods is o conside a ma hema ical
model o he sys em, compa ing he ou pu s o his model
wi h eal ou pu s (measu emen s), and inally ying o
disco e any abno mal si ua ion in he sys em h ough
inding an inconsis ency be ween hese wo signals (Chen
and Pa on, 1999).
No mally, a signal called esidual is gene a ed based
on he di e ence be ween he eal and model ou pu s,
which in he ideal case is ze o bu becomes nonze o in he
aul y case. Unknown inpu s and modeling e o s cause
his signal o be nonze o in heal hy condi ions, so he
∗Co esponding au ho
esidual should be compa ed wi h a h eshold ins ead o
ze o. Due o he e ec o unknown inpu and modeling
e o s on he esidual gene a o sys em, he obus ness
o aul diagnosis me hods is an impo an issue o be
conside ed.
Robus ness can be aken in o accoun a he
esidual gene a ion s age using se e al app oaches, such
as unknown inpu obse e s (UIOs) in he wo k o
Guan and Sai (1991), o using adap i e h esholds in
he esidual e alua ion s age (Mon es de Oca e al.,
2012a). Obse e -based me hods o aul diagnosis and
unknown inpu obus ness in hese me hods ha e been
well es ablished o LTI sys ems (see he monog aph by
Ding (2008) and he e e ences he ein). Some o he
app oaches a e based on he UIO (Guan and Sai , 1991)
o on obus obse e s in he sense o he H∞-no m (Hou
and Pa on, 1996). Hamdi e al. (2012b) and Yousse
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46
A.H. Hassanabadi
e al.
e al. (2014) designed p opo ional in eg al ype UIOs
which make i possible o es ima e unknown inpu s in
he sys em. In addi ion o obus ness, he esidual should
p esen goodsensi i i y o aul s. Va ious me hods such as
model ma ching he esidual wi h a e e ence model (F isk
and Nielsen, 2006; Maza s e al., 2008; Ichalal e al.,
2009; Habib e al., 2010; Ahmadizadeh e al., 2014) and
using he H−index condi ion o di ec ly o mula e aul
sensi i i y (Wang e al., 2007; Aouaouda e al., 2015; Zhai
e al., 2014) we e p oposed o ensu ing aul sensi i i y
in obse e -based me hods.
Time delay occu s in many sys ems ei he because
o delayed ac ua ion o measu emen o because o
delay in in e nal s a es o he sys em. Delay in
sys em dynamics usually deg ades he s abili y and
pe o mance o he sys em. Con ol and diagnosis o
ime delay sys ems equi e mo e in ol ed condi ions
han he sys ems wi hou delay and a e s ill ac i e
esea ch a eas. The cu en esea ch in ime delay
sys ems aims a de eloping condi ions o a ious con ol
p oblems such as s abiliza ion o obse e design ha
a e less conse a i e han he exis ing ones. Saadni
e al. (2005) and Liu e al. (2016) p o ided delay
dependen condi ions o s abiliza ion, while Ka imi and
Chadli (2012) in oduced less conse a i e condi ions o
obse e design. In he case o s a e delay sys ems, UIO
design was conside ed by Fa ouh e al. (1999) and Fu
e al. (2004), while an H∞ aul de ec ion il e o obus
aul de ec ion was s udied by Bai e al. (2006). In he
case o inpu delay sys ems, a UIO which is used o
aul de ec ion and isola ion was p oposed by Koenig
e al. (2005). A geome ic app oach o aul de ec ion
in ime-delay sys ems was pu o wa d by Meskin and
Kho asani (2009). In he wo k o Ahmadizadeh e al.
(2014), a UIO was designed o aul de ec ion in
unce ain ime delay sys ems.
Desc ip o sys ems a e a mo e gene al class o
sys ems compa ed wi h s anda d s a e space ones since
hey allow modeling sys ems, including s a ic and
dynamic equa ions, simul aneously (Dai, 1989; Duan,
2010). They a e also called di e en ial-algeb aic
equa ions (DAEs) o hei modeling powe and singula
sys ems o he singula na u e o he equa ions ha
a ise om he modeling. Singula sys ems ha e been
success ully used in modeling and analysis o many
obo ic sys ems (Samiei and Sha iee, 2010; Sada i e al.,
2013) and also in o he sys ems such as elec ical (Hamdi
e al., 2012a; Zhai e al., 2014) and economic ones
(Koumboulis e al., 2011). A lo o e o has been
de o ed o s abili y analysis, obse e design and con ol
o hese sys ems (see he monog aph by Duan (2010)
and he e e ences he ein). Howe e , aul diagnosis
in desc ip o sys ems is s ill in i s de elopmen al s age.
An eigens uc u e app oach was used o design a UIO
by Duan e al. (2002). A UIO was p oposed by Yeu
e al. (2005) o aul de ec ion and isola ion, and a UIO
was designed o aul es ima ion by Koenig (2005). A
singula UIO and a aul de ec ion il e o diagnosis
pu poses in hese sys ems we e designed espec i ely by
Sha e Pasand e al. (2010) and Yao e al. (2011).
In he case o linea singula sys ems which ha e
delayed dynamics, sca ce esea ch has been done o aul
de ec ion and isola ion. Two obus aul de ec ion il e
schemes o hese sys ems we e in oduced by Chen e al.
(2011) and Zhai e al. (2014). Fo nonlinea singula
delayed sys ems, o he bes o he au ho s’ knowledge,
no wo k has been ca ied ou add essing aul diagnosis.
Simul aneous p esence o nonlinea i y and a delayed e m
added by he singula na u e o hese sys ems inc eases
he demand o a pa icula ea men .
