Preliminary Independent-Window Algorithm Al p

Alp is a preliminary and conservative IWA. Alp proposes to use sliding time win-

dows for diagnosis. A sliding time window moves along an observation flow by one observation at a time. Alp begins at the i-th observation, and the size of each time

window is k. Let the length of a time window be k. The finite delay d between two time windows is 1. The time windows of Alpare defined as follows.

TAlp(i,k, 1) ={[i,i+k−1],[i+1,i+k],[i+2,i+k+1]. . .} For example, wheni=1, the first two time windows are visualised as follows.

time window 1 z }| { o1 | {z } time window 2 o2. . . .okok+1. . . .

Input Time Window Diagnosis Output x,c,c,b,b,b,b,b (x,c,c,b) F F(precise) (c,c,b,b) Alpmay stop (c,b,b,b) Alpmay stop (b,b,b,b) Alpmay stop (b,b,b,b) Alpmay stop x,c,c,c,b,b,b,b (x,c,c,c) N N(imprecise) (c,c,c,b) N (c,c,b,b) N (c,b,b,b) N (b,b,b,b) N

Table 4.2: Examples ofAlp running on M4in Fig. 4.2 where i=1 andk=4

Example of Alp Tab. 4.2 shows a set of results of running Alp on M4 where iis 1

andkis 4. In this case,

TAlp(1, 4, 1) ={[1, 4],[2, 5],[3, 6],[4, 7],[5, 8]}.

The first output is precise while the second output is not precise. Given the first observation sequence x,c,c,b,b,b,b,b, the time windows are (x,c,c,b), (c,c,b,b),

(c,b,b,b),(b,b,b,b), and(b,b,b,b). For the first time window(x,c,c,b), there is only one trace that is compatible with this time window, i.e. A−→x C−→c D−→c D−→v G−→b G. In particular, L(G) evaluates to F. This trace is faulty, and therefore there is no nominal explanation. Thus, the diagnostic result for the first time window is F, i.e.

W1,4(M,o)evaluates to F. Therefore, the first observation sequence is diagnosed as

F.

Given the second observation sequence x,c,c,c,b,b,b,b, the time windows are:

(x,c,c,c),(c,c,c,b),(c,c,b,b),(c,b,b,b), and(b,b,b,b).

• For the first time window(x,c,c,c), there exists a nominal trace that is compat- ible with this time window, i.e. A −→x C −→c D −→c D −→c D. In particular, L(D)

evaluates to N. Thus, the diagnostic result of the first time window is N, i.e. W1,4evaluates to N.

• For the second time window(c,c,c,b), there exists a nominal trace that is com- patible with this time window, i.e. A−→c B−→c B−→c B −→u A−→b A. In particular, L(A)evaluates toN. Thus, the diagnostic result of the second time window is N, i.e. W2,5 evaluates to N.

• For the third time window (c,c,b,b), there exists a nominal trace that is com- patible with this time window, i.e. A−→c B−→c B →−u A−→b A−→b A. In particular, L(A)evaluates to N. Thus, the diagnostic result of the third time window isN, i.e.W3,6 evaluates to N.

• For the fourth time window(c,b,b,b), there exists a nominal trace that is com- patible with this time window, i.e. A−→c B−→u A→−b A−→b A−→b A. In particular, L(A)evaluates to N. Thus, the diagnostic result of the fourth time window is N, i.e. W4,7 evaluates to N.

• For the fifth time window (b,b,b,b), there exists a nominal trace that is com- patible with this time window, i.e. A −→c B −→u A −→b A −→b A −→b A −→b A. In particular, L(A) evaluates to N. Thus, the diagnostic result of the fifth time window isN, i.e. W5,8evaluates to N.

Therefore, the second observation sequence is diagnosed as N, i.e. ∀[i,j]∈ TAlp such thatWi,j(M,o)evaluates toN. However, this observation should lead toFusing exact

diagnosis∆. Therefore, the diagnostic result of the second time window is imprecise.

Alp is a very conservative IWA. Its time windows moves along an observation

sequence by one observation at a time. This preliminary IWA attempts to diagnose a fault as early as using exact diagnosis. However,Alp has two major disadvantages.

• Alp is inefficient to run in terms of time. Suppose the number of observations

is|obs|, and the time window size isk. Since Alp moves by one observation at

§4.2 Four Implementations of IWAs 73

k+1). This leads to a large amount of time windows to diagnose, and the computational time will be long.

• Although Alp has very conservative selections of time windows, the precision

of using Alp varies between DES models. The above example of Alp demon-

strates that it is imprecise w.r.t. M4. In general, Alpcannot handle the situations

where some particular events must be included to one time window in order to produce a precise diagnostic result. The precision of Alpw.r.t. a DES model

will be examined in Section 4.3.

In summary, Alp is a preliminary and very conservative IWA. Alp inspires the

other IWAs, namely Al1, Al2, and Al3, which will be explained in the following

sub-sections.