Environmental Engineering Reference
In-Depth Information
q
+
m
−
n
X
ρ
r
=
,
(2.34)
q
where
q
and
m
are, respectively, the number of state equations and observation equa-
tions, and
n
X
the number of state variables. This index varies between 0 and 1 from
the overall minimal observability to the case where all the states are measured di-
rectly or indirectly. The determination of an unmeasured state observability may be
adifficult task for non-linear
f
and
g
functions. Let's have a look at the linear case.
The linear case.
The system of equations containing the plant information is
MX
=
0
,
(2.35)
CX
=
Z
,
It can be globally rewritten as
Ψ
X
=
Φ
Z
.
(2.36)
A state variable
x
i
is observable if there is at least a subset of equations in (2.36)
that allows the calculation of
x
i
when
Z
is known. The state vector would be observ-
able if the rank of the matrix Ψ is
n
X
, the number of state variables. If the process
is globally at minimal observability, then Ψ is an invertible matrix (regular matrix).
Redundancy equations.
Another way of looking at redundacy is to eliminate the
state variable
X
from the system (2.32) (or (2.35)) in the linear case). The remaining
set of equations is
R
(
Z
r
)=
0or
RZ
r
=
0 in the linear case.
(2.37)
R
is the set of redundant equations, and
Z
r
is here the generic vector of the mea-
sured process variables which are redundant (
Z
or a subset of
Z
). The number of
equations it contains is usually
m
(.)
+
q
−
n
X
,where
n
X
=(
+
) ×
n
1
p
in the linear
(
+
)
case with
components and
p
streams. The number of redundant equations is
thus directly related to the redundancy degree of the reconciliation problem . When
replacing
Z
r
by the measurement values
Y
r
, the system (2.37) is no longer verified
because of the unavoidable measurement errors. The resulting vector is a residual,
that is a vector of a space called parity space, physically related to nodes or joint
nodes imbalances. It is a function of the measurement errors and model uncertain-
ties insofar as stationary conditions are assumed. It vanishes when the uncertainties
have zero values:
n
1
R
(
Y
r
)=
r
=
S
(
e
r
)+
T
(
ε
)
or
RY
r
=
Re
r
+
T
εin the linear case.
(2.38)
X
m
. Let us consider, as an illustrative case,
the situation where the measured variables are state variables (
X
m
), and the station-
ary conservation constraints are linear, and gathered into
The linear stationary case
with
Z
=
MX
=
ε
.
(2.39)
The vector
X
can be reorganized as
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