class notes 870

8/11/2019 Class Notes 870

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Fuzzy Logic and Control

A. Homaifar

Autonomous Control & Information Technology(ACIT) Center

Department of Electrical & Computer Engineering

North Carolina A&T State University


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OUTLINE

Fundamentals

Fuzzy Sets, Operators,

GMP and GMT

Fuzzy Engines

Hybrid Fuzzy-PID Controllers

Generalized Sugeno Controllers

Applications


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1. A B(u) = max { A (u), B(u) }

2. A B(u) = min { A (u), B(u) }

3. A (u) = 1.0 - A (u)

Set Theoretic Operations


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Union Of Fuzzy Sets

0

1

A

X

AB(X)

B

Union of fuzzy sets A & B


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Intersection Of Fuzzy Sets

0

1

A

X

AB(X)

B

Intersection of fuzzy sets A & B


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Complement Of Fuzzy Set

0

1

A

X

A

A(X)

A(X) = 1 - A(X)


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Law of Contradiction

0

1

A

X

A

Fuzzy AA


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Law Of Excluded Middle

0

1

A

X

A

Fuzzy A A X


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Definition oft -norms

t-normsare two-valued functions from [0,1]x[0,1]

that satisfy the following conditions:

vity)(associati(x))(x)),(x),t(t((x))),(x)t((x),t(4.

vity)(commutati(x))(x),t((x))(x),t(3.

ity)(monotonic(x)(x)and(x)(x)if

(x))(x),t((x))(x),t(2.

Xx(x),(x))t(1,(x),1)t(0;t(0,0)1.

C~

B~

A~

C~

B~

A~

A~

B~

B~

A~

D~

B~

C~

A~

D~

C~

B~

A~

A~

A~

A~


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Definition oft -conormst-conormsor s-normsare associative,

commutative, and monotonic two-placed functions

sthat map from [0,1]x[0,1] into [0,1] that satisfy the

following conditions:

vity)(associati(x))(x)),(x),s(s((x))),(x)s((x),s(4.

vity)(commutati(x))(x),s((x))(x),s(3.

ity)(monotonic(x)(x)and(x)(x)if

(x))(x),s((x))(x),s(2.

Xx(x),(x))s(0,(x),0)s(1;s(1,1)1.

C~

B~

A~

C~

B~

A~

A~

B~

B~

A~

D~B~C~A~

D~

C~

B~

A~

A~

A~

A~


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Relationship betweent -norms

andt -conorms

t-normsand t-conorms are related in a sense of

logical duality.

t-conorm

as a two-placed function s mapping from[0,1] x [0,1] in [0,1] such that the function t defined

as

(x))-1(x),-s(1-1(x))(x),t(B

~

A

~

B

~

A

~

Is at-norm.


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R (x,y) defines a relation between x and y:Y1 Y2 Y3

X1 0.2 1 0.4

X2 0 0.6 0.3X3 0 1 0.8

Composition of Rand S

R oS = { [(u,w), sup V (R (u,v) * S (v,w) ], u U,v U, w W}

Fuzzy Relation


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Example:

R = R (x,y) = 0.7 0.5

0.8 0.4

S = S (y,z) = 0.9 0.6 0.2

0.1 0.7 0.5

T = T (x,z) = V y Y(R (x,y) S (y,z) )

= 0.7 0.6 0.5

0.8 0.6 0.4For Example:

T (x,z) = max [ min (0.7,0.9), min (0.5, 0.1) ]

= max [ 0.7, 0.1 ] = 0.7


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Max ProductT = T (x,z) = V y Y(R (x,y) . S (y,z) )

= 0.63 0.42 0.25

0.72 0.48 0.24For Example:

T (x,z) = max [ (0.7)*(0.9) , (0.5)*(0.1) ]

= max [ 0.63, 0.05 ] = 0.63


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Max Average

T = T (x,z) = 0.5V y Y(R (x,y) + S (y,z))

= 0.8 0.65 0.5

0.85 0.7 0.5

For Example:

T (x,z) = 0.5 max [ (0.7)+(0.9) , (0.5)+(0.1) ]

