6.2.2 Progression Modeling
From the assumptions made in the previous section, an indicator progression model is constructed. For normal component behavior, the model produces a d with constant value and a r with constant gain. In response to a component replacement, the model produces an abrupt change in the value of d. To emulate progression patterns associated with mechanical degradation, the model produces random transitions in the value of d and in the gain of r.
The variations in d are obviously not random for a given failure mode. However, given a failure mode which is unknown or not well understood, these .uctuations will appear as random. As this is the case for most failure modes to which a helicopter transmission system is susceptible, the fault signature model is made without assumptions of a priori knowledge of progression patterns.
The two main components for the indicator time series is the deterministic
6.2. PROGRESSION ANALYSIS
process d and the random process r (Eq. 6.1).
The
component
d is itself made up of a step component b and a trend component c.
d(t)= b(t)+ c(t) (6.2)
The component b is recursive with initial value b(0) equal to dcc. This constitutes the initial value of b and thus the initial expected value of the in-dicator. The last term of the expression contains a boolean expression which returns 1 when true, and 0 otherwise. This results in a step of amplitude ab at position pb. The parameters ab and pb can be arrays of several elements,
(k)(k)
ab and pb , in which case several steps will be generated and step index is denoted by k.
(k)(k)
b(t)= b(t . 1) + [t == pb ].ab (6.3)
The trend component is made up of a sum of weighted sigmoids, where ac constitutes amplitude, qc slope and pc position in time. The variable k is the sigmoid index, and Kc is the number of sigmoids in the sum.
(k)ac
c(t)= (6.4)
(k)(k)
c (t.pc )
1+ e.q
k∈Kc
The random component r is itself made up of two processes, the outlier component s and the white noise process w.
r(t)= s(t)+ w(t) (6.5)
The white noise component is constructed from a gaussian process rg with variable gain gw (Eq. 6.6).
The
gain
function
gw is given by a sum of sigmoids
over
a
constant
(Eq.
6.7).
w(t)= gw(t).rg (6.6)
(k)aw
gw(t)= dcw + (k)(k) (6.7)
.qw (t.pw )
1+ e
k∈Kw
The outlier component is constructed from a gaussian distribution rg (Eq. 6.9)
with
gain
gs times the gain of w. Any point smaller than two standard deviations
are
then
set
to
zero
(Eq.
6.8),
so
that
only
the
peak
values
are
kept.
s(t)= s .(t).[s .(t) > 2gs.gw(t)] (6.8)
s .(t)= rg.gs.gw(t) (6.9) This gives a total of 13 con.guration parameters controlling the character-
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