Fuzzy systemsFuzzy systems manipulate fuzzy sets to model the
worldMost fuzzy systems are rule based
If water temperature is low and flow rate is low, then open the warm water tap slightly
antecedents
consequent
fuzzy set
Consequent can be a fuzzy set, a numberor another model (e.g. a linear model)
Antecedents describe a conditionin which a rule is valid locally
linguistic variable
A numerical variable takes numerical values:
Age = 65
A linguistic variables takes linguistic values:
Age is old
A linguistic value is a fuzzy set.
All linguistic values form a term set:
T(age) = {young, not young, very young,
middle aged, not middle aged,
old, not old, very old, more or less old,
not very young and not very old, ...}
0 10 20 30 40 50 60 70 80 90 1000
0.2
0.4
0.6
0.8
1
Age
Mem
bers
hip
Gra
des
(a) Primary Linguistic Values
OldYoung
0 10 20 30 40 50 60 70 80 90 1000
0.2
0.4
0.6
0.8
1
Age
Mem
bers
hip
Gra
des
(b) Composite Linguistic Values
Not Young and Not Old
More or Less Old Extremely Old
Young butNot Too Young
CON A A( ) = 2
DIL A A( ) .= 0 5
INT AA x
A xA
A
( ), ( ) .
( ) , . ( )=
≤ ≤¬ ¬ ≤ ≤
2 0 0 5
2 0 5 1
2
2
µµ
Concentration:
Dilation:
Contrast
intensification:
0 1 2 3 4 5 6 7 8 9 100
0.2
0.4
0.6
0.8
1
X
Mem
bers
hip
Gra
d es
Effects of Contrast Intensifier
Fuzzy if-then rulesGeneral format:
If x is A then y is BExamples:
If pressure is high, then volume is smallIf restaurant is expensive, then order small dishesIf a tomato is red, then it is ripeIf the speed is high, then apply the brake a little
Fuzzy implication function:mR( x , y )= f ( mA( x ) ,mB( y ))= f ( a ,b )
0 0.5 10
0.5
10
0.5
1
a
(a) Min
b
0 10 200
10
200
0.5
1
xy
0 0.5 10
0.5
10
0.5
1
a
(b) Algebraic Product
b
0 10 200
10
200
0.5
1
xy
0 0.5 10
0.5
10
0.5
1
a
(c) Bounded Product
b
0 10 200
10
200
0.5
1
xy
0 0.5 10
0.5
10
0.5
1
a
(d) Drastic Product
b
0 10 200
10
200
0.5
1
xy
A entails BBoolean fuzzy implication (based on )
Zadeh's max-min implication (based on )
Zadeh's arithmetic implication (based on )
Goguen's implication
mR( x , y )=max (1−mA( x ) ,mB( y ))
¬A∨B
¬A∨( A∧B)
mR( x , y )=max (1−mA( x ) ,min (mA( x ) ,mB( y )))
¬A∨BmR( x , y )=min (1−mA( x )+mB( y ) ,1)
mR( x , y )=min (mB( x )/mA( y ) ,1)
Material implication
0 0.5 10
0.5
10
0.5
1
a
(a) Zadeh's Arithmetic Rule
b
0 10 200
10
200
0.5
1
xy
0 0.5 10
0.5
10
0.5
1
a
(b) Zadeh's Max-Min Rule
b
0 10 200
10
200
0.5
1
xy
0 0.5 10
0.5
10
0.5
1
a
(c) Boolean Fuzzy Implication
b
0 10 200
10
200
0.5
1
xy
0 0.5 10
0.5
10
0.5
1
a
(d) Goguen's Fuzzy Implication
b
0 10 200
10
200
0.5
1
xy
Constructing fuzzy relationsPremise: Young people make long GSM calls
X = {18, 20, 22, 25, 30} [years]
Y = {1, 3, 5, 7, 10, 20} [min./call]
2.05.08.011)(
3025222018
young(x)
x
x
µ
19.05.02.01.00)(
20107531
long(y)
y
y
µ
20107531
y
young(x) into X x Y
Compute cylindrical extensions
2.030
5.025
8.022
120
118
)(xx µ
long(y) into X x Y
19.05.02.01.00)(
20107531
y
y
µ
2.02.02.02.02.02.0
5.05.05.05.05.05.0
8.08.08.08.08.08.0
111111
111111
19.05.02.01.00
19.05.02.01.00
19.05.02.01.00
19.05.02.01.00
19.05.02.01.00
Compute the aggregation of the two cylindrical extensions. Since “young” and “long” go together, you can use a conjunctive operator (e.g. minimum).
