Fuzzy Modeling and Control

736 Pages·2001·19.01 MB·English

by  Prof. Andrzej Piegat (auth.)

Fuzzy Modeling
and Control
Studies in Fuzziness and Soft Computing
Editor-in-chief
Prof. Janusz Kacp17.yk
Systems Research Institute
Polish Academy of Sciences
ul. Newelska 6
01-447 Warsaw, Poland
E-mail: [email protected]
http://www.springer.deJcgi-bin/search_book.pl?series = 2941
Further volumes of this series can Val. 58. D.S. Malik and J.N. Mardeson
be found at our homepage. Fuzzy Discrete Structures, 2000
ISBN 3-7908-1335-4
Val. 59. T. Furuhashi, S. Tano and
Val. 47. E. Czogalat and J.1:.eski
H.-A. Jacobsen (Eds.)
Fuzzy and Neuro-Fuzzy Intelligent Systems, 2000
Deep Fusion of Computational
ISBN 3-7908-1289-7
and Symbolic Processing, 2001
Val. 48. M. Sakawa ISBN 3-7908-1339-7
Large Scale Interactive Fuzzy Multiobjective
Val. 60. K.J. Cias (Ed.)
Programming, 2000
Medical Data Mining and Knowledge Discovery,
ISBN 3-7908-1293-5
2001
Val. 49. L.I. Kuncheva ISBN 3-7908-1340-0
Fuzzy Classifier Design, 2000 Val. 61. D. Driankav and A. Saffiatti (Eds.)
ISBN 3-7908-1298-6
Fuzzy I..ogic Techniques for Autonomous Vehicle
Val. 50. F. Crestani and G. Pasi (&Is.) Navigation, 2001
Soft Computing in Information Retrieval, 2000 ISBN 3-7908-1341-9
ISBN 3-7908-1299-4 Val. 62. N. Baba. L.C. Jain (Eds.)
Val. 51. J. Fadar, B. Oe Baets and P. Pemy (Eds.) Computational lntelligence in Games, 2001
ISBN 3-7908-1348-6
Preferences and Decisions under Incomplete
Knowledge, 2000 Val. 63. O. Castilla, P. Melin
ISBN 3-7908-1303-6 Soft Computing for Control of Non-Linear
Dynamical Systems, 2001
Val. 52. E.E. Kerre and M. Nachtegael (Eds.)
ISBN 3-7908-1349-4
Fuzzy Techniques in Image Processing, 2000
ISBN 3-7908-1304-4 Val. 64. I. Nishizaki, M. Sakawa
Fuzzy and Multiobjective Games for Conflict
Val. 53. G. Bardagna and G. Pasi (Eds.)
Resolution, 2001
Recent Issues on Fuzzy Databases, 2000
ISBN 3-7908-1341-9
ISBN 3-7908-1319-2
Val. 65. E. OrIawska. A. Szalas (&Is.)
Val. 54. P. Sinclik and J. Vasclik (Eds.)
Relational Methods for Computer Science
Quo Vadis Computationallntelligence?, 2000 Applications, 2001
ISBN 3-7908-1324-9 ISBN 3-7908-1365-6
Val. 55. J.N. Mardesan, D.S. Malik Vol. 66. R.J. Hawlett, L.C. Jain (Eds.)
and S.-C. Cheng Radiol Basis Function Networks 1, 2001
Fuzzy Mathematics in Medicine, 2000 ISBN 3-7908-1367-2
ISBN 3-7908-1325-7
Val. 67. R.J. Hawlett, L.C. Jain (Eds.)
Val. 56. L. Palkowski, S. Tsumota and T. Y. Lin (Eds.) Radial Basis Function Networks 2, 2001
Rough Set Methods and Applications, 2000 ISBN 3-7908-1368-0
ISBN 3-7908-1328-1
Val. 68. A. Kandel, M. Last and H. Bunke (&Is.)
