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Soft Computing and Its Applications, Volume Two: Fuzzy Reasoning and Fuzzy Control PDF

459 Pages·2014·12.677 MB·English
by  RayKumar S.
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SOFT 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. CRC Press Apple Academic Press, Inc Taylor & Francis Group 3333 Mistwell Crescent 6000 Broken Sound Parkway NW, Suite 300 Oakville, ON L6L 0A2 Boca Raton, FL 33487-2742 Canada © 2014 by Apple Academic Press, Inc. Exclusive worldwide distribution by CRC Press an imprint of Taylor & Francis Group, an Informa business No claim to original U.S. Government works Version Date: 20140905 International Standard Book Number-13: 978-1-4822-5793-9 (eBook - PDF) This book contains information obtained from authentic and highly regarded sources. Reason- able efforts have been made to publish reliable data and information, but the author and publisher cannot assume responsibility for the validity of all materials or the consequences of their use. The authors and publishers have attempted to trace the copyright holders of all material reproduced in this publication and apologize to copyright holders if permission to publish in this form has not been obtained. If any copyright material has not been acknowledged please write and let us know so we may rectify in any future reprint. Except as permitted under U.S. Copyright Law, no part of this book may be reprinted, reproduced, transmitted, or utilized in any form by any electronic, mechanical, or other means, now known or hereafter invented, including photocopying, microfilming, and recording, or in any information storage or retrieval system, without written permission from the publishers. For permission to photocopy or use material electronically from this work, please access www. copyright.com (http://www.copyright.com/) or contact the Copyright Clearance Center, Inc. (CCC), 222 Rosewood Drive, Danvers, MA 01923, 978-750-8400. CCC is a not-for-profit organiza- tion that provides licenses and registration for a variety of users. For organizations that have been granted a photocopy license by the CCC, a separate system of payment has been arranged. Trademark Notice: Product or corporate names may be trademarks or registered trademarks, and are used only for identification and explanation without intent to infringe. Visit the Taylor & Francis Web site at http://www.taylorandfrancis.com and the CRC Press Web site at http://www.crcpress.com For information about Apple Academic Press product http://www.appleacademicpress.com © 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.

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