Inter
national
J
our
nal
of
Electrical
and
Computer
Engineering
(IJECE)
V
ol.
8,
No.
4,
August
2018,
pp.
2157
–
2171
ISSN:
2088-8708
2157
I
ns
t
it
u
t
e
o
f
A
d
v
a
nce
d
Eng
ine
e
r
i
ng
a
nd
S
cie
nce
w
w
w
.
i
a
e
s
j
o
u
r
n
a
l
.
c
o
m
A
New
Hybrid
Rob
ust
F
ault
Detection
of
Switching
Systems
by
Combination
of
Obser
v
er
and
Bond
Graph
Method
Mohammad
Ghasem
Kazemi
and
Mohsen
Montazeri
Department
of
Electrical
and
Computer
Engineering
Shahid
Beheshti
Uni
v
ersity
,
A.C.,
T
ehran,
Iran
Article
Inf
o
Article
history:
Recei
v
ed
August
4,
2017
Re
vised
March
15,
2018
Accepted:
April
5,
2018
K
eyw
ord:
Switching
system
Rob
ust
f
ault
detection
F
ault
sensiti
vity
Disturbance
attenuation
Bond
Graph
ABSTRA
CT
In
this
paper
,
the
problem
of
rob
ust
F
ault
Detection
(FD)
for
continuous
time
switched
system
is
tackled
using
a
h
ybrid
approach
by
combination
of
a
switching
observ
er
and
Bond
Graph
(BG)
method.
The
main
criteria
of
an
FD
system
including
the
f
ault
sen-
siti
vity
and
disturbance
attenuation
le
v
el
in
the
presence
of
parametric
uncertainties
are
considered
in
the
proposed
FD
system.
In
the
first
stage,
an
optimal
switching
observ
er
based
on
state
space
representation
of
the
BG
model
is
designed
in
which
simultaneous
f
ault
sensiti
vity
and
disturbance
a
ttenuation
le
v
el
are
satisfied
using
H
=H
1
inde
x.
In
the
second
stage,
the
Global
Analytical
Redundanc
y
Relations
(GAR
Rs)
of
the
switch-
ing
system
are
deri
v
e
d
based
on
the
output
estimation
error
of
the
observ
er
,
which
is
called
Error
-based
Global
Analytical
Redundanc
y
Relations
(EGARRs).
The
paramet-
ric
uncertainties
are
included
in
the
EGARRs,
which
define
the
adapti
v
e
thresholds
on
the
residuals.
A
constant
term
due
to
the
ef
fect
of
disturbance
is
also
considered
in
the
thresholds.
In
f
act,
a
tw
o-stage
FD
system
is
proposed
wherein
some
criteria
may
be
considered
in
each
stage.
The
ef
ficienc
y
of
the
proposed
method
is
sho
wn
for
a
tw
o-tank
system.
Copyright
c
2018
Institute
of
Advanced
Engineering
and
Science
.
All
rights
r
eserved.
Corresponding
A
uthor:
M.
G.
Kazemi
Shahid
Beheshti
Uni
v
ersity
East
V
af
adar
Blvd.,
T
ehranpars,
T
ehran,
Iran
+989177428946
Email
mg
kazemi@sb
u.ac.ir
1.
INTR
ODUCTION
F
ault
diagnosis
has
become
an
important
topic
in
industrial
applications,
which
further
leads
to
more
attention
in
research
community
.
Model
based
f
ault
detection,
isolation
and
identification
has
recei
v
ed
much
attention
in
the
literature
[1],[2],[3],[4],[5],[6].
Generally
,
obtaining
a
proper
model
of
the
system
is
crucial
for
f
ault
diagnosis
design.
Presenting
a
proper
model
by
considering
multi
ph
ysics
nature
and
the
interaction
between
continuous
and
discrete
dynamics
of
the
system
is
a
challenging
task
in
the
model
based
FD
system
design
for
h
ybrid
systems.
Bond
Graph
(BG)
method
may
be
considered
as
a
con
v
enient
tool
for
modeling
and
f
ault
diagnosis
of
dynamic
systems.
Dif
ferent
applications
of
the
BG
method
can
be
found
in
the
literature
for
modeling
[7],[8],[9],[10],[11],[12]
and
f
ault
diag-
nosis
[12],[13],[14],[15],[16]
applications.
Because
of
its
ef
ficienc
y
in
modeling
and
f
ault
diagnosis
of
continuous
dynamic
systems,
the
BG
method
is
de
v
eloped
to
represent
switching
phenome
n
on
in
h
ybrid
dynamic
systems,
which
leads
to
numerous
applications
of
the
method
for
modeli
ng
and
f
ault
diagnosis
of
h
ybrid
systems
in
the
literature.
Due
to
the
adv
ancement
of
computer
technology
in
control
systems,
switching
systems
ha
v
e
dra
wn
much
attention
in
recent
years.
Switching
systems
may
be
assumed
as
an
important
class
of
h
ybrid
systems
wherein
a
finite
number
of
dynamical
subsystems
and
a
logic
rule
define
the
o
v
erall
dynamic
of
the
system.
The
dynamic
of
the
subsystems
and
thus,
the
o
v
erall
system
may
be
continuous
or
discrete,
linear
or
nonlinear
and
so
on.
In
this
current
paper
,
the
linear
continuous
case
of
the
switching
system
with
a
v
erage
dwell
time
is
considered.
F
ault
detection
of
the
switching
systems
dra
ws
much
attention
in
recent
years.
In
[17],
f
ault
detection
J
ournal
Homepage:
http://iaescor
e
.com/journals/inde
x.php/IJECE
I
ns
t
it
u
t
e
o
f
A
d
v
a
nce
d
Eng
ine
e
r
i
ng
a
nd
S
cie
nce
w
w
w
.
i
a
e
s
j
o
u
r
n
a
l
.
c
o
m
,
DOI:
10.11591/ijece.v8i4.pp2157-2171
Evaluation Warning : The document was created with Spire.PDF for Python.
2158
ISSN:
2088-8708
problem
for
a
class
of
linear
switched
time
v
arying
delay
system
is
formulated
as
H
1
filtering
problem
with
ADT
in
LMI
form.
A
filter
based
f
ault
detection
approach
for
continuous
time
switching
systems
with
ADT
is
considered
in
[18].
