Computers and Chemical Engineering 26 (2002) 1201– 1211
www.elsevier.com/locate/compchemeng
Measurement bias detection in linear dynamic systems
Derrick K. Rollins *, Sriram Devanathan, Ma. Victoria B. Bascuñana
Department of Chemical Engineering and Statistics, Iowa State Uni6ersity, Ames, IA 50011, USA
Received 22 May 2000; received in revised form 28 February 2002
Abstract
A new method to detect the existence of biased measured variables in dynamic processes is presented. Hence, this work presents
a new Dynamic Global Test (DGT) and test procedure for dynamic gross error detection (GED) that brings to light certain of
its attributes which have not hitherto (to our knowledge) been presented in GED literature. Recognition of these attributes leads
to a scheme that enables identification of the type of biased measurement (e.g. flow or level). This approach is not computationally
intensive and is applicable in the case of process leaks and multiple biased variables. Simulation results for the identification of
the type of biased measurement (e.g. flow or level) and the estimation of the time of occurrence (ETOC) are given. The
performance study in this work specifically varied the size of measurement bias (œi ), the bias location (i ), the bias true time of
occurrence (TTOC), the significance level (h), and the sample size (N). This study shows the proposed approach to be accurate
in identifying the type of biased variable and its TTOC. The performance of the proposed scheme improves as N and œi increase.
© 2002 Elsevier Science Ltd. All rights reserved.
Keywords: Dynamic systems; Gross error detection; Fault detection; Sensor validation
Nomenclature
0
AVGD
ETOC
TTOC
D, B
E[q]
F
I
k
M
m
N
n
p
q
qi
Qi
a n × 6 null matrix
average difference of the estimated TOC from the true TOC
estimated TOC of the bias
true TOC of the bias
constraint matrices in generalized dynamic system
the expected value of estimator q
modified incidence matrix
a n× n identity matrix
the current time instant
incidence matrix
number of time instants in each moving window (m= 4 in this study)
sample size
number of nodes
number of trials in which the type of bias is correctly identified
number of successive windows in which Ho must be rejected twice or more for conclusion of flow bias
(q= 5 in this study)
vector of measurement errors in Qi
vector of flow measurements at time instant i
* Corresponding author. Tel.: + 1-515-294-7642; fax: + 1-515-294-2689.
0098-1354/02/$ - see front matter © 2002 Elsevier Science Ltd. All rights reserved.
PII: S 0 0 9 8 - 1 3 5 4 ( 0 2 ) 0 0 0 3 6 - 4
D.K. Rollins et al. / Computers and Chemical Engineering 26 (2002) 1201–1211
1202
Qi*
Ri
STDD
V
VQ
VW
6
wi
W*
i
x*i
Xi
X*
i
vector of true values at time instant i
vector obtained by transformation of measurement model
standard deviation of STDD
covariance matrix of measurement errors
covariance matrix of measurement errors on flows Q
covariance matrix of measurement errors on volumes W
number of flows
vector of measurement errors on Wi
vector of true total mass values at time instant i
a (n+6)× 1 vector of true values of unknown variables at time instant i
a 2(n +6)×1 vector of measured values at time instant i
a 2(n +6)×1 vector of true values
Greek letters
h
level of significance
ii
power function
œW,j
vector of measurement biases at time instant j
œQ,j
vector of measurement biases at time instant j
Dj
vector of measurement biases at time instant j
i
vector of measurement errors at time instant i
W,i
vector of measurement errors at time instant i for total mass variables
Q,i
vector of measurement errors at time instant i for total flow variables
Erri
vector of measurement errors
Li
vector of process leaks at time instant i
¦Ri
vector comprising the elements of the expected value of Ri
J
noncentrality parameter
SRi
the variance– covariance matrix of Ri
S
the variance matrix
F
constraint matrix
the upper (100h)th percentile of the  2 distribution
 2n,h
Other symbols
‘is distributed’
N2(n + 6) a 2(n +6) variate normal distribution
Superscript
T
transpose
1. Introduction
A gross measurement error is made when the measurement of a variable deviates far from its true value.
This article addresses situations when large dynamic
systematic deviations (i.e. measurement biases) are the
cause of the gross errors. Causes of biased measurement
include
instrument
malfunction
and
miscalibration.
When measurements are significantly biased, data
reconciliation (DR) (the adjustment of process variables
to improve their accuracy to satisfy material and energy
balance constraints) may possibly give estimates (of
process variables) that are more inaccurate than the
measured values. Hence, detecting the presence of biased measurements and their true time of occurrence
(TTOC) are important steps in obtaining accurate reconciled values.
