|
|
|
Networking, Sensing and Control, London, UK, 15-17 April 2007 |
|
|
|
|
Written by Administrator
|
|
Monday, 09 August 2004 |
Wireless Sensor Network Modeling Using Modified Recurrent
Neural Networks: Application to Fault Detection
Proceedings of the 2007 IEEE International Conference on
Networking, Sensing and Control, London, UK, 15-17 April 2007
MonA02
Abstract—This paper presents a dynamic model of wireless
sensor networks (WSNs) and its application to a sensor node
fault detection. Recurrent neural networks (RNNs) are used to
model a sensor node, its dynamics, and interconnections with
other sensor network nodes. The modeling approach is used for
sensor node identification and fault detection. The input to the
neural network is chosen to include delayed output samples of
the modeling sensor node and the current and previous output
samples of neighboring sensors. The model is based on a new
structure of backpropagation-type neural network. The input
to the neural network and topology of the network are based on
a general nonlinear dynamic sensor model. A simulation
example has demonstrated effectiveness of the proposed
scheme.
I. INTRODUCTION
Wireless sensor networks (WSNs) consist of a set of
sensor nodes that can communicate with each other, sensors
that measure desired physical quantity, and the system base
station for data collection, processing, and connection to the
wide area network. Modern wireless sensor nodes have
microprocessors for local data processing, networking, and
control purposes [1]. WSNs enabled numerous advanced
monitoring and control applications in environmental,
biomedical, military, and other applications.
Sensors in such networks have their own dynamics,
often nonlinear, and modeling such a sensor network is not
trivial. Since recurrent neural networks (RNNs) consist of
interconnected dynamical nodes, we explore its similarities
with WSNs and exploit that in modeling. This paper
presents modeling of WSNs using modified dynamic RNN.
The real motivation for a WSN modeling stems from the
need for intelligent fault detection in such complex
distributed sensory systems. Since sensor networks often
operate in potentially hostile and harsh environments, most
applications are mission critical. Sensors are often used to
compute control actions [2], [3], where sensor faults can
cause catastrophic events. Recently, NASA was forced to
abort the launch of space shuttle Discovery due to a failure
on one of the sensors in the sensor network in the shuttle’s
This work was supported in part by the NSF EPSCoR Pfund Grant
# NSF/LEQSF(2005)-PFUND-19 and LaSPACE NASA Grant #822.
external tank (failure was discovered by human inspection).
Components such as sensors and actuators have
significantly higher fault rates than the traditional integrated
semiconductor circuits-based systems. Multi-sensor systems
need feedback information about the health status of their
nodes in order to recover and heal from the eventual faults.
Such a system would have improved reliability over existing
sensor networks. Since external and internal malfunctions or
excessive noise can occur, sensor readings are somewhat
uncertain in a sense that no existing sensor will deliver
accurate readings at all times. It is desired to develop a WSN
that will have capability of fault detection, isolation, and
accommodation. Efficiency in converting data to features
and consistency with the uncertainty in the measurements
are key issue for diagnosis of sensor faults [4], [5].
Fault-tolerance emerged to be very essential and urgent
for modern sensory systems [5]. The traditional way of
achieving fault-tolerance in dynamical systems is through
hardware redundancy such as use of multiple sensors. But
multiplication of sensor devices adds a cost, complexity, and
power consumption to the sensor node and whole network.
Most of present research efforts have been concentrated on
an analytical redundancy [7], [8] in which sensor
measurements are processed analytically and mathematical
models are compared with physical measurements.
Koushanfar et al. proposed a heterogeneous fault-tolerance
technique [9], where one type of resources can replace
another type. Considered resources are computing, storage,
communication, sensing and actuating. For example, when
the available power is limited, one can rely more on
computing using microcontrollers, resulting in transmitting
less data to the base station or other sensor nodes (radio
transmission consumes the most energy).
Most of the techniques for fault detection and diagnosis
are based on the comparison of the actual sensor model with
the nominal model [10]. In addition, a comparison with the
fault models (system with faults) allows one to determine
what the type of faults have occurred [10], [11].