One o he usual me hodologies o ea nonlinea
sys ems is o conside special classes, such as hose
sa is ying he Lipschi z condi ion. Howe e , his
condi ion limi s he ange o applicabili y o he ob ained
esul s. Modeling nonlinea sys ems wi h some linea
ones a di e en ope a ing poin s is a common idea ha
is used o ea nonlinea sys ems. Based on his idea,
Shamma (1988) p oposed linea pa ame e a ying (LPV)
sys ems wi h a linea s uc u e bu wi h he s a e space
ma ices depending on some exogenous pa ame e s.
In he wo k o Boko and Balas (2004), a geome ic
app oach was used o design he aul de ec ion il e o
LPV sys ems. Hen y e al. (2009) used he H∞/H−
app oach o add ess simul aneous aul sensi i i y and
obus ness o a esidual in LPV sys ems. A aul
ole an con olle in LPV sys ems based on a UIO aul
es ima o was designed by Mon es de Oca e al. (2012b).
UIO design based on pe ec and obus unknown inpu
decoupling o singula LPV sys ems was conside ed
espec i ely by Hamdi e al. (2012b) and Habib e al.
(2010). Faul diagnosis in hese sys ems in he case o
unmeasu able scheduling pa ame e s was discussed by
Lopez-Es ada e al. (2014). Fo he case o singula LPV
sys ems wi h delay, he p oblem o aul de ec ion has no
been conside ed ye .
The main con ibu ion in his pape is o add ess
he p oblem o obus aul de ec ion o singula delayed
LPV sys ems including mul iple ime a ying s a e delays
and dis u bances. The sys em unde conside a ion also
includes ac ua o aul s. A e con e ing he sys em
o poly opic ep esen a ion, a UIO is p oposed. The
p esence o unknown inpu s in he sys em in ol es
simul aneous obus ness and aul sensi i i y ea men in
he aul de ec ion p ocess. Unknown inpu a enua ion
is conside ed wi h a bounded eal lemma (BRL) o
delayed LPV sys ems. The aul sensi i i y issue is
conside ed by means o a e e ence model which models
he ideal esidual beha io wi h espec o a aul (Maza s
e al., 2008). The esidual e o compa ed wi h he
ou pu o his e e ence model is compu ed and he
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Robus aul de ec ion o singula LPV sys ems wi h mul iple ime- a ying delays
47
aul sensi i i y objec i e is e o mula ed in o aul e ec
a enua ion on he esidual e o . Then, his objec i e
can be o mula ed in a s anda d amewo k using a BRL
o he esidual e o augmen ed sys em. In pa icula ,
he ade-o be ween obus ness and aul sensi i i y is
add essed in a mul i-objec i e op imiza ion p oblem wi h
LMI cons ain s. To illus a e he pe o mance o he
p oposed app oach, a pa o he Ba celona sewe ne wo k
is conside ed as a case s udy.
Sewe ne wo ks a e complex la ge-scale sys ems
which combine sani a y and ain wa e lows wi hin he
same ne wo k in mos ci ies a ound he wo ld (Puig
e al., 2009; Ocampo Ma ínez e al., 2013). They
beha e as open- low channel sys ems. As discussed
by Bolea e al. (2014), hese sys ems p esen dynamic
beha io ha can be ep esen ed by means o a delayed
LPV model. When a ne wo k o open low channels
is conside ed, a singula delayed LPV model should
be used o ake in o accoun he mass balances in he
nodes. Mo eo e , du ing ain s o ms, was ewa e lows
can easily o e load he sewe ne wo k capaci y, he eby
causing ope a o s o dump he excess o wa e in o he
nea es ecei e en i onmen ( i e s, s eams o sea).
Highly sophis ica ed supe iso y-con olsys ems a e used
o ensu e ha high pe o mance can be achie ed and
main ained unde ad e se condi ions. The need o ope a e
in ad e se me eo ological condi ions in ol es, wi h a high
p obabili y, senso and ac ua o mal unc ions ( aul s).
This p oblem calls o he use o an on-line aul de ec ion
and isola ion (FDI) sys em able o de ec such aul s and
co ec hem (i possible) by ac i a ing aul ole ance
mechanisms (Puig and Blesa, 2013).
The emainde o he pape is o ganized as ollows:
In Sec ion 2, he p oblem is o mula ed. In Sec ion 3, he
s uc u e o a UIO o he sys ems unde conside a ion
is p oposed. In Sec ion 4, simul aneous obus ness and
aul sensi i i y is o mula ed wi h he aid o a e e ence
model. The ade-o is conside ed in a mul i-objec i e
op imiza ion p oblem wi h LMI cons ain s. In Sec ion
5, a pa o he Ba celona sewe sys em is discussed as a
eal applica ion case s udy, whe e he me hod p oposed
in his pape is applied o ac ua o aul de ec ion and
dis u bance decoupling. Sec ion 6 d aws he main
conclusions.
No a ion. Th oughou his pape , he ollowing no a ion is
used: Ris he se o eal numbe s; Inis he n-dimensional
iden i y ma ix; o a ma ix X,XTsigni ies i s anspose;
X−1is he in e se and X+is he pseudoin e se o X;
o a symme ic ma ix, ∗is used o show he elemen s
induced om symme y; o a symme ic ma ix X,X>
0(X<0) shows ha i is posi i e (nega i e) de ini e. Fo
a squa e in eg able unc ion x( ), i s L2-no m is de ined
as x( )2=∞
0x( )Tx( )d .
2. P oblem o mula ion
This pape conside s singula delayed LPV sys ems
including dis u bances and ac ua o aul s ha can be
o mula ed in s a e-space o m as ollows:
⎧
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⎪
⎪
⎩
E˙x( )=
s

