Fuzzy Controller

Fuzzy Controller
Fuzzy Inference System
Basic Components of Fuzzy Inference System
‣ Rule based system: Contains a set of fuzzy rules
‣ Data base dictionary: Defines the membership functions
used in the rules base system
‣ Defuzzification system: A defuzzifier to provide a crisp
result from the output membership functions.
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Fuzzy Inference System
Basic Components of Fuzzy Inference System
Knowledge
Base
Fuzzy or
Crisp Input
Fuzzification
Defuzzification
Crisp
Output
Fuzzy
Decision-Making
Logic
(Decision Rules)
Fuzzy
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Fuzzy Controller
Step1:
Values of the
input variables
Step4:
Determine the
consequence of each
rule
Step2:
Fuzzify inputs
Step5:
Aggregate the
consequences
Step3:
Calculate the firing
strength
Step6:
Defuzzify the
output
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Fuzzy Controller
Basic Components of Fuzzy Inference System
‣ 1. Input – vocabulary, fuzzification (creating fuzzy sets)
‣ 2. Fuzzy propositions – IF X is Y THEN Z (or Z is A) … there
are four types of propositions
‣ 3. Combination and evaluation – computation of the results
given the inputs
‣ 4. Action - defuzzification
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Fuzzy Controller
Fuzzification
‣ Transforming measurement (input) data into valuation of
subjective values (It is mapping from an observed input
space to labels of fuzzy sets)
‣ Input data are usually crisp; or it might be fuzzy sets
A A’
A
µ
µ
x
x
x
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Fuzzy Rule-Base
‣ It is the collection of fuzzy IF-THEN rules in which the
preconditions and consequences are linguistic terms
‣ Fuzzy rules relate inputs to outputs (control logic)
‣ Rules are formed using linguistic variables, so it is not
precise
‣ Output is also a linguistic value representing a fuzzy set
‣ Determine degree of match of fuzzy input with rule
antecedent and assign this to the rule conclusion
‣ Antecedent is the intersection or union of fuzzy inputs
‣ It is also called the degree of truth
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Fuzzy Rule-Base
Example
‣ Assume two fuzzy linguistic variables Speed and Position
‣ Speed (fast, 0.65 and Medium , 0.35)
‣ Position (centered, 0.4 and right , 0.6)
‣ Find the membership for the conclusion of the rules:
If Speed = Fast and Position = Centered
then Change in speed = Faster
If Speed = Fast and Position = Right
then Change in speed = Zero
If Speed = Medium and Position = Centered
then Change in speed = Faster
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Fuzzy Rule-Base
Example
‣ If Speed = Fast (0.65) and Position = Centered (0.4)
then Change in speed = Faster (0.4)
‣ If Speed = Fast (0.65) and Position = Right (0.6)
then Change in speed = Zero (0.6)
‣ If Speed = Medium (0.35) and Position = Centered (0.4)
then Change in speed = Faster (0.35)
‣ Apply “Union” or “Intersection” according to “And or “Or”
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Fuzzy Rule-Base
Example
‣ If rules have the same conclusion with different degree of
truth, then take the maximum for the degree of truth for the
conclusions (union of results)
‣ Apply this for the previous example yields:
If Speed = Fast (0.65) and Position = Centered (0.4)
then Change in speed = Faster (0.4)
If Speed = Medium (0.35) and Position = Centered (0.4)
then Change in speed = Faster (0.35)
‣ Change in Speed: Faster = max (0.4, 0.35) = 0.4
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Fuzzy Inference
Combining and Decomposition of Fuzzy Sets
‣ If x is A then y is B
Fact
rule
x is A
If x is A then y is B
Consequence y is B
‣ Given an input of A’ fuzzy set or crisp value A’
Fact x is A’
rule
If x is A then y is B
Consequence y is B’
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Fuzzy Inference
