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Non-parametric modelling of a rectangular flexible plate structure
Intan Z. M. Darus
n
, Ali A. M. Al-Khafaji
Department of System Dynamics & Control, Faculty of Mechanical Engineering, Universiti Teknologi Malaysia (UTM), 81310 Skudai, Johor Bahru, Malaysia
article info
Article history:
Received 20 October 2010
Received in revised form
15 August 2011
Accepted 1 September 2011
Available online 29 September 2011
Keywords:
Active vibration control
ANFIS
Flexible plate
Neural network
System identification
abstract
This research investigates the performance of dynamic modelling using non-parametric techniques for
identification of a flexible structure system for development of active vibration control. In this paper,
the implementation details are described and the experimental studies conducted in this research are
analysed. The input–output data of the system were first acquired through the experimental studies
using National Instruments (NI) data acquisition system. A sinusoidal force was applied to excite the
flexible plate and the dynamic response of the system was then investigated. Non-parametric
modelling of the system were developed using several artificial intelligent methodologies namely
Adaptive Elman Neural Networks (ENN), Backpropagation Multi-layer Perceptron Neural Networks
(MLPNN) and Adaptive Neuro-Fuzzy Inference System (ANFIS). The performance of all these meth-
odologies were compared and discussed. Finally, validation and verification of the obtained model was
conducted using One Step Ahead (OSA) prediction, mean squared error (MSE) and correlation tests. The
prediction ability of the model was further observed with unseen data. The results verified that the
MLPNN converge to an optimum solution faster and the dynamic model obtained described the flexible
plate structure very well. The non-parametric models of the flexible plate structure thus developed and
validated will be used as the representation of the transfer function of the system in subsequent
investigations for the development of active vibration control strategies for vibration suppression in
flexible structures.
&2011 Elsevier Ltd. All rights reserved.
1. Introduction
In the past three decades, the use of flexible structure systems
has been growing quickly in many engineering applications. The
elements for flexible structure such frames, shells, beams and
plates are extensively used in a wide range of manufacturing
applications and particularly in mechanical, civil, marine, aero-
nautical, aerospace and other areas of practical attention, for
example, flexible manipulators of satellites, solar panels, etc.
Plates with different shapes, boundary conditions at the edges
and various complicated effects have often found applications in
different structures such as aerospace, machine design, telephone
industry, nuclear reactor technology, naval structures and earth-
quake-resistant structures. Particularly, the dynamic behaviour of
flexible, flat, thin, rectangular plates has received huge attention in
recent years because of its technical importance (Chakraverty, 2009).
The flexible thin rectangular plates structures are the most
commonly used in the industrialised world and in abroad range of
engineering applications, for examples, electronic circuit board
design, solar panels and bridge decks. The stability of the plate,
where it is subjected to loading, would be associated with a range
of physical effects that lead to high vibration. The high vibration
of flexible structure systems cause noise, fatigue, wear, destruc-
tion, human discomfort and reduced system effectiveness. That is
why the vibration of flexible structure needs to be controlled. Due
to its multiple practical problems and applications, the vibration
of the elastic plates has been treated widely from researchers
with different boundary conditions, both from theoretical and
experimental points of view (Chakraverty, 2009). It is necessary to
find an approximate or accurate model of the plate structure to
control the vibration of a plate well. Suitable modelling of a
dynamic system, for instance a flexible plate, would result in good
control (Tavakolpour et al., 2010).
In the initial stages, results were available for some simple cases,
namely a limited set of boundary conditions and geometries, in
which the analytical solution could be found. With the advent of fast
computers and various efficient numerical methods, there has been
a big increase in the amount of research done for getting better
accuracy in the results. Numerical methods offer reasonable and
accepted solution but with complex shapes of plate sometimes lead
to inaccuracies and more computing time (Chakraverty, 2009).
To predict the physical system behaviour under different
operating conditions or to control it, a model can be created
using an approach called system identification.
Contents lists available at SciVerse ScienceDirect
journal homepage: www.elsevier.com/locate/engappai
Engineering Applications of Artificial Intelligence
0952-1976/$ - see front matter &2011 Elsevier Ltd. All rights reserved.
doi:10.1016/j.engappai.2011.09.009
n
Corresponding author. Tel.: þ60167616716.
E-mail address: intan@fkm.utm.my (I. Z. M. Darus).
Engineering Applications of Artificial Intelligence 25 (2012) 94–106
In the present decade, system identification techniques have
become potential candidates to many control application. Para-
metric and non-parametric system identification methods used to
find approximate or accurate models of dynamic systems depend
on observed inputs and outputs (Mat Darus, 2004). The major aim
of system identification is to locate approximate or accurate
models of dynamics systems depend on observed inputs and
outputs. A number of researchers have applied techniques to
solve the problems related to system identification. Several
methods have been devised to find out models that describe
input output behaviour of a system well (Ismail et al., 2006a).
Ismail et al. (2006a) have reported identification algorithms of
flexible structure using Neural Networks. The research reported a
study into the development of system identification methods for
dynamic modelling and characterisation of flexible plate struc-
tures. The research uses Least Squares and Recursive Least
Squares to find linear parametric model of the system. In addition,
non-parametric models of the system are developed using Elman
Neural Networks (ENN) and Multi-layer Perceptron Neural Net-
works (MLPNN; Ismail et al., 2006a).
