Process Observers and Data Reconciliation Using Mass and Energy Balance Equations - Advanced Control and Supervision of Mineral Processing Plants - page 52

a NLP is required to find the optimal values of F 1 and F 2 , as shown in Figure 2.10.
The optimal values are given in Table 2.1.
This hierarchical technique has been extended to trilinear and quadrilinear prob-
lems, i.e. , problems involving slurry, ore and water, particle size distributions, and
chemical assays within particle size classes. Sub-optimal solutions have been ap-
plied to investigate the path followed by precious metals in a base metal sulphide
plant [59], in a uranium ore grinding circuit [59], and in gold ore comminution and
leaching circuits [28].

2.9 Performance of Data Reconciliation Methods

When the reconciliation calculation is achieved, it is worthwhile to estimate the
reliability of the results in comparison with those of the raw measurements. This
is performed by a sensitivity analysis of the propagation of the measurement errors
and rate of accumulation uncertainties through the reconciliation process [34] and
[44, 88]. In other words, the variance-covariance matrix of the reconciled states has
to be calculated for an assessment of the reconciliation procedure benefits.
There are two approaches to calculate the variance matrix of the reconciled states,
V X , i.e. , the variance of the estimation error. One can explicitly formulate the rec-
onciled states as functions of Y and then calculate the variance V X as a function of
V and V ε . When the reconciliation method leads to a linear estimator, i.e. , when the
reconciled states are linear functions of Y , then the usual linear algebra of variance
calculation can be applied. Otherwise, one can linearize the expression around the
process steady-state values and apply linear variance algebra. The second method
consists in randomly generating synthetic values of Y according to a normal distribu-
tion N

Y

and repeating the reconciliation procedure. If this is done a sufficiently
large number of times, the statistical properties of X can be subsequently estimated.
This method is known as a Monte-Carlo sensitivity analysis.
When the X estimator is linear, one can calculate the variance of X using the
relationship between X and Y . For instance, in the linear steady-state case, Equation
2.65 gives

( , V )

X

= Γ Y , (2.105)

where Γ is given by

Π −

1

M Π −

1 M T

) −

1 M Π − 1

C T V −

1

Γ = [ I − M ( ] . (2.106)

Therefore X variance is

Γ V Γ T

= .

(2.107)
Since X , in the assumed Gaussian context, is a maximum likelihood estimate, it
is obvious that this estimate is such that necessarily the diagonal terms of V X are
lower than the diagonal terms of the variance of reconciled values obtained by any
other estimator. It has been shown [44] that the variance of the reconciled measured

V X