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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