A method to remove chattering alarms using median ﬁlters

Research article
A method to remove chattering alarms using median filters
Yongkui Sun a, c
, Wen Tan b, *
, Tongwen Chen c
a
School of Electrical Engineering, Southwest Jiaotong University, Chengdu, Sichuan, PR China
b
School of Control & Computer Engineering, North China Electric Power University, Beijing, PR China
c
Department of Electrical & Computer Engineering, University of Alberta, Edmonton, Alberta, Canada
a r t i c l e i n f o
Article history:
Received 31 March 2016
Received in revised form
17 May 2017
Accepted 11 December 2017
Available online 27 December 2017
Keywords:
Chattering alarms
Median filters
Alarm systems
a b s t r a c t
Chattering alarms are the most found nuisance alarms that will probably reduce the usability and result
in a confidence crisis of alarm systems for industrial plants. This paper addresses the chattering alarm
reduction using median filters. Two rules are formulated to design the window size of median filters. If
the alarm probability is estimated using process data, one rule is based on the probability of alarms to
satisfy some requirements on the false alarm rate, or missed alarm rate. If there are only historical alarm
data available, the other rule is based on percentage reduction of chattering alarms using alarm duration
distribution. Experimental results for industrial cases testify that the proposed method is effective.
© 2017 ISA. Published by Elsevier Ltd. All rights reserved.
1. Introduction
Alarm systems form the essential part of the operator interfaces
in large modern industrial facilities such as chemical plants, power
stations and oil refineries. An alarm indicates potential problems,
and directs the operator's attention towards plant conditions
requiring immediate assessment or corrective action. Therefore,
alarm systems play an important role in preventing, controlling and
mitigating the effects of abnormal situations [1,2].
According to EEMUA [1], every alarm presented to an operator
should be useful and an operator should not receive more than six
alarms per hour during normal operation of a plant. However, this
is the rare case in practice. Alarm variables, on the one hand, are
very easily implemented in modern industrial plants that are
usually equipped with computerized monitoring systems such as
distributed control systems and supervisory control and data
acquisition systems. Generally the more alarm variables are
configured, the more safety is deemed to be improved. Therefore,
the number of configured alarm variables has increased exponen-
tially [3], and there are tens, hundreds or even thousands of alarms
raised per hour. As a result, many existing industrial alarm systems
have far too many alarms to be handled by plant operators. Per-
formance metrics of existing industrial alarm systems are
summarized in Table 1 [4], which are based on a study of 39 in-
dustrial plants ranging from oil and gas, petrochemical, power and
other industries. The benchmarks in EEMUA [1] are provided for
reference. Obviously, the actual alarm rates in the various in-
dustries exceed the recommended number of alarms in the EEMUA
[1] benchmarks.
Alarms can be classified into nuisance alarms and correct
alarms. A nuisance alarm is one that does not require a specific
action or response from operators [1,4]. An alternative definition of
a nuisance alarm is an alarm that annunciates excessively, unnec-
essarily, or does not return to normal after the correct response is
taken [2]. Hence, nuisance alarms are the ones that do not affect the
process, even if these alarms are ignored by operators. A correct
alarm is the one that requires operators to pay attention or to take
action in prompt manner [4].
However, according to industrial surveys provided by Rothen-
berg [4] and Bransby and Jenkinson [5], a majority of alarms that
operators of industrial plants received are nuisance alarms.
Therefore, nuisance alarms considerably weaken the usability of
alarm systems, and often lead to a confidence crisis of alarm
systems.
Chattering alarms, which are caused by noise or disturbance, are
the most found nuisance alarms and may account for 70% of alarm
occurrences [3]. A chattering alarm is an alarm that is activated and
cleared many times within a short time period. Rothenberg [4]
defines chattering alarms as the ones that activate ten or more
times within 1 min, and alarms repeating more than three or more
times in 1 min are chattering alarms in the ISA standard [2].
* Corresponding author.
E-mail addresses: yksun@swjtu.edu.cn (Y. Sun), wtan@ncepu.edu.cn (W. Tan),
tchen@ualberta.ca (T. Chen).
Contents lists available at ScienceDirect
ISA Transactions
journal homepage: www.elsevier.com/locate/isatrans
https://doi.org/10.1016/j.isatra.2017.12.012
0019-0578/© 2017 ISA. Published by Elsevier Ltd. All rights reserved.
ISA Transactions 73 (2018) 201e207

