Internati
o
nal
Journal of Ele
c
trical
and Computer
Engineering
(IJE
CE)
Vol.
5, No. 6, Decem
ber
2015, pp. 1396~
1
406
I
S
SN
: 208
8-8
7
0
8
ļ²
1
396
Jo
urn
a
l
h
o
me
pa
ge
: h
ttp
://iaesjo
u
r
na
l.com/
o
n
lin
e/ind
e
x.ph
p
/
IJECE
Using H
y
b
r
i
d
Automata f
o
r Di
agn
osis of
Hybrid Dynamical
Systems
L
o
t
f
i
M
h
am
di
*, L
o
bn
a B
e
l
k
acem
*,
Hedi
Dh
oui
b
i
*
, Z
i
neb Si
meu
Ab
a
z
i
**
* LARATSI, National Eng
i
neeri
ng School of
Mo
nastir, Tunisia
** G-SCOP, Grenoble- IPN/UJF- Grenoble 1, CN
RS, France
Article Info
A
B
STRAC
T
Article histo
r
y:
Received Apr 3, 2015
Rev
i
sed
Ju
l 15
,
20
15
Accepte
d Aug 1, 2015
P
h
y
s
ica
l
s
y
s
t
em
s
can fa
il.
F
o
r t
h
is
reas
on th
e
problem
of iden
tif
ying
and
reac
ting to f
a
ul
t
s
has
receiv
e
d a
large
att
e
ntion
i
n
the con
t
rol an
d com
puter
science com
m
unities
.
In th
is pap
e
r we stud
y
th
e
fault d
i
agnosis p
r
oblem
and
modeling of H
y
brid D
y
namical
S
y
stems
(HDS).
Generally
speaking,
HDS is
a s
y
stem mixing
continuous
and
discrete
b
e
hav
i
ors that canno
t
be faithfully
modeled neith
er b
y
using formalism
with
continu
ous dy
n
a
mics only
nor b
y
a
formalism including only
discr
e
te
d
y
namics.
We use the well known
framework of hy
brid
automata
for m
odeling h
y
brid s
y
stems, because th
ey
combine the co
ntinous and dis
c
retes pa
rts on
the same structure. H
y
brid
autom
a
ton is
a s
t
at
es
-trans
i
t
i
ons
graph, wh
ose d
y
n
a
mic
evolution
is
represented b
y
discretes and
continous
steps
alternations, also, con
tinous
evolution happ
ens in the automaton ap
exes,
while discrete evolution is
realized
b
y
trans
itions crossing (
a
rcs) of
the grap
h. Th
eir
simulation presents
man
y
problems mainly
the s
y
nchron
isation between
th
e two models.
Stateflow
,
used
to describ
e
the
discret
e
m
odel, i
s
co-ordinat
ed
with Matlab
,
used to des
c
ribe the
continuous
model.
Th
is article is a descrip
t
ion of a
cas
e
stud
y
,
which
is
a two tanks s
y
stem.
Keyword:
Diagnosis
Hy
bri
d
dy
nam
i
cal
sy
st
em
Hyb
r
i
d
au
to
m
a
ta
Copyright Ā©
201
5 Institut
e
o
f
Ad
vanced
Engin
eer
ing and S
c
i
e
nce.
All rights re
se
rve
d
.
Co
rresp
ond
i
ng
Autho
r
:
Lo
tfi M
h
am
d
i
,
LAR
A
TSI
,
Na
t
i
onal
E
ngi
neer
i
ng Sch
o
o
l
o
f
M
onast
i
r
,
Un
i
v
ersity of
Mo
n
a
stir,
M
onast
i
r
, T
uni
si
a.
Em
ail: lotfienim
@
yahoo.fr
1.
INTRODUCTION
In m
oder
n
co
m
p
l
e
x sy
st
em
s cont
i
n
uo
us a
n
d di
scret
e
dy
n
a
m
i
cs i
n
t
e
ract
. Thi
s
i
s
t
h
e case of wi
de
m
a
nufact
uri
n
g
pl
ant
s
,
age
n
t
s
s
y
st
em
s, robotic
s and
physical plants.
Thi
s
ki
n
d
of sy
st
em
s, cal
l
e
d h
y
b
ri
d i
n
t
h
ei
r
b
e
havi
ou
r,
nee
d
s a sp
eci
ļ¬
c formalis
m
to be a
n
alysed.
In
or
der
t
o
m
ode
l
and
s
p
eci
fy
hy
b
r
i
d
sy
st
em
s i
n
a
f
o
rm
al way
,
t
h
e
not
i
on
o
f
hy
bri
d
aut
o
m
a
t
a
has
bee
n
in
tr
odu
ced [1
], [2
].
In
t
u
itiv
ely, a h
y
b
r
id
au
to
m
a
t
o
n
is a ā
ļ¬
n
ite-state au
to
m
a
to
n
ā
with
co
n
tinu
o
u
s
v
a
riab
les th
at ev
o
l
ve
according to dynamics chara
c
terizing eac
h discrete state.
In the last years, a wi
de spectrum
of
m
odeling
form
al
is
m
[3
]
an
d
algorith
m
i
c tech
n
i
qu
es has b
een
stud
ied
in
th
e con
t
rol an
d
co
m
p
u
t
er scien
ce co
mmu
n
ities
t
o
s
o
l
v
e t
h
e
pr
obl
em
s of
si
m
u
l
a
t
i
on,
ve
ri
ļ¬
catio
n
an
d con
t
ro
l syn
t
h
e
sis for h
ybr
id systems [4
],
[5
].
The c
ont
r
o
l
a
l
go
ri
t
h
m
s
are gene
ral
l
y
dev
e
l
ope
d co
nsi
d
eri
n
g t
h
at
t
h
e
sy
st
em
wor
k
s i
n
n
o
rm
al
situ
atio
n
,
i.e. is n
o
t
fau
lty. Un
fo
rt
u
n
a
tely,
wh
en
fail
u
r
es
o
ccur, t
h
ese alg
o
rith
m
s
b
ecome in
efficien
t
an
d
ev
en
d
a
ng
erou
s
for
th
e system
i
t
self or its env
i
ron
m
en
t. In
or
d
e
r
to
r
each
h
i
gher
perf
or
m
a
n
ces and
m
o
r
e
r
i
go
rou
s
security s
p
ecifi
cations, a Failure
Detection a
n
d
Isol
at
i
o
n
(F
D
I)
[
6
]
sy
st
em
has t
o
be i
m
pl
em
ent
e
d.
In
th
is p
a
p
e
r
we con
cen
trate ou
r atten
tion
t
o
t
h
e
p
r
ob
lem
o
f
fau
lt d
e
tecti
o
n an
d iso
l
ation
for
h
ybrid
syste
m
s.