Simultaneous
F
ault
Detecti
on
and
Control
(SFDC)
for
continuous
and
discrete
time
linear
switched
systems
with
ADT
based
on
dynamic
filter
is
tackled
in
[19]
whe
rein
a
f
ault
sensiti
vity
is
achie
v
ed
for
a
gi
v
en
disturbance
attenuation
le
v
el
based
on
H
=H
1
criteria.
The
rob
ustness
of
an
FD
system,
which
is
defined
as
the
considering
the
ef
fects
of
uncertainties,
distur
-
bances
and
noises
in
the
system,
may
be
re
g
arded
as
the
most
important
criteria
in
an
FD
system.
Ho
we
v
er
,
it
is
impossible
to
completely
remo
v
al
the
ef
fects
of
uncertainties,
disturbances
and
noises
in
an
FD
system.
Thus,
man
y
studies
are
de
v
oted
to
attenuate
the
ef
fects
of
disturbances,
uncertainties
and
noises
in
an
FD
system.
In
other
hand,
the
attenuation
of
disturbance
ef
fects
may
lead
to
reduce
the
f
ault
sensiti
vity
in
an
FD
system.
Therefore,
the
f
ault
sensiti
vity
must
also
be
considered
in
an
FD
system
design.
The
ef
fects
of
uncertainties
in
the
model
may
l
ead
to
missed
or
f
alse
alarms
in
the
FD
system.
T
o
w
ard
this
end,
tw
o
approaches
are
proposed
in
the
literature
including
acti
v
e
and
passi
v
e
approaches
[20].
In
acti
v
e
approach,
the
rob
ustness
of
the
FD
system
is
defined
in
the
residual
generation
stage,
while
the
rob
ustness
is
considered
in
the
residual
e
v
aluation
stage
in
the
passi
v
e
approach
[20].
In
[21],
the
rob
ust
f
ault
detection
filter
design
of
continuous
time
switched
delay
system
s
is
considered.
Design
of
observ
er
-based
rob
ust
po
wer
system
stabilizers
by
considering
parametric
uncertainties
is
noticed
in
[22].
The
parametric
uncertainty
representation
in
the
BG
method
in
Linear
Fractional
T
r
ansformation
(LFT)
form
is
presented
in
[23]
and
noticed
in
thereafter
studies
for
rob
ust
f
ault
diagnosis
[24],[25].
Rob
ust
f
ault
diagnosis
of
ener
getic
system
with
parameter
uncertainties
is
tackled
in
[24].
The
nonlinear
modeling,
structural
analysis,
residual
generation
with
adapti
v
e
thresholds
and
sensiti
vity
analysis
are
done
using
the
BG
method.
The
proposed
method
is
then
used
for
a
boiler
system.
The
modeling
and
f
ault
diagnosis
of
a
DC
motor
relying
on
the
BG
method
is
gi
v
en
in
[16].
The
parameters
of
the
BG
method
are
obtained
using
the
real
data
of
the
system.
The
a
v
erage
v
alues
of
the
parameters
and
their
standard
de
viations
are
assumed
as
the
nominal
v
alues
and
uncertain
parts
of
the
parameters,
respecti
v
ely
.
In
[25],
the
residuals
and
thresholds
generation
in
presence
of
parameter
uncertainties
in
LFT
form
is
considered
and
is
applied
for
mechatronic
systems.
Rob
ust
f
ault
diagnosis
and
prognostic
s
of
a
hoisting
mechanism
based
on
the
BG
method
considering
the
LFT
form
of
the
parametric
uncertainties
is
addressed
in
[26].
Se
v
eral
studies
are
concentrated
on
the
f
ault
diagnosis
of
switched
system
and
rob
ust
f
ault
diagnosis
of
continuous
time
systems
as
well,
while
there
are
fe
w
w
orks
on
the
rob
ust
f
ault
diagnosis
of
switched
systems.
Incremental
BG
approach
is
gi
v
en
for
h
ybrid
systems
in
[27],
which
is
an
e
xtension
of
the
proposed
method
in
[28]
for
continuous
time
dynamic
systems.
The
adapti
v
e
thresholds
are
obtained
considering
the
parametric
uncertainties
in
the
system,
which
are
dependent
on
the
modes
of
the
h
ybrid
system
as
well.
In
[29],
a
rob
ust
f
ault
detection
and
isolation
on
the
basis
of
pseudo
BG
model
for
h
ybrid
systems
is
presented.
The
parametric
uncertainties
are
gi
v
en
in
the
LFT
form
and
are
assumed
2%
of
their
nominal
v
alues
at
maximum.
Rob
ust
FD
systems
based
on
the
BG
method
may
be
considered
as
the
passi
v
e
approach,
since
some
thresholds
are
assumed
in
the
FD
system
to
detect
or
isolate
the
occurred
f
aults.
In
this
paper
,
based
on
the
BG
method,
a
ne
w
acti
v
e
rob
ust
FD
system
is
proposed
in
which
the
disturbance
is
attenuated,
the
ef
fects
of
uncertainties
are
considered
and
the
f
ault
sensiti
vity
is
enhanced
as
well.
The
proposed
method
has
the
benefits
of
both
the
BG
and
the
observ
er
method.
In
summary
,
the
main
contrib
utions
of
the
paper
may
be
stated
as
follo
ws:
A
ne
w
rob
ust
acti
v
e
f
ault
detection
based
on
the
combination
of
the
observ
er
and
BG
method
for
linear
switched
system
is
presented,
which
simultaneously
attenuates
disturbance
le
v
el
and
enhances
f
ault
sensi-
ti
vity
.
Error
form
of
the
GARRs
is
used
as
the
residuals
which
is
based
on
the
output
estimation
error
of
the
observ
er
.
The
ef
fects
of
disturbance
and
parametric
uncertainties
in
the
thresholds
are
considered
as
a
fix
ed
threshold
and
an
adapti
v
e
threshold,
respecti
v
ely
.
The
remainder
of
the
paper
is
or
g
anized
as
follo
ws.
The
problem
formulation
and
some
lemmas
and
re-
marks
based
on
these
lemmas
are
gi
v
en
in
section
2.
The
Research
Method
of
the
paper
is
gi
v
en
in
fi
v
e
subsections
of
section
3
as
disturbance
attention
le
v
el,
enhanced
f
ault
sensiti
vity
,
simultaneous
optimal
f
ault
sensiti
vity
and
disturbance
attenuation
le
v
el,
the
GARRs
in
error
form
based
on
the
B
G
method
and
rob
ustness
of
the
FD
system
ag
ainst
parametric
uncertainties.
