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Fuzzy Logic:

1 Fuzzy Logic


2 Introduction Application areas Fuzzy Control Subway trains Cement kilns Washing Machines Fridges

Fuzzy Sets:

3 Fuzzy Sets Extension of Classical Sets Not just a membership value of in the set and out the set, 1 and 0 but partial membership value, between 1 and 0

Example: Height:

4 Example: Height Tall people: say taller than or equal to 1.8m 1.8m , 2m, 3m etc member of this set 1.0 m, 1.5m or even 1.79999m not a member Real systems have measurement uncertainty so near the border lines, many misclassifications

Member Functions:

5 Member Functions Membership function better than listing membership values e.g. Tall(x) = {1 if x >= 1.9m , 0 if x <= 1.7m else ( x - 1.7 ) / 0.2 }

Example: Fuzzy Short:

6 Example: Fuzzy Short Short(x) = {0 if x >= 1.9m , 1 if x <= 1.7m else ( 1.9 - x ) / 0.2 }

Fuzzy Set Operators:

7 Fuzzy Set Operators Fuzzy Set: Union Intersection Complement Many possible definitions we introduce one possibility

Fuzzy Set Union:

8 Fuzzy Set Union Union ( f A (x) and f B (x) ) = max (f A (x) , f B (x) ) Union ( Tall(x) and Short(x) )

Fuzzy Set Intersection:

9 Fuzzy Set Intersection Intersection ( f A (x) and f B (x) ) = min (f A (x) , f B (x) ) Intersection ( Tall(x) and Short(x) )

Fuzzy Set Complement:

10 Fuzzy Set Complement Complement( f A (x) ) = 1 - f A (x) Not ( Tall(x) )

Fuzzy Logic Operators:

11 Fuzzy Logic Operators Fuzzy Logic: NOT (A) = 1 - A A AND B = min( A, B) A OR B = max( A, B)

Fuzzy Logic NOT:

12 Fuzzy Logic NOT

Fuzzy Logic AND:

13 Fuzzy Logic AND

Fuzzy Logic OR:

14 Fuzzy Logic OR

Fuzzy Controllers:

15 Fuzzy Controllers Used to control a physical system

Structure of a Fuzzy Controller:

16 Structure of a Fuzzy Controller


17 Fuzzification Conversion of real input to fuzzy set values e.g. Medium ( x ) = { 0 if x >= 1.90 or x < 1.70, (1.90 - x)/0.1 if x >= 1.80 and x < 1.90, (x- 1.70)/0.1 if x >= 1.70 and x < 1.80 }

Inference Engine:

18 Inference Engine Fuzzy rules based on fuzzy premises and fuzzy consequences e.g. If height is Short and weight is Light then feet are Small Short( height) AND Light(weight) => Small(feet)

Fuzzification & Inference Example:

19 Fuzzification & Inference Example If height is 1.7m and weight is 55kg what is the value of Size(feet)


20 Defuzzification Rule base has many rules so some of the output fuzzy sets will have membership value > 0 Defuzzify to get a real value from the fuzzy outputs One approach is to use a centre of gravity method

Defuzzification Example:

21 Defuzzification Example Imagine we have output fuzzy set values Small membership value = 0.5 Medium membership value = 0.25 Large membership value = 0.0 What is the deffuzzified value

Fuzzy Control Example:

22 Fuzzy Control Example

Input Fuzzy Sets:

23 Input Fuzzy Sets Angle:- -30 to 30 degrees

Output Fuzzy Sets:

24 Output Fuzzy Sets Car velocity:- -2.0 to 2.0 meters per second

Fuzzy Rules:

25 Fuzzy Rules If Angle is Zero then output ? If Angle is SP then output ? If Angle is SN then output ? If Angle is LP then output ? If Angle is LN then output ?

Fuzzy Rule Table:

26 Fuzzy Rule Table

Extended System:

27 Extended System Make use of additional information angular velocity:- -5.0 to 5.0 degrees/ second Gives better control

New Fuzzy Rules:

28 New Fuzzy Rules Make use of old Fuzzy rules for angular velocity Zero If Angle is Zero and Angular vel is Zero then output Zero velocity If Angle is SP and Angular vel is Zero then output SN velocity If Angle is SN and Angular vel is Zero then output SP velocity

Table format:

29 Table format

Complete Table:

30 Complete Table When angular velocity is opposite to the angle do nothing System can correct itself If Angle is SP and Angular velocity is SN then output ZE velocity etc


31 Example Inputs:10 degrees, -3.5 degrees/sec Fuzzified Values Inference Rules Output Fuzzy Sets Defuzzified Values

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