Computing with Words

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Computing with Words
ECE 1001

• Professor Marian S. Stachowicz
• Laboratory for Intelligent Systems
• University of Minnesota Duluth

October 7, 2004
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What are Fuzzy Sets?
Problem 1 Given the set U 1,2,3,4,5,6,7,8,9,1
0,11,12, describe the set of prime numbers.
A u in U u is a prime number
The elements of the set are defined unequivocally
as A 2,3,5,7,11
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Problem 2 Now using the same set U, suppose we
want to describe the set of small numbers.
M u in U u is a small number
Now, it is not so easy to define the set. We can
use a sharp transition like the following,
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Or an alternative way to define the set would be
to use a smooth transition.
Characteristic Function
A U --gt 0,1
Membership Function
M U --gt 0,1
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The set of small numbers which we described
earlier are represented as a fuzzy set in the
following manner in the Fuzzy Logic Pack.
Small FuzzySet1,1,2,1,3,.9,4,.6,5,.4
,6,.3, 7,.2,8,.1,9,0,10,
0,11,0,12,0,
UniversalSpace -gt 0,12
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Fuzzy set theory provides a strict mathematical
framework in which vague conceptual
phenomena can be precisely and rigorously studied.
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Fuzzy Areas of Application

• Fuzzy Logic Control
• Fuzzy Modeling
• Fuzzy Arithmetic

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Fuzzy Logic Control
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Fuzzy Modeling
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Fuzzy Arithmetic
A1, A2, A1A2
A1, A2, A1-A2
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Where is Fuzzy Logic Used?

• Linear and Nonlinear Process Control
• Robotics, Automation, Tracking
• Consumer Electronics
• VCRs, Digital High Definition Television,
  Microwave Ovens, Cameras,etc.
• Pattern Recognition
• Image Processing, Machine Vision
• Decision Making
• Sensor Fusion, Risk Analysis
• Financial Systems
• Information Systems
• Data Base Management, Information Retrieval
• Data Analysis
• Meteorology
• Art and Music

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Fuzzy Systems

• Why fuzzy systems?
• What are fuzzy systems?
• Where are fuzzy systems used and how?

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Two kinds of justification for fuzzy system
theory

• We need a theory to formulate human knowledge in
  a systematic manner and put it into engineering
  systems.

• The real world is too complicated for precise
  descriptions to be obtained.

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Fuzzy Systems

• Fuzzy Systems are knowledge-based or rules-based
  systems.

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A fuzzy IF-THEN rule is statement in which some
words are characterized by continuous membership
function (MF).
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Example 1.1

• IF the speed of car is high, THEN apply less
  force to the accelerator.Where the words high
  and less are characterized by membership
  functions (MF).

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Fuzzy Systems

• Fuzzy systems are knowledge-based or rules-based
  systems.
• A fuzzy systems is constructed from a collection
  of fuzzy IF-THEN rules.

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Example 1. 2

• Problem
• We want to design a controller to automatically
  control the speed of a car.

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SolutionTwo approaches to designing such a
controller

• use conventional control theory,
• for example, designing a PID controller.

• to emulate human drivers, that is, converting the
  rules used by human drivers into an automatic
  controller.

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Knowledge-based or rules-based.

• IF speed is low, THEN apply more force to the
  accelerator,
• IF speed is medium, THEN apply normal force to
  the accelerator,
• IF speed is high, THEN apply less force to the
  accelerator.
• Where the words low, medium, high and
• more, normal, less are characterized by
  membership functions (MF).

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Note In this example, the rules are control
instructions, that is, they represent what a
human driver does in typical situations.
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Example 1. 3

• Problem.
• A person pumping up a balloon wishes to know how
  much air he could add before it burst. With the
  balloon there are three key variables
• the air inside the balloon,
• the amount it increases, and
• the surface tension

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Solution

• IF the amount of air is small and it is increased
  slightly THEN the surface tension will increase
  slightly,
• IF the amount of air is small and it is increased
  substantially THEN the surface tension will
  increase slightly,
• IF the amount of air is large and it is increased
  slightly THEN the surface tension will increase
  moderately,
• IF the amount of air is large and it is increased
  substantially THEN the surface tension will
  increase substantially.
• Where the words small, slightly,
  substantially, etc., are characterized by
  membership functions.

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Summary

• The starting point of constructing a fuzzy system
  is to obtain a collection of
• fuzzy IF-THEN rules from human experts or based
  on domain knowledge.
• The next step is to combine these rules into a
  single system.

