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Fault Isolability Analysis Based on Steady State Fault
Signatures
Ketan P. DetrojaDepartment of Electrical Engineering
Indian Institute of Technology Hyderabad
Yeddumailaram, [email protected]
Ravindra D. GudiDepartment of Chemical Engineering
Indian Institute of Technology Bombay
Mumbai, [email protected]
Abstract Once a fault has occurred, it is highly desirable
to
quickly detect and reliably isolate the fault. Reliable
fault
isolation is a key to ensure correct remedial action;
therefore
fault isolation module must determine the current fault
scenario
unambiguously and uniquely. However, it may be difficult to
differentiate between all possible fault situations due to
various
practical and physical limitations. The task of a fault
isolation
module is to generate a set of candidate faults, which might
have
occurred such that misclassification is minimized while no
candidate fault scenarios are missed. Fault isolation
schemes
essentially rely on fault signatures, either during transient or
at
steady state. Our main focus is on methods relying on steady
state
fault signatures, such as clustering based algorithms. These
fault
isolation schemes may sometimes misclassify if another
historical
event match the current fault scenario. It is therefore
necessary
to i) analyze the process, ii) analyze how different faults
manifest
themselves and iii) determine which faults can exhibit
similar
steady state fault signatures. Specifically, it may be very
important and useful to determine which faults can be
uniquely
isolated from given process input-output data. In this paper
we
propose a simple fault isolability analysis based on steady
state
process model. A steady state process model can be
obtainedeither based on first principles model or based on process
input-
output data. The proposed fault isolability analysis has
been
validated by obtaining simple steady state process model of
the
benchmark quadruple tank process for normal and fault
operation. Fault isolability predictions, obtained using the
proposed fault isolability analysis, are found to be
accurate.
Keywords-Fault isolability, steady state fault signature,
fault
diagnosis
I. INTRODUCTIONAny deviation from normal/ acceptable behavior of
a plant
is considered a fault situation. A plant operation can
deviatefrom normal/acceptable region due to numerous reasons,
suchas sensor biases, actuator biases, process disturbances,
etc.Further, these faults can manifest themselves with
differentintensities. Thus, Fault diagnosis essentially involves
threetasks, namely fault detection, fault isolation and
faultidentification[1-3]. The task of fault detection, determining
ifsomething is wrong with the plant, is a relatively simple task
ascompared to fault isolation, determining what exactly is
wrongwith the plant. In order to perform fault isolation, any
faultdiagnostic framework should be able hypothesis all
possiblefault scenarios and then determine the current plant
operation
status uniquely and unambiguously. It should however benoted
that if there is no observable difference between faultsignatures
of more than one fault, it may be difficult to isolatethem
irrespective of the algorithm used. Various fault isolationschemes
look at fault signatures from different perspective, i.e.steady
state fault signatures or transient fault signatures. While
diagnostic schemes based on each of them may have
theiradvantages and disadvantages, both are prevalent and
havegenerated considerable interest in academia and industry
alike.
The aim of this work is two folds: i) Generate a pool offaults
that are likely to result in similar fault signatures and ii)Assist
fault diagnosis schemes and/or operators in makinginformed decision
on possible root cause(s). This is the firststep towards hybrid
fault diagnostic framework that looks atsome a priori knowledge of
how different faults manifestthemselves and historical data based
approached for faultdiagnosis. Our main focus is on methods that
look at steadystate fault signatures for fault isolation, e.g.
Principlecomponents analysis (PCA), Fisher discriminant
analysis
(FDA), clustering based algorithms, etc. While these [4,
5]methods have been quite successful in both academia as well
asindustrial practices, it is important to note that faults
havingnon-unique signatures are always difficult to discriminate in
anunambiguous way. This will happen primarily because some
processes may exhibit similar system response under influenceof
different fault cases and there is no observable difference
between these fault signatures. Even for an ideal FDI method,it
may not be possible to always uniquely isolate such
faultsignatures. Therefore, it is important to find out which
faultscan be uniquely isolated. Although some work has been doneto
predict fault isolability [6, 7], such detailed analysis may
not
be required to determine fault isolability when historical
databased diagnostic schemes are employed. In this paper we
propose a simple process analysis method based on steady
statemodel is proposed. Although, developing detailed first
principles based model may be difficult for large-scale
systemsit is relatively easy to obtain steady state gains for
variousinput-output pairing, i.e. steady state model of the
process. Thissimple yet elegant approach can be very useful when
steadystate process model of the plant is available. Such
faultisolability analysis could be a prerequisite task before
whichfault diagnosis steps are carried out so as to highlight
potentially similar fault scenarios. Fault isolability analysis
willgroup a set of faults that have similar fault signatures.
Theoperators would therefore know that the set of faults
presented
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III. FAULT ISOLABILITY ANALYSIS FORQUADRUPLE TANKPROCESS
The Quadruple tank process is a multivariable laboratoryprocess
with an adjustable zero [8]. This process consists offour
interconnected water tanks, two pumps and associatedvalves. A
schematic diagram of the process is shown in Fig. 1.