Instead of using additional hardware in a form of
multiple sensors, we propose to use computational resources
for the intelligent fault detection. The dynamic model of a
sensor node is formed based on information from
Wireless Sensor Network Modeling Using Modified Recurrent
Neural Networks: Application to Fault Detection
Azzam I. Moustapha and Rastko R. Selmic
Department of Electrical Engineering
College of Engineering and Science
Louisiana Tech University
Arizona Avenue, Nethken Hall 229
Ruston, LA 71272, USA
Tel: 318-257-4641 Fax: 318-257-4922
Email:
This e-mail address is being protected from spam bots, you need JavaScript enabled to view it
This e-mail address is being protected from spam bots, you need JavaScript enabled to view it
Proceedings of the 2007 IEEE International Conference on
Networking, Sensing and Control, London, UK, 15-17 April 2007
MonA02
1-4244-1076-2/07/$25.00 ©2007 IEEE 313
neighboring nodes in the network. Recurrent neural
networks have been applied for a WSN model due to
topological similarities with WSNs. Communication
uncertainties are modeled using “confidence factors” which
are based on the received signal strength. More detailed
communication models can be applied, but this is not a topic
of this paper.
There are many techniques for nonlinear dynamical
system identification using NNs. Bernieri et al. [10], [11]
compared output signals of a NN model and the sensor to
detect faults. Once the fault has been detected, the
parameters of the NN identifier are compared as well in
order to isolate a fault. Narendra and Parthasarathy [12]
demonstrated that NNs can effectively be used for the
identification and control of nonlinear dynamical systems.
Ahmed [13] presented a rapid neural network for
identification of unknown nonlinear dynamic systems when
the inputs and outputs are accessible for measurements.
Straub and Shroder [14] presented a new approach of
identifying nonlinear dynamic systems which is based on a
general regression NN.
In addition to neural networks, the identification of
nonlinear dynamics was studied using other techniques.
Gallman and Narendra [15] used an iterative algorithm for
obtaining the dynamics from finite length input and noisy
output data records which is shown to converge for a class
of inputs including colored Gaussian processes. Shiavo and
Luciano [16] presented a new powerful and flexible fuzzy
algorithm for nonlinear dynamic system identification
The rest of the paper is organizes as follows. In Section 2
we cover briefly background on RNNs and their function
approximation property. In Section 3, a modified RNN and
its modeling of a dynamic sensor node is introduced
including a result that shows how neighboring nodes can be
used in sensor node modeling. Section 4 describes how such
a tool can be used in sensors failure detection in a
distributed sensor networks. Numerical simulations are
given in Section 5 to demonstrate the effectiveness of the
proposed modeling scheme.
II. BACKGROUND
Artificial recurrent neural networks (RNNs) have the
ability to capture and model dynamic properties of the
nonlinear systems. The RNN nodes have their own
dynamics with interconnecting weights between the nodes –
similarly to the wireless sensor networks where each sensor
node has its own dynamics. Compared with standard neural
networks, recurrent networks include feedback loops as well
[12], [17], [18].
In this paper we use a nonlinear version of the auto-
regressive moving average model, namely, nonlinear auto-
regressive with exogenous inputs (NARX) model which was
presented by a seminal work of Narendra [12] and is given
by
))(,),2(),1(
),(),(,),2(),1(()(
nkukuku
kumkykykyFky NN
−−−
−−−=
K
K
(1)
where y(k) is the NN output, y(k-i) are previous NN outputs,
u(k-i) are inputs including previous inputs. The nonlinear
function FNN is computed using feedforward neural net
given in a matrix form by
)()( xVWxF TTNN σ= (2)
where x is the NN input, V is the first-layer weights, W is the
second layer weighs, and σ(⋅) is the neural net activation
function, usually chosen as standard sigmoid function.
Output activation function is chosen as a linear function.
The structure of the NN is given in Figure 1.
y(k)
1 1
1
2
L
V
Wσ
σ
σ
Z-1
Z-1
.
.
.
Z-1
...
y(k-1)
y(k-m)
Z-1
Z-1
...
u(k)
u(k-1)
u(k-n)
y(k)
1 1
1
2
L
V
Wσ
σ
σ
Z-1
Z-1
.
.
.
Z-1
...
y(k-1)
y(k-m)
Z-1
Z-1
...
u(k)
u(k-1)
u(k-n)
Figure 1. Two-layer, auto-regressive with exogenous inputs
neural net.
