Environmental Engineering Reference
In-Depth Information
2.6.1 Observability and Redundancy
A process state x i is said to be observable or estimable if it can be calculated using
simultaneously the measurement values and the conservation constraints, or part of
them. Therefore the problem is to find a unique estimate of the variables x i of the
vector X that satisfies the following system:
f
(
X
)=
0
,
(2.32)
(
)=
,
g
X
Z
where Z is known and is the exact value of the measured variables. Equation 2.32
represents the constraint and observation equations of (2.30) where the uncertainty
variables are set to their most probable values, i.e. , zero. A state variable, such as a
metal concentration, which is directly measured, is obviously observable since one
possible estimate is its measured value. Hence, the concept of observability is only
important for state variables that are not directly measured.
When, in system (2.32), there is at least one of the equations (state or measure-
ment equation) that cannot be removed without losing x i observability, x i is said to
be non-redundant. When there is more than one possible way to estimate the value
of a state, using different equations of the system (2.32), this state variable is said
to be redundant. When the state is directly measured, it is redundant when it is still
possible to estimate its value in the case the measurement is unavailable. Because
of the inherent uncertainties of the Y values, the estimate value obtained from the
following system for a redundant state x i :
f
(
X
)=
0
,
(2.33)
g
(
X
)=
Y
,
depends on the subset of equations that is kept for calculating the state variable.
When all the state variables are estimable, the process is said to be observable.
When at least one state variable is redundant the process information is said to be re-
dundant. When all the state variables are observable and non-redundant, the system
is said to be of minimal observability.
One can define the process information overall degree of redundacy as the largest
number of equations that can be eliminated from the system without losing process
observability. Usually it is related to the number of equations minus the minimum
number of required equations to obtain minimal observability. Redundancy degrees
for individual states can also be defined [64].
The redundancy degree is strongly coupled to the data reconciliation perfor-
mance: the higher the redundancy, the higher the reconciled value reliability (for
state estimate reliability, see Section 2.9). Moreover, the higher the redundancy, the
higher the robustness of the observer. This means roughly that the number of possi-
ble sensor failures that do not hinder process observability increases with the degree
of redundancy [64]. Assuming that all the variables are observable, a redundancy
degree can be defined as
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