Historically, dynamic DR can be traced back to the
early 1960s and 1970s (Kalman, 1990; Gertler & Almasy, 1973; Willsky & Jones, 1974). Some of the most
widely used DR and gross error detection (GED) techniques for dynamic processes are based on the Kalman
filtering (KF) technique (Kalman). KF has been used
D.K. Rollins et al. / Computers and Chemical Engineering 26 (2002) 1201–1211
for data smoothing and parameter estimation (Bellingham & Lee, 1977; Newman, 1982; Watanabe & Himmelblau, 1982). These techniques have been developed
for linear dynamic systems using a weighted leastsquares objective function. In using these approaches,
the parameters and measured inputs are estimated hypothetical states and so it is difficult to obtain suitable
values for such a system. In addition, sensor malfunction detection methods based upon KF have been used
(Bellingham & Lee; Newman; Watanabe & Himmelblau) for linear dynamic systems. However, these approaches have not been extended to handle
measurement biases in a general (unrestrictive) fashion.
A significant contribution to current methods in
GED is the generalized likelihood ratio (GLR) approach (Willsky & Jones, 1974). The GLR method has
been used for GED in steady state (Narasimhan &
Mah, 1987) and dynamic processes (Narasimhan &
Mah, 1988). Narasimhan and Mah used their GLR
method to identify various causes of gross errors. However, their work was restricted to pseudo steady state
conditions (i.e. variation around a nominal point) and
does not seem to be applicable during a transition state
(i.e. a period of change from one steady state to a new
steady state).
Kao, Tamhane, and Mah (1992) have presented a
composite procedure for detecting and identifying gross
errors in serially correlated data for pseudo steady state
processes. Measurements are serially correlated when
their errors are related to past values. This phenomenon is common in modeling chemical processes
because of feedback loops, material recycling, etc. They
proposed a prewhitening step to validate the assumptions of Gaussian measurement and process noises,
followed by statistical process control chart techniques.
It involves the use of statistical tests to detect the
presence of gross error, followed by the application of
GLR method to identify and estimate the magnitudes
of the gross errors. However, the method is restricted to
pseudo steady state and its performance in the presence
of multiple biases is unclear. It also appears that the
autocorrelated structures of the data, which can be
different for each variable, must be known.
The measurement error reconciliation method
(Khuen & Davidson, 1961; Swenker, 1964) has been
generalized to transient conditions by Almasy (1990).
This method, which has been used for dynamic DR,
has been termed dynamic balancing, and is based upon
linear conservation equations to reconcile the measured
states. Estimates are obtained for flow and inventory
variables by applying the KF to the balance model.
Almasy, however, does not address the problem of
GED.
Darouach and Zasadzinski (1991) developed a recursive optimal solution technique in weighted leastsquares for DR of transient systems characterized by a
1203
generalized linear dynamic model (i.e. singular model).
A unique feature of this modeling approach is that KF
is not applicable. Rollins and Devanathan (1993)
showed that the Darouach’s and Zasadzinski’s estimates could be very accurate but computationally intensive. In addition, Darouach and Zasadzinski do not
address the topic of GED, and their approach does not
appear to be extendable to this situation. Motivated by
these limitations, Rollins and Devanathan developed a
constrained least-squares DR approach that provides
accurate and unbiased estimates for process variable
and is computationally less intensive than the
Darouach and Zasadzinski approach.
Ramamurthi, Sistu, and Bequette (1991), Kim, Leibman, and Edgar (1991), Leibman, Edgar, and Lasdon
(1992) have presented schemes for nonlinear dynamic
DR (NDDR). Ramamurthi et al. presented a successively-linearized horizon-based estimation (SLHE)
strategy for the estimation of variables and physical
parameters in transient systems. They showed the
SLHE strategy to be significantly more accurate than
the extended KF and computationally more efficient
than the nonlinear programming (NLP) approach of
Leibman et al.. Kim et al., developed a sequential
error-in-variables method for DR and parameter estimation. They compared their nonlinear dynamic errorin-variables method (NDEVM) to other conventional
least-square techniques and to estimation using orthogonal collocation and showed improved performance.
The NDEVM does not appear to be rigorous in estimating process variables when multiple biases are
present and it does not address detection and identification of biased measurements.
Leibman et al. (1992) have presented a new method
for NDDR based on NLP techniques. They showed
this method to be superior to KF, both in its ability to
cope with inequality constraints and in nonlinear situations. Leibman et al. commented that the main disadvantages of the approach are the requirement of an
accurate process model and the problem of intensive
computations. In their approach, biased measurements
are treated as parameters to be estimated. In addition,
the approach does not seem to be designed to handle
multiple biases. Ramamurthi et al. (1991), Kim et al.
(1991) and Leibman et al. seem to have made significant progress in DR for transient processes but only
moderate progress in GED.
While it is important, from a practical point of view,
to develop GED strategies for transient processes,
steady state techniques are perhaps the foundation
from which to build methods suitable under a wider
range of operating conditions. Traditionally, methods
used to detect biases in steady state systems have
involved a Global Test (GT). A GT conclusion that no
biases are present will obviate location identification.
Thus, a GT is not designed to make conclusions about
D.K. Rollins et al. / Computers and Chemical Engineering 26 (2002) 1201–1211
1204
the location of any biased measurement that is present.
