Introduction to Fuzzy Logic. James K. Peckol

      1 Chapter 1Figure 1.1 Crisp and fuzzy circles.Figure 1.2 Membership in subset H.Figure 1.3 Membership functions for “close to 7.”Figure 1.4 Bottles of liquids – front side.Figure 1.5 Bottles of liquids – back side.

      2 Chapter 2Figure 2.1 Algebraic AND and OR operations.Figure 2.2 Truth‐table proof.Figure 2.3 Single‐variable truth table.Figure 2.4 Two‐variable truth table.Figure 2.5 Verifying De Morgan's Law.Figure 2.6 De Morgan's tea vending machine.Figure 2.7 Basic logic devices.Figure 2.8 Logic gates.Figure 2.9 Logic circuit.Figure 2.10 Logic circuit.Figure 2.11 Tea circuit.Figure 2.12 Basic truth tables.Figure 2.13 Basic truth table.Figure 2.14 Four‐variable K‐Map.Figure 2.15 Four‐variable K‐Map.Figure 2.16 Four‐variable K‐Map.Figure 2.17 Four‐variable K‐Map.Figure 2.18 Three‐input system.Figure 2.19 BCD to Excess 3 decoder.Figure 2.20 BCD to Excess 3 encoder block diagrams.

      3 Chapter 3Figure 3.1 Three sets.Figure 3.2 Crisp set and universe of discourse.Figure 3.3 Two sets of light bulb states.Figure 3.4 Four basic logical operations.Figure 3.5 Motor control subsets.Figure 3.6 Basic crisp set symbols.Figure 3.7 Basic universe of discourse.Figure 3.8 Viewing a movie.Figure 3.9 Expressing an opinion‐0.

      4 Chapter 4Figure 4.1 Real‐world problems fuzzy logic solution.Figure 4.2 Universe of discourse and three subsets.Figure 4.3 Crisp and fuzzy membership values.Figure 4.4 Membership functions for “close to 7.”Figure 4.5 Fuzzy set membership vs. probability.Figure 4.6 Information from the backs of the bottles.Figure 4.7 Basic fuzzy set symbols.Figure 4.8 Four basic logical operations.Figure 4.9 Basic universe of discourse with two subsets.Figure 4.10 Viewing a movie.Figure 4.11 Expressing an opinion‐1, 2.Figure 4.12 Expressing an opinion‐3.Figure 4.13 Expressing an opinion‐4.Figure 4.14 Simple crisp and fuzzy adult membership functions.Figure 4.15 Linear increasing and decreasing fuzzy membership functions.Figure 4.16 Degree of membership.Figure 4.17 Linear increasing and decreasing fuzzy membership functions.Figure 4.18 Increasing and decreasing fuzzy membership functions.Figure 4.19 Basic linear and bell‐shaped fuzzy membership functions around 5...Figure 4.20 Basic linear membership functions around and restricted domains....Figure 4.21 Universe of discourse.Figure 4.22 Set of support.Figure 4.23 Sigmoid membership curves growth and decline.Figure 4.24 PI membership curve.Figure 4.25 Beta membership curve.Figure 4.26 Gaussian membership curves.Figure 4.27 Simple membership functions.Figure 4.28 Using a membership function.Figure 4.29 Using a membership function.Figure 4.30 Using a membership function.Figure 4.31 Using membership functions.

      5 Chapter 5Figure 5.1 Evolution of language.Figure 5.2 Simple communication.Figure 5.3 Complex communication.Figure 5.4 No communication.Figure 5.5 Set membership.Figure 5.6 Subset membership.Figure 5.7 Boolean union/OR.Figure 5.8 Boolean intersection/AND.Figure 5.9 The variable number.Figure 5.10 Subsets of the set number.Figure 5.11 Fuzzy set.Figure 5.12 Graph of red color linguistic variable.Figure 5.13 Tall membership function.Figure 5.14 Tall and very tall membership functions.Figure 5.15 Tall and more or less tall membership functions.

      6 Chapter 6Figure 6.1 Membership function graph for resistor size.Figure 6.2 Membership function for wine Quality vs. Aging.Figure 6.3 Entailment and containment.Figure 6.4 Set, subset, member hierarchy.Figure 6.5 Entailment and containment.Figure 6.6 A union.Figure 6.7 Not a union.Figure 6.8 The union of two fuzzy subsets.Figure 6.9 Intersection of two fuzzy subsets.Figure 6.10 Universe of discourse S.Figure 6.11 Basic universe of discourse S.Figure 6.12 Membership function for transistor leakage.Figure 6.13 Cartesian cross product A × B.Figure 6.14 Conditioned fuzzy subsets.Figure 6.15 Applying Relation R to Set X.Figure 6.16 MAX‐MIN composition of a fuzzy subset and a fuzzy relation.Figure 6.17 MAX‐MIN composition of two fuzzy subsets.Figure 6.18 Max‐product composition of two fuzzy subsets.Figure 6.19 Approximately Equal over a Set.

