Course ECSE 6710

Fuzzy Sets and Expert Systems in Computer Engineering


Tentative Detailed Course Outline (for Fuzzy Set Part)

  1. Administrative Matters Introduction, Motivation, Overview.

  2. Fuzzy Sets Theory

    1. Definition of a fuzzy set
    2. Membership (characteristic) function
    3. Cardinality
    4. Set operations (union, intersection, complementation)
    5. Algebraic Properties
    6. Measure of Fuzziness, Entropy
    7. Level Sets
    8. Extension Principle
    9. Relations, Lattice
    10. Fuzzy relations
    11. Fuzzy Compositions (theory and implementation)
    12. Possibilistic interpretation
    13. Possibility and Probability
    14. Consistency Principle
    15. Possibility and Necessity Measures

  3. Fuzzy Logics

    1. Boolean Logic
    2. Extended Boolean Logics
    3. Multi-Valued Logics ( Kleen, Lukasiewicz )
    4. Fuzzy-Valued Logics
    5. Approximate Reasoning

  4. Linguistic Approach and Linguistic Approximation

    1. Syntax, Semantics and Pragmatics
    2. Programming Languages and Natural Languages
    3. Chomsky classification of formal languages Semantic Networks
    4. Language as a fuzzy mapping
    5. Linguistic variables, linguistic values
    6. Modifiers and Relations
    7. Linguistic Approximation: Feature Space, Final Label Selection
    8. Saaty's Cardinal Ratio scale

  5. Fuzzy Numbers

    1. Definitions
    2. Extension Principle and Computer Implementation
    3. Sampling and Parametrization
    4. Table of Close-formed Formulae - assumptions for closure
    5. Properties

  6. Fuzzy Set Applications: Decision Making

    1. Decision Theory and Game Theory
    2. Certainty, Risk, Uncertainty
    3. One-stage, Binary Choice, Single Criterion
    4. One-stage, Multi-choices, Single Criterion
    5. One-stage, Multi-choices, Multi-Criteria
    6. Optimization under fuzzy constraints: Fuzzy Linear Programming

  7. Pattern Recognition and Cluster Analysis

    1. Pattern Recognition:

      1. Statistical/Structural (Syntactical),
      2. Feature Extraction/String Parsing,
      3. Learning (Supervised, Unsupervised)/Grammatical Inference

    2. Cluster Analysis (Graph Theoretical Approach):

      1. Similarity Relations
      2. Transitive Closures of Fuzzy Relations
      3. Extended Warshall's Algorithm

    3. Cluster Analysis (Functional Minimizer Approach)

      1. Fuzzy ISODATA Algorithm
      2. Convergency and Implementation
      3. Case study

  8. Fuzzy Algorithms

    1. Fuzzy Production Rules
    2. Case Study

  9. Fuzzy Logic Control

    1. Comparison of FLC with Conventional Controller

      1. Assumptions regarding models, sensors, and processors
      2. Proportional Integral Controllers (PI) and Sliding mode vs Fuzzy PIs
      3. Common denominator: control surface, deterministic mapping, undistinguishability of design methodology, memory in state vector definition
      4. Design Parameters: Gain vectors vs. scaling factors, termsets, and rule sets; Relationship between gains and ratios of scaling factors

    2. A Knowledge-Based Software View of FLCs

      1. A higher level language for the synthesis of Non-Linear Controllers
      2. Software Engineering cascade: Development, Compilation, Run-time Phase

    3. FLC Development Phase

      1. Interpreter: Knowledge Representation, Inference, Control of Inference

        1. Knowledge Representation:

          • Fuzzy Knowledge Base: scaling factors, termsets, rules
          • Input Representation (Quantization or Fuzzification):

        2. Inference: Fuzzy mapping and Generalized Modus Ponens
          • Left Hand Side evaluation of fuzzy rules
          • Possibility Measure and Intersection operators
          • Rule firing or detachment (Modus Ponens)
          • Aggregation of rule outputs

        3. Control of inference
          • Defuzzification Methods (Aggregate-and-defuzzify vs Defuzzify-and-Aggregate)
          • Tradeoff between performance and cost

      2. Synthesis: KB development (manual, automatic via self-organizing architecture or induction); KB Tuning (manual changes of scaling factors, termsets, rules vs. Automatic tuning via gradient descent and other NN algorithms.
      3. Analysis: Visualization (control surface, phase-plane trajectories), Stability, Robustness

    4. FLC Compilation Phase

      1. Exact Methods: Software Architecture and Hardware realizations (Coordinate Generation, Partitions and Pointers, Rules, State Termsets, Output Termset)
      2. Approximate Methods (with no run-time evaluation): Software Architecture and Hardware realizations

    5. Run-time Phase (Crisp input and Fuzzy input)
      1. Run-Time engine

    6. Hierarchical Control (Supervisory Control)
      1. Mode Switching vs Mode Melding
      2. Mediating conflicting goals
      3. Similarity as a Type 3 System

    7. Industrial FLC Applications

      1. Power Electronics
      2. Steam Turbine Start-up and Load Following
      3. Locomotive Wheel Slip Control
      4. LV100 (Turbo-shaft Aircraft Engine) Supervisory Control

    8. Exercise in KB Development, Tuning, Analysis

      1. Problem definition for the exercise: simulation model
      2. KB Development: determining scaling factors, termsets, rules for Fuzzy PI
      3. Selecting the interpreter (rule firing, defuzzification)
      4. Analyzing the results: Phase plane
      5. Tuning the FLC controller

    9. Conclusions: FLC Cost Analysis

DETAILED DESCRIPTION OF INTERPRETER

Knowledge Representation (Interpreter) Fuzzy Knowledge Base: scaling factors, termsets, rules Semantics: Scaling Factors Impact on stability Termsets to describe linguistic values Number of terms, (parametric) shape, support and peak locations Sampled vs parametric representation Syntactic Mapping: Fuzzy Rules Difference between Boolean and fuzzy rules Type 1 fuzzy rules: Monotonic non-decreasing function Type 2 fuzzy rules: Fuzzy sets Type 3 fuzzy rules: Linear functions in the state space Input Representation (Quantization or Fuzzification): Crisp input Fuzzy Input

Inference (Interpreter) Rule Sets: Disjunctive Interpretations Crisp Function (Mapping) Crisp Rule Set (Cartesian Product) Fuzzy Mapping (Compatibility Relation) Fuzzy Cartesian Product Modus Ponens From Cartesian product to Tabular representation Inference Process Using Tabular Representation Left Hand Side evaluation of fuzzy rules Possibility Measure and Intersection operators Degree of rule applicability (lambda) Rule firing or detachment (Modus Ponens) Aggregation of rule outputs

Control of inference (Interpreter) - Defuzzification of rule outputs Aggregate-and-defuzzify Mean of Maxima Center of Gravity Defuzzify-and Aggregate Height Method - Equivalence to simplified Type 3 Local Mean of Maxima Area Method Tradeoff between performance and cost



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bonisson
Thu Aug 21 22:43:36 EDT 1997