The two-layer NN in Figure 1 consists of two layers of
tunable weights and thresholds and has a hidden layer and
an output layer. The hidden layer has L neurons, and the
input layer is a combination of delayed input u(k) and the
output y(k).
Many well-known results indicate that any sufficiently
smooth function can be approximated arbitrary closely on a
compact set using a two-layer NN with appropriate weights
[19], [20]. Both layer weights, V and W can be tuned. The
NN universal approximation property says that any
continuous function f can be approximated arbitrarily well
using a linear combination of sigmoidal functions, namely
)()()( xxVWxf TT εσ += , (3)
where )(xε is the NN approximation error. The
reconstruction error is bounded on a compact set S by
Nx εε <)( . Moreover, for any Nε one can find a NN such
that Nx εε <)( for all Sx∈ .
III. MODIFIED RECURRENT NEURAL NETS IN SENSOR
NETWORK MODELING
Dynamic RNNs consist of a set of dynamic nodes that
provide internal feedback to their own inputs, see Figure 1.
They can be used to model dynamic systems such as a
network of sensors. WSNs consist of a large number of
sensors, which in turns have their own dynamics. They
interact between themselves and the base station which
controls the network. In a multi-hop WSN, information hops
314
from one node to another, and finally to the network
gateway or the base station.
To dynamically model such sensors, without a loss of
generality, we assume that there is one sensor per sensor
node. More sensors per node will just increase the size of the
RNNs.
Sensor nodes can be viewed as small dynamic systems
with memory-like features. Output of one node forwards the
information to the next node (for example node 3 provides
the input to node 5, Figure 2). While the standard RNN is
structured in layers, we introduce an ad-hoc RNN analogous
to WSN systems with confidence factors ( 10 << ijC )
between nodes i and j. It is assumed that communication
links are symmetric, i.e., if the sensor node i communicates
with the sensor node j, then opposite is also true with the
same confidence factor. The confidence factor depends on
the signal strength and data quality in communication links
between the nodes. For instance, in tuning node 2, valuable
inputs are coming from nodes 1, 3, 4, and 5 providing that
corresponding confidence factors are close to 1. If node 7 is
not in the coverage area of node 2, then confidence factor is
0 and node 7 will not influence node 2 directly.
RNN Nodes
Confidence Factors
11
33
55
22
66
77
44
C13
C35
C23
C12 C25
C56
C67
C45
C47
C24
RNN Nodes
Confidence Factors
11
33
55
22
66
77
44
C13
C35
C23
C12 C25
C56
C67
C45
C47
C24
Figure 2. Ad hoc recurrent neural network with topology of a
wireless sensor network.
Note that the confidence factors do not provide
stochastic modeling of the communication channel. The
overall modeling process can be divided into two phases: the
learning phase where the neural network (NN) adjusts its
weights that correspond to the healthy and N faulty models,
where N is the number of fault types, and the production
phase where the current output of the sensor node is being
compared with the output of the NN. The difference
between these two signals is used as a measure of a sensors
health status. In case of a fault, NN weights (model) are
compared with the faulty models to isolate the fault. If there
is no similar fault model, then the fault bank model is
updated with the new type of fault and the corresponding
parameters. The whole process is being repeated during the
production phase.
Consider a nonlinear dynamical sensor model given by
))(),(,),2(),1(()( kumkykykyfky iiiiii −−−= K (4)
where )(kui , )(kyi are the sensor input and output at
sample k, and if ’s are unknown nonlinear functions. In
order for sensor to be operational and user to be able to
determine the real sensor input, the function if has to be
invertible
))(),(,),2(),1(()(
1
kymkykykyfku iiiiii −−−=
−
K . (5)
Expression (5) indicates that in order to determine the
physical input at the sample k, one has to know the present
and past m sensor outputs. More general dynamical sensor
model is given by [16]
))(),(,),2(),1(()( kumkykykyfky iiiiii rK −−−= , (6)
where )(kuir is a vector of input data
)](,),1(),([)( nkukukuku iiii −−= Kr . Similarly, it is
assumed that the nonlinear function is invertible with respect
to input signal arguments
))(),(,),2(),1(()( kymkykykyfku iiiiii −−−= Kr . (7)
Such sensor models correspond to the general sensor
model given in [11] i.e., Hammerstein-Wienner nonlinear
feedback dynamic sensor model (Figure 3) which involves
linear dynamic block surrounded by three nonlinear static
blocks [21].
