NEURO-FUZZY SYSTEM WITH INCREASED ACCURACY SUITABLE FOR
HARDWARE IMPLEMENTATION
Except where reference is made to the work of others, the work described in this thesis is
my own or was done in collaboration with my advisory committee. This thesis does not
include proprietary or classi ed information.
Kannan Govindasamy
Certi cate of Approval:
Vishwani Agrawal
James J. Danaher Professor
Electrical and Computer Engineering
Bogdan Wilamowski, Chair
Professor
Electrical and Computer Engineering
Michael Baginski
Associate Professor
Electrical and Computer Engineering
George T. Flowers
Dean
Graduate School
NEURO-FUZZY SYSTEM WITH INCREASED ACCURACY SUITABLE FOR
HARDWARE IMPLEMENTATION
Kannan Govindasamy
A Thesis
Submitted to
the Graduate Faculty of
Auburn University
in Partial Ful llment of the
Requirements for the
Degree of
Master of Science
Auburn, Alabama
May 09, 2009
NEURO-FUZZY SYSTEM WITH INCREASED ACCURACY SUITABLE FOR
HARDWARE IMPLEMENTATION
Kannan Govindasamy
Permission is granted to Auburn University to make copies of this thesis at its
discretion, upon the request of individuals or institutions and at
their expense. The author reserves all publication rights.
Signature of Author
Date of Graduation
iii
Vita
Kannan Govindasamy, son of Govindasamy Krishnasamy and Karthiga Govindasamy,
was born on May 29, 1984 in Madurai, Tamil Nadu, India. He graduated in 2005 with a
Bachelor of Engineering degree in Electronics and Communication Engineering from Thi-
agaraja College Engineering, Madurai, India. In the pursuit of enhancing his academic
quali cation he joined the M.S. Program at Auburn University in the Department of Elec-
trical and Computer Engineering in August 2006.
iv
Thesis Abstract
NEURO-FUZZY SYSTEM WITH INCREASED ACCURACY SUITABLE FOR
HARDWARE IMPLEMENTATION
Kannan Govindasamy
Master of Science, May 09, 2009
(B.E., Anna University, 2005)
62 Typed Pages
Directed by Bogdan Wilamowski
Fuzzy controllers are easy to design for complex control surfaces but produce rough con-
trol surfaces which might lead to unstable operation. On the other hand neural controllers
are hard and complex to train but they produce very accurate output control surfaces com-
pared to that of fuzzy controllers.The Neuro-Fuzzy controller proposed in this thesis exploits
the fuzzy systems nature of utilizing expert knowledge and also produces smooth surfaces
by implementing it in neural networks.
Monotonic sigmoid membership function is used to make the neural implementation
an easy task. Levenberg-Marquardt algorithm(LM) which is used for feed forward networks
is implemented to train the neurons. A defuzzi cation with trigonometric approximation
algorithm using LUT-Lookup Table is developed to implement in low cost microcontrollers
to make the control system highly cost e ective. It is shown through extensive simulations
that the proposed model produces accurate and smooth control surfaces.
v
Acknowledgments
I take this opportunity to thank my academic advisor Dr. Bogdan Wilamowski, for
his guidance in my research throughout my Masters degree program here in Auburn. I also
have had the privilege of Dr. Vishwani Agrawal and Dr. Michael Baginski taking their
precious time to be in my graduate committee and receiving from them valuable feedback
on my work.
I also would like to thank all my friends who have been helpful throughout my degree.
I wish to dedicate this degree and work to my family who made this possible and for all
their support throughout my life.
vi
Style manual or journal used IEEE
Computer software used Latex together with the departmental style- le aums.sty.
vii
Table of Contents
List of Figures x
1 Introduction 1
2 Neural Network 3
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
2.2 Historical Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
2.2.1 A Promising and Emerging Technology . . . . . . . . . . . . . . . . 3
2.2.2 Period of Frustration and Disrepute . . . . . . . . . . . . . . . . . . 4
2.2.3 Re-Emergence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
2.3 Human Brain Working Model . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2.4 Arti cial Neuron . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2.4.1 Components of Neuron . . . . . . . . . . . . . . . . . . . . . . . . . 6
2.5 Types of Neural Network . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2.5.1 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2.5.2 Architecture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2.5.3 Learning Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
2.6 Error Back-Propagation Algorithm . . . . . . . . . . . . . . . . . . . . . . . 9
2.7 Advantages of Neural Network . . . . . . . . . . . . . . . . . . . . . . . . . 9
2.8 Limitations of Neural Network . . . . . . . . . . . . . . . . . . . . . . . . . 10
2.9 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
3 Fuzzy Logic 13
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
3.2 Historical Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
3.3 Fuzzy System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
3.4 Membership Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
3.5 Mamdani and TSK Fuzzy Controllers . . . . . . . . . . . . . . . . . . . . . 15
3.6 Advantages of Fuzzy Logic . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
3.7 Limitations of Fuzzy Logic . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
3.8 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
4 Proposed Neuro-Fuzzy Model 19
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
4.2 Limitations of FPGA Implementaion . . . . . . . . . . . . . . . . . . . . . . 23
4.3 Neuro-Fuzzy Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
4.4 Neural Fuzzi cation Using Monotonic Functions . . . . . . . . . . . . . . . 24
4.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
viii
5 LUT Based Nonlinear Mapping 29
5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
5.2 Defuzzi cation with Trignometric Approximation Algorithm . . . . . . . . . 29
5.3 Application of Algorithm to 1-Dimension Problem . . . . . . . . . . . . . . 33
5.4 Extension of Algorithm to 2-Dimension and more . . . . . . . . . . . . . . . 34
6 Simulation Results 38
7 Conclusion 42
Bibliography 43
Appendices 45
A Matlab Code 46
A.1 1-Dimensional LUT algorithm . . . . . . . . . . . . . . . . . . . . . . . . . 46
A.2 2-Dimensional LUT algorithm . . . . . . . . . . . . . . . . . . . . . . . . . 48
ix
List of Figures
2.1 A neuron with n inputs, weights and threshold value . . . . . . . . . . . . . 12
3.1 Membership Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
3.2 Fuzzy System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
3.3 Block Diagram Mamdani Fuzzy Controller . . . . . . . . . . . . . . . . . . . 15
3.4 TSK Fuzzy Controller . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
4.1 Required Control Surface . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
