Advanced Fuzzy Systems Design and Applications

Fuzzy rule systems have found a wide range of applications in many fields of science and technology. Traditionally, fuzzy rules are generated from human expert knowledge or human heuristics for relatively simple systems. In the last few years, data-driven fuzzy rule generation has been very active. Compared to heuristic fuzzy rules, fuzzy rules generated from data are able to extract more profound knowledge for more complex systems. This book presents a number of approaches to the generation of fuzzy rules from data, ranging from the direct fuzzy inference based to neural net­ works and evolutionary algorithms based fuzzy rule generation. Besides the approximation accuracy, special attention has been paid to the interpretabil­ ity of the extracted fuzzy rules. In other words, the fuzzy rules generated from data are supposed to be as comprehensible to human beings as those generated from human heuristics. To this end, many aspects of interpretabil­ ity of fuzzy systems have been discussed, which must be taken into account in the data-driven fuzzy rule generation. In this way, fuzzy rules generated from data are intelligible to human users and therefore, knowledge about unknown systems can be extracted.

1. Fuzzy Sets and Fuzzy Systems.- 1.1 Basics of Fuzzy Sets.- 1.1.1 Fuzzy Sets.- 1.1.2 Fuzzy Operations.- 1.1.3 Fuzzy Relations.- 1.1.4 Measures of Fuzziness.- 1.1.5 Measures of Fuzzy Similarity.- 1.2 Fuzzy Rule Systems.- 1.2.1 Linguistic Variables and Linguistic Hedges.- 1.2.2 Fuzzy Rules for Modeling and Control.- 1.2.3 Mamdani Fuzzy Rule Systems.- 1.2.4 Takagi-Sugeno-Kang Fuzzy Rule Systems.- 1.2.5 Fuzzy Systems are Universal Approximators.- 1.3 Interpretability of Fuzzy Rule System.- 1.3.1 Introduction.- 1.3.2 The Properties of Membership Functions.- 1.3.3 Completeness of Fuzzy Partitions.- 1.3.4 Distinguishability of Fuzzy Partitions.- 1.3.5 Consistency of Fuzzy Rules.- 1.3.6 Completeness and Compactness of Rule Structure.- 1.4 Knowledge Processing with Fuzzy Logic.- 1.4.1 Knowledge Representation and Acquisition with IFTHEN Rules.- 1.4.2 Knowledge Representation with Fuzzy Preference Models.- 1.4.3 Fuzzy Group Decision Making.- 2. Evolutionary Algorithms.- 2.1 Introduction.- 2.2 Generic Evolutionary Algorithms.- 2.2.1 Representation.- 2.2.2 Recombination.- 2.2.3 Mutation.- 2.2.4 Selection.- 2.3 Adaptation and Self-Adaptation in Evolutionary Algorithms.- 2.3.1 Adaptation.- 2.3.2 Self-adaptation.- 2.4 Constraints Handling.- 2.5 Multi-objective Evolution.- 2.5.1 Weighted Aggregation Approaches.- 2.5.2 Population-based Non-Pareto Approaches.- 2.5.3 Pareto-based Approaches.- 2.5.4 Discussions.- 2.6 Evolution with Uncertain Fitness Functions.- 2.6.1 Noisy Fitness Functions.- 2.6.2 Approximate Fitness Functions.- 2.6.3 Robustness Considerations.- 2.7 Parallel Implementations.- 2.8 Summary.- 3. Artificial Neural Networks.- 3.1 Introduction.- 3.2 Feedforward Neural Network Models.- 3.2.1 Multilayer Perceptrons.- 3.2.2 Radial Basis Function Networks.- 3.3 Learning Algorithms.- 3.3.1 Supervised Learning.- 3.3.2 Unsupervised Learning.- 3.3.3 Reinforcement Learning.- 3.4 Improvement of Generalization.- 3.4.1 Heuristic Methods.- 3.4.2 Active Data Selection.- 3.4.3 Regularization.- 3.4.4 Network Ensembles.- 3.4.5 A Priori Knowledge.- 3.5 Rule Extraction from Neural Networks.- 3.5.1 Extraction of Symbolic Rules.- 3.5.2 Extraction of Fuzzy Rules.- 3.6 Interaction between Evolution and Learning.