The
simulation
results
and
discussion
for
a
tw
o-tank
system
are
gi
v
en
in
section
4,
follo
ws
by
conclusion
in
section
5.
IJECE
V
ol.
8,
No.
4,
August
2018:
2157
–
2171
Evaluation Warning : The document was created with Spire.PDF for Python.
IJECE
ISSN:
2088-8708
2159
2.
PR
OBLEM
FORMULA
TION
AND
PRELIMIN
ARIES
Consider
the
continuous
time
linear
switching
system
state
space
representation
in
the
form
of
_
x
(
t
)
=
A
i
x
(
t
)
+
B
i
u
(
t
)
+
B
di
d
(
t
)
+
B
f
i
f
(
t
)
(1)
y
(
t
)
=
C
i
x
(
t
)
+
D
di
d
(
t
)
+
D
f
i
f
(
t
)
(2)
which
can
be
gi
v
en
as
Eq.(3)
-
Eq.(4)
based
on
the
BG
model
of
the
switching
system.
_
x
(
t
)
=
A
(
a;
)
x
(
t
)
+
B
(
a;
)
u
(
t
)
+
B
d
(
a
)
d
(
t
)
+
B
f
(
a
)
f
(
t
)
(3)
y
(
t
)
=
C
(
a;
)
x
(
t
)
+
D
d
(
a
)
d
(
t
)
+
D
f
(
a
)
f
(
t
)
(4)
where
x
2
R
n
is
the
states,
y
2
R
p
is
the
outputs,
u
2
R
k
is
the
inputs,
d
2
R
q
is
disturbances
and
f
2
R
r
is
f
aults
in
the
switched
system.
A
,
B
and
C
are
respecti
v
ely
system,
input
and
output
matr
ices
with
appropriate
dimensions
denote
the
system
,
input
and
output
matrices,
respecti
v
ely
.
.
B
d
,
B
f
,
D
d
and
D
f
are
the
matrices
of
disturbance
and
f
ault
distrib
ution
on
the
states
and
on
the
outputs
of
the
system,
respecti
v
ely
.
The
i;
i
2
l
=
1
;
2
;
:
:
:
;
N
inde
x
is
considered
to
define
modes
of
the
switched
system.
The
a
=
[
a
1
a
2
:
:
:
a
z
]
is
the
state
of
the
controlled
junctions
in
the
BG
model,
which
will
define
the
acti
v
e
mode
of
the
switching
system.
The
number
of
controlled
junctions
in
the
BG
model
is
assumed
as
z
.
The
system
matrices
are
dependent
on
the
parameters
of
the
BG
model
(
R
,
C
,
L
,
.
.
.
)
which
represented
as
in
the
model.
The
switching
Luenber
ger
observ
er
is
assumed
in
the
form
of
Eq.(5).
_
^
x
(
t
)
=
A
i
^
x
(
t
)
+
L
i
(
y
(
t
)
^
y
(
t
))
(5)
^
y
(
t
)
=
C
i
^
x
(
t
)
where
L
i
is
the
observ
er
g
ain
and
is
dependent
on
the
acti
v
e
mode
of
the
BG
model.
The
observ
er
g
ain
is
designed
for
each
mode
in
such
a
w
ay
that
some
criteria
are
satisfied.
The
output
estimation
error
,
which
is
defined
as
Eq.(6),
is
used
in
the
ne
w
form
of
the
GARRs.
e
y
(
t
)
=
y
(
t
)
^
y
(
t
)
(6)
The
asymptotical
stability
of
the
observ
er
is
in
v
estig
ated
on
the
error
dynamic
of
the
observ
er
as
Eq.(7).
_
e
(
t
)
=
(
A
i
L
i
C
i
)
e
(
t
)
+
(
B
f
i
L
i
D
f
i
)
f
(
t
)
+
(
B
di
L
i
D
di
)
d
(
t
)
=
A
cl
i
e
(
t
)
+
B
cl
f
i
f
(
t
)
+
B
cl
di
d
(
t
)
(7)
The
output
estimation
error
is
achie
v
ed
as
Eq.(8).
e
y
(
t
)
=
C
i
e
(
t
)
+
D
f
i
f
(
t
)
+
D
di
d
(
t
)
(8)
In
this
paper
,
tw
o
criteria
are
considered
for
the
output
estimation
error
of
the
observ
er
for
disturbance
attenuation
le
v
el
and
f
ault
sensiti
vity
,
which
is
presented
in
Eq.(9)
and
Eq.(10),
respecti
v
ely
.
sup
jj
e
y
(
t
)
jj
2
jj
d
(
t
)
jj
2
<
;
>
0
(9)
inf
jj
e
y
(
t
)
jj
2
jj
f
(
t
)
jj
2
>
;
>
0
(10)
The
follo
wing
definitions
and
lemmas
are
used
in
the
current
paper
.
Definition
.
Let
N
(
t
1
;
t
2
)
stand
for
the
s
witching
number
of
(
t
)
on
the
interv
al
[
t
1
;
t
2
)
for
a
gi
v
en
switching
signal
(
t
)
and
an
y
t
2
>
t
1
>
t
0
.
If
Eq.(11)
is
satisfied
for
N
0
0
and
a
>
0
,
then
a
is
called
the
ADT
and
N
0
is
chatter
bound
[30].
N
(
t
1
;
t
2
)
N
0
+
(
t
2
t
1
)
=
a
(11)
Lemma
1
:
F
or
the
continuous
time
switchi
ng
system
Eq.(1)
and
Eq.(2),
le
t
>
0
,
1
and
i
>
0
;
8
i
2
l
be
constant
scalars.
Suppose
there
e
xists
positi
v
e
definite
C
1
function
V
(
t
)
:
R
n
!
R
;
2
l
with
V
(
t
0
)
(
x
(
t
0
))
=
0
such
that
[31]:
_
V
i
(
x
t
)
V
i
(
x
t
)
y
T
t
y
t
+
2
i
u
T
t
u
;
8
i
2
l
(12)
A
Ne
w
Hybrid
Rob
ust
F
ault
Detection
of
Switc
hing
Systems
by
...
(Mohammad
Ghasem
Kazemi)
Evaluation Warning : The document was created with Spire.PDF for Python.
