Process Observers and Data Reconciliation Using Mass and Energy Balance Equations - Advanced Control and Supervision of Mineral Processing Plants - page 61

To add time links between successive values of stream variables included in
Equations 2.122, it has been proposed to add empirical evolution models for each
flowrate [36]. Random walk, for instance, can be used:

f ( t + 1 )= f ( t )+ ω ( t ) ω ∼ N ( 0 , V ω ), (2.124)

where V ω is considered to be a diagonal matrix. This model is rather a rough and
arbitrary approximation since it does not take account of the physical links that must
exist between the input and output flows of a process. Stationary autoregressive
models should be preferred for describing the time evolution of the variables.
Instead of adding empirical information to the mass conservation constraint of
Equation 2.122, one may add parts of the phenomenological model. For instance,
the assumption of perfect mixing creates a link between the outlet streams and the
inventories, therefore limiting the problems generated by the difficulties measuring
inventories.
The various models and sub-models presented above must now be coupled with
measurement values to subsequently perform data reconciliation. The next section
describes some of the options for on-line and dynamic data reconciliation.

2.10.4 Reconciliation Methods

The state variables are usually not directly measurable. As in the steady-state and
stationary methods, two main options can be used: either reconstructing pseudo-
measurements of state variables by combining primary measured variables, or using
the actually measured variables and expressing them as functions of the state vari-
ables. The second approach has the drawback of generating non-linear observation
equations but the merit of using directly the raw information. As usual the measure-
ment equation is

Y ( t )= g [ X ( t )] + ee ∼ N ( 0 , V ), (2.125)

where e is the measurement error and X the state variables, either f i and m i in the
linear state equation case, or f 0 and c i ,totalmassflowrates and component concen-
trations, in the bilinear case.
A variety of reconciliation methods are possible depending on the model and cri-
terion used in the filter. The process equations can be those of a full model or a sub-
model, empirical or phenomenological, steady-state, stationary, or fully dynamic.
The reconciliation criterion usually contains weighted quadratic terms derived from
the maximum likelihood estimation of the state variables X from the measurement
Y , in a Gaussian context for model uncertainties, measurement errors, and driving
white noise. The criterion can be instantaneous, or expressed in a finite time win-
dow, or launched at time zero. Depending on the consistency between the modeling
assumptions and the criterion formulation and resolution, the resulting filter can be
optimal or sub-optimal.