The inputs are the currents supplied to the control valves
so
as to manipulate the flow output of the fixed speed pumps andthe
outputs are the water level measurements of the two bottomtanks.
The flow outlet of the control valve is split before it goes
to the individual tanks and 1
and 2
are the fraction of the
total flow going to the bottom tanks. The nonlinear modelderived
based on the first principles is given in (7).
dh1
dt=
a1
A1
2gh1
+
a3
A1
2gh3
+
1
A1
f1
dh2
dt=
a2
A2
2gh2
+
a4
A2
2gh4
+
2
A2
f2
dh3
dt=
a3
A3
2gh3
+
1 2( )
A3
f2
dh4
dt=
a4
A4
2gh4
+
1 1
( )A
4
f1
(7)
For tanki,Ai is the cross-section of the tank, ai is the
cross-section of the outlet hole and hi is the water level. The
currentapplied to the valve i, determines the total flow fi.
Theacceleration of gravity is denoted byg.
The proposed fault isolability analysis is validatedanalytically
as well as using experimental data. The analytical
validation is based on the first principles based model of
thequadruple tank process, while experiments were carried out onthe
quadruple tank process system available at Indian Instituteof
Technology Bombay.
As discussed earlier, fault isolability analysis can be
carriedout to check which fault scenarios would result in
differentsignatures. From the first principles model given in , the
steadystate model can be derived as
(8)
where,
Ti
=
Ai
ai
2hi
SS
gand
hi
SS
is steady state heightfori
thtank during normal operation.
The first principles model given in (8) is valid even when
asensor is biased, however the output equation will now be
y1
= h1
+ fsensor
1
y2
= h2
+ fsensor
2(9)
where,fsensor
1
andfsensor
2
are sensor bias magnitudes forbias in Y1 and Y2 respectively and
only one of them is non-zeroat a time (single fault
hypothesis).
Thus, the steady state model given in (8) will be changed tothe
following under sensor bias condition.
(10)
The first principles model for actuator bias condition can
bewritten as follows:
dh1
dt=
a1
A1
2gh1
+
a3
A1
2gh3
+
1
A1
f1
+ factuator
1( )
dh2
dt=
a2
A2
2gh2
+
a4
A2
2gh4
+
2
A2
f2
+factuator
2( )
dh3
dt
=
a3
A 3
2gh3
+
1 2
( )
A 3
f2
+factuator
2( )
dh4
dt=
a4
A4
2gh4
+
1 1
( )A
4
f1
+ factuator
1( )
(11)
where,factuator
1
andfactuator
2
are sensor bias magnitudes forbias in U1 and U2 respectively and
only one of them is non-zeroat a time (single fault
hypothesis).
Actuator bias affects the plant in a different way throughsteady
state process gain matrix. The steady state model foractuator bias
is given by
(12)
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It is clear from (10) and (12) that sensor and actuator
biasescan be easily discriminated in the original larger
dimensionalspace. Statistical method used for the data compression
andmodel building can be used to generate a lower
dimensionalsubspace where these fault signatures can be
discriminated.
We next consider disturbances in the process that can affectthe
steady state plant in many different ways. We will nowcheck if leak
in tank 1 can be isolated from sensor and actuator
biases. If we assume that the amount of leakage is equivalent
to
discharge from an orifice having cross sectional area ofl1,
the
first principles model given in (8) will become:
dh1
dt=
a1
+ l1
( )A
1
2gh1
+
a3
A1
2gh3
+
1
A1
f1
dh2
dt=
a2
A2
2gh2
+
a4
A2
2gh4
+
2
A2
f2
dh3
dt=
a3
A 3
2gh3
+
1 2
( )A 3
f2
dh4
dt=
a4
A4
2gh4
+
1 1
( )A
4
f1
(13)
The steady state model for leak in tank 1 can then bewritten as
given below:
(14)
where, Tl
=
A1
al
2h1
SS
g.
It is clear from (10) & (14) that leak in tank 1 is
equivalent
to a bias of 1T12
A1T
1+T
l( )
U1
ss+ 1
2( )T
1
2
A1T
1+T
l( )
U2
ssmagnitude in
sensor 1. For small magnitude leak in one of the lower tanks,the
effect of this fault on steady state plant is similar to that
ofsensor bias in one of the sensors. Thus, leak in the tank and
bias in sensor generate similar fault signatures and it will
bedifficult to isolate these faults for the quadruple tank process.
Itshould however be noted that both these faults have similarsteady
state fault signatures and it may be necessary to isolatethese
faults using some model based techniques that
analyzetransients.
IV. EXPERIMENTALVALIDATION OF FAULT ISOLABILITYANALYSIS
In the previous section, it was analytically shown thatsensor
and actuator biases could be easily discriminated for thequadruple
tank process. It was also found that somedisturbances, e.g. leak in
tank 1 and bias in sensor 1, couldresult in similar fault
signature. It has been shown that sensorand actuator biases for the
quadruple tank process could bediscriminated by CA based FDI scheme
proposed in [9]. Wewill therefore focus on the case when there is a
bias in sensor 1and leak in tank 1.
