Fuzzy Logic

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Fuzzy Logic
Lotfi Zadeh created in 1965
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• reference
• Matlab Fuzzy Logic Tutorial

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Fuzzy Logic Process
Crisp Input
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Fuzzification

• The First Step...

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How tall is Kevin?

• Very Tall?
• Tall?
• Average?
• Short?
• Very Short?

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How tall is Kevin?

• Very Tall (7 feet)?
• Tall (6 feet)?
• Average (5 feet)?
• Short (4 feet)?
• Very Short (3 feet)?

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Fuzzification Rules
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Some Examples

• If you are 5 feet
• Very tall - 0
• Tall - 0
• Average - 100
• Short - 0
• Very Short - 0
• Same as Boolean logic (so far)

• Very Tall (7 feet)?
• Tall (6 feet)?
• Average (5 feet)?
• Short (4 feet)?
• Very Short (3 feet)?

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Some Examples

• If you are 5½ feet
• Very tall - 0
• Tall - 50
• Average - 50
• Short - 0
• Very Short - 0
• NOT Boolean logic (Whoa. Cool!)

• Very Tall (7 feet)?
• Tall (6 feet)?
• Average (5 feet)?
• Short (4 feet)?
• Very Short (3 feet)?

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How tall is Kevin?
Kevin is 6 2

• Very Tall -
• Tall -
• Average -
• Short -
• Very Short -

• Very Tall - 16
• Tall - 84
• Average - 0
• Short - 0
• Very Short - 0

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The Second Step...
Fuzzy Logic the FAM
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Fuzzy Logic Process
Crisp Input
fuzzy associative matrices
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Solar Pool Heater Example

• suppose we measure the pool water temp and the
  wind speed and we want to adjust the valve that
  sends water to the solar panels
• we have two input parameters temp wind_speed
• we have one output parameter change_in_valve

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Solar Pool Heater Example

• set up membership functions for the inputs
• for each input, decide on how many categories
  there will be and decide on their membership
  functions

cold
nominal
warm
cool
calm
hot
strong
brisk
calm brisk strong
4 12 20 mph
60 70 80 90
100 F
wind_speed
temp
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Solar Pool Heater Example

• set up the rules
• if (temp is hot) AND (wind_speed is calm)then
  (change_in_valve is big_negative)
• if (temp is warm) AND (wind_speed is brisk)
• then (change_in_valve is small_negative)
• if (temp is nominal) OR (temp is warm)then
  (change_in_valve is no_change)

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Solar Pool Heater Example

• fuzzify the inputs

cold
nominal
warm
cool
calm
hot
0
0.6
0.35
strong
0
brisk
0
0.4
0
0.55
calm brisk strong
4 12 20 mph
60 70 80 90
100 F
wind_speed 9 mph
temp 87F
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Solar Pool Heater Example
cold
nominal
warm
cool
hot
0
0.6
0.35
0
0

• fire the rules
• if (temp is hot) AND (wind_speed is calm)then
  (change_in_valve is big_negative)
• if (temp is warm) AND (wind_speed is brisk)
• then (change_in_valve is small_negative)
• if (temp is nominal) OR (temp is warm)then
  (change_in_valve is no_change)

calm
strong
brisk
0.4
0
0.55
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Solar Pool Heater Example
bigneg.
nochange
smallpos.
bigpos.
smallneg.
0.6
0
0
0
0.4

• fire the rules
• if (temp is hot) AND (wind_speed is calm)then
  (change_in_valve is big_negative)
• if (temp is warm) AND (wind_speed is brisk)
• then (change_in_valve is small_negative)
• if (temp is nominal) OR (temp is warm)then
  (change_in_valve is no_change)

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Solar Pool Heater Example
bigneg.
nochange
smallpos.
bigpos.
smallneg.

• now that we have this info, howdo we get the
  output?
• what is output value forchange_in_valve ?
• NEXT - defuzzify the output(s)

0.6
0
0
0
0.4
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Fuzzy Logic Process
Crisp Input
fuzzy associative matrices
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Defuzzify the output

• set up membership functions for the output(s)
• for each output, decide on how many categories
  there will be and decide on their membership
  functions

bigneg.
nochange
smallpos.
bigpos.
smallneg.
big neg sm. neg. no ch. sm. pos.
big pos.
-10 -5 0 5
10 degrees
change_in_valve
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Solar Pool Heater Example
bigneg.
nochange
smallpos.
bigpos.
smallneg.

• NEXT - defuzzify the output(s)

0.6
0
0
0
0.4
0 0.4 0.6 0.0 0.0
big neg sm. neg. no ch. sm. pos.
big pos.
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Solar Pool Heater Example
bigneg.
nochange
smallpos.
bigpos.
smallneg.