4.2 Control Surface Using Triangular TSK Fuzzy system . . . . . . . . . . . . . 21
4.3 Control Surface Using Guassian Memberhsip TSK Fuzzy system . . . . . . 22
4.4 Piecewise Linear . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
4.5 Exponential Sigmoid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
4.6 Parabolic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
4.7 A Membership Function Implemented Using Exponential Functions F1 and F2 26
4.8 A Membership Function Implemented Using Exponential Functions F1 and F2 27
4.9 Exponential Sigmoid Membership Functions . . . . . . . . . . . . . . . . . . 27
5.1 Defuzzi cation Block . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
5.2 Defuzzi cation with Sinusoidal Approximation . . . . . . . . . . . . . . . . 30
5.3 Illustration of Approximation Algorithm . . . . . . . . . . . . . . . . . . . . 31
5.4 Double Space Linear Approximation . . . . . . . . . . . . . . . . . . . . . . 34
5.5 Single Space Linear Approximation . . . . . . . . . . . . . . . . . . . . . . . 35
5.6 Double Space Sine Approximation . . . . . . . . . . . . . . . . . . . . . . . 35
x
5.7 Single Space Sine Approximation . . . . . . . . . . . . . . . . . . . . . . . . 36
5.8 Error Observed Using LUT Algorithm in 1-D . . . . . . . . . . . . . . . . . 36
5.9 Application of Algoritm to 2-Dimensional Problem . . . . . . . . . . . . . . 37
5.10 Application of Algoritm to 2-Dimensional Problem . . . . . . . . . . . . . . 37
6.1 A plot of the Stored data in LUT . . . . . . . . . . . . . . . . . . . . . . . . 38
6.2 parabolic with Traditonal TSK Fuzzy system . . . . . . . . . . . . . . . . . 39
6.3 Exponential with Traditional TSK Fuzzy system . . . . . . . . . . . . . . . 39
6.4 Nonlinear Mapping Using Single Space Linear Approximation . . . . . . . 40
6.5 Nonlinear Mapping Using Sinusoidal Approximation . . . . . . . . . . . . . 40
6.6 Error plot of Nonlinear Mapping Using Sinusoidal Approximation . . . . . . 41
xi
Chapter 1
Introduction
Control systems for nonlinear system [1] still remains a challenge in modern control
theory, when compared to control system for linear objects. The nonlinear system becomes
more di cult to understand and analyze when the system becomes a function of time. A
nonlinear object is usually linearized before a controller system can be designed for the
system. This is more often done by adding a reverse nonlinear function to compensate
the nonlinear behavior and make the input-output relationship more linear. An adaptive
nonlinear system that has nonlinear characteristics which change with time is better man-
aged by methods of computational intelligence such as neural network and fuzzy systems
[1][3][4][5]. The adaptive nature of these systems enables them to model any dynamic
non-linear behavior of a control system.
Fuzzy systems utilize expert knowledge and perception based information in the form of
set of rules. The dynamics of a system is generally complicated, but sometimes its behavior
can be de ned more conveniently using linguistic terms in which case fuzzy logic is the best
option to model the system. Moreover, fuzzy systems are easy to design and implement in
hardware. But, the major drawback of fuzzy controllers is that the control surfaces obtained
from these systems are rough, which can cause unstable operation.
Arti cial neural systems, on the other hand are known by their property of perform-
ing complex nonlinear mappings. They also produce smooth control surfaces unlike the
fuzzy systems. Even though neural networks are widely implemented in software many
applications require the hardware implementation. But, these systems require the compu-
tation of tangent hyperbolic activation functions. This involves division algorithms which
are often too complex for simple microprocessors. So, a suitable combination of the neural
architecture and the fuzzy architecture is proposed in this thesis. It exploits the property
1
of producing  ne control surfaces from neural architecture and utilizing expert knowledge
from fuzzy logic.
The Neuro-Fuzzy controller proposed in this work uses a fuzzy logic with monotonic
sigmoid membership functions that makes the neural implementation a very easy task.
Levenberg-Marquardt algorithm(LM) [6] which is used for feed forward networks is imple-
mented to train the neurons. Since, the implementation of neural networks in low cost
microcontrollers is a challenging task, a defuzzi cation with trigonometric approximation
algorithm using LUT [7][8] is developed to implement neurons in low cost microcontrollers
to make the control system highly cost e ective.
The rest of the thesis is organized as follows.
Chapter 2 provides the relavent background information about the arti cial neural
network. The advantages and limitations of using arti cial neural networks, their types,the
structural information and history of neural network are discussed.
Chapter 3 provides the relavent background information about the fuzzy control system.
The two major fuzzy controllers, the Mamdani fuzzy controller and TSK fuzzy controller are
discussed. The advantages and limitations of using fuzzy control system are also discussed.
Chapter 4 provides information about the proposed neuro-fuzzy architechture. The
usage of monotonic membership functions is discussed.
Chapter 5 provides information about the proposed LUT based defuzzi cation algo-
rithm. The extension of the algorithm to higher dimension is also explained.
Chapter 6 shows the simulations results and chapter 6 concludes the work.
2
Chapter 2
Neural Network
2.1 Introduction
An Arti cial Neural Network (ANN) is an interconnected group of nodes similar to
the network of neurons in brain. The information processing system of neural network is
inspired by the functioning model behavior of biological nervous systems, such as the brain.
A key aspect of neural network is that, it has to be trained. The neural network by virtue
does not have the expert knowledge to solve a problem. An arti cial neural network much
akin to the human nervous system learns by example. In biological system learning involves
by adjustments to the synaptic connections that exist betweens the neurons. This is true
of arti cial neural network as well. An arti cial neural network is con gured for speci c
function [15],[16].
2.2 Historical Background
Neural network simulations appear to be a recent development. However, this  eld was
established before the advent of computers, and has survived at least one major setback
and several eras.
The history of neural networks consist of the following three periods [15].
2.2.1 A Promising and Emerging Technology
McCulloch and Pitts (1943) gave the  rst successful simulation model of the logic
based on the working model of biological neurons. These models were based on how simple
neurons worked and how they were akin to simple logic elements with thresholds.
