Fault Detection and Diagnostics for Rooftop Air Conditioners

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As a result of these considerations, a new fault detection classifier, which does not
require a faulty operation covariance matrix, was developed and is described in the next
section.
1.1.1.2 Normalized Distance Fault Detection Classifier
The attached ASHRAE paper (Li & Braun, 2003) and Deliverables 2.1.3 & 2.1.4
(2002) present details of a normalized distance fault detection classifier that can be used
for both individual and multiple-simultaneous faults simply. The classifier evaluates the
following inequality.
ω1:
Normal
T −1
≤ 2 −1
( Y − M
normal
) Σnormal
( Y − M
normal)
( χ ) ),
>
−1
normal
T
where ( Y − M ) Σ ( Y − M )
normal
normal
ω2:
Faulty
{(1
−α
m}
(1-2)
2 −1
is the normalized distance, ( χ ) {(1
−α),m}
2 −1
is the threshold of normalized distance for normal operation, ( χ ) {, } is the inverse of
the chi-square cumulative distribution function, α is the false alarm rate, and m is the
degree of freedom or dimension which is equal to the number of chosen state variables.
Due to modeling error M
normal
is not exactly zero, so equation (1-2) takes modeling error
into account to statistically evaluate whether Y is zero or not.
The above fault detection scheme can be illustrated using Figure 1-3. The residual
distribution of normal operation can be characterized in terms of the covariance matrix
Σ
normal
and mean vector M
normal
and depicted in the residual space plane as in Figure 1-3.
In the residual space plane, any operating states (points) outside the normal operating
region are classified as faulty while those inside the normal operation region are
classified as normal. The normal operating ellipse is the fault detection boundary.
Practically, normal operation information, such as the mean and covariance
matrix, is more accessible and more reliable, compared to faulty operation data. In
addition, this scheme is intuitive in that the opposite of normal operation is abnormal
operation. If the current operation point is not inside the normal operation region at a
certain confidence according to reliable prior information, it should be classified as faulty
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