Mohd Hashim et al. (2004) have reported non-linear dynamic
modelling of flexible beam structures using Neural Networks. The
research investigated the utilisation of neural network (NNs)
backpropagation for modelling flexible beam with fixed-free
mode. Comparative analysis of the performance of the Recursive
Least Squares scheme and intelligent Neural Networks model in
characterizing the system was carried out in the frequency and
time domains. Simulated results have shown that by Neural
Networks the system is modelled better than with the conven-
tional linear modelling method (Mohd Hashim et al., 2004).
Mat Darus et al. (2008) have reported Adaptive Neuro-model-
ling of a twin rotor system. The research investigated the utilisa-
tion of Adaptive Neural Networks (NNs) for dynamic modelling
and identification of a highly non-linear TRMS system. An
adaptive Elman neuro-model is designed to characterise a twin
rotor multi-input multi-output system (TRMS) in vertical motion
based on one step-ahead (OSA) prediction. The results obtained,
in both frequency and time domains, are compared to the
identification using the conventional adaptive technique of Recur-
sive Least Squares (RLS). Simulations indicate the superiority of an
adaptive neuro-modelling technique over RLS algorithm in mod-
elling and identification of the TRMS (Mat Darus et al., 2008).
Ismail et al. (2006b) have reported dynamic characterisation of
flexible vibrating structures using adaptive neuro-fuzzy inference
system (ANFIS). In this research ANFIS was used to develop a
model characterizing the vibration of the plate. The input/output
data used in this research was obtained from a simulation of a
square, flat, flexible plate with all edges clamped using finite
difference (FD) algorithm (Ismail et al., 2006b).
Toha et al. (2008) have reported ANFIS modelling of a twin
rotor system. An Adaptive Neuro-Fuzzy Inference System (ANFIS)
network design is deployed and used for modelling a twin rotor
multi-input multi-output system (TRMS). It is demonstrated
experimentally that ANFIS can be effectively used for modelling
the system with highly accurate results. The accuracy of the
modelling results is demonstrated through validation tests
including training and test validation and correlation tests (Mat
Darus and Tokhi, 2006).
System identification is a broad idiom used to describe algo-
rithms and mathematical tools that build dynamical models from
measured data. Over the last two decades system identification
has received a lot of attention. System identification methods are
widely used as a fundamental requirement in scientific applica-
tions and engineering. The practical application domains include
the Boolean function generation, symbolic regression and pattern
recognition and time-series prediction. The problem of finding an
approximate or accurate model for dynamical systems occurs
often in engineering applications. System identification is one
way to solve this problem (Ismail et al., 2006a).
The major aim of system identification is to find approximate
or accurate models of dynamic systems depend on observed
inputs and outputs. When a model of the physical system is
obtained, it can be used for solving different problems; such as to
predict its behaviour under different operating conditions or
control the physical system. Numerous researchers have applied
techniques to solve the system identification problems. A number
of methods have devised to obtain models that best describe
input output behaviour of a system.
The reason of this study is to develop a model characterizing
vibration of two-dimensional flexible rectangular plate structures
using non-parametric identification techniques as soft computing.
In this work, a thin rectangular plate with all edges clamped is
considered. Prior to this, a dynamic model of the plate structure
based on laboratory experiments characterizing the flexible plate
structure is developed. Finally, the validity of the obtained model
was investigated using correlation tests. The procedure of system
identification can be represented as shown in Fig. 1.
2. Experimental setup
From theoretical and experimental points of view, the vibra-
tion of the flexible plates has been treated extensively. The
vibration of a plate can be excited and detected with a suitable
experimental setup. Accurate understanding of the results allows
us to achieve useful information. Several researchers made
experimental studies with different types of setup and instru-
mentation to measure the vibration parameters and to control the
plate (Shimona et al., 2005;Shimon and Hurmuzlu, 2007;Qiu
et al., 2009).
In this investigation, the input–output data of the system were
first acquired through the experimental studies using National
Instruments (NI) data acquisition system. To provide experimental
data, a rectangular plate with dimensions of 1 m 1.5 m 0.003 m
was investigated. To allow a 0.04 m width clamped boundary on all
four sides, the rectangular plate was cut as 1.58 m 1.08 m. The
experimental arrangement developed for this study was established
as shown in Fig. 2.
To acquire the perfect conditions of clamped boundaries, four
steel bars with rectangular cross sections were used to clamp the
plate edges giving 40 mm as the thickness of the clamped
Fig. 1. Schematic description of the system identification procedure.
