fuzzy logic with engineering application 3rdedition

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FUZZY LOGIC WITH ENGINEERING APPLICATIONS Third Edition Fuzzy Logic with Engineering Applications, Third Edition © 2010 John Wiley & Sons, Ltd ISBN: 978-0-470-74376-8 Timothy J Ross FUZZY LOGIC WITH ENGINEERING APPLICATIONS Third Edition Timothy J Ross University of New Mexico, USA A John Wiley and Sons, Ltd., Publication This edition first published 2010 © 2010 John Wiley & Sons, Ltd First edition published 1995 Second edition published 2004 Registered office John Wiley & Sons Ltd, The Atrium, Southern Gate, Chichester, West Sussex, PO19 8SQ, United Kingdom For details of our global editorial offices, for customer services and for information about how to apply for permission to reuse the copyright material in this book please see our website at www.wiley.com The right of the author to be identified as the author of this work has been asserted in accordance with the Copyright, Designs and Patents Act 1988 All rights reserved No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, except as permitted by the UK Copyright, Designs and Patents Act 1988, without the prior permission of the publisher Wiley also publishes its books in a variety of electronic formats Some content that appears in print may not be available in electronic books Designations used by companies to distinguish their products are often claimed as trademarks All brand names and product names used in this book are trade names, service marks, trademarks or registered trademarks of their respective owners The publisher is not associated with any product or vendor mentioned in this book This publication is designed to provide accurate and authoritative information in regard to the subject matter covered It is sold on the understanding that the publisher is not engaged in rendering professional services If professional advice or other expert assistance is required, the services of a competent professional should be sought Library of Congress Cataloging-in-Publication Data Ross, Timothy J Fuzzy logic with engineering applications / Timothy J Ross.–3rd ed p cm Includes bibliographical references and index ISBN 978-0-470-74376-8 (cloth) Engineering mathematics Fuzzy logic I Title TA331.R74 2010 620.001 511313–dc22 2009033736 A catalogue record for this book is available from the British Library ISBN: 978-0-470-74376-8 Set in 10/12pt Times Roman by Laserwords Pvt Ltd, Chennai, India Printed in Singapore by Fabulous Printers Pte Ltd This book is dedicated to my brother Larry, my cousin Vicki Ehlert and my best friends Rick and Judy Brake, all of whom have given me incredible support over the past years Thank you so much for helping me deal with all my angst! CONTENTS About the Author Preface to the Third Edition Introduction The Case for Imprecision A Historical Perspective The Utility of Fuzzy Systems Limitations of Fuzzy Systems The Illusion: Ignoring Uncertainty and Accuracy Uncertainty and Information The Unknown Fuzzy Sets and Membership Chance Versus Fuzziness Sets as Points in Hypercubes Summary References Problems Classical Sets and Fuzzy Sets Classical Sets Operations on Classical Sets Properties of Classical (Crisp) Sets Mapping of Classical Sets to Functions Fuzzy Sets Fuzzy Set Operations Properties of Fuzzy Sets Alternative Fuzzy Set Operations Summary References Problems xiii xv 10 13 14 14 16 18 20 20 21 25 26 28 29 32 34 35 37 40 41 42 42 viii CONTENTS Classical Relations and Fuzzy Relations Cartesian Product Crisp Relations Cardinality of Crisp Relations Operations on Crisp Relations Properties of Crisp Relations Composition Fuzzy Relations Cardinality of Fuzzy Relations Operations on Fuzzy Relations Properties of Fuzzy Relations Fuzzy Cartesian Product and Composition Tolerance and Equivalence Relations Crisp Equivalence Relation Crisp Tolerance Relation Fuzzy Tolerance and Equivalence Relations Value Assignments Cosine Amplitude Max–Min Method Other Similarity Methods Other Forms of the Composition Operation Summary References Problems Properties of Membership Functions, Fuzzification, and Defuzzification Features of the Membership Function Various Forms Fuzzification Defuzzification to Crisp Sets λ-Cuts for Fuzzy Relations Defuzzification to Scalars Summary References Problems Logic and Fuzzy Systems Part I Logic Classical Logic Proof Fuzzy Logic Approximate Reasoning Other Forms of the Implication Operation Part II Fuzzy Systems Natural Language Linguistic Hedges 48 49 49 51 52 52 53 54 55 55 55 55 62 63 64 65 68 69 71 71 72 72 73 73 89 90 92 93 95 97 98 110 111 112 117 117 118 124 131 134 138 139 140 142 CONTENTS Fuzzy (Rule-Based) Systems Graphical Techniques of Inference Summary References Problems Development of Membership Functions Membership Value Assignments Intuition Inference Rank Ordering Neural Networks Genetic Algorithms Inductive Reasoning Summary References Problems Automated Methods for Fuzzy Systems Definitions Batch Least Squares Algorithm Recursive Least Squares Algorithm Gradient Method Clustering Method Learning From Examples Modified Learning From Examples Summary References Problems Fuzzy Systems Simulation Fuzzy Relational Equations Nonlinear Simulation Using Fuzzy Systems Fuzzy