3
This was followed by Rosenblatt?s (1958) perceptron. The perceptron was a three lay-
ered architecture consisting of an input layer, middle layer and output layer. The middle
layer was also known as the hidden layer or the associate layer. The perceptron had the
ability to learn and associate a set of inputs to output. The other important development
during this initial period of interest in neural network was adaptive linear element (ADA-
LINE) which was developed in 1960 by Widrow and Ho . The ADALINE used an LMS
(Least-Mean-Square) learning rule and was an analog system [15],[16].
2.2.2 Period of Frustration and Disrepute
The period of initial enthusiasm and high funding was followed by a period of frus-
tration and disrepute. Minsky and Papert published "Perceptrons: An Introduction to
Computational Geometry" (1969) in which they highlighted the limitations of single layer
Perceptrons to multilayer systems, in other word a perceptrons could only solve linear sepa-
rable functions but not in particular like XOR or XNOR logic. The publication disenchanted
researchers in the  eld of neural network and was followed by a period of corresponding low
funding [15],[16].
2.2.3 Re-Emergence
Currently signi cant process has been made and works have surpassed the limitations
highlighted by Minsky and Papert. The adaptive resonance theory (ART) networks by
Steve Grossberg and Gail Carpenter in 1988, the associative techniques of Anderson and
Kohonen and the seminal error back propagation learning algorithm by Paul Werbos 1974,
infused life into neural networks. Since then this  eld has enjoyed a period of high funding
and signi cant commercial applications for industry and a  nancial solution have been
developed [15],[16].
4
2.3 Human Brain Working Model
The human brain is still a mystery and it is unclear how the human nervous sys-
tem functions and process a information. A typical neuron consists of three main parts -
dendrites, axons and synapses. A neuron has the amazing capability of transferring electro-
chemical signals. A branch like projections from the cell body collects information in the
form of signals known as dendrites. These signals are carried over the length of cell through
a long cable like projection known as axon. The axon which carries the signal end into
numerous branch like projection called synapse. The synapse converts the signal into elec-
trical activity. When the electrical activity is of su cient strength the electrical activity
is conveyed to the next neuron. Learning is involved by the adjustment of these synaptic
connections [15],[16].
2.4 Arti cial Neuron
Neural network is composed of a large number of highly interconnected processing
elements (neurons) working together to solve speci c problems. A single neuron could be
modeled by the equation (1). It is diagrammatically represented as shown in Fig 2.1. The
output of the neuron is given by the sigmoid transfer function. Neural networks cannot be
programmed to a speci c task but trained to solve by using various algorithms
net =
nX
i=1
Wixi +Wn+1 (1)
An arti cial neuron has many inputs and one output. The neuron operates in two
modes the using mode and the training mode. In training mode the neuron is trainied to
either  re or not for a particular speci ed input. In using mode when a speci ed pattern is
detected at the input the neuron responds by  ring the associated output of the speci ed
value. If the input pattern detected does not match any of the one pattern in the speci ed
list the  ring rule is used to determine whether the neuron  res or not [15],[16].
5
2.4.1 Components of Neuron
The behavior of a neural network depends on the two components, the weights and
the input-output function (transfer function) that is speci ed for the units. The transfer
function mainly falls into three categories:
 Linear Transfer Function
For linear units, the output activity is proportional to the total weighted output.
 Threshold
For threshold units, the output is set at one of two levels, depending on whether the
total input is greater than or less than some threshold value.
 Sigmoid
For sigmoid units, the output varies continuously but not linearly as the input changes.
Sigmoid units bear a greater resemblance to real neurons than do linear or threshold
units, but all three must be considered rough approximations.
To make a neural network that performs some speci c task, it must be choosen how
the units are connected to one another and how to set the weights on the connections
appropriately. The connections determine whether it is possible for one unit to in uence
another. The weights specify the strength of the in uence.
A three-layer network can be trained to perform a particular task by using the following
procedure:
1. The network is fed with training examples, which consist of a pattern of activities for
the input units together with the desired pattern of activities for the output units.
2. The di erence between the actual output of the network and the desired output is
found.
3. The weights of each connection is changed accordingly so that the network produces
a better approximation of the desired output.
6
2.5 Types of Neural Network
Neural Network types can be classi ed based on following attributes [15],[16]:
2.5.1 Applications
Neural network applications can be grouped in the following categories [15],[16]:
 Clustering:
A clustering algorithm explores the similarity between patterns and places similar
patterns in a cluster. Best known applications include data compression and data
mining [19].
 Classi cation/Pattern recognition:
The task of pattern recognition is to assign an input pattern (like handwritten symbol)
to one of many classes. This category includes algorithmic implementations such as
associative memory.
 Function approximation:
The tasks of function approximation is to  nd an estimate of the unknown function
f() subject to noise. Various engineering and scienti c disciplines require function
approximation.
 Prediction:
The task is to forecast some future values of a time-sequenced data. Prediction has
a signi cant impact on decision support systems. Prediction di ers from function
approximation by considering time factor.
2.5.2 Architecture
 Feed-forward networks
7
Arti cial neural networks that has signals that travel only from inputs to output are
called feed-forward networks. The signals travel only in one direction and there is
no feedback (loops). The output of any layer does not a ect that same layer. Feed-
forward arti cial neural networks are straight forward networks that associate inputs
with outputs [15],[16].
 Feedback networks
Arti cial neural networks that have loops in the network making the signals traveling
in both directions are referred as feed back networks. Feedback networks are very
powerful and can get extremely complicated. The loops make the feedback network
very dynamic. The feedback network changes their state continuously until they reach
an equilibrium point. The network then remains in the equilibrium point until the
input changes and a new equilibrium is found. Single layer feedbacks architectures are
also referred to as recurrent networks; multilayer feedback are referred to as interactive
networks [15],[16].
2.5.3 Learning Methods
 Supervised:
Supervised training has both inputs and the desired outputs. The output of the
training network for the input is compared against the desired outputs to adjust
weights which control the network. The network is adjusted till the resultant outputs
of the network match the desired network.
 Unsupervised:
In unsupervised training the desired outputs are not provided. The system then on
its own must decide to self organize to group the input data. Unsupervised networks
are referred to as self-organizing or adaptive networks.
