A COMBINED APPROACH TO FAULT DIAGNOSIS IN
DYNAMIC SYSTEMS
Application to the Three-Tank Benchmark
Luís Palma, Fernando Coito, Rui Silva
Departamento de Engenharia Electrotécnica, Faculdade de Ciências e Tecnologia
Universidade Nova de Lisboa, Monte de Caparica, 2829-516, Portugal
Keywords: Fault diagnosis, classical and soft computing methods, nonlinear hybrid dynamic systems
Abstract: This paper presents a combined approach to fault diagnosis in discrete-
time dynamic systems. The approach
integrates classical and soft computing techniques. The typical methods based on signal models, and process
models for residual generation are considered: parity equations, observers and parameter estimation. The
role of integration of classical and intelligent techniques is enhanced. The performance of the proposed
approach is analysed with application to a typical nonlinear feed-water system – the three-tank benchmark.
The three typical fault scenarios (actuator and component faults) defined in the benchmark problem are
tackle in this work.
1 INTRODUCTION
Modern supervision and control systems are
becoming more and more sophisticated. The issues
of reliability, operating safety, availability, cost
efficiency, and environment protection are of great
importance. For safety-critical systems, the
consequences of faults can be extremely serious in
terms of human fatalities, environment impact or
economic loss. There is a growing need for on-line
supervision and fault diagnosis (FDI) to increase the
reliability of such safety-critical systems (Chen and
Patton, 1999). For systems that are not safety-
critical, on-line FDI techniques can be use to
improve reliability, plant efficiency, availability, and
maintainability.
Since the beginning of the 1970’s, research in
faul
t diagnosis has been gaining increasing
consideration world-wide in both theoretical and
application areas (Chen and Patton, 1999; Frank, et.
al., 1999; Gertler, 1998; Isermann, 1997; Patton, et.
al., 2000). This development was (and still is)
mainly stimulated by the trend of automation
towards more complexity and the growing demand
for higher security and availability of supervision
and control systems. The great progress of computer
technology made feasible the use of powerful
techniques of modern and intelligent control theory
applied to the FDI problems, like mathematical
modelling, state estimation and parameter
identification.
The main purpose of fault diagnosis is the
det
ermination of kind, size, location and time
occurrence of a fault (Isermann, 1997). Many
approaches to FDI in the time as well in the
frequency domain have been proposed (Chen and
Patton, 1999; Frank, et. al., 1999; Gertler, 1998;
Isermann, 1997; Patton, et. al., 2000). From the
multiple model-based methods in the literature, the
three main groups are parity equations, observers,
and parameter estimation. Fault detection based on
signal processing and on signal models is also
possible (Isermann, 1997).
A “fault” is an unexpected change of a system
fu
nction, although it may not represent physical
failure or breakdown (Chen and Patton, 1999).
Another definition is (Isermann, 1997): a “fault” is a
non-permitted deviation of a characteristic property
that leads to the inability to fulfil the intended
purpose.
163
Palma L., Coito F. and Silva R. (2004).
A COMBINED APPROACH TO FAULT DIAGNOSIS IN DYNAMIC SYSTEMS - Application to the Three-Tank Benchmark.
In Proceedings of the First International Conference on Informatics in Control, Automation and Robotics, pages 163-171
DOI: 10.5220/0001144101630171
Copyright
c
 SciTePress

Figure 1: The three-tank benchmark.
Modern fault diagnosis systems are model-based,
instead of the traditional approach based on
“hardware (or physical/parallel) redundancy. Two
kinds of models can be used to perform the FDI task
(Chen and Patton, 1999; Frank, et. al., 1999; Gertler,
1998; Isermann, 1997; Patton, et. al., 2000):
quantitative (analytical) models, or qualitative
models (knowledge-based models: fuzzy models,
neural networks, etc). Model-based fault diagnosis is
define as the determination of faults of a system
from the comparison of available system
measurements with a priori information represented
by the system’s mathematical model, through
generation of residual signals and their analysis. A
residual is a fault indicator or an accentuating signal
that reflects the faulty situation of the monitored
system (Chen and Patton, 1999).