Associative Memories (FAMS) Summary References Problems Decision Making with Fuzzy Information Fuzzy Synthetic Evaluation Fuzzy Ordering Nontransitive Ranking Preference and Consensus Multiobjective Decision Making Fuzzy Bayesian Decision Method Decision Making Under Fuzzy States and Fuzzy Actions Summary ix 145 148 159 161 162 174 175 175 176 178 179 189 199 206 206 207 211 212 215 219 222 227 229 233 242 242 243 245 250 251 255 264 265 266 276 278 280 283 285 289 294 304 317 REFERENCES 571 in the first place Then, the assumption of a strong truth-functionality (for a fuzzy logic) could be viewed as a computational device that simplifies calculations, and the resulting solutions would be presented as ranges of values that most certainly form bounds around the true answer if the assumption is not reasonable A choice of whether a fuzzy logic is appropriate is, after all, a question of balancing the model with the nature of the uncertainty contained within it Problems without an underlying physical model; problems involving a complicated weave of technical, social, political, and economic factors; and problems with incomplete, ill-defined, and inconsistent information where conditional probabilities cannot be supplied or rationally formulated perhaps are candidates for fuzzy logic applications Perhaps, then, with additional algorithms like fuzzy logic, those in the technical and engineering professions will realize that such difficult issues can now be modeled in their designs and analyses REFERENCES Altunok, E., Reda Taha, M.M., and Ross, T.J (2007) A possibilistic approach for damage detection in structural health monitoring ASCE J Struct Eng., 133 (9), 1247–1256 Blockley, D (1983) Comments on ‘Model uncertainty in structural reliability,’ by Ove Ditlevsen J Struct Safety, 1, 233–235 Dempster, A (1967) Upper and lower probabilities induced by a multivalued mapping Ann Math Stat., 38, 325–339 Donald, S (2003) Development of empirical possibility distributions in risk analysis PhD dissertation University of New Mexico, Department of Civil Engineering, Albuquerque, NM Dubois, D and Prade, H (1988) Possibility Theory, Plenum Press, New York Gaines, B (1978) Fuzzy and probability uncertainty logics Inf Control , 38, 154–169 Joslyn, C (1997) Measurement of possibilistic histograms from interval data Int J Gen Syst., 26, 9–33 Kim, J.J (2009) Uncertainty quantification in serviceability of reinforced concrete structures PhD dissertation Department of Civil Engineering, University of New Mexico, Albuquerque, NM Klir, G and Folger, T (1988) Fuzzy Sets, Uncertainty, and Information, Prentice Hall, Englewood Cliffs, NJ Klir, G and Smith, R (2001) On measuring uncertainty and uncertainty-based information: recent developments Ann Math Artif Intellig., 32, 5–33 Klir, G and Yuan, B (1995) Fuzzy Sets and Fuzzy Logic, Prentice Hall, Upper Saddle River, NJ Klir, G and Wierman, M (1999) Uncertainty Based Information: Elements of Generalized Information Theory, Physica-Verlag/Springer, Heidelberg Shafer, G (1976) A Mathematical Theory of Evidence, Princeton University Press, Princeton, NJ Wallsten, T and Budescu, D (1983) Manage Sci , 29 (2), 167 Yager, R (1993) Aggregating fuzzy sets represented by belief structures J Intellig Fuzzy Syst , (3), 215–224 Yager, R and Filev, D (1994) Template-based fuzzy systems modeling J Intellig Fuzzy Syst , (1), 39–54 Zadeh, L (1978) Fuzzy sets as a basis for a theory of possibility Fuzzy Sets Syst., 1, 3–28 Zadeh, L (1984) Review of the book A mathematical theory of evidence, by Glenn Shafer AI Mag., 81–83 Zadeh, L (1986) Simple view of the Dempster–Shafer theory of evidence and its implication for the rule of combination AI Mag., 85–90 572 BELIEF, PLAUSIBILITY, PROBABILITY, AND POSSIBILITY PROBLEMS 15.1 In structural dynamics, a particular structure that has been subjected to a shock environment may be in either of the fuzzy sets “damaged” or “undamaged,” with a certain degree of membership over the magnitude of the shock input If there are two crisp sets, functional (F) and nonfunctional (NF), then a monotone measure would be the evidence that a particular system that has been subjected to shock loading is a member of functional systems or nonfunctional systems Given the evidence from two experts shown here for a particular structure, find the beliefs and plausibilities for the focal elements Focal elements m1 m2 F NF F ∪ NF 0.3 0.5 0.2 0.3 0.6 0.1 bel1 bel2 pl1 pl2 15.2 Suppose you have found an old radio (vacuum tube type) in your grandparents’ attic and you are interested in determining its age The make and model of the radio are unknown to you; without this information you cannot find in a collector’s guide the year in which the radio was produced Here, the year of manufacture is assumed to be within a particular decade You have asked two antique radio collectors for their opinion on the age The evidence provided by the collectors is fuzzy Assume the following questions: Was the radio produced in the 1920s? Was the radio produced in the 1930s? Was the radio produced in the 1940s? Let R, D, and W denote subsets of our universe set P – the set of radio-producing years called the 1920s (Roaring 20s), the set of radio-producing years called the 1930s (Depression