2160
ISSN:
2088-8708
V
i
(
x
tk
)
V
j
(
x
tk
)
;
8
(
i;
j
)
2
l
l
;
i
6
=
j
then
the
switching
system
is
Globally
Uniformly
Asymptotically
Stable
(GU
AS)
and
will
satisfied
Eq.(9),
which
is
called
H
1
performance,
with
inde
x
no
greater
than
max(
i
)
for
an
y
switching
signal
with
ADT
as
Eq.(13).
a
>
a
=
l
n
(
)
=
(13)
V
is
called
the
L
yapuno
v
function.
Remark
1
:
In
order
to
use
the
Lemma
1
in
this
paper
,
the
u
and
y
are
substituted
by
d
(
t
)
and
e
y
(
t
)
,
respecti
v
ely
,
which
indicate
the
disturbance
attenuation
le
v
el
on
the
output
estimation
error
of
the
observ
er
.
Lemma2
:
F
or
the
continuous
time
switching
system
Eq.(1)
and
Eq.(2),
let
>
0
,
1
and
i
>
0
;
8
i
2
l
be
constant
scalars.
Suppose
there
e
xists
positi
v
e
definite
C
1
function
V
(
t
)
:
R
n
!
R
;
2
l
with
V
(
t
0
)
(
x
(
t
0
))
=
0
,
which
is
called
L
yapuno
v
function,
such
that
[32]:
_
V
i
(
x
t
)
V
i
(
x
t
)
+
y
T
t
y
t
2
i
u
T
t
u
;
8
i
2
l
(14)
V
i
(
x
tk
)
V
j
(
x
tk
)
;
8
(
i;
j
)
2
l
l
;
i
6
=
j
then
the
switching
system
is
GU
AS
and
will
satisfied
H
performance
as
Eq.(10)
with
inde
x
no
smaller
than
min(
i
)
for
an
y
switching
signal
with
ADT
as
Eq.(13).
Remark
2
:
In
order
to
use
the
Lemma
2
in
this
paper
,
the
u
and
y
are
substituted
by
f
(
t
)
and
e
y
(
t
)
,
respecti
v
ely
,
which
indicate
the
f
ault
sensiti
vity
on
the
output
estimation
error
of
the
observ
er
.
Lemma
3
[33]:
F
or
a
gi
v
en
m
m
symmetric
matrix
Z
2
S
m
and
tw
o
matrices
U
and
V
of
column
dimension
m
,
there
e
xists
matrix
X
that
is
unstructured
and
will
satisfy
Eq.(15).
U
T
X
V
+
V
T
X
T
U
+
Z
<
0
(15)
if
and
only
if
the
follo
wing
inequalities
as
Eq.(16)
and
Eq.(17)
with
respect
to
X
are
satisfied.
N
T
U
Z
N
U
<
0
(16)
N
T
V
Z
N
V
<
0
(17)
N
U
and
N
V
are
arbitrary
matrices
that
their
columns
are
a
basis
for
null
spaces
of
U
and
V
,
respecti
v
ely
.
3.
RESEARCH
METHOD
In
this
section,
the
LMI
formulation
for
f
ault
detection
problem
of
li
near
continuous
time
switched
system
Eq.(1)-Eq.(2)
is
gi
v
en.
The
o
v
erall
results
are
gi
v
en
in
fi
v
e
subsections.
3.1.
Disturbance
attenuation
perf
ormance
in
the
output
estimation
err
or
of
the
switched
obser
v
er
The
switching
L
yapuno
v
function
for
disturbance
attenuati
on
criterion
of
the
observ
er
is
considered
as
Eq.(18),
which
its
deri
v
ati
v
e
must
be
ne
g
ati
v
e
definite
as
Eq.(19).
V
(
t
)
=
e
(
t
)
T
P
i
e
(
t
)
>
0
(18)
_
V
(
t
)
=
_
e
(
t
)
T
P
i
e
(
t
)
+
e
(
t
)
T
P
i
_
e
(
t
)
<
0
(19)
The
switching
L
yapuno
v
function
deri
v
ati
v
e
may
be
obtained
as
Eq.(20)
in
f
ault
free
case.
_
V
(
t
)
=
(
e
(
t
)
T
A
T
cl
i
+
d
(
t
)
T
B
T
cl
di
)
P
i
e
(
t
)
+
e
(
t
)
T
P
i
(
A
cl
i
e
(
t
)
+
B
cl
di
d
(
t
))
(20)
which
can
be
written
as
Eq.(21)
for
simplicity
.
_
V
(
t
)
=
e
(
t
)
T
(
A
T
cl
i
P
i
+
P
i
A
cl
i
)
e
(
t
)
+
2
e
(
t
)
T
P
i
B
cl
di
d
(
t
)
(21)
According
to
Remark
1,
we
need
e
y
(
t
)
T
e
y
(
t
)
for
using
lemma
1,
which
can
be
gi
v
en
as:
e
y
(
t
)
T
e
y
(
t
)
=
e
(
t
)
T
C
T
i
C
i
e
(
t
)
+
2
e
(
t
)
T
C
T
i
D
di
d
(
t
)
+
d
(
t
)
T
D
T
di
D
di
d
(
t
)
(22)
IJECE
V
ol.
8,
No.
4,
August
2018:
2157
–
2171
Evaluation Warning : The document was created with Spire.PDF for Python.
IJECE
ISSN:
2088-8708
2161
Thus,
Eq.(12)
may
be
obtained
as
follo
ws
using
Lemma
1.
e
(
t
)
T
(
A
T
cl
i
P
i
+
P
i
A
cl
i
)
e
(
t
)
+
2
e
(
t
)
T
P
i
B
cl
di
d
(
t
)
+
V
i
(
t
)
+
e
y
(
t
)
T
e
y
(
t
)
2
d
(
t
)
T
d
(
t
)
<
0
(23)
Equation
(23)
can
be
gi
v
en
as
Eq.(24)
by
re
g
arding
Eq.(18)
and
Eq.(22),
which
may
be
represented
in
matrix
inequality
form
as
Eq.(25).
e
(
t
)
T
(
A
T
cl
i
P
i
+
P
i
A
cl
i
+
P
i
)
e
(
t
)
+
2
e
(
t
)
T
P
i
B
cl
di
d
(
t
)
+
e
(
t
)
T
C
T
i
C
i
e
(
t
)+
2
e
(
t
)
T
C
i
D
di
d
(
t
)
+
d
(
t
)
T
D
T
di
D
di
d
(
t
)
2
d
(
t
)
T
d
(
t
)
<
0
(24)
A
T
cl
i
P
i
+
P
i
A
cl
i
+
P
i
P
i
B
cl
di
0
+
C
T
i
C
i
C
T
i
D
di
D
T
di
D
di
2
I
<
0
(25)
One
may
obtain
Eq.(26)
by
some
simplifications
in
Eq.(25).