Static
Nonlinearity Linear Dynamics
Static
Nonlinearity
Static
Nonlinearity
Physical
Input
Measured
OutputStatic
Nonlinearity Linear Dynamics
Static
Nonlinearity
Static
Nonlinearity
Physical
Input
Measured
Output
Figure 3. A linear dynamic block surrounded by three static
nonlinear blocks representing a Hammerstein-Wiener dynamic
sensor model.
It is assumed that all sensors have models of the same
order. If that is not the case the analysis can still be carried
out with slight modification.
Assumption 1: Sensor nodes have a nonlinear model of
the same order given by (4).
Assumption 2: Functions if ’s are globally Lipschitz
functions with Li’s being their Lipschitz constant
respectively.
While wireless sensor nodes are distributed in the field,
it is assumed that the neighboring nodes have bounded
difference in measured physical quantity. Mathematically
the assumption is given as follows.
Assumption 3: Neighboring sensor nodes have
measurement events that differ by a bounded constant, i.e.,
for a sensor nodes neighbors a and b,
)()()( kekuku abba =− , (8)
and eke
ab <)( .
The next result shows how to model a wireless sensor
network using a recurrent neural network and how to use
such tool in a failure detection of sensor nodes.
315
Theorem 1 (Wireless sensor network model using RNNs):
Having a model of a sensor node i (4), Assumptions 1-3, and
the node neighbors that include nodes i1, i2,..., iNi (see
Figure 4), the output of the sensor node can be approximated
using recurrent neural network with inputs consisting of the
previous outputs from node i and its neighboring nodes
cmkykyky
mkykykyRNNky
jijiji
iiiii
+−−
−−−=
))(,),1(),(
),(,),2(),1(()(
K
K
(9)
where j=1, 2,..., Ni, and c is a small bounded constant.
i
i1 i2
iNi
i
i1 i2
iNi
Figure 4. Sensor node i and its neighboring
sensors i1, i2, ..., iNi.
Proof:
From Assumption 3 it follows that
)()()( kekuku ij
jii += , (10)
where j=1, 2,..., Ni . Equivalently, the input )(kui is given
by
∑
= +=
iN
j ijji
i
i keku
Nku 1 )()(
1
)( . (11)
Therefore one has
))()(1
),(,),2(),1(()(
1∑= +
−−−=
iN
j ijji
i
iiiii
keku
N
mkykykyfky K
(12)
Using expression (5) one has
))())(),(,),2(),1((1
),(,),2(),1(()(
1
1
∑
=
−
+−−−
−−−=
iN
j ijjijijijiji
i
iiiii
kekymkykykyf
N
mkykykyfky
K
K
(13)
Knowing that the function fi is Lipschitz yields
dmkykyky
mkykykygky
jijiji
iiiii
+−−
−−−=
))(,),1(),(
),(,),2(),1(()(
K
K
(14)
where j=1, 2,..., Ni , and )max( jLed ≤ .
Using NN function approximation property, there is a
recurrent neural network that approximates the unknown
function gi such that
)()(
))(,),1(),(
),(,),2(),1((
xxRNN
mkykyky
mkykykyg
ii
jijiji
iiii
ε+=
−−
−−−
K
K
(15)
where the vector x is given by
)](,),1(),(
),(,),2(),1([
mkykyky
mkykykyx
jijiji
iii
−−
−−−=
K
K
(16)
The bounded constant c is then given by
)max( jNi Lec +=ε . (17)
This completes the proof.
This shows that the sensor node output can be
approximated as a RNN with inputs as m previous output
samples of a same node, and present and m previous output
samples of neighboring sensors.
Previous result assumes ideal communication links. In
case that there are communications link uncertainties, the
exact value of )(ky
ji is not available. Instead, we use output
values of the neighboring sensor nodes combined with
confidence factors, i.e., )(kyC
jiji . Then, the recurrent neural
net sensor node models are given by
cmkyCkyCkyC
mkykykyRNNky
jijijijijiji
iiiii
+−−
−−−=
))(,),1(),(
),(,),2(),1(()(
K
K
(18)
Confidence factors for sensor node i are proportional to the
signal strength between the node i and its neighbors. More
detailed communication modes can be included, but it is not
a topic of this paper.