years), and the set of radio-producing years called the 1940s (War years), respectively The radio collectors provide beas as given in the accompanying table Collector bel1 pl1 Collector Focal elements m1 R D W R∪D R∪W D∪W R∪D∪W 0.05 0.05 0.8 0.15 0.15 0.85 0.1969 0.1 0.1 0.1 0.1 0 0 0.2 0.35 0.25 0.5 0.2677 0.05 0.05 0.1 0.05 0.5 0.4 m2 bel2 pl2 Combined evidence m12 bel12 (a) Calculate the missing belief values for the two collectors (b) Calculate the missing plausibility values for the two collectors pl12 PROBLEMS 573 (c) Calculate the missing combined evidence values (d) Calculate the missing combined belief and plausibility values 15.3 The quality control for welded seams in the hulls of ships is a major problem Ultrasonic defectoscopy is frequently used to monitor welds, as is X-ray photography Ultrasonic defectoscopy is faster but less reliable than X-ray photography Perfect identification of flaws in welds is dependent on the experience of the person reading the signals An abnormal signal occurs for three possible types of situations Two of these are flaws in welds: a cavity (C) and a cinder inclusion (I); the former is the more dangerous Another situation is due to a loose contact of the sensor probe (L), which is not a defect in the welding seams but an error in measuring Suppose we have two experts, each using a different weld monitoring method, who are asked to identify the defects in an important welded seam Their responses in terms of beas are given in the table Calculate the missing portions of the table Expert bel1 Expert Focal elements m1 pl1 C I L C∪I C∪L I∪L C∪I∪L 0.3 0.3 0.85 0.2 0.2 0.05 0.05 0.1 0.1 0.05 0.05 0.05 0.05 0.2 0.55 0.15 0.45 0.05 0.4 0.95 0.05 0.3 0.05 0.5 0.15 0.3 0.3 0.3 m2 bel2 Combined evidence pl2 m12 bel12 0.4 0.15 0.4 0.15 0.16 0.71 pl12 15.4 You are an aerospace engineer who wishes to design a bang–bang control system for a particular spacecraft using thruster jets You know that it is difficult to get a good feel for the amount of thrust that these jets will yield in space Gains of the control system depend on the amount of the force the thrusters yield Thus, you pose a region of three crisp sets that are defined with respect to specific gains Each set will correspond to a different gain of the control system You can use an initial estimate of the force you get from the thrusters, but you can refine it in real time utilizing different gains for the control system You can get a force estimate and a belief measure for that estimate for a specific set Suppose you define the following regions for the thrust values, where thrust is in pounds: A1 applies to a region 0.8 ≤ thrust value ≤ 0.9 A2 applies to a region 0.9 ≤ thrust value ≤ 1.0 A3 applies to a region 1.0 ≤ thrust value ≤ 1.1 Two expert aerospace engineers have been asked to provide evidence measures reflecting their degree of belief for the various force estimates These beas along with calculated belief measures are given here Calculate the combined belief measure for each focal element in the table 574 BELIEF, PLAUSIBILITY, PROBABILITY, AND POSSIBILITY Expert Expert Focal elements m1 bel1 m2 bel2 A1 A2 A3 A1 ∪ A2 A1 ∪ A3 A2 ∪ A3 A1 ∪ A2 ∪ A3 0.1 0.05 0.05 0.05 0.05 0.1 0.6 0.1 0.05 0.05 0.2 0.2 0.2 0.05 0.1 0.05 0.15 0.05 0.6 0.05 0.1 0.1 0.25 0.2 15.5 Consider the DC series generator shown in Figure P15.5 Let Ra = armature resistance, Rs = field resistance, and R = load resistance The voltage generated across the terminal is given by Vl = Eg − (Is Rs + Ia Ra ) Note: If there is no assignment for R, that is, if the value of R is infinity, then the generator will not build up because of an open circuit Also Ra can have a range of values from a low value to a high value To generate different load voltages required, we can assign values for R, Ra , and Rs in different ways to get the voltage They are very much interrelated and the generated voltage need not have a unique combination of R, Ra , and Rs Hence, nesting of focal elements for these resistances does have some physical significance Ia Ra Va Load R Il Vl Armature Rs Is Id FIGURE P15.5 Let the bea for the elements of universe X {R, Ra , Rs } be as shown in the accompanying table: PROBLEMS X m1 m2 Ra R Rs Ra ∪ Rs R ∪ Ra Rs ∪ R Ra ∪ R ∪ Rs 0.1 0.1 0.3 0.1 0.3 0.1 0.2 0.4 0.3 0 0.1 575 (a) Does m1 or m2 represent a possibility measure? (b) If either of the evidence measures (or both) is nested, find the possibility distributions 15.6 A general problem in biophysics is to segment volumetric MRI data of the head given a new set of MRI data We use a “model head” that has already had the structures in the head (mainly brain and brain substructures) labeled We can use the model head to help in segmenting the data from the new head by assigning beas to each voxel (a voxel is a three-dimensional pixel) in the new MRI data set, based on what structures contain, or are near, the corresponding voxel in the model head In this example, a nested subset corresponds to the physical containment of a head structure within another structure We will select a voxel for which the beas form a consonant body of evidence X = {structures in the MRI data} H = head B = brain N = neocortex L = occipital lobe C = calcarine fissure Thus, we have C ⊂ L ⊂ N ⊂ B ⊂ H For voxel