• defuzzify the output(s)

0.6
0
0
0
0.4
0 0.4 0.6 0.0 0.0
big neg sm. neg. no ch. sm. pos.
big pos.
-10 -5 0 5
10 degrees
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Solar Pool Heater Example
bigneg.
nochange
smallpos.
bigpos.
smallneg.

• defuzzify the output(s)

0.6
0
0
0
0.4
0 0.4 0.6 0.0 0.0
big neg sm. neg. no ch. sm. pos.
big pos.
-10 -5 0 5
10 degrees
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Solar Pool Heater Example
bigneg.
nochange
smallpos.
bigpos.
smallneg.

• defuzzify the output(s)

0.6
0
0
0
0.4
0 0.4 0.6 0.0 0.0
big neg sm. neg. no ch. sm. pos.
big pos.
-10 -5 0 5
10 degrees
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Solar Pool Heater Example
bigneg.
nochange
smallpos.
bigpos.
smallneg.

• defuzzify the output(s)

0.6
0
0
0
0.4
0 0.4 0.6 0.0 0.0
big neg sm. neg. no ch. sm. pos.
big pos.
-10 -5 0 5
10 degrees
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Solar Pool Heater Example
bigneg.
nochange
smallpos.
bigpos.
smallneg.

• defuzzify the output(s)

0.6
0
0
0
0.4
0 0.4 0.6 0.0 0.0
big neg sm. neg. no ch. sm. pos.
big pos.
-10 -5 0 5
10 degrees
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Solar Pool Heater Example
bigneg.
nochange
smallpos.
bigpos.
smallneg.

• defuzzify the output(s)

0.6
0
0
0
0.4
0 0.4 0.6 0.0 0.0
big neg sm. neg. no ch. sm. pos.
big pos.
-10 -5 0 5
10 degrees
output is -1.28 degrees
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• fuzzy inference is a method that interprets the
  values in the input vector, and based on some set
  of rules, assigns values to the output vector

general case
specific example
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• topics
• fuzzy membership functions
• logical operations
• if-then rules
• fuzzy inference systems

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• classical sets black and white

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• fuzzy sets the truth of a statement becomes a
  matter of degree

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• fuzzy membership functions

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• fuzzy membership functions(another example)

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• a fuzzy membership function is a curve that
  defines how each point in the input space is
  mapped to a membership value (degree of
  membership) between 0 and 1

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• non fuzzy vs. fuzzy

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• the fuzzy logic toolbox includes 11 built-in
  membership function types

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• summary of membership functions
• Fuzzy sets describe vague concepts (fast runner,
  hot weather, weekend days).
• A fuzzy set admits the possibility of partial
  membership in it. (Friday is sort of a weekend
  day, the weather is rather hot).
• The degree an object belongs to a fuzzy set is
  denoted by a membership value between 0 and 1.
  (Friday is a weekend day to the degree 0.8).
• A membership function associated with a given
  fuzzy set maps an input value to its appropriate
  membership value.

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• topics
• fuzzy membership functions
• logical operations
• if-then rules
• fuzzy inference systems

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• We need logical operations in order to put
  together necessary if conditions
• if x AND y then
• if x OR y then
• if NOT x then

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• boolean logic
• fuzzy logic

0.2 0.6 0.2 0.2 0.6 0.6
0.2 0.8
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• topics
• fuzzy membership functions
• logical operations
• if-then rules
• fuzzy inference systems

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• a single fuzzy if-then rule assumes the formif
  x is A then y is B

antecedent or premise
consequent or conclusion
If service is good then tip is average.
if (service good) then (tip average)
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• summary of if-then rules
• Interpreting if-then rules is a three-part
  process
• Fuzzify inputs Resolve all fuzzy statement in
  the antecedent to a degree of membership between
  0 and 1
• Apply fuzzy operator to multiple part
  antecedents If there are multiple parts to the
  antecedent, apply fuzzy logic operators and
  resolve the antecedent ot a single number between
  0 and 1. This is the degree of support for the
  rule.
• Apply implication method Use the degree of
  support for the entire rule to shape the output
  fuzzy set. The consequent of a fuzzy rule
  assigns an entire fuzzy set to the output. If
  the antecedent is only partially true, then the
  output fuzzy set is truncated according to the
  implication method.

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• topics
• fuzzy membership functions
• logical operations
• if-then rules
• fuzzy inference systems

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• 5 steps to the fuzzy inference process
• fuzzification of the inputs
• application of the fuzzy AND/OR in the antecedent
• implication from the antecedent to the consequent
• aggregation of the consequents across the rules
• deffuzification

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• fuzzification of inputs

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• apply fuzzy operator

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• apply implication method

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• aggregate all outputs

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• deffuzify

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