FACTA UNIVERSITATIS
Series:
Automatic Control and Robotics Vol. 9, No 1, 2010, pp. 79 - 86
FAULT DETECTION IN A THREE-TANK SYSTEM BASED ON
SEQUENTIAL HYPOTHESIS TESTING
UDC 621.642.3 519.87 53.088.7
Aleksandra Marjanović, Željko Đurović, Branko Kovačević
Department of Signals and systems, Faculty of Electrical Engineering, University of
Belgrade, Bul. Kralja Aleksandra 73, 11000 Belgrade, Serbia
E-mail: amarjanovic@etf.rs
Abstract.
The task of fault detection implies discovering and locating the failure in the
system. This type of autonomous fault diagnostics reduces further damage and also saves
time and cost in repairing the system. This paper presents an online way of retrieving a
leak in a Three-tank system. The method being used is the Wald’s sequential hypothesis
testing. This is a model based technique that includes residual generation and evaluation.
Some conventional fault detection methods have problems with the number of “false
alarm“ decisions, which is reduced using the proposed technique.
Key words:
fault detection and isolation, sequential hypothesis testing, residual,
model-based method, three-tank system
1. INTRODUCTION
A fault is any kind of unexpected behavior in the system that is an exception to the
regular behavior. In some cases the consequences can be catastrophic. As one of the
priorities in industry is the safety of the plants and people, error detection and diagnostics
are of great importance. The goal is to detect the malfunctions and to locate the broken
components in the system. For a long time the idea was to have a hardware redundancy by
duplicating sensors at critical parts of the system. This realization depends on the room
and money available. That is why nowadays new techniques have been developed which
can use either the model of the system (analytical methods) [1,4,5,6] or the knowledge
about the system (heuristic methods) [7] for error detection. The performance of such
methods depends on several terms such as robustness, time between the failure and
detection, and sensitivity, meaning the possibility of finding failures of small intensity.
Some of these techniques use the analytical redundancy which consists of comparing the
behavior of the model and the system. This can be done by using the residuals generation
as proposed by J. Gertler, M. Staroswiecki and M. Shen [4]. In the ideal conditions, these
Received November 18, 2010
80
A. MARJANOVIĆ, Ž. ĐUROVIĆ, B. KOVAČEVIĆ
two should be the same, which is rarely the case. One possible way of managing this is the
Wald sequential hypothesis test [2]. This is a sequential technique because at each step it
is necessary to make a decision whether to accept or decline the hypothesis, or to continue
the experiment by taking more observations. Making a decision depends on the error
probabilities that must be set in advance. The bigger probability error is allowed, the less
time is needed for decision making. Therefore, a compromise must be made. One of the
main problems of modern methods is finding an optimal solution. Wald sequential test
offers a way of doing so, by minimizing the number of needed measurement for the preset
error probabilities. In this paper the algorithm is demonstrated on a well-known Three-
tank system with simulated leaks on the pipelines between the tanks. The described
method is used on a closed-loop system controlled by a PID regulator.
2. SEQUENTIAL TESTING
There are many practical applications where it is necessary to make a real time
decision based on the measured data. Sequential testing is a mathematical approach used
in these cases and it helps saving money and time by stopping an experiment when there
is enough evidence to come to a conclusion. When using these methods a compromise
must be made between the time needed for decision making and the probability error.
This problem is successfully solved by using the Wald sequential test. Wald sequential
test consists of making one of three possible decisions: (1) to accept the hypothesis, (2) to
reject the hypothesis and (3) to continue making observation. If one of the first two
decisions is made, the test stops. In the case of the third decision the test goes on until
either the first or the second decision is made. Therefore the number of observations
required depends on the outcome of the observation and is not predetermined.