8
2.6 Error Back-Propagation Algorithm
The most common method of training a neural network is error back-propagation
algorithm. The main technique of error back-propagation is to gradually adjust the neural
network to produce the correct result for an input. This is accomplished through an iterative
process by which the weights between the various nodes are altered until the network as a
whole performs correctly.
In order to train a neural network to perform some task, the weights of each unit must
be adjusted in such a way that the error between the desired output and the actual output
is reduced. This process requires that the neural network compute the error derivative of
the weights (EW). In other words, it must calculate how the error changes as each weight
is increased or decreased slightly. The back propagation algorithm is the most widely used
method for determining the EW.
The basic steps involved in the algorithm are [15],[16]:
1. Assertion of random weights to the network.
2. An input is applied to the network and the results for all nodes are obtained.
3. The errors for each node are calculated by comparing the desired result with obtained
result(starting from the last node and propagating the error backward).
4. The weights of each node are updated based on the error calculated.
5. The steps 1 to 4 are repeated until the desired output is gotten.
2.7 Advantages of Neural Network
A neural network can be assumed to be as expert in the category of information that
it has been trained. The neural network has the ability to interpret meaningful information
from imprecise data, interpret patterns and trends which are very hard and too complex
to be done by other computing techniques. The following are the important features of
arti cial neural networks [15],[16].
9
 Parallel Processing:
The parallel processing capability of neural networks allows it to solve problems where
multiple constraints have to be met, which is very much similar to the real biological
nervous system.
 Self-Organization:
A neural network has the ability to self organize while learning. The self-organization
helps in the visualization of low dimensional view of high dimensional data.
 Graceful Degradation:
An important feature of neural network is that partial removal of components only
causes corresponding degradation in the accuracy of output rather than the failure of
the system.
 Continuous Adaptivity:
The neural network has the capability to learn from the sample space of the input
vectors.
2.8 Limitations of Neural Network
Neural network is the not the perfect solution of every problem. Neural network can
only be considered as the expert only in the category of information it has been trained and
tasked. Neural networks are limited by problems like the following:
 Accuracy: Neural network is not an hundred percent accurate system. The accuracy
of neural network depends on the amount of training and the size of the network.
Systems which require 100 percent accuracy cannot depend on neural networks.
 Black Box:
Akin to the nervous system the neural network system is like a black box. The system
learns based on training and experience but cannot justify the decision.
10
 Training:
The amount of training needed by neural network is an obstacle, but this problem
has been overcome with the vast amount of data available in the digital world.
Neural network is more of art rather than science, but an art whose results can be
observed and measured.
2.9 Conclusion
The computing world has a lot to gain from neural networks. The ability of neural
networks to learn by example makes them very  exible and powerful. The self-learning and
self-tuning capabilities makes it ideal for industrial application. Furthermore there is no
need to write an algorithm in order to perform a particular task; i.e. there is no need to
understand the internal mechanisms of that task. They are also very well suited for real
time systems because of their fast response and computational times which are due to their
parallel architecture.
Neural networks also contribute to other areas of research such as neurology and psy-
chology. They are regularly used to model parts of living organisms and to investigate the
internal mechanisms of the brain.
Perhaps the most exciting aspect of neural networks is the possibility that some day
conscious networks might be produced. There is a number of scientists arguing that con-
sciousness is a mechanical property and that conscious neural networks are a realistic pos-
sibility.
11
Figure 2.1: A neuron with n inputs, weights and threshold value
12
Chapter 3
Fuzzy Logic
3.1 Introduction
"Fuzzy logic is basically a multi-valued logic that allows intermediate values to be
de ned between conventional evaluations like yes/no, true/false, black/white, etc. Notions
like rather warm or pretty cold can be formulated mathematically and processed by comput-
ers." - Bauer et al. Fuzzy logic provides a means of calculating intermediate values between
absolute true and absolute false with resulting values ranging between 0 and 1. With fuzzy
logic, it is possible to calculate the degree to which an item is a member [13],[14],[17].
3.2 Historical Background
The concept of Fuzzy logic was introduced by Professor Zadeh at the University of
California at Berkeley in the 1960?s. His goal was to develop a model that could more closely
describe the natural language process. Fuzzy logic can be compared to the human decision
making process. Conventional or classic logic is more of boolean conditions (true/false).
Fuzzy logic is a superset of boolean logic, having its own similarities and di erences with
boolean algebra [13],[14],[17].
3.3 Fuzzy System
The structure of a Fuzzy system consists of:
 Rule base: selects the set of fuzzy rules
 Database: de nes the membership functions used in the fuzzy rules
13
 Reasoning mechanism: performs the inference procedure (derive a conclusion from
facts and rules)
3.4 Membership Function
The membership function represents the degree of truth as an extension of valuation.
The degree of truth is often confused with probabilities. Probability is the likelihood that
something is true. Fuzzy logic is the degree to which something is true (or within a mem-
bership set)
Figure 3.1: Membership Function
The three major types of membership functions are gaussian, triangular and trape-
zoidal.
The basic rules of membership functions are
 Each point of the input should at least belong to one membership function.
 The sum of two overlapping membership function should not exceed 1.
 The accuracy of the fuzzy system can be raised by increasing the number of member-
ship functions, but this a ects the stability of the control system.
14
3.5 Mamdani and TSK Fuzzy Controllers
The most important application of fuzzy logic is the fuzzy control system which directly
uses the fuzzy theory. The Mamdani fuzzy controller originally used three steps to create a
fuzzy controlled system.
1. Fuzzi cation: The process of using membership functions to graphically describe a
situation
2. Rule evaluation: The process of applying the fuzzy rules to fuzzi ed inputs.
3. Defuzzi cation: The process of obtaining the crisp or actual results.
The Madmani fuzzy controller consists of the following block shown in  g 3.3,
Figure 3.2: Fuzzy System
Figure 3.3: Block Diagram Mamdani Fuzzy Controller
15
A membership function speci es the e ect of the particular variable on the  nal output.
The block diagram of a typical fuzzy system as proposed by Takagi-Sugeno-Kang (TSK) is
shown in Fig 3.4.
