Process Observers and Data Reconciliation Using Mass and Energy Balance Equations - Advanced Control and Supervision of Mineral Processing Plants - page 50

could confirm this assumption. In that case a model for estimating the losses could
be used to generate a virtual measure of the heat loss that can be added to the rec-
onciliation problem.

2.8.2 An Example of Hierarchical Methods: BILMAT™ Algorithm

The BILMAT™ algorithm has been developed mainly for bilinear problems [22]
but it can be extended to trilinear systems [53, 59]. The basic principle is that, when
the main phase flowrates are known (upper level of the hierarchy), the species mass
conservation constraints become linear functions of the species mass fractions (see
Section 2.7.3). Therefore the idea is to define a lower level reconciliation problem
where the criterion contains the mass fraction part of the data, and the constraints
are written only for the species conservation. This problem, being LQ, has a direct
analytical solution (see Section 2.7.3). At the upper level, only the flowrate variables
are manipulated to minimize the overall criterion. Formally the problem is thus split
into two minimization problems defined by

min
X m

J f

min
X f

J ( X f , X m )= min

X f

J m ( X m | X f )+ , (2.97)

, X m

where X f and X m are, respectively, the flowrate state variables and the species mass
fractions variables, and J m and J f the J parts containing the measured mass fractions
and flowrates, respectively. Since the upper minimization level is non-linear, one can
apply the substitution/PNL method to decrease the number of search variables. This
method is depicted in Figure 2.10.
To illustrate the method, let us again consider the introductory example of Sec-
tion 2.1.1. Flowrates F 1 and F 2 can be selected as independent flowrates of the upper
level, since F 3 can be deduced from

F 3 = F 1 − F 2 . (2.98)

Giving values to these three variables leads to a linear structure for the copper
and zinc mass conservation equations:

F 1 c 1 − F 2 c 2 − F 3 c 3 = 0;

(2.99)

F 1 z 1 − F 2 z 2 − F 3 z 3 = 0 .

The calculation of the reconciled metal mass fractions is then a LQ problem given
the values of the flowrates. The solution to this problem is obtained by forming the
Lagrange function:

L = J c + 2λ c ( F 1 c 1 − F 2 c 2 − F 3 c 3 )+ 2λ z ( F 1 z 1 − F 2 z 2 − F 3 z 3 ), (2.100)

where J c is the part of the criterion which contains the metal assays: