Fuzzy Inference Systems

9/4/2001

Fuzzy Rules and Fuzzy Reasoning


Fuzzy Inference System a.k.a.: - fuzzy rule based system - fuzzy expert system

Fuzzy Inference Systems (Chapter 4)

- fuzzy model - fuzzy associative memory - fuzzy logic controller

Kai Goebel, Bill Cheetham GE Corporate Research & Development

- fuzzy system

[email protected] [email protected]

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The Big Picture

Outline

Inference systems consist of three components:

Introduction/Motivation Mamdani Model

• rule base • data base (defines membership functions)

Tsukamoto Model

• reasoning mechanism (aggregation)

et cetera

TSK Model

• defuzzification

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Motivation

Fuzzy Inference System

Fuzzy Reasoning left us with something similar to: x is A1

C’ z is C’

w1

y is B1

Z

Input x

Real world:

x is A2 ... x is Ar

Sensor measurements are crisp (albeit imprecise) Output to controller has to be crisp

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w2

wr

y is B2

Aggregator

Defuzzify

y

y is Br

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Design of IS

Mamdani Model

Consistency: no rules with the same antecedents but different consequents

-Coherent modeling of inference system -Appropriate defuzzification strategy

Completeness: for any x ∈ X there is at least one rule j s.t. µ A

j ,i

≠0

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Defuzzification Requirements

Defuzzification: COA Center of Area

• Intuition:

+ intuitive C’

A crisp value should represent the fuzzy

+ smooth Z

set from an intuitive point of view (e.g.,

- comp. burden

zCOA

max. membership grade) • Computational Burden: simple (real-time constraints) • Continuity: small changes in fuzzy sets

zCOA =

should not result in large changes of z

∫µ

A

( z )zdz

z

∫µ

A

( z )dz

z

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Defuzzification: BOA

Defuzzification: MOM

Bisector of area:

Mean of maximum:

C’

C’

Z

Z

zBOA z BOA

∫µ

α

A

(z )dz =

zMOM

β

∫µ

A

(z )dz

z MOM =

z BOA

α = min {z z ∈ Z } 11

∫ zdz

Z’

∫ dz

Z’

β = max{z z ∈ Z }

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{

}

Z ’ = z µ A ( z ) = max µ A ( z )

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SISO: max-min composition centroid defuzzification

Center Average Defuzzifier

If X is small then Y is small If X is medium then Y is medium If X is large then Y is large

Approximation of COA Defuzzifier by average of center of areas of fuzzy sets s.t. n

z CA =

∑z w j =1 n

* j

∑w j =1

j

j

where z*j center of area of jth fuzzy set B wj height of jth fuzzy set B 13

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MISO: max-min composition centroid defuzzification

Sugeno Model (TSK) Combines fuzzy sets in antecedents with crisp function in output:

If X is small and Y is small then Z is negative large If X is small and Y is large the Z is negative small

IF x is A AND y is B THEN z=f(x,y)

If X is large and Y is small the Z is positive small

(Does not follow compositional rule of inference) IF x is small THEN

If X is large and Y is large then Z is positive large

Y=4 IF X is medium THEN Y=-0.5X+4 IF X is large THEN Y=X-1 15

16 (sug1_1.m)



Sugeno: MISO

Tsukamoto Model

IF X is small AND Y is small THEN z=-x+y+1 IF X is small AND Y is large THEN z=-y+3

- Consequent of rule is represented by monotonical MF

IF X is large and Y is small THEN z=-x+3

- crisp output is induced by firing strength - overall output: weighted average of rule output

IF X is large and Y is large THEN z=x+y+2

- BUT: not as transparent

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Tsukamoto Example

Summary

IF X is small THEN Y is C1 IF X is medium THEN Y is C2 IF X is large THEN Y is C3 (a) Antecedent MFs

(b) Consequent MFs

large Me m b e rs hip G ra de s

medium

1 0.8 0.6 0.4 0.2

Y

0 -10

C1

C2

- Sugeno functional relation of output

C3

0.4 0.2

10

0

12

12

10

10

8

8

6

6

5 10 Y (d) Overall Input-Output Curve

- Tsukamoto

4

2 0 -10

via appropriate method

0.6

0 -5 0 5 X (c) Each Rule’s Output

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1 0.8

Y

Me m b e rs hip G ra de s

small

Fuzzy Inference: - Mamdami Defuzzification derived from area

2 -5

0 X

5

10

0 -10

-5

0 X

5

10

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last slide

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