V, we have a basic distribution of m = (μC , μL , μN , μB , μH ) = (0.1, 0.1, 0.2, 0.2, 0.4) Find the corresponding possibility distribution and draw the nesting diagram 15.7 A test and diagnostics capability is being developed for a motion control subsystem that consists of the following hardware: a motion control IC (integrated circuit), an interconnect between motion control IC, an H-switch current driver, an interconnect between H-switch current driver, a motor, and an optical encoder The elements of the motion control subsystem are as follows: x1 x2 x3 x4 x5 x6 = motion control IC = interconnect = H − switch current driver = interconnect = motor = optical encoder 576 BELIEF, PLAUSIBILITY, PROBABILITY, AND POSSIBILITY If a motion control subsystem failure exists, a self-test could describe the failure in the following bea: m = (0.2, 0, 0.3, 0, 0, 0.5) This nested structure is based on the level of hardware isolation of the diagnostic software This isolation is hierarchical in nature You first identify a motion control subsystem failure m(A6 ) that includes a possibility of any component failure (x1 , x2 , x3 , x4 , x5 , x6 ) The test then continues and, due to isolation limitations, a determination can be made of the failure possibility consisting of m(A3 ), subset (x1 , x2 , x3 ), followed by the ability to isolate to an x1 failure if x1 is at fault Beas are constructed from empirical data and experience Find the associated possibility distribution and draw the nesting diagram 15.8 Design of a geometric traffic route can be described by four roadway features: a corner, a curve, a U-turn, and a circle The traffic engineer can use four different evaluation criteria (expert guidance) to use in the design process: m1 m2 m3 m4 = criteria:fairly fast, short distance, arterial road, low slope points = criteria:slow, short distance, local road, low slope points = criteria:fast, long distance, ramp - type road, medium slope points = criteria:very fast, medium distance, highway, medium slope points Corner Curve U-turn Circle m1 m2 m3 m4 0 0 0 1 1 1 1 0 1 1 0 0 1 1 1 0 1 0 1 0 1 1 1 1 1 0.1 0.3 0.1 0.2 0.1 0.1 0.1 0 0.1 0.2 0.1 0.3 0.1 0.1 0.1 0.2 0 0.2 0.5 0 0.1 0 0 0 0.1 0 0.4 0.3 0 0 0.2 0 0 Using the 15 (24 − 1) focal elements shown in the accompanying table, determine which, if any, of the four evidence measures (m1 –m4 ) results in an ordered possibility distribution 15.9 Given a communication link with a sender, receiver, and interconnecting link, an error in a message could occur at the sender, receiver, or on the interconnecting link Combinations such as an error on the link that is not corrected by the receiver are also possible Let S, R, and L represent sources of error in the sender, receiver, and link, respectively If E is the universe of error sources, then P(E) = (Ø, {S}, {R},{L},{S,R},{S,L},{R,L},{S,R,L}) PROBLEMS 577 Now assume each source has its own expert and each of these provides their basic assignment of the actual source of an error as follows: S R L mS mR mL 0 0 1 1 0 1 0 1 1 1 0.4 0.2 0.2 0.1 0.1 0.2 0.1 0.5 0.2 0 0 0.1 0.4 0.4 0.1 Indicate which experts, if any, have evidence that is consonant For each of these, the following: (a) Determine the possibility distribution (b) Draw the nesting diagram (c) Give the physical significance of the nesting 15.10 There are a number of hazardous waste sites across the country that pose significant health risk to humans However, due to high costs involved in exposure analysis only a limited amount of information can be collected from each site to determine the extent of contamination Suppose it is determined that one of the sites is contaminated by a new carcinogenic chemical identified as Tox The table shows the results from the chemical analysis of the groundwater samples collected from one of the sites Given these sparse data, determine the possibility distribution of exposure concentrations for the chemical Tox Assume the observation weights are identical Observation Concentration (mg l−1 ) [0.01, 0.12] [0.03, 0.24] [0.03, 0.15] [0.008, 0.06] 15.11 Due to their excellent self-healing properties, rock salt caverns are used to store nuclear waste from various nuclear plants One of the properties useful in determining the suitability of a cavern for nuclear waste storage is the creep rate of salt; salt creeps very slowly with time This creep rate determines the strength of the cavern and the duration that the cavern can be accessible to human operations The table shows the strain rate results from creep tests conducted on rock salt cores from four locations of the waste repository Given these data, determine the strain rate interval that is 80% possible (possibilistic weight = 0.8) Assume the observation weights are identical Also, find the degree of confirmation 578 BELIEF, PLAUSIBILITY, PROBABILITY, AND POSSIBILITY Observation Strain rate (s−1 ) [6.0E−10, 9.0E−10] [8.0E−10, 1.2E−9] [9.0E−10, 3.0E−9] [5.0E−10, 10.0E−10] 15.12 Predicting interest rates is critical for financial portfolio management and other investment decisions Based on historical variations and other factors, the following interest rates are predicted for the next two months Assume the observation weights are identical Observation Interest rate [0.75, 