Table Of ContentSOFT COMPUTING AND
ITS APPLICATIONS
Volume II: Fuzzy Reasoning and Fuzzy Control
© 2014 by Apple Academic Press, Inc.
SOFT COMPUTING AND
ITS APPLICATIONS
Volume II: Fuzzy Reasoning and Fuzzy Control
Kumar S. Ray, PhD
Apple Academic Press
TORONTO NEW JERSEY
© 2014 by Apple Academic Press, Inc.
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© 2014 by Apple Academic Press, Inc.
ABOUT THE AUTHOR
Kumar S. Ray, PhD
Kumar S. Ray, PhD, is a Professor in the Electronics and Communication Science Unit
at the Indian Statistical Institute, Kolkata, India. He is an alumnus of the University
of Bradford, UK. He was a visiting faculty member under a fellowship program at
the University of Texas, Austin, USA. Professor Ray was a member of the task force
committee of the Government of India, Department of Electronics (DoE/MIT), for the
application of AI in power plants. He is the founder and member of the Indian Society
for Fuzzy Mathematics and Information Processing (ISFUMIP) and a member of the
Indian Unit for Pattern Recognition and Artificial Intelligence (IUPRAI). In 1991, he
was the recipient of the K. S. Krishnan memorial award for the best system-oriented
paper in computer vision. He has written a number of research articles published in in-
ternational journals and has presented at several professional meetings. He also serves
as a reviewer of several International journals. His current research interests include
artificial intelligence, computer vision, commonsense reasoning, soft computing, non-
monotonic deductive database systems, and DNA computing.
He is the co-author of two edited volumes on approximate reasoning and fuzzy
logic and fuzzy computing, and he is the co-author of Case Studies in Intelligent Com-
puting; Achievements and Trends. He has is also the author of Polygonal Approxima-
tion and Scale-Space Analysis of Closed Digital Curves, published Apple Academic
Press, Inc.
© 2014 by Apple Academic Press, Inc.
CONTENTS
List of Abbreviations .........................................................................................ix
Preface ............................................................................................................xiii
1. Fuzzy Reasoning...............................................................................................01
2. Fuzzy Reasoning Based on Concept of Similarity ......................................139
3. Fuzzy Control .................................................................................................233
4. Concluding Remarks .....................................................................................415
References .......................................................................................................435
Index ................................................................................................................449
© 2014 by Apple Academic Press, Inc.
LIST OF ABBREVIATIONS
AARS Approximate analogical reasoning scheme
AI Artificial intelligence
ANN Artificial neural network
AR Approximate reasoning
ART Adaptive resonance theory
CBD Case-based design
CBR Case-based reasoning
CDR Consequent dilation rule
CMI Compatibility modification inference
CRI Compositional rule of inference
DDC Direct digital computer
DFI Decomposed fuzzy implication
DID Dynamic importance degree
DSM Design structure matrices
EC Evolutionary computation
FARMA Fuzzy autoregressive moving average model
FSMC Fuzzy sliding mode control
GMC Generalized method-of-case
GMP Generalized modus ponens
GPC Generalized predictive control
LQR Linear quadratic regular design
MEDI Methodology for estimation of design intent
MFI Multidimensional fuzzy implication
MFR Multiple fuzzy reasoning
MIMO Multi-input-multi-output
MIQ Machine intelligence quotient
NARMA Nonlinear autoregressive moving average
OSF Output scaling factor
PGF Pressure gradient force
PID Proportional-integral-derivative
PTOS Proximate time-optimal servomechanism
QFD Quality function deployment
SBR Similarity based reasoning
SCAD Soft computing aided design
SIRMs Single input rule modules
SOPSS Self-organizing power system stabilizer
SPC Smith predictor control
SSR Solid state relay
© 2014 by Apple Academic Press, Inc.
Dedicated to:
Dhira Ray (wife)
Aratrika Ray (daughter)
© 2014 by Apple Academic Press, Inc.
PREFACE
At present the notion of soft computing is well established in various fields of science
and engineering. The journey of soft computing started in the early 90s when Zadeh
first coined the term ‘soft computing’. Since then the topic soft computing has passed
through growth and development through advanced features of fuzzy reasoning and
its application to fuzzy control. Though the notion of vagueness was sensed from the
period of Bertrand Russell and Max Black and continued through the multivalued
concept of Lukasiewicz and till today, the formal notion of soft computing has been
sensed by Zadeh, with his spirit and intuition about vagueness and its flexible repre-
sentation to handle real life problem. Normally, science models real life; but to make
the model accurate science crosses a certain threshold of precision for which the model
itself becomes very complex or sometimes becomes impossible to represent. Under
such circumstances the soft computing approach replaces the complexity of model-
ling and describes the real world in a more cost-effective manner using implicit model
(fuzzy IF-THEN rules). Where our experience is concerned we understand that from
car parking to moon landing, from life science to physical and earth science, from
medical science to management science, there exists a high degree of complexity that
can be easily represented by fuzzy IF-THEN models of soft computing. Hence the
tool soft computing can be a landmark paradigm of computation with cognition which
directly or indirectly tries to replicate rationality of a human being. Even though today
we cannot quantify in a specific manner the term human intelligence for machine
implementation, still there are several attempts to do so by some intelligent methods
of computing which in our view is basically rational computing.
