Aerospace and Electronic Systems Magazine March 2017 - 9

Leung and Rife

Figure 3.

Figure 4.

chain intersects with another fault chain or with a modifier. Where
two fault chains combine, an OR gate is implied. Where a fault
chain intersects a mitigation, a mitigation gate is implied. Similarly, where a fault chain intersections some other modifier, the
appropriate gate associated with that modifier is implied.
Two examples of CSL networks are shown in Figure 3. In
Figure 3(a), the network consists of four fault chains. Fault
chains must be combined with an OR gate, so each node (black
circle) in this graph is an OR gate. In Figure 3(b), the right side
of the tree shows two fault chains, implicitly combined by an OR
operation. The output of the OR operation is another fault chain,
which is combined with a promotion event. Implicitly, the node
combining the promotion event with the fault chain must be a
promotion gate.
In order to compute probability distributions in CSL analysis,
we must define a mathematical model for each gate. The model
for the OR gate is identical to that for fuzzy logic, as described by
(9) and Table 5, above. The mitigation, promotion, upper threshold, and lower threshold gates are defined below. In each case, the
variable M is used to indicate the modifier state. The variable A
indicates the input state of the fault chain and C indicates the output state of the fault chain. Each gate is associated with an operator
that maps values of A and M to a particular value of C. When the
modifier is a promotion, the mapping is

where Amax corresponds to the highest possible value on the range
of A. When the modifier is a mitigation, the maping instead reduces severity, such that

Two CSL networks including (a) a case with four fault chains and no
modifiers and (b) a case where a promotion event, such as nighttime
conditions, aggravates the severity of a pair of independent fault chains.


C min  A  M , Amax  ,

MARCH 2017

(11)

UAS creating a 3D image of a building using synthetic aperture radar
(SAR).


C max  A  M , Amin  ,

(12)

where Amin corresponds to the lowest value on the range of A. The
minimum and maximum operators are introduces to make sure that
the output state stays in the desired range. For instance if A, M,
C ∈ {0,1,2,3,4}, then Amin = 0, and Amax = 4. Whereas the promotion
and mitigation gaits shift severity, the threshold gates cap severity.
If the modifier is a lower threshold, then the mapping is
C  max  A, M  .

(13)

When the modifier is an upper threshold, the mapping is
C  min  A, M  .

(14)

For comparison with gates used in conventional and fuzzy
analysis, it is useful to convert the CSL gate mappings into tabular
form. For the case of aviation severity levels, the mappings for the
promotion, mitigation, lower threshold, and upper threshold gates,
given by (11)-(14), are expanded in Tables 7-10. For each gate, the
output probability distribution P(C) can be obtained by combining
the appropriate table for the gate with (3) and (6). The last two
equations must be very slightly modified, with M substituted for B.

IEEE A&E SYSTEMS MAGAZINE

9



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