2_fuzzylog_sbic_en

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Introduction to Fuzzy LogicIntroduction to Fuzzy Logicand Fuzzy Controland Fuzzy Control

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Fuzzy logic and fuzzy control 2

Fuzzy Logic allows obtaining conclusions from vague,ambiguous or imprecise information

Linguistic variable

A variable whose values are words or phrases in a natural or syntheticlanguage

Knowledge is expressed through subjective concepts Tall, short, cold, hot, positive, zero, big,

Fuzzy rule: IF THEN

A rule in which the antecedent and the consequent are propositionscontaining linguistic variables

Fuzzy logic


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Rule based, linguistic, symbolic, qualitative,

The users view (interface)

Fuzzy control

Fuzzy control, users view

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Non linear, Quantitative, ...

The process view

Fuzzy control

Fuzzy control, process view


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The reality (the process to be controlled) is non linear

Linear systems are, in general, a good local approximation of anon linear system

Linear control is, in general, sufficient (PID, for example)

However, there is an increasing need of non linear control Increasing functionality/complexity

Fast production changes

Higher precision

Larger operating ranges

Non linear control

Larger operating range of aninverted pendulum

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Non linear process

Non linear dynamics

Non linear dynamics + constraints on the inputs, states, outputs

Non linear specifications

Time-optimal control

Constraints

Small signal vs large signal behaviour

Need of non linear control


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In the controller

Feedback path

Direct path

Non linearities in control

In the process model Basis for model based

controller design

Part of a model basedcontroller

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Difficult representation

Different approaches (parameterizations)

Look-up tables

Splines

Fuzzy systems

Neural networks

Wavelets Analytic functions

Radial basis functions

Logic and selection

Non linear mappings


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Introduced by Lotfi Zadeh in the early 1960s

Fuzzy sets theory / logic for dealing with non probabilisticuncertainties

Application domains

Control systems, supervision, maintenance Decision support systems

Data classification, pattern recognition, computer vision

Knowledge based systems

First application automatic control

Abe Mamdani, Queen Mary College, 1974

Steam turbine and boiler

Fuzzy logic

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Non linear connection

Manual control is typically non linear, e.g. on-off control

Fuzzy logic based control: motivation

Conceptual evolution of fuzzy logic based control


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Conventional or fuzzy

Model-based control

Stages for implementation of model based control

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Heuristic fuzzy control

Stages for implementation of heuristic based fuzzycontrol


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The set

Crisp sets

{ }AxxA = |:

The characteristic function

( )1,

0,

A

x Ax

otherwise

=

Set operations:

Intersection

Union

Complement

Subset

Definition of crisp sets andmembership values of their elements

Membershipvalue

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Fuzzy sets.Membership functions and membership value

Membership function and membership value

[ ]1,0: xA

( )( ){ }UxxxA A = |,

Fuzzy sets definition

Membershipvalue


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Typical membership functions

M

embershipvalue

Gaussian

Triangular/Trapezoidal/Rectangular

Membershipvalue

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Typical membership functions

Singleton

Membership function example: singleton

Mem

bershipvalue


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Support, nucleous and -cut

Characterization and properties of a fuzzy set

A(x)

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Fuzzy set operations

Intersection: AND

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

BA

Union: OR BA

( ) ( ) ( )( )xxx BABA = ,max

Complement: NOT 'A( ) ( )xx AA = 1'


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Intersection operators

Probabilistic AND (product operator):

( ) ( ) ( )xxx BABA =

Lukasiewicz AND (bounded difference):

( ) ( ) ( )( )1,0max += xxx BABA

Other T-Norm operators ...

[ ] [ ] [ ]1,01,01,0

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Union operators

Probabilistic OR:

Lukasiewicz OR (bounded sum):

( ) ( ) ( ) ( ) ( )xxxxx BABABA +=

Other T-Conorm operators ...

[ ] [ ] [ ]1,01,01,0

( ) ( ) ( )( )xxx BABA += ,1min


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Fuzzy logic

Generalization of ordinary boolean logic

Propositions have truth values between 0 and 1, the membershipvalues in the fuzzy sets

Truth values of simple propositions

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Fuzzy logic

AND, OR and NOT connect simple propositions into compoundpropositions

Fuzzy inference is expressed by:

IF THEN

1 1 2 2( is ) AND ( is ) OR ...

i ix A x A

1 1 2 2 1 1 2IF ( is ) AND ( is ) OR ... THEN ( is ) AND ( ...)

i i ix A x A u B u


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Fuzzy logic

A very large body of theory has been developed

Very little of this theory is needed to understand/use fuzzycontrol

Fuzzy set theory and its use in control

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Architecture of a fuzzy control system

A fuzzy system is consituted by:

Knowledge Base, or Rule Base

Inference mechanism

Interfaces

Fuzzification: numbers - > fuzzy sets

Defuzzification: fuzzy sets -> numbers


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Architecture of a fuzzy control system

Architecture of a fuzzy system with its input and output interfaces

Which of the subsystems most influence the controllerperformance?

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Inference systems

Mamdani inference system:

Prototype of a rule, being A and Bfuzzy sets:

IF is THEN isx A u B

Takagi-Sugeno inference system:

Prototype of a rule, being A a fuzzy set:

IF is THEN ( )x A u f x=

A fuzzy set, the f(x) function, is, in general, a linear

combination of the inputs:

nn xlxllxf +++= ...)( 110


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Steps of a Mamdani inference system

1. Input fuzzy set evaluation

i. Temperature is Cold with a truth value of 0.7

ii. Temperature is OK with a truth value of 0.3

iii. Temperature is Hot with a truth value of 0.0

Fuzzy set evaluation, depending on the input crisp values

Members

hipvalue

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2. Evaluate the firing level of each rule:

IF Tempis LowAND Pressureis AverageTHEN ...

