This book is both a unified textbook and to some degree an edited volume. It aims
at the college audience in engineering. Each chapter contains detailed homework
problems though some of the later research chapters contain fewer problems than
do earlier chapters. And each chapter fits in the larger topics scheme that runs from
theory to applications and the topological scheme that runs from feedforward systems
to feedback systems. I worked on some chapters with my graduate students and asked
colleagues to prepare a third of the chapters and wrote the rest myself. We worked
on the book for over two years and both students and colleagues had to endure my
editorial control. In that time I added and deleted many chapters to best fit and
extend the shifting frontiers of fuzzy engineering. The result has the collaborative
strength of an edited volume and the structure and tone that only a sole author can
give.
Author(s): Bart Kosko
Publisher: Prentice Hall
Year: 1996
Language: English
Pages: 575
Kosko, Bart Fuzzy engineering (PH, 1996) ......Page 3
Copyright ......Page 4
Contents ......Page 6
FUZZY ENGINEERING: FUZZY LOGIC AS FUNCTION APPROXIMATION ......Page 15
ADDITIVE FUZZY SYSTEMS AND THE STANDARD ADDITIVE MODEL ......Page 18
FUZZY EPISTEMOLOGY AND THE CONCEPTUAL ANARCHY ......Page 17
THE CURSE OF DIMENSIONALITY AND LEARNING: PATCH THE BUMPS ......Page 21
ABOUT THIS TEXT ......Page 23
PART I: INTRODUCTION 1 ......Page 25
1.1 Fuzziness as Vagueness: Everything is a Matter of Degree 3 ......Page 27
1.2 The History of Vague Logic 5 ......Page 29
1.3 Fuzzy Sets and Systems 8 ......Page 32
1.4 Fuzzy Systems in Commercial Products 12 ......Page 36
1.5 Adaptive Fuzzy Systems and the Search for Better Rules 15 ......Page 39
1.6 Overview of Fuzzy Logic, Set Theory, and Systems 18 ......Page 42
References 25 ......Page 49
Homework Problems 26......Page 50
PART II: FUZZY FUNCTION APPROXIMATION 39 ......Page 63
Preview 41 ......Page 65
2.1 Fuzzy Function Approximation as a Fuzzy Cover 42 ......Page 66
2.2 The Curse of Dimensionality: Rule Explosion and Optimal Rules 45 ......Page 69
2.3 Additive Fuzzy Systems: The Standard Additive Model 48 ......Page 72
2.4 Set Functions in SAMs 54 ......Page 78
2.5 The “Center of Gravity” Method as a Constant-Volume SAM 59 ......Page 83
2.6 Generalized SAMs and the TSK Model: Convex Sums of Nonlinear Systems 65 ......Page 89
2.7 Generalized SAMs as Set Mappings and Correlators: The Set SAM 70 ......Page 94
2.8 Combining Fuzzy Systems: Convex Sums of Expert Responses 75 ......Page 99
2.9 Fuzzy Systems as Conditional Means: The Probability Connection 80 ......Page 104
2.10 Function Approximation with Additive Fuzzy Systems 89 ......Page 113
2.11 Function Representation with SAMs 95 ......Page 119
2.12 Learning in SAMs: Unsupervised Clustering and Supervised Gradient Descent 97 ......Page 121
2.13 Optimal Additive Systems: Patch the Bumps 105 ......Page 129
2.14 Feedback SAMs and Fuzzy Cognitive Maps 112 ......Page 136
References 126 ......Page 150
Homework Problems 129 ......Page 153
3 Ellipsoidal Fuzzy Systems 139 ......Page 163
3.1 Learning with Ellipsoid Rules 140 ......Page 164
3.2 Additive Fuzzy Systems 141 ......Page 165