The analog inputs are converted to a set of n fuzzy values by the fuzzi er block.
This process of mapping a single value into an n dimensional space is called fuzzi cation.
Fuzzi cation is done by using membership functions; n membership functions are required
to map to n dimensional space. Each analog input belongs to at least one and preferably
two membership functions (overlapping).
Figure 3.4: TSK Fuzzy Controller
The accuracy of the system can be increased by increasing the number of membership
functions. But, very dense functions can lead to frequent controller action (also known as
hunting) and sometimes may lead to instability. The fuzzi ed values are processed by fuzzy
logic blocks which implement MIN and MAX operations. MIN and MAX operations are
analogous to AND and OR operations in Boolean logic. Boolean logic can be considered as
a subset of fuzzy logic since fuzzy logic is applicable for all the values from 0 to 1, unlike
Boolean logic which operates on 0 or 1 only. TSK architecture does not require a MAX
operator. The MAX operator is replaced by a weighted average which is applied directly to
the values of MIN operation [13],[14],[17].
16
3.6 Advantages of Fuzzy Logic
 Fuzzy logic mimics human decision making to handle vague concepts.
 Computation is faster due to intrinsic parallel processing nature.
 Fuzzy logic has the ability to deal with imprecise information [13],[14],[17].
 Modeling of complex, non-linear problems.
 Fuzzy logic is popular in engineering because of its ability to deal with real-world,
ambiguous problems. Fuzzy logic is mathematical oriented, but also at the same time
emphasizes ambiguity and uncertainty. The main popularity is due to its practicality,
not because of its mathematical rigor. It was developed to address practical problems,
and the math came later. Fuzzy was developed in an atmosphere of application
[13],[14],[17].
 Fuzzy logic does not preclude the idea of absolutes - after all, membership functions
can be equal to 0 or 1. Fuzzy rather adds to the idea of absolutes more levels of
belonging [13],[14],[17].
 Fuzzy logic uses membership functions and rules to approximate any continuous func-
tion to a precision of any degree.
 Fuzzy logic simpli es design complexity by using experience and knowledge in simple
linguistic like rules. The input - output relations are not modeled by complex math
equations and can be even used by non-experts [13],[14],[17].
3.7 Limitations of Fuzzy Logic
 Highly abstract and heuristic.
 Need experts for rule discovery (data relationships).
 lack of self-organizing and self-tuning mechanisms of neural network [13],[14],[17].
17
 The most important drawback of fuzzy logic is that it?s not always accurate. The
results are perceived as guesses, so the results cannot be as widely trusted as that of
classical logic.
 Fuzzy logic is often confused with probability theory, and the terms are used in-
terchangeably. While they are similar concepts, they do not say the same things.
Probability is the likelihood that something is true. Fuzzy logic is the degree to which
something is true [13],[14],[17].
 The traditional low respectability of the fuzzy logic is the biggest problem. Though
fuzzy logic is the superset of all logic, it?s not been widely accepted as classical logic
because of the lack of precision.
3.8 Conclusion
Fuzzy logic is an alternative design methodology for control system which is simpler
and faster. Fuzzy logic reduces the complexity of the design for control system and reduces
the design time. The tuning of the control system is done by adjusting the membership
function rather than redesigning the control system. Fuzzy logic is a better solution to the
non-linear control system because it has better control performance than linear, peiceise
linear and lookup table techniques.
18
Chapter 4
Proposed Neuro-Fuzzy Model
4.1 Introduction
Control systems for nonlinear system [1],[18] still remains a challenge in modern control
theory, when compared to control system for linear systems. An adaptive nonlinear system
that has nonlinear characteristics which change with time is better managed by methods of
computational intelligence such as neural network and fuzzy systems. Fuzzy systems utilize
expert knowledge and perception based information in the form of a set of rules. The
dynamics of a system is generally complicated, but sometimes its behavior can be de ned
more conveniently using linguistic terms in which case fuzzy logic is the best option to model
the system. Moreover, fuzzy systems are easy to design and implement in hardware. But,
the major drawback of fuzzy controllers is that the control surfaces obtained from these
systems are rough, which can cause unstable operation [18].
Fuzzy set system theory was introduced by Zadeh [9]. The fact that a decision (output)
might depend on several factors (variables) is implemented using membership functions.
A membership function speci es the e ect of the particular variable on the  nal output.
The block diagram of a typical fuzzy system as proposed by Takagi-Sugeno-Kang (TSK) is
shown in Fig 3.4.
The analog inputs are converted to a set of n fuzzy values by the fuzzi er block.
This process of mapping a single value into an n dimensional space is called fuzzi cation.
Fuzzi cation is done by using membership functions; n membership functions are required
to map to n dimensional space. Each analog input belongs to at least one and preferably
two membership functions (overlapping) [18].
19
Figure 4.1: Required Control Surface
20
Figure 4.2: Control Surface Using Triangular TSK Fuzzy system
21
Figure 4.3: Control Surface Using Guassian Memberhsip TSK Fuzzy system
22
The accuracy of the system can be increased by increasing the number of membership
functions. But, very dense functions can lead to frequent controller action (also known as
hunting) and sometimes may lead to instability.
The fuzzi ed values are processed by fuzzy logic blocks which implement MIN and
MAX operations. MIN and MAX operations are analogous to AND and OR operations in
Boolean logic [18]. Boolean logic can be considered as a subset of fuzzy logic since fuzzy
logic is applicable for all the values from 0 to 1, unlike Boolean logic which operates on
0 or 1 only. TSK architecture does not require a MAX operator. The MAX operator is
replaced by a weighted average which is applied directly to the values of MIN operation. Fig
4.1 depicts a required control surface, Fig 4.2, 4.3 gives a control surface from traditional
triangular and Gaussian membership function [18].
4.2 Limitations of FPGA Implementaion
The TSK defuzzi cation architecture is not suitable for FPGA implementation because
it requires normalization, which is very computational intensive. Mamdani architecture is
also not suitable because its gives a very rough control surface and the architecture requires
division in defuzzi cation which makes it a tough task to implement [18].