2.02.02.02.02.02.0
5.05.05.05.05.05.0
8.08.08.08.08.08.0
111111
111111
19.05.02.01.00
19.05.02.01.00
19.05.02.01.00
19.05.02.01.00
19.05.02.01.00
20107531
y
30
2522
20
18
x
2.02.02.02.01.00
5.05.05.02.01.00
8.08.05.02.01.00
19.05.02.01.00
19.05.02.01.00
min
R
19.05.02.01.00)(Pr
2.02.02.02.02.01.002.030
5.05.05.05.02.01.005.025
8.08.08.05.02.01.008.022
119.05.02.01.00120
119.05.02.01.00118
)(Pr19.05.02.01.00)(\)(
20107531
R
Ryxx
y
Y
Xµµ
long
young
Max-min compositionThe max-min composition of two fuzzy relations R
(defined on X and Y) and S (defined on Y and Z) is
The result is the combined relation defined on X and Z
)],(),([),( zyyxzx SRy
SR µµµ ∧=∨
example
=
2.02.00
5.05.00
8.05.00
15.00
15.00
15.00
9.05.00
5.05.00
2.02.00
1.01.00
000
2.02.02.02.01.00
5.05.05.02.01.00
8.08.05.02.01.00
19.05.02.01.00
19.05.02.01.00
R S R°S
People who make long GSM calls give a lot of money to clothing
Young people give a lot of money to clothing
Compositional rule of inferenceMax-min composition of a fuzzy set A on X and a
fuzzy relation R on X x Y
Returns the image of A transformed through the relation R
The composition of A and R can also be seen as the shadow of R on A (for every y)
μ A∘R( y )=maxx∈X
{μ A( x )∧μ R( x , y )} ∀ y∈Y
Derivation of y = b from x = a and y = f(x):
a and b: points
y = f(x) : a curve
a
b
y
xx
y
a and b: intervals
y = f(x) : an interval-valued
function
a
b
y = f(x) y = f(x)
Crisp case Fuzzy case
a is a fuzzy set and y = f(x) is a fuzzy relation
02
46
810
0
10
20
30
400
0.5
1
X
(a) Fuzzy Relation F on X and Y
Y
Mem
bers
hip
Gra
des
02
46
810
0
10
20
30
400
0.5
1
X
(b) Cylindrical Extension of A
Y
Mem
bers
hip
Gra
des
02
46
810
0
10
20
30
400
0.5
1
X
(c) Minimum of (a) and (b)
Y
Mem
bers
hip
Gra
des
02
46
810
0
10
20
30
400
0.5
1
X
(d) Projection of (c) onto Y
Y
Mem
bers
hip
Gra
des
[ ] [ ]5.05.05.02.01.00
2.02.02.02.01.00
5.05.05.02.01.00
8.08.05.02.01.00
19.05.02.01.00
19.05.02.01.00
01000 =
[ ] [ ]8.08.05.02.01.00
2.02.02.02.01.00
5.05.05.02.01.00
8.08.05.02.01.00
19.05.02.01.00
19.05.02.01.00
02.014.00 =
• Rule: if x is A then y is B• Fact: x is A’• Conclusion: y is B’
Graphic Representation
)(
)())]()(([)( ''
yw
yxxy
B
BAAxB
µµµµµ
∧=∧∧∨=
A
X
w
A’ B
Y
x is A’
B’
Y
A’
Xy is B’
• Rule: if x is A and y is B then z is C• Fact: x is A’ and y is B’• Conclusion: z is C’Graphic Representation
A B T-norm
X Y
w
A’ B’ C2
Z
C’
ZX Y
A’ B’
x is A’ y is B’ z is C’
Multiple RulesFuzzy rules like A → B are represented as fuzzy
relationThe collection of all rules is represented as an
aggregated relation using the union operator, i.e.