j=0
Aj(θ( ))x( −τj( ))
+B(θ( ))u( )+R(θ( ))d( )
+F(θ( )) ( ),
y( )=Cx( ),
τ0( )=0,
0≤τj( )≤τm,j=1,...,s,
˙τj( )≤μj,j=1,...,s,
x( )=φ( )−τm< <0,
(1)
whe e x( )∈Rn,u( )∈Rku,y( )∈Rm,d( )∈
Rkdand ( )∈Rk a e he s a e ec o , inpu ec o ,
ou pu ec o , exogenous dis u bances and ac ua o aul s,
espec i ely. In (1), E∈Rn×nis a cons an squa e
ma ix ha may ha e ank de iciency ( ank(E)= ≤
n). Aj(θ( )) o j=0,...,s ,B(θ( )),R(θ( )) and
F(θ( )) a e ma ices wi h he app op ia e dimensions
which depend a inely on he ime a ying pa ame e
θ( )∈Rl ha is eal ime measu able. Cis a cons an
ma ix wi h app op ia e dimensions. He e τj( ) o
j=1,...,s a e ime a ying delays and τ0( )=0.
Fu he mo e, τmis he maximum bound on all delay
alues, and μj o j=1,...,s a e he maximum
bounds on delay de i a i e alues. φ( )is a con inuous
ec o - alued ini ial unc ion. The ime a ying pa ame e
ec o θ( )is assumed o be bounded in a hype box
(Hamdi e al., 2012b), i.e.,
θm
k≤θk( )≤θM
k,k=1,...,l. (2)
De ini ion 1. (Dai, 1989; Duan, 2010) The ma ix pencil
(E,A)is egula i de (sE −A)is no iden ically ze o.
De ini ion 2. (Dai, 1989; Duan, 2010) The ma ix pencil
(E,A)is impulse- ee i deg(de (sE −A)) = ank(E).
De ini ion 3. (Li and Zhang, 2012) Sys em (1) is egula
and impulse- ee i he ma ix pencil (E,A0(θ( )) is
egula and impulse- ee o he whole domain o θ( )
de ined in (2).
De ini ion 4. (Li and Zhang, 2012) The sys em (1) is ad-
missible i i is egula , impulse ee and s able.
Rema k 1. (Li and Zhang, 2012) Regula i y and
impulse- eeness o he sys em (1) gua an ee a unique
con inuous solu ion wi hou impulsi e beha io in he
case o a compa ible ini ial unc ion φ( ) o he
sys em (1).
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48
A.H. Hassanabadi
e al.
Assump ion 1. The singula delayed LPV sys em
conside ed in his pape is admissible acco ding o he
de ini ions p o ided.
Thus, admissibili y, s abiliza ion and con ol o his
sys em a e no add essed in his pape . The eade
in e es ed in hese issues is e e ed o he wo ks o Li
and Zhang (2012) as well as Zhang and Zhu (2012).
The sys em (1) can be ans o med o a poly opic
ep esen a ion. In he poly opic desc ip ion, h=2
l
subsys ems a e conside ed in h e ices o he hype box
in he θ( )space and he whole sys em is de ined as a
con ex combina ion o hese subsys ems. Acco ding o
he heo y o con ex se s, each poin in he θ( )space can
be ep esen ed by a se o subsys em weigh s de ined on h
e ices o he hype box in his space. These weigh s a e
conside ed he e by ρ(θ( )) ∈Rhand sa is y
0≤ρi(θ( )) ≤1,i=1,...,h, (3)
h