= 0.5 max [ 1.6, 0.6 ] = 0.8


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Typical dual pairs of

nonparameterized t-norms and t-conorms


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))x(),x((B~

A~

otherwise..............................0

1)}x(),x(max{if)}...x(),x(min{ B~A~B~A~

drastic product

drastic sumsw ))x(),x(( B~A~ =

otherwise...............................10)}x(),x(min{if)}...x(),x(max{ B~A~B~A~

}1)()(,0{max))(),((t ~~~~1 xxxxBABA

bounded different

tw =

)}()(,1min{))(),(( ~~~~1 xxxxs BABA bounded sum


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))x(),x(( B~A~ t1.5 =)]x().x()x()x([2

)x().x(

B~

A~

B~

A~

B~

A~

Einsteinproduct

s1.5 ))x(),x(( B~A~ =

Einstein

sum)x().x(1

)x()x(

B~

A~

B~

A~

t2 ))x(),x(( B~A~ =

s2 ))x(),x(( B~A~ =

)x().x(B~

A~

)x().x()x()x(B~

A~

B~

A~

Algebraic product

Algebraic sum


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t2.5 ))x(),x(( B~A~ =)x().x()x()x(

)x().x(

B~

A~

B~

A~

B~

A~

Hamacher

product

s2.5 ))x(),x(( B~A~ =)x().x(1

)x().x(2)x()x(

B~

A~

B~A~B~A~

Hamacher

sum

t3 ))x(),x(( B~A~ =

s3 ))x(),x(( B~A~ =

min

max

)}x(),x({B~

A~

)}x(),x({B~

A~

minimum

maximum


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Theorem 1:

All t-norm operators, are bounded below by drastic product tw,

and bounded above by t3

twt1t1.5t2t2.5t3

Theorem 2:

Alls-norm operators, are bounded below bys3, and bounded

above by drastic sumsw

sws1s1.5s2s2.5s3


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Fuzzy Logic & Approximate Reasoning

1- Generalized Modus Ponens (GMP):

Premise 1: x is A (meaning not exactly A),Premise 2: if x is A then y is B,

consequence: y is B (i.e. not exactly B)


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Fuzzy Logic & Approximate Reasoning

1- Generalized Modus Tollens (GMT):

Premise 1: y is B (meaning not exactly B),Premise 2: if x is A then y is B,

consequence: x is A (i.e. not exactly A)


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Fuzzy Logic Controller

Fuzzification Interface

Measures the values of input variables,

Performs a scale mapping (if necessary) that

transfers the range of values of input variables intocorresponding universes of discourse

Performs the function of fuzzification that

converts the crisp real world input data into suitable

linguistic values, which may be viewed as labels of

fuzzy sets.


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Knowledge Base

Fuzzy rules consist of a premise with one or

more antecedents, and a conclusion with one

or more consequences. The individual rules in

the rule base are connected through the

operator also.


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Decision-making Logic

The two most common types are Min of

Mamdani and product of Larson [lee,1990].

The Min operator takes the minimum of allfuzzy membership values in the "if-side" for the

rule being evaluated, and clips the corresponding

output membership at this level.

The product operator scales the output

membership as opposed to clipping it.


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Defuzzification Interface

The defuzzification interface performs the

following functions:

a scale mapping, which converts the range ofvalues of output variables into corresponding

universes of discourse

defuzzification which yields a nonfuzzy controlaction from an inferred fuzzy control action.


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Defuzzification Interface

Two methods of defuzzification are most often

used:

maximum membership: chooses the output value

corresponding to the maximum degree of

membership in the output fuzzy set.

centroid or center of gravity: is the most

commonly used defuzzification method. In thecase of a discrete universe, the centroid method

yields a weighted sum of the output values [Lee,

1990].


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KB

Fuzzification

Interface

Defuzzificatio

Interface

Decision

Making

LogicFuzzy Fuzzy

Process Output & State Actual Control

Non-Fuzzy

Controled

System

(Process)

Block Diagram


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Hybrid Fuzzy PID


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HYBRID FUZZY-PID CONTROLLER

A Hybrid Fuzzy-PID controller is a type of fuzzycontroller in which the fuzzy engine is placed above a

conventional PID controller in the control hierarchy. This

is shown by the picture below.

