Slovak University of Technology
Faculty of Material Science and Technology in Trnava
Intelligent Control
Methods
Lecture 11: Fuzzy control 2
Fuzzy system uses rules of the type:
rule ::= If <antecedent> then <consekvent>
antecedent ::= <atomic fuzzy assertion>
{and|or <atomic fuzzy assertion>}
consequent ::= <atomic fuzzy assertion>
<atomic fuzzy assertion> ::= <variable> is
<linguistic value>
Example:
if (v is small) and (d is medium) then (F is small)
2
Linguistic variable:
Variable, which has a verbal value.
Example: (speed – small, medium, big)
Definition:
Linguistic variable is a tetrad (x, Lx, Ux, Mx), where:
x – the name of the variable (speed, v, e)
Lx – set of verbal values (small, medium, big)
Ux – definition scope (possible values of the variable
(physical interval of the speed expressed by numbers,
e.g. 0 – 120))
Mx – function, which expresses the verbal values by fuzzy
sets in definition scope
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Linguistic variable example:
v – the name of linguistic variable „speed“
Lx – the set of verbal values
(N – nearly null, M - mini, S - medium, V - big)
Ux – universe (definition scope) <0 – 120>
Mx – function, which maps the verbal values in universe by
fuzzy sets, e.g.:
120
N   L(v,10,30) / v
120
M   (v,10,30,40,60) / v
0
0
120
120
S   (v,40,60,70,90) / v
0
V   (v,70,90) / v
0
4
Combination of assertions:
<atomic fuzzy assertion> ::= <variable> is <linguistic value>
Konjunction:
Let p and q are atomic fuzzy assertions p: „x is A“ and q: „y is B“, where A
and B are defined in the same universe.
The value of assertion (x is A) and (y is B) is given by konjunction of
fuzzy sets A and B, i.e. by value AB = min (A,B).
Disjunction:
The value of assertion (x is A) or (y is B) is given by disjunction of fuzzy
sets A and B, i.e. by value AB = max (A,B).
Negation:
The negation of the assertion „x is A“ (i.e. „x is not A“) is given by
complement A’ of fuzzy set A, i.e. by value A’ = 1- A.
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Combination of assertions in different
universes (1):
Let a and b are linguistic variables defined in universes Ua, Ub.
Let p and q are assertions „a is F1“ and „b is F2“.
F1 

Ua
F1
(a) / a
F2    F2 (b) / b
Ub
The cylindric extension of sets F1 a F2 into cartesian product Ua x Ub is
necessary before the assertion combination.
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Combination of assertions in different
universes (2):
Let a and b are linguistic variables defined in definition scopes Ua, Ub.
Let p and q are assertions „a is F1“ and „b is F2“.
Conjunction:
Combined assertion (a is F1) and (b is F2) is given by fuzzy relation
defined on cartesian product Ua x Ub
 r ( a, b) 
 min( 
F1
(a),  F2 (b)) /( a, b)
U a xU b
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Combination of assertions in different
universes (3):
Let a and b are linguistic variables defined in universes Ua, Ub.
Let p and q are assertions „a is F1“ and „b is F2“.
Disjunction:
Combined assertion (a is F1) or (b is F2) is given by fuzzy relation defined
on cartesian product Ua x Ub
 r ( a, b) 
 max( 
F1
(a),  F2 (b)) /( a, b)
U a xU b
8
Fuzzy implication:
Fuzzy implication is a construction if (fuzzy assertion) then (fuzzy
assertion), where (fuzzy assertion) is atomic or combined one.
„if (a is F1) then (b is F2)“
Boolean implication:
pq = (not p) or q
((not p) is given as 1-p, or is calculated as max after cylindric extension)
Rb  ce( F1 ' )  ce( F2 ) 
 max( 1  
F1
(a),  F2 (b)) /( a, b)
U a xU b
 R (a, b)  max( 1   F (a),  F (b))
b
1
2
9
Fuzzy implication (2):
Boolean implication is not used. It is replaced by implication by
Lukasziewicz, Zadeh, Larsen and by others. The most used implication
is expression, which is not an implication (it is only called implication) –
implication by Mandami.
pq = p and q (!!!)
