BULETINUL INSTITUTULUI POLITEHNIC DIN IAŞI - Universitatea ...

(6)
⎧
⎪
⎨
⎪
⎩⎪
FR 1
FR 2
FR 3
Bul. Inst. Polit. Iaşi, t. LVI (LX), f. 2, 2010 5
⎧
⎪
⎪
⎪
⎫ ⎡
⎪ ⎢
⎬ = ⎢
⎪ ⎢
⎭⎪
⎣⎢
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
A17 A27 A37 A18 A28 A38 ⎤ ⎪
⎥ ⎪
⎥ ⎨
⎥ ⎪
⎦⎥
⎪
⎪
⎪
⎪
⎩
One can see that DP1, DP2 and DP3 are the only possible contributing
parameters in Eq. (4). As for the design of Eq. (5), just DP4, DP5 and DP6 are
significant. At last, in what concerns to Eq. (6), only DP7 and DP8 contribute. In
other words, each one of the above-defined coexisting designs can fulfil all the
FRs of Eq. (3) using entirely different subsets of the DPs included in the latter
equation, in such a manner that all the DPs are taken into account.
Now, one can figure out that the design of Eq. (3) could be achieved by
merging the designs of Eq. (4), Eq. (5) and Eq. (6).
The design matrices of Eq. (4) and Eq. (5) could be reduced to block
matrices of size m x m by eliminating the zero elements of their design matrices.
Therefore, the surrogate “condensed design equations” could be obtained
from those equations by eliminating the non-relevant DPs, and by using the all
the existing FRs and the non-zero block matrices. For example, Eq. (7) is the
condensed equation that is obtained from Eq. (5).
The same treatment could be done to Eq. (6), but in this case we would
obtain a non-square block matrix of size (n mod m) x m.
⎧ FR ⎫ ⎡
1 A14 A15 A ⎤ ⎧
16 DP ⎫
4
⎪ ⎪ ⎢
⎥ ⎪ ⎪
(7)
⎨ FR2 ⎬ = ⎢ A24 A25 A26 ⎥ ⎨ DP5 ⎬ .
⎪ FR ⎪ ⎢
⎩⎪
3 ⎭⎪
A34 A35 A
⎥ ⎪
⎣⎢
36 ⎦⎥
DP ⎪
⎩⎪
6 ⎭⎪
As a result, the redundant design of Eq. (3) could be considered “suitable
design” if the condensed equations obtained from Eq. (4), Eq. (5) and Eq. (6)
are not coupled. The coupled condition is excluded in Eq. (4) and Eq. (5) if the
relevant DPs are chosen so that their non-zero block matrices are either
triangular or diagonal.
As for Eq. (6), the coupled condition is excluded if its non-zero block
matrix is populated in such a manner that the condensed equations correspond
to uncoupled or decoupled designs. If this is not the case, one can selectively
“freeze” as many DPs as required, so that the condensed design become
uncoupled or decoupled, as exemplified elsewhere [3].
DP 1
DP 2
DP 3
DP 4
DP 5
DP 6
DP 7
DP 8
⎫
⎪
⎪
⎪
⎪
⎪
.
⎬
⎪
⎪
⎪
⎪
⎪
⎭