Because different types of process variables have different time
scales in terms of variation dynamics, there is not a unanimous
definition for chattering alarms in the literature. In this context,
considering the sampling period of process variables and general-
izing the rule of thumb from the ISA standard [2], for a process
variable with the sampling period T, chattering alarms are defined
as alarms that repeat more than three times in the duration of sixty
times of the sampling period T.
Clearly, eliminating the chattering alarms can improve the ef-
ficiency of an alarm system. Consequently, how to reduce chatter-
ing alarms has recently been receiving much attention among
engineers and researchers. EEMUA [1] and ISA [2] suggest that
some methods be used in practice, such as, filters, deadbands, delay
timers, and shelving, to remove chattering alarms. Izadi et al. [6]
presented a procedure based on the Receiver Operating Charac-
teristic (ROC) framework to design optimal filters, deadbands, delay
timers by considering two major trade-offs between the false alarm
rate and missed alarm rate, and between latency and accuracy.
Hugo [7] proposed an adaptive alarm deaband method to reduce
the number of chattering alarms via an ARIMA model based on
process variable measurement. Naghoosi et al. [8] studied the
relation between alarm deabands and optimal alarm limits, and
estimated the optimal threshold with respect to the deadband and
history of the process variable. In order to quantify the degree of
chattering, Kondaveeti et al. [9,10] defined a chattering index based
on the run lengths of alarms, and Naghoosi et al. [11] developed a
method to estimate the chattering index based on the statistical
properties of the process variable as well as alarm parameters, and
derived an alarm chattering formula for alarm deadbands to aid in
optimal design. Wang and Chen [12] revised the chattering index
and proposed an online method to detect the repeating alarms due
to oscillation and exploited two mechanisms to reduce the number
of chattering alarms. To overcome the drawbacks of the above-
mentioned chattering indexes, Wang and Chen [13] formulated
two rules to detect chattering and repeating alarms using alarm
durations and intervals, and proposed an online approach to
remove chattering alarms, using three performance indices,
namely, the false alarm rate (FAR), missed alarm rate (MAR) and
averaged alarm delay. An efficient design strategy was provided to
select an appropriate delay timer using the historical alarm data
[14].
Median filters are nonlinear digital filtering techniques and have
been widely used to remove noises in the field of signal processing
[15]. Median filters outperform linear filters in environments
where the assumed statistics deviate from Gaussian models and are
possibly contaminated with outliers [16]. A method to remove
chattering alarms using median filters is proposed in this paper.
Two rules to design the window size for median filters are
formulated. If the past samples of the process variable in the
normal and abnormal conditions are available, according to the
maximum probability of alarms in 60T (T is the sampling period), a
window size is chosen for a median filter to satisfy the re-
quirements on FAR, or MAR. If only alarm data is available, ac-
cording to the cumulative distribution function of alarm duration
distribution, a window size is chosen to satisfy the reduction per-
centage of alarms. The advantages of the proposed graphic design
method are its simplicity and ease in implementation, which are
desirable for field engineer.
The rest of this paper is arranged as follows: Section 2 in-
troduces some preliminaries for the paper. Section 3 gives a brief
introduction to median filters and discusses its performance. Sec-
tion 4 addresses the design steps of median filters using process
data, and presents the simulated results. Section 5 gives a design
method based on alarm data, and presents the design example.
Finally, some concluding remarks are given in Section 6.
2. Preliminaries
This section introduces some terminology used later in the
paper.
An abnormal situation is defined as an event disturbing a pro-
cess that requires the operators to intervene to supplement the
control system. This definition specifically is used to distinguish
among normal, abnormal and emergency situations from the
perspective of console operations [1,2].
Assume a process variable xðtÞ has a probability distribution
function (PDF) pðxÞ at normal situation, a PDF qðxÞ at abnormal
situation, and the high-alarm trip point xtp is shown in Fig. 1, then
the definition of the FAR and MAR are [6]:
FAR ¼
Z∞
xtp
pðxÞdx (1)
MAR ¼
Zxtp
À∞
qðxÞdx (2)
A chattering index j is an index to measure chattering alarms,
which is presented by Kondaveeti et al. [9,10] based on run length
distributions:
j ¼
X
r2N
Pr
1
r
(3)
where r is a run length, which is the time difference between two
consecutive alarms on the same tag. Pr represents the probability
and Pr ¼ nr
,
P
r2N
nr
; c r2N, where nrrepresents the alarm count
Table 1
Cross-industry activation study.
EEMUA Oil & Gas PetroChem Power Other
Average alarms per day 144 1200 1500 2000 900
Peak alarms per 10 min 10 50 100 65 35
Average alarms per 10 min 1 6 9 8 5
Fig. 1. Probability density functions of normal and abnormal data and high-alarm trip
point.
Y. Sun et al. / ISA Transactions 73 (2018) 201e207202

