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Fuzzy Logic
Conception
Introduced by Lotfi Zadeh in 1960s at BerkleyWanted to expand crisp logic
Why Fozzy?
Real world not BooleanUncertainty of natural language
Set Theory
Degrees of truth
Set Theory
Crisp vs. Fuzzy Venn Diagrams for Complements
using Membership Functions
Set Theory
CrispComplement is negationLogical AND: A /\ B is intersection of sets A and BLogical OR: A \/ B is union of sets A and B
FuzzyComplement is mS(x) – 1Intersect (/\) is given by MIN operationUnion (\/) is given by MAX operation
Fuzzy Relations
Ordered pairs showing connection between two setsRelations are sets themselvesExpressed as matrices
Fuzzy Relations
Value of the membership function, mR(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 OperationsDot-product (or MAX-MIN composition): MAX( MIN( mR1(x, y), mR2(y, z) ) )
Cross product (or MAX-PROD):MAX( mR1(x, y) * mR2(y, z) )
MAX-AVE composition:½ * MAX( mR1(x, y) + mR2(y, z) ).
Fuzzy Inference
Modus PonensCrisp: A => B (~A \/ B)Fuzzy: use membership functionsA = mA(x),
AC = 1 – mA(x),
B = mB(y).
Fuzzy Modus Ponens
“OR” == “MAX” in fuzzy logicA => B is equivalent to MAX( 1 – mA(x), mB(y) ).
A => B if and only if mA(x) >= mB(y). (Implication Rule—premise must be larger than or equal to the conclusion)
Calculating Relational Matrices
The most popular methods are:MIN implication:mA=>B(x, y) = MIN( mA(x), mB(y) )
And Product implication:mA=>B(x, y) = mA(x) * mB(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 inputCrisp input is first transformed into a vector of membership degrees through the process of fuzzificationInput 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 defuzzificationThere are multiple methods of defuzzification in use today, each with its own advantages.
Defuzzification
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 speedDeciding and maintaining a target speedStopping accurately at a target position
Intelligent Control System
Adopted because it makes qualitative decisions, based on membership functions for variable dataDecides 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 takenHas model of the motor and break to predict the next state of speed, stopping point, and running time input variablesController 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