Evaluation Warning : The document was created with Spire.PDF for Python.
I
J
ECE
I
S
SN
:
208
8-8
7
0
8
ļ²
Using
H
y
b
r
i
d
Au
toma
ta
f
o
r
Diag
no
sis
o
f
Hyb
r
id Dyna
mica
l S
y
stems (Lo
t
fi Mh
amd
i
)
1
397
Th
e literature i
n
th
at
field
is
ab
und
an
t and
d
i
fferen
t
so
l
u
tio
n
s
h
a
v
e
b
e
en p
r
op
osed
fo
r
ex
am
p
l
e th
e
app
r
oach
of
[
7
]
,
t
h
i
s
m
e
t
hod o
f
m
odel
-
ba
sed F
D
I al
go
ri
t
h
m
s
i
s
t
o
co
m
p
are t
h
e exp
ect
ed be
havi
or
of t
h
e
sy
st
em
, gi
ve
n
by
a m
odel
wh
i
c
h i
s
m
odel
e
d
by
hy
b
r
i
d
a
u
t
o
m
a
t
a
, wi
t
h
i
t
s
act
ual
beha
vi
o
r
,
k
n
o
w
n t
h
ro
u
g
h
t
h
e
onl
i
n
e
o
b
ser
v
a
t
i
ons.
AR
R
-
ba
sed resi
du
al
s are i
ndi
cat
ors
of
beha
vi
o
r
s a
n
d
t
hus m
a
y
be used f
o
r
FD
I:
t
h
ey
are
eq
u
a
l to
zero
in
no
rm
al (n
o
-fau
lt) situ
atio
n
s
an
d
d
i
ff
eren
t fro
m
zero
wh
en
th
e fau
lts they are sen
s
itiv
e to
,
o
ccur.
A stru
ct
u
r
ed
set of resid
u
a
l, i.e. a set
o
f
residu
als that are no
t sen
s
itiv
e to
th
e same su
b
s
ets
o
f
fau
lts,
m
a
y
be use
d
t
o
i
s
ol
at
e t
h
e
fa
ul
t
s
.
In th
e con
t
ex
t
o
f
ou
r
work,
we con
s
id
er a d
i
ag
no
stic system
b
a
sed
on con
t
ro
l of t
h
e execu
tio
n tim
e
o
f
th
e task
s during
th
e
o
p
e
ratio
n of t
h
e syst
e
m
; g
i
v
e
n th
at
th
e system
o
p
e
ratio
n correspon
d
s
to th
e ex
ecu
tion
of al
l
t
a
sks of
t
h
e pr
ocess f
o
r w
e
l
l
-
de
fi
ne
d t
i
m
e
i
n
t
e
rval
s. Thi
s
m
e
t
h
o
d
i
s
based
on
a general
m
odel
i
n
g
ap
pro
ach
u
s
i
n
g
h
y
b
r
i
d
au
tomata. Th
is tem
p
o
r
al
m
o
d
e
l
is u
s
ed
to
fau
lt d
e
tectio
n an
d iso
l
atio
n the faster
p
o
s
sib
l
e an
d wh
o d
i
ag
no
ses
m
o
re p
r
ecisely as po
ssi
b
l
e b
y
find
ing
fau
lt syste
m
co
m
p
o
n
en
ts will
b
e
presen
ted
by
u
s
i
n
g St
at
efl
o
w c
ont
r
o
l
l
er.
In
dee
d
,
i
f
t
h
e
di
ag
n
o
sis is fast and
t
h
e
failed
co
mp
on
en
t is id
en
tified
,
m
a
i
n
t
e
nance
o
p
erat
i
o
ns ca
n
b
e
m
a
de
m
o
re q
u
i
c
kl
y
.
The pape
r
i
s
or
gani
ze
d
as fol
l
ows
.
Sect
i
o
n 2 desc
ri
bes t
h
e
h
y
b
ri
d
dy
nam
i
cal
sy
st
em
m
odel
i
n
n
o
rm
al
ope
rat
i
n
g co
nd
i
t
i
on by
hy
b
r
i
d
aut
o
m
a
t
a
. In
Sect
i
on 3,
we
expl
ai
n t
h
e
o
b
ject
i
v
e of
ou
r
di
agn
o
st
i
c
ap
pr
oac
h
base
d
on
hy
bri
d
m
odel
.
The
t
w
o t
a
n
k
s sy
st
em
i
s
co
nsi
d
e
r
ed i
n
sect
i
o
n
4 t
o
s
h
o
w
t
h
e
effect
i
v
e
n
ess
o
f
ou
r
diagnosis approach.
At the e
n
d
,
a con
c
lusion is presen
ted
with
so
m
e
p
e
rspectiv
es.
2.
MODELLING OF HYBRID SYSTE
M
Tw
o cl
asses o
f
hy
bri
d
m
odel
are di
st
i
n
gui
s
h
ed [
8
]
.
The
first class, known as integrated form
alis
m
,
ex
tend
s
o
n
e
of th
e m
o
d
e
ls (discrete o
r
con
tin
uou
s) in
or
de
r t
o
s
p
eci
fy
a
n
d
descri
be t
h
e
sy
st
em
. The se
con
d
class o
f
m
o
d
e
ls co
-o
rd
in
ate th
e d
i
screte m
o
d
e
l an
d
th
e con
tin
uou
s on
e; th
is is th
e ap
pro
a
ch
th
at we
h
a
v
e
taken. T
h
is c
h
oice is due t
o
the fact that usi
ng a
m
odel for each com
p
one
n
t reta
ins the s
p
ecification
potential
of eac
h d
o
m
a
in. C
o
nt
i
n
uo
us
and
di
scret
e
as
pect
s cor
r
es
po
nd t
o
t
w
o di
f
f
e
r
ent
w
o
rl
ds p
r
e
s
ent
i
n
g t
w
o di
f
f
ere
n
t
views
of a
syste
m
.
In t
h
i
s
sect
i
o
n
,
hy
bri
d
a
u
t
o
m
a
t
a
m
odel
bel
ongi
ng t
o
t
h
e se
con
d
cl
ass i
s
descri
be
d. T
h
e con
s
t
r
uct
i
o
n
of t
h
e di
a
g
n
o
si
s i
s
based o
n
t
h
i
s
no
rm
al
operat
i
ng m
odel
whi
c
h rep
r
ese
n
t
s
t
h
e sy
st
em
i
n
norm
a
l
sit
u
at
i
ons,
whe
n
no
faul
t
i
s
prese
n
t
.
Pr
oced
u
r
es base
d o
n
t
h
i
s
m
odel
are a pri
o
ri
abl
e
onl
y
t
o
det
ect
faul
t
s
.