for any run length r.
The chattering index j of an alarm can be perceived as the mean
frequency of annunciation of that alarm. The higher the chatter
index is, the higher the amount of chattering alarms. According to
ISA standard [2], a reasonable cutoff on j to identify the worst
chattering alarms is jcutoff ¼ 0:05.
Suppose that the process variable xðtÞ is with high alarm trip
point xtp, then the alarm signal xaðtÞ is generated as [12].
xtp ¼
&
1; if xðtÞ ! xtp
0; if xðtÞ < xtp
(4)
The alarm duration, denoted as L, is the time duration of adja-
cent ‘1's for xaðtÞ in (4), that is
L ¼ t2 À t1 þ 1 (5)
where xaðt1 À 1Þ ¼ 0, xaðt2 þ 1Þ ¼ 0,
Pt2
t¼t1
xaðt2 þ 1Þ ¼ t2 À t1 þ 1,
for t2 > t1.
3. Median filters
3.1. Definition of median filters
For a set of n numbers fx1; x2; /; xng, if they are arranged in
order such that
xð1Þ xð2Þ/ xðnÞ (6)
Then xðiÞ is the ith-order statistic (i ¼ 1; /; n) of fx1; x2; /; xng.
The median of n numbers x1; x2; /; xn is the ðv þ 1Þth-order
statistic, where v ¼ ðn À 1Þ=2, if n is odd. It is denoted by
xðvþ1Þ ¼ mediannðx1; x2; /; xnÞ: (7)
For even n, v ¼ n=2or v ¼ n=2 À 1, the definition differs only
slightly, which will not be discussed in this paper.
Let fxigði ¼ 1; 2; /Þ be sample data of a process variable xðtÞ at
sample instant t ¼ 1T; 2T; /, where T is the sampling period. A
median filter is a discrete-time dynamic system with fxig and
output fyig, where
yi ¼ median
n
ðxiÀ2v; /; xiÀv; /; xiÞ; ði ¼ 1; 2; /Þ (8)
The median filter defined in (8) is slight different from the one in
the literature, e.g., [17]. Because (8) is causal and is easy to realize in
practice.
3.2. Performance of median filters
Consider a process signal xðtÞ with high-alarm trip point. That is,
an alarm raises if xðtÞ exceeds or equals xtp, and clears if xðtÞ is
smaller than xtp. Without loss of generality, the following analysis
assumes that the process signal xðtÞ ¼ x0ðtÞ þ nðtÞ, and where x0ðtÞ
is a deterministic signal, and x0ðtÞ is always less than xtp, nðtÞ is the
noise or disturbance signal, and nðtÞ is independent. There is only
nðtÞ to activate the alarms, thus, all the alarms are false alarms.
We suppose each of sample data is independent variable with
cumulative distribution function Fxð,Þ and xiði ¼ 1; 2; /Þ denotes
the sampling value of a process signal xðtÞ at time instant i, let
process signal xi be great than or equal to the trip point xtp with
probability p i.e., p ¼ 1 À FxðxtpÞ, and the median filter (8) is applied
to signal xi, then, there is no alarm for the output yi if and only if the
number of alarms within the window n is less than half of window
size v ¼ ðn À 1Þ=2, that is, less than or equal to v, i.e. the probability
of yi xtp has a binomial distribution.
P
À
yi xtp
Á
¼
Xv
i¼0

