UNIT IV

1UNIT-4

FUZZY LOGIC CONTROL

www.Vidyarthiplus.com

Fuzzy Membership Functions

Fuzzy Operations

Fuzzy Union

Fuzzy Intersection

Fuzzy Complement

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Some info for LAB

Work on an m-file (open m-file for each task,

write your programme, save the file (e.g.,

lab2task1), then execute the file. Now, this file

has become a function in MATLAB). (see the first

weeks slides - Week 1).

Use help (e.g., help newfis) if you

dont know how to use the function. It gives you

information about how to use the function and

what parameters it requires

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Fuzzy Membership Functions

One of the key issues in all fuzzy sets is how to determine fuzzy membership functions

The membership function fully defines the fuzzy set

A membership function provides a measure of the degree of similarity of an element to a fuzzy set

Membership functions can take any form, but there are some common examples that appear in real applications

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Membership functions can

either be chosen by the user arbitrarily, based on the users experience (MF chosen by two users could be different depending upon their experiences, perspectives, etc.)

Or be designed using machine learning methods (e.g., artificial neural networks, genetic algorithms, etc.)

There are different shapes of membership functions; triangular, trapezoidal, piecewise-linear, Gaussian, bell-shaped, etc.

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Triangular membership function

a, b and c represent the x coordinates of the three vertices

of A(x) in a fuzzy set A (a: lower boundary and c: upper

boundary where membership degree is zero, b: the centre

where membership degree is 1)

=

cxifcxbif

bcxc

bxaifabax

axif

xA

0

0

)(

a b cx

A(x)1

0

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Gaussian membership function

c: centre s: width m: fuzzification factor (e.g., m=2)

A(x)

=

m

As

cxmscx

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exp),,,(

0 1 2 3 4 5 6 7 8 9 100

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

x

c=5

s=2

m=2

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c=5

s=0.5

m=20 1 2 3 4 5 6 7 8 9 10

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 1 2 3 4 5 6 7 8 9 100

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

c=5

s=5

m=2

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c=5

s=2

m=0.2

c=5

s=5

m=5

0 1 2 3 4 5 6 7 8 9 100.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 1 2 3 4 5 6 7 8 9 100

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

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Fuzzy Operations

(Fuzzy Union, Intersection, and Complement)

Fuzzy logic begins by barrowing notions from crisp logic, just as fuzzy set theory borrows from crisp set theory. As in our extension of crisp set theory to fuzzy set theory, our extension of crisp logic to fuzzy logic is made by replacing membership functions of crisp logic with fuzzy membership functions [J.M. Mendel, Uncertain Rule-Based Fuzzy Logic Systems, 2001]

In Fuzzy Logic, intersection, union and complement are defined in terms of their membership functions

This section concentrates on providing enough of a theoretical base for you to be able to implement computer systems that use fuzzy logic

Fuzzy intersection and union correspond to AND and OR, respectively, in classic/crisp/Boolean logic

These two operators will become important later as they are the building blocks for us to be able to compute with fuzzy if-then rules

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Logical AND () Logical OR (U)Truth Table

A B A B

0 0 0

0 1 0

1 0 0

1 1 1

Truth TableA B A UU B0 0 00 1 11 0 11 1 1

Crisp UnionCrisp Intersection

AAB

B

Classic/Crisp/Boolean Logic

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Fuzzy Union

The union (OR) is calculated using t-conorms t-conorm operator is a function s(.,.) Its features are

s(1,1) = 1, s(a,0) = s(0,a) = a (boundary) s(a,b) s(c,d) if a c and b d (monotonicity) s(a,b) = s(b,a) (commutativity) s(a,s(b,c)) = s(s(a,b),c) (associativity)

The most commonly used method for fuzzy union is to take the maximum. That is, given two fuzzy sets A and B with membership functions A(x) and B(x)

))(),(max()( xxx BAAUB =12


Fuzzy Intersection

The intersection (AND) is calculated using t-norms. t-norm operator is a function t(.,.) Its features

t(0,0) = 0, t(a,1) = t(1,a) = a (boundary) t(a,b) t(c,d) if a c and b d (monotonicity) t(a,b) = t(b,a) (commutativity) t(a, t(b,c)) = t(t(a,b),c) (associativity)

The most commonly adopted t-norm is the minimum. That is, given two fuzzy sets A and B with membership functions A(x) and B(x)

))(),(min()( xxx BABA =

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Fuzzy Complement

To be able to develop fuzzy systems we also have to deal with NOT or complement.

