Fuzzy Logic Models and Fuzzy Control. An Introduction

409 Pages·2017·11.37 MB·English

by  D. S. Hooda

Fuzzy Logic Models
and Fuzzy Control
An Introduction
D. S. Hooda
Vivek Raich
α
Alpha Science International Ltd.
Oxford, U.K.
Fuzzy Logic Models and Fuzzy Control
An Introduction
408 pgs.
D. S. Hooda
Former PVC, Kurukshetra Univesity
Honorary Professor (Mathematics)
GJ University of Science & Technology Hisar
and adviser (R) to ABV university Bhopal
Vivek Raich
Department of Mathematics
Govt. Holkar Science College
Indore
Copyright © 2017
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Preface
Fuzzy control has increased tremendous interest in applications over the past few years and
also among control equipment. The present book titled “Fuzzy Logic Models and Fuzzy Con-
trol: An Introduction” has been written to meet these inspirations. It consists of total nine chap-
ters: First three chapters are related to fuzzy set theory, fuzzy logic, fuzzy systems and models,
while next six chapters deal with design and analysis methodology of fuzzy control. However,
it should be clear that such a universal theory does not exist for conventional control engineer-
ing either, so we have to proceed from a few isolated spots where we already know exactly how
to design a fuzzy control algorithm to clusters of problems and related design methodologies.
Thus, we have tried to structure this book to make it a text book for control engineering cover-
ing just the relevant part of the theory and focussing on the principles of fuzzy control rather
than particular applications or tools. Starting with relevant mathematics including fuzzy set
theory, fuzzy logic and models, we go to design parameters and choices and discuss fuzzy
control in control engineering terms such as linear, non-linear, adaptive control and stability
criteria.
The book explicitly aimed at readers from B. Tech./ M.Tech engineering students who want
to study fuzzy control as elective course. It contains the following nine chapters: Chapter 1
Fuzzy Set and Fuzzy Numbers; Chapter 2 Fuzzy logic and Fuzzy Systems; Chapter 3 Fuzzy
System Models; Chapter 4 Fuzzy Control; Chapter 5 Fuzzy Knowledge Base Control (KFBC);
Chapter 6 Adaptive Fuzzy Control; Chapter 7 Non-linear Fuzzy Control; Chapter-8 Stability of
Fuzzy Control Systems, and Chaper 9 Fuzzy Control System Model.
D. S. Hooda
Vivek Raich

