Fuzzy Logic

Conception Introduced by Lotfi Zadeh in 1960s at Berkley Wanted to expand crisp logic

Why Fozzy? Real world not Boolean Uncertainty of natural language

Set Theory

Degrees of truth

Set Theory Crisp vs. Fuzzy Venn Diagrams for Complements using Membership Functions

Set Theory Crisp Complement is negation Logical AND: A /\ B is intersection of sets A and B Logical OR: A \/ B is union of sets A and B Fuzzy Complement is mS(x) – 1 Intersect (/\) is given by MIN operation Union (\/) is given by MAX operation

Fuzzy Relations Ordered pairs showing connection between two sets Relations are sets themselves Expressed as matrices

Fuzzy Relations Value of the membership function, m R (x, y), for an element (x, y) of the relation R is the value at row x and column y in the relational matrix Shows degree of correspondence between x-qualities (color) and y- qualities (ripeness)

Fuzzy Relations Matrices Color – ripeness relation for tomatoes

Fuzzy Relations Matrices Ripeness - taste relation for tomatoes

Fuzzy Relations Matrices Color - taste relation for tomatoes

Matrix Operations Dot-product (or MAX-MIN composition): MAX( MIN( m R1 (x, y), m R2 (y, z) ) ) Cross product (or MAX-PROD): MAX( m R1 (x, y) * m R2 (y, z) ) MAX-AVE composition: ½ * MAX( m R1 (x, y) + m R2 (y, z) ).

Fuzzy Inference Modus Ponens Crisp: A => B (~A \/ B) Fuzzy: use membership functions A = m A (x), A C = 1 – m A (x), B = m B (y).

Fuzzy Modus Ponens “OR” == “MAX” in fuzzy logic A => B is equivalent to MAX( 1 – m A (x), m B (y) ). A => B if and only if m A (x) >= m B (y). (Implication Rule—premise must be larger than or equal to the conclusion)

Calculating Relational Matrices The most popular methods are: MIN implication: m A=>B (x, y) = MIN( m A (x), m B (y) ) And Product implication: m A=>B (x, y) = m A (x) * m B (y) This is how the matricies are calculated

Defuzzification There are two Fuzzy Set types: Normal: maximal degree of belonging cannot be greater than 1 (typically the set of input variables) Not-normal: maximal degree of belonging can be greater than 1 (typically the set of output variables)

Defuzzification

Fuzzification To use fuzzy logic: creating input Crisp input is first transformed into a vector of membership degrees through the process of fuzzification Input typically forms a normal fuzzy set since it is derived from a crisp set

Defezzification After fuzzy inputs are processed, often the outputs are Not- normal fuzzy sets In practical uses a decision maker needs a crisp output signal a procedure for transforming the fuzzy output value into a crisp output value is necessary

Defuzzification Transformation from fuzzy output back to crisp output is called defuzzification There are multiple methods of defuzzification in use today, each with its own advantages.

Defuzzification

Fastest method:first-of-maxima method Smoother: center-of-area Most practical: center-of-area for singletons (faster and simpler than center-of-area method, though not as smooth)

Real Applications: Some Guy I am choosing to conduct a research project on the Sendai subway, in particular its fuzzy control operation. I have been a fan of subway systems since the mid 1970's when I was only three or four years old. My first subway experience was the Tokyo subway, when I visited with my family in 1974 (I was only two years old then). Since then I have been on systems as diverse as London, Paris, Berlin, Munich, Moscow, San Francisco, and St. Louis (Missouri). [from internet research paper, really]

The Sendai Subway Application of fuzzy logic; Sendai, Japan

Sendai Subway Development The subway in Sendai, Japan uses a fuzzy logic control system developed by Serji Yasunobu of Hitachi. It took 8 years to complete and was finally put into use in 1987.

Control System Based on rules of logic obtained from train drivers so as to model real human decisions as closely as possible Controls the speed at which the train takes curves as well as the acceleration and braking systems of the train

Capabilities Capable of determining: Rate of acceleration given a target speed Deciding and maintaining a target speed Stopping accurately at a target position

Intelligent Control System Adopted because it makes qualitative decisions, based on membership functions for variable data Decides from a set of control rules what actions to take Useful in representing degrees of state, such as “high” or “slightly high”

 This system is still not perfect; humans can do better because they can make decisions based on previous experience and anticipate the effects of their decisions This led to…

Predictive Fuzzy Control Can assess the results of a decision and determine if the action should be taken Has model of the motor and break to predict the next state of speed, stopping point, and running time input variables Controller selects the best action based on the predicted states.

The results of the fuzzy logic controller for the Sendai subway are excellent!! The train movement is smoother than most other trains Even the skilled human operators who sometimes run the train cannot beat the automated system in terms of smoothness or accuracy of stopping