Fault Detection and Diagnostics for Rooftop Air Conditioners

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So, the input-output relationship of the system after being incorporated with a
normal operation model can be described approximately as follows,
where,
Y = F(X )
(1-1)
X
⎡x1⎤
⎢ ⎥
⎢
x2
= ⎥ , Y
⎢ M ⎥
⎢ ⎥
⎣x n ⎦
⎡ y
⎢
⎢
y
=
⎢ M
⎢
⎣
1
2
y m
⎤
⎥
⎥ , and
⎥
⎥
⎦
⎡ f
⎢
⎢
f
F(
X ) =
⎢
⎢
⎣ f
1
2
m
( x1,
x2,
L,
xn
) ⎤
( x , , , )
⎥
1
x2
L xn
⎥ .
M ⎥
⎥
( x1,
x2,
L,
xn
) ⎦
X is the fault vector with each entry
x
i
representing a measure of the fault level for fault
i (for example x
1
could characterize the level of compressor valve leakage, say, 20%).
Y is the state variable residual vector, with each entry
y
i
representing a particular statevariable
residual (for example y
1
is discharge line temperature residual
∆Tdis
and its
value is temperature variation, say, 10 F). F (X ) is a nonlinear function vector with each
individual nonlinear function f x , x , L , x ) defining the relationship between different
i( 1 2 n
faults at different levels and the state-variable residuals Y . n is the number of fault types
considered, and m is the number of chosen state variables.
1.1.1 Fault Detection
Fault detection, which is to indicate whether the system is normal or not, can be
done by evaluating whether the resulting Y in equation (1-1) is zero or not in a statistical
sense.
1.1.1.1 Original SRB Fault Detection Classifier
Rossi & Braun (1997) proposed a way to evaluate whether Y is zero indirectly by
evaluating the overlap (see Figure 1-2) of the actual distribution and the expected
distribution of the residual(s). When the overlap of the actual distribution and the
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