i=1
ρi(θ( )) = 1,i=1,...,h. (4)
Thus, he poly opic ep esen a ion o (1) is
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⎪
⎩
E˙x( )=
h

i=1
ρi(θ( ))s

j=0
Ajix( −τj( ))
+Biu( )+Rid( )+Fi ( ),
y( )=Cx( ),
τ0( )=0,
0≤τj( )≤τm,j=1,...,s,
˙τj( )≤μj,j=1,...,s,
x( )=φ( ),−τm< <0.
(5)
In (5), Aji,Bi,Riand Fia e ma ices ela ed o he
i- h subsys em (i=1,...,h) loca ed a he i- h e ex o
he hype box.
3. UIO s uc u e
Conside ing he poly opic sys em (5), he ollowing
obse e is p oposed:
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˙z( )=
h

i=1
ρi(θ( ))s

j=0 Njiz( −τj( ))
+Ljiy( −τj( ))+Giu( ),
ˆx( )=z( )+H2y( ),
ˆy( )=Cˆx( ),
( )=T(y( )−ˆy( )),
z( )=0,−τm< <0.
(6)
whe e ˆx( )∈Rn,ˆy( )∈Rmand z( )∈Rna e he s a e
es ima e, ou pu es ima e and obse e s a e, espec i ely.
Nji,Lji,Gi,Tand H2a e obse e ma ices wi h
app op ia e dimensions ha will be de e mined wi h he
p oposed design me hodology la e on. He e ( )∈Rk
is he esidual signal which is used o de ec ing he aul s
in he sys em. The s a e es ima ion e o is
e( )=x( )−ˆx( ).(7)
The e o dynamics will be designed so ha he e o
con e ges o ze o unde a bi a y ini ial condi ions and
any inpu u( ). Acco ding o (6), he e o e( )in (7) is
e( )=x( )−z( )−H2Cx( )
=(In−H2C)x( )−z( ).(8)
I H1∈Rn×ncan be ound such ha he condi ion
H1E=In−H2C, (9)
is sa is ied (Hamdi e al., 2012b) hen
e( )=H1Ex( )−z( ).(10)
Thus, he e o dynamics a e desc ibed by means o
˙e( )=H1E˙x( )−˙z( ).(11)
Subs i u ing (5) and (6) in (11) esul s in
˙e( )=
h

i=1
ρi(θ( ))s

j=0 H1Ajix( −τj( ))
−Njiz( −τj( )) −LjiCx( −τj( ))
+H1Biu( )+H1Rid( )
+H1Fi ( )−Giu( )
(12)
and, acco ding o (10), obse ing ha
−Njiz( −τj( ))
=Njie( −τj( )) −NjiH1Ex( −τj( )),
(12) can be w i en as ollows:
˙e( )
=
h

i=1
ρi(θ( ))s

j=0 Njie( −τj( ))
+(H1Aji −LjiC−NjiH1E)x( −τj( ))
+(H1Bi−Gi)u( )+H1Rid( )+H1Fi ( ).
(13)
I he ollowing condi ions a e sa is ied o i=
1,...,hand j=0,...,s:
H1E+H2C=In,(14)
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Robus aul de ec ion o singula LPV sys ems wi h mul iple ime- a ying delays
49
H1Aji −NjiH1E=LjiC, (15)
Gi=H1Bi,(16)
he esidual dynamic sys em can be ans o med in o
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˙e( )=
h

i=1
ρi(θ( ))s

j=0
Njie( −τj( ))
+H1Fi ( )+H1Rid( ),
( )=TCe( ).
(17)
Now, a lemma is in oduced o obus s abili y o
a delayed LPV sys em ha will be used o obus aul
de ec ion o he sys em (5).
Lemma 1. Conside he ollowing delayed LPV sys em:
⎧
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⎪
⎪
⎩
˙e( )=
h

i=1
ρi(θ( ))
×s

j=0
¯
Ajie( −τj( )) + ¯
Biw( ),
z( )= ¯
Ce( )+ ¯
Dw( ),
(18)
in which w( )is a L2-no m bounded exogenous inpu and
z( )is he measu ed ou pu , and all ma ices ha e com-
pa ible dimensions. Fo a gi en γ>0,i he eexis
P>0and Qj>0 o j=1,...,s such ha he ol-
lowing ma ix inequali y holds o i=1,...,h:
⎡
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎣
¯
AT
0iP+P¯
A0i+Q1+···+QsP¯
A1i···
∗−(1 −μ1)Q1···
.
.
..
.
....
∗ ∗ ···
∗ ∗ ···
∗ ∗ ···
P¯
Asi P¯
Bi¯
CT
000
.
.
..
.
..
.
.
−(1 −μs)Qs00
∗−γ2I¯
DT
∗∗−I
⎤
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎦
<0,
(19)
hen he sys em (18) is asymp o ically s able o w( )=0
and z( )2<γw( )2 o ze o ini ial condi ions.
P oo . The ollowing Lyapuno –K aso skii unc ional is
conside ed:
V( , e )=eT( )Pe( )
+
s

j=1 
−τj( )
eT(λ)Qje(λ)dλ, (20)
in which Pand Qj(j=1,...,s)a e symme ic posi i e
de ini e ma ices and e := e( +ω),whe eω∈[−τm,0].
Conside he index
J2=∞
0z( )Tz( )−γ2w( )Tw( )d . (21)
Then he dis u bance a enua ion condi ion z( )2<
γw( )2in his lemma is equi alen o J2<0. Thus,
o p o ing he lemma, i is jus necessa y o show ha
J2<0 o nonze o w( )wi h ze o ini ial condi ions and
˙
V( , e )<0 o ze o w( ).J2can be w i en as ollows:
J2=∞
0z( )Tz( )−γ2w( )Tw( )+ ˙
V( , e )d
+V( , e )| =0 −V( , e )| =∞.
(22)
Unde ze o ini ial condi ions, V( , e )| =0 =0by
he de ini ion o he Lyapuno –K aso skii unc ional and
V( , e )| =∞≥0owing o he posi i e de ini eness o
his unc ional, and i ollows ha
J2≤∞
0z( )Tz( )−γ2w( )Tw( )+ ˙
V( , e )d
=∞
0eT( )¯
CT¯
Ce( )+eT( )¯
CT¯
Dw( )
+wT( )¯
DT¯
Ce( )+wT( )¯
DT¯
Dw( )
−γ2w( )Tw( )d
+∞
0
h

