Ming-Feng Yeh 2-1

4. 4. General Fuzzy SystemsGeneral Fuzzy Systems

A fuzzy system is a static nonlinear mapping between its inputs and outputs (i.e., it is not a dynamic system).

Ming-Feng Yeh 2-2

Universe of DiscourseUniverse of Discourse

The “universe of discourse” for ui or yi since it provides the range of values (domain) of Ui or Yi that can be quantified with linguistic and fuzzy sets.

An “effective” universe of discourse [, ].

Width of the universe of discourse:

22 ,

44 ,

20,20

Ming-Feng Yeh 2-3

Linguistic VariablesLinguistic Variables

Linguistic expressions are needed for the inputs and outputs and the characteristics of the inputs and outputs.

Linguistic variables (constant symbolic descriptions of what are in general time-varying quantities) to describe fuzzy system inputs and outputs.

Linguistic variables: is to described the input

is to described the output

for example, “position error”

iu~ iu

iy~ iy

1~u

Ming-Feng Yeh 2-4

Linguistic ValuesLinguistic Values

Linguistic variables and take on “linguistic values” that are used to describe characteristic of the variables.

Let denote the jth linguistic value of the linguistic variable defined over the universe of discourse Ui.

For example, “speed”

iu~ iy~

iu~j

iA~

},...,2,1:~

{~

ij

ii NjAA 1

~u

}~

,~

,~

{~

""~

,""~

,""~

31

21

111

31

21

11

AAAA

fastAmediumAslowA

Ming-Feng Yeh 2-5

Linguistic RulesLinguistic Rules

The mapping of the inputs to the outputs for a fuzzy system is in part characterized by a set of condition action rules, or in modus ponens (If-Then) form: If premise Then consequent.

Multi-input single-output (MISO) rule:

the ith MISO rule:

Multi-input multi-output (MIMO) rule:

pqq

lnn

kj BisyAisuAisuAisu~~~~,...,

~~~~2211 ThenandandIf

srlnn

kj BisyBisyAisuAisuAisu 22112211

~~~~~~,...,~~~~ andThenandandIf

iqplkj ),;,...,,(

Ming-Feng Yeh 2-6

Number of Fuzzy RulesNumber of Fuzzy Rules

If all the premise terms are used in very rule and a rule is formed for each possible combination of premise elements, then there are rules in the rule-base.

n

ini NNNN

121 ...

Ming-Feng Yeh 2-7

Fuzzy Quantification of RFuzzy Quantification of Rules: Fuzzy Implicationsules: Fuzzy Implications

Multi-input single-output (MISO) rule:

Define:

These fuzzy sets quantify the terms, in the premise and the consequent of the given If-Then rule, to make a “fuzzy implication” (which is a fuzzy relation).

