fuzzy logic basics vk

7/31/2019 Fuzzy Logic Basics VK

1/49

5diih ;^VXR 1PbXRb5diih ;^VXR 1PbXRb5diih ;^VXR 1PbXRb5diih ;^VXR 1PbXRb5diih ;^VXR 1PbXRb5diih ;^VXR 1PbXRb5diih ;^VXR 1PbXRb5diih ;^VXR 1PbXRb

Vojislav KECMAN, Department of Mechanical Engineering, The University of Auckland, NZ

Slides accompanying The MIT Press book:: Learning and Soft Computing


2/49

The slides shown here are developed around the basic notion of

Embedding Structured Human Knowledge into workable mathematical models

by Fuzzy Logic

as presented in the book

LEARNING AND SOFT COMPUTINGSupport Vector Machines, Neural Networks and Fuzzy Logic Models

Author: Vojislav KECMAN

The MIT Press, Cambridge, MA, 2001

ISBN 0-262-11255-8

608 pp., 268 illustrations, 47 examples, 155 problems

They are intended to support both the instructors in the development and deliveryof course content and the learners in acquiring the ideas and techniques

presented in the book in a more pleasant way than just reading.


3/49

What is Fuzzy Logic ?What is Fuzzy Logic ?

FL is a Tool for Embedding Human Structured

Knowledge (Experience, Expertise, Heuristic)

Why is it Fuzzy ?Why is it Fuzzy ? Because - human knowledge is fuzzy : expressed in Fuzzy

Linguistic Terms - Young, Old, Big, Cheap are FUZZY words

Temperature is expressed as Cold, Warm or Hot. Noquantitative meaning.


4/49

5diih ;^VXR \Ph QT eXTfTS Pb P5diih ;^VXR \Ph QT eXTfTS Pb P5diih ;^VXR \Ph QT eXTfTS Pb P5diih ;^VXR \Ph QT eXTfTS Pb PQaXSVT êTa cWT TgRTbbXeT[h fXSTQaXSVT êTa cWT TgRTbbXeT[h fXSTQaXSVT êTa cWT TgRTbbXeT[h fXSTQaXSVT êTa cWT TgRTbbXeT[h fXST

VP_ QTcfTT] cWT _aTRXbX^] ÛVP_ QTcfTT] cWT _aTRXbX^] ÛVP_ QTcfTT] cWT _aTRXbX^] ÛVP_ QTcfTT] cWT _aTRXbX^] Û

R[PbbXRP[ RaXb_ [^VXR P]S cWTR[PbbXRP[ RaXb_ [^VXR P]S cWTR[PbbXRP[ RaXb_ [^VXR P]S cWTR[PbbXRP[ RaXb_ [^VXR P]S cWT

X\_aTRXbX^] Û Q^cW cWT aTP[ fâ[SX\_aTRXbX^] Û Q^cW cWT aTP[ fâ[SX\_aTRXbX^] Û Q^cW cWT aTP[ fâ[SX\_aTRXbX^] Û Q^cW cWT aTP[ fâ[S

P]S Xcb Wd\P] X]cTa_aTcPcX^]P]S Xcb Wd\P] X]cTa_aTcPcX^]P]S Xcb Wd\P] X]cTa_aTcPcX^]P]S Xcb Wd\P] X]cTa_aTcPcX^]?PaP_WaPbX]V ; IPSTW?PaP_WaPbX]V ; IPSTW?PaP_WaPbX]V ; IPSTW?PaP_WaPbX]V ; IPSTW


5/49

FL attempts to model the way of

reasoning that goes in the human brain. Almost all of human experience is stored

in the form of the IF - THEN rules. Human reasoning is pervasively

approximate, non-quantitative, linguistic,and dispositional (meaning, usuallyqualified).


6/49

FL attempts to model the way of

reasoning that goes in the human brain. Almost all of human experience is stored

in the form of the IF - THEN rules. Human reasoning is pervasively

approximate, non-quantitative, linguistic,and dispositional (meaning, usuallyqualified).

Why is that way ?


