Process Observers and Data Reconciliation Using Mass and Energy Balance Equations - Advanced Control and Supervision of Mineral Processing Plants - page 37

X m
X um

X = , (2.40)

where X m is the set of measured variables, i.e. , the vector Z of this particular case,
and X um the set of unmeasured variables. The matrix M can also be decomposed
into two sub-matrices, one for X m , and the other one for X um :

M =( M m , M um ). (2.41)

This allows Equation 2.39 to be written as

M m X m + M um X um = ε . (2.42)

Providing that the system is at minimal observability, i.e. , that the measured val-
ues are properly placed on the state vector, M um is invertible and X um calculable
as

) − 1 X um =( ( − ).

M um ε M m X m (2.43)

− =

This last situation corresponds to n X m q ,and X um is estimated by setting

ε = 0and X m = Y .

q , there are more unmeasured variables than conservation con-
straints. The process is not fully observable (some states or even all states are non-
estimable).
When n X

When n X − m >

q , there are more equations than unmeasured states. It is highly
improbable that, due to the measurement uncertainties, a value of X um that would
simultaneously satisfy all the conservation constraints could exist. Since the number
of equations in Ψ, m

− m <

q , is larger than the number of states to be estimated n X ,the
observation system is redundant. The elimination of X um from (2.42) leads to the
linear redundancy equations:

+

RX mr = T ε , (2.44)

where X mr is the vector of the redundant measured states. By replacing X mr by the
measurement Y r , one obtains the values of the parity vector:

r = RY r = Re r + T ε . (2.45)

This equation clearly shows that the redundancy equations residuals are not zero
because of the uncertainties prevailing in the measurement and mass conservation
constraints. To manage this problem, one possibility could be to remove redundant
measurements. But this is the wrong approach, since experimental information is
lost and, furthermore, the estimate values would depend upon the data that would
have been removed. The right approach is to reconcile the information by the pro-
cedures discussed in this chapter.
Classification of the process variables:. The condition n x

q is not sufficient
to ensure process observability. It may happen that the process observability is only
partial, i.e. that some states are non-observable. It may also happen that, though the
system is not redundant, some variables are observable redundant, while, as a conse-

− m ≤