A Thesis Presented to the Faculty of the Fritz J. and Dolores H. Russ College of Engineering and Technology Ohio University in P
Total Page：16
File Type：pdf, Size：1020Kb
A Thesis Presented to The Faculty of the Fritz J. and Dolores H. Russ College of Engineering and Technology Ohio University In Partial Fulfillment of the Requirement for the Degree Master of Science by Brian 0.Bush June 1996 0 1996 Brian 0. Bush All Rights Reserved wasted views that's all they see blue hot blood guild optic nerve with the right attitude you will succeed blue seIfabusive recluse too late for me -Skinny Puppy, Worlock, a never ending backdrop to this work I would like to thank my mother and father, Pascal and especially Ryoko. ABSTRACT v Natural language is perhaps the most powerful form of conveying infonnation for any given problem or situation. Combining natural language and numerical infonnation into fuzzy systems provide the framework to represent knowledge, constraints and inference procedures. Fuzzy systems are advantageous in the developtnent of systenls solutions that perform tasks such as automatic modeling, prediction, pattern recognition, and optimal decision making, control and planning. Thus, fuzzy systems are an essential tool for industrial and tnanufacturing systems engineering. Traditionally fuzzy systenl design is carried out by a time-consuming trial-and-error process, in order to incorporate expert knowledge into a fuzzy model. To date there is no systematic procedure to design fuzzy systems that has been met with wide-acceptance and usage. In this research, a fonnal methodology has been developed for the design and adaptation of effective fuzzy systems displaying both robustness and computational efficiency. The goal was to improve upon a functional fuzzy system by modifying the fuzzy sets of fuzzy rules using evolutionary computation principles. An experimental implementation was developed. The relative influence of selected factors such mutations and crossover schemes, mutations rates, and fuzzy rule structure upon the tnethodology's effectiveness within the experimental context were also addressed. Finally, the application of the methodology to function approximation, classification, regression, and the prediction of chaotic dynamics were accomplished to demonstrate the computational characteristics, complexity and inherent strength of the developed methodology. Acknowledgements .................................................... iv Abstract ............................................................. v Table of Contents ...................................................... vi Listofsymbols ....................................................... xii ListofAcronyms ...................................................... xiii Chapter 1 Introduction .......................................................... 1 1. I Fuzzy Systems .............................................. 1 1.2 Statement of Problem .......................................... 6 1.3Approach ................................................... 7 1.4 Outline of the Thesis .......................................... 9 Chapter 2 Background and Literature Review ....................................... 11 2.1 Background ................................................. 12 2.1.1 Fuzzy Algebraic Concepts ............................. 13 2.1.2 Fuzzy Systems ...................................... 16 2.1.2.1 A Basic Fuzzy Control Algorithm ................ 19 2.1.2.2 The Fuzzifier ................................ 20 2.1.2.3 The Fuzzy Inference Engine .................... 27 2.1.2.4 The Defuzzifier .............................. 30 The Maximum Membership Method ............. 30 The Centroid Defuzzification Method ............ 32 The Weighted Average (Singleton) Method ......... 33 2.1.2.5 The Fuzzy Rule Base .......................... 34 2.1.3 Evolutionary Computation ............................. 36 2.1.3.1 A Simple Genetic Algorithm .................... 38 2.1.3.2 Genetic Operators ............................ 38 Crossover ................................... 39 Mutation ................................... 41 2.1.3.3 Models of Evolution .......................... 42 Parent Selection Techniques ..................... 43 Offspring Insertion Techniques .................. 44 2.1.3.4 Underlying Theory of Genetic Algorithms ......... 45 2.2 Traditional Methods ......................................... 46 2.2.1 Straight Forward Brute-Force Approach .................. 46 2.2.2 The Wang and Mendel Methodology ..................... 47 2.3 Soft-Computing Methods ..................................... 48 2.3.1 Artificial Neural Network Fuzzy Design Methods .......... 48 2.3.2 Fuzzy System Synthesis and Optimization using Genetic Algorithms ........................................ 54 2.4Summary .................................................. 