However, a GT can also serve as a final verification of
global closure after process variables are corrected.
Thus, there is strong motivation to have a GT technique for accurate detection when conditions are
dynamic.
Specifically, this article presents a GT to determine
the bias type (flow or inventory variable) and the
TTOC of the bias in dynamic systems. To our knowledge there has been no prior work dealing with GT for
dynamic systems. For simplicity, this article deals with
the situations where the assumptions of no process
leaks and no drifting biases are valid (i.e. the proposed
technique is not restricted by these assumptions). We
also assume in this particular situation that all relevant
variables are measured. The purpose of this article is to
present a novel scheme for accurate detection of measurement biases. We are hopeful that this scheme can
possibly be extended to enable identification of the
specific locations of measurement biases. We introduce
the proposed technique by first presenting the process
models when measurement biases are present and the
equations used. Following this discussion, we present
the results of a Monte Carlo simulation study to evaluate the proposed approach.
2. Process models
The physical model, presented first, represents material balance constraints. This is followed by the measurement model, a statistical expression for the
measurements. A detailed development of these models
is given in Appendix A, for a simple process network,
containing 2 nodes (interconnecting units) and 5 flows.
At the ith time instant, a total mass balance on each
of the n nodes (Rollins & Devanathan, 1993) gives
n × 6 *6 × 1
×1
−W*n
+W*
Qi
i
i − 1 +M
= −[In × n − Mn × 6]
n
= −Dx*+
Bx*=
[ −D B]
i
i
= FXi*= Li,
n
Wi*
W*i − 1
+[In × n 0n × 6]
Q*i
Q*i − 1
n
xi*
x*i − 1
(1)
where M is the process constraint matrix, i= 2, …, k
and
n × (n + 6)
D
=[I − M],
Bn × (n + 6) = [I 0],
+ 6) × 1
x*(n
=
i
n
W*
i
,
Qi*
(2)
(3)
(4)
Fn × 2(n + 6) = [D B],
+ 6) × 1
X*2(n
=
i
(5)
n
xi*
.
x*
i−1
(6)
Wi* is a n×1 vector of true and unknown total mass in
n nodes at time instant i, Qi* is a 6× 1 vector of true
and unknown total mass flow rates for the 6 streams at
time instant i, Li is a n× 1 vector of process leaks, and
k is the current time instant.
The measurement model that applies to Eq. (1) is,
+ 6) × 1
X*2(n
= X*+
Erri,
i
i
(7)
where i =2, …, k and
n
Æ W,i Ç
à  à i
Erri = Ã q,i Ã
,
Ýw,i − 1 à i − 1
È q,i − 1 É
(8)
Var(Erri )= S2(n + 6) × 2(n + 6),
(9)
j N(n + 6)(œj, V),
V(n + 6) × (n + 6) =
(10)
n
VW 0
,
0 VQ
(11)
Var(w, j )= VW,
(12)
Var(q, j )= VQ,
(13)
+ 6) × 1
œ (n
=
j
n
œ nW,×j1
,
œ 6Q,×j1
(14)
where j= 1, …, k. Note that VW and VQ are assumed to
be known although this is not a restrictive assumption.
When VW and VQ are unknown, the sample estimates
can be used (if the sample size is large). However, the
distribution of the test statistic for the GT (given
below) would be different (see Rollins & Davis, 1993).
Also, the sample size is assumed to be one for convenience. If the sample size is greater than one, the
measurement vector contains measurement means and
the measurement variance–covariance matrix is taken
to be V/n.
Note that,
Xi N2(n + 6)(Xi*+ œi, S),
(15)
and
Æ œW,i Ç
Ã
à œ
+ 6) × 1
œ 2(n
= Ã Q,i Ã,
i
ÜW,i − 1 Ã
È œQ,i − 1 É
(16)
where N2(n + 6) is used to represent a 2(n + 6) variate
normal distribution.
D.K. Rollins et al. / Computers and Chemical Engineering 26 (2002) 1201–1211
3. The transformed measurement model
The vector Ri given below (also known as the transformed vector of measurements) plays a central role in
the formulation of the Dynamic Global Test (DGT)
and this section gives its distributional properties. Let,
Rni × 1 =FXi = FX*+
FErri =Li +FErri,
i
(17)
with FX* =L by Eq. (1). Therefore,
E[Ri ]=¦Ri =Li +Fœi,
(18)
and
Var(Ri )= FSFT =SRi.
(19)
Thus,
Ri Nn (¦Ri, SRi ).