3.3.1 Ellipsoids as Fuzzy Patches 144 ......Page 168
3.3.2 Weighting Ellipsoidal Rules 147 ......Page 171
3.4.1 Unsupervised Competitive Learning 151 ......Page 175
3.4.2 Fuzzy Rule Learning 153 ......Page 177
3.4.3 Histogram Density Estimation 154 ......Page 178
3.5 Supervised Learning of Fuzzy Rule Patches 157 ......Page 181
3.5.1 Gradient Descent Algorithm 158 ......Page 182
3.5.2 Supervised Function Approximation 160 ......Page 184
3.6 Hybrid System: Unsupervised and Then Supervised Ellipsoidal Learning 162 ......Page 186
3.7 Optimal Function Approximation 164 ......Page 188
3.8 Conclusions 166 ......Page 190
References 168 ......Page 192
Appendix 3.A: Unsupervised Competitive Learning 169 ......Page 193
Homework Problems 172 ......Page 196
Software Problems 173 ......Page 197
PART III: FUZZY CONTROL AND CHAOS 175 ......Page 199
Preview 177 ......Page 201
4.1 Platoons of Smart Cars 178 ......Page 202
4.2.1 Platoon Maintenance 180 ......Page 204
4.2.2 Platoon Maneuvers 183 ......Page 207
4.3 Brake Controller 186 ......Page 210
4.4.2 Brake Models 190 ......Page 214
4.4.3 Fuzzy Controller Results 191 ......Page 215
4.5 Hybrid Ellipsoidal Function Learning 193 ......Page 217
4.6 Control Surface Learning 195 ......Page 219
4.7 Gap Control Test 197 ......Page 221
4.7.1 Longitudinal Car Model 198 ......Page 222
4.7.2 Simulation Results 199 ......Page 223
4.7.4 Gap Controller with Throttle and Brake 200 ......Page 224
4.8 Conclusions 206 ......Page 230
References 208 ......Page 232
Homework Problems 209 ......Page 233
Preview 213 ......Page 237
5.2 Fuzzy System Design 214 ......Page 238
5.2.1 Fuzzy System Architecture 215 ......Page 239
5.2.2.1 Partitioning Criterion 216 ......Page 240
5.2.2.2 Recursive Partitioning of the Product Space 217 ......Page 241
5.2.2.3 Defining a Fuzzy Rule Patch 219 ......Page 243
5.2.2.4 Adapting to New Sample Data 220 ......Page 244
5.2.4 Properties of the Algorithm 223 ......Page 247
5.2.4.2 Direction of Search 224 ......Page 248
5.3.1 Approximation of a Two-Dimensional Curve 225 ......Page 249
5.4 Approximating Dynamical Systems and Chaos 227 ......Page 251
5.4.1 A Dynamical System as a Combination of Functions 230 ......Page 254
5.4.3 The Van der Pol Oscillator 237 ......Page 261
5.4.4 The Lorenz and Rossler Attractors 240 ......Page 264
5.5 Conclusions 242 ......Page 266
References 243 ......Page 267
Homework Problems 244 ......Page 268
PART IV: FUZZY SIGNAL PROCESSING 249 ......Page 273
Preview 251 ......Page 275
6.1 Filtering Impulsive Noise 252 ......Page 276
6.2.1 Stable Distribution and Its Properties 253 ......Page 277
6.2.2 Covariation and Pseudo-covariation 255 ......Page 279
6.3 Additive Fuzzy Systems 256 ......Page 280
6.3.1 Additive Fuzzy Systems 257 ......Page 281
6.3.2 Covariation Rules and Mahalanobis Sets: Product Space Clustering and Projection 258 ......Page 282
6.4 Fuzzy Prediction with Covariation Rules 261 ......Page 285
6.5 Fuzzy Filters with Covariation Rules 265 ......Page 289
6.6 Conclusion 268 ......Page 292
6.7 Appendix 6. A: Competitive AVQ Algorithm for Local Means and Covariations 273 ......Page 297
Homework Problems 274 ......Page 298
Preview 277 ......Page 301
7.1 Tilings of the Time-Frequency Plane 278 ......Page 302
