Page 420 - Design of Simple and Robust Process Plants

9.8 Appendix 407
The constraint minimization problem is:
T
min (X ± X¢) W(X±X¢)
s.t. AX + BY + C
2
with W = diag(1/r i )
The constrained problem can be transformed into an unconstrained one using
Lagrange formulation:
T
T
min L= (X ± X¢) W(X±X¢)+2k (AX + BY + C)
X,Y, k
where k is a (p ” 1)matrix (Lagrange multipliers)
Stationarity conditions are:
dL =WX+A k =WX¢

T
dX

dL =+B k =0
T
dY

dL =AX + BY =±C
d
A square matrix is defined (size m + n + p)an array V and an array D

T

T
W0 A
M ˆ
00B
AB0
0
X WX

V ˆ Y D ˆ 0

k

C
The solution of the optimization/validation can be expressed as:
±1
±1
V= M DM is the sensitivity matrix
Both X and Y arrays appears as linear combinations of measured values X¢
If we note N = M ±1 we will have V = ND
The sensitivity matrix allows the evaluation of the validated values of a variable
from all measured variables
p
X i = P m‡n‡p N† i, j D j = P m N i; j W jj X P
0
jˆ1 jˆ1 j N i; n‡m‡k C k
kˆ1
m‡n‡p m p
Y i = P N n‡i; j D j ˆ P N n‡i; j W j; j X P N n‡i; n‡m‡k C k
0
j
jˆ1 jˆ1 kˆ1
The variance of a linear combination Z of several variables X j is known from
literature,
m m
2
Z= P a j X j Var (Z)= P a X j †
j
jˆ1 jˆ1