Process Observers and Data Reconciliation Using Mass and Energy Balance Equations - Advanced Control and Supervision of Mineral Processing Plants - page 60

C

R

F

Figure 2.15 An empirical model of a flotation single unit (F, R and C stand for feed, concentrate
and reject streams, and G for transfer functions, s is species separation coefficient, and ξs are white
noise)

2.10.3 Sub-models

The minimal set of equations that can be used for designing model-based data fil-
tering methods is the mass conservation constraints of Equations 2.114 written for
components that are not transformed:

dm i
dt =

M i f i i ∈{ 0 ... n + 1 } (2.121)

or

dm
dt =

Mf (2.122)

using the stacked vectors f and m , and the block-diagonal matrix of the M i s. This
equation does not contain any parameters to be calibrated or uncertainties to be es-
timated. Obviously these would be ideal conditions for data reconciliation. Unfor-
tunately the number of equations q ( = n n ( n + 1 ) ) is low compared with the number

n X of variables to be estimated. Hence - even if all the process

variables are measured - this creates a low redundancy level, which in turn leads to
limited improvements of the process variable reliability. In addition, the inventories
m j are usually not available for measurement.
To cope with the problem of inventory unavailability, it is possible to consider the
stationary model described in Section 2.7.2 for instantaneous data reconciliation,
while adding time correlation through the statistical behavior of the accumulation
rates (node imbalances):

(= ( p + n n )( n + 1 ))

Mf = εε ∼ N ( 0 , V ε ( τ )), (2.123)

where τ is the time lag used for calculating V which is the autocovariance of

ε. This stationary dynamic model degenerates into the stationary model of Section
2.7.1 when V ε

( τ )

( )= > =

τ 0forτ 0, or into the steady-state mass balance when V ε 0.