I.Z.M. Darus, A.A.M. Al-Khafaji / Engineering Applications of Artificial Intelligence 25 (2012) 94 –106 95
boundaries. The opposing faces and the inside edges of each bar
were milled flat and straight. The plate with the clamped
boundaries was then fixed firmly to a main frame. To tie up the
frame pieces jointly, bolts of type M9 were used, with shoulder
bolts being used to ensure precise alignment of the opposing
frame sections upon final assembly. The test structure was
excited by a sinusoidal force generated by a magnetic shaker
and applied at the excitation point as shown in Fig. 3. To sense the
plate response at the desired detection and observation points, a
piezo-beam type accelerometer (Kistler-8636C5) with sensitivity
of 1004 mV/g was used as shown in Fig. 4. For direct connection
of the piezo-type accelerometers, the acceleration signal was
acquired through a National Instruments (NI) compact-data
acquisition unit as shown in Fig. 5, which is equipped with NI-
9234 module (with 24-bit resolution) as shown in Fig. 6. The
necessary signal conditioning circuits such as anti-aliasing filter
have been built-in into the data acquisition system. The acquired
signal was then analysed using Intel Core TM Duo Processor and
LabVIEW software.
PC+LabVIEW Shaker DAQ
Function
generator
Piezo actuator
amplifier
Power amplifier Piezoelectric actuator
Fig. 2. Experimental setup.
Fig. 3. Electromagnetic shaker.
Fig. 4. Piezo-beam type accelerometer.
Fig. 5. NI compact-data acquisition.
I.Z.M. Darus, A.A.M. Al-Khafaji / Engineering Applications of Artificial Intelligence 25 (2012) 94 –10696
3. Model structure
A range of model structures are available to assist in modelling
a system. The choice of model structure depends on an insight
into and understanding of the system undergoing identification
and understanding of the system identification method. Non-
linear autoregressive moving average model with exogenous
inputs (NARMAX) is the most renowned for non-linear models
(Mat Darus, 2004;Lennart, 1999). It is obvious from the literature
that if the plant’s input and output data are obtainable, the
NARMAX model is an appropriate option with standard back-
propagation learning algorithms for modelling non-linear sys-
tems. Mathematically the model is given by Eq. (1)
^
y¼fðuðt1Þ,...,uðtn
u
Þ,yðt1Þ,...,yðtn
y
Þ,eðt1Þ,...,eðtn
e
ÞÞ ð1Þ
where ^
yrepresents the output vector determined by the past
values of the system input vector, output vector and noise. n
u
,n
y
and n
e
represent model orders. f( ) represent the system mapping,
which can be constructed through non-parametric methods with
a suitable learning algorithm. If the model is acceptable to
identify the system without noise term incorporated or the noise
is considered as additive term at the output, the model can be
represented in the NARX form (Mat Darus, 2004;Lennart, 1999).
This research used the NARX model. Mathematically the model in
Eq. (1) can be written in discrete form as in Eq. (2):
y¼fðuðk1Þ,...,uðkn
u
Þ,yðk1Þ,...:,yðkn
y
ÞÞþeðtÞð2Þ
4. Non-parametric identification
For non-parametric estimation, most popular members of non-
parametric models such as fuzzy logic (FL) and neural networks
(NNs) are normally utilised. Neural networks have a variety of
attractive characteristics such as generalisation ability, distributed
representation and computation, adaptability massive parallelism
and inherent contextual information processing. NNs used widely
in a range of identification and control applications owing to the
proficient nature of their working principles and other attractive
features. Amongst the diverse types of NNs, the Radial-Basis Function
(RBF), Multi-Layered Perceptron Neural Network (MLPNN), and
Elman recurrent Neural Network (ENN) are usually used in identifi-
cation and control of dynamic systems. The most attractive applica-
tions propose a suitable grouping of two methods, NN and FL
resulting in a hybrid system, ANFIS, which both operate on linguistic
descriptions of the variables and the numeric values through a
parallel and fault tolerant architecture (Mat Darus, 2004).
4.1. Artificial Neural Networks
An Artificial Neural Network is composed of a massive amount
of extremely interconnected processing elements called neurons
working as one to solve specific problems. It is an information
processing concept which is stimulated by the way biological
nervous systems, for instance the brain. Artificial Neural Net-
works, like people, learn by example. An Artificial Neural Network
is constructed for a specific application, for instance data classi-
fication or pattern recognition, through a learning process. Learn-
ing in biological systems involves adjustments to the synaptic
connections that exist between the neurons (Aleksander and
Morton, 1990).
4.1.1. Structure of Neural Networks (NNs)
In Neural Networks, nodes or computational models are joined
through weights that are modified during use to improve perfor-
mance. The key idea is to obtain best performance through large
interconnection of simple computational elements. The simple
node provides a linear combination of Nweights w
1
,w
2
,y,w
n
and
Ninputs x
1
,x
2
,y,x
n
and passes the result through a nonlinearity
F
, as shown in Fig. 7. Models of NNs are specified by the net
topology, node characteristics and training or learning rules. NNs
are specified by
1. Node: normally a sigmoid function.
2. Layer: a set of nodes at the same hierarchical level.
3. Connection: constant weights or weights as a linear dynamical
system, feedfoward or recurrent.
4. Architecture: an arrangement of interconnected neurons.
5. Mode of operation: analogue or digital.
From Fig. 7, the equation y¼
^
ðP
i
x
i
w
i
þw
o
Þis a mathematical
explanation of a neuron where the input vector is given by x¼[x
1
,
x
2
,y,x
n
,1]
T
while w¼[w
1
,w
2
,y,w
N
,w
0
]
T
represents the weight
vector of a neuron.