Firing level of each rule Operators for the antecedent of a rule


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Depends of the definition of each operator ...

IF ... THEN uis Zero

3. Evaluate the fuzzy output of each rule:

Determining the consequent of a rule

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4. Fuzzy output aggregation

Fuzzy output aggregation, including all active rules, using the max operator


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Other choices: Centre of sums, Mean of maximum, ...

5. Defuzzification. Defuzzifier: Fuzzy set B-> number y

Common choice: Center of Gravity

( )

( )

=

U U

U U

duu

duuuu

=

=

=

ni iU

ni iUi

u

uuu

1

1

)(

)(

Deffuzification with Centre of Gravity COG, for continuous anddiscrete fuzzy sets.

(u

)

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Defuzzification methods: Centre of gravity (left) and Mean of maximum (right),alternative: first maximum

5. Defuzzification (cont.)


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Mamdani inference systems: internal vision

Architecture of a Mamdani controller

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Mamdani inference systems: external vision

Mamdani generic control surface (non linear)

Example: Mamdani controller for two input variables

ZX, Y


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Piecewise constant systems with extensive interpolation

Overlap of the input membership functions define the interpolationregions

Very little reason to use anything but singletons as outputmembership functions

Mamdani inference systems: look-up tables

Mamdani system: inputs, look-up table and control surface

ZX, YControl

(Z)

Control

(Z)

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Takagi-Sugeno inference systems

Rules for a Takagi-Sugeno inference system:

IF is THEN ( )x A u f x=

A fuzzy set, the f(x) function is, in general, a linear combinationof the inputs:

0 1 1( ) ... n nf x b b x b x= + + +

The steps of the inference system are:

1. Input fuzzy set evaluation

2. Calculation of the firing degree (weight) of each rule

3. Calculation of the output of each rule

4. Calculation of the output by weighted average


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Takagi-Sugeno inference systems

The global output of a Takagi-Sugeno inference system is:

Takagi-Sugeno controller: rules firing level (weights) and rule output

=

=

=

Ni i

Ni ii

w

uwu

1

1

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Takagi-Sugeno inference systems: internal vision

Architecture of a Takagi-Sugeno controller


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Takagi-Sugeno inference systems: external vision

Generally, the Takagi-Sugeno controller can be viewed as a set ofhyper surfaces with weighted connections

Control surface (non linear) of aTakagi-Sugeno controller

ZX, Y

R1: IF Xis NAND Yis NTHEN

u1=b01+b11X+b12Y

R2: IF Xis NAND Yis PTHEN

u2=b02+b21X+b22Y

R3: IF Xis PAND Yis NTHEN

u3=b03+b31X+b32Y

R4: IF Xis PAND Yis PTHEN

u4=b04+b41X+b42Y

Example:

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Takagi-Sugeno inference systems: look-up tables

Piecewise linear systems with interpolation

Operating regions interpolation regions

Gain-scheduling

Takagi-Sugeno system: inputs, look-up tables and control surface

ZX, Y


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Summary

Introduction to fuzzy logic and fuzzy logic based controllers

Linguistic variables e linguistic values

Linguistic rules

Membership functions

Fuzzy sets, operations with fuzzy sets

Fuzzification, Inference and Defuzzification

Fuzzy logic controller structures

Mamdani

Takagi-Sugeno

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References

D. Driankov, H. Hellendorn, M. Reinfrank, An Introduction toFuzzy Control. Springer-Verlag, Berlin, 1993

L. Reznik, Fuzzy Controllers. Newnes, 1997

K. M. Passino, S. Yurkovich, Fuzzy Control. Addison-Wesley,Menlo Park, 1998


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Appendix: Fuzzy relations

The classical relation between two universes, Uand Vis defined as:

1 1 1 1

2 1 1 1

a b c

R U V

= =

( ){ }, ,U V u v u U v V =

It combines any element of Uwith any element of Vin orderedpairs, and makes associations without restrictions between uand v

The strength of the ordered pairs of the elements of each universe ismeasured by the characteristic function, with values of 1 (complete relation)or 0 (no relation)

Example:

( )1,2U = ( ), ,V a b c= ( ) ( ) ( ) ( ) ( ) ( ){ }1, , 1, , 1, , 2, , 2, , 2,U V a b c a b c =

In a matrix representation:

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Fuzzy relations

A fuzzy Relation, R, maps elements of one universe into the otheruniverse through the cartesian product of the two universes

The relation strength between the pairs is measured by a membershipfunction:

( ) ( ) ( )

( ) ( ) ( )

min 0.2,0.3 min 0.2,0.5 min 0.2,1 0.2 0.2 0.2=

min 0.9,0.3 min 0.9,0.5 min 0.9,1 0.3 0.5 0.9R

=

[ ]: 0,1R U V

Example, using the min operator:

( ) ( ) ( ) ( )( ), , min ,R A B A Bu v u v u v = =

( )1 0.2,0.9A = ( )2 0.3,0.5,1A = [ ]1 20.2

0.3 0.5 10.9

R A A

= =


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Fuzzy relations

Ris a fuzzy relation between universes Uand V. Sis a fuzzyrelation between universes Vand W. The fuzzy relation T, relatingthe elements of Ucontained in R, with the elements of Wcontainedin Sis given by the composition:

( ) ( )( ){ }max min , , ,R Sv V

T R S u v v w

= =

0.6 0.8;

0.7 0.9R

=

0.3 0.1;

0.2 0.8S

=

Example, using min-max, the relation T=RoS:

0.3 0.8

0.3 0.8R S

=

The element T(1,1) is obtained by: max{min(0.6, 0.3), (0.8, 0.2)}

R S S R It is verified that:

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