n
i

pi
ð1 À pÞnÀi
(9)
For simplicity, let Pðyi xtpÞbQðn; pÞ, the false alarms rate (FAR)
is given by
FAR ¼ 1 À Qðn; pÞ (10)
Eq. (10) relates FAR to the window size and the alarm probability
of the process signal xðtÞ, FAR for some different values of window
size n and alarm probability p are shown in Fig. 2.
It is observed from Fig. 2:
(1) FAR gradually decreases and tends to zero when window size
increases, if the process variable xðtÞ is in the alarm state
with probability p 0:5.
(2) FAR gradually increases and tends to one when window size
increases, if the process variable xðtÞ is in the alarm state
with probabilityp 0:5.
(3) FAR equals 0.5 regardless of window size if the process var-
iable xðtÞ is in the alarm state with probability p ¼ 0:5.
Remark 1. The median filter can be used to remove the chattering
alarm on condition that the process variable is in the alarm state
with probability p 0:5. For example p ¼ 0:3, window size n ¼ 15,
FAR ¼ 0.05.
Remark 2. According to Eq. (9), given an alarm duration length L,
this alarm can be removed if the window size n is chosen to be at
least twice of the alarm duration length L, for odd n, n ¼ 2L þ 1.
Remark 3. If the process variable xðtÞ is in the alarm state for a
majority of time, p 0:5, the trip point can be considered as a low-
alarm trip point, i.e., it is reasonable to accept that the process
variable should be in the alarm state. If the median filter is applied
to signal xðtÞ, the output value yðtÞ at time instant t will be in the
alarm state, if and only if the number of alarms within the window
centered at time instant is more than half of number of points in the
window, that is, more than v ¼ ðn À 1Þ=2, i.e. the probability of
yðtÞ xtp in the window is given by
Fig. 2. FAR for different p and window size n.
Y. Sun et al. / ISA Transactions 73 (2018) 201e207 203

P
À
yðtÞ xtp
Á
¼
Xn
i¼vþ1

n
i

pi
ð1 À pÞnÀi
(11)
Let PðyðtÞ xtpÞbQ
0
ðn; pÞ, the missed alarms rate (MAR) is given
by
MAR ¼ 1 À Q
0
ðn; pÞ (12)
where Q
0
ðn; pÞ ¼
Pn
i¼vþ1

