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### Fuzzy Logic

11/6/2001

Agenda

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

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

Applications

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

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)).

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

Rule Base

- 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}

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}

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

RulesAir Conditioning Controller Example:

- IF Cold then Stop
- If Cool then Slow
- If OK then Medium
- If Warm then Fast
- IF Hot then Blast

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