This is the same in fuzzy logic as for Boolean logic

For a fuzzy set A, A denotes the fuzzy complement of A

Membership function for fuzzy complement is

)(1)( xx AA

=

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4.2 Knowledge Base

1. The database provides necessary

definitions that are used to define

linguistic variables and fuzzy data

manipulation in the FLC.

2. The rule base characterizes the

control goals and control policy by

means of a set of linguistic control

rules.


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5.2 Knowledge Base

Fig. 4(a) Membership functions for Error

- 2 - 1 .5 - 1 - 0 .5 0 0 .5 1 1 .5 2

0

0 .2

0 .4

0 .6

0 .8

1

E R R O R

D

e

g

r

e

e

o

f

m

e

m

b

e

r

s

h

i

p

N L N M N S Z E P S P M P L


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5.2 Knowledge Base Cont.

Fig. 4(b) Membership functions for Change in Error

- 0 . 5 - 0 . 4 - 0 . 3 - 0 . 2 - 0 . 1 0 0 . 1 0 . 2 0 . 3 0 . 4 0 . 5

0

0 . 2

0 . 4

0 . 6

0 . 8

1

C H A N G E I N E R R O R

D

e

g

r

e

e

o

f

m

e

m

b

e

r

s

h

i

p



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5.2 Knowledge Base Cont.

Fig. 4(c) Membership functions for Change in Output

- 0 .0 3 - 0 .0 2 -0 .0 1 0 0 .0 1 0 .0 2 0 .0 3

0

0 .2

0 .4

0 .6

0 .8

1

C H A N G E IN O U T P U T

D

e

g

r

e

e

o

f

m

e

m

b

e

r

s

h

i

p



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4.3 Decision Making Logic

Based on the fuzzified inputs and rule

base, fuzzy output is determined by

applying the rules of Boolean Algebra

(Union and Intersection).


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Genetic Algorithms (GA) OVERVIEW

A class of probabilistic optimization algorithms

Inspired by the biological evolution process

Uses concepts of Natural Selection and Genetic Inheritance (Darwin 1859)

Originally developed by John Holland (1975)

Particularly well suited for hard problems where little is known about the underlying search space

Widely-used in business, science and engineering


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Classes of Search TechniquesSearch Techniqes

Calculus Base

TechniqesGuided random search

techniqes

Enumerative

Techniqes

BFSDFS Dynamic

Programming

Tabu Search Hill Climbing Simulated

Anealing

Evolutionary

Algorithms

Genetic

Programming

Genetic

Algorithms

Fibonacci Sort


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A genetic algorithm maintains a population of

candidate solutions for the problem at hand,

and makes it evolve by

iteratively applying a set of stochastic

operators.


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Stochastic operators

Selection replicates the most successful solutions

found in a population at a rate proportional to

their relative quality

Recombination (Crossover) decomposes two

distinct solutions and then randomly mixes their

parts to form novel solutions

Mutation randomly perturbs a candidate solution


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Simple Genetic Algorithm{

initialize population;evaluate population;while TerminationCriteriaNotSatisfied{

select parents for reproduction;perform recombination and mutation;evaluate population;

}}


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The Evolutionary Cycle

selection

population evaluation

modification

discard

deleted

members

parents

modified

offspring

evaluated offspring

initiate &

evaluate


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GA Cycle


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Crossover


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Mutation


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