(20)
4. The dynamic globe test scheme (DGTS)
The GT is a test designed to determine if any œj,
j= 1, …, n+ 6 is nonzero and is developed by using the
equations in the transformed measurement model. For
a null hypothesis of Ho: ¦Ri =0 and an alternate hypothesis of Ha: ¦Ri "0, the following test is used: reject
Ho in favor of Ha if and only if (see Mardia et al.,
1979),
RTi SR−i 1Ri ]  2n,h,
(21)
where i denotes the time instant, n is the number of
nodes and  2n,h is the upper (100h)th percentile of the  2n
distribution. The probability of rejecting Ho when Ho is
false (i.e. the power function for the test) is given by,
1− ii = P[RTi SR−i 1Ri ] 2n,h ¦Ri ]
= P[noncentral  2n ]  2n,h J2i ],
(22)
with
J2i =¦ TRi(SR−i 1)¦Ri,
(23)
where Ji is called the noncentrality parameter. From
Eq. (18), we see that the DGT can also be used to detect
the presence of process leaks. However, in this article,
we restrict our attention to situations where there are
no process leaks.
Note that when Li =0,
¦Ri =Fœi = − œW,i +MœQ,i +œW,i − 1,
(24)
by Eqs. (5), (16) and (18). Thus, it appears that, under
dynamic conditions, this test can be used to determine
the time measurements become biased. Also, Eq. (24)
indicates that it appears possible to distinguish biases in
W from Q. This distinction appears to be possible
because œW,i can cancel out of Eq. 52, since − œW,i +
œW,i − 1 can be zero if œW remains constant over two
consecutive time instants. Also, note from Eq. (24) that
1205
while the procedure address biases in only flow and
inventory variables (and when these two types of biases
do not occur simultaneously) there is no limit to the
number of biases that can be detected of each type. To
our knowledge, a dynamic condition GT has not been
presented before this work. Furthermore, an additional
significant contribution of this DGT is that it has the
potential to indicate the type of biased variables: flow
or inventory.
We exploited the canceling attribute of Eq. (24) and
developed the following procedure to differentiate between a nonzero œW and a nonzero œQ (this study
assumes that only one measured variable can become
biased at any time instant). This procedure tests the
null hypothesis (at each time instant) in a moving
‘window’ of m successive time instants, starting with the
first m time instants. That is, each window is size m and
contains the results of hypothesis tests of the past m−1
time instants and the current one. If the null hypothesis
is rejected at least twice in each of q consecutive windows, the conclusion is that there is a bias in a flow
measurement. The estimated time of occurrence
(ETOC) is concluded to be the time instant of the first
rejection in the first of the q windows meeting this
condition of the conclusion of having a flow bias. On
the other hand, if Ho is rejected only once in two
consecutive time windows, then the conclusion is a bias
in a level measured variable. If no rejection of Ho
occurs, then the conclusion is no bias in any measured
variable. For each analysis, the algorithm first tests for
flow biases, if none are found, it evaluates the presence
of a level bias.
From preliminary studies, we selected m=4 and
q= 5 (not shown for space considerations), and held
these values constant throughout this simulation study.
In these preliminary studies the parameters m, q, and
the number of rejections required within a window were
all varied in a systematic manner. Consequently, the
effect of the changes on the ability of the DGTS to
detect the bias type and estimate the TOC was noted.
Thus, these are the parameter values for which (for the
given process) the performance appeared to be the best.
In Appendix B, we have explained how an investigator
may determine the parameter values for his/her problem. To illustrate this approach (with m= 4 and q=5),
consider the hypothetical example given in Table 1.
First, suppose that a bias occurred in a flow variable at
TTOC = 4. Since there are two or more rejections in
each window from the time instant 5 onwards, the
conclusion is that a flow variable is biased and that it
occurred at time instant 4 (since this is the time instant
that the first detection occurred in the windows from
time instants 5 to 9). Hence, for this case, the ETOC
and the TTOC agree. Now consider a second case.
Suppose there is a rejection of Ho at time instant 4
(denoted by R*) and that none of the windows had two
D.K. Rollins et al. / Computers and Chemical Engineering 26 (2002) 1201–1211
1206
or more rejections; the conclusion for this case would
be that one of the level variables is biased and that the
bias occurred at time instant 4. Thus, as in the first
case, ETOC and TTOC are the same.
This study assumes that once a measurement bias has
occurred. It will continue to exist (with the same magnitude) until it is identified and removed. Thus, this
procedure may distinguish biases in inventory variables
(i.e. in W’s) from biases in flow variables (i.e. in Q’s),
since œW,i can cancel out of Eq. (24) after one time
instant has elapsed. Therefore, allowing for incorrect
conclusions by hypothesis testing, the scheme assumed
that two rejections (of Ho) separated by several time
instants indicate a bias in W, and that, several rejections close together (in time) indicate a bias in Q. We
will refer to this technique and its use of our DGT as
the DGTS.
5. Performance studies
This section examines the accuracy of the DGTS in
determining the type of biased measurement and the
TOC. Below we define measures of performance in the
simulation study.