R tot=∪i=1
N
Ri=∨i=1
N
Ri
3
Fuzzy inference systemMultiple namesFuzzy rule-based systemFuzzy expert systemFuzzy modelFuzzy associative memoryFuzzy logic controllerFuzzy system (simply)
Building blocksFuzzifierRule baseInference engineDefuzzifier
Fuzz
i fie
r
Def
uz z
ifie
rRule base(knowledge base)
Inference engine
input
outp
ut
FuzzifierInterface between the outside world and the
fuzzy system (input)Transform a measurement in a fuzzy setMultiple methods:
singleton methodtriangular methodGaussian function method...
Fuzzificationsingleton method
= 1 when x = a, otherwise 0
X = {a,b,c,d}a { (a,1), (b,0), (c,0), (d,0) }→
triangular function method = max(0,|x-a|/s)
(Here a is the measurement)
mA( x )
mA( x )
Rule baseHeart of the knowledge base of the fuzzy systemEncodes the general relation between the inputs
and the outputsRules can be examples, rules of thumb, encoded
experience, qualitative relations between variables, etc.
Rules are often represented as if-then statements
Inference engineThe reasoning mechanism of the fuzzy system (to
infer: to reason/deduce)Combines actual inputs with the information
encoded in the rule base to compute the fuzzy output of the system
Usually implements the compositional rule of inference or some equivalent computation
Not as context-independent as the inference engine of an expert system
DefuzzifierInterface between the fuzzy systems and the
outside world (output)Needed when a crisp output is required (e.g. a
final decision, a control action, a final advice, etc.)Computes a number/symbol that represents the
output fuzzy setEnhances the interpolation properties of the
fuzzy system
Mamdani fuzzy modelsFive major steps:FuzzificationDegree of fulfillmentInferenceAggregationDefuzzificationComputations from Mamdani reasoning are
mathematically equivalent to the compositional rule of inference
FuzzificationCalculate the membership degrees of the
(measured) inputs to the linguistic terms in the fuzzifier
Crisp inputs Fuzzy inputs
Degree of fulfillmentWhen the antecedent (if-part) of a rule contains
multiple variables, the total match between all inputs and the rule antecedent must be determined
Degree of fulfilment determines to what degree the rule is valid
Take minimum of all fuzzified values in the rule antecedent
InferenceComputes the output of each fuzzy rule in the rule base,
given the degree of fulfilment of the rule baseModifies the output fuzzy set depending on the degree
of fulfilment
Clip the output fuzzy set at the height corresponding to the degree of fulfillment
AggregationMultiple rules may become active in a fuzzy system
Combined output of the fuzzy system is the combined output of all active rules
Corresponds to calculating the total relation from the relations of individual rules
Calculate maximum of individual fuzzy rule outputs
DefuzzificationDetermines the crisp output of the fuzzy system
Replaces the output fuzzy set with a representative number
Compute the defuzzified value using an agreed upon defuzzification operator (e.g. center of area)
Defuzzification rulesCentroid-of-area
Bisector of area
Mean of maximum
Smallest of maximum
Largest of maximum
∫∫=
Z A
Z A
dzz
dzzzz
)(
)(*
µ
µ
∫ ∫∞−∞
=*
*)()(
z
z AA dzzdzz µµ
*})(|{',*
'
' µµ ===∫∫
zzZdz
dzzz A
Z
Z
zZz '
min∈
zZz '
max∈
Mamdani - single input
-10 -8 -6 -4 -2 0 2 4 6 8 100
0.2
0.4
0.6
0.8
1
1.2
X
Me
mb
ers
hip
Gra
de
s small medium large
0 1 2 3 4 5 6 7 8 9 100
0.2
0.4
0.6
0.8
1
1.2
Y
Me
mb
ers
hip
Gra
de
s small medium large
X is Small →Y is Small