n
i

pið1 À pÞnÀi
¼
Pv
i¼0

n
i

pnÀið1 À pÞi
.
4. Design of median filters using process data
This section will propose a design method for median filters to
reduce chattering alarms by using the process data, while the re-
quirements on FAR, or MAR are satisfied. The chattering index in
Section 2 is used to quantify chattering alarms. The following are
assumed to be hold:
A1. The past samples of the process variable xðtÞ in the normal and
abnormal conditions are available.
A2. The alarm signal is independently, identically distributed (IID)
and there exist chattering alarms.
A3. The alarm signal xaðtÞ is in the non-alarm state for the ma-
jority of time, i.e.,p 0:5
The past samples of xaðtÞ in assumption A1 are used to estimate
the probability of chattering alarms. Assumption A2 is a statistical
requirement for (9)e(10). Many cases satisfy this assumption in
practice, e.g., the process variable xðtÞ ¼ x0ðtÞ þ nðtÞ is disturbed by
IID noise nðtÞ, where x0ðtÞ is a deterministic signal. Assumption A3
can be verified because the alarm state is often less than normal
state in practice. If assumption A3 does not hold, the proposed
method requires minor modifications; see Remark 6 presented
later in this section.
The proposed method consists of five steps:
Step 1. Calculate the chattering index j of an alarm signal xaðtÞ.
Step 2. If the chattering index j is more than the cutoff threshold
0.05, estimate the maximum probability of alarm in sixty times
of sampling period. Otherwise, stop.
Step 3. Choose the minimum window size n to satisfy the re-
quirements on FAR according to (10).
Step 4. Filter the process variable using the median filter with
the chosen window size n and obtain the alarm signal x
0
aðtÞ of
the filtered process variable xðtÞ.
Step 5. Calculate the chattering index j of alarm signal x
0
aðtÞ, if
j 0:05, stop and n is window size of median filter, Otherwise,
increment n by 2 and turn to step 2.
There are three remarks for the above steps.
Remark 4. In step 2, we estimate the maximum probability of
alarms in sixty times sampling period. This choice is based on
chattering alarm defined in Section 1, which considers the sampling
period and generalizing the rule of thumb from ISA standard.
Remark 5. The proposed method needs step 5. (10) only relates
the window size of standard median filters to FAR. Since the
moving median filters are applied to the process variable, alarms
close to each other may still remain.
Remark 6. If assumption A3 does not hold, i.e. p 0:5, the alarm
signal xaðtÞ is in the alarm state for a majority of time, the proposed
method needs to be modified as follows: The complement of alarm
data xaðtÞ is used to calculate the chattering index in step 2, i.e. ‘1' of
xaðtÞ is substituted by ‘0’, and ‘0’ of xaðtÞ by ‘1’. The window size n to
be chosen is to satisfy the requirement on the MAR as in (12).
One simulation example and one industrial example are pre-
sented here to illustrate the proposed method.
Example 1. Consider the continuous stirred tank heater (CSTH)
with closed loop control [18]. The CSTH of configuration is shown in
Fig. 3 and the temperature in tank is the process variable xðtÞ, which
is measured by a type J metal sheathed thermocouple. The simu-
lation data N ¼ 10000.
Let the temperature xðtÞ be disturbed by a random noise with
uniform distribution in ½0:1; 0:2Š, and the alarm probability of xðtÞ
equal to p ¼ f0:1; 0:2; 0:3g, that is the alarm is triggered by noise.
Fig. 4(a)-(c) show the temperature xðtÞ, the alarm xaðtÞ corre-
sponding to high-alarm trip point xtp ¼ 10:6 and the run length
distribution of alarms for the normal situation. The chattering in-
dex is j ¼ 0:01, so the alarm is not chattering.
Fig. 5(a)-(c) present the run length distribution of alarm xaðtÞ for
the three probabilities. When p increases, there are more and more
alarms with short run lengths and the chattering indexes are
increasing, i.e. j1 ¼ 0:15, j2 ¼ 0:23, j3 ¼ 0:26 for p1 ¼ 0:1,
p2 ¼ 0:2, p3 ¼ 0:3, respectively.
If the requirement on the FAR in step 3 is FAR 0.05, according
to (10), the median filter window size N1 ¼ 3, N2 ¼ 7, N3 ¼ 17 are
chosen for p1 ¼ 0:1, p2 ¼ 0:2, p3 ¼ 0:3, respectively. For p3 ¼ 0:3,
window size N3 ¼ 21 is chosen instead of N3 ¼ 17, see Remark 5 in
this section. The alarm x
0
aðtÞ is generated by the filtered data, and
Fig. 6(a)-(c) shows the run length distribution of alarm signal x
0
aðtÞ.
From Fig. 6, it is clear that there are a few alarms with short run
lengths, and the distribution is almost uniform for each existing run
lengths. The chattering indexes j
0
1 ¼ 0:04, j
0
2 ¼ 0:049, j
0
3 ¼ 0:037
are less than jcutoff ¼ 0:05.
For comparison, the chattering indexes for moving average fil-
ters using the same windows are given in Table 2. The indexes for
median filters are less than jcutoff ¼ 0:05, and the indexes for
moving average filters are more than jcutoff ¼ 0:05, that is, the
chattering alarms caused by noise are not completely removed.
Suppose the temperature xðtÞ is disturbed by an auto-correlated
noise, which is generated by the random noise passing through a
linear time invariant system s2
s2þ
ffiffiffi
2
p
sþ1
. Median filters are used and
the chattering indexes are in Table 3. Compared with the inde-
pendent noise, the chattering indexes caused by auto-correlation
noise are smaller except alarm probability p ¼ 0:1. For alarm
Fig. 3. The continuous stirred tank heater.

