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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
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