I
A
T
cl
i
0
B
T
cl
di
P
i
P
i
P
i
0
I
0
A
cl
i
B
cl
di
+
0
C
T
i
I
D
T
di
2
I
0
0
I
0
I
C
i
D
di
<
0
(26)
By
assuming
Eq.(26)
as
N
T
U
Z
N
U
,
we
ha
v
e[32]:
I
A
T
cl
i
0
0
B
T
cl
di
I
2
4
P
i
+
C
T
i
C
i
P
i
C
T
i
D
di
0
0
2
I
+
D
T
di
D
di
3
5
:
2
4
I
0
A
cl
i
B
cl
di
0
I
3
5
<
0
(27)
Thus,
it
can
be
concluded
that:
2
4
P
i
+
C
T
i
C
i
P
i
C
T
i
D
di
0
0
2
I
+
D
T
di
D
di
3
5
<
0
(28)
Therefore,
U
may
be
achie
v
ed
as
[
A
cl
i
I
B
cl
di
]
taking
into
account
that
N
U
columns
are
basis
for
the
null
space
of
U
.
Furthermore,
according
to
[32],
N
V
and
V
are
assumed
as:
N
V
=
2
4
I
0
0
0
0
I
3
5
V
=
[0
I
0]
By
using
abo
v
ementioned
results,
Eq.(15)
in
Lemma
3
may
be
obtained
as:
2
4
A
T
cl
i
I
B
T
cl
di
3
5
X
i
[0
I
0]
+
2
4
0
I
0
3
5
X
i
[
A
cl
i
I
B
cl
di
]
+
2
4
P
i
+
C
T
i
C
i
P
i
C
T
i
D
di
0
0
2
I
+
D
T
di
D
di
3
5
<
0
(29)
which
can
be
cast
into
one
LMI
as
Eq.(30).
LM
I
(1)
:
2
4
11
12
13
22
23
33
3
5
<
0
(30)
where
11
=
P
i
+
C
T
i
C
i
+
A
T
i
X
i
+
X
T
i
A
i
N
i
C
i
C
T
i
N
T
i
12
=
P
i
X
T
i
+
A
T
i
X
i
C
T
i
N
T
i
13
=
C
T
i
D
di
+
X
T
i
B
di
N
i
D
di
22
=
X
i
X
T
i
23
=
X
T
i
B
di
N
i
D
di
33
=
D
T
di
D
di
I
=
2
;
N
i
=
X
T
i
L
i
A
Ne
w
Hybrid
Rob
ust
F
ault
Detection
of
Switc
hing
Systems
by
...
(Mohammad
Ghasem
Kazemi)
Evaluation Warning : The document was created with Spire.PDF for Python.
2162
ISSN:
2088-8708
3.2.
F
ault
sensiti
vity
perf
ormance
in
the
output
estimation
err
or
of
the
switched
obser
v
er
By
considering
the
L
yapuno
v
function
as
Eq.(31)
for
f
aulty
case,
its
deri
v
ati
v
e
may
be
obtained
as
Eq.(32).
V
(
t
)
=
e
(
t
)
T
P
f
i
e
(
t
)
(31)
_
V
(
t
)
=
_
e
(
t
)
T
P
f
i
e
(
t
)
+
e
(
t
)
T
P
f
i
_
e
(
t
)
=
e
(
t
)
T
(
A
T
cl
i
P
f
i
+
P
f
i
A
cl
i
)
e
(
t
)
+
2
e
(
t
)
T
P
f
i
B
cl
f
i
f
(
t
)
(32)
According
to
Remark
2,
we
need
e
y
(
t
)
T
e
y
(
t
)
for
using
lemma
1,
which
can
be
gi
v
en
as
Eq.(33):
e
y
(
t
)
T
e
y
(
t
)
=
e
(
t
)
T
C
T
i
C
i
e
(
t
)
+
2
e
(
t
)
T
C
T
i
D
f
i
f
(
t
)
+
f
(
t
)
T
D
T
f
i
D
f
i
f
(
t
)
(33)
According
to
Remark
2
and
Lemma2,
one
may
obtain
that:
e
(
t
)
T
(
A
T
cl
i
P
f
i
+
P
f
i
A
cl
i
)
e
(
t
)
+
2
e
(
t
)
T
P
f
i
B
cl
f
i
f
(
t
)
+
e
(
t
)
T
P
f
i
e
(
t
)
e
(
t
)
T
C
T
i
C
i
e
(
t
)
2
e
(
t
)
T
C
T
i
D
f
i
f
(
t
)
f
(
t
)
T
D
T
f
i
D
f
i
f
(
t
)
+
2
f
(
t
)
T
f
(
t
)
<
0
(34)
which
can
be
arranged
into
Eq.(35).
e
(
t
)
T
(
A
T
cl
i
P
f
i
+
P
f
i
A
cl
i
+
P
f
i
)
e
(
t
)
+
2
e
(
t
)
T
P
f
i
B
cl
f
i
f
(
t
)
e
(
t
)
T
C
T
i
C
i
e
(
t
)
2
e
(
t
)
T
C
T
i
D
f
i
f
(
t
)
+
f
(
t
)
T
(
D
T
f
i
D
f
i
+
2
I
)
f
(
t
)
<
0
(35)
The
matrix
inequality
representation
of
Eq.(35)
is
gi
v
en
as
Eq.(36).
A
T
cl
i
P
f
i
+
P
f
i
A
cl
i
+
P
f
i
P
f
i
B
cl
f
i
0
+
C
T
i
C
i
C
T
i
D
f
i
2
I
D
T
f
i
D
f
i
<
0
(36)
Equation
Eq.(36)
is
gi
v
en
as
Eq.(37)
in
form
of
N
T
U
Z
N
U
.
I
A
T
cl
i
0
0
B
T
cl
f
i
I
2
4
P
f
i
C
T
i
C
i
P
f
i
C
T
i
D
f
i
0
0
0
0
2
I
D
T
f
i
D
f
i
3
5
2
4
I
0
A
cl
i
B
cl
f
i
0
I
3
5
<
0
(37)
Thus,
Z
matrix
may
be
obtained
as
follo
ws.