probability p ¼ f0:1; 0:2g, the chattering index for the filtered xðtÞ is
more than jcutoff ¼ 0:05, and the chattering alarms are totally
removed for the p ¼ 0:3. Fig. 7 and Fig. 8 show the run length
distribution for unfiltered and filtered xðtÞ, respectively. It is shown
that median filters can still reduce the chattering index under auto-
correlated noise.
Example 2. An industrial data was studied in Ref. [13]. The pro-
cess variable xðtÞ is a temperature of stator outlet pipes for a power
generator, which is measured by 54 temperature sensors. xtp ¼ 8
degree is a high-alarm trip point. Due to the existing noise, there
are many chattering alarms in the temperature alarm signal xaðtÞ.
Fig. 9 shows the process variable xðtÞ with 1155 alarms in 24h; and
the run length distribution is given in Fig. 9(c). There are many
alarms with short run length and the chattering index j ¼ 0:11.
The maximum alarm probability of temperature samples in sixty
times of sampling period is estimated as ~pmax ¼ 0:23, thus, N ¼ 7 is
the window size of median filter for stratifying FAR 0.05. The
result of filtered x
0
ðtÞ is shown in Fig. 10.
From Fig. 10(c), there are only 20 alarms left and the run length
distribution is uniform with one alarm count for each run length,
and the chattering index j
0
¼ 0:02.
5. Design of median filters using alarm data
Sometimes it is not possible to obtain enough process data to
estimate the alarm probability in industrial practice for the
following reasons: Firstly, the abnormal operation conditions in
reality are very rare; even if an alarm is raised, it is difficult to
distinguish if the alarm is caused by normal or abnormal
Fig. 4. (a) Temperature xðtÞ(solid) and alarm trip point xtp(dash), (b) alarm signal xaðtÞ,
(c) run length distribution for xaðtÞ.
Fig. 5. Run length distributions for xðtÞ with p ¼ 0:1 (a), p ¼ 0:2 (b), p ¼ 0:3 (c)
(unfiltered).
Fig. 6. Run length distributions for xðtÞ p ¼ 0:1 (a), p ¼ 0:2 (b), p ¼ 0:3 (c) (filtered).
Table 2
Chattering indices for different filters.
Alarm probability window size median moving average
p ¼ 0:1 3 0.04 0.06
p ¼ 0:2 7 0.049 0.089
p ¼ 0:3 21 0.037 0.1
Table 3
Chattering indexes for CSTH with auto correlation noise.
alarm probability unfiltered median
p ¼ 0:1 0.18 0.07 (window size ¼ 3)
p ¼ 0:2 0.19 0.07 (window size ¼ 7)
p ¼ 0:3 0.23 0 (window size ¼ 21)
Fig. 7. Run length distributions for xðtÞ with auto correlated noise p ¼ 0:1 (a), p ¼ 0:2
(b), p ¼ 0:3 (c) (unfiltered).