4.1.2. Multi-layer Perceptron Neural Networks (MLPNN)
MLPNN is considered as possibly the most frequently used
member of the neural network family. The major reason for this is
its capability to model simple functional relationship in addition
to very complex one. This has been demonstrated throughout a
large amount of practical applications including system identifi-
cation, speech and natural language processing, prediction and
control, function approximation and pattern recognition. An
MLPNN is able to representing the Boolean functions and forming
arbitrary decision boundaries (Demuth et al., 1992).
A MPLNN needs a set of data, to be presented as inputs to the
input node element layer. The outputs from this layer are fed to
Fig. 6. NI-9234 module.
Fig. 7. Connectives within a node (Mat Darus, 2004).
I.Z.M. Darus, A.A.M. Al-Khafaji / Engineering Applications of Artificial Intelligence 25 (2012) 94 –106 97
the first hidden layer as weighted inputs, and subsequently the
outputs from the first layer are fed, as weighted inputs, to the
second hidden layer. This process continues until the output layer
is reached. In these networks it is assumed that the network can
be made up of any number of layers with reasonable number of
neurons in each layer, based on the nature of the particular
application.A generalised structure of MLP with its basic function
is shown in Fig. 7.
The input layer is formed from one layer of nodes and then the
second layer of nodes forms the output layer, with a number of
intermediate or hidden layers existing between them. In detail,
the input layer is the layer to which the input data is supplied and
the output layer is the layer from which the output is taken. All
other intermediate layers are called hidden layers. Usually, one,
two or even no hidden layers are employed. Fig. 8 depicts m
inputs and moutputs, it is not necessary for these values to be
equal. The layers are completely interconnected, that means each
neuron is connected to every neuron in the previous and succeed-
ing layers. However, the neurons in the same layer are not
connected to each other. A neuron performs two functions;
combining and activation. Several type of activation function
such as sigmoid, piecewise linear, threshold, Gaussian and tan-
sigmoid are used for activation.
The backpropagation (BP) algorithm, which is the most com-
monly adopted MLP learning algorithm, is a gradient descent
algorithm. The design of the BP learning algorithm for the MLPNN
is a landmark in the development history of neural networks.
Actually, the powerful properties of neural networks have been
well recognised after the introduction of BP learning algorithm.
The backpropagation algorithm was created by generalising the
Widrow and Hoff learning rule to multiple layer networks and
non-linear differentiable transfer functions. The network is
trained, using input vectors and the corresponding target or
output vectors, until it can associate input vectors with specific
output vectors, approximate a function or classify input vectors in
an appropriate way as defined by the user. Standard backpropa-
gation is a gradient descent algorithm, as is the Widrow and Hoff
learning rule, in which minimising the sum squared error
between the actual output and desired output. Through the BP
learning algorithm the mean squared error of the network is
minimised by continually adjusting the weights and biases in the
direction of the steepest descent with respect to the error. This is
known as gradient descent procedure. Any function with a finite
number of discontinuities can be approximated using Networks
with biases, with at least one sigmoid neuron layer, and a linear
output neuron layer. The best number of neurons and hidden
layers to be selected in NN can be obtained by a simple trial and
error or by optimisation technique. Correctly trained BP networks
tend to give reasonable answers when presented with inputs that
they have never seen. Typically, a new input leads to an output
similar to the correct output for input vectors used in training
that are similar to the new input being presented. This general-
isation property makes it possible to train a network on a
representative set of input target pairs and get good results
without training the network on all possible input output pairs
(Shaheed and Tokhi, 2001). Rules of the backpropagation algo-
rithm for the connection weights between hidden and output
layers are clearly described in Shaheed and Tokhi (2001).
This research studies the utilisation of backpropagation
MLPNN for modelling a single-input single-output flexible plate
system. Fig. 9 shows the diagrammatic representation of the
neural network algorithm.
4.2. Elman Neural Network (ENN)
Recurrent Neural Networks (RNNs) are the classes of Neural
Networks which contain cycles or feedback connections. An RNN
can take arbitrary topology as any node in the network may be
linked with any other node (including itself), while the set of
topologies of feedforward networks is fairly constrained. The
recurrent network developed by Elman has a simple architecture;
this network has been proved to be effective for modelling linear
systems not higher than the first order (Mandic and Chambers,
2001). Elman networks are two-layer backpropagation networks
with the addition of a feedback connection from the output of the
hidden layer to its input. This feedback path allows Elman
networks to recognise, to learn and to generate temporal patterns,
as well as spatial patterns.
Actually, the Elman neural network comprises four layers,
namely the input layer, hidden layer, output layer and context
layer that can store internal states. So, it belongs to special type of
feedforward neural network with additional memory neurons
and local feedback (Ismail et al., 2006a). Fig. 10 shows the
architecture of an Elman neural network.
Fig. 8. Multiple layers of feedfoward neural network (Ismail et al., 2006a).
Fig. 9. Diagrammatic representation of the neural network modelling algorithm.
Fig. 10. Structure of the Elman Neural Networks model.