X is Medium →Y is Medium
X is Large →Y is Large
Mamdani - single input
-10 -8 -6 -4 -2 0 2 4 6 8 100
1
2
3
4
5
6
7
8
9
10
X
Y
-1 0 -8 -6 -4 -2 0 2 4 6 8 1 00
0 . 2
0 . 4
0 . 6
0 . 8
1
1 . 2
X
Memb
ersh
ip Gr
ades
s m a l l m e d iu m la rg e
0 1 2 3 4 5 6 7 8 9 1 00
0 . 2
0 . 4
0 . 6
0 . 8
1
1 . 2
Y
Memb
ersh
ip Gr
ades
s m a l l m e d iu m la rg e
Mamdani - double input
-5 -4 -3 -2 -1 0 1 2 3 4 50
0 . 5
1
X
Mem
bersh
ip Gr
ades
s m a ll la rg e
-5 -4 -3 -2 -1 0 1 2 3 4 50
0 . 5
1
Y
Mem
bersh
ip Gr
ades
s m a ll la rg e
-5 -4 -3 -2 -1 0 1 2 3 4 50
0 . 5
1
Z
Mem
bersh
ip Gr
ades
la rg e n e g a ti ve s m a ll n e g a ti ve s m a ll p o s i t i ve la rg e p o s i t i v e
X is Small and Y is Small Z is →negative Large
X is Small and Y is Large Z is →negative Small
X is Large and Y is Small Z is →positive Small
if X is Large and Y is Large Z →is positive Large
Mamdani - double input
-5 -4 -3 -2 -1 0 1 2 3 4 50
0 . 5
1
X
Mem
bersh
ip Gr
ades
s m a ll la rg e
-5 -4 -3 -2 -1 0 1 2 3 4 50
0 . 5
1
Y
Mem
bersh
ip Gr
ades
s m a ll la rg e
-5 -4 -3 -2 -1 0 1 2 3 4 50
0 . 5
1
Z
Mem
bersh
ip Gr
ades
la rg e n e g a ti ve s m a ll n e g a ti ve s m a ll p o s i t i ve la rg e p o s i t i v e
- 5
0
5
- 5
0
5
- 3
- 2
- 1
0
1
2
3
XY
Z
Takagi-Sugeno fuzzy modelsFuzzy antecedents, crisp consequentsConsequent is a crisp function of inputs
if x is A and y is B then z = f(x,y)Zero-order Sugeno: constant consequent
if x is A and y is B then z = cFirst-order Sugeno: linear consequent
if x is A and y is B then z = ax+by+cOverall output is a weighted
average of individual rule outputs∑
∑=
==Ni i
Ni ii z
z
1
1*β
β
Fuzzy vs. crisp rule set
-10 -5 0 5 100
0.2
0.4
0.6
0.8
1
1.2
X
Mem
bers
hip
Gra
des
small medium large
(a) Antecedent MFs for Crisp Rules
-10 -5 0 5 100
2
4
6
8
X
Y
(b) Overall I/O Curve for Crisp Rules
-10 -5 0 5 100
0.2
0.4
0.6
0.8
1
1.2
X
Mem
bers
hip
Gra
des
small medium large
(c) Antecedent MFs for Fuzzy Rules
-10 -5 0 5 100
2
4
6
8
X
Y
(d) Overall I/O Curve for Fuzzy Rules
Sugeno - double inputIf X is Small and
Y is Small thenz = -x+y+1
If X is Small and Y is Large thenz = -y+3
If X is Large andY is Small thenz = -x+3
If X is Large andY is Large thenz = x+y+2
-5 -4 -3 -2 -1 0 1 2 3 4 50
0.2
0.4
0.6
0.8
1
1.2
X
Mem
bers
hip
Gra
des
Small Large
-5 -4 -3 -2 -1 0 1 2 3 4 50
0.2
0.4
0.6
0.8
1
1.2
Y
Mem
bers
hip
Gra
des
Small Large
Approximation capabilityFuzzy systems are general function
approximators (c.f. neural networks)You can increase the accuracy of a mapping by
increasing the number of rules (examples) in the rule base
Best results are obtained when the number of linguistic terms in the input and the output are increased (a finer partition)
Using too many linguistic terms diminishes the transparency of fuzzy systems
Interpolation propertiesMultiple rules in a fuzzy system may fire (become
active) because fuzzy sets overlapFuzzy rules represent typical cases or examples of
the relation between two quantitiesThe reasoning mechanism interpolates between
the rules to determine the system output
18 20 22 24 26 28 30 32 34 36 380
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1young, middle-aged, old people
age
mem
bers
hip
0 2 4 6 8 10 12 14 16 18 200
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1short, medium, long GSM calls
duration
mem
bers
hip
Interpolation propertiesConsider a fuzzy system with three rulesYoung people make long GSM callsMiddle aged people make short GSM callsOld people make medium-long GSM calls
example
age duration