1.5] [1.0, 1.25] [0.75, 1.25] [1.5, 2.0] [1.75, 2.25] (a) What is the possibility that the interest rates will be higher than 2%? (b) Give the reason for your choice of consonant intervals (c) Find the degree of confirmation INDEX Accuracy, Adjacency matrix, cognitive mapping, 510 Agent-based models, 10, 520 crisp, 522 fuzzy, 522–524 Aggregation operators, averaging, 41 ordered weighted averaging, 41 Algebra, abstract, 264 linear, 264 mapping, 264–265 α-cut (see λ-cut), 67 Ambiguity, ANFIS, 265 Antecedents, 120 disjunctive, 147, 253 fuzzy, 136 Approaching degree, maximum, 375–376 similarity, 375 weighted, 379 Approximate solution, 7–8 Approximate reasoning, 117, 134–138, 249 Arithmetic, fuzzy, 418 Associativity, 30 Atomic terms, natural language, 140–144 Attribute data, statistical process control, 465 fuzzy, 474 traditional, 472 Averaging operations, 41 Ball-box analogy, evidence theory, 547 Basic evidence assignment, definition of, 535 joint, 537 Fuzzy Logic with Engineering Applications, Third Edition © 2010 John Wiley & Sons, Ltd ISBN: 978-0-470-74376-8 Batch least squares, rule generation, 211, 215–219 Bayesian, decision making, 278, 294–318 inference, updating, problems with, 278 Belief, monotone measures, 532 Binary relation, 50 Binomial distribution, statistical process control, 473 Body of evidence, 537 consonant, 540 Boundary, crisp sets, 25–26 fuzzy sets, 26, 90–91 Cardinality, classification, 342, 350 consensus relations, 287 possibility distributions, 554 sets, 27 Cartesian product, 49 classical sets, 49 fuzzy sets, 55–56 Certainty (also necessity), monotone measures, 546 average, 286 Chance, 16–17 games of, Characteristic function, 32, 119 Chi-square distribution, statistical process control, 466, 481 Classical sets, operations, 28 properties, 29 Timothy J Ross 580 INDEX Classification, definition, 332–333 equivalence relations, 333–339 fuzzy, 335–336 metric, 357 Cluster centers, 341 Clustering, 333 method, rule generation, 227 c-means, clustering, 33 fuzzy, 333, 349 hard (crisp), 333, 341 weighting parameter, 352 Cognitive mapping, 508 conventional, 508 fuzzy, 510 genetically evolved, 520 indeterminate, 513 Comb’s method of rapid inference, 524 Combination, rule of, 537, 548 Combinatorial explosion, Commutativity, 29 Comparison matrix, 284–285 Complement, classical, 28 relations, 52, 55 standard fuzzy operation, 35, 40–41 Complex system, Complexity, 246 Composite terms, natural language, 140 Composition, 53 chain strength analogy, 54 fuzzy, 55–57, 135 max-min, 53 max-product, 53 other methods, 72 relations, 64 Concentration, linguistic hedges, 142 Conclusion, 120 Conjunction, 119, 133 Consensus, degree of, 285 distance to, 288 types of, 286–288 Consequent, 120 fuzzy, 136 Consistency, condition, 569 principle, 550 Consonant measure, 542 consistent sets, 552, 561 non-consonant, 552 possibility distribution, 542, 549 Containment, 33, 52, 55 Continuity, 111 Continuous valued logic, Contradiction, 125–126, proof by, 128 Contrapositive, 126 Contrast enhancement, image recognition, 391–392 Control limits, statistical process control, 468 Control surface, 440, 445 Controller, Control systems, graphical simulation, 446–453 industrial process, 478–479 multi-input, multi-output (MIMO), 461 PID vs fuzzy, 480 single-input, single-output (SISO), 453 Control, adaptive, 479 comparisons of classical and fuzzy, 479–481 conventional methods, 439, 455–457 disturbance-rejection, 439, 454, 461, 463, 479 economic examples, 438 feedback, 438–439, 453 nonadaptive, 441 regulatory, 438–439 set-point tracking, 439, 454, 457, 461–464, 479 stability and optimality, 441, 481 Converse, 126 Convexity, membership functions, 91 Core, membership function, 91 Covariance matrix, 221 Credibility, 246 Crossover, genetic algorithms, 189, 191 Decision, optimum, 290 fuzzy states, fuzzy actions, 304 independence axiom, 278, 317 rational, 278 Decomplexify, 110 Deduction, 246 fuzzy rule-based, 145–146 shallow knowledge, 145 Defocus, image processing, 393 Deductive, logic, 9–10 reasoning, 129 Defuzzify, 90 Defuzzification, 90, 257–261 Center, of largest area, 106 -average, 214 of sums, 105 centroid, 99 correlation-minimum, 138, 464 first (or last) maxima, 106 fuzzy relations, 97 λ-cut sets, 95, 433 maximum membership principle, 99 mean-max membership, 100 measure criteria, 111 nearest neighbor classifier, 360 properties, 98 scalars, 98 weighted average, 99 INDEX Degree of, attainment, 232–233 confirmation, possibility distributions, 557–558 disconfirmation, 557 Delta functions, 214, 216 DeMorgan’s principles, 30–32, 36, 38, 121, 178 relations, 52 Dempster’s rule, evidence theory, 537 Difference, operator, 28–29 classical, 121 Dilations, linguistic hedges, 142–143 Disambiguity, 111 Disjunction, axiomatic, 119 Distributitivity, 30 DSW algorithm, 426, 428 Dissonance, Dissonance, evidence theory, 542 Dual, 126 El Farol problem, Entropy minimization, inductive reasoning, 200–205 Equivalence, relations, graphical analog, 62–68 axiomatic, 119 logical, 120 properties of, 125–126 classification, 333 Error, surface, 223 Euclidean distance, 340 norm, 343, 353 Evidence, perfect, 551 Evidence theory, 537 Evolutionary, genetic, 520 Excluded middle axiom, 11, 36, 570 applications of, 570 axiomatic basis, 570 contradiction, axiom, 30 