Let us consider the independent equally distributed random observation vectors
X1,…,
Xm. Now let us form the negative logarithm of likelihood ratio
∑ =
-
=
-
=
m
i
i
i
m
m
m
Xf
Xf
X
Xf
X
Xf
s
1
2
1
1
2
1
1
)(
)(
ln
)
,...,
(
)
,...,
(
ln
(1)
where
f1 and
f2 represent probability distributions. The main idea of Wald sequential test
is for parameter
m to be variable. The test stops when
sm reaches some predefined values:
sm ≤
a, accept the hypothesis,
sm ≥
b, decline the hypothesis and
a <
sm <
b, take another
measurement, where
a and
b are
2
1
1
ln
ε
ε
-
-
=
a
(2)
2
1
1
ln
ε
-
ε
-
=
b
(3)
It is important to say that Wald sequential test ends with the probability of 1. Also,
this method minimizes the number of observation needed for predefined probability
errors.
Fault Detection in a Three-Tank System Based on Sequential Hypothesis Testing
81
3. THREE-TANK SYSTEM
The three-tank system considered is this paper is shown in Fig.1.
Fig. 1. Three-tank system
The system consists of three liquid tanks which are interconnected by the pipes with
valves. In general, there can be two pumps for delivering liquid to the system, but in this
case it is assumed that only the pump that drives liquid to the first tank is active. The
liquid flow of the pump can be manipulated from the flow of 0 to a maximum flow, Qmax.
And that is the manipulated input variable. The liquid level in each tank can be measured
by level sensors and one of the goals of this research is to maintain a constant level in the
second tank, which is the controlled variable. All three tanks have the same physical
features such as the same height, hmax, and cross-sectional area, S. Also, the cross-sections
of the pipes is the same, Sp. Using simple laws of physics such as conversation of mass in
tanks (4) and Torricelli’s law (5) a mathematical model can be derived.
)
(
1
)
(
)(
ij
ki
i
ij
ki
i
q
q
s
dt
dh
q
q
dt
tdm
-
=
→
-
ρ
=
(4)
j
i
j
i
pi
ij
hhg
hh
sign
S
q
-
-
μ
=
2)
(
(5)
where
qij is the flow between the
i-th and
j-th tank and hi is the height in the
i-th tank.
The exact values of the system parameters are given in Table1.
Table 1. System parameters
Symbol
Meaning
Value
S
cross-section of the tanks
0.0154 m2
Sp
cross-section of the pipes
0.0050 m2
g
gravity constant
9.81 m/s2
hmax
height of the tanks
1m
µ1
flow coefficient for the first pipe
0.6836
µ2
flow coefficient for the second pipe
0.4819
µ3
flow coefficient for the third pipe
0.4339
umax
maximum input flow
0.001 m3/s
82
A. MARJANOVIĆ, Ž. ĐUROVIĆ, B. KOVAČEVIĆ
The model of the system is
3
3
3
2
3
2
2
3
3
2
3
2
2
2
1
2
1
1
2
2
1
2
1
1
1
2
2)
(
(
1
)
2)
(
2)
(
(
1
)
2)
(
(
1
gh
S
hhg
hh
sign
S
S
h
hhg
hh
sign
S
hhg
hh
sign
S
S
h
hhg
hh
sign
S
u
S
h
p
p
p
p
p
μ
-
-
-
μ
=
-
-
μ
-
-
-
μ
=
-
-
μ
-
=
q
q
q
(6)
In this paper, a closed-loop system is considered. Before describing the algorithm for
failure detection, let us first design a PID controller. The controlled variable is the height
in the second tank and therefore the input of the PID controller is the difference between
the reference and the measured signal. Nominal value for h2 is set to be 0.7m. This gives
nominal value of 0.856 m for h1, 0.386 m for h3 and 0.005975 m3/s for u. For these
conditions the parameters of the PID regulator are
Kp = 0.001
Ki = 0.0001
Kc = 0.0002
4. FAULT DETECTION IN A THREE-TANK SYSTEM
The idea of this paper is to detect the mechanical failure on the pipes between the
tanks and to locate which interconnection has the malfunction. The leakage is simulated
by changing the value of the parameter μ
i.