Single Rule with Single Antecedent
‣ µ B’ (y) = U [µ A’ (x) ^µ A (x) ] ^ µ B (y)
= ω ^ µ B (y)
ω is called the rule firing strength
B
A A’
B’
ω
y
x
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Fuzzy Inference
Single Rule with Single Antecedent
‣ Assume crisp input (fuzzification)
‣ µ B’ (y) = ω ^ µ B (y)
B
A
B’
ω
y
A’
x
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Fuzzy Inference
Single Rule with Multiple Antecedent
‣ Rule: If x is A and y is B then z is C
Fact: x is A’ and y is B’
Conclusion: z is C’
‣ µ C’ (z) = {U [µ A’ (x)^ µ A (x)]} ^ {U [ µ B’ (y)^ µ B (y) ]}^µ C (z)
= (ω1 ^ ω2) ^ µ C (z)
(ω1 ^ ω2) is called the rule firing strength
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Fuzzy Inference
Single Rule with Multiple Antecedent
A A’
B B’
C
ω1
ω2
x
y
C’
z
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Fuzzy Inference
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Fuzzy Inference
Multiple Rules with Multiple Antecedent
‣ Rule:
If x is A1 and y is B1 then z is C1
If x is A2 and y is B2 then z is C2
Fact: x is A’ and y is B’
Conclusion: z is C’
‣ C’ = [(A’ x B’) o R1] U [(A’ x B’) o R2]
= C1’ U C2’
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Fuzzy Inference
Multiple Rules with Multiple Antecedent
A1 A1’
B1 B1’
ω2
ω1
x
y
A2 A2’
B2 B2’
ω3
ω4
x
y
C1
C1’
C2
C2’
z
z
C2’
C1’
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Fuzzy Inference
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Fuzzy Inference
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Fuzzy Inference
Combining and Decomposition of Fuzzy Sets
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Fuzzy Inference
Combining and Decomposition of Fuzzy Sets
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Fuzzy Inference
User
input
Value of
variable 1
WF = 90
Value of
variable 2
SE = 3
Value of
variable 3
UP = 6
Value of
output
R =
Rule 1
.
.
.
.
Rule 6
.
.
.
.
.
Rule 27
1.0
y
L
ω1=.2
0
0 50 100
Area1
y
y
y
1.0
M
1.0
H
1.0
I
ω2=.67
Area6
ω3=.33
0 0 0
x
1 8
5 8 500 1500
Area27
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Fuzzy Inference
Overall Membership Function
0.7
0.6
0.5
Envelop enclosing all
areas (area1 to area27)
0.4
0.3
Area6
0.2
0.1
0.0
0 500 1000 1500 2000 2500 3000 3500 4000 4500
Overall output =
Center of Area=2100
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Fuzzy Inference
Defuzzification
‣ Defuzzification represents the way a crisp value is extracted
from a fuzzy set as a representative value
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Fuzzy Inference
Defuzzification
‣ Z COA = ∫µ A (z)Z dz / ∫µ A (z) dz
µ A (z) is the aggregated membership function
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Fuzzy Inference
Defuzzification
‣ Mean of maximum
Z mom = (Z 1 + Z 2 ) /2
‣ Smallest of maximum
Z som is the minimum of the
maximums of Z = Z 1
‣ Largest of maximum
Z Lom = Z 2
Z 1
Z 2
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Fuzzy Controller
Example
H
Fuzzy logic
controller
T
Cooling rate
C
Air
conditioner
Temperature Humidity
sensor sensor
Room

Fuzzy Controller
Example
Low (LW)
High (HG)
Low (LW)
High (HG)
Temperature
T
Humidity
H
Negative high
Negative low
Positive low
Positive high
(NH)
(NL)
(PL)
(PH)
C
Cooling Rate
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Fuzzy Controller
Example: If-then rules
‣ Rule 1: If T is HG and H is HG then C is PH
‣ Rule 2: If T is HG and H is LW then C is PL
‣ Rule 3: If T is LW and H is HG then C is NL
‣ Rule 4: If T is LW and H is LW then C is NH
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Fuzzy Controller
Example
High
High
Positive
high
T
H
C
High
Low
Positive
low
T
H
C
Low
High
Negative
low
Low
T
Low
H
Negative
high
C
T
H
C
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Fuzzy Controller
Example: Overall Output Membership Function
C
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