The performance measure for false conclusion for
unbiased variables is the average type I error (AVTI). It
is defined as,
Table 1
Example illustrating how the TOC is determined
Time instant
Result in each
window
Conclusion
1
2
3
N/A
N/A
N/A
None
None
None
4
5
6
7
8
9
(A
(A
(A
(R
(R
(R
10
A
A
R
R
R
R
A
R
R
A
R
R
R*)
R)
A)
R)
R)
A)
(R R A R)
Continue
Continue
Continue
Continue
Continue
Bias in flow (TOC=4)
Stop
This table is used to illustrate how the ETOC is determined for biased
flow and inventory variables. First, a flow bias occurs at time instant
4, note that there are two or more rejections of Ho in each of 5 ( = q)
successive windows starting at the window at time instant 5. Thus, the
estimated TOC is the time instant of the first rejection in the first
window that had two or more rejections, which is time instant 4. The
second case is for a biased level variable. When the DGTS finds that
the criterion for a biased flow is not met, it will mark the first time Ho
is rejected as the estimated TOC. For example, suppose that the first
rejection occurred at time instant 4 (denoted by R*) and that none of
the windows had two or more rejections. Then the conclusion would
be that a level bias occurred at time instant 4. Here, A represent the
case when the hypothesis that œW and œQ are equal to zero is not
rejected; and R denotes the case when the hypothesis that œW and œQ
are equal to zero is rejected.
AVTI
=
c of trials identifying the wrong type of bias
.
c of simulation trials
(25)
One simulation run1 consisted of 1000 simulation trials,
where each trial consisted of a complete set of artificially (stochastically) generated data from the same set
of conditions (1000 trial runs were chosen because we
found this size to be sufficiently large enough to give
exceptional accuracy). In this study, either a flow or
level measurement becomes biased at the same time
during each simulation trial. For each trial, one of the
following three conclusions were made: a level bias was
present, a flow bias was present, or neither a flow nor
level bias was present. Thus, the AVTI can range from
0 to 1.
The performance measure for correct detection of
biased variables is the overall power (OP) and is defined
as,
OP =
c of biased variables correctly identified
.
c of biased variables simulated
(26)
Note that, since either a level or flow variable will be
biased for 1000 trials in each run, when the type for the
run is always identified as either flow or level bias, OP
and AVTI will add to one for that run.
The third and fourth quantities that we determine in
this study give a measure of DGTS’s ability to estimate
accurately the TOC of the bias. The mean difference
between the estimated TOC and the true TOC is determined by the simulation study. This quantity is called
the average difference (AVGD)2 and is given by,
%pi = 1 (ETOCi − TTOC)
AVGD =
p
,
(27)
where the ETOCi ’s are the estimated TOC’s, and p is
the number of trials in which the type of bias has been
identified correctly. The standard error of AVGD is
called STDD and is determined by,
1
For example, one such trial could be generating values for the
measured variables under the following conditions: (1) flow measurement bias; (2) h =0; (3) sample size of one; (4) bias located in stream
c2; (5) magnitude of bias = 6. These values of the measured variables are generated 1000 times under these exact conditions and that
constitutes one run.
2
The AVGD may be positive or negative. Suppose TTOC = 45
and ETOC= 35. A large number of such values for TTOC would
lead to a negative AVGD. This does not mean that the GTS predicts
in advance the occurrence of the bias. This simply means that the
GTS leads us to believe that the bias occurred at time instant 35,
whereas, the bias actually occurred at time instant 45.
D.K. Rollins et al. / Computers and Chemical Engineering 26 (2002) 1201–1211
Fig. 1. Process network.
Table 2
Effect of œQ (flow measurement is biased); h= 0.1, N = 1, location =
2
œQ
TTOC
AVTI
OP
AVGD
STDD
6
8
10
5
5
5
0
0
0
1
1
1
−0.14
−0.36
−0.39
1.24
0.97
0.94
Table 3
Effect of œW (level variable is biased); h= 0.1, N= 1, location = 2
œW
TTOC
AVTI
OP
AVGD
STDD
6
8
10
5
5
5
0.07
0.07
0.07
0.93
0.93
0.93
2.95
0.30
−0.39
6.45
4.04
2.94
Table 4
Effect of measurement bias location (œQ, flow measurement biases),
N= 1, TTOC=5
œQ
Location
h
AVTI
OP
AVGD
STDD
6
6
6
6
1
2
3
4
0.10
0.10
0.10
0.10
0
0
0
0
1
1
1
1
−0.32
−0.14
−0.16
−0.19
1.93
1.24
1.23
1.13
6
6
6
6
5
6
7
8
0.10
0.10
0.10
0.10
0
0
0
0
1
1
1
1
−0.51
−0.21
−0.17
−1.12
2.05
1.14
1.17
2.52
10
10
10
10
1
2
3
4
0.05
0.05
0.05
0.05
0
0
0
0
1
1
1
1
−0.14
−0.17
−0.19
−0.20
0.61
0.56
0.56
0.62
10
10
10
10
5
6
7
8
0.05
0.05
0.05
0.05
0
0
0
0
1
1
1
1
−0.17
−0.21
−0.21
−0.13
0.65
0.66
0.66
0.73
STDD =
D
.
mined, consider the following hypothetical examples
with conditions:
“ number of simulation trials is 100;
“ type of bias simulated is flow.