I.Z.M. Darus, A.A.M. Al-Khafaji / Engineering Applications of Artificial Intelligence 25 (2012) 94 –10698
Symbol z
1
represents unit delay. It can be seen from Fig. 10
that in Elman Neural Networks Model, besides the input layer,
hidden layer, and output layer, also exists a context layer. Each
two adjacent layers are adjusted by connection weights. The input
and output layers interact with the outside environment, while
the hidden and context layers do not. The input layer is only
buffer layer that passes the signals without changing them. The
output layer is linear and it sums the signal fed to it. The hidden
layer can have linear or non-linear activation functions. The
context layer is used only to memorise the previous activations
of the hidden layer and can be considered to function as one-step
delay. Generally, it can be considered as a special type of
feedfoward neural network with additional memory neurons
and local feedback. The distinct self-connections of the context
nodes in Elman network make it sensitive to the history of input
data, which is essentially useful in modelling dynamic system
(Pham and Liu, 1995).
The research reported in (Ismail et al., 2006a) investigated the
utilisation of backpropagation Elman Neural Networks for mod-
elling a SISO flexible, flat, plate system. The diagrammatic
representation of an Elman neural network for system identifica-
tion is similar to MLPNN algorithms as shown in Fig. 9.
4.3. Adaptive Neuro-Fuzzy Inference System (ANFIS)
Neural networks and fuzzy inference systems are the most
popular members of the non-parametric methods. Fuzzy logic
based mechanisms employ the verbal power whereas neural
networks provide the mathematical power of the brain. The
largely attractive applications offer a suitable combination of
these two methods resulting in a hybrid system. The hybrid
system operates on both linguistic descriptions of the variables
and the numeric values through a parallel and fault tolerant
architecture (Mat Darus, 2004).
The term ANFIS, created by Jang, 1993 represents Adaptive-
Network-Based Fuzzy Inference System. It is classified as a hybrid
neuro-fuzzy model, constructed by combination of a fuzzy system
and a neural network into a uniform architecture (Nauck, 1999).
ANFIS can integrate human expertise in addition to adapt itself
through repeated learning. This architecture has verified a high
performance in many applications (Toha et al., 2008).
The network-type structure of ANFIS is similar to that of a
neural network; it can be used to interpret the input/output map.
ANFIS maps inputs through input membership functions and
associated parameters, and then through output membership
functions and associated parameters to outputs.
4.3.1. ANFIS architecture
ANFIS architecture uses Sugeno-type fuzzy system or Takagi and
Sugeno’s IF-THEN rules with appropriate membership functions
implant into adaptive networks. Fig. 11 presents type-3 fuzzy
reasoning where Takagi and Sugeno’s if-then rules are used
(Ismail et al., 2006b). Fig. 12 depicts the equivalent type-3 fuzzy
reasoning ANFIS architecture for system with two inputs xand y
(Jang, 1993). The rules for ANFIS structure are described in details in
(Mat Darus, 2004;Ismail et al., 2006b;Mat Darus and Tokhi, 2006).
Neural-adaptive learning methods present a technique for the
fuzzy modelling process to learn information about a data set, so
as to compute the membership function parameters that best
allow the associated fuzzy inference system to follow the given
input/output data. This learning method works likewise to that of
neural networks. To use ANFIS for identification problem, the
following steps are needed:
1. A Sugeno FIS appropriate for identification problem is design.
2. The FIS, with given actual input identification data is optimised.
3. Training and testing matrices composed of inputs and the
desired identification corresponding to those inputs are set up.
4. The ANFIS algorithm is run on the training data.
5. The results are tested using the testing data.
During the identification using ANFIS structure, input/output
data set are used to constructs a Fuzzy Inference System (FIS). The
membership function parameters in FIS are adjusted using either
a combination of a backpropagation algorithm and a Least
Squares type of method, or backpropagation algorithm alone.
This allows the fuzzy systems to learn from the data given.
In ANFIS, backpropagation learning is used to learn the para-
meters related to membership functions and least mean square
estimation to determine the consequent parameters. In the
learning procedure, every step includes two parts. The input sets
are propagated, and the optimal consequent parameters are
estimated by an iterative least mean square procedure. Through
w1
w2
A1
A2
B1
B2
xy
X
X
Y
Y
f1= p1x + q1y+r1
f2= p2x + q2y+r2
2211
21
2211 fwfw
ww
f wfw
f+=
+
+
=
Fig. 11. Type-3 fuzzy reasoning (Ismail et al., 2006b).
2
B
1
B
2
A
1
A
x
y
∏
∏
x
x
y
1
w
2
w
1
w
f
layer 1
layer 2 layer 3
layer 4
layer 5
Ν
Ν
∑
2
w
y
Fig. 12. Structure of type-3 ANFIS (Jang, 1993).
I.Z.M. Darus, A.A.M. Al-Khafaji / Engineering Applications of Artificial Intelligence 25 (2012) 94 –106 99
the training set, the premise parameters are assumed fixed for the
current cycle. The set is propagated again, and in this epoch,
backpropagation is used to adjust the premise parameters while
the consequent parameters remain fixed (Mandic and Chambers,
2001).