i=1
ρi(θ( ))eT( )P¯
A0ie( )
+eT( )¯
AT
0iPe( )+
s

j=1 eT( )P¯
Ajie( −τj( ))
+eT( −τj( )) ¯
AT
jiPe( )+eT( )P¯
Biw( )
+wT( )¯
BT
iPe( )+
s

j=1 eT( )Qje( )
−(1 −˙τj( ))eT( −τj( ))Qje( −τj( ))d .
Since
h

i=1
ρi(θ( )) = 1
and ˙τj≤μj o j=1,...,s, he p e ious exp ession can
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50
A.H. Hassanabadi
e al.
be bounded as ollows:
J2≤∞
0
h

i=1
ρi(θ( )) ⎡
⎢
⎢
⎢
⎢
⎢
⎣
e( )
e( −τ1( ))
.
.
.
e( −τs( ))
w( )
⎤
⎥
⎥
⎥
⎥
⎥
⎦
T
Θ
×⎡
⎢
⎢
⎢
⎢
⎢
⎣
e( )
e( −τ1( ))
.
.
.
e( −τs( ))
w( )
⎤
⎥
⎥
⎥
⎥
⎥
⎦
d , (23)
whe e
Θ=⎡
⎢
⎢
⎢
⎢
⎢
⎣
ΞP¯
A1i···
∗−(1 −μ1)Q1···
.
.
..
.
....
∗ ∗ ···
∗ ∗ ···
P¯
Asi P¯
Bi+¯
CT¯
D
00
.
.
..
.
.
−(1 −μs)Qs0
∗¯
DT¯
D−γ2I
⎤
⎥
⎥
⎥
⎥
⎥
⎦
,
Ξ= ¯
AT
0iP+P¯
A0i+¯
CT¯
C+Q1+···+Qs.
Then, by using he Schu complemen lemma, (19) can be
ob ained which is a su icien condi ion o J2<0unde
ze o ini ial condi ions. The easibili y o he LMI:
⎡
⎢
⎢
⎢
⎣
¯
AT
0iP+P¯
A0i+Q1+···+QsP¯
A1i···
∗−(1 −μ1)Q1···
.
.
..
.
....
∗ ∗ ···
P¯
Asi
0
.
.
.
−(1 −μs)Qs
⎤
⎥
⎥
⎥
⎦
<0
(24)
is di ec ly deduced om (19), which insu es ˙
V( , e )<0
o ze o w( ), so he sys em (18) is asymp o ically s able
unde no ac ua ion. 
Rema k 2. The BRL in oduced in Lemma 1 o obus
s abili y o he delayed LPV sys em (18) is dependen
on delay de i a i es bu independen o delay alues.
The e a e some esul s in he li e a u e which p esen
delay dependen condi ions (see he wo ks o Saadni e al.
(2005), Wu e al. (2010) and Liu e al. (2016) o delay
dependen s abili y and s abiliza ion esul s). Howe e ,
he goal o Lemma 1 is o in oduce a obus s abili y
esul o he sys em (18) which can be di ec ly used
in he emaining pa s o he pape , o he design o
a obus aul de ec ion obse e o a singula delayed
LPV sys em in he LMI o ma . O he obus s abili y
esul s in he li e a u e canno be di ec ly used since he
LMI o mula ion o he p esen me hod in ol es dealing
wi h some nonlinea e ms in ma ix inequali ies, which is
beyond he scope o his pape .
4. Robus aul de ec ion sys em design
Now, he design o he obse e (6) is conside ed such ha
he condi ions (14)−(16) a e me and he e o dynamics
(17) will be obus ly s able while ha ing maximum
sensi i i y o he ac ua o aul based on Lemma 1.
Assump ion 2. Fo he sol abili y o (14), i is assumed
ha
ank E
C=n. (25)
Unde Assump ion 2, he solu ion o he ma ix
equa ion (14) is (Da ouach and Bou ayeb, 1995; Hamdi
e al., 2012b)
H1H2=E
C+
,(26)
whe e
E
C+
∈Rn×(n+m)
is he pseudoin e se o [E
C]calcula ed om
E
C+
=E
CTE
C−1E
CT
.(27)
Fo ex ending he solu ion domain o (14), a e m is
added in he null space o his equa ion, so ha
H1H2
=E
C+
+K(In+m−E
CE
C+
),(28)
whe e K∈Rn×(n+m)is a gain ac o ha can be eely
chosen o sa is y o he es ic ions on he p oblem. Now
(28) is ew i en as ollows:
H1=H10 +KX1,(29)
H2=H20 +KX2,(30)
whe e H10 ∈Rn×nand H20 ∈Rn×ma e cons uc ed
om he i s ncolumns and las mcolumns o [E
C]+
espec i ely. In a simila way, X1∈R(n+m)×nand
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Robus aul de ec ion o singula LPV sys ems wi h mul iple ime- a ying delays
51
X2∈R(n+m)×ma e made om he i s ncolumns and
las mcolumns o
X=In+m−E
CE
C+
,
espec i ely.
Un il now, he condi ion (14) has been conside ed
o obse e exis ence. On he o he hand, he condi ion
(15) should be me such ha he e o dynamics will ha e
he desi ed cha ac e is ic o being obus o dis u bances
and sensi i e o aul s. Now, some new a iables a e
in oduced o j=0,...,sand i=1,...,h:
Kji =Lji −NjiH2.(31)
Wi h hese new a iables, he condi ion (15) can be
ew i en as
Nji =H1Aji −KjiC. (32)
Subs i u ing (29) in (32) esul s in
Nji =H10Aji +KX1Aji −KjiC. (33)
Wi h his me hodology, whene e Kand Kji a e
ound, Nji can be calcula ed om (33) and hen,
acco ding o (31),
Lji =Kji +NjiH2.(34)
Subs i u ing (33) in (17) esul s in
⎧
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎨
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎩
˙e( )=
h

