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source of flow (MSf) representing the relation
d t
the BP of positive deviation in
x
(
t
) leads identify-
ing the damage causes as reduction in either
k
or
D
. The FP for reductions in
k
and
D
leads to the
damage signature 00+ (see table 3).
Consider the MDOF structure of Figure4. An
example is shown in Figure 7(a) for a deviant
displacement
x
1
(
t
) being above normal. From the
TCG of Figure 4(c), an increase in x1(t) initiates
backward propagation along
f
( ) (
= +
(Figure 4(b)). Thus, for the
displacement sensor measuring the displacement
of the first mass e7n-2= dm, e7n-1 = dt and e7n
= B1. The C-element (
C
=1
) integrates the true
velocity to give the true displacement. This leads
to
e
d t
B
m
t
( ) (
= +
1
. In
case of drift, the bias quantity B1 is replaced
with a time drift function (i.e.,
d t
=
d
+
or
d t
B
d t
B
7
n
−
2
7
n
−
1
1
m
t
dt
and
implies that the first derivative of f3 is above
normal (f3+). The step along
f
→
x
3
1
= +
). Each sensor fault is
represented by separate BG and TCG blocks.
d t
d
t
( )
( )
( )
m
t
drift
+
→
implies
1
e
3
3
←
1
e3 above normal (e3+), and
e
implies
e
3
2
k
e2 below normal and along
e
implies
k1 below normal, and so on. Eventually, we arrive
at k1,
k
2
,
k
3
,
D
1
,
D
2
,
D
3
below normal due to
x
1
(
t
)
being above normal.
The forward propagation starts from one of
the damage causes identified in the previous step
by propagating the change in this parameter and
tracing its signature on the system measurements.
An example of
k
1
- (
k
1
below normal) is shown in
Figure 7(b). The signature of k1- on x1 is derived
by assuming that k1 decreases and following the
path shown in Figure 7(b), extracted from Figure
4(c). A decrease in k1 implies a decrease in e2
from the relation
e
←
f
2
2
7. EXTRACTION OF DAMAGE
SIGNATURES FROM TEMPORAL
CAUSAL GRAPH
This section describes the extraction of the damage
signature matrix that lists the qualitative effects
of changes in system components on system
response measurements. This step is performed
off-line using the TCG model before the collection
of sensor data. The TCG is derived as described
in Section 5. The TCG links damages to their
causal effects on measurements, called damage
signatures, which represent 0th through k
th
order
derivative changes on a measurement residual at
the instant of damage occurrence. They provide
the discriminatory power in the damage isolation
approach.
The damage signature derivation consists of
two steps: backward propagation (BP) and forward
propagation (FP). First, the TCG is used to per-
form a BP from possible deviant measurements to
generate possible damage causes. This identifies a
set of parameters that could be the reason for the
damage (e.g.
k
and
D
in the SDOF example). It also
determines whether the cause is above normal (+)
or below normal (-). In the FP step we use each of
these damage causes to derive its signatures on the
system measurements (e.g. effect of decrease in
k
or
D
on
x
). The damage signature of the SDOF
structure of Figure 4 is given in Table 3. Herein,
=
∫
and is indicated
with a downward arrow. This implies an increase
in e3 since
e e e e e
2
k
f dt
2
1
2
= − − −
(shown with an
upward arrow). The increase in e3 implies an
increase in the first derivative of f3 from the rela-
1
3
4
5
t
∫
tion
f
=
( /
1
m e dt
)
(shown with two upward
3
3
−∞
arrows), and so on. Finally, this forward propaga-
tion implies an increase in the second derivative
of x1 which is indicated by three upward arrows
and the signature is 00+. The signatures of
k
2
,
k
3
,
D
1
,
D
2
,
D
3
,
B
1
,
B
2
,
B
3
,
dr
1
,
dr
2
and
dr
3
on the sys-
tem responses
x
1
,
x
2
and
x
3
are extracted following
the same procedure and are shown in Table 4.
In Table 4, damage signatures are shown up
to the first nonzero direction of change. Damage
scenarios which produce discontinuities on the
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