IV. APPLICATION TO A SENSOR NODE FAULT DETECTION
Previous results provide a tool to approximate a wireless
sensor node output using recurrent neural networks. The
method can be applied to a wide range of nonlinear dynamic
models. A motivation for the above results stems from the
need to detect faults in a network of distributed, wireless
network of sensor nodes.
In order to detect possible sensor node faults, we
compare the real output and the recurrent neural net (RNN)
approximation model. If such a difference is larger than a
threshold then there is a fault at the sensor. Having a sensor
node i, its real output )(kyi , and a RNN model output
)(kRNNi , if iii kykRNN η≥− )()( , then there is a fault at
the sensor node i.
Figure 5.a and Figure 5.b show the structure of the
modified recurrent network with its inputs consisting of the
delayed output signals of the same NN, and the previous and
current modified output signals from neighboring sensors. It
is initially assumed that all confidence factors between node
i and the neighboring nodes are equal to 1. Figure 5.a shows
the topology during the learning phase and Figure 5.b during
the production phase, where a fault analyzer detects the
difference between the sensor and modified RNN.
316
Modified
Recurrent
Neural
Network
)(
)1(
)(
1
1
1
mky
ky
ky
i
i
i
−
−
M
)(
)1(
)(
2
2
2
mky
ky
ky
i
i
i
−
−
M
)(
)1(
)(
mky
ky
ky
N
N
N
i
i
i
−
−
M
M
Sensor 1i
Sensor
Sensor
2i
Ni
Sensor i)(kui )(kyi
)(kRNNi
LEARNING
ALGORITHM
iiC1
iiNC
iiC2
M
)(
)2(
)1(
mkRNN
kRNN
kRNN
i
i
i
−
−
−
M
Modified
Recurrent
Neural
Network
)(
)1(
)(
1
1
1
mky
ky
ky
i
i
i
−
−
M
)(
)1(
)(
2
2
2
mky
ky
ky
i
i
i
−
−
M
)(
)1(
)(
mky
ky
ky
N
N
N
i
i
i
−
−
M
M
Sensor 1i
Sensor
Sensor
2i
Ni
Sensor i)(kui )(kyi
)(kRNNi
LEARNING
ALGORITHM
iiC1
iiNC
iiC2
M
)(
)2(
)1(
mkRNN
kRNN
kRNN
i
i
i
−
−
−
M
Figure 5.(a) Block diagram of the system identification in the
learning phase.
Modified
Recurrent
Neural
Network
)(
)1(
)(
1
1
1
mky
ky
ky
i
i
i
−
−
M
)(
)1(
)(
2
2
2
mky
ky
ky
i
i
i
−
−
M
)(
)1(
)(
mky
ky
ky
N
N
N
i
i
i
−
−
M
M
Sensor 1i
Sensor
Sensor
2i
Ni
Sensor i
FAULT
ANALYZER
)(kui )(kyi
)(kRNNi
Threshold
Fault
Alarm
iiC1
iiNC
iiC2
M
)(
)2(
)1(
mkRNN
kRNN
kRNN
i
i
i
−
−
−
M
Modified
Recurrent
Neural
Network
)(
)1(
)(
1
1
1
mky
ky
ky
i
i
i
−
−
M
)(
)1(
)(
2
2
2
mky
ky
ky
i
i
i
−
−
M
)(
)1(
)(
mky
ky
ky
N
N
N
i
i
i
−
−
M
M
Sensor 1i
Sensor
Sensor
2i
Ni
Sensor i
FAULT
ANALYZER
)(kui )(kyi
)(kRNNi
Threshold
Fault
Alarm
iiC1
iiNC
iiC2
M
)(
)2(
)1(
mkRNN
kRNN
kRNN
i
i
i
−
−
−
M
Figure 5.(b) Block diagram of the system identification in the
production phase.
V. SIMULATION RESULTS
We have simulated a sensor network with 15 sensors
nodes and one sensor per node. Each sensor has 2 or 3
“visible” neighbors. Of course, if sensor i is a neighbor of
j then the opposite is also true.