The control inference of the HFPID is of the following:IF e is bigand de/dt issmalland edt issmallTHEN P is bigand D is

smalland I is medium


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Sugeno Engines


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GENERALIZED SUGENO CONTROLLER

A Generalized Sugeno Controller (GSC) is a fuzzy

controller which maps the input space to the output

space by the following fuzzy control

inference:

Ri: IF x1is A1iand ... xnis An

iTHEN y = Pi(x); 1


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PROBLEM STATEMENT

The Generalized Sugeno Controller (GSC) requires thedetermination of several unknown parameters for its design

and implementation. The search for these parameters can

be quite exhaustive.

Therefore, the purpose of this research is to develop asimple approach to determine the unknown parameters of a

Generalized Sugeno Controller.

Our approach is applied to the following systems:

A Hybrid Fuzzy-PID controlled robot manipulator arm

An approximation of an optimal feedback control law

for a ship tracking problem


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GSC vs. HFPID

The GSC has more desirable properties than the HFPID

The GSC is easier to design since the consequence

of each rule is caluclated automatically.

Rule evaluation and defuzzification is easier for theGSC than the HFPID.

It is easier to analyze the GSC in a qualitative

manner for stablility, controllability, observability,

and other issues of control systems.


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APPLICATIONS


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Applicability of Using A Fuzzy Controller

for DC-DC Converters


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Dynamic Control of Paralleled

DC-DC Converters(Current Sharing)

Desirable Characteristic:

Steady State Load Sharing

Transient Load Sharing


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BASIC CELL STRUCTURE OF DC-DC POWER MODULE

Vref+

-

VinIL

d

VC+-Hi Hv

Vo

Basic Cell

I V

+

-


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DIAGRAM OF TWO PARALLELED MODULES WITH MSC

WITH A DEDICATED MASTER

Module I

Module n

Cell #1

Cell #2

LO

AD

VoIo1

Io2

Vref1+

-Hv

+

+

+

-Hv

+

-Hi

Vref2

Module #1

Module #2

CS_Bus


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BLOCK DIAGRAM OF #N PARALLELED

CONVERTERS WITH MASTER-SLAVE CONTROL

Cell #1

Cell #n

Module #1

Module #n

L

OA

D

VoVinIo1

Ion

Vref1

+

++-Hv +

-Hi

++

+-Hv

+-Hi

Vrefn

CS Bus


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Classical Control

Complexity of analysis:

Huge number of loops and transfer

functions

Stability analysis is almost impossible


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Fuzzy Control

Linguistic Rules:

Ease of analysis

Ease of implementation


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Fuzzy Implementation

Input and output selection for the FLC

Membership function definitionFuzzy rules definition

(If x is A and y is B, then z is C)


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INPUT

N NS Z PS P

OUTPUT

N NS Z PS P

ATTRIBUTES OF THE MEMBERSHIP FUNCTION

IN DESIGNING FLC OF LOAD SHARING

NB: Negative Big NS: Negative SmallZ: Zero

PS: Positive Small PB: Positive Big


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BASIC MEMBERSHIP FUNCTION OF INPUT AND

OUTPUT FOR THE CSC FUZZY LOGIC CONTROLLER

(err) NB NS Z PS PB

-2 -1 0 1 2 err


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The centroid method is determined from the

fuzzy set as follow:

i

iic

i

iici

z

zz

z

)(

)(

0

Defuzzification


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If the error (difference of output currents)is PB,

then the duty cycle change of slave should be PB

If the error is NS, then the duty cycle change ofslave should be NS

If the error is PS and its derivative is NB, then the

duty cycle change of slave should be Z

Example of Fuzzy Control Rules

master-slave problem


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Block Diagram


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DEFFERENCE BETWEEN TWO OUTPUT CURRENTS

OUTPUT VOLTAGE VO

SYSTEM USING A CLASSICAL CSC SYSTEM USING A FUZZY LOGIC CSC

two paralleled buck converters output waveforms

Results


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DEFFERENCE BETWEEN TWO OUTPUT CURRENTS

OUTPUT VOLTAGE VO

SYSTEM USING A CLASSICAL CSC SYSTEM USING A FUZZY LOGIC CSC

STEP LOAD RESPONSES FOR 50% OF THE LOAD

Results

1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8 3

x 10-3

-1.5

-1

-0.5

0

0.5

1

T

OutputCurrent1-OutputCurrent2

1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8 3

x 10-3

13

14

15

16

17

18

19

T

Outp

utVoltage

class notes 870

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