Fuzzy Logic

1
Fuzzy Logic

• 11/6/2001

2
Agenda

• General Definition
• Applications
• Formal Definitions
• Operations
• Rules
• Fuzzy Air Conditioner
• Controller Structure

3
General Definition

• Fuzzy Logic - 1965 Lotfi Zadeh, Berkely
• superset of conventional (Boolean) logic that has
  been extended to handle the concept of partial
  truth
• central notion of fuzzy systems is that truth
  values (in fuzzy logic) or membership values (in
  fuzzy sets) are indicated by a value on the range
  0.0, 1.0, with 0.0 representing absolute
  Falseness and 1.0 representing absolute Truth.
• deals with real world vagueness

4
Applications

• ABS Brakes
• Expert Systems
• Control Units
• Bullet train between Tokyo and Osaka
• Video Cameras
• Automatic Transmissions

5
Formal Definitions

• Definition 1 Let X be some set of objects, with
  elements noted as x.
• X x.
• Definition 2 A fuzzy set A in X is characterized
  by a membership function mA(x) which maps each
  point in X onto the real interval 0.0, 1.0. As
  mA(x) approaches 1.0, the "grade of membership"
  of x in A increases.
• Definition 3 A is EMPTY iff for all x, mA(x)
  0.0.
• Definition 4 A B iff for all x mA(x) mB(x)
  or, mA mB.
• Definition 5 mA' 1 - mA.
• Definition 6 A is CONTAINED in B iff mA ? mB.
• Definition 7 C A UNION B, where mC(x)
  MAX(mA(x), mB(x)).
• Definition 8 C A INTERSECTION B where mC(x)
  MIN(mA(x), mB(x)).

6
Operations

A
B
A ? B A ? B
?A
7
Controller Structure

• Fuzzification
• Scales and maps input variables to fuzzy sets
• Inference Mechanism
• Approximate reasoning
• Deduces the control action
• Defuzzification
• Convert fuzzy output values to control signals

8
Rule Base

• Air Temperature
• Set cold 50, 0, 0
• Set cool 65, 55, 45
• Set just right 70, 65, 60
• Set warm 85, 75, 65
• Set hot ?, 90, 80

• Fan Speed
• Set stop 0, 0, 0
• Set slow 50, 30, 10
• Set medium 60, 50, 40
• Set fast 90, 70, 50
• Set blast ?, 100, 80

9
Rules
Membership function is a curve of the degree of
truth of a given input value
default The truth of any statement is a matter
of degree

• Air Conditioning Controller Example
• IF Cold then Stop
• If Cool then Slow
• If OK then Medium
• If Warm then Fast
• IF Hot then Blast

10
Fuzzy Air Conditioner
11
Mapping Inputs to Outputs