Some process plants are complex systems, like
nuclear reactors, chemical plants, aircrafts, power
plants, feed-water plants, etc. For that cases, an
efficient FDI approach must combine (integrate)
different FDI methods: parity equations, observers,
and parameter estimation. These methods can be
implemented using classical or intelligent soft
computing techniques. The three-tank benchmark
used in our work (Figure 1), developed during the
COSY (control of complex systems) programme of
the European Science Foundation, is a typical hybrid
complex system (Heiming and Lunze, 1999). The
main reasons are: a) each water tank is a nonlinear
system; b) is a hybrid system in the sense that has
continuous and discrete sensors; c) has eight
different operating modes; d) the models build to
represent the system behaviour usually have a
significant uncertainty associated.
The main contribution of this paper is the
integrated approach proposed to deal with nonlinear
FDI problems in hybrid nonlinear dynamic systems,
where nonlinear neural observers play an important
role. Section 2 describes, briefly, the three-tank
benchmark. Section 3 details the proposed combined
approach for FDI, and the application to the
benchmark. The simulation results are in section 4.
The final section presents the conclusions and the
future work.
2 THE THREE-TANK
BENCHMARK
The benchmark problem concerns the three coupled-
tanks depicted in Figure 1 (Heiming and Lunze,
1999). The aim is to provide a continuous water
flow
to a consumer by maintaining a desired
level in the central tank
. Pipes, which can be
controlled by several valves, connect the water
tanks. All valves can only be completely opened or
completely closed. Water can be let into the left and
right tank using two identical pumps (P1 and P2).
Measurements available from the process are the
continuous water levels
in each tank, and two
discrete levels
from two proximity switches
attached to the central tank (
). For the middle tank
, the qualitative values are: low = [0..9] cm,
medium = [9..11] cm, and high = [11..60] cm.
N
Q
3
T
i
h
d
h
3
T
3
T
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164

PROCESS
FEATURE
GENERATION
PROCESS
MODEL
CHANGE
DETECTION
ACTUA-
TORS
SEN-
SORS
Faults
Analytical Symptoms
Features
Normal
behaviour
d
u
y
∆ s
s
Model based
fault detection
Figure 2: General model-based scheme for fault detection.
In the fault-free situation, only the left tank
and the middle tank
are used. A continuous PI-
controller or other type of controller can be used to
control the level around 0.5 m at tank
. A
switching (on-off) controller opens and closes valve
thus maintaining the level around 0.1 m at
tank
; the water level in this middle supply-tank
has therefore to be maintained at a
level
. All other valves are closed, and
the right tank
is empty. The three standard fault
scenarios considered are in this work: a) fault F1,
valve V1 is closed and blocked; b) fault F2, valve
V1 is open and blocked; c) fault F3, valve V1L is
open (simulating a leak in tank T1).
1
T
3
T
1
T
1
V
3
T
mediumh =
3
2
T
The 3-tank benchmark was developed mainly for
FDI and for controller reconfiguration tasks. The
main problem is to find a new control strategy if a
fault in the technical plant has occurred. In this
work, only the fault diagnosis problem is considered.
The fault-tolerant control problem will be analysed
in a future work.
3 THE COMBINED FAULT
DIAGNOSIS APPROACH
3.1 The General Model-Based
Scheme for FDI
A general scheme of process-model-based fault
detection is depicted in Figure 2 (Isermann, 1997).
Based on measured input signals
u
and output
signals
, the detection methods generate features
y
s
(residuals
r
, parameter estimates or state
estimates
). By comparison with the normal
features, changes of features are detected, leading to
analytical symptoms
∧
θ
∧
x
s
∆
.
After fault detection, the fault isolation task must
be performed and consists in symptom evaluation
and decision-making, in order to decide the location
of the fault – a sensor fault, an actuator fault or a
component fault, and the time of occurrence.
3.2 The Combined FDI Approach
The combined approach proposed in this paper, to
solve fault diagnosis (FDI) problems in nonlinear
dynamic systems, is based on a combination of
different FDI approaches detailed in the next sub-
sections.
The neural observer proposed plays an important
role in the combined approach, since it’s able to
works simultaneously as a state and outputs
observer.
First one must define the type of faults (additive
or multiplicative, and their location - on the sensors,
on the actuators or on the process components) that
are to be detected, and then use these elements as a
guideline to build the process models and signal
models for FDI.
A COMBINED APPROACH TO FAULT DIAGNOSIS IN DYNAMIC SYSTEMS - Application to the Three-Tank
Benchmark
165

f
m
z
-1
K
n
()
x^(k+1)
y(k)
g
m
x^(k)
u(k)
Control input
y^(k)
-
Plant model
Figure 3: Structure of a nonlinear observer.