7.2.1 Subband Filters in Two Dimensions 279 ......Page 303
7.2.2 Quadrature Mirror Filters 281 ......Page 305
7.2.3 Fully Decomposed SBC of Images with Vector Quantization 283 ......Page 307
7.3 Fuzzy Virtual Tilings for Subband Coding 286 ......Page 310
7.3.1 Additive Fuzzy Systems 287 ......Page 311
7.3.2 Fuzzy Virtual Tilings 288 ......Page 312
7.4 Simulation Results 297 ......Page 321
7.5 Conclusion 298 ......Page 322
References 299 ......Page 323
Homework Problems 301 ......Page 325
PART V: FUZZY COMMUNICATION 305 ......Page 329
Preview 307 ......Page 331
8.1 Frequency Hopping Spread Spectrum 308 ......Page 332
8.2.1 Inputs to the Fuzzy System 309 ......Page 333
8.2.2 Fuzzy Rules 311 ......Page 335
8.2.3 Adaptive Rule Generation 313 ......Page 337
8.2.5 Defuzzification 314 ......Page 338
8.3.1 Output Fuzzy Sets as Conditional Densities 318 ......Page 342
8.4 Simulation Results 320 ......Page 344
8.5.2 System Comparison 322 ......Page 346
8.6 Conclusion 323 ......Page 347
Appendix 8. A: Derivation of the Centroid as the Conditional Mean for the Frequency-Hopping SAM 324 ......Page 348
Homework Problems 325 ......Page 349
Preview 329 ......Page 353
9.2 Source Model 330 ......Page 354
9.3 Scalar Channel Model 331 ......Page 355
9.4 Fuzzy Detector 333 ......Page 357
9.4.1 Reflexive and Contextual Fuzzy Rules 334 ......Page 358
9.4.2 If-Part Fuzzy Sets 336 ......Page 360
9.4.3 Detection in the Fuzzy Cube 337 ......Page 361
9.4.4 Computing the Conditional Densities 339 ......Page 363
9.4.5 Clarity of a Fuzzy Set 341 ......Page 365
9.4.6 Combined Additive Output Fuzzy Set: Combining Throughputs 343 ......Page 367
9.4.7 Mode Defuzzification and Detection 344 ......Page 368
9.5 Simulation Results 345 ......Page 369
9.6 Conclusion 347 ......Page 371
References 349 ......Page 373
Appendix 9. A: Functional and Geometrical View of Fuzzy Sets 350 ......Page 374
Appendix 9.B: Fuzziness and Clarity Measures 351 ......Page 375
Homework Problems 352 ......Page 376
PART VI: FUZZY HARDWARE 355 ......Page 379
Preview: SAMs in Silicon 357 ......Page 381
10.1.1 Function Approximation with Electrons 358 ......Page 382
10.1.2 Rule Explosion 360 ......Page 384
10.2 The SAM Architecture and VLSI 361 ......Page 385
10.2.1.1 Gaussian Set Functions 362 ......Page 386
10.2.1.2 Cauchy Set Functions 363 ......Page 387
10.2.1.3 Exponential Set Functions 364 ......Page 388
10.3.1 Set SAMs 365 ......Page 389
10.4.2 SAM Supervised Learning Algorithms 368 ......Page 392
10.4.2.2 Cauchy Set Learning 369 ......Page 393
10.4.3 Learning Rates 370 ......Page 394
10.5.2 Algorithms 371 ......Page 395
10.5.2.2 Evaluation Phase 372 ......Page 396
10.5.2.3 Update Phase 373 ......Page 397
10.5.3 Silicon Architecture 374 ......Page 398
10.6.1 Data Path Architecture 375 ......Page 399
10.6.2 Control Architecture 377 ......Page 401
10.7 ASAM Operation 378 ......Page 402
10.7.1 Evaluation Task 379 ......Page 403
10.7.2 Update Task 380 ......Page 404
10.7.3 Performance 381 ......Page 405
References 382 ......Page 406
Homework Problems 383 ......Page 407
11.1 Fuzzy Computation: Electronics in Time Versus Optics in Space 387......Page 411
11.2 Parallel Fuzzy-Set Function Optical Processor 388 ......Page 412
