Aerospace and Electronic Systems Magazine April 2017 - 8

Airplane Flight Safety Using Error-Tolerant Data Stream Processing
overview of the error detection method
based on different error signatures is
summarized in Figure 3.
The DDDAS PILOTS approach
includes five steps: (1) error function
model development, (2) error signature
pattern analysis, (3) likelihood estimation of anomalies against stochastic errors, (4) dynamic mode estimation, and
(5) application-based data correction
based on logical redundancy in environmental and system conditions.
1. Error functions: An error function is a function that computes a
numerical value from independently
measured input data. It is used to examine the validity of redundant data. Figure 3.
Error signature-based fault detection method.
If the value of an error function is
zero, we interpret it as no error in the
given data. For the speed vector example shown in Figure 2a,
we can define an error function as follows [22].

(2)
The values of input data are sampled periodically from corresponding data streams. Thus, an error function e changes its value
as time proceeds, represented as e(t).
2. Error signatures: A error signature is a set of constrained
mathematical functions that is used to capture the characteristic
pattern of an error function e(t) under a specific fault condition.
Using a vector of constants K = 〈k1, k2, ..., kn〉, a function f(t, K ),

    

 

and a set of constraint predicates P K  p1 K ,, pl K ,
the error signature S is defined as follows:

     

 

 

S f t , K , P K  f | p1 K    pl K .

(3)

For example, an interval error signature SI can be defined using a
set of constraint predicates PI K , A, B = {a1 ≤ k1 ≤ b1, ..., am ≤ km
≤ bm} denoting intervals, as follows:



  




S I S f t , K , PI K , A, B



 f | a1  k1  b1,, am  km  bm .



(4)



For example, letting K 
k , f t, k 
t  k, A 
2 and B  5 ,
the error signature S1 contains all linear functions with slope 1, and
crossing the Y-axis at values from 2 to 5 as shown in Figure 4.
3. Mode likelihood vectors: Given a vector of error signatures
〈S0, S1, ..., Sm〉, we calculate δi(t), the distance between the
monitored error function e(t) and each error signature Si by:


 i  t  min g t Si

8



t

t 

e  t   g  t  dt ,

(5)

Figure 4.

Error signature S1 = S(f(t, 〈k〉) = t + k, {2 ≤ k ≤ 5}).

where ω is a window size parameter. Note that our convention is to
capture normal conditions as signature S0. The smaller the distance
δi, the closer the measured data is to the theoretical signature Si.
We define the mode likelihood vector as L(t) = 〈l0(t), l1(t), ..., lm(t)〉
where li(t) is defined as:

1,
if  i  t   0

li  t    min  0  t  ,,  m  t 
.
, otherwise

t


i 





(6)

Observe that each li ∈ (0,1] represents the ratio of the likelihood of
signature Si being matched with respect to the likelihood of the best
matching signature.
4. Mode estimation: Using the mode likelihood vector, the final mode, which corresponds to the most likely signature, is

IEEE A&E SYSTEMS MAGAZINE

APRIL 2017



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