4.3 Neuro-Fuzzy Model
The proposed neuro-fuzzy architecture consists of two blocks. The  rst block does the
fuzzi cation and is implemented using a neural network. The second block which does the
processing of fuzzi ed values and defuzzi cation is done through a LUT based sinusoidal
approximation algorithm, which makes a FPGA or microcontroller implementation a simple
and easy task [18].
23
Figure 4.4: Piecewise Linear
4.4 Neural Fuzzi cation Using Monotonic Functions
The control surface obtained using the triangular, trapezoidal and Gaussian member-
ship functions are not smooth and accurate. And moreover, these membership functions are
di cult to be modeled and trained in a neural network. So, a neuro-fuzzy model with mono-
tonic membership functions is proposed and used in this work which is easier to implement
in a neural network [18].
Three monotonic membership functions are used in this model. One of them is a
exponential sigmoid function Fig 4.5. and is given by the formula:
f(net) = 1=(1 +exp( net)) (2)
The other two monotonic functions are piecewise linear Fig 4.4. and parabolic sigmoid
Function Fig 4.6.
The reasons for choosing monotonic functions are:
 A sigmoid with a exponential function is the characteristic transfer function of bipolar
di erential transistor pair.
 A MOS di erential transistor has similar characteristics as  rst half of Fig 4.6.
24
Figure 4.5: Exponential Sigmoid
Figure 4.6: Parabolic
25
 A piecewise linear function could be easily generated in a microcontroller and DSP
chips.
Hence a single neuron could be could be easily modeled in hardware using these mono-
tonic functions. Now membership functions can be constructed by mirror imaging the
sigmoid monotonic functions as shown in Fig 4.8. The membership function is gotten by
subtracting two functions F1 and F2 , where F1 and F2 are two sigmoid monotonic functions
[18].
The membership functions are constructed from these basic functions. This member-
ship function can be implemented using two neurons. One of the neurons output gives the
 rst half of the membership function and the negated output of the second neuron imple-
ments the second half of the function. Extending this idea a set of neurons is trained to
give the required number of sigmoid membership functions. The output of a trained neural
network for a sample application is shown in 4.9.
Figure 4.7: A Membership Function Implemented Using Exponential Functions F1 and F2
All the neurons use unipolar activation function and if designed properly any fuzzy
system can be easily modeled. It can be observed that there was no training process
required for this architecture. If training is allowed the network architecture is signi cantly
simpli ed. LM (Lavenberg-Marquardt) [6] is used in this work to train the network. This
algorithm is implemented in MATLAB. Cross layer weights have been introduced to make
26
Figure 4.8: A Membership Function Implemented Using Exponential Functions F1 and F2
Figure 4.9: Exponential Sigmoid Membership Functions
27
the feed forward network [6] more accurate. Thus the above idea shows, how a fuzzi er
block can be implemented through neural network by consciously choosing the membership
functions [18].
4.5 Conclusion
Thus the limitation of implemnting a fuzzy control system identifed can be succesfully
negotiated by using a neural network to implement a fuzzy system by consciously choosing
the membership Function.
28
Chapter 5
LUT Based Nonlinear Mapping
5.1 Introduction
Using the established methods for defuzzi cation results in rougher control surface
also hardware implementation of defuzzi cation by a neural network is very expensive and
requires more processing capability. Micro controllers like Motorola HC 11 which is 8-bit
processor is incapable of doing such processing. These limitations warrant the use of a LUT
based solution for high precision and cost e ective control system [18].
Figure 5.1: Defuzzi cation Block
5.2 Defuzzi cation with Trignometric Approximation Algorithm
A LUT based model which is suitable for FPGA implementation is shown in Fig 5.1.
The inputs to the system are the digital values of the fuzzi ed values from the  rst neural
fuzzi cation block. The digital values are obtained using an analog to digital (A/D) con-
verters in the FPGA. In this approach, the model consists of a ROM and address of which
29
is determined by the 3 most signi cant bits of the fuzzi ed values. Weighted average is
done using the values stored in the LUT [18].
This method results in higher precision and smoother control surface. However the
major disadvantage is if the number of input is increased the size of LUT increases expo-
nentially. To handle this problem the size of LUT is kept constant and least signi cant bits
are used for  ner calculation, but essentially the size of LUT is kept constant. In this model
we use a (4,4) LUT for the simulation.
The advantage of this method is the e cient usage of a very small lookup table (LUT),
which minimizes the hardware requirement; hence, easier implementation and also the error
produced by this approach is very minimal resulting in a very smooth control surface [18].
The values from the LUT are defuzzi ed by a second order sinusoidal approximation
algorithm which is better than the Second order defuzzi cation algorithm (SODA) [12][13].
The algorithm is explained by Fig 5.2. The single space approximation is a linear approx-
imation; the double space approximation needs the value of ?y? from the LUT, but gives
higher resolution output. The single space sinusoidal approximation is the one which this
thesis deals with. The value of ?y? is found by  xing a sine curve.
Figure 5.2: Defuzzi cation with Sinusoidal Approximation
Consider the equation
30
Figure 5.3: Illustration of Approximation Algorithm
y(x) = ax+b+csin(  x=x) (3)
Considering 4 points
separated by x1;x2;x3 and x4 distance x. Suppose x is a point between x2 and x3 at
a distance  x from x2 .The corresponding Y values are y1;y2;y3 and y4 which are gotten
respectively from the LUT
At x2 and x3 the value of sine is zero
y(x2) = ax2 +b (4)
y(x3) = ax3 +b (5)
Solving equation (4) and equation (3) gives 0a0 and 0b0
The  rst derivative is
 y= x = a+ccos(  x=x) (6)
31
At x2 and x3 the value of cos is +1 and -1 respectively
Therefore,
y;(x2) = a c (7)
y;(x3) = a+c (8)
Equation(8) - equation(7) gives:
y;(x3)  y;(x2) = 2c (9)
The slopes are determined by
Slope of y at x2
y31 = y3  y1=2x (10)
Slope of y at x3
y42 = y4  y2=2x (11)
Slope of y at x
y32 = y3  y2=x (12)
Substituting equation (10) and equation (11) in equation (9)
c = 0:25(y4 +y1  y2  y3) (13)
The second derivative
32
 y1 = y31  y32=2 (14)
 y2 = y42  y32=2 (15)
Substituting equation(13), equation (14), equation (15) in equation(3)
y(x) = y2 + (y3  y2) x+sin(  x=x) y1 +sin(  x=x) y2 (16)
The  rst two terms of equation (16) gives the linear approximated value. These sin
values are also retrieved from the LUT.