7/49

The World is Not Binary!Gradual Transitions & Ambiguities at the

Boundaries

Good, Day, Young, Healthy,

YES,

True, Happy, Tall, 0

Bad, Night, Old, Ill

NO

False, Sad, Short, 1


8/49

Criteria: When and Why to Apply FL


9/49


Human (Structured)Knowledge Available


10/49



Mathematical Model

Unknown orImpossible to Obtain


11/49



Mathematical Model


Process SubstantiallyNonlinear

Lack of Precise

Sensor Informations


12/49



Mathematical Model


Process SubstantiallyNonlinear

Lack of Precise

Sensor Informations

At the HigherLevels of

HierarchicalControl Systems

In GenericDecision MakingProcess


13/49

How to Transfer HumanKnowledge Into the Model?

Knowledge must be structured !

Possible shortcomings:

Knowledge is very subjective category Experts bounce between some extreme poles:

Have problems with structuring the knowledge, or

Too aware in his/hers expertise, or Tend to hide knowledge, or ...

Solution: Find a Good Expert! There are always

some around.


14/49

10B82B10B82B10B82B10B82B

01>DC01>DC01>DC01>DC 2A8B?2A8B?2A8B?2A8B?

0=30=30=30=3 5DIIH5DIIH5DIIH5DIIH

B4CBB4CBB4CBB4CB


15/49

Fuzzy SetsFuzzy SetsCrisp SetsCrisp Sets

Venn Diagrams


16/49

11

0 0

- membership degree, possibility distribution, grade of belonging

Fuzzy SetsFuzzy SetsCrisp SetsCrisp Sets


17/49

Modeling or Approximating A Function:Modeling or Approximating A Function:

Curve or Surface FittingCurve or Surface Fitting

More Different Names in Different Disciplines:

Regression (L or NL), Estimation, Identification, Filtering


18/49

Standard Mathematical Procedure of Curve FittingResults in a More or Less Acceptable Solution

Modeling A FunctionModeling A Function


19/49

Curve Fitting by Using Fuzzy Rules (Patches)

When There Are More Inputs We Try to Approximate

A Surface (2 Inputs) or Hyper-Surface (3 or More Inputs)


Small Number of Rules - Big Patches or Rough Approximation


20/49

Fuzzy Patches


More Rules - More Smaller Patches and Better Approximation

Origin of the patches and how do they work?


21/49

x

y

Consider modeling two different functions by fuzzy rules (books Fig 1.3.)

Less rules leads to the approximation accuracy decrease. An increase in a number ofrules increases the precision at the cost of a computation time needed to process morerules. This is the most classical soft computing dilemma that trades off between the

imprecision and uncertainty on one hand and low solution cost, tractability androbustness on the other. The appropriate rules for functions in Fig 1.3 are:

Left graph Right graph

IF xis lowTHEN yis high. IF xis lowTHEN yis high.IF xis mediumTHEN yis low. IF xis mediumTHEN yis medium.IF xis bigTHEN yis high. IF xis bigTHEN yis low.

These rules define three large rectangular patches that cover our functions. They are

shown in the next slide (Fig 1.4 in the book) together with two possible approximators foreach function.

x

y


22/49

x

y y

x

Modeling two different functions by fuzzy rules (books Fig 1.4.)

Figure 1.4 Two different functions (solid lines in both graphs) covered by three patches producedby IF-THEN rules and modeled by two possible approximators(dashedand dotted curves).

COMENTS: Note that humans do not (or only rarely) think in terms of nonlinear functions.

We do not try to draw these functions in our mind. We neither try to see them as

geometrical artifacts. In general, we do not process geometrical figures, curves, surfaces orhypersurfaces while performing some tasks or expressing our knowledge. Even more, ourexpertise or understanding of some functional dependencies is very often not a structuredpiece of knowledge at all. We typically perform very complex tasks without being able toexpress how are we executing them.

The curious reader should try, for example, to explain to his/hers colleague, in theform of IF-THEN rules, how s/he is riding the bike, recognizing numerals or surfing.


23/49

Fuzzy Control of the Distance Between Two CarsGeneral approach in all fuzzy modeling is always the same.

i) Define the variables of relevance, interest or importance:

In engineering we use to call them Input and Output Variables,

ii) Define the subsets intervals:

Small - Medium, or Negative - Positive, or

Left - Right (labels are variables dependant)

iii) Choose the Shapes and the Positions of Fuzzy Subsets i.e.,

Membership Functions i.e., Attributes

iv) Set the Rule Basis i.e., IF - THEN Rules:

v) Perform calculations and (if needed) tune (learn, adjust, adapt) the

positions and the shapes of both the input and the output attributes o

the model.