Whe
n
i
n
f
o
rm
at
i
on i
s
pr
o
v
i
d
e
d
ab
out
whi
c
h
part
o
f
t
h
e no
rm
al op
erat
i
on m
odel
i
s
un
veri
ļ¬
e
d
in presence
of
faults,
i
s
ol
at
i
on (l
ocat
i
o
n
)
of
t
h
e faul
t
y
com
ponent
becom
e
s
p
o
ssi
bl
e.
2.
1. Hy
bri
d
A
u
to
ma
ta
Hyb
r
i
d
au
t
o
mata [9
], [10
]
can
b
e
seen
as an
ex
tension
of ti
m
e
d
au
to
m
a
ta with
m
o
re g
e
n
e
ral
dy
nam
i
cs. A cl
ock
x
is a con
tin
uou
s
v
a
riab
le with
tim
e
d
e
ri
v
a
tiv
e equal to
1
,
t
h
at is
1
x
ļ½
ļ¦
. I
n
a hy
bri
d
aut
o
m
a
t
on, t
h
e
cont
i
n
uo
us va
ri
abl
e
s
x
can evol
ve accordi
n
g to som
e
m
o
re
gene
ral diffe
rential equations, for
ex
am
p
l
e
.
Th
is
allo
ws
h
ybrid
au
to
m
a
ta to
cap
ture no
t
o
n
l
y
th
e ev
o
l
u
tio
n
o
f
tim
e b
u
t
also
th
e
ev
o
l
u
tio
n
of a
wid
e
rang
e
o
f
p
h
y
sical en
tities. Th
e
d
i
scret
e
d
y
n
a
m
i
cs o
f
h
ybrid
au
to
m
a
ta can
also
b
e
m
o
re
com
p
l
e
x
an
d d
e
scri
be
d wi
t
h
m
o
re
gene
ral
c
onst
r
ai
nt
s. I
n
t
h
e
f
o
l
l
o
wi
n
g
, we prese
n
t
a
c
o
m
m
onl
y
used
ver
s
i
o
n
of
hy
b
r
i
d
aut
o
m
a
t
a
. Di
ffere
nt
f
o
rm
s of c
onst
r
ai
nt
s
resu
l
t
i
n
di
f
f
ere
n
t
vari
a
n
t
s
of t
h
i
s
m
odel
.
A
hy
b
r
i
d
au
to
m
a
to
n
co
n
s
ists of a
fin
i
te set Q of
d
i
screte states and
a set
o
f
n
c
o
nt
i
n
u
o
u
s
vari
a
b
l
e
s ev
ol
vi
ng
i
n
a
continuous state space
. In eac
h
discrete state
,
the e
v
olution of the c
onti
n
uous
va
riables a
r
e
go
ve
rne
d
by
a
di
ffe
re
nt
i
a
l
eq
uat
i
on:
. T
h
e i
n
vari
a
n
t
o
f
a
di
s
c
ret
e
st
at
e
i
s
d
e
fi
ne
d as
a s
u
b
s
et
of
. The
co
n
d
i
t
i
ons
fo
r s
w
i
t
c
h
i
ng
bet
w
ee
n
di
scret
e
st
at
es ar
e speci
fi
e
d
by
a set
of
g
u
ar
ds
suc
h
t
h
at
f
o
r
each
discrete transition
.
A state (q,
)
of
can c
h
ange i
n
two
ways as follows:
1
)
b
y
a co
n
tinu
o
u
s
evo
l
u
tion, th
e co
n
tinuou
s state
evolves accordi
ng t
o
the
dynam
i
c
s
fq while
th
e d
i
screte state q
rem
a
in
s con
s
tan
t
;
2) by
a di
scret
e
evol
ut
i
o
n,
x
satisfies th
e g
u
a
rd
of an
ou
tgo
i
n
g
tran
sitio
n, th
e syste
m
ch
an
g
e
s d
i
screte
state b
y
tak
i
ng
th
is transitio
n
.
Let u
s
con
s
id
er th
e
h
ybrid
au
t
o
m
a
ta g
i
v
e
n
i
n
Figure
1
.
This a
u
tom
a
ta has t
h
ree
discrete states
and
.
The co
nt
i
n
uo
us e
vol
ut
i
on i
n
t
h
e st
at
es i
s
repre
s
ent
e
d
resp
ect
i
v
el
y
by
.
((
)
)
x
fx
t
ļ½
ļ¦
A
n
X
ļ
ī
qQ
ļ
((
)
)
q
xf
xt
ļ½
ļ¦
q
q
ļ
X
q
T
x
A
x
12
,
qq
3
q
11
2
2
3
3
()
,
(
)
a
n
d
(
)
x
fx
x
f
x
x
f
x
ļ½ļ½
ļ½
īī
ī
Evaluation Warning : The document was created with Spire.PDF for Python.
ļ²
I
S
SN
:
2
088
-87
08
IJECE
Vol. 5, No. 6, D
ecem
ber
2015 :
1396 ā
1406
1
398
The i
nva
riant i
n
the
state
and
a
re re
spectivel
y
,
and
.
Th
e i
n
itial stat
e of th
is
system
is represen
ted
b
y
an inp
u
t
arc in th
e
o
r
i
g
in state
.
Fi
gu
re
1.
Hy
br
i
d
A
u
t
o
m
a
t
a
Th
rou
gh th
e
u
s
e of
t
h
e
h
ybr
id au
t
o
m
a
ta o
n
e
m
u
st
create a norm
al ope
ration m
odel
whic
h re
prese
n
ts
th
e syste
m
in
n
o
rm
al s
itu
ati
o
n
s
, wh
en
n
o
fau
lt is p
r
esent. Th
is
m
o
d
e
l is o
b
t
ain
e
d
fro
m
id
en
tificatio
n
o
f
diffe
re
nt possi
ble states
of
the system
, the evol
ution e
quati
ons in
each sta
t
e and the
nece
ssary c
o
nditions for
t
h
e t
r
a
n
si
t
i
ons
fr
om
one st
at
e
t
o
an
ot
her.
T
h
i
s
dy
nam
i
c
m
o
del
i
s
nei
t
h
er
m
o
re no
r l
e
ss
a co
py
of
t
h
e
p
r
o
g
ram
cont
rol
-
c
o
m
m
and
sy
st
em
t
o
di
ag
no
se, t
o
whi
c
h i
s
a
d
de
d t
i
m
e i
n
form
at
i
on,
suc
h
as
t
h
e
du
rat
i
o
n
of t
h
e
di
ffe
re
nt
st
ep
of
ope
rat
i
o
n a
nd
dat
e
o
f
occ
u
r
r
ence e
v
e
n
t
s
. Pr
oced
u
r
es b
a
sed o
n
t
h
i
s
m
odel
are a
pri
o
ri
abl
e
onl
y
t
o
det
ect
faul
t
s
.