i=1
ρi(θ( ))
×s

j=0
(H10Aji +KX1Aji −KjiC)
×e( −τj( )) + H1Fi ( )+H1Rid( ),
( )=TCe( ).
(35)
Now, he esidual signal is exp essed in in e nal o m,
ha is, in e ms o dis u bances and aul s:
( )=
h

i=1
ρi(θ( ))(Gi
dd( )+Gi
( )),(36)
whe e, by abuse o no a ion, he ollowing de ini ions a e
in oduced:
Gi
d := ΛH1Ri
TC 0,(37)
Gi
:= ΛH1Fi
TC 0,(38)
which ep esen he ans e om dis u bance d( )
and aul ( ) o esidual ( )in he i- h subsys em,
espec i ely, whe e
Λ=
s

j=0
(H10Aji +KX1Aji −KjiC)e−τjs.
This no a ion o Eqns. (37) and (38) was in oduced
he e jus o ease o ep esen a ion, and all he ollowing
calcula ions will be ca ied ou in he ime domain. Fo
a enua ing he dis u bance e ec on he esidual signal,
Lemma 1 can be di ec ly used such ha he esidual
dynamic sys em is obus ly s able in he p esence o
unknown inpu s. On he o he hand, o ha e an accep able
le el o esidual aul sensi i i y, a e e ence model Wi
e
is in oduced o each subsys em which cha ac e izes he
esidual ideal beha io in he p esence o aul s in he
sys em (Maza s e al., 2008):
Wi
e =Ai
e Bi
e
Ci
e Di
e .(39)
These e e ence models ha e n e s a es, k aul s as
inpu and k ideal esiduals as ou pu s, and hei s a e
space ma ices a e wi h app op ia e dimensions. The
e e ence models a e chosen om s able ans e ma ices
wi h he ollowing p ope y:

Wi
e 
−=in
w∈R(σWi
e (jw))≥1,(40)
whe e ·−deno es he H−index and σ(·)is he
minimum singula alue. The e e ence models wi h he
condi ion (40) a e sui able o aul de ec ion because hey
ha e no aul a enua ion in any equency ange (Ichalal
e al., 2009). The condi ion (40) can be es ed wi h he
ollowing lemma.
Lemma 2. (Maza s e al., 2008) Le he e e ence model
be Wi
e as de ined in (39); he condi ion (40) holds i and
only i he e exis s a symme ic ma ix P e ∈Rn e ×n e
such ha
P e Ai
e +Ai
e
TP e +Ci
e
TCi
e
Bi
e
TP e +Di
e
TCi
e
P e Bi
e +Ci
e
TDi
e
Di
e
TDi
e −Ik ≥0.
(41)
Maza s e al. (2008) p esen a linea iza ion p ocedu e
o sa is y he condi ion (41) o e e ence model selec ion
in s a e space sys ems, bu p o ide a subop imal solu ion
due o i s linea iza ion p ocedu e. In he case o ime
delay sys ems, Ahmadizadeh e al. (2014) sugges ed a
me hodology o selec ing he e e ence model. The
me hod o Ahmadizadeh e al. (2014) is based on
sa is ying a ela ed H−index o design he e e ence
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52
A.H. Hassanabadi
e al.
model. Howe e , his me hod is no applicable since
i canno be ex ended o ou sys em because he e m
D ( )is no p esen in he sys em ou pu equa ion (1).
Fu he esea ch may be conduc ed o p esen a me hod o
designing an app op ia e e e ence model o he singula
delayed LPV sys em (1).
Now he esidual e o wi h espec o hese ideal
e e ence models is de ined:
e ( )= ( )−
h

i=1
ρi(θ( ))Wi
e ( ).(42)
Wi h his o mula ion, he goal o enhancing esidual
aul sensi i i y can be eplaced by an equi alen one o
a enua ing he aul e ec on he esidual e o . Then, by
using Lemma1, his p oblem can also be add essed. Thus,
he obus aul de ec ion p oblem has h ee objec i es:
1. The esidual dynamic sys em (35) should be s able.
2. The e ec o unknown inpu s on he esidual signal
should be minimized: 1