F C l f h


24/49

D

v

B

Fuzzy Control of the

Distance Between Two CarsINPUTS: D = DISTANCE, v = SPEED

OUTPUT:B = BRAKING FORCE

Lets analyze the rules for some given distance

D and for different velocities v i.e., B = f(v)


25/49

IFtheVelocity is Low, THEN the Braking Force is Small.

Low Medium High

1

10 120 (km/h)

Small Medium Big

1

0 100 (%)

Velocity Braking Force

IF theVelocity is Medium, THEN the Braking Force is Medium.

IF theVelocity is High, THEN the Braking Force is Big.


26/49

Velocity

Braking Force

The Fuzzy PatchesThe Fuzzy Patches

Small MediumHigh10 120

Big

Medium

Low

1100

0(%)

1


27/49

Velocity

Braking Force


Small MediumHigh10 120

Big

Medium

Low

1100

0(%)

1


28/49

Velocity

Braking Force


Note the overlapping of

fuzzy subsets that leads

to a smooth approximation

of the function between the

Velocity & Braking Force

Small MediumHigh

10 120

Big

Medium

Low

1100

0(%)

1


29/49

Velocity

Braking Force

The Fuzzy Patches Define the FunctionThe Fuzzy Patches Define the Function

Small MediumHigh

10 120

Big

Medium

Low

1100

0(%)

1

Three possible

dependencies

between the v and B.Each of us drives

differently, and we all

drive in different way

each time.

)81&7,21$/ '(3(1'(1&( 2) 7+( 9$5,$%/(6


30/49

)81&7,21$/ '(3(1'(1&( 2) 7+( 9$5,$%/(6

SURFACE O F KN O W LEDGE

020

40

60

80

100

050

100

150

20010

2030

40

50

60

70

80

90

DistanceSpeed

Braking

Force

V.K.Fuzzy Control of the Distance Between Two Cars again

now, however,2 INPUTS: D and v, and 1 OUTPUT B visualization still possible.

For more inputs everything remains same but no longer visualization.


31/49

Second ExampleSecond Example

FuzzyFuzzy

RoomRoom

TemperatureTemperature

ControlControl


32/49

Room TemperatureRoom Temperature Fan SpeedFan Speed

Cold Warm Hot

1

10 30 (oC)

Slow Medium Fast

1

0 100 (%)


33/49

Fan SpeedFan Speed

If Room Temperature is Cold then Fan Speed is Slow

Cold Warm Hot

1

10 30 (oC)

Slow Medium Fast

1

0 100 (%)

Room TemperatureRoom Temperature


34/49

Fan SpeedFan Speed


If Room Temperature is Warm then Fan Speed is Medium

Cold Warm Hot

1

10 30 (oC)

Slow Medium Fast

1

0 100 (%)



35/49

Fan SpeedFan Speed



If Room Temperature is Hot then Fan Speed is Fast

Cold Warm Hot

1

10 30 (oC)

Slow Medium Fast

1

0 100 (%)



36/49

RoomTemperature

The Fuzzy PatchesThe Fuzzy PatchesFan Speed

Cold Warm Hot10 30 (oC)

Fast

Medium

Slow

1100

0

(%)

1


37/49

RoomTemperature

The Fuzzy PatchesThe Fuzzy PatchesFan Speed


Fast

Medium

Slow

1100

0

(%)

1


38/49

RoomTemperature

Fan Speed


Note the overlappingof fuzzy subsets again - it leadsto smooth approximationof the function between the

Fan Speed & Temperature


Fast

Medium

Slow

1100

0

(%)

1


39/49

There must be some overlapping of the input

fuzzy subsets (membership or characteristicfunctions) if we want to obtain a smooth model

0 2 4 6 8 10 120

2

4

6

8

10

12

VS S M B VB

VB

B

M

S

VS

I N P U T

OU

T

P

U

T


40/49



0 2 4 6 8 10 120

2

4

6

8

10

12

VS S M B VB

VB

B

M

S

VS

I N P U T

OU

T

P

U

T

Because,

if there is no

overlapping

one obtains

the stepwisefunction as

shown next!