If we go by the fundamental slogan, ‘Man is a rational animal’ and if we consider
human perception as a basic element of rationality, then our target of soft computing
should be to mimic such cognitive process so that machine can also behave in a ratio-
nal manner and it becomes indistinguishable from human rationality. At present it may
appear that the above idea is over projected; still we should be optimistic about our
promise on soft computing.
Keeping these ideas in mind, the present volume on soft computing essentially
handles several advanced features of fuzzy reasoning and its application to fuzzy con-
trol which would be very attractive to the research communities and to the industries
who are keen to make the technology more advanced to face the challenge of the real
world. This book is very valuable for academic purposes and also for industry. It con-
tains several real life applications to convince readers about the utility and potentiality
of soft computing. The book is balanced between theory and practice.
— Kumar S. Ray, PhD
© 2014 by Apple Academic Press, Inc.
CCHHAAPPTTEERR 11
FUZZY REASONING
1.1 INTRODUCTION
Human reasoning is basically a cognitive process. To mimic the cognitive process of
human reasoning and its implementation through machine, we pass through a passage
of a set of propositions, represented as premises, to a further proposition, taken as the
conclusion (consequence) with some degree of confidence, which link the conclusion
to the premises. Reasoning is approximate when some of the propositions are impre-
cise and the rules for derivation are inexact in nature.
Approximate reasoning is considered to be a powerful tool to study the remarkable
human ability to understand real-world activities in terms of computational entities
with immense confidence. As for instance, crossing the railway track safely even after
seeing a train approaching with high speed or avoidance of a particular road and use of
by-lanes to beat the office-hour rush are typical examples of approximate reasoning.
Approximate reasoning is defined as the process or processes by which an approxi-
mate conclusion may be deduced from a set of possibly imprecise information using
some inexact rule for the derivation. It was first formally introduced by Zadeh. Since
its inception in 1973, significant theoretical advances have established approximate
reasoning as an important field of research. Different mechanisms of approximate rea-
soning with applications have been proposed and discussed in the literature. We have
witnessed among many other things the birth of fuzzy logic controller, soft computing
approach to pattern classification and object recognition, weather forecasting, and so
on.
Zadeh’s concept of approximate reasoning is based on the fuzzy logic and the
theory of fuzzy sets.
In order to have an adequate understanding of the theory of approximate reason-
ing, some basic concepts are studied in the following:
© 2014 by Apple Academic Press, Inc.
2 Soft Computing and Its Applications
1.2 MODEL OF APPROXIMATE REASONING
In 1979, Zadeh introduced a theory of approximate reasoning. It provides a powerful
framework for reasoning in the face of imprecise and uncertain information. Central
to this theory is the representation of crisp statements as statements assigning fuzzy
sets as values to variables.
Suppose, we have two interactive variables x and y , and a causal relation-
ship between x and y is completely known. Namely, we know that y is a function of
x, that is y = f(x). ∈ X ∈ Y
Then, we can make inferences easily:
"y=f(x)"and"x=x "→"y=f(x )" (1.1)
1 1
This inference rule says that if we have y = f(x), for all x X and we observe that
x = x then y takes the value f(x). More often than not, we do not know the complete
1 1
causal link f between x and y, only we know the values of ∈f(x) for some particular
values of x, that is:
R: If x = x then y = y
1 1 1
R: If x = x then y = y
1 2 2
.
.
.
.
R :if x=x theny=y (1.2)
n n n
If we are given an x’ X and want to find an y’ Y which corresponds to x’ under
the rule baseR={R ,.....R } then we have to solve an interpolation problem.
1 ∈n ∈
In this section, we present an analogy between approximate reasoning and the
method interpolation for a large class of problems. We first describe the analogy and
then illustrate it through several simple yet concrete examples. The results obtained
through the method of interpolation are compared with those obtained by the applica-
tion of the existing method of approximate reasoning. Thus, we show that approximate
reasoning can also be realized effectively by the method of interpolation.
The celebrated concept of approximate reasoning has been tremendously used
in different fields of science and engineering for solving problems having uncertain,
imprecisem, and incomplete information. But, for performing approximate reasoning
using compositional rule of inference, we have to approximate the linguistic vague-
nesses by finite fuzzy sets. In most cases, it is also possible to represent the universe of
those fuzzy sets by subsets of the real number systems. In Section 3.9 of Chapter 3 of
© 2014 by Apple Academic Press, Inc.