5.3 Application of Algorithm to 1-Dimension Problem
To compare the performance of the new algorithm with SODA, consider the equation:
y(x) = xsin(x) (17)
Linear approximation is applied and the result is shown in Fig 5.4. Results obtained
by the new algorithm are depicted in Fig 5.6.and Fig 5.7. Error pattern observed by the
new algorithm is shown in Fig 5.8. The total error is less than the total error of SODA by
25
The di erence between the single space approximation and the double space approxima-
tion is that the double space algoritm uses a (4,4) LUT while the single space approximation
uses a (8,8) LUT.
33
Figure 5.4: Double Space Linear Approximation
5.4 Extension of Algorithm to 2-Dimension and more
The same approach is extended for the 2-dimensional problem. 4 values from the LUT
are required to  nd the linear approximation shown in Fig 5.9.
The algorithm is extended to 2-dimension by solving both x-axis and y-axis. The Fig
5.10 depicts the approximation is done along the four direction and the average is found.
12 adjacent points are required to  nd the value of 0y0. In the same manner, the algorithm
could also be extended to more than 2-dimension problems [18].
34
Figure 5.5: Single Space Linear Approximation
Figure 5.6: Double Space Sine Approximation
35
Figure 5.7: Single Space Sine Approximation
Figure 5.8: Error Observed Using LUT Algorithm in 1-D
36
Figure 5.9: Application of Algoritm to 2-Dimensional Problem
Figure 5.10: Application of Algoritm to 2-Dimensional Problem
37
Chapter 6
Simulation Results
In this section the performance of the proposed membership functions are compared
with traditional triangular and gaussian membership functions. Fig 6.1 shows the given data
points from which the control surface has to be constructed. Fig 6.2- Fig 6.5 shows the
control surfaces constructed using various methods. The TSK with the parabolic monotonic
functions has better control surface Fig 6.2. It can be observed that the exponential Fig
6.3 gives an even more smoother control surface when compared to triangular, Gaussian,
piecewise linear and parabolic membership functions.
Fig. 6.5 shows the control surface constructed using trigonometric approximation al-
gorithm. This algorithm produces a surface which is very close to the expected one and
demonstrated to be better than all the other membership functions [18].
Figure 6.1: A plot of the Stored data in LUT
38
Figure 6.2: parabolic with Traditonal TSK Fuzzy system
Figure 6.3: Exponential with Traditional TSK Fuzzy system
39
Figure 6.4: Nonlinear Mapping Using Single Space Linear Approximation
Figure 6.5: Nonlinear Mapping Using Sinusoidal Approximation
40
Figure 6.6: Error plot of Nonlinear Mapping Using Sinusoidal Approximation
41
Chapter 7
Conclusion
In this thesis a simple yet accurate implementation of a neuro-fuzzy system was pro-
posed. Sigmoid membership functions were used in fuzzy logic so that they could be im-
plemented using neural networks. A LUT (Lookup Table) algorithm was introduced to
implement it in the FPGA. This LUT algorithm in itself can be used to model a non-linear
control surface with su cient accuracy. It was demonstrated using MATLAB simulations
that the proposed neuro-fuzzy model produces accurate and smooth control surfaces avoid-
ing the instability problem in fuzzy systems
42
Bibliography
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with feedback linearization Neural Networks, IEEE Transactions on On page(s): 1409-
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[2] Deutsch, S. and Deutsch, A., Understanding the NerwusSjmem: An Engineering Per-
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[3] Zurada, J., Intrudiiction to Artifkid Naira1 Systems, West Publishing Company, 1992.
[4] Wilamowski, B. M. Jaeger, R. C. and Kaynak, M. 0., "NewFuzzy Architecture for
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pp. 1132-1136, Dec. 1999.
[5] Wilamowski, B. M. Chen, Y.,"E cient Algorithm for Training Neural Networks with
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[6] Tan H. Sandige, R. and Wilamowski, B. M., "Hardware implementation of PLD based
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[7] Tikk, D. and Baranyi, P., "Comprehensive Analysis of a New Fuzzy Rule Interpolation
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[8] Zadeh, L. A, Fuzzy sets. Information and control, New York, Academic Press vol 8,
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[9] Takagi, T. and Sugeno, S, Derivation of Fuzzy Control Rules from Human Operator?s
