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 Inference Mechanism Defuzzification
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 Air Conditioning Controller Example: IF Cold then Stop
default:
The truth of any statement is a matter of degree
Membership function is a curve of the degree of truth of a given input value
Rules
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