      7 Chapter 7Figure 7.1 A simplified fuzzy system.Figure 7.2 Possible membership values in a crisp logic system.Figure 7.3 Possible membership values in a fuzzy logic system.Figure 7.4 Membership functions.Figure 7.5 Velocity degree of membership.Figure 7.6 Trapezoidal membership function.Figure 7.7 Membership function features.Figure 7.8 Fuzzy system – high‐level viewFigure 7.9 Mamdani's implication function.Figure 7.10 Larsen's implication function.Figure 7.11 Winter driving dilemma.Figure 7.12 Winter driving dilemma continued.Figure 7.13 Winter driving dilemma continued max criterion defuzzification....Figure 7.14 Winter driving dilemma continued mean of maximum criterion defuz...Figure 7.15 Winter driving dilemma continued center of gravity defuzzificati...Figure 7.16 Grand prix driver dilemma.Figure 7.17 A simplified controller system.Figure 7.18 A temperature controller block diagram.Figure 7.19 Traditional temperature control membership function.Figure 7.20 A temperature controller block diagram.Figure 7.21 Temperature control membership functions.Figure 7.22 Temperature control membership functions.Figure 7.23 Queuing and delay at a turnstile.Figure 7.24 Membership functions.Figure 7.25 Output distribution.Figure 7.26 Block diagram – automatic transmission design.Figure 7.27 Membership functions.Figure 7.28 Gear selection.Figure 7.29 Block diagram – automatic transmission design.

      8 Chapter 8Figure 8.1 System model – high level external view.Figure 8.2 A simplified fuzzy system.

      9 Chapter 9Figure 9.1 Threshold logic gate.Figure 9.2 Threshold gate disguised as an AND gate.Figure 9.3 Threshold gate disguised as an OR gate.Figure 9.4 Classic logic diagram.Figure 9.5 Logical function.Figure 9.6 Threshold gate final diagram.Figure 9.7 Basic combinational logic circuits.

      10 Chapter 10Figure 10.1 Basic threshold device.Figure 10.2 High‐level basic neuron schematic.Figure 10.3 Basic artificial neuron schematic.Figure 10.4 AND, OR, NOT, NOR, inhibit gate configurations.Figure 10.5 AND, OR, NOT, NOR, INHIBIT weighted gate configurations.Figure 10.6 Simple McCulloch–Pitts neuron network.Figure 10.7 Basic perceptron schematic.Figure 10.8 Single and multi‐layer perceptrons.Figure 10.9 Basic perceptron models with and without bias.Figure 10.10 Linear activation function effect of a threshold signal.Figure 10.11 Linear.Figure 10.12 Threshold.Figure 10.13 Unit step.Figure 10.14 Piecewise linear.Figure 10.15 ReLU.Figure 10.16 Single‐neuron system schematic.Figure 10.17 Closed‐loop feedback control system.Figure 10.18 Perceptron weight layer.Figure 10.19 Summed input signals.Figure 10.20 Activation function.Figure 10.21 Feedback support.

      11 Appendix AFigure A.1 The system and its environment.Figure A.2 The interface between the customer and the design process.Figure A.3 The system and its environment – Step 0.Figure A.4 The system and its environment – Step 1.Figure A.5 Refining the system specificationFigure A.6 The system and Its environment – Step 2.Figure A.7 The system as an aggregation of componentsFigure A.8 The customer, the requirements, the design, and the engineer.

      12 Appendix BFigure B.1 The UML use case diagram.Figure B.2 Use case diagram for a simple data acquisition system.Figure B.3 Writing a use case description.Figure B.4 Class diagram.Figure B.5 UML inheritance diagram.Figure B.6 Representing an interface.Figure B.7 Representing the aggregation relationship.Figure B.8 Representing the composition relationship.Figure B.9 The call and return interaction diagram.Figure B.10 The create and destroy interaction diagram.Figure B.11 The send interaction diagram.Figure B.12 Principal components of the UML sequence diagram.Figure B.13 Sequence diagram for making a time interval measurement.Figure B.14 UML fork and join diagram..Figure B.15 UML branch and merge diagram.Figure B.16 UML activity diagram.Figure B.17 UML events.Figure B.18 Possible transitions in an UML state chart.Figure B.19 Transitions with guard conditions in a UML state chart.Figure B.20 Composite states in a UML state chartFigure B.21 Expressing a history substate in a UML state chart.Figure B.22 Expressing concurrent substates in a UML state chart.Figure B.23 Expressing a data source or sink.Figure B.24 Expressing a data store.Figure B.25 Capturing the data