41/49



0 2 4 6 8 10 120

2

4

6

8

10

12

VS S M B VB

VB

B

M

S

VS

I N P U T

OU

T

P

U

T

Calculation of the output: Fuzzification, Inference and Defuzzifaction


42/49

Fan SpeedFan Speed

R1: If Room Temperature is Cold Then Fan Speed is Slow

R2: If Room Temperature is Warm Then Fan Speed is Medium

R3: If Room Temperature is Hot Then Fan Speed is Fast

After the fuzzy modeling is done there is an operation phase: Calculate the

Fan Speed when Room Temperature = 22 oC .

NOTE! 22 oC belongs to the Subsets Warm and Hot

Cold Warm Hot

1

10 30 (oC)

Slow Medium Fast

1

0 100 (%)22


Fuzzification and Inference


43/49

Fan SpeedFan Speed

If Room Temperature is Cold Then Fan Speed is Slow

If Room Temperature is Warm Then Fan Speed is Medium

If Room Temperature is Hot Then Fan Speed is Fast

Cold Warm Hot

1

10 30 (oC)

Slow Medium Fast

1

0 100 (%)22

0.6


S



44/49



If Room Temperature is Hot then Fan Speed isFast

Cold Warm Hot

1

10 30 (oC)22

0.6

0.3

Fan SpeedFan Speed

Slow Medium Fast

1

0 100 (%)


S

F S d



45/49



If Room Temperature is Hot then Fan Speed isFast

THE QUESTION NOW IS: WHAT IS THE OUTPUT VALUE?

Cold Warm Hot

1

10 30 (oC)22

0.6

0.3

Fan SpeedFan Speed

Slow Medium Fast

1

0 100 (%)


D f ifi iD f ifi ti


46/49

DefuzzificationDefuzzification

The Result of the Fuzzy Inference is a Fuzzy Subset Composed ofthe Slices of Fan Speed: Medium (green) and Fast (red)

How to Find a Crisp (for the real world application useful) Value?

One out of several different methods SHOWN ABOVE is:

the Centre of Area Formula to Obtain a Crisp Output

60

Fan Speed

1

0 100 (%)


47/49

Example given in program LEARNSC from theExample given in program LEARNSC from thebook:book:

i) Vehicle Turning Problemi) Vehicle Turning Problem

Generic Fuzzy Logic Controller

- Developed in Matlab

- User Friendly

- Multiple Inputs

Many Other Commercial Applications Possible

C fi i f h V hi l T i P bl


48/49

Configuration of the Vehicle Turning ProblemConfiguration of the Vehicle Turning Problem

= Angle of Car (INPUT 1)

d = Distance From Center Line (INPUT 2)

max = /4 :Upper Bound of Steering Angle (OUTPUT)

v = 10.0 m/s

Finish

Start

d

& 2 1 & / 8 6 , 2 1 6& 2 1 & / 8 6 , 2 1 6& 2 1 & / 8 6 , 2 1 6& 2 1 & / 8 6 , 2 1 6& 2 1 & / 8 6 , 2 1 6& 2 1 & / 8 6 , 2 1 6& 2 1 & / 8 6 , 2 1 6& 2 1 & / 8 6 , 2 1 6


49/49

& 2 1 & / 8 6 , 2 1 6& 2 1 & / 8 6 , 2 1 6& 2 1 & / 8 6 , 2 1 6& 2 1 & / 8 6 , 2 1 6& 2 1 & / 8 6 , 2 1 6& 2 1 & / 8 6 , 2 1 6& 2 1 & / 8 6 , 2 1 6& 2 1 & / 8 6 , 2 1 6

Fuzzy Logic Can Be Implemented Wherever There is StructuredHuman Knowledge, Expertise, Heuristics, Experience.

Fuzzy Logic is not needed whenever there is an analytical closed-

form model that, using a reasonable number of equations, can solve

the given problem in a reasonable time, at the reasonable costs and

with higher accuracy.

POSSIBLE PROBLEMS:

Finding good(accountable) expert,

Right choice of the variables,

Increasing the number ofinputs, as well as the number of fuzzy subsets perinput variable, the number of rules increases exponentially (curse of

dimensionality)

Good news is, that there are plenty of real life problems and situations that

can be solved with small number of rules only.

fuzzy logic basics vk

Documents