Z
=
2
4
P
f
i
C
T
i
C
i
P
f
i
C
T
i
D
f
i
0
0
2
I
D
T
f
i
D
f
i
3
5
(38)
U
=
[
A
cl
i
I
B
cl
f
i
]
;
V
=
[0
I
0]
Using
Eq.(15)
in
Lemma
3,
we
ha
v
e:
2
4
A
T
cl
i
I
B
T
cl
f
i
3
5
X
i
[0
I
0]
+
2
4
0
I
0
3
5
X
i
[
A
cl
i
I
B
cl
f
i
]
+
2
4
P
f
i
C
T
i
C
i
P
f
i
C
T
i
D
f
i
0
0
2
I
D
T
f
i
D
f
i
3
5
<
0
(39)
Altogether
,
a
linear
matrix
inequality
as
Eq.(40)
is
achie
v
ed
by
assuming
=
2
.
LM
I
(2)
:
2
4
11
12
13
22
23
33
3
5
<
0
(40)
where
11
=
P
f
i
C
T
i
C
i
+
A
T
i
X
i
+
X
T
i
A
i
N
i
C
i
C
T
i
N
T
i
12
=
P
f
i
X
T
i
+
A
T
i
X
i
C
T
i
N
T
i
13
=
C
T
i
D
f
i
+
X
T
i
B
f
i
N
i
D
f
i
22
=
X
i
X
T
i
23
=
X
T
i
B
f
i
N
i
D
f
i
33
=
I
D
T
f
i
D
f
i
in
which,
,
N
i
,
P
f
i
and
X
i
are
unkno
wn
v
ariables
of
the
problem.
IJECE
V
ol.
8,
No.
4,
August
2018:
2157
–
2171
Evaluation Warning : The document was created with Spire.PDF for Python.
IJECE
ISSN:
2088-8708
2163
3.3.
Simultaneous
Optimal
F
ault
sensiti
vity
and
Disturbance
Attenuation
Le
v
el
According
to
Eq.(30)
and
Eq.(40),
the
disturbance
attenuation
le
v
el
and
f
ault
sensiti
vity
are
defined
in
tw
o
separate
LMI
problems,
which
must
be
considered
simultaneously
in
the
observ
er
design
at
a
gi
v
en
disturbance
attenuation
le
v
el
or
f
ault
sensiti
vity
.
In
this
paper
,
by
defining
a
weighted
LMI
optimization
problem,
the
optimal
v
alues
of
the
disturbance
attenuation
and
f
ault
sensiti
vity
are
obtained.
The
theorem
is
gi
v
en
as
follo
ws.
Theor
em
:
Consider
the
continuous
time
switched
system
as
Eq.(1)-Eq.(2)
with
ADT
along
with
the
switched
observ
er
a
s
Eq.(5).
The
state
estimation
error
dynamic
Eq.(7)
is
asymptotically
stable
and
satisfies
the
performance
criteria
as
Eq.(9)
and
Eq.(10)
for
an
y
nonzero
d
(
t
)
2
l
2
[0
;
1
)
,
if
there
be
the
scalars
v
alues
>
0
,
>
0
and
the
matrices
P
i
>
0
Q
i
>
0
X
i
N
i
such
that
the
subsequent
weighted
LMI
optimization
problem
has
solution.
min
(
P
i
;Q
i
;X
i
;N
i
;;
)
w
(1
w
)
s:t:LM
I
(1)
<
0
;
LM
I
(2)
<
0
P
i
<
P
j
;
P
f
i
<
P
f
j
where
0
w
1
is
the
weighting
f
actor
for
simultaneous
f
ault
sensiti
vity
and
disturbance
attenuation
perfor
-
mances
and
its
v
alue
is
defined
by
minimization
of
the
follo
wing
cost
function:
min
w
Lastly
,
the
g
ains
of
the
observ
er
for
dif
ferent
modes
of
the
switched
system,
optimal
disturbance
attenuation
le
v
el
and
optimal
f
ault
sensiti
vity
of
the
output
estimation
error
of
the
observ
er
are
achie
v
ed
as:
L
i
=
X
T
i
N
i
;
=
p
;
=
p
3.4.
Global
Analytical
Redundancy
Relations
in
Err
or
F
orm
Using
BG
method
The
generic
form
of
a
GARR
may
be
gi
v
en
in
Eq.(41)
[34].
GAR
R
:
(
y
(
n
)
;
:
:
:
;
y
;
u
(
m
)
;
:
:
:
;
u;
;
a
)
(41)
where
y
,
u
and
are
denoted
for
outputs,
inputs
and
parameters
of
the
system,
respecti
v
ely
.
As
in
Eq.(3)-Eq.(4),
a
=
[
a
1
a
2
:
:
:
a
z
]
is
a
binary
v
ector
that
determine
the
acti
v
e
mode
of
the
switched
system..
The
GARRs
are
dif
ferential
equations,
which
their
orders
are
dependent
on
the
sensor
locations
in
the
system.
The
lo
wer
order
GARRs
are
more
ef
ficient
for
monitoring
of
a
system.
The
generic
form
of
the
GARRs
may
be
presented
in
matrix
form
by
maximum
second
order
assumption
of
deri
v
ati
v
es.
GAR
R
:
(
•
y
;
_
y
;
y
;
•
u;
_
u;
u;
;
a
)
=
M
1
(
;
a
)
•
y
(
t
)
+
M
2
(
;
a
)
_
y
(
t
)
+
M
3
(
;
a
)
y
(
t
)+
Z
1
(
;
a
)
•
u
(
t
)
+
Z
2
(
;
a
)
_
u
(
t
)
+
Z
3
(
;
a
)
u
(
t
)
(42)
The
ef
fects
of
disturbances,
noises
and
parametric
uncertainties
on
the
GARRs
may
lead
to
f
alse
or
missed
alarms
in
the
FD
system.
In
this
paper
,
a
ne
w
method
by
combination
of
the
BG
method
and
observ
er
method
is
proposed
to
o
v
ercome
these
problems
in
the
FD
system
based
on
the
BG
method,
which
further
leads
to
some
benefits
in
the
FD
system.
In
normal
case
i.e
.
in
the
case
that
there
are
no
disturbances,
f
aults,
noises
and
uncertainties
in
the
system,
the
estimated
states
and
outputs
of
the
observ
er
con
v
er
ge
to
the
states
and
outputs
of
the
system.