Suppose that these 100 simulation trials resulted in the
following conclusions:
“ 80 trials identified as flow bias;
“ 15 trials identified as level bias;
“ five trials with no bias detected.
Thus, for this example, the
“ AVTI = 15/100 = 0.15; and the
“ OP = 80/100 =0.80.
We now discuss the study to evaluate the DGTS. With
the conditions for each run held constant from run to
run; this study created data by varying the following
conditions: (1) the type of bias; (2) the magnitude of the
bias; (3) the TTOC; (4) the bias location; (5) the level of
significance (h); and (6) the sample size (N). (As stated
previously each simulation run consisted of an evaluation
of 1000 trials of simulated data.) This study used true
values from Darouach and Zasadzinski (1991) for the
inventory (i.e. level) variables and set all the measurement variances for W’s and Q’s to 1.0. Note that
increasing N has the same effect as decreasing the
measurement variance. Finally, it examined flow and
level measurement biases separately and determined
AVTI, OP, AVGD and STDD for each run. Fig. 1 shows
the process network used in the study taken from
Darouach and Zasadzinski, and used by Rollins and
Devanathan (1993) in a previous study.
6. Results of the simulation study
%pi = 1 (ETOCi −TTOC)2
p
1207
(28)
As discussed earlier using the example in Table 1, the
best value of AVGD and STDD is 0.0 for both flow and
level bias, i.e. when ETOCi =TTOC for all i.
To illustrate how the AVTI and the OP are deter-
This section presents results of the study to evaluate
the DGTS. Tables 2 and 3 contain the results for varying
measurement bias in a flow and level variable, respectively. Three values for œW and œQ were used: 6, 8 and
10. Table 2 shows that AVGD is close to 0.0 and that
STDD decreases with increasing œQ as expected. Table
3, in case with varying œW, shows similar performance
characteristics. Specifically, both AVGD and STDD
decrease with increasing œW. Comparing Tables 2 and 3,
one sees that the cases with nonzero œQ perform better
(compare the STDD’s) than the cases with nonzero œW.
Next, the effect of bias location is investigated. Tables
4 and 5 show the effect of measurement bias location for
each variable. When œQ = 6, Table 4 reveals, STDD is
higher when variable 1, 5 or 8 is biased; all three of these
streams are associated with only one node, as opposed
to the other streams which are associated with two nodes.
In contrast, for all other cases in Tables 4 and 5, the
location does not seem to cause a significant difference
in performance.
The effect of TTOC is best revealed in Tables 6 and
7. As shown, both STDD and the absolute value of
D.K. Rollins et al. / Computers and Chemical Engineering 26 (2002) 1201–1211
1208
AVGD increase as TTOC increases. In other words, the
ETOC’s are closer to TTOC when TTOC is relatively
small. Thus, it appears that the accuracy of the ETOC
depends on the number of observations collected. The
more data the analysis used, the more accurate the
ETOC.
Table 5
Effect of measurement bias location (œW, level measurement biases),
N= 1, TTOC= 5
œW
Location
h
AVTI
OP
6
6
1
2
0.10
0.10
0.07
0.07
0.92
0.93
1.62
1.95
6.14
6.45
6
6
3
4
0.10
0.10
0.07
0.07
0.93
0.93
1.34
1.44
5.70
5.80
10
10
1
2
0.10
0.10
0.08
0.07
0.92
0.93
−0.44
−0.39
1.32
1.94
10
10
3
4
0.10
0.10
0.08
0.08
0.92
0.92
−0.49
−0.46
0.97
1.27
AVGD
STDD
Table 6
Effect of measurement bias TTOC (œQ, flow measurement biases),
h=0.1, N =1, location =2
œQ
TTOC
AVTI
OP
AVGD
STDD
6
6
6
5
25
45
0.0
0.0
0.2
1.0
1.0
0.8
−0.14
−2.22
−5.32
1.24
7.24
14.46
10
10
10
5
25
45
0.0
0.0
0.0
1.0
1.0
1.0
−0.39
−2.05
−4.32
0.94
6.50
12.88
Table 7
Effect of measurement bias TTOC (œW, level measurement biases),
h= 0.1, N =1, location= 2
œW
TTOC
AVTI
OP
AVGD
STDD
6
6
6
5
25
45
0.07
0.08
0.07
0.93
0.92
0.93
1.93
−14.29
−33.81
6.45
9.14
13.21
10
10
10
5
25
45
0.08
0.08
0.07
0.92
0.92
0.93
−0.39
−14.54
−33.48
1.94
8.66
13.03
Table 8
Effect of h-level (œQ, flow measurement biases), TTOC = 5, N =1,
location= 2
œQ
h
AVTI
OP
AVGD
STDD
6
6
0.10
0.05
0
0
1
1
−0.14
−0.04
1.24
1.47
10
10
0.10
0.05
0
0
1
1
−0.39
−0.17
0.94
0.56
Table 9
Effect of h-level (œW, level measurement biases), TTOC = 5, N =1,
location =2
œW
h
AVTI
OP
6
6
0.10
0.05
0.06
0.08
0.93
0.95
1.95
5.30
6.45
10.45
10
10
0.10