Contents
Preface ...............................................................................................................................iii
Chapter-1 Fuzzy Sets and Fuzzy Numbers .............................................................................1.1
1.1 Basic Concepts and Definitions ...................................................................................1.1
1.2 Types of Fuzzy Sets .....................................................................................................1.9
1.3 Operations on Fuzzy Sets ...........................................................................................1.15
1.4 Fuzzy Number ............................................................................................................1.30
1.5 Arithmetic Operations on Intervals and Fuzzy Numbers ..........................................1.37
Exercise ......................................................................................................................1.49
Bibliography ..............................................................................................................1.52
Chapter-2 Fuzzy Relations and Fuzzy Logic ..........................................................................2.1
2.1 Introduction ..................................................................................................................2.1
2.2. Features of Membership Function ...............................................................................2.1
2.3 Fuzzification .................................................................................................................2.3
2.4 Fuzzy Relations and their Graphs ..............................................................................2.11
2.5 Classical and Fuzzy Logic .........................................................................................2.31
Exercise ......................................................................................................................2.55
Bibliography ..............................................................................................................2.58
Chapter-3 Fuzzy Systems and Fuzzy Models .........................................................................3.1
3.1 Defuzzification .............................................................................................................3.1
3.2 Fuzzy Systems .............................................................................................................3.8
3.3 Fuzzy Modelling ........................................................................................................3.18
3.4 Modelling of Static Fuzzy Systems ...........................................................................3.20
3.5 Modelling of Dynamics Fuzzy Systems ....................................................................3.25
Exercise ......................................................................................................................3.33
References ..................................................................................................................3.36
vi Contents
Chapter-4 Fuzzy Control ..........................................................................................................4.1
4.1 Introduction ..................................................................................................................4.1
4.2 Conventional Control System Design ..........................................................................4.3
4.3 Fuzzy Control From an Industrial Perspective ............................................................4.9
4.4 Fuzzy Control System Design ...................................................................................4.18
4.5 Conclusions Conclusions Conclusions ......................................................................4.23
Exercise ......................................................................................................................4.25
Bibliography ..............................................................................................................4.26
Chapter-5 Fuzzy Knowledge Base Control (FKBC) ..............................................................5.1
5.1 Introduction ..................................................................................................................5.1
5.2 Classical Fuzzy Control in Inverted Pendulum ...........................................................5.1
5.3 Fuzzy and Human Control an Introduction .................................................................5.7
5.4 Rules........... ...............................................................................................................5.11
5.5 Fuzzy Quantification of Knowledge ..........................................................................5.13
5.6 Fusion of Fuzzy and PID ...........................................................................................5.18
5.7 Inversion of Fuzzy Systems .......................................................................................5.21
5.8 Summary and Conclusions ........................................................................................5.30
Exercise ......................................................................................................................5.32
Bibliography ..............................................................................................................5.35
Chapter-6 Adaptive Fuzzy Control .........................................................................................6.1
6.1 Introduction ..................................................................................................................6.1
6.2 Conventional PID Controllers ......................................................................................6.2
6.3 Design of the PI Controller in Frequency Domain ......................................................6.9
6.4 Fuzzy PID Controller .................................................................................................6.17
6.5 Fuzzy PD Controller ..................................................................................................6.18
6.6 Fuzzy PI Controller ....................................................................................................6.30
6.7 Fuzzy PI+D Controller ...............................................................................................6.33
6.8 Stability Analysis of Fuzzy PID Controller ...............................................................6.47
Exercise ......................................................................................................................6.61
Bibliography ..............................................................................................................6.62
Chapter-7 Non-linear Fuzzy Control .......................................................................................7.1
7.1 Introduction ..................................................................................................................7.1
7.2 The Control Problem (Crisp Logic) ............................................................................7.3
Contents vii
7.3 The Control Problem (Fuzzy Logic) Model Free Approach ........................................7.8
7.4 Examples of Model-Free Fuzzy Controller Design ...................................................7.22
7.5 The Control Problem (Fuzzy Logic) Model-Based Approach ...................................7.32
Exercise ......................................................................................................................7.41
Bibliography ..............................................................................................................7.45
Chapter-8 Stability of Fuzzy Control System .........................................................................8.1
8.1 Introduction ..................................................................................................................8.1
8.2 The State Space Approach ...........................................................................................8.4
8.3 Stability and Robustness Indices ...............................................................................8.12
8.4 Input-Output Stability ................................................................................................8.25
8.5 The Cricle Criterion ..................................................................................................8.26
8.6 The Conicity Criterion ...............................................................................................8.32
Exercise ......................................................................................................................8.41
Bibliography ..............................................................................................................8.44
Chapter-9 Fuzzy Control System Models ...............................................................................9.1
9.1 Introduction ..................................................................................................................9.1
9.2 Health Monitoring Systems ........................................................................................9.1
9.3 Monitoring System for Diabetes ................................................................................9.12
9.4 Air Conditioner Controller .........................................................................................9.18
9.5 Blood Pressure Controller During Anesthesia ...........................................................9.22
9.6 Low Cost Temperature Control ..................................................................................9.24
9.7 Aircraft Landing Control Problem .............................................................................9.27
Exercise ......................................................................................................................9.33
Bibliography ..............................................................................................................9.35
Index ...... .....................................................................................................................I .1

Chapter-1
Fuzzy Sets and Fuzzy Numbers
1.1 BASIC CONCEPTS AND DEFINITIONS
Only in 20th century, mathematicians defined the concepts of sets and functions to represent
problems. In many circumstances the solution using this concept are meaningless. This
difficulty was overcome by the fuzzy concept. Almost all mathematical, engineering,
medicine etc. concepts have been redefined using fuzzy set. It essentially prepares the teacher
as well as student in the use of fuzzy sets. Basic definition of crisp sets properties of α-cuts,
representation of fuzzy sets, operations on fuzzy sets. These help in providing a more general
appreciation of what has been done with fuzzy sets and applied in problems related to various
areas. It also helps in providing an integrated view of extension principle for fuzzy sets. In
this chapter we are going to introduce the Basic Definition of Fuzzy set, types of Fuzzy sets
and Notation, Operations on Fuzzy sets, α-Cuts and Properties, Interval valued Fuzzy Set and
Arithmetic of Fuzzy numbers for more details refer to [Klier and Yuan (1995) and Lee
(2005)].
In real world, the complexity generally arises from uncertainty in the form of ambiguity. The
probability theory has been an old age and effective tool to handle uncertainty, but it can be
applied only to situations whose characteristics are based on random processes, i.e., processes
in which the occurrence of events is strictly determined by chance. Uncertainty may arise due
to partial information about the problem, or due to information which is not fully reliable, or
due to inherent imprecision in the language with which the problem is defined or due to
receipt of information from more than one source. Fuzzy set theory is an excellent
mathematical tool to handle the uncertainty arising due to vagueness. In 1965, Lotfi A. Zadeh
propounded the fuzzy set theory in his paper.
A set A can be equivalently represented by its characteristic function i.e. A mapping  from
A
the universe of discourse (i.e. region of consideration is a large set) containing A to the set
0,1; equivalently, xA iff  (x) = 1.
A
In “fuzzy” case “belonging to” relation  (x)between x and A is no longer “0 or 1” But it
A
has a degree of “belonging to” i.e. membership degree such as 0.6. Therefore, the range has
to extend from set {0, 1} to interval [0, 1].