Introduction to Fuzzy Sets, Fuzzy Logic, and Fuzzy Control Systems

Table of Contents
1. Fuzzy Set Theory................................................................................ 1
I. Classical Set Theory.................................................................... 1
A. Fundamental Concepts........................................................... 1
B. Elementary Measure Theory of Sets...................................... 3
II. Fuzzy Set Theory ........................................................................ 5
III. Interval Arithmetic ...................................................................... 9
A. Fundamental Concepts........................................................... 9
B. Interval Arithmetic................................................................. 12
C. Algebraic Properties of Interval Arithmetic........................... 13
D. Measure Theory of Intervals .................................................. 17
E. Properties of the Width of an Interval.................................... 20
F. Interval Evaluation................................................................. 22
G. Interval Matrix Operations..................................................... 25
H. Interval Matrix Equations and Interval Matrix Inversion ...... 30
IV. Operations on Fuzzy Sets ............................................................ 37
A. Fuzzy Subsets ........................................................................ 37
B. Fuzzy Numbers and Their Arithmetic.................................... 42
Problems............................................................................................. 54
References .......................................................................................... 56
2. Fuzzy Logic Theory ........................................................................... 57
I. Classical Logic Theory................................................................ 58
A. Fundamental Concepts........................................................... 58
B. Logical Functions of the Two-Valued Logic ......................... 61
II. The Boolean Algebra................................................................... 62
A. Basic Operations of the Boolean Algebra.............................. 62
B. Basic Properties of the Boolean Algebra ............................... 63
III. Multi-Valued Logic..................................................................... 65
A. Three-Valued Logic............................................................... 65
B. n-Valued Logic ...................................................................... 65
IV. Fuzzy Logic and Approximate Reasoning .................................. 66
V. Fuzzy Relations ........................................................................... 69
VI. Fuzzy Logic Rule Base................................................................ 75
A. Fuzzy IF-THEN Rules ........................................................... 75
B. Fuzzy Logic Rule Base .......................................................... 77
C. Interpretation of Fuzzy IF-THEN Rules ................................ 80
D. Evaluation of Fuzzy IF-THEN Rules..................................... 82
Problems............................................................................................. 84
References .......................................................................................... 87
3. Fuzzy System Modeling ..................................................................... 89
I. Modeling of Static Fuzzy Systems .............................................. 90
A. Fuzzy Logic Description of Input-Output Relations.............. 90
B. Parameters Identification in Static Fuzzy Modeling .............. 96
II. Discrete-Time Dynamic Fuzzy Systems and Their Stability
Analysis....................................................................................... 102
A. Dynamic Fuzzy Systems without Control.............................. 103
B. Dynamic Fuzzy Systems with Control................................... 109
III. Modeling of Continuous-Time Dynamic Fuzzy Control
Systems ....................................................................................... 114
A. Fuzzy Interval Partitioning .................................................... 115
B. Dynamic Fuzzy System Modeling......................................... 119
IV. Stability Analysis of Continuous-Time Dynamic Fuzzy
Systems ....................................................................................... 124
V. Controllability Analysis of Continuous-Time Dynamic
Fuzzy Systems............................................................................. 129
VI. Analysis of Nonlinear Continuous-Time Dynamic Fuzzy
Systems ....................................................................................... 133
Problems............................................................................................. 136
References .......................................................................................... 138
4. Fuzzy Control Systems....................................................................... 139
I. Classical Programmable Logic Control....................................... 140
II. Fuzzy Logic Control (I): A General Model-Free Approach........ 145
A. A Closed-Loop Set-Point Tracking System........................... 147
B. Design Principle of Fuzzy Logic Controllers......................... 150
C. Examples of Model-Free Fuzzy Controller Design ............... 160
III. Fuzzy Logic Control (II): A General Model-Based Approach.... 170
Problems............................................................................................. 178
References .......................................................................................... 182
5. Fuzzy PID Controllers........................................................................ 183
I. Conventional PID Controllers: Design........................................ 183
II. Fuzzy PID Controllers Design .................................................... 192
A. Fuzzy PD Controller .............................................................. 193
B. Fuzzy PI Controller................................................................ 207
C. Fuzzy PI+D Controller........................................................... 209
III. Fuzzy PID Controllers: Stability Analysis .................................. 223
A. BIBO Stability and the Small Gain Theorem ........................ 223
B. BIBO Stability of Fuzzy PD Control Systems....................... 226
C. BIBO Stability of Fuzzy PI Control Systems ........................ 229
D. BIBO Stability of Fuzzy PI+D Control Systems ................... 231
E. Graphical Stability Analysis of Fuzzy PID Control
Systems .................................................................................. 232
Problems............................................................................................. 236
References .......................................................................................... 237
6. Adaptive Fuzzy Control ..................................................................... 239
I. Fundamental Adaptive Fuzzy Control Concept .......................... 240
A. Operational Concepts............................................................. 240
B. System Parameterization........................................................ 242
C. Adjusting Mechanism ............................................................ 243
D. Guidelines for Selecting an Adaptive Fuzzy Controller ........ 245
II. Gain Scheduling .......................................................................... 246
III. Fuzzy Self-Tuning Regulator ...................................................... 252
IV. Model Reference Adaptive Fuzzy Systems................................. 255
V. Dual Control................................................................................ 257
VI. Sub-Optimal Fuzzy Control ........................................................ 258
A. SISO Control Systems ........................................................... 259
B. MIMO Control Systems......................................................... 260
Problems............................................................................................. 266
References .......................................................................................... 269
7. Some Applications of Fuzzy Control ................................................. 271
I. Health Monitoring Fuzzy Diagnostic Systems............................ 271
A. Fuzzy Rule-Based Health Monitoring Expert Systems.......... 272
B. Computer Simulations ........................................................... 276
C. Numerical Results.................................................................. 277
II. Fuzzy Control of Image Sharpness for Autofocus Cameras........ 281
A. Basic Image Processing Techniques ...................................... 282
B. Fuzzy Control Model ............................................................. 283
C. Computer Simulation Results ................................................ 286
III. Fuzzy Control for Servo Mechanic Systems ............................... 291
A. Fuzzy Modeling of a Servo Mechanic System ...................... 292
B. Fuzzy Controller of a Servo Mechanic System...................... 294
C. Computer Simulations and Numerical Results ...................... 295
IV. Fuzzy PID Controllers for Servo Mechanic Systems .................. 300
A. Fuzzy PID Controller of a Servo Mechanic System .............. 300
B. Adaptive Fuzzy Controller of a Servo Mechanic System ...... 301
V. Fuzzy Controller for Robotic Manipulator.................................. 302
A. Fuzzy Modeling of a 2-Link Planar Manipulator................... 304
B. Fuzzy Controller of a 2-Link Planar Manipulator.................. 307
C. Numerical Simulations........................................................... 307
Problems............................................................................................. 311
References .......................................................................................... 313
Index................................................................................................... 315

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