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Model-Based Fault Diagnosis
4.6 MBD on the Power Supply
Nr. x hA a hB b hC c hD y p
1 1 0 1 0 1 0 1 1 0 0.00000099
2 1 0 1 0 1 1 0 0 0 0.00000099
3 1 0 1 1 0 0 0 0 0 0.00000099
4 1 0 1 1 0 1 1 1 0 0.00970299
5 1 1 0 0 0 0 0 0 0 0.00000099
6 1 1 0 0 0 1 1 1 0 0.00970299
7 1 1 0 1 1 0 1 1 0 0.00970299
8 1 1 0 1 1 1 0 0 0 0.00970299
Table 4.1: Mode catalog of the 4-inverter model, with x=1 and y=0.
where p i is the probability that health vector i is the actual candidate given a specific set of observations.
Formula 4.10 is a cost function to estimate the expected cost of identifying the actual
candidate. It is constructed as follows. The cost of locating a particular candidate is proportional to
log p −1
i (so, a binary search through p −1
i objects). The expected costs of identifying one candidate
is the multiplication of the costs to locate it (log p −1
i ) and the probability that the candidate is the
actual candidate (p i ). The entropy H, defined by Formula 4.10, adds the costs of all candidates
(∑ p i log p −1
i = −∑ p i log p i ).
A mode catalog is a table that specifies all possible observations for each health vector, and
can be derived from the model. The used 4-inverter model, as listed in Appendix C.1.4, is a strong
model, and the fault mode of 1 inverter is defined by: ¬h => (o = i). The mode catalog of this
4-inverter model contains 8 entries, when none of the observables a, b, and c are observed. This
information, the existence of 8 health vectors, can be stored in 3 bits, in contrast to the 4 bits needed
for storing the information that there are 16 health vectors. This does not imply that the entropy
gain is 4 − 3 = 1 bit, because entropy also depends on the probability of each health candidate (the
entropy gain of measuring the input x and output y is 0.064). Table 4.1 shows the 4-inverter model,
when x=1 and y=0. The entropy of this mode catalog (no additional observation are made), is 0.260.
This means that the uncertainty of the outcome of the MBD engine is 0.260.
Consider that a is being measured. There are two possibilities: a=0 or a=1. In either case
another set of entries is consistent with the observations. These possible new mode catalogs define
the cost to locate the actual candidate. The expected entropy of observing a is 0.046. The entropy
gain is defined as the entropy a priori model minus the expected entropy of observing a: 0.260 −
0.046 = 0.213. The entropy gain of observing b and c is respectively 0.221 and 0.213. Thus, the
entropy gain of observing b is highest, and is the best choice for to measure.
4.6 MBD on the Power Supply
This section presents a real-life application of MBD on the power supply example. The model is
shown in Appendix C.2.1. The model specifies six systems, namely Cable, Fuse, Low_ voltage_
power_ supply, PDU, Unit, and Power_ Supply. The specification of the behavior is very abstract.
The model describes the power supply example as a digital system. The used variables are all in
the boolean domain. A value of 1 denotes that there is current, and a value of 0 denotes that there
is no current. For example, the model of a Cable specifies that when a cable is healthy it conducts
current, using the following definition:
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