When
i
n
fo
rm
ati
on i
s
pr
ovi
ded a
b
out
w
h
i
c
h
par
t
of t
h
e no
rm
al
operat
i
o
n
m
odel
i
s
un
va
ri
ed i
n
pre
s
ence
of
fa
ul
t
s
,
i
s
ol
at
i
o
n
(l
oca
t
i
on)
o
f
t
h
e
fa
u
l
t
y
co
m
pon
ent
becom
e
s p
o
ssi
bl
e.
Fi
gu
re
2.
R
e
p
r
esent
a
t
i
o
n
o
f
t
h
e
dy
nam
i
c
m
ode
l
B
o
t
h
co
nt
i
n
uo
us an
d di
scret
e
vari
abl
e
s are
necessary
t
o
descri
be t
h
e b
e
havi
or
of a h
y
b
ri
d sy
st
em
.
Th
e tim
e ev
olu
tio
n
of th
e
syste
m
resu
lts in
a
su
ccessio
n
of m
o
d
e
s. Each m
o
d
e
is
ch
aracterized
b
y
a
m
o
d
a
lity
o
f
th
e d
i
screte state, a
set
o
f
equ
a
lity c
o
n
s
t
r
ain
t
s (equ
atio
ns o
f
state) and
d
e
fi
n
itio
n
of do
m
a
in
ad
m
i
ssi
b
ility
(written
b
y
in
eq
u
a
lity
co
n
s
t
r
ain
t
s: invarian
t). A d
i
screte tran
sition
o
f
a
m
ode t
o
anot
h
e
r m
ode occu
r
s
whe
n
cert
a
i
n
l
ogi
cal
co
nditions a
r
e satisfied. T
h
ese cha
nges
m
a
y be caused by
di
scret
e
eve
n
t
s
t
h
at
are ge
ner
a
t
e
d by
di
sc
ret
e
act
uat
o
rs
or
sens
ors
.
The a
c
t
i
v
i
t
y
tim
e of an associ
at
e
d
m
ode
state is specific to a give
n s
ituation. T
h
ere
f
ore we sa
y t
h
at th
e system is in
n
o
rm
al
m
o
d
e
, if th
e m
o
d
e
activation time is noted i
n
the in
terval
, if the activation tim
e
ex
ceeds
term
inal
the
syste
m
is
co
nsid
ered
fau
lty. It is rep
r
esen
ted
b
y
12
,
qq
3
q
1
()
in
v
q
2
()
in
v
q
3
()
in
v
q
1
q
1
,(
)
in
it
q
12
(M
,
...
)
n
M
M
n
T
m
I
max
i
M
T
Evaluation Warning : The document was created with Spire.PDF for Python.
I
J
ECE
I
S
SN
:
208
8-8
7
0
8
ļ²
Using
H
y
b
r
i
d
Au
toma
ta
f
o
r
Diag
no
sis
o
f
Hyb
r
id Dyna
mica
l S
y
stems (Lo
t
fi Mh
amd
i
)
1
399
Figure
3
.
F
o
r e
ach m
ode,
we
defi
ne t
w
o t
i
m
e val
u
e
s
, as
f
o
l
l
ows:
1)
The
m
i
nim
u
m
t
i
m
e
necessa
ry
f
o
r
co
rrect
e
x
ecut
i
o
n
o
f
m
ode
, n
o
t
e
d
by
2
)
Th
e m
a
x
i
mu
m
ti
me to
lerated
for th
e ex
ecu
tio
n of m
o
d
e
, no
ted
b
y
Th
us we
defi
n
e
t
h
e no
rm
al
operat
i
n
g i
n
t
e
r
v
al
s not
ed
,
=
and t
h
at
t
h
e fa
u
l
t
y
m
o
d
e
not
e
d
:
.
Fi
gu
re
3.
Act
i
v
at
i
on t
i
m
e and
m
ode of
o
p
erat
i
o
n
Th
e u
tility u
s
ed
for d
y
n
a
m
i
c
m
o
d
e
ls is
th
e h
ybrid
au
t
o
m
a
ta, we sh
all also
see th
at th
is to
o
l
h
a
s th
e
di
sad
v
a
n
t
a
ge
o
f
i
n
c
r
easi
n
g t
h
e si
ze o
f
t
h
e
m
odel
s
cons
i
d
erab
ly
with
the co
m
p
lex
ity o
f
th
e system
stu
d
i
ed
(p
articu
l
arly ind
u
s
t
r
ial syste
m
s). Th
us, we mu
st first an
swe
r
the question "
I
s what
all m
odeled traject
orie
s are
actually achievable in the syste
m
Ā» An
swer that question bac
k
to the
reach
a
b
ility analysis of states in the
g
r
aph
wh
ich lead
s to red
u
c
ed m
o
d
e
l o
f
h
ybrid
au
t
o
m
a
ta ca
lled
attain
ab
le
au
to
m
a
to
n
wh
i
c
h
we m
o
d
e
led
th
e
p
o
s
sib
l
e ev
o
l
u
tio
n
s
of th
e syste
m
fo
r a
g
i
v
e
n
in
itial co
nd
itio
n
.
3.
OBJECTI
V
E OF
O
U
R DIA
G
NO
SIS AP
P
R
O
A
C
H
BA
S
E
D
O
N
HYB
R
ID
M
O
DEL
In t
h
i
s
pape
r
we base
ou
rsel
ves o
n
hy
bri
d
m
odel
s
, whi
c
h
pr
op
ose t
h
e c
o
m
p
i
l
a
t
i
on of
a di
ag
nose
r
from
a hybrid autom
a
ta
m
o
del of th
e system
. In Figure
4, we illustra
te the global schem
a
of diagnos
e
r
co
nstru
c
tion
.
Fi
gu
re
4.
Et
ape
o
f
c
onst
r
uct
i
o
n
Di
ag
no
ser
i
M
mi
n
i
M
T
i
M
ma
x
i
M
T
m
I
mi
n
m
a
x
,
ii
MM
TT
ļ©
ļ¹
ļ«
ļ»
ma
x
,
i
M
T
ļ¹ļ©
ļ«ļ„
ļ»ļ«
Evaluation Warning : The document was created with Spire.PDF for Python.
