M odels for Inexact Reasoning Fuzzy Logic Lesson 8 Fuzzy - - PowerPoint PPT Presentation

M odels for Inexact Reasoning Fuzzy Logic – Lesson 8 Fuzzy Controllers

M aster in Computational Logic Department of Artificial Intelligence

Fuzzy Controllers

• Fuzzy Controllers are special expert systems

– KB expressed in terms of fuzzy inference rules – Use of an inference engine

• FC can use knowledge elicited from human
• perators (conversely to classical control)

– This is crucial in many control problems due to

• Difficult or impossible to build precise mathematical models
• Acquired models are difficult or expensive to use

– Nonlinearities, – Time-varying processes – Difficulty to obtain precise measurements from sensors, etc.

Fuzzy Controllers

• Knowledge of an experienced human can

substitute precise, mathematical models – Often this knowledge is difficult or impossible to

express in precise terms

– However, an imprecise linguistic description can

be easily articulated by the operator

• Example:

IF the temperature is very high AND the pressure is slightly low THEN the heat change should be slightly negative

Typical Architecture of a FC

Operative Cycle of a FC

• A FC operates by repeating the cycle:

1. M easurements are taken of all relevant variables 2. Fuzzification: conversion of the measurements into fuzzy sets to express their uncertainties 3. Fuzzified measurements are used by the inference engine to evaluate the control rules

• The result of this evaluation is one or more fuzzy sets

defined on the universe of possible actions

4. Defuzzification: the outcome is converted into a single crisp value (best representative of the fuzzy set)

Design of Fuzzy Controllers

• Step 1: Identification of I/ O variables

– Range their values – Select meaningful linguistic states for each

variable

• Use fuzzy numbers to carry out this task
• Examples:

– Approximately zero – Positive small – Negative small, etc.

Step 1: Example

• Let us suppose we have identified the

following variables: [ ] [ ] [ ]

Input: , , , Output: , e a a e b b v c c ∈ − ∈ − ∈ − &

• We have identified seven linguistic states for

each of the three variables

NL – Negative Large PL – Positive Large NM – Negative M edium PM – Positive M edium NS – Negative Small PS – Positive Small AZ – Approximately Zero

Step 1: Example

• Possible fuzzy quantization of the range [-a, a]

by triangular-shaped fuzzy numbers

Step 2: Fuzzification

• In this step a fuzzification function is introduced

for each input variable

[ ]

: ,

e

f a a − → ℜ

• ℜ denotes the set of all fuzzy numbers
• fe(x0) is a fuzzy number chosen by fe as a fuzzy

approximation to measurement e = x0

• In some applications measurements are not

fuzzified

• They are used directly as facts

Fuzzification: Example

ε is a parameter to be tuned Other shapes can be used instead of triangular functions

Step 3: Design of the KB

• In this phase, control knowledge is formu-

lated in terms of fuzzy inference rules

• Two possibilites:

– Elicit the rules from experienced human operators – Learn the rules from empirical data by suitable

learning methods

• M ost often using artificial neural networks
• Canonical form of rules:

If and , then , , are fuzzy numbers chosen from the linguistic states e A e B v C A B C = = = &

Example: Design of the KB

Step 4: Design of the Inference Engine

• Rules in the KB can be expressed as:

( ) ( ) ( )

( )

( )

If , is then is , where: , min , , ( )

A B A B e e

e e A B v C x y x y e e f x f y µ µ µ

×

× = = × & &

• Then, we obtain the following:

( ) ( )

1 1 1 2 2 2

Rule 1: If , is , then is Rule 2: If , is , then is Rule n: If , is , then is Fact: , is

n n n e e

e e A B v C e e A B v C e e A B v C e e f x f y × × × × ===== & & K K K K K K K K K K K K K K K K K K & & Conclusion: is v C ==========================

• Inferences: Interpolation method

Step 5: Defuzzification

• Purpose: convert each conclusion (fuzzy set)

into a single real number (crisp)

• The resulting number defines the action to be

taken by the fuzzy controller

• Different defuzzification methods

– Center of area (aka center of gravity) – Center of maxima – M ean of maxima

Example: Stabilization of an Inverted Pendulum

Knowledge Base (tuned)

Fuzzy Rule Base

Inference with Crisp Observations

Inference with Fuzzified Observations

Real World Applications

• Sendai Railway (Japan)

– Use of FC to control accelerating, braking and

stopping

• Industrial and Consumer Applications

– Vacuum cleaners (M atsushita) – Washing machines (AEG) – CCD Autofocus Camera (Canon) – “ Intelligent” Dishwasher (M aytag)