Through the learning process, the parameters associated with
the membership functions will be changed. The parameters’
adjustment is facilitated by a gradient vector, which provides a
measure of how well the fuzzy inference system is modelling the
input/output data for a given set of parameters. After obtaining
the gradient vector, any of several optimisation routines could be
applied to adjust parameters that will reduce some error measure
(usually defined by the sum of the squared differences between
actual and desired response; Sutton and Craven, 1998;Minghui
et al., 2009).
In this research the utilisation of Neuro-fuzzy model is
investigated using ANFIS structure to model the flexible, flat,
plate system. The diagrammatic representation of an ANFIS net-
work for system identification is similar to MLPNN algorithms as
shown in Fig. 9.
5. Model validation
After obtaining the model of the system, it is necessary to
validate whether the model is sufficient to represent the system
or no. The procedures that considered for sensing the sufficiency
of a fitted model are called Model validity tests. The principle of
model validation is:
Compare model simulation or prediction with real data in time
domain.
Compare estimated model’s frequency response and spectral
analysis result in frequency domain.
Perform statistical test on prediction errors.
Numerous validation tests are existing in the literature, some
of which are mean squared error, correlation error, model
predicted output and one step-ahead prediction (Mat Darus,
2004). In this research one step-ahead prediction, mean squared
error and correlation test are used to validate the model.
5.1. One step-ahead prediction
The One Step-Ahead (OSA) prediction of the system output is a
familiar measure of predictive accuracy used in system identifica-
tion and control. This is expressed as
^
y¼fðuðtÞ,uðt1Þ,...,uðtn
u
Þ,yðt1Þ,...,yðtn
y
ÞÞ ð3Þ
where uand yare the inputs and outputs respectively. f()isa
non-linear function. The error or prediction is given as shown in
Eq. (4).
eðtÞ¼yðtÞ^
yðtÞð4Þ
If the model is biased, ^
ywill be a quite good prediction of y(t)
over the estimation set because the model was estimated by
minimising the prediction errors (Tokhi and Veres, 2002).
5.2. Mean Squared Error (MSE)
MSE is one of the most common methods of validations. The
MSE is different between the real output y(n) of the system and
the predicted output ^
yðnÞproduced from the input to the system
and the optimised parameters as shown in Eq. (5).
mse ¼1
NX
N
t
ðyðtÞ^
yðtÞÞ
2
ð5Þ
5.3. Correlation test
Correlation test is a statistical test that shows the degrees of
the relationship between two variables. There are two types of
correlation test:
1. Autocorrelation test is representing as a vector.
2. Cross correlation test is representing as matrix.
Correlation test is a more convincing method of model valida-
tion. It is the usual statistical method to validating identified non-
linear models. It has been shown that a suitable prediction
through different data sets is produced only if the model is
unbiased. The prediction error sequence e(t) should be uncorre-
lated with all linear and non-linear combinations of past inputs
and outputs (unbiased) when the model structure and the
estimated parameters are correct. This will hold if and only if
the following conditions are satisfied (Billings and Voon, 1986):
f
ee
ð
t
Þ¼E½eðt
t
ÞeðtÞ¼
d
ðtÞ
f
ue
ð
t
Þ¼E½uðt
t
ÞeðtÞ¼0,8
t
f
u
2
e
ð
t
Þ¼E½ðu
2
ðt
t
Þu
2
ðtÞÞeðtÞ¼0,8
t
ð6Þ
f
u
2
e
2
ð
t
Þ¼E½ðu
2
ðt
t
Þu
2
ðtÞÞe
2
ðtÞ¼0,8
t
f
ðeÞðeuÞ
ð
t
Þ¼E½eðtÞeðt1
t
Þuðt1
t
Þ¼0
t
Z0
where
f
ue
(
t
) indicates the cross-correlation function between u(t)
and e(t), eu(
t
)¼e(tþ1)u(tþ1),
d
(t) is an impulse function.
Actually, the correlation will never be precisely zero for all lags
but the model is considered as satisfactory if the correlation tests
lay within 95% confidence limits, defined as 1:96=ffiffiffiffi
N
p,whereNis
the data length. Autocorrelation of the error also will never be as an
ideal delta function but will be considered as sufficient if the
autocorrelation plot enters the 95% confidence limits before lag one.
6. Implementation and results
For experimental purposes, a rectangular, thin, flat aluminium
plate with all edges clamped (C–C–C–C) boundary condition is
considered. Table 1 shows the properties of the plate.
For the purpose of the development of active vibration control
(AVC), which will be conducted for the future work, a new
arrangement of the experimental rig is shown in Fig. 13. A setup
strategic location for shaker (for input signal) at point (X)is
identified appropriately on the plate. The location of sensor (for
detection signal) at point (Y) and the location of sensor (for
observation signal) at point (Z) are chosen so that it is far enough
from the nodal lines defined by the first five natural frequencies of
the plate.
Table 1
Plate specifications.
Parameter Value
Length (a) (m) 1.58
Width (b) (m) 1.08
Thickness (h) (m) 0.003
Density (
r
) (Kg m
3
) 2690
Modulus of elasticity (E)(Nm
2
) 6.831010
Poisson ratio (u) 0.34
I.Z.M. Darus, A.A.M. Al-Khafaji / Engineering Applications of Artificial Intelligence 25 (2012) 94 –106100
These arrangements are as such as to ensure the best perfor-
mance of AVC for vibration reduction at the observation point (Z),
will be achieved. Nevertheless, the best AVC can only be achieved
Fig. 14. Lateral deflection detected at x¼0.75a, and y¼0.75bpoint H(detection
point).