pqq

lnn

kj BisyAisuAisuAisu~~~~,...,

~~~~2211 ThenandandIf

}:))(,{( 111111

UuuuA jA

j

}:))(,{( 22222

2UuuuA kA

k

}:))(,{(

}:))(,{(

nqqBqp

q

nnnAnln

YyyyB

UuuuA

pq

ln

Ming-Feng Yeh 2-8

Fuzzy ImplicationsFuzzy Implications

A fuzzy implication:the fuzzy set quantifies the meaning of the linguistic statement “ “, and quantifies the meaning of “ “.

Two general properties of fuzzy logic rule-bases1. Completeness whether there are conclusions for every possible fuzzy controller input.2. Consistency whether the conclusions that rules make conflict with other rules’ conclusions.

pq

ln

kj BAAA ThenandandIf ,...,21jA1

jAisu 11

~~ pqB

pqq Bisy

~~

Ming-Feng Yeh 2-9

FuzzificationFuzzification

Fuzzification: convert its numeric inputs ui Ui into fuzzy sets.

Let denote the set of all possible fuzzy sets that can be defined on Ui. Given ui Ui, fuzzification transforms ui to a fuzzy set denoted by defined on the universe of discourse Ui.

Fuzzification operation:where

iU

fuziA

ii UUF :fuz

ii AuF ˆ)(

Ming-Feng Yeh 2-10

Singleton FuzzificationSingleton Fuzzification

When a singleton fuzzification is used, which produces a fuzzy set with a membership function defined by

Any fuzzy set with this form for its membership function is called a “singleton”.Other fuzzification methods haves not been used very much because they are complexity.

ifuz

i UA

otherwiseux

xu iA fuz

i ,0,1

)(ˆ

Ming-Feng Yeh 2-11

Inference MechanismInference Mechanism

Two basic tasks –(1) matching: determining the extent to which each rule is relevant to the current situation as characterized by the inputs ui, i = 1, 2, …, n.(2) inference step: drawing conclusions using the current input ui and the information in the rule-base.

Ming-Feng Yeh 2-12

MatchingMatching

Assume that the current inputs ui, i = 1, 2, …, n, and fuzzification produces the fuzzy sets representing the inputs.

Step 1: combine inputs with rule premises

Step 2: determine which rules are on

fuzn

fuzfuz AAA ˆ,...,ˆ,ˆ21

)()()()()( 11ˆ1ˆ11ˆ11111

uuuuu jjfuzjj AAAAA

)()()()()( ˆˆˆ nAnAnAnAnAuuuuu l

nln

fuzn

ln

ln

)()()(

)()()(),...,,(

21

ˆ2ˆ1ˆ21

21

21

nAAA

nAAAni

uuu

uuuuuu

ln

kj

ln

kj

Ming-Feng Yeh 2-13

Rule CertaintyRule Certainty

We use to represent the certainty that the premise of rule i matches the input information when we use singleton fuzzification.

An additional “rule certainty” is multiplied by i. Such a certainty could represent our a priori confidence in each rule’s applicability and would normally be a number between zero and one.

),...,,( 21 ni uuu

Ming-Feng Yeh 2-14

Inference StepInference Step

Alternative 1: determine implied fuzzy setsFor the ith rule, the “implied fuzzy set” with membership function

Alternative 2: determine the overall implied fuzzy sets. The “overall implied fuzzy set” with membership function

iqB

)(),...,,()( 21ˆ qBniqByuuuy p

qiq

qB

)()()()( ˆˆˆˆ 21 qBqBqBqByyyy R

qqqq

Ming-Feng Yeh 2-15

Compositional Rule of InfereCompositional Rule of Inferencence

The overall implied fuzzy set:

whereSup-star compositional rule of inference:“sup” corresponds to the operation, and the “star” corresponds to the operation. Sup (supremum): the least upper boundZadeh’s compositional rule of inference:maximum is used for and minimum is used for .

)(),...,,()( 21ˆ qBniqByuuuy p

qiq

)()()()( ˆˆˆˆ 21 qBqBqBqB

yyyy Rqqqq

Ming-Feng Yeh 2-16

Defuzzification:Defuzzification:Implied Fuzzy SetsImplied Fuzzy Sets

Center of gravity (COG): using the center of are and area of each implied fuzzy set

Center-average: using the centers of each of the output membership functions and the maximum certainty of each of the conclusions represented with the implied fuzzy sets

R

i Y qqB

Ri Y qqB

qicrisp

q

qiq

qiq

dyy

dyyby

1 ˆ

1 ˆ

)(

)(

R

i qBy

Ri qBy

qicrisp

qy

yby

iqq

iqq

1 ˆ

1 ˆ

)}({sup

)}({sup

Ming-Feng Yeh 2-17

Defuzzification:Defuzzification:Overall Implied Fuzzy SetsOverall Implied Fuzzy Sets

Max criterionA crisp output is chosen as the point on the output universe of discourse Yq for which the overall implied fuzzy set achieves a maximum

“arg supx{(x)}” returns the value of x that results in the supremum of the function being achieve.Sometimes the supremum can occur at more than one point in Yq. In this case you also need to specify a strategy on how to pick one point for (e.g., choosing the smallest value)

crispqy

qB

)}({suparg ˆ qBY

crispq yy

qq

crispqy

Ming-Feng Yeh 2-18

Defuzzification:Defuzzification:Overall Implied Fuzzy SetsOverall Implied Fuzzy Sets

Mean of maximum (MOM)A crisp output is chosen to represent the mean value of all elements whose membership in is a maximum.Define as the supremum of the membership function of over Yq. Define a fuzzy set with the following membership function

crispqy

qBmaxqb

qBqq YB ˆ

q q

q qq

q

Y qqB

Y qqBqcrispq

qqBqB dyy

dyyyy

othereise

byy

)(

)(

,0

ˆ)(,1)(

ˆ

ˆmax

ˆ

ˆ

Ming-Feng Yeh 2-19

Defuzzification:Defuzzification:Overall Implied Fuzzy SetsOverall Implied Fuzzy Sets

Center of area (COA)A crisp output is chosen as the center of area for the membership function of the overall implied fuzzy set .For a continuous output universe of discourse Yq, the center of area output is defined by

crispqy

qB

q q

q q

Y qqB

Y qqBqcrispq dyy

dyyyy

)(

)(

ˆ

ˆ

Ming-Feng Yeh 2-20

Functional Fuzzy SystemsFunctional Fuzzy Systems

Standard fuzzy system:

Functional fuzzy system:

The choice of the function gi(·) depends on the application being considered. The function gi(·) can be linear or nonlinear.Defuzzification:

pqq

lnn

kj BisyAisuAisuAisu~~~~,...,