Since it is necessary to detect the leak as soon as possible, the real-time data
acquisition is done using the Wald sequential test. Therefore, the first step is to make a
decision whether the system is working properly or not. Once the fault has been detected,
the second step is to determine the nature of the failure. Here, the classifier is reset and
the second Wald test is started. Since the possible fault will be visible at the output of the
system, structured residuals can be used.
The residuals are generated using
hhh
ˆ
-
=
∆
(7)
where h(h1, h2, h3) is a 3-dimensional measurement vector, )ˆ,ˆ,ˆ(ˆ3
2
1
hhhh
is the estimated
height vector and Δ
h (Δ
h1, Δ
h2, Δ
h3).
Before starting the test it is necessary to say the Gaussian distribution is considered for
the residual vectors. Also, the parameters of these distributions must be determined in
three cases: when there are no leaks
f ~N(M, Σ), when there is a leak on the first pipe
f1
~N (M1, Σ1) and when there is a leak on the second pipe
f2
~N (M2, Σ2). These constants
are calculated using (8) and (9)
N
h
M
N
i
i
∑ =
∆
=
1
(8)
T
N
i
i
i
Mh
Mh
N
∑
∑
=
-
∆
-
∆
=
1
)
)(
(
1
(9)
Fault Detection in a Three-Tank System Based on Sequential Hypothesis Testing
83
When there are no failures, the residuals are only the disturbances on the output. The
parameters in this case are
│
│
│
⌋
⌉
│
│
│
⌊
⌈
-
= -
1430.0
1360.0
0633.0
10 4
M
│
│
│
⌋
⌉
│
│
│
⌊
⌈
-
-
-
-
=
∑ -
0967.0
0011.0
0013.0
0011.0
1037.0
0006.0
0013.0
0006.0
1005.0
10 4
Next, in a case of a failure on the first pipe, changes in the liquid level of the first tank
are visible, and the calculated values are
│
│
│
⌋
⌉
│
│
│
⌊
⌈
∙
-
∙
-
=
-
-
5
5
1
10
3554.1
10
3334.1
0263.0
M
│
│
│
⌋
⌉
│
│
│
⌊
⌈
-
-
-
-
=
∑
-
0967.0
0011.0
0014.0
0011.0
1035.0
0005.0
0014.0
0005.0
1004.0
10
1
4
Similar, when there is a leak on the second pipe, there are some significant changes in
the levels of the first and the second tank.
│
│
│
⌋
⌉
│
│
│
⌊
⌈
∙
-
-
-
=
-5
2
10
3367.1
0812.0
0812.0
M
│
│
│
⌋
⌉
│
│
│
⌊
⌈
-
-
-
-
=
∑
-
0967.0
0011.0
0014.0
0011.0
1037.0
0005.0
0014.0
0005.0
1006.0
10
2
4
Once all three residual vectors have been described, sequential hypothesis testing
method can begin. When trying to detect a failure two residual distributions are
considered in (1). One, when system is working in normal mode, already calculated and
the other, in case of a failure, when joint contribution is considered (10).
f12 = 0.5
f1 + 0.5
f2
(10)
When trying to determine a type of failure,
f1 and
f2 are considered in (1). Using (2) and
(3) and predefined probability errors, boundaries can be calculated. If the sum (1) is out of
the provided boundaries, decision will made, otherwise, another observation is taken.
84
A. MARJANOVIĆ, Ž. ĐUROVIĆ, B. KOVAČEVIĆ
5. RESULTS
Let us first simulate the work of the classifier when there are no faults in the system.
Fig. 2. System without a fault
As expected, the residual is very close to zero, and the classifier decides that there is
no error. Classifier was unsure for several times, but never did he make a wrong decision.
Now, let us see what happens in case of a leak. Both types of leakage are
demonstrated. First there is a leak on the first pipe. After that, the system is in a normal
state for some time. Then the second kind of leak happens. The figure shows that the
classifier has detected both errors and determined the type of error correctly in most
cases. In these simulations, the error probabilities are set to ε1 = ε2 = 0.001. It is important
to say that in case of the greater error probabilities it would take less time to make a
decision, but also there would be more errors in the process of classification. Therefore, it
is necessary to make a compromise between the time needed for decision making and the
probability of error.