Therefore,
the
GARRs
by
using
estimated
outputs
may
be
gi
v
en
as:
GAR
R
h
:
(
•
^
y
;
_
^
y
;
^
y
;
•
u;
_
u;
u;
;
a
)
=
M
1
(
;
a
)
•
^
y
(
t
)
+
M
2
(
;
a
)
_
^
y
(
t
)
+
M
3
(
;
a
)
^
y
(
t
)+
Z
1
(
;
a
)
•
u
(
t
)
+
Z
2
(
;
a
)
_
u
(
t
)
+
Z
3
(
;
a
)
u
(
t
)
(43)
By
subtracting
Eq.(43)
from
Eq.(42),
the
GARRs
in
the
error
form,
which
is
called
EGARRs,
are
obtained.
E
GAR
R
:
M
1
(
;
a
)
•
e
y
(
t
)
+
M
2
(
;
a
)
_
e
y
(
t
)
+
M
3
(
;
a
)
e
y
(
t
)
(44)
The
output
est
imation
errors
in
Eq.(44)
come
from
the
observ
er
,
which
is
designed
in
such
a
w
ay
that
be
more
sensiti
v
e
to
f
aults
and
simultaneously
the
ef
fect
of
disturbance
is
attenuated.
Therefore,
the
ef
fects
of
the
disturbances
are
reduced
in
the
GARRs,
while
the
f
ault
sensiti
vity
is
enhanced.
A
Ne
w
Hybrid
Rob
ust
F
ault
Detection
of
Switc
hing
Systems
by
...
(Mohammad
Ghasem
Kazemi)
Evaluation Warning : The document was created with Spire.PDF for Python.
2164
ISSN:
2088-8708
Figure
1.
The
tw
o-tank
system
3.5.
Rob
ustness
of
the
FD
system
against
parametric
uncertainties
The
f
ault
detection
and
isolation
are
based
on
the
EGARRs,
which
are
dependent
on
the
parameters
of
the
BG
model.
In
order
to
obtain
a
rob
ust
FD
system
ag
ainst
parametric
uncertainties,
the
proposed
met
hod
in
[23]
is
considered
in
this
paper
.
In
rob
ust
f
ault
detection
based
on
the
BG
method,
a
GARR
can
be
decoupled
into
nominal
(GARRn)
and
uncertain
part
(UGARR)
as
follo
ws.
GAR
R
=
GAR
R
n
(
a;
n
;
u;
y
)
+
U
GAR
R
(
a;
;
u;
y
)
(45)
where
n
is
the
nominal
part
and
is
relati
v
e
de
viation
compared
to
the
nominal
v
alue
of
the
parameter
.
The
nominal
part
is
the
obtained
GARRs,
which
contain
the
nominal
v
alues
of
the
parameters
and
made
the
residuals.
The
uncertain
part
contains
the
uncertain
parameters
of
the
nominal
GAR
Rs,
which
further
deter
-
mines
the
upper
and
lo
wer
bounds
of
the
residuals.
An
EGARR
may
be
gi
v
en
as
Eq.(46)
re
g
arding
this
f
act
that
the
EGARRs
are
not
directly
dependent
on
the
inputs
and
their
distrib
ution
matrices.
E
GAR
R
=
E
GAR
R
n
(
a;
n
;
e
y
)
+
U
E
GAR
R
(
a;
;
e
y
)
(46)
The
adapti
v
e
thresholds
on
the
residuals
are
defined
as
UEGARRs,
which
are
dependent
on
the
parametric
uncertainty
(
),
mode
of
the
system
(
a
)
and
the
output
estimation
error
of
the
observ
er
(
e
y
).
As
mentioned
abo
v
e,
the
proposed
method
is
not
directly
dependent
on
the
inputs
and
distrib
ution
matrices
of
the
inputs
(
Z
1
to
Z
3
).
Thus,
the
rob
ustness
of
the
proposed
method
may
be
impro
v
ed.
Finally
,
the
residual
e
v
aluation
function
and
thresholds
for
f
ault
detection
are
assumed
as
Eq.(47)
and
Eq.(48),
respecti
v
ely
.
J
L
(
t
)
=
jj
r
k
jj
2
=
(
l
0
+
L
X
l
0
r
T
k
r
k
)
0
:
5
(47)
J
th
=
sup
d
(
t
)
;u
(
t
)
2
l
2
;f
(
t
)=0
J
L
(
t
)
+
jj
k
jj
2
(48)
in
which
J
L
(
t
)
is
the
residual
e
v
aluation
function,
J
th
is
the
threshold
on
the
residual
e
v
aluation
function,
l
0
is
the
initial
ti
me,
L
is
the
windo
w
length
for
residual
e
v
aluation
function
calculations
and
k
is
the
ef
fect
of
parametric
uncertainties
on
the
residual,
which
is
obtained
from
UEGARR.
Finally
,
the
f
aulty
or
f
ault
free
conditions
are
declared
by
comparison
between
the
residual
e
v
aluation
function
and
considered
threshold.
4.
RESUL
TS
AND
DISCUSSION
Consider
the
tw
o-tank
system
in
Fig.1,
which
in
v
olv
e
a
flo
w
source,
tw
o
output
v
alv
es
named
as
R
1
and
R
2
and
one
interconnection
v
al
v
e
(
R
3
).
The
v
alv
es
could
be
in
open
or
closed
state,
which
determined
the
acti
v
e
dynamic
of
the
switched
system.
Three
configurations
as
T
able
I
are
assumed
[35].
The
BG
models
of
the
understudy
tw
o-tank
system
in
inte
gral
and
deri
v
ati
v
e
causality
in
20-sim
softw
are
are
sho
wn
as
Fig.2
and
Fig.3,
respecti
v
ely
.
As
mentioned
earlier
,
the
v
alv
es
could
be
in
open
or
closed
state.
The
coef
ficients
a
1
,
a
2
and
a
3
are
used
to
sho
w
the
v
alv
e
states
of
R
1
,
R
2
and
R
3
,
res
pecti
v
ely
,
which
are
set
to
”0”
in
closed
state
and
”1”
in
open
state.
IJECE
V
ol.
8,
No.
4,
August
2018:
2157
–
2171
Evaluation Warning : The document was created with Spire.PDF for Python.