Val. 57. V. Navlik and I. Perfilieva (Eds.) Data Minining and Computational Intelligence,
Discovering the World with Fuzzy I..ogic, 2001 2001
ISBN 3-7908-1330-3 ISBN 3-7908-1371-0
Andrzej Piegat
Fuzzy Modeling
and Control
With 680 Figures
and 96 Tables
Springer-Verlag Berlin Heidelberg GmbH
Prof. Andrzej Piegat
Faculty of Computer Science and Information Systems
Technical University of Szczecin
Zolnierska Street 49
71-210 Szczecin
Poland
Andrzej [email protected]
ISSN 1434-9922
ISBN 978-3-7908-2486-5 ISBN 978-3-7908-1824-6 (eBook)
DOI 10.1007/978-3-7908-1824-6
Cataloging-in-Publication Data applied for
Die Deutsche Bibliothek - CIP-Einheitsaufnahme
Piegat, Andrzej: Fuzzy modeling and control: with 96 tables I Andrzej Piegat. - Heidelberg; New York:
Physica-Verl., 2001
(Studies in fuzziness and soft computing; Vol. 69)
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© Springer-Verlag Berlin Heidelberg 2001
Originally published by Physica-Verlag Heidelberg in 2001.
Softcover reprint ofthe hardcover 1st edition 2001
The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply,
even in the absence of a specific statement, that such names are exempt from the relevant protective laws
and regulations and therefore free for general use.
Hardcover Design: Erich Kirchner, Heidelberg
To my Family
Foreword
Since their inception in the mid-1960s by Professor Lotfi A. Zadeh from the
University of California, Berkeley, fuzzy sets have triggered mixed feeling in
the scientific community. On the one hand, there has been a growing number
of devotees who have early recognized potentials of fuzzy sets to model and
solve many real world problems. On the other hand, however, there have been
quite a considerable number of opponents - quite often prominent scholars
and researchers - who have fiercely fought against this emerging new tools.
One of their arguments has been a lack of applications.
The situation had changed since the mid 1980s when the so-called "fuzzy
boom" occurred, primarily in Japan, but then also in Korea and Europe,
much less so in the USA. Basically, the turning point was the launching
on the market of fuzzy logic control based appliances and other equipment
exemplified by subway trains, cranes, elevators, etc. They had primarily been
successful applications of fuzzy logic control originated by Mamdani, Sugeno,
Takagi and other people.
FUzzy logic control has since then been a benchmark application of fuzzy
sets, and even to many people synonymous with applications. Many excellent
books have been published on this topic. However, some of them have been
authored by people from outside of the control community. One of conse
quences of this situation has been, in my view, too high emphasis on logical,
relational, etc. aspects of fuzzy control, with too little emphasis on more
control specific issues and aspects.
One of those control specific aspects is modeling. In fact, modeling is prob
ably much more relevant than control itself, and is of much greater applica
bility as a general tool and technique for problems in virtually all domains.
Unfortunately, modeling has not received adequate attention in the fuzzy lit
erature though developments in juzzy modeling have been quite considerable.
The present book is probably the most comprehensive coverage of fuzzy
modeling and control in the literature. First of all, it presents in depth the
crucial field of fuzzy modeling, in all its aspects and fiavors. The book dis
cusses virtually all better known and more popular techniques exemplified by
rule based modeling, logical models, and also hybrid models like neuro-fuzzy.
The author discusses in a clear and simple yet formal and strict way issues
VIII Foreword
related to fuzzy modeling that can be of much use for readers interested in
both more theoretical and practical aspects.
Then, after that comprehensive discussion of fuzzy modeling, the author
proceeds to the presentation of fuzzy control. He starts with a more tradi
tional approach of what might be called a purely fuzzy logic control that is
not model based, and then proceeds to modern, more promising approaches
that rely on fuzzy models of plants and systems under control, and also of
more sophisticated control structures like adaptive or multivariate control.
Finally, the author discusses issues related to the stability of fuzzy control
systems, and again, the scope of analysis is rarely encountered in the existing
literature.
To summarize, in my opinion this is an extraordinary books, like almost
no ones in the literature. It gives a great, comprehensive coverage of fuzzy
modeling and control, written in a manner that is both understandable and
acceptable by the control community. The writing of such a book has required
not only great knowledge but also a considerable scholarly and research ma
turity to be able to find proper, most promising tools and techniques in a
vast literat ure.
Professor Piegat deserves thanks and appreciation of the whole fuzzy com
munity for writing such an exceptional books that should be read by all
interested in modern approaches to fuzzy modeling and control.