3.3 FDI based on Signal Processing
and Signal Models
In practice, the most frequently used diagnosis
method is to monitor the value (or trend) of a
particular signal, and taking action when the signal
reached a given threshold (Chen and Patton, 1999).
This method of limit checking, whilst simple to
implement, has at least two main drawbacks: a) the
possibility of false alarms in the event of noise, and
the change of operating point; b) a single fault could
cause many signals to exceed their limits and appear
as multiple faults.
In our work, the nominal qualitative level in tank
T3, controlled by a switching (on-off) controller,
must be medium. The discrete sensors (in
conjunction to other residuals) can be used to detect
the occurrence of a fault (F1, or F2) on the system.
A signal
was defined based on the discrete
sensors information:
)(3 kdh
⎪
⎩
⎪
⎨
⎧
=⇐−
=⇐
=⇐
=
lowh
mediumh
highh
kdh
3
3
3
1
0
1
)(3
(1)
More advanced methods based on signal models,
like the determination of autocorrelation functions,
the Fast Fourier Transform (FFT), etc, can also be
used to perform fault diagnosis (Isermann, 1997).
3.4 FDI via Parity Equations
In the early development of fault diagnosis, the
parity equation approach was applied to static or
parallel redundancy schemes, which may be
obtained directly from measurements or from
analytical relations. There are typically two cases for
arranging hardware redundancy, one is the use of
sensors having identical or similar functions to
measure the same variable, another is the use of
dissimilar sensors to measure different variables but
with their outputs being relative to each other. The
basic idea of the parity equation method is to
provide a proper check of the parity (consistency) of
the measurements of the monitored system (Chen
and Patton, 1999; Gertler, 1998).
This type of approach is not used in this work,
but it can be used to diagnose, for example, additive
faults on sensors.
3.5 FDI based on Observers
The basic idea behind the observer or filter-based
classical approaches is to estimate the outputs of the
system from the measurements by using either
Luenberger observer(s) in a deterministic setting, or
Kalman filter(s) in a stochastic setting (Friedland,
1996; Chen and Patton, 1999). For a nonlinear
system, the structure of the observer is not nearly
obvious as it is for a linear system (Friedland, 1996).
Let’s assume a nonlinear stochastic dynamic model
for a nonlinear plant:
⎩
⎨
⎧
=
=+
))(),(),(()(
))(),(),(()1(
kRkukxgky
kQkukxfkx
m
m
(2)
where
is the state, is the input
vector,
is the system output vector, and
and are nonlinear functions. The
matrices
and are the process and
measurement noises. Assuming known the noise
characteristics, an Extended Kalman Filter (EKF)
can be used as a nonlinear observer; in practice the
n
Rkx ∈)(
r
Rku ∈)(
m
Rky ∈)(
(...)
m
f (...)
m
g
)(kQ )(kR
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166

spectral densities matrices and are
hardly ever known to be better than an order of
magnitude. For a deterministic system, assuming
and , a general nonlinear
observer in discrete-time is depicted in Figure 3, and
can be expressed by:
)(kQ )(kR
0)( =kQ 0)( =kR
⎪
⎪
⎪
⎩
⎪
⎪
⎪
⎨
⎧
−=
−=−=
Κ+=+
∧
∧∧
∧∧
)()()(
))(),(()()()()(
)())(),(()1(
kxkxke
kukygkykykykr
krkukxfkx
m
nm
(3)
In (3),
is the observed state,
is the input vector, is the
system output vector, and
and are
nonlinear functions. The residual is expressed by
, and is the estimation error. By proper
choice of the nonlinear function
, the error
equation can be made asymptotically stable.
n
Rkx ∈
∧
)(
r
Rku ∈)(
m
Rky ∈)(
(...)
m
f (...)
m
g
)(kr
)(ke
(...)
n
K
In our work, observers based on neural networks
were used to estimate the system outputs (Palma, et.