IJECE
ISSN:
2088-8708
2165
T
able
1.
Considered
modes
of
the
tw
o-tank
system
Mode
R
1
R
2
R
3
A
=
[
a
1
a
2
a
3
]
Mode
1
of
f
on
of
f
[0
1
0]
Mode
2
on
on
of
f
[1
1
0]
Mode
3
of
f
on
on
[0
1
1]
Figure
2.
BG
model
of
tw
o-tank
system
in
inte
gral
causality
.
In
h
ydraulic
domain,
the
ef
fort
and
flo
w
v
ariables
are
re
g
arded
as
pressure
and
flo
w
,
respecti
v
ely
.
The
subscript
c
is
considered
to
determine
the
c
o
nt
rolled
junctions
of
the
model.
Three
controlled
junctions
are
assumed
for
three
v
alv
es.
In
BG
theory
of
h
ybrid
systems,
the
flo
w
v
ariable
of
the
connected
bonds
to
the
i
th
1-type
controlled
junction
is
multiplied
by
a
i
.
T
w
o
pressure
sensors
are
used
to
measure
pressures
of
the
tw
o
tanks
which
is
gi
v
en
as
P
1
and
P
2
.
The
state
space
representation
of
a
h
ybrid
system
may
be
obtained
in
compact
form
using
the
BG
model.
The
st
ates
of
the
system
are
considered
as
generalized
momentum
of
inertia
elements
(
p
I
)
and
generalized
dis-
placement
of
capaciti
v
e
elements
(
q
C
).
Hence,
the
state
v
ector
of
the
tw
o-tank
system
is
achie
v
ed
as
[
q
2
q
9
]
T
.
The
flo
w
through
the
v
alv
es
is
assumed
laminar
.
This
assumption
leads
to
linear
equations
of
the
system
[35].
Using
the
state
space
deri
v
ation
method
in
[34],
the
go
v
erning
equations
in
compact
form
are:
_
q
2
=
(
a
1
R
1
C
T
1
a
2
R
2
C
T
1
)
q
2
+
a
2
R
2
C
T
2
q
9
+
q
in
(49)
_
q
9
=
a
2
R
2
C
T
1
q
2
+
(
a
2
R
2
C
T
2
a
3
R
3
C
T
3
)
q
9
y
1
=
1
C
T
1
q
2
;
y
2
=
1
C
T
2
q
9
The
parameters
definition
and
their
v
alues
are
gi
v
en
in
T
able
II.
It
is
also
noted
that
the
tank
capacity
is
defined
as
C
T
=
A=g
wherein
g
=
9
:
81
m=s
2
is
the
acceleration
of
gra
vity
.
Thus,
the
system
matrices
for
three
considered
modes
of
the
tw
o-tank
system
are:
A
1
=
0
:
0007
0
:
0011
0
:
0007
0
:
0011
;
C
1
=
0
:
1962
0
0
0
:
3270
Figure
3.
BG
model
of
tw
o-tank
system
in
deri
v
ati
v
e
causality
.
A
Ne
w
Hybrid
Rob
ust
F
ault
Detection
of
Switc
hing
Systems
by
...
(Mohammad
Ghasem
Kazemi)
Evaluation Warning : The document was created with Spire.PDF for Python.
2166
ISSN:
2088-8708
T
able
2.
P
arameters
of
tw
o-tank
system
V
ariable
Definition
V
alue
A
1
Cross
sectional
area
of
tank
1
50
m
2
A
2
Cross
sectional
area
of
tank
2
30
m
2
R
1
Output
pipe
of
tank
1
resistance
300
s=m
2
R
2
Interconnection
pipe
resistance
300
s=m
2
R
3
Output
pipe
of
tank
1
resistance
100
s=m
2
0
0.2
0.4
0.6
0.8
1
0.18
0.2
0.22
0.24
0.26
0.28
0.3
0.32
weight
Disturbance attenuation level
0
0.2
0.4
0.6
0.8
1
0.34
0.36
0.38
0.4
0.42
0.44
weight
Fault sensitivity
Figure
4.
Disturbance
attenuation
le
v
el
and
f
ault
sensiti
vity
v
ersus
weighting
f
actor
A
2
=
0
:
0013
0
:
0011
0
:
0007
0
:
0011
;
C
2
=
0
:
1962
0
0
0
:
3270
A
3
=
0
:
0007
0
:
0011
0
:
0007
0
:
0044
;
C
3
=
0
:
1962
0
0
0
:
3270
B
1
=
B
2
=
B
3
=
1
0
The
other
matrices
of
the
system
as
in
Eq.(1)
and
Eq.(2)
are
assumed
as
follo
ws.
B
d
1
=
B
d
2
=
B
d
3
=
B
1
;
B
f
1
=
B
f
2
=
B
f
3
=
1
0
0
1
D
d
1
=
D
d
2
=
D
d
3
=
0
:
1
0
:
1
;
D
f
1
=
D
f
2
=
D
f
3
=
0
:
3
0
0
:
1
0
:
5
The
GARRs
of
the
system
may
be
obtained
using
the
Causality
In
v
ersion
Method
(CIM)
in
[34].
GAR
R
1
=
C
T
1
dP
1
dt
+
a
2
R
2
(
P
1
P
2
)
+
a
1
R
1
P
1
q
in
=
0
(50)
GAR
R
2
=
C
T
2
dP
2
dt
a
2
R
2
(
P
1
P
2
)
+
a
3
R
3
P
2
=
0
The
matrices
M
i
and
Z
i
can
be
calculated
as:
M
1
=
0
;
M
2
=
C
T
1
0
0
C
T
2
;
Z
1
=
Z
2
=
0
M
3
=
a
2
R
2
+
a
1
R
1
a
2
R
2
a
2
R
2
a
2
R
2
+
a
3
R
3
;
Z
3
=
1
0
The
upper
bound
of
parameter
uncertainty
is
assumed
as
2%
of
their
nominal
v
alues
[29].
Solving
the
weighted
LMI
optimization
problem
for
=
0
:
08
,
=
1
:
2
and
=
0
:
9
,
the
disturbance
attenuation
le
v
el
and
f
ault
sensiti
vity
v
ersus
weighting
f
actor
are
gi
v
en
in
Fig.4.
By
minimization
the
cost
function
of
the
Theorem,
we
ha
v
e:
w
=
0
:
8
IJECE
V
ol.
8,
No.
4,
August
2018:
2157
–
2171
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