~~~~2211 ThenandandIf

)(~~,...,

~~~~2211 ii

lnn

kj gbAisuAisuAisu ThenandandIf

Ri i

Ri iib

y1

1

Ming-Feng Yeh 2-21

Takagi-Sugeno fuzzy system:

If ai,0=0, then the gi(·) mapping is a linear mapping.

If ai,00, then the gi(·) mapping is called “affine.”Suppose n = 1, R = 2.

nniiiii uauauaagb ,22,11,0,)(

.2~~

11111 ubAisu ThenIf

.1~~

122

11 ubAisu ThenIf

21

2211

bb

y

Takagi-Sugeno Fuzzy SystemTakagi-Sugeno Fuzzy System

Ming-Feng Yeh 2-22

Singleton O/P Fuzzy SystemSingleton O/P Fuzzy System

If gi= ai,0, then Takagi-Sugeno fuzzy system is equivalent to a standard fuzzy system that uses center-average defuzzification with singleton output membership function at ai,0.

The corresponding fuzzy rule is of the form:

where bi is a real number.

ilnn

kj bAisuAisuAisu ThenandandIf~~,...,

~~~~2211

Ri i

Ri iib

y1

1

Ming-Feng Yeh 2-23

Consequent FormsConsequent Forms

Type 1: a crisp value (singleton output)

Type 2: a fuzzy number (standard fuzzy system)

Type 3: a function (functional fuzzy system)

ilnn

kj bAisuAisuAisu ThenandandIf~~,...,

~~~~2211

pqq

lnn

kj BisyAisuAisuAisu~~~~,...,

~~~~2211 ThenandandIf

)(~~,...,

~~~~2211 ii

lnn

kj gbAisuAisuAisu ThenandandIf

Ming-Feng Yeh 2-24

Universal Approximation PropertyUniversal Approximation Property

Suppose that we use center-average defuzzification, product for the premise and implication, and Gaussian membership functions. Name this fuzzy system f(u). Then, for any real continuous (u) defined on a closed and bounded set and an arbitrary > 0, there exists a fuzzy system f(u) such that

: Psi, : Epsilon

)()(sup uufu