- 3.7 Summary.- 4. Conventional Data-driven Fuzzy Systems Design.- 4.1 Introduction.- 4.2 Fuzzy Inference Based Method.- 4.3 Wang-Mendel’s Method.- 4.4 A Direct Method.- 4.5 An Adaptive Fuzzy Optimal Controller.- 4.6 Summary.- 5.Neural Network Based Fuzzy Systems Design.- 5.1 Neurofuzzy Systems.- 5.2 The Pi-sigma Neurofuzzy Model.- 5.2.1 The Takagi-Sugeno-Kang Fuzzy Model.- 5.2.2 The Hybrid Neural Network Model.- 5.2.3 Training Algorithms.- 5.2.4 Interpretability Issues.- 5.3 Modeling and Control Using the Neurofuzzy System.- 5.3.1 Short-term Precipitation Prediction.- 5.3.2 Dynamic Robot Control.- 5.4 Neurofuzzy Control of Nonlinear Systems.- 5.4.1 Fuzzy Linearization.- 5.4.2 Neurofuzzy Identification of the Subsystems.- 5.4.3 Design of Controller.- 5.4.4 Stability Analysis.- 5.5 Summary.- 6. Evolutionary Design of Fuzzy Systems.- 6.1 Introduction.- 6.2 Evolutionary Design of Flexible Structured Fuzzy Controller..- 6.2.1 A Flexible Structured Fuzzy Controller.- 6.2.2 Parameter Optimization Using Evolution Strategies...- 6.2.3 Simulation Study.- 6.3 Evolutionary Optimization of Fuzzy Rules.- 6.3.1 Genetic Coding of Fuzzy Systems.- 6.3.2 Fitness Function.- 6.3.3 Evolutionary Fuzzy Modeling of Robot Dynamics.- 6.4 Fuzzy Systems Design for High-Dimensional Systems.- 6.4.1 Curse of Dimensionality.- 6.4.2 Flexible Fuzzy Partitions.- 6.4.3 Hierarchical Structures.- 6.4.4 Input Dimension Reduction.- 6.4.5 GA-Based Input Selection.- 6.5 Summary.- 7. Knowledge Discovery by Extracting Interpretable Fuzzy Rules.- 7.1 Introduction.- 7.1.1 Data, Information and Knowledge.- 7.1.2 Interpretability and Knowledge Extraction.- 7.2 Evolutionary Interpretable Fuzzy Rule Generation.- 7.2.1 Evolution Strategy for Mixed Parameter Optimization.- 7.2.2 Genetic Representation of Fuzzy Systems.- 7.2.3 Multiobjective Fuzzy Systems Optimization.- 7.2.4 An Example: Fuzzy Vehicle Distance Controller.- 7.3 Interactive Co-evolution for Fuzzy Rule Extraction.- 7.3.1 Interactive Evolution.- 7.3.2 Co-evolution.- 7.3.3 Interactive Co-evolution of Interpretable Fuzzy Systems.- 7.4 Fuzzy Rule Extraction from RBF Networks.- 7.4.1 Radial-Basis-Function Networks and Fuzzy Systems.- 7.4.2 Fuzzy Rule Extraction by Regularization.- 7.4.3 Application Examples.- 7.5 Summary.- 8. Fuzzy Knowledge Incorporation into Neural Networks.- 8.1 Data and A Priori Knowledge.- 8.2 Knowledge Incorporation in Neural Networks for Control.- 8.2.1 Adaptive Inverse Neural Control.- 8.2.2 Knowledge Incorporation in Adaptive Neural Control.- 8.3 Fuzzy Knowledge Incorporation By Regularization.- 8.3.1 Knowledge Representation with Fuzzy Rules.- 8.3.2 Regularized Learning.- 8.4 Fuzzy Knowledge as A Related Task in Learning.- 8.4.1 Learning Related Tasks.- 8.4.2 Fuzzy Knowledge as A Related Task.- 8.5 Simulation Studies.- 8.5.1 Regularized Learning.- 8.5.2 Multi-task Learning.- 8.6 Summary.- 9. Fuzzy Preferences Incorporation into Multi-objective Optimization.- 9.1 Multi-objective Optimization and Preferences Handling.- 9.1.1 Multi-objective Optimization.- 9.1.2 Incorporation of Fuzzy Preferences.- 9.2 Evolutionary Dynamic Weighted Aggregation.- 9.2.1 Conventional Weighted Aggregation for MOO.- 9.2.2 Dynamically Weighted Aggregation.- 9.2.3 Archiving of Pareto Solutions.- 9.2.4 Simulation Studies.- 9.2.5 Theoretical Analysis.- 9.3 Fuzzy Preferences Incorporation in MOO.- 9.3.1 Converting Fuzzy Preferences into Crisp Weights.- 9.3.2 Converting Fuzzy Preferences into Weight Intervals.- 9.4 Summary.- References.