ļ²
I
S
SN
:
2
088
-87
08
IJECE
Vol. 5, No. 6, D
ecem
ber
2015 :
1396 ā
1406
1
400
Acco
r
d
i
n
g t
o
t
h
e m
odel
of t
h
e sy
st
em
t
o
di
agn
o
se, t
h
e e
v
ent
be
ha
vi
o
r
s
,
t
e
m
poral
an
d
di
ffe
rent
i
a
l
equat
i
o
ns
o
f
t
h
e
pr
ocess a
r
e
i
d
ent
i
f
i
e
d. T
h
us, t
h
e m
odel
of t
h
e sy
st
em
i
s
sup
p
o
se
d t
o
be "c
om
pl
et
e" i
n
t
h
e
sen
s
e it
h
a
s
p
r
esen
t th
e no
rm
al an
d failing
b
e
h
a
v
i
or
o
f
the p
r
o
cess b
y
t
h
e too
l
FMEA (Failure Modes and
Effects Analysis).
An
d t
h
ere
f
ore
we wa
nt
t
o
bui
l
d
t
h
e di
ag
nosi
s
of t
h
i
s
sy
st
em
, i
n
t
h
e form
of a hy
b
r
i
d
m
odel
.
The r
o
l
e
o
f
th
e
d
i
agno
sis is to
in
fer t
h
e ex
isten
ce no
n- ob
serv
ab
le fau
lts b
a
sed
o
n
ob
serv
ab
le ev
en
ts and
th
e ti
m
e
elapsed bet
w
ee
n the
s
e e
v
ents.
A system is sai
d
to
b
e
fau
lty wh
en
th
e actual b
e
h
a
vi
or i
s
n
o
t
co
nsi
s
t
e
nt
w
i
t
h
one
nom
i
n
al
t
r
aject
ory
.
Eve
n
i
f
fa
ul
t
s
are rel
a
t
e
d t
o
phy
si
cal
c
o
m
pone
nt
s
(act
uat
o
rs
, se
ns
ors
,
a
n
d
sy
st
em
com
ponent
s)
t
h
e
y
m
a
y
be
classi
ļ¬
ed
with
respect t
o
their effects:
fau
lts
may eith
er affect th
e con
tinu
o
u
s
-tim
e ev
o
l
u
tio
n in
a
g
i
v
e
n
m
o
d
e
(so-called
con
tin
uo
us fau
lts)
o
r
m
a
y affect th
e
d
i
screte trajecto
r
y (so-called
d
i
screte
fau
l
ts) [11
]
.
3.
1. Contin
uous
F
a
ults
C
ont
i
n
u
ous
fa
u
l
t
s
are rel
a
t
e
d
t
o
a
gi
ve
n m
ode
. T
h
ey
m
a
y
be of
t
w
o t
y
pes:
1)
Fau
lt th
at co
rru
p
t
s th
e equ
a
lity co
nstrain
t
s.
2)
Fau
lt th
at co
rru
p
t
s th
e i
n
equ
a
lity co
n
s
train
t
s.
3.2. Discrete
F
a
ults
These
fa
ul
t
s
pe
rt
ur
b t
h
e
di
scre
t
e
evol
ut
i
o
n. T
h
ree
ki
nds
o
f
f
a
ul
t
s
m
a
y
be consi
d
ere
d
:
1)
The sy
st
em
i
s
m
ovi
ng
fr
om
one m
ode i
to an
ot
he
r o
n
e
whi
c
h i
s
n
o
t
refe
rre
d as a pos
si
bl
e
success
o
r if the
syste
m
works norm
ally, th
at is
to say a s
u
cces
s
o
r whic
h does not belo
ng to the pre
d
iction
gra
p
h
of l
e
vel
1 ass
o
ci
at
ed
wi
t
h
m
ode i
.
2)
The system
is
m
oving
from one m
ode i to a
s
u
ccess
o
r
that belong to the pre
d
iction
gra
ph
asso
ciated
with m
o
d
e
i bu
t th
e tran
sition
cond
itio
n
is no
t
v
e
ri
ļ¬
ed.
3)
The sy
st
em
i
s
st
ay
i
ng i
n
a
m
ode e
v
en t
h
o
u
gh a sp
o
n
t
a
ne
ous
or f
o
rced
swi
t
c
hi
n
g
co
n
d
i
t
i
on i
s
validated.
To b
e
t
t
e
r u
nde
rst
a
n
d
t
h
e
di
ffe
rent
phase
s o
f
di
ag
no
si
s (c
on
st
ruct
i
o
n o
f
t
h
e
dy
nam
i
c
m
odel
,
det
ect
i
o
n
p
h
a
se, l
o
catio
n) we will
d
e
scrib
e
in
m
o
re
d
e
t
a
il th
ese id
eas th
rou
g
h
a h
y
d
r
au
lic syste
m
with
two
tan
k
s.
4.
APPLI
CATI
O
N: TWO
T
A
NKS
SYSTEM
4.1. Description
of the Sys
t
e
m
The t
w
o t
a
nks
sy
st
em
depi
ct
ed i
n
Fi
g
u
re
5 i
s
co
nsi
d
e
r
ed t
o
i
l
l
u
st
rat
e
t
h
e
di
agn
o
si
s m
e
t
h
o
dol
ogy
.
The
syste
m
co
n
s
ists of:
1)
2 t
a
n
k
s R
1
a
n
d
R
2
,
w
hose
sec
t
i
ons a
r
e e
qual
S1 =
S
2
=
0.
0
1
5
4
m
2
. These
t
a
nks a
r
e l
i
n
ke
d
by
a
l
o
we
r pi
pe
C
2
an
d an upp
er pip
e
C
3
. T
h
e
ļ¬
ow
thro
ugh
p
i
p
e
C
2
can
b
e
i
n
terru
p
t
ed
with
a switch
i
ng
v
a
lve
V
2
,
2)
On
e pu
m
p
P that d
e
liv
ers a liqu
i
d
ļ¬
ow
Q
p
t
h
at fall in
to
tan
k
R1
,
3)
4
s
w
i
t
c
hi
n
g
val
v
es V
1
, V
2
, V
3
and
V
4
allo
w to
co
n
t
ro
l t
h
e
ļ¬
ows
Q
1
, Q
2
, Q
3
and
Q
4
,
4)
Tw
o l
e
vel
se
ns
ors
S
h1et
Sh
2
(
r
espect
i
v
el
y
t
o
m
easu
r
e th
e lev
e
l in
bo
th
tank
s R
1
an
d R2),
5)
An
o
v
e
rflo
w
s
e
ns
or
D
h
1
in t
h
e
reser
v
oir R
1
.
Fi
gu
re 5.