0.05
0.07
0.01
0.93
0.98
−0.39
0.21
1.95
3.53
AVGD
STDD
Table 10
Effect of N (œQ, flow measurement biases), h = 0.1, TTOC= 5,
location =2
œQ
N
AVTI
OP
AVGD
STDD
6
6
6
1
5
10
0
0
0
1
1
1
−0.20
−0.13
−0.05
1.30
0.38
0.23
10
10
10
1
5
10
0
0
0
1
1
1
−0.46
0.00
0.00
1.06
0.00
0.00
Tables 8 and 9 show the effect of h on AVTI, OP,
AVGD, and STDD. When flow variables are biased, as
h decreases, AVGD and STDD decrease. The result
appears reasonable because, for a lower value of h, there
is a smaller probability of misidentifying situations with
no biased measurement. However, when the magnitude
of the bias is smaller (œQ = 6) and thus, more difficult to
detect, larger values for h give better AVGD and STDD
performance as supported by Table 8.
For a fixed bias in a level measurement (Table 9),
AVTI appears to be unaffected or weakly affected by
small values of œW and more affected by larger values of
œW. The large value affect appears to be valid since
decreasing h decreases the probability of two (or more)
false rejections (of Ho) occurring close together in time.
In addition, Table 9 shows that the ETOC appears to be
more variable (as related to the TTOC) for smaller h.
The final effect that this study addressed was the
sample size (N). The N measurements for each time
instant for each variable were averaged. As stated
earlier, increasing N has the same effect as lowering
sampling variance. The results are given in Tables 10
and 11. Table 10 shows that, when a flow bias is
present, larger sample size gives a better AVGD and a
smaller STDD, as expected. When a level bias is
present, Table 11 shows a quick approach to high
power (OP=1.0) and to an accurate ETOC (i.e. small
AVGD and STDD) as N increases. Thus, this analysis
demonstrates that the technique can be very accurate
for both types of variables when measurement error
variances are low or N is sufficiently large. Additionally, the analysis presents evidence to indicate that the
estimators used are reasonable ones (effect of large
sample size). Based on this study, the DGTS appears to
D.K. Rollins et al. / Computers and Chemical Engineering 26 (2002) 1201–1211
be an effective way to detect the presence of measurement biases and to distinguish level biases from flow
biases for dynamic processes. Performance improves
with large l, larger N, and longer time periods after the
bias has occurred.
1209
To illustrate the derivation of the linear dynamic
model, consider the simple process given above. A
material balance around node 1 (i.e. tank 1) gives the
following equation:
q1*+ q2*− q3*=
Dw*
dw*
1
1
=
.
dt
Dt
(A1)
With Dt = 1, and Dw1 = w1,i
* − w*
1,i − 1, we get two balance equations corresponding to the two nodes:
7. Closing remarks
In this article, the DGTS, a technique for GED for
dynamic processes, was developed for the case when
flow and inventory measurement biases are present.
The technique appears to have much promise in accurately detecting biased measurements, distinguishing biased flow and inventory variables, and determining the
time variables become biased.
Future challenges could be extending this approach
to identification of specific nodes and variables that are
biased and to situations where constraints are nonlinear. We are currently evaluating these conditions in
research. In addition, we are also studying ways to
accurately estimate process variables once the biases
have been identified since an important task is DR.
− w*
1,i +w*
1,i − 1 + q*
1,i +q*
2,i − q*
3,i = 0,
−w2,i
* +w*
* +q4,i
* − q5,i
* = 0,
2,i − 1 + q3,i
where w*
j,i represents the true value of the total mass in
tank j at time instant i, and w*
j,i represents the true value
of the mass flow rate in stream j at time instant i. In the
presence of process leaks, the above equations can be
modified as,
− w1,i
* +w*
* +q2,i
* − q3,i
* = u1,i,
1,i − 1 + q1,i
− w*
2,i +w*
2,i − 1 + q*
3,i +q*
4,i − q*
5,i = u2,i,
(A3)
where uj,i represents the magnitude of the leak in node
j at time instant i.
Then, with
8. Uncited reference
W*=
i
Scheffe, 1959
(A2)
Æ q*
1,i Ç
à q* Ã
à 2,i Ã
Qi*= Ã q3,i
* Ã,
à q* Ã
à 4,i Ã
È q*
5,i É
n
w1,i
*
,
w*
2,i
Acknowledgements
Li =
n
u1,i
,
u2,i
(A4)
and with
We are grateful to the National Science Foundation
for partial support of this research under Grant No.