As stated earlier, experimental setup is available atAutomation
laboratory, in the Department of ChemicalEngineering of IIT Bombay.
The system has a multivariableadjustable zero and we selected the
operating conditions suchthat the system exhibited non-minimum
phase behavior. The
presence of right half plane multivariable zero andconsequently
inverse response makes it a difficult system tocontrol. The
experimental studies on the quadruple tank
process setup were conducted under closed-loop. Levels intank-1
and tank-2 were controlled by manipulating the control
valves, i.e. the process becomes a 2x2 system. The
operatingrange for levels in the bottom tanks is 0-55 cm and for
totalflow from the control valve is 20-250 cm
3/s. The setup has a
provision to introduce leak of different magnitudes in all
thetanks. The levels in tank-1 and tank-2 were controlled
bymanipulating flow rates f2 and f1 respectively. Single loop
PIcontrollers were implemented for both the loops with
K1,Ti1( ) = 8.2,0.0368( ) , K2,Ti2( ) = 0.5,0.0284( ) . The
experiments were carried out for normal operating condition
aswell as biases in each of the sensors and actuators and also
forleak in tank 1. Various sensor and actuator bias situations
werecreated by a computer program. For instance, for generating
asensor bias during online plant operation, a constant value
wasadded to the measured value that reaches to the computerthrough
data acquisition module. The setup has a provision tointroduce leak
of different magnitudes in all the tanks.Interfacing of the data
acquisition module (Advantech ADAM-5000) with the computer and
online controller implementationwas done using MATLAB
.
To validate the fault isolability predictions obtained usingthe
proposed method, we carried out the followingexperimental case
study. We first introduced a leak equivalentto an orifice of cross
sectional area 0.05 cm
2in tank 1. The
plant data corresponding to this leak fault was collected.
Nextwe calculated equivalent bias magnitude for a leak of
crosssectional area 0.05 cm
2. The calculations can be carried out
from (14) and we found that leak of cross sectional area
0.05cm
2is equivalent to a bias of -3.16652 cm in sensor 1. Thus,
we introduced a bias of -3.16652 cm in sensor 1 and
collecteddata for sensor bias. The system responses are compared
forleak in tank 1 and bias in sensor 1. As can be seen from Fig.
2and Fig. 3, the quadruple tank process exhibits similar
systemresponse under both the fault situations. Further, it is
clear that(see Fig. 4) although during transients input-output
signaturesare different; steady state fault signatures for these
faults areidentical. Hence, leak in tank 1 and bias in sensor 1
cannot beisolated uniquely if their magnitudes are such that they
result in
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similar system response. However, it must be noted that
ifdifferent magnitudes of these faults are chosen, it may result
indistinct system response and appear as distinct clusters.
Whenviewed from fault line perspective [5], leak in tank 1 and
biasin sensor 1 will be on the same fault line. Hence, out
analyticalcalculations and experimental results, both corroborate
the factthat leak in tank 1 and bias in sensor 1 are difficult to
isolate ifwe rely on historical data based methods.
500 1500 2500 3500 4500 5500120
140
160
180
200
220
Time
u1
500 1500 2500 3500 4500 5500160
170
180
190
200
210
Time
u2
Bias in Sensor 1Leak in Tank 1
Figure 2: Validation of the proposed fault isolability
analysis
500 1500 2500 3500 4500 550023
24
25
26
27
28
29
Time
y1
500 1500 2500 3500 4500 5500171819202122
232425
y2
Bias in Sensor 1Leak in Tank 1
Figure 3: Validation of the proposed fault isolability
analysis
500 1500 2500 3500 4500 5500-5-4-3-2-10123
Time
Dee
ny
500 1500 2500 3500 4500 5500-10
-505
1015202530
Time
Dee
nu
Difference in y1Difference in y2
Difference in u1Difference in u2
Figure 4: Difference in inputs (u) and outputs (y) for leak in
Tank 1 andbias in Sensor 1.
Thus, whether any disturbance or fault will generate adistinct
signature from sensor and actuator biases depends onthe process and
also the kind of disturbance introduced.Further, care must be taken
when fault isolation is performedand the proposed fault isolability
analysis can provide suchinformation a priori.
V. CONCLUSIONS AND FUTURE WORKWe have proposed a simple yet
elegant approach to fault
isolability study based on steady state process model. It
hasbeen shown that even for a nonlinear and highly
interactingsystem, the proposed fault isolability analysis can
correctly
predict if a fault will have distinct signature. In case of
thequadruple tank process, fault in one of the lower tanks
issimilar to corresponding sensor bias. This was
validatedanalytically as well as using experimental data.
Here we have proposed a steady state model based
faultisolability analysis method and validated it on the
benchmarkquadruple tank process. However, further work is needed
toextend the presented work to large-scale systems where first
principles based models are difficult to obtain. The main
task
there is to obtain steady state process model from
historicaldata. Extension of the proposed methodology to multiple
faultscenarios could also be a future area of research.
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