Tw
o t
a
nks
sy
st
em
Th
e pu
m
p
is co
n
t
ro
lled
i
n
onā
off-co
n
t
ro
l so as to
m
a
in
tain
in
a fix
e
d
i
n
terv
al. Th
e log
i
c o
f
th
e
pum
p i
s
as
f
o
l
l
o
ws:
1)
Th
e
pu
m
p
is in
itiall
y tu
rn
ed
on
,
2
h
Evaluation Warning : The document was created with Spire.PDF for Python.
I
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ECE
I
S
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8-8
7
0
8
ļ²
Using
H
y
b
r
i
d
Au
toma
ta
f
o
r
Diag
no
sis
o
f
Hyb
r
id Dyna
mica
l S
y
stems (Lo
t
fi Mh
amd
i
)
1
401
2)
She st
o
ppe
d
w
h
en
,
3)
It is switch
e
d on
wh
en
The pum
p
flow is
m
easured, the state of the pum
p (
on
or
of
f) i
s
not
co
ns
i
d
ere
d
t
h
erea
ft
er.
When t
h
e
pum
p i
s
st
o
p
p
e
d
, ze
r
o
fl
ow
(
Q
p
= 0)
,
wh
en oper
a
tin
g Q
p
= Q
0
= 0.001
m
3
/s.
The sy
st
em
m
a
y
be
m
odel
e
d by
consi
d
eri
n
g
5 di
scret
e
st
at
es. The
ļ¬
r
s
t
on
e i
s
t
h
e st
at
e of
pi
pe C
3
th
at
may
tak
e
th
e t
w
o
m
o
d
a
lities:
E
m
p
t
y (E) o
r
Fu
ll (F). Th
e 4
o
t
h
e
r d
i
screte states are th
e states o
f
v
a
lv
e V
1
,
val
v
e V
2
, V
3
an
d v
a
lv
e
V
4
that
m
a
y tak
e
the m
o
d
a
lities Op
en
ed
(O)
or
Clo
s
ed
(C).
As a co
n
s
equ
e
nce 32
m
o
d
e
s allo
ws
to
represen
t
all p
o
ssib
l
e
no
rmal situ
ati
o
n
s
. Each m
o
d
e
is ch
aracterized
b
y
a m
o
d
a
lity o
f
th
e
discrete state vector (V
3
, V
1
, V
2
, V
3
, a
nd
V
4
), a set o
f
con
t
in
uo
us state eq
u
a
tio
ns and
ineq
u
a
lity co
n
s
t
r
ain
t
s.
The f
o
ur
val
v
e
s
V
1
, V
2
, V
3
and V
4
a
r
e c
ont
r
o
l
l
e
d m
a
nual
l
y
. They
m
a
y
be ope
ne
d o
r
cl
o
s
ed by
t
h
e o
p
er
at
or at
any
t
i
m
e
. W
e
con
s
i
d
er t
h
at
t
h
e sy
st
em
i
s
used i
n
a
gi
ve
n expl
oi
t
a
t
i
on m
ode i
n
w
h
i
c
h
V
1
and V
2
are always
ope
ne
d. O
n
l
y
pi
pe C
3
and
va
l
v
e V
4
ar
e op
erated
. Th
e two
co
rr
esp
ond
ing
actio
n
s
(o
p
e
n
an
d
clo
s
e)
cor
r
e
sp
ond
in
th
e fo
llo
wi
ng
to
ev
en
ts e
1
and e
2
. T
h
e e
v
ents e
1
and e
2
(res
p
ect
i
v
el
y
t
h
e t
i
m
e
of ope
ni
n
g
an
d cl
osi
ng t
h
e
val
v
e V
4
)
are
c
ont
rol
l
e
d;
e
1
o
c
cu
rs at ti
m
e
t = 24
0 s,
wh
ile e
2
o
c
cu
rs at tim
e
t = 38
0 s.
The
ļ¬
ow
s Q
1
, Q
2
, Q
3
an
d Q
4
are gi
ve
n by
:
1)
2)
3)
4)
W
i
t
h
:
; o
Ć¹
:
an
d
is grav
ity.
Q
1
an
d Q
4
: th
e
o
u
t
flows
resp
ectiv
ely R1
and
R2
tank
s.
Q
2
an
d Q
3
:
t
h
e
out
fl
o
w
s
fr
om
t
h
e rese
r
voi
r
R
1
t
o
R
2
t
h
r
o
ug
h
pi
pes
C
2
and
C
3
, res
p
ectivel
y.
The
hybri
d
aut
o
m
a
ton that re
prese
n
ts the sy
stem
unde
r norm
a
l conditions
in
this exploit
a
tion m
ode
is p
a
rt
of th
e co
m
p
lete au
to
mato
n
and
is
g
i
ven
b
y
Figure
6
.
Fi
gu
re
6.
Hy
br
i
d
aut
o
m
a
t
a
:
norm
a
l
beha
vi
o
r
From
dy
nam
i
c m
odel
const
r
u
c
t
i
on
we
go t
o
ex
pl
ai
n t
h
e
di
ffe
rent
ph
ases
of t
h
e
di
ag
no
st
i
c
wi
t
h
t
o
ol
Matlab
/ Sim
u
l
i
n
k
/
Stateflow.
4.
2.
C
o
ns
truct
i
on
of
the
Di
a
g
n
o
ser w
i
th
St
ate
f
l
o
w
C
o
n
t
r
o
l
o
f
H
y
b
r
i
d
P
o
w
e
r
P
l
a
n
t
s
t
a
t
e
f
l
o
w
i
s
a
t
o
o
l
i
n
t
e
g
r
a
t
e
d
i
n
t
h
e
M
A
T
L
A
B
e
n
v
i
r
o
n
m
e
n
t
and
use
d
fo
r
t
h
e devel
opm
ent
and t
h
e s
i
m
u
l
a
t
i
on of
com
p
l
e
x react
i
v
e sy
st
em
s. It
uses a va
ri
ant
of t
h
e
fin
ite state
m
a
ch
in
e.
Sp
ecifically, it u
s
es th
e h
y
b
r
i
d
St
ate charts
form
alis
m
.
It provi
des a bloc
k that
can be
in
clu
d
e
d
i
n
a
Si
m
u
lin
k
m
o
del. Ad
d
ition
a
lly, it en
ab
les th
e rep
r
esen
t
a
tio
n
o
f
h
i
erarch
y,
p
a
rallelism
an
d
2
0.2
hm
ļ³
2
0.
1
hm
ļ£
11
Qh
ļ”
ļ½
21
2
1
2
(h
)
Qs
i
g
n
h
h
h
ļ”
ļ½ļ
ļ
31
1
(h
0
.
5
)
0
.
5
Qs
i
g
n
h
ļ”
ļ½ļ
ļ
42
Qh
ļ”
ļ½
2
A
g
ļ”
ļ½
52
3.