CTS-9310095.
M=
1
0
1
0
−1
1
0
1
0
−1
n
(A5)
we have
Appendix A
− Wi*+ W*
i − 1 +MQi*= Li
= − [I2 × 2 − M2 × 5]
n
n
Wi*
W*
i−1
+ [I2 × 202 × 5]
.
Q*
Q
*
i
i−1
(A6)
Now, using the notation,
Table 11
Effect of N (œW, level measurement biases), h= 0.1, TTOC= 5,
location= 2
D= [I − M],
œW
N
AVTI
OP
STDD
Eq. (A5) can be written as,
6
6
6
1
5
10
0.07
0.00
0.00
0.93
0.51
0.41
2.09
0.00
0.00
6.91
0.00
0.00
− DX*+
BX*
i
i − 1 =[D B]
10
10
10
1
5
10
0.08
0.00
0.00
0.92
1.00
1.00
−0.32
0.00
0.00
2.07
0.00
0.00
= FXi*= Li,
AVGD
where
B= [I 0],
X*=
i
n
Wi*
.
Q*
i
(A7)
n
Xi*
X*
i−1
(A8)
n
D.K. Rollins et al. / Computers and Chemical Engineering 26 (2002) 1201–1211
1210
F = [− D B],
Xi* =
X*i
.
Xi*− 1
(A9)
Appendix B
The following illustrates how the relationship between the parameters (window size m, number of rejections in each window t, and the number of windows q)
and the performance of the DGTS can be obtained. We
show the derivation for the case of t = 2. Thus, the
DGTS concludes that there is a bias in one or more
flow variables when (call this event Z) there are at least
two rejections (of Ho: ¦Ri =0 in favor of Ha: ¦Ri " 0) in
each of q consecutive windows of size m. Let Ak = {at
least two rejections of Ho in window k, k= 1, …, q}.
Let Bi,k = {no rejection of Ho at any time within window i }, B2,k = {exactly one rejection in window i },
Cj = {no rejection at time j, where j = 1, …, m}. Note
that Cj is independent of Ck, for all j" k; j, k=
1, …, m}. Additionally, we use the convention notation
of h =Ô[Type I error] and i =Ô[Type II error], Thus,
Ô[Cj Ho is true] = 1 −h, and Ô[Cj Ho is false]= i.
Case 1 (Truth: No bias in any of the flow 6ariables). The
probability of concluding that there is at least one flow
variable that is biased is given by Ô[Z] =
Ô[A1 S A2 S···S Aq ]. We know that P[Ak ]= 1 − P[A c3].
Here, P[A ck]= P[B1,k @ B2,k ]= P[B1,k ]+ P[B2,k ] (since
B1,k and B2,k are mutually exclusive events). Now, note
that,
Ô[B1,k ]=Ô[C1 S C2 S ···S Cm ]= Ô[C1]Ô[C2]…Ô[Cm ]
= (1 − h)m.
(B1)
Now, B2,k can occur in ‘m’ ways because the one
rejection can be in any one of the ‘m’ slots (time
instants) in the ‘k’th window. Thus,
Ô[B2,k ]=mh(1−h)m − 1.
Ô[B1,k ]= Ô[C1 SC2 S···SCm ]
= Ô[C1]Ô[C2]…Ô[Cm ]= i m,
Ô[B2,k ]= m(1− i)i m − 1,
(B5)
and
P[A c3]= i m + m(1− i)i m − 1.
Therefore, using Bonferroni’s inequality we get,
Ô[Z] ]1− Ô[A1]+ Ô[A2]+ ···+ Ô[Aq ]
= 1− q{i m + m(1− i)i m − 1}.
(B6)
The relationship between i and the magnitude of the
bias (l) is given by
1− i= P[RTi SR−i 1Ri ]  2n,h¦Ri ]
= P[noncentral  2n ]  2n,hJ2i ],
(B7)
with
J2i = ¦ TRiSR−i 1¦Ri,
(B8)
where Ji is called the noncentrality parameter and ¦Ri
and l are related as shown in Eq. (24). Thus, it is
possible to study the effect of varying parameters t, q, m
and l on the performance of the DGTS. However, for
any given process we suggest, if possible, that the
investigator determine the ‘best’ values of these parameters by simulation study as that would be simpler. This
is as what we have done as described in the body of this
paper.
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(B3)
A1, A2, …, Aq are not independent events and so for
convenience we use Bonferroni’s inequality (see Bain &
Engelhardt, 1992) to get,
Ô[Z]]1 −{Ô[A1]+Ô[A2]+ ···+Ô[Aq ]}
=1− q{(1−h)m + mh(1 − h)m − 1}.
Ô[Cj j] g]= i,
(B2)
Hence,
P[A c3]= (1− h)m +mh(1 −h)m − 1.
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(B4)
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