al., 2003; Palma, et. al., 2004). The nonlinear neural
observer used in this work obeys the model (3), and
has a recurrent dynamic structure. For our case, the
state variables are the measured output variables of
the system (the levels at the two coupled tanks
and
): . In that case the FDI
residual is equal to the residual of the neural
observer,
. For the state
variable
, the neural observer is
expressed by:
1
T
3
T
⎥
⎦
⎤
⎢
⎣
⎡
=
)(
)(
)(
3
1
kh
kh
kx
)(1 koer
)()()(
1
11
khkhkr
∧
−=
)()(
11
khkx =
)(
))1(),(,()1(
11
1
1
},,{
1
krK
kqkhWNNkh
n
cba
+
+−=+
∧∧
(4)
In a similar way, for tank T3, the residual
is equal to the residual of the neural
observer,
. For the state
variable
, the neural observer is
expressed by:
)(3 koer
)()()(
3
33
khkhkr
∧
−=
)()(
33
khkx =
)(
))(),(),(,()1(
33
11
3
},,{
3
krK
kVkhkhWNNkh
n
cba
+
+=+
∧∧
(5)
In equations (4) and (5),
represents a multi-layer perceptron feed-forward
neural network (MLP-FF-NN) with weight matrix
(Hagan, 1995; Palma, et. al., 2003). The structure
(...)
},,{ cba
NN
W
}1,4,2{
=
=
=
cba
defines the number of neurons in
each layer, respectively, the input layer, the hidden
layer, and the output layer. The train of the MLP-FF-
NN neural network was done off-line in a set-point
range varying between 0.35 and 0.5 m, using the
Levenberg-Marquardt backpropagation optimization
algorithm. The continuous levels in each tank are
, and . The flow from pump )(
1
kh )(
3
kh 1
P
is
, and the switch control signal acts on
the switching controller. The nonlinear function
is a design parameter that adjusts the
observer dynamics and guarantees the stability. In
the experiments, this nonlinear gain functions were
defined as
)(
1
kq )(
1
kV
)(
1
kK
n
)()( krKkK
iini
=
, for constants
2.0
1
=
K and 5.0
3
=
K . These values were tested in
simulations, in order to obtain a stable and slow
dynamics.
3.6 FDI via Parameter Estimation
Model-based FDI can also be achieved by the use of
system identification techniques. In most practical
cases, the process parameters are not known at all, or
are not known exactly enough. Then they can be
determined with parameter estimation methods by
measuring input and output signals if the basic
structure of the model is known (Isermann, 1997).
This approach is based on the assumption that the
faults are reflected in the physical system parameters
such as resistance, capacitance, viscosity, friction,
etc. The basic idea of the detection method is that
the parameters of the actual process are repeatedly
estimated on-line using well known parameter
estimation methods (RLS, Kalman filter, etc), and
the results are compared with the parameters of the
reference model obtained under the faulty-free
condition. Any substantial discrepancy is due to a
fault. This approach normally uses the input-output
mathematical model of a system in the following
form (Chen and Patton, 1999):
))(,()( kuPfky
=
(6)
A COMBINED APPROACH TO FAULT DIAGNOSIS IN DYNAMIC SYSTEMS - Application to the Three-Tank
Benchmark
167

where,
P
is the model coefficient vector which is
directly related to physical parameters of the system.
The function
can be linear or nonlinear. To
generate residuals using this approach, an on-line
parameter identification algorithm should be used.
The residual can be defined in either of the
following ways (Chen and Patton, 1999):
(.,.)f
)()()(
0
kPkPkr −=
∧
(7)
))(),1(()()( kukPfkykr −−=
∧
(8)
In this work, an adaptive residual generator
based on an ARX model was used. A Kalman filter
was used as a parameter estimator for identification
of the parameters of the ARX model (Soderstrom
and Stoica, 1989). It was assumed that the dynamics
of the tank
is modelled, in steady-state, by an
autoregressive
1
T
)2,1,2(
=
=
== dnknbnaARX
model, assuming
is a white Gaussian noise
(Soderstrom and Stoica, 1989):
)(ke
)()()()()(
11
kekuzBzkyzA
d
+=
−−−
02
12
2
1
1
1
)(;1)( bbzBzazazA ==++=
−−−−
(9)
A residual was constructed based on (7), for the
static gain of the model (9), as defined by (Palma, et.
al., 2004):
kbkka
kLksgksgkrg
kbkkbsgksgkrg
≤≤
⇐+−−=
>
⇐
−=
...)):1(1()(1)(1
)(1)(1)(1
µ
(10)
3.7 Thresholds and Symptoms Values
The thresholds ( – high, and – low) for
each residual were computed according to a
r
ThrH
r
ThrL
r
σ
3
(standard deviation) limit around the mean value
r
µ
; these statistical values were computed in a
nominal region (Palma, et. al., 2003; Palma, et. al.,
2004).