counterexamples, 160 evidence theory, 533 principle of, 160 probability measure, 541 relations, 52 Exclusive-nor, 126–127 Exclusive-or, 28, 118, 126–127 Extension principle, 72, 408, 411 definition of, 408–409 Falsity set, 118 Feature analysis, pattern recognition, 369–371 Fitness-function, genetic algorithms, 195 Forgetting factor, automated methods of rule generation, 220 Function-theoretic, 32 Fuzzification, 93 Fuzziness, 16–17 average, 286 fuzzy algebra, maximum, 19 Fuzzy, associative, memories, 255–256 cognitive maps, 10 mapping, input-output, 409 measure theory, 531 number, definition, 92, 418–420 triangular, 476 Ranking, 282 relational equations, 252–253 relations, cardinality, 55 operations and properties, 54–55 sets, fuzzy, 34 convex, 91 noninteractive, 41 notation, 34 orthogonal, 302 system(s), 118, 139 transfer relation, 253 vectors, 372, 411 definition of, and complement, 373–374 product, inner and outer, 373 similarity, 374 transform, 411–412 weighting parameter, classification, 352 Generalized information theory, 569 Genetic algorithms, control, 479 binary bit-string, 190 crossover, 189, 191 fitness values, 190 function, 195 modified learning from examples, 242 mutation, 189, 191 reproduction, 189, 191 Gradient method, rule generation, 222 Grammar, formal, 398–399 Graphical inference, 256–264 Hardening fuzzy c-Portition, 360–361 max membership, 360 nearest center, 360 Height, membership functions, 92 Hidden layers, neural networks, 180 Hypercubes, 18 Hypothesis, 120 Indicator function, 15, 32 Idempotency, 30 Identity, 30 IF-THEN rules, 249 control, 441 581 582 INDEX Ignorance, incoherent data, 247 monotone measures, 533 total, 545 Implication, axiomatic, 119, 133 Brouwerian, 139 classical, 120 conditional, 134 correlation-product, 139 decision making, 290 Lukasiewicz, 138–139 Mamdani, 138 other techniques, 138 Impossible, 6, 546, 550 Imprecision, Inclusive-or, 118 Involution, 30 Independence, axiom of, 278 Indeterminacy, cognitive mapping, 509 Induction, 10, 246 deep knowledge, 145–146 laws of, 199–200 Inductive reasoning, 199, 206 entropy, 200–205 partitioning, 201 probability, 200–202 threshold value, 201 Inclusive-or, logical connectives, 119 Inference, deductive, 129–130 defuzzification, 150 centroidal, 151 weighted average, 153 fuzzy, 148 graphical methods, 148 implication, max-min, 148 max-product, 149 Mamdani, 148, 152–159 min-max, cognitive mapping, 511 Sugeno, 152–159 Takagi-Sugeno, 214 Tsukamoto, 153–159 Information, distinction between fuzzy and chance, 16 fuzzy, 301–302 imperfect, 298 new, 297 organized data, 247 uncertainty in, 5, 12 value of, decision making, 298 Input-output data, 215 Intensification, linguistic hedges, 142 image recognition, 392 Interpolative reasoning, 117, 249 control, 441 Intersection, classical, 28 relations, 52, 55 standard fuzzy operation, 35 Intervals, analysis, types of, possibility distributions, 420, 551 expected, 564 sets, 560 Intuitionism, logic, 160 Inverse, 126 Isomorphism, fuzzy systems, Iterative optimization, classification, 344, 352 Knowledge, conscious, 242, 246, 250 deep, 145 information, 245–247 shallow, 145 subconscious, 242, 250 Lambda (λ)-cuts, 95, 335–339, 477 optimization, 503 relations, 96 Language, fuzzy (see also grammar), 398 Laplace transforms, control, 455 Learning, shallow, 246 from examples, rule generation, 229 neural networks, 189 Least squares, 217, 219–221 statistical process control, 481 Length, possibility distribution, 543 Likelihood values, decision making, 297 Linguistic variables, 140 concentration, 142–143 dilations, 142–143 hedges, 140–142 precedence, 144 intensification, 142 natural language, 142 rule, 145 Logic, Aristotelian, 1, 120 classical (binary, or two-valued), 159 constructive, 161 fuzzy, 131 linear, 160 paradox, 131 Sorites, 131 Logical, connectives, 120 negation, 120 proofs, 127–129 propositions, 120 empty set, 132 -or, 28, 119 universal set, 132 Logics, multivalued, 118 Mamdani inference, 141, 148, 471, 476 Mapping, 32, function-theoretic, 411–416 set-theoretic, 32 simulation, 264 INDEX Matrix norm, classification, 353 Maximal fuzziness, decision making, 285–286 Maximum, fuzziness, 19 membership, criterion of, 371–372 operator, 33, 54 Measure, decision, 321–322 Measurement data, statistical process control, 504–510 fuzzy, 507 traditional, 505 Membership function, 174 automated generation, definitions for, 212 boundaries, 91 convex, 91 core, 90 crossover points, 92 dead band, 464 definition of, 15–16 delta function, 213 Gaussian, 212 generalized, 92–93 genetic algorithms, 189–199 height, 92 inductive reasoning, 199–205 inference, 176–178 interval-valued, 93 intuition, 175–176 neural networks, 179–189 normal, 91 ordering, 178–179 ordinary, 93 orthogonal, 302 properties of, 15–16, 90–92 prototype, 91 rank ordering, 178 shoulder function, 153, 158 smoothness, 153 support, 90 triangular, 213, 230 tuning, 242 type-2, 93 Membership, classical (binary) sets, 14–16 fuzzy sets, 14–16 unshared, (shared), 358 MIN and MAX, extended operations, 283 Minimum, operator, 33, 54 Model-free methods, 246 Models, abstraction, 8–9 Modified learning from examples, rule generation, 233 distance measures, 236–237 Modus ponens, deduction, 123, 137, 251 Modus tollens, deduction, 124 Monotone measures, 531 583 