11.3 Scanning Laser Diode Fuzzy System Processor 392 ......Page 416
11.4 DSTOP Fuzzy System Processor 394 ......Page 418
11.5 Optical Fuzzy System Processing: Present and Future 395 ......Page 419
Homework Problems 398 ......Page 422
PART VII: COMPUTING IN FUZZY CUBES 401 ......Page 425
Preview 403 ......Page 427
12.1 The First Theorem: Addition as Fuzzy Mutual Entropy 404 ......Page 428
12.2 Fuzzy Sets as Points in Hypercubes: Degrees of Subsethood 405 ......Page 429
12.3 Fuzziness and Entropy 409 ......Page 433
12.4 Fuzzy Mutual Entropy 411 ......Page 435
12.5 The Proof: Diffeomaps Between Real Spaces and Fuzzy Cubes 412 ......Page 436
12.6 Information Fields: Shannon Entropy From Fuzzy Mutual Entropy 414 ......Page 438
12.7 The Field Equations of Fuzzy Information: It from Fit 417 ......Page 441
12.8 Conclusions 422 ......Page 446
References 423 ......Page 447
Homework Problems 424 ......Page 448
Preview 429 ......Page 453
13.2 The Measure of Mutual Subsethood: Fuzzy Equivalence 430 ......Page 454
13.3 Mutual Subsethood for Bell-Shaped and Triangular Set Functions 436 ......Page 460
13.4.1 The Adaptive Subsethood Controller 440 ......Page 464
13.4.2 On-line Learning with Adaptive Subsethood 443 ......Page 467
13.5 A Test Case: The Fuzzy Car 451 ......Page 475
13.7 Appendix 13. A 456 ......Page 480
References 460 ......Page 484
Homework Problems 462 ......Page 486
PART VIII: FEEDBACK IN FUZZY CUBES 465 ......Page 489
Preview 467 ......Page 491
14.1.1 When Is a Dog Not a Dog? 468 ......Page 492
14.1.4 Error-Based Learning 470 ......Page 494
14.2 ART and Fuzzy Cubes 471......Page 495
14.3 ART Dynamics 472......Page 496
14.4.1 Fast-Learn Slow-Recode and Complement Coding Options 475 ......Page 499
14.5 Fuzzy ARTMAP 476 ......Page 500
14.6.4 Category Choice 478 ......Page 502
14.6.7 Fast-Commit and Slow-Recode 480 ......Page 504
14.6.8 Normalization by Complement Coding 481 ......Page 505
14.6.10 A Comparison of Fuzzy ARTMAP, NGE, and FMMC 484 ......Page 508
14.7.1 ARTa and ART*, 485 ......Page 509
14.7.4 Map Field Learning 486 ......Page 510
14.8 Fuzzy ARTMAP Simulation: Circle-in-the-Square 487 ......Page 511
14.9 ART Applications 491 ......Page 515
References 492 ......Page 516
Homework Problems 495 ......Page 519
Preview 499 ......Page 523
15.1 Fuzzy Virtual Worlds 500 ......Page 524
15.2 Fuzzy Cognitive Maps 502 ......Page 526
15.2.1 Simple FCMs 504 ......Page 528
15.2.3 Augmented FCMs 506 ......Page 530
15.2.4 Nested FCMs 507 ......Page 531
15.3 Virtual Undersea World 510 ......Page 534
15.3.1 Augmented Virtual World 512 ......Page 536
15.3.2 Nested FCMs for Fish Schools 516 ......Page 540
15.4 Adaptive Fuzzy Cognitive Maps 519 ......Page 543
15.5 Conclusions 522 ......Page 546
References 524 ......Page 548
Homework Problems 527 ......Page 551
A.1 The Fuzzy Function Approximation Software 530 ......Page 554
A.1.1 Instructions for the MS-DOS Version 531 ......Page 555
A.1.2 Instructions for the Windows Version 532 ......Page 556
A.2.2 Bipolar vs. Binary Concept Values 533 ......Page 557
A.2.5 Using the Fuzzy Cognitive Map 534 ......Page 558
A.2.6 File and Help Menu Commands 535 ......Page 559
Index 536......Page 560
List Of Contributors 549 ......Page 573
cover......Page 1