Control Action . Proc. of the IFAC Symp. On Fuzzy Info Knowledg Representation
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[10] Ying, H, " General SISO Takagi-Sugeno Fuzzy Systems with Linear Rule Conse-
quent are Universal Approximators", IEEE Transaction on Fuzzy System, vol.6, no.4,
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44
Appendices
45
Appendix A
Matlab Code
A.1 1-Dimensional LUT algorithm
clc;
clear all;
total=1200;
t=40;
 gure(1);clf;
for i=1:total
p(i)=i-1;
x(i)=(pi*p(i))/180;
y(i)=(i/1000)*sin(x(i));
end
plot(p,y);
 gure(2);
stem(p,y);
for i=1:t:total
a(i)=p(i);
b(i)=y(i);
end
 gure(3);
stem(a,b);
for i=1:total-t
c(i)=p(i);
modx=mod(i-1,t);
if (modx==0)
d(i)=b(i);
else
y1=b(i-modx);
temp=i+(t-modx);
y2=b(temp);
d(i)=y1+((y2-y1)*(1/t)*modx);
end
end
hold on;
 gure(4);
plot(c,d,?g?);
46
for i=1:t
j1(i)=i-1;
temp1=(i-1)*(180/t);
k1(i)=sin(pi*temp1/180);
end
 gure(5);
stem(j1,k1);
for i=1:total-(2*t)
c1(i)=p(i);
modx=mod(i-1,t);
if i<=t
if (modx==0)
d1(i)=b(i);
else
y1=b(i-modx);
temp=i+(t-modx);
y2=b(temp);
d1(i)=y1+((y2-y1)*(1/t)*modx);
end
else
if (modx==0)
d1(i)=b(i);
else
y1=b(i-(modx+t));
y2=b(i-modx);
temp=i+(t-modx);
y3=b(temp);
y4=b(temp+t);
y11=(y3-y1)/(2*t);
y21=(y3-y2)/t;
y31=(y4-y2)/(2*t);
dely=(((y11-y21)/2)-((y31-y21)/2));
dely1=(dely*k1(modx));
dely2=((((y11-y21)/2)+((y31-y21)/2))/2)*k1(modx);
d1(i)=y2+((y3-y2)*(1/t)*modx)+((t/4)+4)*dely1+dely2;
end
end
end
plot(c1,d1,?r?);
for i=1:total-(2*t)
d2(i)=y(i)-d1(i);
d3(i)=y(i)-d(i);
end
 gure(7);
plot(c1,d2);
47
 gure(8);
plot(c1,d3);
A.2 2-Dimensional LUT algorithm
clc;
clear all;
[j,k] = meshgrid(-3:.1:3);
l = j.2 +k:2;
 gure(1);clf;
mesh(j,k,l);
xlabel(?x?);
ylabel(?y?);
zlabel(?z=x2 +y20);
total1=320;
t=40;
v=0;
for i=1:total1
x(i)=i-total1/2;
for j=1:total1
y(j)=j-total1/2;
z(i,j) = x(i).2  y(j):2;
end
end
 gure(1);clf;
mesh(x,y,z);
clc; clear all;
for i=1:20:500
a(i)=i-250;
for j=1:20:500
b(j)=j-250;
c(i,j) = a(i).2 +b(j):2;
end
end
 gure(3);clf;
mesh(a,b,c);
for i=1:t:total1
a(i)=x(i);
v=v+1;
for j=1:t:total1
b(j)=y(j);
c(i,j) = z(i,j);
end
end
 gure(2);clf;mesh(a,b,c);
48
for i=1:t
j1(i)=i-1;
temp1=(i-1)*(180/t);
k1(i)=sin(pi*temp1/180);
end
for i=1:t
j1(i)=i-1;
temp1=(i-1)*(180/t);
k2(i)=sin(2*pi*temp1/180);
end
 gure(3);clf;
plot(j1,k1);
for i=1:(total1-(2*t))
d1(i)=x(i);
for j=1:(total1-(2*t))
e1(j)=y(j);
if i<=t | j<=t
 ag=0;
modx=mod(i-1,t);
mody=mod(j-1,t);
if (modx =0)|(mody =0)
a1=c(i-modx,j-mody);
a2=c(i+(t-modx),j-mody);
a3=c(i-modx,j+(t-mody));
a4=c(i+(t-modx),j+(t-mody));
k=(1/t)*modx;
l=(1/t)*mody;
 ag=1;
end
if  ag==1
f1(i,j)=a1*(1-k)*(1-l) + a3*(l)*(1-k) + a4*(k)*(l) +a2*(k)*(1-l);
else
f1(i,j)=c(i,j);
end
else
 ag=0;
modx=mod(i-1,t);
mody=mod(j-1,t);
if (modx =0)|(mody =0)
a1=c(i-modx,j-mody);
a2=c(i+(t-modx),j-mody);
a3=c(i-modx,j+(t-mody));
a4=c(i+(t-modx),j+(t-mody));
k=(1/t)*modx;
l=(1/t)*mody;
49
a11=c(i-modx-t,j-mody);
a12=c(i-modx,j-mody-t);
a21=c(i+(t-modx),j-mody-t);
a22=c(i+(t-modx)+t,j-mody);
a31=c(i-modx-t,j+(t-mody));
a32=c(i-modx,j+(t-mody)+t);
a41=c(i+(t-modx),j+(t-mody)+t);
a42=c(i+(t-modx)+t,j+(t-mody));
slop11=(a2-a11)/(2*t);
slop21=(a22-a1)/(2*t);
slop31=(a4-a31)/(2*t);
slop41=(a42-a3)/(2*t);
slop12=(a3-a12)/(2*t);
slop22=(a32-a1)/(2*t);
slop32=(a4-a21)/(2*t);
slop42=(a41-a2)/(2*t);
mslop12=(a2-a1)/t;
mslop34=(a4-a3)/t;
mslop13=(a3-a1)/t;
mslop24=(a4-a2)/t;
dely1=(((slop11-mslop12)/2)-((slop21-mslop12)/2))*k1(modx+1);
dely2=(((slop31-mslop34)/2)-((slop41-mslop34)/2))*k1(modx+1);
delz1=(((slop12-mslop13)/2)-((slop22-mslop13)/2))*k1(mody+1);
delz2=(((slop32-mslop24)/2)-((slop42-mslop24)/2))*k1(mody+1);
dely11=(((slop11-mslop12)/2)+((slop21-mslop12)/2))*k2(modx+1);
dely21=(((slop31-mslop34)/2)+((slop41-mslop34)/2))*k2(modx+1);
delz11=(((slop12-mslop13)/2)+((slop22-mslop13)/2))*k2(mody+1);
delz21=(((slop32-mslop24)/2)+((slop42-mslop24)/2))*k2(mody+1);
 ag=1;
end
if  ag==1
y1=a1+((a2-a1)*(1/t)*modx)+(t/4)*dely1;
y2=a3+((a4-a3)*(1/t)*modx)+(t/4)*dely2;
z1=a1+((a3-a1)*(1/t)*mody)+(t/4)*delz1;
z2=a2+((a4-a2)*(1/t)*mody)+(t/4)*delz2;
f1(i,j)=(y1+y2+z1+z2)/4;
avg1=(dely1+dely2+delz1+delz2)/4;
avg2=(dely11+dely21+delz11+delz21)/4;
f1(i,j)=(a1*(1-k)*(1-l) + a3*(l)*(1-k) + a4*(k)*(l) +a2*(k)*(1-l))+(((t/4)+10)*avg1);
else
f1(i,j)=c(i,j);
end
end
end
end
50
 gure(4); clf;
mesh(d1,e1,f1);
error calculation.........................................
for i=1:total1-2*t
for j=1:total1-2*t
err(i,j)=f1(i,j)-z(i,j);
end
end
 gure(5);clf;
mesh(d1,e1,err);
51