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A distinguished feature of the fuzzy systems.

• FS are multi-input-single output mappings from a
  real-valued vector to real valued scalar.
• FS are constructed from human knowledge in the
  form IF- THEN rules.
• FS theory provides a systematic procedure for
  transforming a knowledge base into a nonlinear
  mapping.
• The analysis and design can be performed in a
  mathematically rigorous fashion.

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There are three types of fuzzy systems that are
commonly used in the literature

• pure fuzzy systems,
• Takagi-Sugeno-Kang (TSK) fuzzy systems,
• fuzzy systems with fuzzifier and defuzzifier.

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Configuration of Fuzzy Systems

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Where Are Fuzzy Systems Used ?

• Fuzzy washing machine,
• Digital image stabilizer,
• Fuzzy systems in cars,
• Fuzzy control of a cement kiln,
• Fuzzy control of subway train.

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Digital image stabilizer.

• IF all the points in the picture are moving in
  the same direction, THEN the hand is shaking.
• IF only some points in the picture are moving,
  THEN the hand is not shaking.

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The inverted pendulum system
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IF - THEN

• If x1 is small and x2 is small, then apply a
  slight negative u force
• If x1 is small and x2 is large, then apply a
  negative u force
• If x1 is large and x2 is small, then apply a
  negative u force
• If x1 is large and x2 is large, then apply a
  large negative u force

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Application 1 Matsushita Vacuum

• Dust sensors are used to adjust air-flow
• System is closed-loop
• Fuzzy rules are applied to reduce power
  consumption
• Example of rule
• If there is little dust, then reduce power

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Application 2 Cannon Camera

• Screen is divided into 6 parts
• 2 Inputs per part
• 13 Rules
• 1.1 Kbytes of Memory
• Rules are used to bring the picture into focus
  using less memory that traditional methods.

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Application 3 Mitsubishi Heating/Cooling

• 25 Heating Rules
• 25 Cooling Rules
• Heats/Cools 5x faster
• Reduces power consumption by 24

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Application 4 Maytag Dishwasher

• Measures soil in water, adjusts wash accordingly
• Adjusts for dried-on foods
• Determines optimum wash cycle
• System is open-loop

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Application 5 Smart Traffic Light

• 8 Sensors
• Inputs
• Cycle Time
• Cars Behind Red Light
• Cars Behind Green Lt.
• If/Then Rules
• And/Or Operators
• Example
• If cycle time is medium AND Cars Behind Red is
  low AND Cars Behind Green is medium, then change
  is Probably Not

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Application 6 Sony Palmtop

• Used directly for character recognition
• Each person writes letters slightly differently
• Fuzzy rules account for these differences

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References

• 1 L.A. Zadeh, "Fuzzy sets", Information and
  Control, vol.8, pp. 338-353, 1965
• 2 George S. Klir and Tina Folger, Fuzzy Sets,
  Uncertainty, and Information. Prentice Hall,
  Englewood Cliffs, New Jersey, 1988.
• 3 H.J. Zimmermann, Fuzzy Set Theory and Its
  Applications, Second Edition, Kluwer Academic
  Publishers, Boston, MA, 1991.
• 4 M.S. Stachowicz and Lance E. Beall, Fuzzy
  Logic Pack for use with Mathematica, Wolfram
  Research, Inc. Champaign, IL 61820-7237,1995
• 5 C.M. Charlton, Strange Attractor, CD Album
  of Piano Improvisations, Orange Moon Production,
  Inc. 19672 Stevens Creek Blvd., 178, Cupertino,
  CA 95014

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RESEARCH WITH LIS

• Adilbek Karaguishiyev
• Intelligent Stethoscope

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HOMEWORK - COMPUTING WITH WORDS

• 1. Find five WEB sites that contain valuable
  information about computing with words
  applications. Provide the URL for each of the
  sites. Provide a brief description of what
  information is available on each site.
• 2. Two page limit !!!
• 3. On the top of the first page, provide name and
  e-mail address.
• 4. Typed (Word Processor) is much preferred over
  a hand written submission.
• 5. Due in class on Thursday, October 14.
• 6. Mathematica by WOLFRAM Research has a
  FuzzyLogic Package by Prof. M. S. Stachowicz and
  Lance Beall with some good tutorial information.
  http//www.wolfram.com/fuzzylogic
• Study chapters
• FuzzyLogic - Documentation English
  chapters 0.01, 1.01, 2.01, 2.08 example 1 and 2