Gi
d 
∞<γ
d.(43)
3. The aul e ec on he esidual e o should be
minimized:2

Gi
e 
∞=
Gi
−Wi
e 
∞<γ
.(44)
Now, wi h he ma e ial p o ided so a , a heo em
can be s a ed ha p o ides he solu ion o he obus
aul de ec ion p oblem o singula delayed LPV sys ems
conside ing he p e ious goals.
Theo em 1. Assume ha , o a gi en scala a∈[0,1]
and he e e ence models Wi
e wi h he cons ain (40),
he e exis symme ic posi i e de ini e ma ices P1,P2,
Q1j,Q2j o j=1,...,s, ma ices T,Mand Mji o
j=0,...,sand i=1,...,hand posi i e scala s ¯γ and
¯γdas he solu ion o he ollowing op imiza ion p oblem:
min
P1,P2,Q1j,Q2j,M,Mji,T,¯γd,¯γ
a¯γ +(1−a)¯γd(45)
subjec o he LMI cons ain s
⎡
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎣
Ωi
0Ωi
1··· Ωi
s¯
RiCTTT
∗¯
Q11 ··· 00 0
.
.
..
.
.....
.
..
.
..
.
.
∗ ∗ ··· ¯
Q1s00
∗ ∗ ··· ∗ −¯γdI0
∗ ∗ ··· ∗ ∗ −I
⎤
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎦
<0,(46)
1This c i e ion is o mula ed as  ( )2<γ
dd( )2and, by abuse
o no a ion, w i en as (43).
2This c i e ion is o mula ed as e ( )2<γ
 ( )2and, by
abuse o no a ion, w i en as (44).
⎡
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎣
Ωi
00Ω
i
10··· Ωi
s0
0Γ
i00··· 00
∗∗¯
Q11 0··· 00
∗∗∗¯
Q21 ··· 00
.
.
..
.
..
.
..
.
.....
.
..
.
.
∗ ∗ ∗ ∗ ··· ¯
Q1s0
∗ ∗ ∗ ∗ ··· ∗ ¯
Q2s
∗ ∗ ∗ ∗ ··· ∗ ∗
∗ ∗ ∗ ∗ ··· ∗ ∗
P1H10Fi+MX1FiCTTT
P2Bi
e −(Ci
e )T
00
00
.
.
..
.
.
00
00
−¯γ I−(Di
e )T
∗−I
⎤
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎦
<0
(47)
o all i=1,...,h, and whe e o i=1,...,hand j=
1,...,swe ha e
Ωi
0=P1H10A0i+MX1A0i−M0iC
+(P1H10A0i+MX1A0i−M0iC)T+Q11
+···+Q1s,
Ωi
j=P1H10Aji +MX1Aji −MjiC,
Γi=P2Ai
e +(P2Ai
e )T+Q21 +···+Q2s,
¯
Q1j=−(1 −μj)Q1j,
¯
Q2j=−(1 −μj)Q2j,
¯
Ri=P1H10Ri+MX1Ri.
Then he obus esidual gene a o (6) exis s and can be
de e mined as ollows:
K=P−1
1M, (48)
Kji =P−1
1Mji.(49)
The obse e ma ices Nji,Lji,Giand H2can be cal-
cula ed om (33),(34),(16) and (30), wi h a enua ion
le els ega ding unknown inpu s on esiduals and aul s
on esidual e o s gi en espec i ely by
γd=√¯γd(50)
γ = ¯γ .(51)
P oo . Wi h he no a ion in oduced in (38), he i- h
ans e ma ix om he aul o he esidual e o is
Gi
−Wi
e
=⎛
⎝
Λ0 H10Fi+KX1Fi
0Ai
e Bi
e
TC −Ci
e −Di
e ⎞
⎠.(52)
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Robus aul de ec ion o singula LPV sys ems wi h mul iple ime- a ying delays
53
Choosing symme ic posi i e de ini e block diagonal
ma ices
P=P10
0P2
and
Qj=Q1j0
0Q2j,
he a enua ion o he aul e ec on esidual e o can be
add essed by using Lemma 1. Subs i u ing he associa ed
e ms in (19) esul s in
⎡
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎣
¯
Ωi
00¯
Ωi
10··· ¯
Ωi
s0
0Γ
i00··· 00
∗∗¯
Q11 0··· 00
∗∗∗¯
Q21 ··· 00
.
.
..
.
..
.
..
.
.....
.
..
.
.
∗ ∗ ∗ ∗ ··· ¯
Q1s0
∗ ∗ ∗ ∗ ··· ∗ ¯
Q2s
∗ ∗ ∗ ∗ ··· ∗ ∗
∗ ∗ ∗ ∗ ··· ∗ ∗
P1H10Fi+P1KX1FiCTTT
P2Bi
e −(Ci
e )T
00
00
.
.
..
.
.
00
00
−γ2
I−(Di
e )T
∗−I
⎤
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎦
<0,
(53)
whe e o i=1,...,hand j=1,...,s
¯
Ωi
0=P1H10A0i+P1KX1A0i−P1K0iC
+(P1H10A0i+P1KX1A0i−P1K0iC)T
+Q11 +···+Q1s,
¯
Ωi
j=P1H10Aji +P1KX1Aji −P1KjiC,
Γi=P2Ai
e +(P2Ai
e )T+Q21 +···+Q2s,
¯
Q1j=−(1 −μj)Q1j,
¯
Q2j=−(1 −μj)Q2j.
Ob iously, his is a nonlinea ma ix inequali y
because o p oduc e ms o unknown a iables, so he
ollowing change o a iables is applied:
M:=P1K, (54)
Mji :=P1Kji,(55)
¯γ :=γ2
.(56)
Then he LMI (47) is ob ained. Acco ding o
Lemma 1, dis u bance a enua ion in he esidual sys em
(36) can be add essed in a simila manne . By using P1
and Q1j(j=1,...,s)as posi i e de ini e ma ices o
Pand Qjin Lemma 1 and subs i u ing he associa ed
exp essions o ma ices o he ans e ma ix Gi
d in (19)
and again by using he change o a iables (54)−(55), and
¯γd:= γ2
d,(57)
he nonlinea i y in he ma ix inequali y is esol ed
and he LMI (46) is ob ained. Mo eo e , (48)−(51)