Warsaw, Poland, December 2000 Janusz Kacprzyk
Preface
Conventional mathematics enables processing of precise information, e.g.:
• temperature 39.7 (OC),
• velo city 90 (km/h),
• business pay $12317,
• sea wave amplitude 1.75 (m).
However, in the surrounding world we very often meet imprecise informa
tion such as:
• high temperature,
• high velocity,
• low business pay,
• calm sea,
• pleasant sales person,
• big customer interest,
• small turbidity of the liquid,
• high steel quality, etc.
Imprecise information has been used by people for thousands of years.
However, until quite recently it has not been used at all in methods based
on conventional mathematics. Therefore it has been lost. Because of this,
the efficiency of many design-, control-, modeling-, forecasting- and decision
making methods was considerably limited - all the more, as in some systems
imprecise information is the only accessible information. Besides, each piece
of "precise" information was measured with some (often considerable) error -
so it is also imprecise.
The domain of mathematics dealing with imprecise information was
named fuzzy set theory. This theory in connection with conventional math
ematics enables the processing and use of any information. It discloses new
and very interesting possibilities and prospects for science and technology.
This book provides the reader with essential information of fuzzy set
theory, fuzzy modeling and control. It is based on the latest literature and
on the results of the author's investigations.
Good understanding of a theory is the main condition for its application
and improvement or development of one's own ideas and concepts. To make
this easier, the author illustrated the presented methods with a large number
X Preface
of figures and examples. The author wishes the reader many benefits from
the information contained in this book.
The author would like to thank the following persons:
• Deputy Rector for Seientific Affairs of the Technical University of Szczeein,
Professor Walerian Arabczyk for finaneial support of the book's prepa
ration,
• Dean of Faculty for Computer Seience and Information Systems of the
Technical Un iversity of Szczeein, Professor J erzy Soldek for financial sup
port of the book's preparation,
• Foundation for Development Support of the Technical University of Szcze
ein and especially its Director Krzysztof Leszczynski for finaneial sup
port of the book's preparation,
• Bogdan Grzywacz Ph.D., Stanislawa Lewandowska M.Sc., Ewa
Lisek M.Sc., for the book's translation,
• Richard Stark BA, MA, CELTA (Great Britain) for language correction
of the book,
• Marcin Plucinski Ph.D. for computer setting of the book.
Szczecin, December 2000 A ndrzej Piegat
Contents
1. Introduction.............. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1 Essence of fuzzy set theory .............................. 1
1.2 Development of fuzzy set theory . . . . . . . . . . . . . . . . . . . . . . . . . . 6
2. Basic Notions of Fuzzy Set Theory ....................... 11
2.1 Fuzzy sets. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 11
2.2 Characteristic parameters (indices) of a fuzzy set. . . . . . . . . .. 22
2.3 Linguistic modifiers of fuzzy sets ......................... 27
2.4 Types of membership functions of fuzzy sets . . . . . . . . . . . . . .. 34
2.5 Type 2 fuzzy sets. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 52
2.6 Fuzziness and probability: two kinds of uncertainty . . . . . . . .. 56
3. Arithmetic of Fuzzy Sets ................................. 59
3.1 The extension principle ................................. 60
3.2 Addition of fuzzy numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 67
3.3 Subtraction of fuzzy numbers ... . . . . . . . . . . . . . . . . . . . . . . . .. 75
3.4 Multiplication of fuzzy numbers . . . . . . . . . . . . . . . . . . . . . . . . .. 79
3.5 Division of fuzzy numbers ............................... 95
3.6 Peculiarities of fuzzy numbers ............................ 100
3.7 Differences between fuzzy numbers and linguistic values .. . . . 107
4. Mathematics of Fuzzy Sets ............................... 111
4.1 Basic operations on fuzzy sets ............................ 111
4.1.1 Intersection operation (logical product) of fuzzy sets .. 112
4.1.2 Union (logical sum) of fuzzy sets ................... 126
4.1.3 Compensatory operators .......................... 132
4.2 Fuzzy relations ......................................... 136
4.3 Implication ............................................ 149
5. Fuzzy Models ............................................. 157
5.1 Structure, main elements and operations in fuzzy models .... 157
5.1.1 Fuzzification ..................................... 159
5.1.2 Inference ........................................ 160
5.1.2.1 Premise evaluation ........................ 162