RM  ce( F1 )  ce( F2 ) 
 min( 
F1
(a),  F2 (b)) /( a, b)
U a xU b
 R (a, b)  min(  F (a),  F (b))
M
1
2
10
Implication example „if (a is F1) then (b is F2)“
F1 = 0,1/a1 + 0,4/a2 +0,7/a3 + 1,0/a4
F2 = 0,2/b1 + 0,5/b2 +0,9/b3
ce(F1’)
ce(F1)
b1
b2
b3
a1
0,1
0,1
0,1
a2
0,4
0,4
a3
0,7
a4
1,0
ce(F2)
b1
b2
b3
a1
0,9
0,9
0,9
0,4
a2
0,6
0,6
0,7
0,7
a3
0,3
1,0
1,0
a4
0
Boolean implication:
ce(F1’)  ce(F2):
b1
b2
b3
a1
0,9
0,9
0,9
a2
0,6
0,6
a3
0,3
a4
0,2
b1
b2
b3
a1
0,2
0,5
0,9
0,6
a2
0,2
0,5
0,9
0,3
0,3
a3
0,2
0,5
0,9
0
0
a4
0,2
0,5
0,9
Mandami implication:
ce(F1)  ce(F2):
b1
b2
b3
a1
0,1
0,1
0,1
0,9
a2
0,2
0,4
0,4
0,5
0,9
a3
0,2
0,5
0,7
0,5
0,9
a4
0,2
0,5
0,9
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Valuation of list of rules:
k. (one) rule:
R
(a),  F2 (b))
( k ) (a, b)  min(  F
1
RM
(k )
(k )
M
 min(  F1 (a),  F2
(k )

(k )
(k )
(b)) /( a, b)
U a xU b
Value of all rules:
n
R   R (k )
k 1
 R (a, b)  max  R (a, b)  max min(  F
k
(k )
k
1
(k )
(a),  F2 (b))
(k )
12
Fuzzy systems
Systems, which variables (input, state, output) are defined by linguistic
values (by fuzzy sets)
Structure of (technical) fuzzy system:
Crisp
Input values
Fuzzification modul
Fuzzy
input values
Fuzzy
output values
Inference
engine
and rules
base
Crisp
output values
Defuzzication modul
Database
13
Database and base of rules:
Database consists of data about fuzzy sets of all
fuzzy variables (the definition scope, the form
given by membership function).
Base of rules consists of inference rules (in the
form of Mandami fuzzy implications).
14
Fuzzification:
It transforms the crisp value of variable into fuzzy one.
F(x)
1
F
F(x) = 0.6
x = 1.614
x
v = - 3.67
NB
NM
v = -3.67
ZO
PM
PB
NB
NM
ZO
PM
PB
0.7
0.3
0
0
0
15
Inference engine:
It evaluates the set of rules. The result is a fuzzy set.
16
Inference engine on example of fuzzy PI-controller:
IF e is Ae AND e is Ae THEN u is Bu
E.g.:
IF e is PS AND e is PM THEN u is PB
e
NB
NM
NS
ZO
PS
PM
PB
NB
NB
NB
NB
NB
NM
NS
ZO
NM
NB
NB
NB
NM
NS
Z0
PS
NS
NB
NB
NM
NS
Z0
PS
PM
e ZO
NB
NM
NS
Z0
PS
PM
PB
PS
NM
NS
Z0
PS
PM
PB
PB
PM
NS
Z0
PS
PM
PB
PB
PB
PB
Z0
PS
PM
PB
PB
PB
PB
17
Inference engine on example of fuzzy PIcontroller (2):
Let Ue = <-0.2, 0.2>
NB
NM
NS
NM
NS
-5
PM
ZO
PS
NM
NS
ZO
PS
Uu = <-5.0, 5.0>
PB
e = 0.024
e = 0.016
0,2
PM
e = 0.016
-0,1
NB
PS
e = 0.024
-0,2
NB
ZO
Ue = <-0.1, 0.1>
PB
0,1
PM
e
e
PB
5
u
NB
NM
NS
ZO
PS
PM
PB
e
0
0
0
0,4
0.6
0
0
e
0
0
0
0.3
0.7
0
0
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Inference engine on example of fuzzy PIcontroller (3):
e
NB
NM
NS
ZO
PS
PM
PB
NB
NB
NB
NB
NB
NM
NS
ZO
NM
NB
NB
NB
NM
NS
Z0
PS
NS
NB
NB
NM
NS
Z0
PS
PM
e ZO
NB
NM
NS
Z0
PS
PM
PB
PS
NM
NS
Z0
PS
PM
PB
PB
PM
NS
Z0
PS
PM
PB
PB
PB
PB
Z0
PS
PM
PB
PB
PB
PB
4 active rules (from 49):
IF e is ZO AND e is ZO THEN u is ZO
IF e is ZO AND e is PS THEN u is PS
IF e is PS AND e is ZO THEN u is PS
IF e is PS AND e is PS THEN u is PM
(1)
(2)
(3)
(4)
e = 0.024
e = 0.016
NB
NM
NS
ZO
PS
PM
PB
e
0
0
0
0,4
0.6
0
0
e
0
0
0
0.3
0.7
0
0
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Inference engine on example of fuzzy PIcontroller (4):
Before the inference: The crisp values of input variables (e,
e) are fuzzificated.
