Environmental Engineering Reference
In-Depth Information
Steady-state
2
1
0
0
100
200
300
400
500
600
700
800
900
1000
Stationary
1.5
1
0.5
0
100
200
300
400
500
600
700
800
900
1000
Transient
1.2
1.1
1
400
420
440
460
480
500
520
540
560
580
600
Quasi-stationary
2
1
0
0
100
200
300
400
500
600
700
800
900
1000
Ti m e
Figure 2.3
Typical variations of process variables for various operating conditions
described by the usual statistical features: the mean μ
x
, the variance
V
x
(
0
)
and the
autocovariance
V
x
(
k
)
(or autocorrelation ρ
(
k
)
if normalized by the variance):
E
(
x
)=
μ
x
;
(2.5)
T
E
((
x
(
t
)−
μ
x
)(
x
(
t
−
k
)−
μ
x
)
)=
V
x
(
k
).
The nature of the reconciliation procedure to be applied to filter industrial data
must be adapted to the operating regime that was prevailing during data gathering,
and to the measurement strategy then applied. There is no systematic method to
decide whether a steady-state, stationary or dynamic filter must be applied, but some
hints can be helpful. Some of the conditions for the application of a steady-state
observer are:
•
The process deviation from a theoretical steady-state is of low magnitude com-
pared with measurement error amplitudes (variance matrix
V
). In other words,
the diagonal terms of
V
are large in comparison with the diagonal terms of
V
x
(
0
)
.
•
The process deviation from a theoretical steady-state is significant with respect
to the measurement errors, but the dynamic variations are produced by stationary
disturbances of high frequency spectra in comparison with the natural process
dynamics - in other words, the process variable autocorrelogram time widths
are small in comparison with the width of the cross-correlation between process
inputs and outputs. However, instantaneous measurements data set could be pro-
cessed by the steady-state method providing that the high frequency disturbance
variances are added to the measurement error variances.
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