Each residual signal was converted to the range
{-1;0;+1}. The value “+1” means that the residual
exceeds the upper threshold (
), “-1”
means that
, and “0” means the
residual is bounded (
).
rx
ThrHkr >)(
rx
ThrLkr <)(
rxr
ThrHkrThrL << )(
3.8 The Fault Diagnosis Structure
Based on the 4 symptoms referred, the following
fault isolation structure was build according to
simulation tests:
Table 1: Fault isolation structure.
Faults
F1 F2 F3
rg1(k) +1 -1 -1
r1oe(k) 0 0 -1
r3oe(k) +1 0 0
Symptoms
h3d(k) -1 (low) +1 (high) 0 (med)
3.9 The Adaptive Polynomial Linear
Quadratic (LQ) Controller
Liquid level systems, like the one used in this work,
are typical nonlinear systems. To control the level at
tank
an optimal linear quadratic (LQ) controller,
based on a polynomial approach, was designed and
implemented (Lewis, 1996). For the ARX(2,1,2)
model (9), the obtained control law is defined by
(11).
1
T
The control action
is computed each time
instant based on the on-line identified parameters of
the ARX model:
)(ku
).()(),(),(
0221
kbkbkaka =
The Kalman filter was used as a parameter
estimator. The reference signal is denoted
.
The scalar
is a design parameter used to tune the
closed-loop performance.
)(kw
0
r
))()1()()()(
)1()((
1
)(
212
2
1
2
2
0
21
2
2
0
2
kwkyaakyaa
ku
b
r
ba
b
r
b
ku
−−+−+
+−+−
+
−=
(11)
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168

Figure 4: Signals for fault F1.
4 SIMULATION RESULTS
4.1 Operating Conditions
The simulations were done in a Matlab/Simulink
®
programming environment. The Simulink model of
the three-tank benchmark runs in continuous-time in
a computer, and the supervision Matlab
software
runs on discrete-time on another computer; the link
between the two PC’s is done by serial port
communication.
A sampling time of
was used. All the
values were normalized to the range [0;1].
sT
s
1=
4.2 Simulation Results
In Figure 4, are presented the signals obtained for an
experiment in which is detected and isolated the
fault F1. From top to bottom, the signals in this
figure are: a1) the set-point for tank T1, the level h1,
and the output predicted; a2) the flow q1; a3) the
set-point for tank T3, the level h3, and the output
predicted; a4) the residual rg1; a5) the residual r1oe,
from the neural observer; a6) the residual r3oe, from
the neural observer; a7) the qualitative level h3d; a8)
the fault isolation signal.
The following table shows the detection delay for
each fault, and the results are acceptable since the
system has a slow dynamics.
Table 2: Detection delay
Fault Detection delay [s]
F1 31
F2 32
F3 14
The robustness of the FDI approach against set-
point variations was also tested, for a range between
{0.35;0.5}m, and a good performance (without false
alarms) was obtained.
The experiments done with the faults F2 and F3
are shown in Figure 5, and Figure 6. In these figures,
the same signals described in
Figure 4 can be
observed. The two faults, F2 and F3, were also well
detected and isolated.
As can be observed in all simulations, the
residuals and the signal h3d used for fault isolation
(
Table 1) reveals a good performance for FDI.
A COMBINED APPROACH TO FAULT DIAGNOSIS IN DYNAMIC SYSTEMS - Application to the Three-Tank
Benchmark
169

Figure 5: Signals for fault F2.
Figure 6: Signals for fault F3.
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170

5 CONCLUSIONS
The paper proposes a combined approach to fault
diagnosis (FDI) in dynamics systems. This approach
integrates several FDI classical and intelligent (soft
computing) methods.
All the available information must be use to
perform FDI. An integration of process models and
signals models improves the reliability of the FDI
approach. A robust FDI system, able to be
implemented in a practical problem, should combine
both quantitative (numerical) and qualitative
(symbolic) information. The soft computing
techniques for FDI, like nonlinear neural observers,
are particularly important and efficient as shown in
this work. One great advantage of this type of
approach is that a precise mathematical model is not
required.
The proposed combined approach has been
applied to a simulation model of the three-tank
benchmark (a typical feed-water system), and the
results shown good performance, and robustness
against set-point variation.
The future work will concern to fault-tolerant
control approaches via controller reconfiguration
strategies, and the stability analysis of nonlinear
neural observers.
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