Fig. 15. Lateral deflection detected at x¼0.75a, and y¼0.25bpoint H(observation
point).
Fig. 16. The actual and MLPNN predicted output.
Fig. 13. Schematic description of the plate with all edged clamped boundary
conditions for development of AVC (Tavakolpour et al., 2010).
Fig. 17. Error between actual and MLPNN predicted output.
030 60 90 120 150
10-3
10-2
10-1
100
Best Mean-square error = 0.00017103
10-4
Mean-squared error
Number of Training Passes
Fig. 18. Mean-squared error vs. number of training passes.
I.Z.M. Darus, A.A.M. Al-Khafaji / Engineering Applications of Artificial Intelligence 25 (2012) 94 –106 101
if the model of the system is described accurately. Therefore, an
experimental studies were carried out to obtained the input/
output data (between detection and observation points) and these
data will later used for system identification to obtain the best
model describing the model of the plate. A sinusoidal input force
(F) with an amplitude of [19 Hz, 5 V] at time instance of 4 s was
applied to the excitation point (X) located at x¼0.25a, and
y¼0.25b. The experimental lateral deflection of the plate at
detection point (Y) located at x¼0.75a, and y¼0.75b, in time
domain response is plotted in Fig. 14. The experimental lateral
deflection of the plate at observation point (Z) located at x¼0.75a,
and y¼0.25b, in time domain response is plotted in Fig. 15.
6.1. Multi-layer Perceptron Neural Network (MLPNN) modelling
A coding has been produce base on the Multi-layer Perceptron
using MATLAB environment. An MLPNN with 2 hidden layers, with
6 tansigmoid neurons in first hidden layer, 6 tansigmoid neurons in
second hidden layer, and one output layer with linear neuron. Since
there was not a priori knowledge about a suitable order of the
model for the flexible plate system, the structure realisation was
performed by a trial-and-error method. So, the deflection model was
experimented with different orders.Thedataset,comprising4000
data points, was separated into two sets of 3000 and 1000 data
points. The model was trained using the first set and the model was
-4000 -2000 0 2000 4000
-1
-0.5
0
0.5
1
lag
-4000 -2000 0 2000 4000
-1
-0.5
0
0.5
1
lag
-4000 -2000 0 2000 4000
-1
-0.5
0
0.5
1
lag
-4000 -2000 0 2000 4000
-1
-0.5
0
0.5
1
lag
-4000 -2000 0 2000 4000
-1
-0.5
0
0.5
1
lag
Fig. 19. Correlation tests of MLPNN. (a) Auto–correlation of error, (b) correlation of the input and the error, (c) correlation of the square of input and the error,
(d) correlation of square of the input square of the error and (e) correlation of multiplication of the input by the error and the error.
I.Z.M. Darus, A.A.M. Al-Khafaji / Engineering Applications of Artificial Intelligence 25 (2012) 94 –106102
validated with the whole 4000 points including the 1000 points that
had not been used in the training process. Both output and
estimated outputs are plotted in the Fig. 16. The error between
actual and predicted MLPNN output are plotted in the Fig. 17 and
the mean-squared error vs. number of training passes in Fig. 18.The
best result was achieved with an order 20, which means, n
u
¼n
y
¼10
for 4000 data length was trained to characterize the plate. The
models reached a sum-squared error level of 0.00017103 with 150
training passes for MLPNN modelling.
The correlation tests were carried out to determine the effec-
tiveness of the (MLP) BP-based model. Fig. 19 shows the results of
the correlation tests. The results were also found to be within 95%
confidence level thus confirmed the accuracy of the results.
6.2. Elman Neural Network (ENN) modelling
A coding has been produce based on the Elman Neural Networks
within MATLAB environment. An Elman Neural Network with
2 hidden layers, with 6 tansigmoid neurons in first hidden layer,
6 tansigmoid neurons in second hidden layer, and one output layer
with linear neuron. Since there was no prior knowledge about a
suitable order of the model for the flexible plate system, the
structure realisation was performed by a heuristic method. The
deflection model was tested with different orders. The data set,
comprising 4000 data points, was divided into two sets of 3000 and
1000 data points. The model was trained using the first set and the
model was validated with the whole 4000 data, including the first
1000 data, which had not been used in the training process. Fig. 20
shows the result of the actual and Elman predicted output. Fig. 21
shows the error between actual and predicted Elman output and the
mean-squared error vs. number of training passes in Fig. 22.The
best result was achieved with an order 20, which means, n
u
¼n
y
¼10
for 4000 data length was trained to characterise the plate. The
models reached a sum-squared error level of 0.002 with 150 training
passes for Elman Neural Network modelling.
The correlation tests were carried out to determine the
effectiveness of the ENN-based model. Fig. 23 shows the results
of the correlation tests. The results were also found to be within
95% confidence level thus confirmed the accuracy of the results.