6
1
0
A
m
ļ
ļ½ļ“
2
9.81
/
gm
s
ļ½
Evaluation Warning : The document was created with Spire.PDF for Python.
ļ²
I
S
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088
-87
08
IJECE
Vol. 5, No. 6, D
ecem
ber
2015 :
1396 ā
1406
1
402
hi
st
ory
.
Hi
e
r
ar
chy
e
n
abl
e
s
t
h
e
or
ga
ni
zat
i
o
n
of
com
p
l
e
x
sy
st
em
s by
defi
ni
n
g
a
pa
rent
/
o
ffs
p
r
i
n
g
ob
ject
structure.
4.
2.
1. N
o
rm
al
B
e
ha
vi
or
Sim
u
lation
of t
h
e T
W
O T
A
NKS
SYSTEM
w
ith State
flow is de
picted i
n
Figure
7
.
1)
M
ode
1:
y
o
u
m
u
st
c
o
m
p
l
e
t
e
reserv
oi
r
R
1
t
o
t
h
e l
e
vel
h
1
attain
ed
0
.
5
.
2)
M
ode 2:
t
h
e l
e
vel
h
1
reac
he
d 0.
5, t
h
e t
w
o pi
pes C
2
and C
3
are open, you
m
u
st com
p
lete
reservoir
R2
to th
e lev
e
l
h
2
attain
ed
0
.
5
.
3)
Mo
d
e
3
:
Wh
ere th
e two
lev
e
ls
tan
k
s
ex
ceed
li
mits, th
e v
a
lv
e
V
4
is
o
p
e
n
e
d
to th
e tim
e e
1
.
4)
Mo
d
e
4
:
we m
u
st em
p
t
y th
e two tank
s t
o
level
h
1
<0
.5
.
Fi
gu
re
7.
M
o
d
e
l
i
ng
of
t
h
e
p
r
o
cess wi
t
h
st
at
e
fl
o
w
Evaluation Warning : The document was created with Spire.PDF for Python.
I
J
ECE
I
S
SN
:
208
8-8
7
0
8
ļ²
Using
H
y
b
r
i
d
Au
toma
ta
f
o
r
Diag
no
sis
o
f
Hyb
r
id Dyna
mica
l S
y
stems (Lo
t
fi Mh
amd
i
)
1
403
Fi
gu
re
8.
N
o
r
m
al
behavi
or
The c
o
n
s
t
r
u
c
t
i
o
n
o
f
t
h
e
di
a
g
nosi
s
i
s
base
d
on
t
h
e t
e
m
por
al
kn
o
w
l
e
d
g
e
of t
h
e
pr
ocess;
we
nee
d
t
o
kn
o
w
t
h
e
pr
oc
ess d
u
rat
i
o
n s
u
ch as t
h
e
ope
ni
ng t
i
m
e of t
h
e val
v
es
or t
h
e t
i
m
e of sens
o
r
s
t
at
us cha
nge
. F
i
gu
re
8
illu
strates t
h
e no
m
i
n
a
l b
e
h
a
v
i
or
o
f
t
h
e instru
m
e
n
t
atio
n
p
r
o
cess
of an
o
p
e
ratin
g cycle.
Fro
m
figu
re abo
v
e
, th
e tran
sitio
n
tim
es are de
fine
d for eac
h phase
of the
norm
al proce
ss.
Tabl
e 1.
I
d
ent
i
fi
cat
i
on of
t
e
m
p
o
r
al
pr
ocess
Act
i
ons
Tim
e
in sec
Int
erval of
t
i
m
e
Opening the pu
m
p
Qp.
Closing the pum
p
Qp.
0
20
[
0
, 2
0
]
Opening the valve V
4
Closing the valve
V
4
240
380
[24
0
,
380]
Activating the sensor
Sh1
Deactivating the se
nsor Sh1
7.
7
60
[
7
.7
, 6
0
]
Activating the sensor
Sh2
Deactivating the se
nsor Sh2
20
180
[20 ,
180]
4.
2.
2. C
o
nsi
d
e
red
F
a
i
l
u
res
In
o
r
d
e
r t
o
illustrate th
e po
ssi
b
l
e fau
lt cases stated
at secti
o
n
3
.
1
an
d
3
.
2 (con
tinu
o
u
s
an
d
d
i
screte
fau
lts), we
co
nsid
er
t
h
e fo
llowing
2
p
a
rticu
l
ar
situ
ation
s
:
1)
Th
e
fau
lts t
h
at
p
e
rt
u
r
b
th
e state equ
a
tio
ns (con
tin
uou
s
fau
lts):
a)
Fault
of sensor Sh1 that m
easures
h
1
,
b)
Faul
t
o
f
se
ns
or
Sh
2 t
h
at
m
easures
h
2
.
2)
The faul
t
s
t
h
at
pert
ur
b
t
h
e
pa
s
s
age bet
w
ee
n d
i
ffere
nt
m
odes (di
s
c
r
et
e
fa
ul
t
s
):
a)
Eve
n
t e
1
i
s
co
nt
r
o
l
l
e
d.
It
occ
u
rs t
i
m
e t
= 240s
. I
f
t
h
i
s
e
v
ent
has
n
o
ef
f
ect
on t
h
e di
sc
ret
e
state evolution (valve
V
4
stays b
l
o
c
k
e
d
o
p
en
ed), th
e
syste
m
will s
t
ay
in
m
o
d
e
2
(Th
e
passa
ge m
ode
2 t
o
m
ode 3
i
s
di
sabl
e
d
),
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S
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08
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ecem
ber
2015 :
1396 ā
1406
1
404
b)
Eve
n
t e
2
i
s
cont
rol
l
e
d. It
occ
u
rs at
t
i
m
e
t = 3
80s
. If t
h
i
s
eve
n
t
has n
o
effec
t
on t
h
e di
scret
e
state evolution (valve
V
4
stays b
l
o
c
k
e
d
cl
o
s
ed
), t
h
e syste
m
will s
t
ay
in
m
o
d
e
4
(Th
e
passa
ge m
ode
4 t
o
m
ode 1
i
s
di
sabl
e
d
),
c)
In
fact, if
p
i
p
e
P3
is en
tirely clo
g
g
e
d, th
e
dyn
amics o
f
th
e state v
a
riab
les will co
n
tin
u
e
to
co
rresp
ond
to th
e con
tin
uou
s
state eq
u
a
tion
s
of m
o
d
e
1
ev
en
if th
e lev
e
l
h
1
bec
o
m
e
s l
a
rg
er
t
h
an
0,
5m
(t
he
sy
st
em
rest
i
n
m
ode 1)
.
4.
2.
