Probabilistic Performance Analysis of Fault Diagnosis Schemes

These relationships can be used to form residual signals. For example,
For i = 1,2,3,4, let ε i > 0 and define
r 1 = y 3 − g 3 (y 2 , y 4 ),
r 2 = y 4 − g 4 (y 1 , y 3 ),
r 3 = y 2 − g 2 (y 1 , y 4 ),
r 4 = y 1 − g 1 (y 2 , y 3 ).
⎧
⎨0, if |r i | < ε i ,
s i :=
⎩1, otherwise.
Then, faults can be detected based on the following symptom table [48, §17]:
Symptoms
Fault s 1 s 2 s 3 s 4
1 0 1 1 1
2 1 0 1 1
3 1 1 0 1
4 1 1 1 0
Note that when Sensor i fails (i.e., Fault i occurs), all of the residual except r i are affected.
Hence, this is an example of a generalized residual set. For this example, when two sensors
fail, all the symptoms are present and there is no way to determine which faults have
occurred.
Advantages of analytical redundancy
The key advantage of using analytical redundancy is the reduced physical complexity of the
system. For example, in Figure 2.5, four sensors are used to measure four different quantities
y 1 , y 2 , y 3 , and y 4 . Thus, each sensor is performing a unique useful task and no extraneous
hardware is being used. By moving the redundancy to the software side, the overall system
consumes less space, weight, and power.
Disadvantages of analytical redundancy
In general, analytically redundant configurations are less reliable. Since each component
performs a unique function, the loss of a single component may compromise an entire
subsystem. For example, suppose that Sensor 1 in Figure 2.5 fails. Then, the system no
longer has access to a measurement of the quantity y 1 . At best, the signal ŷ 1 = g (y 2 , y 3 )
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