Each sensor is modeled as a Hammerstein-Wienner [21]
nonlinear feedback dynamic sensor (Figure 3), where the
nonlinearity part is an arctan(·) function and the dynamical
element is given by
1
1
)(
2
++= sssH . Input to the sensor i
during both training and production phases is chosen as
)(
3
2
sin10)( tnkiku ii +⎟
⎠
⎞
⎜
⎝
⎛ +
+= π (19)
where )(tni is a white noise at sensor node i with the
variance of 0.6 and the sampling time is equal to 0.1 sec.
Each sensor is modeled using a modified recurrent
neural network (MRNN) described in a previous section. A
MRNN node has input consisting of delayed output samples
of the same node and current and previous outputs of the
neighboring sensor nodes. At first, we assumed confidence
factors equal to one and later we make a more realistic
assumption with confidence factors less than one.
In the simulation, the recurrent neural network has input
layer with 8 nodes, hidden layer with 10 nodes, and output
layer with one node. The learning algorithm is the standard
backpropagation. The learning rates for the first layer and
the hidden layer are set to 0.01 The learning phase stopped
after the difference between expected and actual NN output
reached a steady-state value. The simulation software used is
Microsoft Visual C++ .NET
For the sensor #1 results are shown in Figure 6. The
output of the MRNN closely approximates the actual output
of the sensor with a small error. The MRNN model can
certainly reproduce the dynamic behavior of the sensor.
Figure 7 shows the discrepancy between actual output of
sensor #1 and its MRNN model after simulation with
confidence factors set to 1.
-0.2
0.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
1.6
1.8
9.00 9.05 9.14 9.26 9.58 9.78 10.00 10.21 10.42 10.74 10.87 10.96 11.00
Input
sensor #1 output
neural model
Figure 6. Actual output of sensor #1 and its modified recurrent
neural network model with confidence factors set to 1.
-0.3
-0.1
0.1
0.3
0.5
0.7
0.9
9.00 9.05 9.14 9.26 9.58 9.78 10.00 10.21 10.42 10.74 10.87 10.96 11.0
Input
sensor output-neural
model
Figure 7. Discrepancy between actual output of sensor #1 and its
modified recurrent neural network model with confidence factors
set to 1.
During the learning phase Figure 8 shows the evolution
of the difference between neural network model and the
actual sensor output. Notice that this error decreases as the
number of iterations increases.
317
-0.7
-0.6
-0.5
-0.4
-0.3
-0.2
-0.1
0
0 50 100 150 200 250 300 350 400 450
k
sensor #1 error
Figure 8. Evolution of the difference e(k) between the MRNN
model and the actual output of sensor #1 with confidence factors
set to 1.
A more realistic assumption is to consider confidence
factors between nodes less than one. Taking
,8.021 =C ,6.031 =C and 95.041 =C , the results for sensor
node 1 are shown in Figure 9.
confidence factors < 1
0.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
1.6
1.8
9.00 9.05 9.14 9.26 9.58 9.78 10.00 10.21 10.42 10.74 10.87 10.96 11.00
Input
sensor #1 output
neural model
Figure 9. Actual output of sensor #1 and its modified recurrent
neural network model with confidence factors less than 1.
Figure 10 shows the sampled output of sensor #1 when
this sensor has a fault (drift) starting at 1.6 seconds. Also
shown is estimated MRNN output when the sensor was in a
normal healthy mode.
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
0.00 2.00 4.00 6.00 8.00 10.00 12.00
time (s)
Faulty Sensor #1
Output
NN Ouput
Figure 10. Output of faulty sensor #1 and its MRNN model.
VI. CONCLUSION
This paper describes a dynamic model of a wireless
sensor network and its application to a sensor failure
detection and identification. A recurrent neural network is
used to model a dynamic behavior of a sensor. The overall
network model corresponds to topology of the wireless
sensor network. The inputs to the NN are taken from the
node that is modeled and neighboring nodes. Simulation is
example includes modeling a WSN with 15 nodes and
successful detection of a sensor drift.
VII. REFERENCES
[1] Crossbow, http://www.xbow.com
[2] Xiao Di, B. K. Gosh, Xi Ning, T. J. Tarn. “Sensor-based hybrid
position/force control of a robot manipulator in an uncalibrated
environment,” IEEE Transactions on Control Systems
Technology. vol. 8, no. 4, pp. 635-645, July 2000.
