Models for Inexact Reasoning

Fuzzy Logic – Lesson 8Fuzzy Controllers

Master 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 operators (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. Measurements 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 bv c c∈ − ∈ −

∈ −&

• We have identified seven linguistic states for each of the three variables

NL – Negative Large PL – Positive Large

NM – Negative Medium PM – Positive Medium

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

[ ]: ,ef 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 tunedOther 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

• Most 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:

( ) ( ) ( )( )( )0 0 0 0

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

0 0

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 Ce e A B v C

e e A B v Ce e f x f y

××

××

=====

&&

KKKKKKKKKKKKKKKKKK&

&

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

– Mean 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 (Matsushita)

– Washing machines (AEG)

– CCD Autofocus Camera (Canon)

– “Intelligent” Dishwasher (Maytag)