Aerospace and Electronic Systems Magazine March 2017 - 7

Leung and Rife
gate. This is true because the fault-free probability can be computed implicitly (one minus fault probability). This tendency to
think of gates as outputting a single probability (rather than a distribution) is underscored by the simplicity of working only with
fault probabilities when those probabilities are small. To see this,
consider the case of small fault probabilities P(A = 1) << 1 and (B
= 1) << 1. In this limit, OR gates effectively sum fault probabilities,
in the sense that
POR  C 
1 P  A 
1 P  B 
1 .

(7)

Thus, the OR gate can be viewed as a summation; the AND gate
can be viewed as a multiplication of input fault probabilities.
(8)

PAND  C 
1 P  A 
1 P  B 
1 .

Although binary analysis is intuitive and convenient, it has
a limitation in that the binary model leads to overconservatism.
For example, a binary state description cannot distinguish between different levels of accident (hazardous as compared with
catastrophic). Instead, all faults of any severity must be lumped
together. To be safe, it is conservative to apply the most severe applicable requirement (e.g., from Table 1). For instance, if the toplevel fault probability might represent either a hazardous or a catastrophic fault, it would be conservative to apply the more stringent
requirement, that for catastrophic faults. This conservatism makes
verification and validation extremely onerous. If, by contrast, the
probabilities for each severity category were tracked separately,
validation would be simpler in that system safety analysis for a
less severe fault could be compared with a less strict regulatory
requirement.

CHALLENGES MODELING AVIATION SEVERITY WITH FUZZY
LOGIC
A straightforward approach to tracking severity categories separately would be to apply fuzzy logic. Fuzzy logic is a well-understood
concept that can generalize binary logic gates to apply to a wider
range of possible state values [9]. In this case, assume that all state
variables can take one of five (nonoverlapping) values from the set
{no fault, minor, major, hazardous, catastrophic}. It is customary in

Table 4.

(9)

The corresponding mapping function for the AND operation is
C  min  A, B  .

(10)

These fuzzy-logic definitions generalize conventional binary logic
described by Table 2 and Table 3. To see this let us apply (9) and
(10) to define equivalent logic tables for the fuzzy OR and AND
gates (Table 5 and Table 6, respectively). Compare the upper left 2
× 2 block from Table 5 with Table 2. They are identical. Similarly,
compare the upper left 2 × 2 block from Table 6 with Table 3. They
are again identical. This parallelism suggests fuzzy logic is a rational extension of binary logic.
Although the fuzzy logic approach introduces multivalued
states, which in principle should reduce overconservatism, a new
problem arises. The problem lies in trying to define severity levels
for individual fault modes. The question is particularly evident in
the context of the AND gate. What should be the severity level for
each of two events if both are required for failure? To help understand this, consider an example involving two faults combined by
an AND gate. The first fault is that the pilot fails to take mitigating

Table 5.

Fuzzy-Logic OR Table
Fuzzy
OR

A=0

A=1

A=2

A=3

A=4

B=0

Ω0

Ω1

Ω2

Ω3

Ω4

B=1

Ω1

Ω1

Ω2

Ω3

Ω4

B=2

Ω2

Ω2

Ω2

Ω3

Ω4

B=3

Ω3

Ω3

Ω3

Ω3

Ω4

B=4

Ω4

Ω4

Ω4

Ω4

Ω4

Fuzzy-Logic AND Table

Integer Value

Fuzzy Logic
Values

No failure

0

A∈[0,0.2)

Minor

1

Major

2

Hazardous
Catastrophic

MARCH 2017

C  max  A, B  .

Table 6.

Severity Levels Subscripts
Severity

fuzzy logic applications to map the discrete levels onto the range
between 0 and 1. In our analysis, however, we simply designate
each state with an integer value, as summarized in Table 4.
The fuzzy-logic mapping function for the OR operation is

Fuzzy
AND

A=0

A=1

A=2

A=3

A=4

B=0

Ω0

Ω0

Ω0

Ω0

Ω0

A∈[0.2,0.4)

B=1

Ω0

Ω1

Ω1

Ω1

Ω1

A∈[0.4,0.6)

B=2

Ω0

Ω1

Ω2

Ω2

Ω2

3

A∈[0.6,0.8)

B=3

Ω0

Ω1

Ω2

Ω3

Ω3

4

A∈[0.8,1.0]

B=4

Ω0

Ω1

Ω2

Ω3

Ω4

IEEE A&E SYSTEMS MAGAZINE

7



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