Environmental Engineering Reference
In-Depth Information
states should be calculated from the variance of the measurement errors of the source
measured variables.
Alternatively, as the state variable selection is not unique, one may also select
the state variables in such a way that they are confounded with directly measured
variables. The drawback of this procedure is that it usually does not allow state
equations to be linear. In summary, selection of the
X
and
Y
sets is not unique, but it
must be made such that the functions
f
and
g
and the
X
and
Y
uncertainties covari-
ance matrices be the least complex possible. Obviously, there is no perfect selection,
since simplifying one function necessarily implies that the other one becomes more
complex.
2.6 General Principles of Stationary and Steady-state Data
Reconciliation Methods
The core of a data reconciliation procedure is a model-based observer that makes
use simultaneously of process models (stationary regime constraints as defined in
Section 2.3.6) and measurements as defined in Section 2.5. It optimally estimates
unmeasured process variables in such a way that the data is reconciled with the pro-
cess model, while respecting the measurement and model uncertainties (see Figure
2.5). The observer is based on the minimization of a reconciliation criterion, which
usually consists of a quadratic sum of residuals
J
containing both the node im-
balances εand the measurement errors
e
. The reconciliation problem is formulated
as
(
X
)
X
T
V
−
1
ε
T
V
−
1
ε
=
arg min
X
[(
Y
−
Z
)
(
Y
−
Z
)+
ε
Z
=
g
(
X
)
(2.30)
Y
=
Z
+
e
;
e
∼
N
(
0
,
V
)
ε
=
f
(
X
)
; ε
∼
N
(
0
,
V
ε
)
subject to
X
min
≤
X
≤
X
max
,
(2.31)
where
X
are the plant states,
Y
the measured values of
Z
,
e
the measurement er-
rors assumed to have zero mean values and known variance matrix
V
,andε the
constraint uncertainty values assumed also to have zero mean values and known
variance matrix
V
ε
. State equation
f
consists of mass conservation constraints
and additional constraints (as well as energy constraints when needed), while obser-
vation equation
g
(.)
relates measured variables to state variables. Finally, estimated
plant states
X
have to be within physically meaningful intervals. For instance, mass
fractions have to be between 0 and 1 and flowrates should have positive values.
The criterion in Equation 2.30 can be viewed as an empirical least-squares pro-
cedure or as the maximum likelihood solution of the state estimation problem if
measurement errors and model uncertainties are Gaussian. The latter might be not
strictly verified since model structure, parameters, and neglected dynamics may not
be Gaussian, as well as measurement error uncertainties, which, obviously, can-
(.)
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