can be di ec ly de i ed om (54)−(57). Finally, he
simul aneous p oblem o obus ness o dis u bances
and sensi i i y o aul s can be add essed by he
mul i-objec i e op imiza ion p oblem (45) wi h he
cons ain s (46) and (47) conside ing ha he pa ame e
a∈[0,1] is used o adjus he objec i e weigh ing. 
Summa y o he me hod. A summa y o he p esen ed
me hod o obus aul de ec ion in singula LPV
sys ems wi h mul iple ime a ying delays is p esen ed in
Algo i hm 1.
Algo i hm 1. Robus aul de ec ion sys em design o a
singula LPV sys em wi h mul iple ime a ying delays.
S ep 1. Check Assump ion 2.
S ep 2. Calcula e H10 and X1 om he i s ncolumns o
E
C+
and X=In+m−E
CE
C+
, espec i ely.
S ep 3. Choose some app op ia e e e ence models Wi
e
sa is ying he cons ain (40) by checking Lemma 2.
S ep 4. Sol e he op imiza ion p oblem (45) unde he
LMI cons ain s (46), and (47) and ob ain ma ices P1,
P2,M,T,Q1j,Q2j o j=1,...,s and Mji o j=
0,...,sand i=1,...,h.
S ep 5. Calcula e Kand Kji o j=0,...,s and i=
1,...,h om (48) and (49), espec i ely.
S ep 6. Calcula e H2,Gand Nji o j=0,...,s and
i=1,...,h om (30), (16) and (33), espec i ely.
S ep 7. Calcula e Lji o j=0,...,sand i=1,...,h
om (34).
5. Applica ion
5.1. Desc ip ion. In o de o show he pe o mance
o he p oposed aul de ec ion app oach, a case s udy
based on he Rie a Blanca ca chmen o he Ba celona
sewe ne wo k is conside ed. This ca chmen has al eady
been used o illus a ing he model p edic i e con ol o
sewe ne wo ks in he wo k o Puig e al. (2009). This
ca chmen can be modeled using he i ual ank app oach
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60
A.H. Hassanabadi
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Robus aul de ec ion o singula LPV sys ems wi h mul iple ime- a ying delays
61
Ami Hossein Hassanabadi was bo n in 1985.
He ecei ed his B.Sc. deg ees in con ol sys ems
enginee ing and elec onics in 2007 and 2009, e-
spec i ely, bo h om he Ami kabi Uni e si y o
Technology (Teh an Poly echnic), Teh an, I an.
He ecei ed his M.Sc. deg ee in con ol engi-
nee ing in 2010 om he same uni e si y, whe e
he is cu en ly pu suing his Ph.D. deg ee in con-
ol sys ems enginee ing. His p esen esea ch in-
e es s include singula sys ems, obo ics, ime-
delay sys ems, and aul diagnosis.
Masoud Sha iee was bo n in 1957. He ecei ed a B.Sc. deg ee in ma h-
ema ics in 1975, an M.Sc. deg ee in ma hema ics and in sys ems engi-
nee ing om W igh S a e Uni e si y, Day on, OH, in 1980 and 1982,
espec i ely, and M.Sc. and Ph.D. deg ees in elec ical enginee ing om
Louisiana S a e Uni e si y, Ba on Rouge, LA, in 1984 and 1987, espec-
i ely. M. Sha iee is cu en ly a ull p o esso in he Depa men o Elec-
ical Enginee ing a he Ami kabi Uni e si y o Technology, Teh an,
I an. Also, he joined he SIC Resea ch Ins i u e in 2014. He is he au-
ho o o e 230 esea ch pape s and 13 books, and a ansla o o 11
books. His esea ch in e es s include mul i-dimensional (M-D) sys ems,
singula sys ems, in o ma ion and communica ions, obo ics, and ne -
wo k s abili y.
Vicenç Puig was bo n in Gi ona, Spain, in
1969, ecei ed he Ph.D. deg ee in con ol engi-
nee ing in 1999 and he elecommunica ions en-
ginee ing deg ee in 1993, bo h om Uni e si-
a Poli ècnica de Ca alunya (UPC), Ba celona,
Spain. He is cu en ly a p o esso o au oma ic
con ol and he leade o he Ad anced Con ol
Sys ems (SAC) Resea ch G oup o he Resea ch
Cen e o Supe ision, Sa e y and Au oma ic
Con ol (CS2AC) a UPC. His main esea ch in-
e es s include aul de ec ion, isola ion and aul - ole an con ol o dy-
namic sys ems. He has been in ol ed in se e al Eu opean p ojec s and
ne wo ks, and has published many pape s in in e na ional con e ence
p oceedings and scien i ic jou nals.
Recei ed: 2 No embe 2014
Re ised: 12 Ap il 2015
Re- e ised: 22 July 2015
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