Fig. 3. System with a leak on the first and the second pipe
Fault Detection in a Three-Tank System Based on Sequential Hypothesis Testing
85
Fig. 4. shows how the number of observations depends on the probability of error. As
expected, the bigger the probability is, the less time is needed to come to a conclusion.
Also, there is a theoretical result concerning this, which is also show in the same Figure.
Fig. 4. Number of observation needed for decision making
as a function of the error probabilities
Also, it is interesting to see how much the estimated error deviates from the real one.
Fig. 5. Relation between the estimated and the observed error probability
Based on the last two figures it can be estimated what is the actual error one should
consider for reaching a decision in a specified time.
6. CONCLUSION
The performance of the proposed method has been evaluated on a three-tank system
with simulated leaks on the interconnections between the tanks. The test is controller
independent. The algorithm provides the information on whether the failure has
86
A. MARJANOVIĆ, Ž. ĐUROVIĆ, B. KOVAČEVIĆ
happened, but also the type of the malfunction. Once the fault has been identified the next
step could be, based on the structured residuals, to find the magnitude and time of fault.
The method gives a result in the minimum time, for the preset error probabilities. There is
also more fault diagnostics to be done, on the failures with smaller magnitude changes.
Acknowledgement.
This work is supported by the European Commission's Seventh Framework
Programme, as part of the PRODI project (INFSO-ICT-224233).
REFERENCES
1. A.Asokan and D.Sivakumar, “Fault Detection and Diagnosis for a Three-tank system using Structured
Residual Approach”,
ICGST-ACSE Journal, volume 7, Issue 2, 2007.
2. A. Wald, “Sequential Tests of Statistical Hypothesis”,
Annals of Mathematical Statistics 117-186,
Columbia University,1945.
3. J.Gertler, “Fault Detection and Diagnosis in Engineering Systems”,
Marcel Dekker, Inc. USA, 2007.
4. J. Gertler, M. Staroswiecki and M. Shen, “Direct design of structured residuals forfault diagnosis in
linear systems”
, American Control Conference,Anchorage, Alaska, 2002.
5. S. Ding, “Model-based Fault Diagnostics Technique, Design Schemes, Algorithms, and Tools”,
Springer-Verlag, Berlin, 2008.
6. J. Gertler, “Residual Generation in Model Based Fault Diagnosis”,
Control- Theory and Advanced
Technology, Vol. 9, pp. 259-285, 1993.
7. B. KÄoppen-Seliger, E. Alcorta-Garca, P. Frank, ”Fault Detection: Differentstrategies for Modelling
Applied to the Three-tank Benchmark - A Case Study”,
European Control Conference, Karlsruhe,
Germany,1999.
8. A. Marjanović, Ž. Đurović, B. Kovačević, “Fault detection in a three tank system based on sequential
hypothesis testing”,
SAUM 2010, Niš, Serbia, pp. 278-281, 2010
PRIMENA SEKVENCIJALNOG TESTIRANJA HIPOTEZA ZA
DETEKCIJU OTKAZA U SISTEMU SA TRI REZERVOARA
Aleksandra Marjanović, Željko Đurović, Branko Kovačević
Rad predstavlja metodu za detekciju i lokalizaciju otkaza u sistemu sa tri rezervoara,
korišćenjem Wald-ovog sekvencijalnog testa koji se primenjuje na sekvencu reziduala dobijenu na
osnovu modela. Primenjeni metod minimizira srednji broj potrebnih odbiraka do donošenja
odluke, a na osnovu zahtevanih verovatnoća greške. Snimljene su zavisnosti broja potrebnih
odbiraka i dobijene verovatnoće greške od zahtevanih verovatnoća greški i ovi su rezultati
uporedjeni sa teorijskim.
Ključne reči:
detekcija i izolacija greške, sekvencijalno testiranje hipoteza, residual, metode na
bazi modela, sistem sa tri rezervoara.