conditions. Secondly, due to limitation of storage media, the pro-
cess data is always stored based on some down sampling technique,
there are few process data at the original sampling frequency of the
controller. Furthermore, most process control systems only store
alarm event data. Finally, field engineers are more concerned with
the number reduction of chattering alarms, rather than its quan-
titative description. In addition, when the median filter is applied to
an alarm signal xaðtÞ, it is a type of generalized delay-timers with k
out of n delay-timer, this is, in the consecutive n samples, an alarm
is to be raised if and only if there are at least v þ 1 alarms, and an
alarm is cleared if and only if there are at most v alarms, where n is
the window size and n ¼ 2v þ 1. According to Remark 2, the alarm
with alarm duration length L can be removed if the window size of
the median filter is at least twice of the he alarm duration length L.
Therefore, the method to design the median filter using historical
alarm data in term of percentage reduction of chattering alarms is
proposed in this section.
The proposed method consists of the following steps:
Step 1. Identify chattering alarms of the historical alarm data.
Step 2. Plot the duration distribution of alarms for each alarm
tag.
Step 3. Plot the cumulative distribution function of the duration
distribution.
Step 4. According to a reduction percentage of alarms, find the
duration length L of the alarm. Then the minimum of window
size nmin is equal to 2L þ 1.
Step 5. Apply the median filter with nmin to the alarm data. If the
reduction percentage of alarms is not satisfied, increase the
window size, and return to step 5 again.
Two industrial examples are presented here to illustrate the
proposed method. Consider Example 2 in Section 4; the red dash
line in Fig. 11 shows the cumulative distribution function repre-
sented by the blue bars. The duration length of all the alarms is less
than or equal to 5 s. If we choose the window size nmin ¼ 11, all
these alarms can be removed. The duration length of over 97%
alarms is less than or equal to 2 s, i.e. we use the median filter with
wind size nmin ¼ 5, there are more than 95% alarms reduced. The
black line in Fig. 11 shows the percentage of reduction in chattering
alarms using the median filter with different window size.
Consider the industrial data with sampling period T ¼ 1 minute
shown in Fig. 12(a), which is used in Ref. [12], Fig. 12(b) shows the
run length distribution. It is shown that there exist many chattering
alarms.
In this case, the cumulative distribution function is represented
by the red dash line, and the percentage reduction in chattering
alarms by median filters with different window sizes is showed in
Fig. 8. Run length distributions for xðtÞ with auto-correlated noise p ¼ 0:1 (a), p ¼ 0:2
(b), p ¼ 0:3 (c) (filtered).
Fig. 9. (a) Process variable xðtÞ(solid) and alarm trip point xtp(dash), (b) alarm signal
xaðtÞ, (c) run length distribution for xaðtÞ.
Fig. 10. (a) Filtered process variable xðtÞ(solid) and alarm trippoint xtp(dash), (b) alarm
signal x
0
aðtÞ, (c) run length distribution for x
0
aðtÞ.
Fig. 11. Design of median filter using the alarm and duration distribution.

black line. More than 94% alarm repeated within 6T can be
considered as chattering alarms, using the black curve in Fig. 13, we
obtain easily that over 93% alarms are reduced using a median filter
with window size n ¼ 13.
6. Conclusion
This paper discussed the performance of median filters in alarm
rationalization and applied them to remove chattering alarms to
improve the performance of alarm systems. This work is mainly
based on the fact that chattering alarms repeatedly and rapidly
change between alarm and normal states in a short time period and
have short alarm duration lengths, and the median filter with
window size n(for odd n) can remove alarms whose duration
lengths are less than or equal to v ¼ ðn À 1Þ=2. Two design methods
of median filters were proposed and two rules for choosing the
window size of median filters are formulated in this paper. If the
past samples of process variable xðtÞ in the normal and abnormal
conditions are available, the maximum alarm probability in sixty
times sampling period can be used to choose the window size to
satisfy the FAR or MAR. Otherwise, if there is only alarm data
available, the alarm duration length distribution can be used to
determine the window size which satisfies a given percentage of
reduction of chattering alarms. The industrial case studies illus-
trated the effectiveness of the proposed method.
Acknowledgments
This work was partially supported by National Natural Science
Foundation of China under Grant 61733015, U1730105, 61433011,
61603316 and 61773323, China Scholarship Council and Natural
Sciences and Engineering Research Council of Canada.
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Fig. 12. (a) The process signal xðtÞ(solid) and its alarm trip point xtp(dash). (b) Run
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Fig. 13. Design of median filter using the alarms and duration distribution.

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