3. Fa
ul
t Di
ag
nosi
s
Usi
n
g Sta
t
e
f
l
o
w
In
order to
det
ect and locate
a fault on the s
t
udied
process
,
a syste
m
for i
n
ject
i
n
g ra
n
d
o
m
defect
s on
the process i
n
s
t
rum
e
ntation was created.
Fi
gu
re 9.
I
n
ject
i
on bl
oc
k ra
nd
om
faul
t
s
5. RES
U
LT
A
PPLIC
ATIO
N
In
th
is
work
we are in
terested
on
ly in
d
e
tecti
ng the o
v
e
rfl
ow
of the
rese
rv
oir R2
(gi
v
e
n
by
h
2
) an
d
we c
onsi
d
er
o
n
e
faul
t
s
rel
a
t
e
d t
o
t
h
e
rese
rv
oi
r:
S
h
2_
st
u
c
k cl
o
s
e (
2
Sh
F
: Does no
t detect th
e lower lev
e
l
(al
w
ay
s
set
t
o
1)
). Thi
s
be
hav
i
or
i
s
re
prese
n
t
e
d by
t
h
e fa
ul
t
y
m
odel
(
F
i
g
ur
e
1
0
)
.
Fi
gu
re
1
0
. M
o
del
o
f
t
h
e fa
ul
t
y
sy
st
em
Our obj
ectiv
e i
s
to
d
e
tect and id
en
tify th
e fau
lts o
c
cu
rring
in
th
e pro
c
ess.
Th
at lead
s to
determin
in
g
th
e way of lo
catin
g
a fau
lt, an
d
t
o
d
e
term
in
in
g
th
e tim
e
of i
t
s
occur
r
e
n
ce
. Fi
g
u
re
1
1
des
c
ri
bes t
h
e
va
ri
at
i
o
n
s
o
f
th
e lev
e
l in
t
h
e two
Reservo
i
rs.
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I
J
ECE
I
S
SN
:
208
8-8
7
0
8
ļ²
Using
H
y
b
r
i
d
Au
toma
ta
f
o
r
Diag
no
sis
o
f
Hyb
r
id Dyna
mica
l S
y
stems (Lo
t
fi Mh
amd
i
)
1
405
Figu
re 1
1
. Dia
g
n
o
sis fo
r fa
ult
Sh
2
_
ST
UCK
CLOSE
To
b
e
tter
under
s
tand
th
e p
r
in
cip
l
e o
f
th
e d
i
ag
no
ser
,
Fig
u
r
e
11
allo
ws u
s
to
co
m
p
ar
e n
o
r
m
al
ope
rat
i
o
n (
p
a
r
t
ab
ove
)
of
t
h
e
pr
ocess
wi
t
h
a
st
at
e of
fa
ul
t
y
ope
rat
i
o
n (
p
a
r
t
bel
o
w
)
.
In
t
h
e
figu
re belo
w, d
e
sp
ite t
h
e lev
e
l
sen
s
or d
e
tects
lower
reservo
i
r R
2
lev
e
l th
at is t
o
say, p
a
ss to
the zero
state (gree
n
si
gnal), i
t
rem
a
ins in st
ate 1
(bl
u
e si
gnal). This
m
o
ment repr
ese
n
ts the
occ
u
rrence of a
failure.
Since t
h
e sensor Sh2
rem
a
ins in state 1 (
h
2
> 0.
2
)
;
t
h
ere
f
o
r
e t
h
e
r
e
i
s
di
st
ur
ban
ce on t
h
e ev
ol
ut
i
on
of
t
h
e co
nt
i
n
u
ous
part
i
s
d
u
e t
o
t
h
e
passa
ge m
ode
3 t
o
m
ode 4
i
s
cancel
e
d
.
5.
CO
NCL
USI
O
N
In t
h
i
s
pa
pe
r we st
u
d
i
e
d t
h
e
faul
t
-
di
a
g
n
o
si
s pr
obl
em
for hy
b
r
i
d
sy
st
em
s.
W
e
use
d
t
h
e
fo
rm
ali
s
m
of
h
ybrid
au
to
m
a
t
a
for m
o
d
e
ling h
ybrid
syste
m
s with
fau
lts an
d to
d
e
ļ¬
n
e
t
h
e no
tio
ns of
d
i
ag
no
sab
ility an
d
tim
e
ab
stract d
i
agno
sab
ility. W
e
fo
cu
sed
ou
r at
ten
tio
n
on
ti
me-ab
s
t
r
act d
i
agn
o
s
ab
ility an
d we d
e
ļ¬
ned a
Faul
t
Diagn
o
sis on
h
ybrid
au
to
m
a
ta with
fau
lts
for
T
W
O
T
A
NK
S S
Y
STE
M
. H
o
we
ve
r,
ou
r a
p
p
r
oach
can
be
i
n
t
e
grat
e
d
wi
t
h
t
h
e pr
o
g
n
o
si
s.
Thi
s
i
s
t
o
be
ca
pabl
e b
o
t
h
o
f
r
e
sp
on
di
n
g
t
o
t
h
e occu
rre
nce
of
a
fa
ul
t
,
b
u
t
a
l
so
to
b
e
ab
le to
an
ticip
ate.
ACKNOWLE
DGE
M
ENTS
The a
u
t
h
ors
w
oul
d l
i
k
e t
o
t
h
ank t
h
e com
m
e
nt
s p
r
o
v
i
d
e
d
by
t
h
e an
o
n
y
m
ous revi
ewe
r
s and e
d
i
t
o
r
,
whi
c
h
hel
p
t
h
e
aut
h
o
r
s i
m
pro
v
e t
h
i
s
pa
pe
r s
i
gni
fi
ca
nt
l
y
. We
wo
ul
d
al
s
o
ac
kn
o
w
l
e
d
g
e
t
h
e
hel
p
f
r
om
Zi
ne
b
Si
m
e
u
-
Ab
azi,
In
stitu
t Nation
a
l Po
lytec
h
n
i
que
d
e
Grenob
le, INPG Fran
ce.
REFERE
NC
ES
[1]
R. Alur, C
.
Cou
r
coubetis, N. H
a
lbwachs,
T. A
.
Henzinger
,
P.
H. Ho, X. Nico
llin, A. Olivero
,
J. Sifakis,
and
S.
Yovine, "
T
he Algorithmic Anal
ysis of H
y
br
id S
y
stems",
T
h
eor
e
ti
cal Com
puter Science 138
, pp
. 3
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.
[2]
O. Maler
,
Z. M
a
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ueli
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y
b
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y
ste
m
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W.
de
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,
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[4]
Branick
y
M.S.
,
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y
br
id s
y
stem
s: Modeling
,
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the
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MI
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ssa
c
huse
tts,
USA, 1995.
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