57 Chapter 3 A Fuzzy System Design Methodology .................................... 58 3.1 Fuzzy System Design Elements ................................ 58 3.1.1 A Variation on a Theme: The Wang and Mendel Method ..... 61 3.1.2 OPTIMIST: An Evolutionary Optimization Method ........ 65 3.1.2.1 Fuzzy System Representation ................... 65 3.1.2.2 Fuzzy System Qualitative Measurement .......... 68 3.1.2.3 Mutation Mechanism .......................... 68 3.1.2.4 Crossover ................................... 73 3.1.2.5 Model of Evolution ........................... 75 Chapter 4 System Design and Implementation Specifics ............................... 77 4.1 Fuzzy System Design ........................................ 78 4.2 Implementation ............................................. 80 4.2.1 Design ............................................. 80 4.2.1.1 Functions ................................... 81 4.2.1.2 Steps in Using DESIGN ........................ 83 4.2.2 OPTIMIST ......................................... 84 4.2.2.1 Functions ................................... 84 4.2.2.2 Steps in Using OPTIMIST ..................... 87 4.2.3 Prototype Characteristics ............................. 87 Chapter 5 Empirical Results and Benchmark Results .................................. 89 5.1 Function Approximation and Prediction .......................... 90 5.1.1 Experiment Series One ................................ 92 5.1.3 Experiment Series Three ............................... 108 viii 5.2 Classification ............................................... 114 5.2.1 The Mechanics of Abductive Modeling ............. 115 5.2.2 Experiments Series One ................................ 118 5.3 Prediction of Boston Housing Prices ............................. 124 5.4 Predicting Chaotic Dynamics ................................... 128 Chapter 6 Conclusions and Further Research ........................................ 136 6.1Conclusions ................................................ 137 6.2 Accomplishments ............................................ 139 6.3 Limitations and Problems ...................................... 139 6.3.1 Problems in general ................................... 140 6.3.2 Problems with DESIGN ............................... 140 6.3.3 Problems with OPTIMIST ............................ 140 6.4 Future Research ............................................. 141 References .......................................................... 143 Appendix A DESIGNSource ..................................................... 154 Appendix B OPTIMISTSource .................................................... 166 Appendix C FUZZY SYSTEM ENGINE Source ...................................... 199 Appendix D Fifth-Degree Polynomial Data Generator Source ............................. 207 Appendix E Fuzzy System Definition Format ........................................ 210 Figure 2.1 An abstract view of a fuzzy system ............................. 18 Figure 2.2 A triangular fuzzy number (TFN) ............................... 22 Figure 2.3 A crisp nutnber viewed as a TFN ............................... 23 Figure 2.4 A trapezoidal fuzzy number (TpFN) ............................ 24 Figure 2.5 The fuzzy set Maximum ...................................... 26 Figure 2.6 (a) Fuzzy reasoning using Mamdani's minimum operation rule as a fuzzy implication function. (b) resulting decision in output space MI .................. 29 Figure 2.7 The centroid defuzzification method ............................. 32 Figure 2.8 The evolutionary computation cycle ............................. 37 Figure 2.9 An exa~npleof single-point crossover ............................ 40 Figure 2.10 An example of two-point crossover ............................. 41 Figure 2.11 An example of single-point mutation ............................ 42 Figure 2.12 Patches representing fuzzy rules ............................... 52 Figure 3.1 Architecture of the DESIGN system ............................. 59 Figure 3.2 Architecture of the OPTIMIST system .......................... 60 Figure 3.3 The division of the input space x into fuzzy regions ................. 62 Figure 3.4 The division of the output space y into fuzzy regions ............... 63 Figure 3.5 Individual structure within population ........................... 67 Figure 3.6 A trapezoidal fuzzy number ................................... 69 Figure 3.7 A linear shift mutation ........................................ 70 Figure 3.8 An expansion mutation ......................................... 70 X Figure 3.9 A nonlinear shift mutation ..................................... 71 Figure 3.10 Mutation depiction in Kosko's representation .................... 73 Figure 3.11 Crossover in fuzzy system reproduction ........................ 74 Figure 4.1 File input and output in DESIGN ..............................