fuzzy sets, difference between, 531, 533 Multifeature, pattern recognition, 378 Multinomial distribution, statistical process control, 474, 481 Multiobjective, decision making, 289–294 Multivalued logic, Mutation, genetic algorithms, 189 rate of, 191 Mutual exclusivity, 119 Natural language, 140 interpretations, 140–141, cognitive, 140 linguistic hedges, 142–144 linguistic variable, 141, 145 Nearest center, pattern recognition, 379 Negation, 119, 133 Nearest neighbor, 227 pattern recognition, 378 Necessity, monotone measures, 542 Nested sets, evidence theory, 542 Nesting diagram, possibility distribution, 544 Neural networks, 179 back-propagation, 181 clustering, 183 cognitive learning, 183 errors, 182, 187–188 inputs and outputs, 180–182, 185 sigmoid function, 180, 186 threshold element, 180, 186 training, 183 weights, 180–182, 186, 188–189 Newton’s second law, 248 Newtonian mechanics, Noninteractive fuzzy sets, 12, 40, 253–254, 381, 414 Nonlinear, simulation, 247 systems, 251–255 Nonrandom errors, 11 Nonspecificity, 13 possibility distribution, 551, 553 Nontransitive ranking, 283–285 Normal, membership function, 91 Null set, 19, 27 evidence theory, 533 Objective function, fuzzy c-means, 352 hard c-means, 343 optimization, 502–503 Optimist’s dilemma, Optimization, fuzzy, 501–507 one-dimensional, 502 Ordering, crisp, 280 fuzzy, 280–283 ordinal, 282 Ordered weighted averaging, 41 584 INDEX Pairwise function, decision making, 283 Paradigm shift, fuzzy control, 437, 479 Partitioning, input and output, 254 classification, 341 p-chart, statistical process control, 465, 472 fuzzy, statistical process control, 474–478 Perfect evidence, possibility distribution, 545 Plausibility, monotone measures, 532 defuzzification, 111 Point set, classification, 341 Possibility, theory, 4, 542 anchoring, 560 distribution, as a fuzzy set, 549 decision making, 318 definition of, 542–543 monotone measures, 532 Power set, 28, 33 fuzzy, 36 Precision, 1, 245 Preference, degree of, 285 importance, 290 measures of, 286 Premise, 145 Principle of incompatibility, 245 Probability, posterior, 297 calculus of, conditional, 297 density functions, 92, 535, 537 evidence theory, 540 marginal, 299 measure, belief as a lower bound, 540, 545 evidence theory, 586 plausibility as an upper bound, 545 monotone measures, 533 of a fuzzy event, 302 prior, 295 singleton, 540 theory, 3–4, history of, Proposition, compound, 120 fuzzy, 132 simple, 118 Propositional calculus, 119 Pseudo-goal, optimization, 503 Quantum mechanics, 248 physics, 530 Quotient set, classification, 334 Random, errors and processes, 11 Rational maximizer, agent-based models, 521 Rational man, concept of, 160 Rationality, bounded, 10 R-chart, statistical process control, 466–468 Reasoning, approximate, 134–137 classical, 124 deductive, 9–10 deep and shallow, imprecise, 117 inductive, 9–10 Recursive least squares, rule generation, 211, 219–222, 238, 240–241 weighted, 220 Regression vector, 220–221 Redistribution factor, possibility distributions, 554 Reflexivity, tolerance relations, 63, 65 Relation(s), binary, 49 cardinality, 51, 55 complete, 52, 55 constrained, 51 equivalence, 62–68, 333 function-theoretic operations, 52 fuzzy, 54, 65 fuzzy preference, 287 identity, 51 matrix, 50 null, 52, 55 properties, 52, 55 reciprocal, 285 similarity, 66, 69 361–362 strength of, 50, 73 tolerance, 62–68 unconstrained, 50 universal, 51 Relational equation, 252 Relativity, function, 283 values, matrix of, (also comparison matrix), 284 Reproduction, genetic algorithms, 189, 191 Risk averse, 277 Robust systems, Rule generation, methods, 211 Rule-base, 118 conjunctive, 146 disjunctive, 147 reduction methods, 524 Rule(s), aggregation, conjunctive and disjunctive, 147 fuzzy IF-THEN, 145 Generation methods, 211 statistical process control, 469 Tsukamoto, 153 Sagittal diagram, 53, 57 Set membership, 13 Set-theoretic, 32 Sets, as points, 18–20 classical, 26 fuzzy, 34 Shoulder, membership function, 153 Sigmoid function, neural networks, 180 INDEX Similarity, classification, 48, 361 relations, cosine amplitude, 69 max-min, 71 other methods, 71 Single-sample identification, pattern recognition, 371 Singleton, crisp, 164 examples, 540 fuzzy, 391 Singular value decomposition, 524 Smoothing, image recognition, 393 Standard fuzzy, intersection, 553 operations, 35, 40, 132 Stationary processes, random error, 11 Statistical, mechanics, Statistical process control (SPC), 464 Statistics, 12 Strong-truth functionality, 11, 571 Subjective probabilities, Subdistributivity, 421 Support, membership function, 90 Sum of mins, 361 Symmetry, tolerance relation, 63, 65 Syntactic recognition, 398 Synthetic evaluation, fuzzy, 278–280 Systematic error, 11 Tautologies, 123–125 modus ponens, 123 modus tollens, 124 Taylor’s approximation, control, 455 t-conorm, 41 Tie-breaking, multiobjective decisions, 291, 293–294 t-norm, 41 product, 214 possibility theory, 553 Triangular norms, 41, 553 Transitivity, 30 equivalence relations, 63, 65, 335–337 Truth, set, 118 half, 131 585 table, 122, 124–125, 127, 129 value, 118, 282 fuzzy, 282 Uncertainty, 3, 13, 246 general, linguistic, 11 random, Union, classical, 28 relations, 52, 55 standard fuzzy operation, 35 Universal approximator, 6, 265 control, 441 fuzzy systems, 6–7 Universe of discourse, 15, 25, 233 Continuous and discrete, 27 monotone measures, 532 Unknown, 13–14 Utility, matrix, 295 maximum expected, 295 rational theory, 278 values, 295 Vagueness, Value set, 20, 33 assignments, 68 Venn diagrams, 28–31, 121–123, 126–127 extended, 36–37 Vertex method, 423 Weighted recursive least squares, 220 Weighting factor, modified learning from examples, 235 Whole set, 19, 27 Wisdom, 246–247 x-bar chart, statistical process control, 466 x-bar-R chart, fuzzy, statistical process control, 471–472 statistical process control, 468, 469, 477 [...]