2 FAULT DETECTION
METHODS FOR UNCERTAIN
PARAMETER SYSTEMS
2.1 Binary Logic Based Method
This technique consists on a test of the signal
amplitude. The adjustment parameters are the
thresholds regulated according to the various
operating assumptions and the desired performances
for detection (Brunet J., 90).
2.2 The Proposed Method Based on
Fuzzy Logic
Observed residuals, written in integral form obtained
when a rectangular fault affects sensors or actuators
in a limited interval, have the following forms:
(a)
(b)
0 1 2 3 4 5 6 7 8 9 10
-2.5
-2
-1.5
-1
-0.5
0
0.5
x 10
7
r3
temps
r3
data 1
(c)
Figure 1: Residual forms in case of a rectangular fault.
The (c) residuals can not be processed in the
same way as the (a) and (b) residuals. In fact for (a)
and (b) residuals, the fault cancellation brings back
the residual to a constant or null value. For the (c)
residual, the fault cancellation does not prevent its
divergence due to the double integration.
2.2.1 (a) and (b) Residuals
We have proposed in (Bouabdallah S. et al., 2005), a
fault detection method based on the fuzzification of
(a) and (b) residuals.
Fuzzy reasoning is composed of the following
stages: attribute fuzzification, application of
inference rules and defuzzification (Bûhler H., 94).
In the Fuzzy Logic Toolbox of Matlab 7.0, there
are five steps of the fuzzy inference process:
Step 1: Fuzzify inputs
It consists in taking inputs and determining the
degree to which they belong to each of the
appropriate fuzzy sets via membership functions. A
membership function is a curve that defines how
each point in the input space is mapped to a
membership value or degree of membership between
0 and 1. The output is then a fuzzy degree of
membership in the qualifying linguistic set.
Step 2: Apply Fuzzy Operator
Once the inputs have been fuzzified, we know the
degree to which each part of the antecedent has been
satisfied for each rule. If the antecedent of a given
rule has more than one part, the fuzzy operator is
applied to obtain one number that represents the
result of the antecedent for that rule. This number
will then be applied to the output function. The input
to the fuzzy operator is two or more membership
values from fuzzified input variables. The output is a
single truth value.
Step 3: Apply Implication method
Every rule has a weight (a number between 0 and 1),
which is applied to the number given by the
antecedent. Once proper weighting has been
assigned to each rule, the implication method is
implemented. A consequent is a fuzzy set
represented by a membership function, which
weights appropriately the linguistic characteristics
that are attributed to it. The consequent is reshaped
using a function associated with the antecedent (a
single number). The input for the implication
process is a single number given by the antecedent,
and the output is a fuzzy set. Implication is
implemented for each rule. Two built-in methods are
supported by fuzzy toolbox of Matlab 7.0, and they
are the same functions that are used by the AND
method: min (minimum), which truncates the output
fuzzy set, and prod (product), which scales the
output fuzzy set.
Step 4: Aggregate All Outputs.
Aggregation is the process by which the fuzzy sets
that represent the outputs of each rule are combined
into a single fuzzy set. The input of the aggregation
process is the list of truncated output functions
1
A FUZZY APPROACH FOR FAULT DETECTION AND ISOLATION OF UNCERTAIN PARAMETER SYSTEMS
AND COMPARISON TO BINARY LOGIC
99