6.3. Adaptive Neuro-Fuzzy Inference System (ANFIS) modelling
A coding has been produce base on the ANFIS Networks A
coding has been produce base on the ANFIS Networks using
MATLAB environment. Since there was not a priori knowledge
about a suitable order of the model for the flexible plate system,
the structure realisation was performed by a trial-and-error
method. So, the deflection model was experimented with differ-
ent orders. The data set, comprising 4000 data points, was divided
into two sets of 3000 and 1000 data points. the model was trained
with The first set and the model was validated with the whole
4000 points, including the 1000 points that had not been used in
the training process. The best result was achieved with an order 4,
which means, n
u
¼n
y
¼2 for 4000 data length was trained to
characterize the plate. The models reached a sum-squared error
level of 0.00039781 with 150 training passes for ANFIS network
modelling. The model has 16 rules for model order n¼4 and two
Gaussian membership functions. Gaussian membership functions
with product inference rule are applied at the fuzzification level.
The fuzzifier outputs the firing strengths for each rule. At the
defuzzification level, the first order sugeno model is utilised.
Fig. 24 shows the result of the actual and ANFIS predicted output,
Fig. 25 shows error between actual and predicted ANFIS output.
The correlation tests were carried out to determine the
effectiveness of the ANFIS-based model. Fig. 26 shows the
results of the correlation tests. The results were also found to
be within 95% confidence level thus confirmed the accuracy of
the results.
Fig. 20. Actual and predicted ENN output in time domain.
Fig. 21. Error between actual and predicted ENN output.
0 30 60 90 120 150
10-2
10-1
100
101
Best Mean-square error = 0.002
10-3
Mean-squared error
Number of Training Passes
Fig. 22. Mean-squared error vs. number of training passes.
I.Z.M. Darus, A.A.M. Al-Khafaji / Engineering Applications of Artificial Intelligence 25 (2012) 94 –106 103
-4000 -2000 0 2000 4000
-1
-0.5
0
0.5
1
lag
-4000 -2000 0 2000 4000
-1
-0.5
0
0.5
1
lag
-4000 -2000 0 2000 4000
-1
-0.5
0
0.5
1
lag
-4000 -2000 0 2000 4000
-1
-0.5
0
0.5
1
lag
-4000 -2000 0 2000 4000
-1
-0.5
0
0.5
1
la
g
Fig. 23. Correlation tests of ENN. (a) Auto–correlation of error, (b) correlation of the input and the error, (c) correlation of the square of input and the error, (d) correlation
of square of the input square of the error and (e) correlation of multiplication of the input by the error and the error.
Fig. 24. Actual and ANFIS predicted output. Fig. 25. Error between actual and predicted output.
I.Z.M. Darus, A.A.M. Al-Khafaji / Engineering Applications of Artificial Intelligence 25 (2012) 94 –106104
7. Comparative assessment
Comparative performance of non-parametric modelling meth-
ods in terms of the mean-squared of error is summarised in
Table 2. It follows from detection and observation mapping as
presented earlier in this study, non-parametric models have
performed very well. Validations through test procedures and
correlation tests have also been performed with the MLPNN, ENN
and ANFIS based models. It is observed from the validation tests
that different modelling methods considered in this study have
performed sufficiently well. Comparing the mean-squared errors
in Table 2, it is noted that for non-parametric identification
technique, MLPNN have performed better than ENN and ANFIS
in characterising the flexible plate structure.
8. Discussion
Results of various modelling methods have been validated with
a range of tests including input/output mapping, mean-squared
-4000 -2000 0 2000 4000
-1
-0.5
0
0.5
1
lag
-4000 -2000 0 2000 4000
-1
-0.5
0
0.5
1
lag
-4000 -2000 0 2000 4000
-1
-0.5
0
0.5
1
lag
-4000 -2000 0 2000 4000
-1
-0.5
0
0.5
1
lag
-4000 -2000 0 2000 4000
-1
-0.5
0
0.5
1
lag
Fig. 26. Correlation tests of ANFIS. (a) Auto–correlation of the error, (b) correlation of the input and the error, (c) correlation of the square of input and the error,
(d) correlation of square of the input square of the error and (e) correlation of multiplication of the input by the error and the error
Table 2
Performance of parametric and non-parametric
modelling.
Method MSE
MLPNN 0.00017103
ENN 0.002
ANFIS 0.00039781
I.Z.M. Darus, A.A.M. Al-Khafaji / Engineering Applications of Artificial Intelligence 25 (2012) 94 –106 105
error and correlation tests. It is observed that all the modelling
methods have performed very well in approximating the system
response.
A comparative assessment of the performance of non-para-
metric approaches in modelling a flexible plate structure has been
carried out. It is demonstrated that MLPNNs and ANFIS perform
better than ENN in modelling and identification of a flexible plate
structure. Thus, the system data can closely be predicted with a
very small prediction error with suitable choice of the input data
structure.
The non-parametric models of the flexible plate structure thus
developed and validated will be used as the transfer function of
the system in subsequent investigations for the development of
active vibration control strategies for vibration suppression in
flexible structures.
Acknowledgement
The authors would like to express their gratitude to the
Universiti Teknologi Malaysia (UTM) for their continuous support
in the research work. This research was fully supported by the
UTM Research University Grants (GUP) using vote no. 00H11.
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