... current and potential applications, case studies, and education in intelligent and fuzzy systems in engineering and related technical fields The problems address the disciplines of computer science, electrical engineering, manufacturing engineering, industrial engineering, chemical engineering, petroleum engineering, mechanical engineering, civil engineering, environmental engineering, and engineering management,... more than 130 publications and has been active in the research and teaching of fuzzy logic since 1983 He is the founding Co-Editor-in-Chief of the International Journal of Intelligent and Fuzzy Systems, the co-editor of Fuzzy Logic and Control: Software and Hardware Applications, and the co-editor of Fuzzy Logic and Probability Applications: Bridging the Gap His sabbatical leaves in 2001–2002 at the University... avoid confusion with any of the terms typically used in probability theory As with the first two editions, this third edition is designed for the professional and academic audience interested primarily in applications of fuzzy logic in engineering and technology Always, I have found that the majority of students and practicing professionals are interested in the applications of fuzzy logic to their... 418 420 422 423 426 428 429 432 433 433 CONTENTS 13 Fuzzy Control Systems Control System Design Problem Control (Decision) Surface Assumptions in a Fuzzy Control System Design Simple Fuzzy Logic Controllers Examples of Fuzzy Control System Design Aircraft Landing Control Problem Fuzzy Engineering Process Control Classical Feedback Control Fuzzy Control Fuzzy Statistical Process Control Measurement Data... and fuzzy sets PREFACE TO THE THIRD EDITION xvii Chapter 2 reviews classical set theory and develops the basic ideas of fuzzy sets Operations, axioms, and properties of fuzzy sets are introduced by way of comparisons with the same entities for classical sets Various normative measures to model fuzzy intersections (t-norms) and fuzzy unions (t-conorms) are summarized Chapter 3 develops the ideas of fuzzy. .. published, in 1995, the technology of fuzzy set theory and its application to systems, using fuzzy logic, has moved rapidly Developments in other theories such as possibility theory and evidence theory (both being elements of a larger collection of methods under the rubric “generalized information theories”) have shed more light on the real virtues of fuzzy logic applications, and some developments in... logic that admits only the opposites of true and false, a logic which does not admit degrees of truth in between these two extremes In other words, Aristotelian logic does not admit imprecision in truth However, Aristotle’s quote is so appropriate today; it is a quote that admits uncertainty It is an admonishment that we Fuzzy Logic with Engineering Applications, Third Edition © 2010 John Wiley & Sons,... this subject since the first edition of 1995, and to others who by their simple association with me have caused me to be more circumspect about the use of the material contained in the book Three colleagues at Los Alamos National Laboratory have continued to work with me on applications of fuzzy set theory, fuzzy logic, and generalized uncertainty theory: Drs Greg Chavez (who wrote much of Chapter 7),... their invitations and travel support, have enabled me to train numerous South American scientists and engineers in fuzzy logic applications in their own fields of work, most notably nuclear waste management and risk assessment My discussions with them have given me ideas about where fuzzy logic can impact new fields of inquiry Some of the newer end-of-chapter problems of the third edition came from a... on the relationships between fuzzy control and classical control; Figure 13.41 of this text comes from his perspectives of this matter One individual deserves my special thanks and praise, and that is Prof Mahmoud Taha, my colleague in Civil Engineering at the University of New Mexico In the last five years Prof Taha has become an expert in fuzzy logic applications and applications using possibility .. .FUZZY LOGIC WITH ENGINEERING APPLICATIONS Third Edition Fuzzy Logic with Engineering Applications, Third Edition © 2010 John Wiley & Sons, Ltd ISBN: 978-0-470-74376-8 Timothy J Ross FUZZY LOGIC. .. Timothy J Fuzzy logic with engineering applications / Timothy J Ross.–3rd ed p cm Includes bibliographical references and index ISBN 978-0-470-74376-8 (cloth) Engineering mathematics Fuzzy logic. .. engineering, manufacturing engineering, industrial engineering, chemical engineering, petroleum engineering, mechanical engineering, civil engineering, environmental engineering, and engineering management,