Process Observers and Data Reconciliation Using Mass and Energy Balance Equations - Advanced Control and Supervision of Mineral Processing Plants - page 44

where Π is defined in Equation 2.66. Equation 2.77 gives Equation 2.65 when V ε is
set to 0.
Tuning of the accumulation rate covariance matrix. Three methods are proposed
to estimate the V ε covariance matrix:
• Measurement of the state variables by a suitable combination of instrumentation

for a sufficiently long period of time. This may require transient installation of
instruments on the streams. Since it may happen that not enough sensors are
available to simultaneously measure all the state variables, the different parts of
the circuit may be sequentially studied, assuming that the statistical properties of
the operation do not change much from a test on a plant sub-network to a test on
another sub-network. For such a subset of X , the measurement equation is

Y = X + e . (2.78)

Multiplying Equation 2.78 by the incidence matrix M of the sub-network corre-
sponding to the fully instrumented part of the plant, one obtains

MY = MX + Me = ε + Me . (2.79)

Assuming that there is no correlation between the state variables and their mea-
surement errors, the variance-covariance matrix is given by:

M T MV M T MVar ( Y ) = V ε + , (2.80)

where Var is the measurement variance-covariance matrix of Y , the elements

of which can be evaluated by the usual statistical estimation methods. V ε can then
be calculated by

( Y )

M T MV M T V ε = MVar ( Y ) − . (2.81)

The total V ε matrix can therefore be constructed by consistently assembling the
various terms of the partial variance matrices. Obviously some covariance terms
will not be evaluated, but since they necessarily correspond to distant streams in
the plant, they can be neglected.

• Alternatively, one may evaluate the V ε matrix by an approximation of the plant

dynamics, using for instance first order transfer functions with rough evaluation
of their time constants. The process dynamics are subsequently gathered into a
plant state-space model (see Section 2.10 for more detailed explanations). This
model allows an estimation of V x , the state variable variance characterizing the
underlying unknown random dynamic variations of the process. V ε can finally be
estimated by

MV x M T

V ε = . (2.82)

• A third option is to use V ε as a tuning factor that can be adjusted heuristically fol-

lowing an evaluation of the data reconciliation performance in comparison with
a desired behavior. In fact, this is the subjective method that is mostly adopted,
for example, when tuning model uncertainty relative variances and measurement
uncertainties in Kalman filtering techniques.