Inference = Evaluation of active rules:
1. Each rule is evaluated independently (there are
allways two fuzzy assertions connected by AND in
the antecedent, the Mandami implication (min) is
used)
2. Partiall results from active rules are aggregated by
operation or (i.e. max is used, see slade 12). Fuzzy
set is a result.
After the inference: The obtained fuzzy set is defuzzificated
to crisp number.
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Inference engine on example of fuzzy PIcontroller (5):
NB
NM
NS
ZO
PS
PM
PB
0.4
NB
-0,2
NB
e=0.024
NM
NS
ZO
PS
0.2
PM
e=0.016
MIN
PB
NB
NM
NS
ZO
PS
0.1
PM
NS
ZO
PS
PM
PB
e
0.3
-5
0.3
-0,1
NM
5
u
IF e is ZO AND e is ZO THEN u is ZO
e
PB
0.4
NB
-0,2
NB
e=0.024
NM
NS
ZO
PS
0.2
PM
NM
NS
ZO
PS
PM
PB
e
MIN
PB
0.4
0.7
-5
-0,1
e=0.016
0.1
e
5
u
IF e is ZO AND e is PS THEN u is PS
21
Inference engine on example of fuzzy PIcontroller (6):
NB
NM
NS
ZO
PS
PM
PB
0.6
NB
-0,2
NB
e=0.024
NM
NS
ZO
PS
0.2
PM
e=0.016
MIN
PB
NB
NM
NS
ZO
PS
0.1
PM
NS
ZO
PS
PM
PB
e
0.3
-5
0.3
-0,1
NM
5
u
IF e is PS AND e is ZO THEN u is PS
e
PB
0.6
-0,2
NB
e=0.024
NM
NS
ZO
PS
0.2
PM
NB
e
MIN
PB
NM
NS
ZO
PS
PM
PB
0.6
0.7
-5
-0,1
e=0.016
0.1
e
5
u
IF e is PS AND e is PS THEN u is PM
22
Inference engine on example of fuzzy PIcontroller (7):
Resultant fuzzy set:
NB
1
NM
NS
ZO
PS
PM
PB
0.7
0.6
0.4
0.3
0
-5
5
u
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Defuzzification:
The result of inference is a fuzzy set. The goal of defuzzification is to obtain a
crisp value from this fuzzy set.
There are more methods of defuzzification.
Centre of Area (centre of Gravity) - COA (COG):
y* 
 y
F
( y)dy
y

y
F
( y)dy
y*
y
24
Defuzzification (2):
Center of Sum (COS), reflects the overlapped areas:
r
y* 
 y 
y
k 1
k
( y )dy
r

y k 1
k
( y )dy
y*
y
25
Defuzzification (3):
FoM, SoM (First of Maximum, Smallest of Maximum):
y* = inf {yY/F(y) = hgt(F)}
hgt(F)
y
LoM (Last of Maximum, Largest of Maximum):
y* = sup {yY/F(y) = hgt(F)}
y*
FoM
MoM
LoM
MoM (Middle of Maxima):
y* = (y1+y2)/2
26