[3] S. E. Lysheyski, “Smart flight control surfaces with
microelectromechanical systems,” Aerospace and Electronic
Systems, IEEE Transactions on Control Systems Technology, vol.
38, no. 2, pp. 543-552, Apr. 2002.
[4] Pouliezos A. D. and G. S. Stavrakankis, Real Time Fault
Monitoring of Industrial Processes, Kulwer Academic Publishers,
1994.
[5] Zhirabok, O. V. Preobragenskaya, “Instrument fault detection in
nonlinear dynamic systems,” Proc. Systems, Man and
Cybernetics, 1993.
[6] C. B Weinstock, W. L Heirnerdinger, H. Ihara, S. B. Johnson, H.
D. Kirrmann, J. J. Stiffler, L. Yount, “The state of the practice in
fault tolerant systems,” 22nd international Symposium on Fault-
Tolerant Computing, July 1992.
[7] S. C. Lee, “Sensor Value Validation Based on Systematic
Exploration of the sensor Redundancy for Fault Diagnosis”, IEEE
Trans. on systems, Man and Cybernetics, vol.24, no. 4, pp. 594-
605, Apr. 1994.
[8] M. L. Leushen, J. R. Cavallaro, I. D. Walker, “Robotic fault
detection using analytical redundancy”, in Proc. IEEE Conf.
Robotics and Automation, 2002, pp. 456-463.
[9] F. Koushanfar, M. Potkonjak, A. Sangiovanni-Vincentelli, “Fault
tolerance techniques for wireless ad hoc sensor networks,” in
Proc. IEEE Conf. Sensors, 2002, pp. 1491-1496.
[10] A. Bernieri, M. D’Apuzzo, L. Sansone, and M. Savastano, “A
neural network approach for identification and fault diagnosis on
dynamic systems,” IEEE Trans. on Instrumentation and
Measurement, vol. 43, no. 6, pp. 867-873, Dec. 1994.
[11] A. Bernieri, G. Betta, A. Pietrosanto, and C. Sansone, “A neural
network approach to instrument fault detection and isolation”, in
Proc. IEEE Conf. Instrumentation and Measurement, 1994, pp.
139-144.
[12] K. S. Narendra and K. Parthasarathy, “Identification and control
of dynamical systems using neural networks”, IEEE Transactions
on Neural Networks, vol. 1, no. 1, pp. 4-27, March. 1990.
[13] R. S. Ahmed, “Identification of nonlinear dynamic systems using
a rapid neural network”, the 27th Annual Conference of the IEEE,
vol. 3, Dec. 2001, pp. 1734-1739.
[14] S. Straub, D. Shroder, “Identification of nonlinear dynamic
systems with recurrent neural networks and Kalman filter
methods”, 1996 IEEE International Symposium, 12-15, May
1996, pp. 341-344.
[15] P. G. Gallman and K. S. Narendra, “Identification of nonlinear
systems using a Uryson model”, Automatica, Nov. 1976.
[16] A. Lo Shiavo and A. M. Luciano, “Powerful and flexible fuzzy
algorithm for nonlinear dynamic system identification”, IEEE
Transactions on Fuzzy Systems, vol. 9, no. 6, pp. 828-835, Dec.
2001.
[17] S. Haykin, Neural Networks – A Comprehensive Foundation
(second ed.), Prentice-Hall, Upper Saddle River, NJ, 1998.
[18] F. L. Lewis, S. Jagannathan, and A. Yesilidrek, Neural Network
Control of Robot Manipulators and Nonlinear Systems, Taylor
and Francis, Philadelphia, PA, 1999.
[19] G. Cybenko, “Approximation by superpositions of a sigmoidal
function,” Math. Contr. Signals, Syst., vol. 2, no. 4, pp. 303- 314,
1989.
[20] K. Funahashi, “On the approximate realization of continuous
mappings by neural networks,” Neural Networks, 2. 183-192.
[21] Dennis S. Bernstein, “Sensor Performance Specifications”, IEEE
Control Systems Magazine, pp 9-18, Aug. 2001.
Fault Instant
318 |
|
Last Updated ( Sunday, 20 April 2008 )
|
|
Moustapha Home 
Moustapha Blog 
