Slovak University of Technology
Faculty of Material Science and Technology in Trnava
Intelligent Control
Methods
Lecture 10: Fuzzy Control (1)
Introduction

Classical control theory:
 mathematical
description of processes
(differential equations)

Fuzzy control:
 normally
used by people, based on experiences and
expert-knowledge, (which are described by linguistic
tools, not by equations, mathematical tools are
replaced by fuzzy logic).
 Examples:

reverted pendulum (described by 4 non-linear differential
equations!)

car parking: turn the driving wheel just a bit to the left and
turn back (not: turn the wheel 16o33“ and drive back 2,675 m)
2
h(t)
e(t)
R(s)
u(t)
S(s)
y(t)
-
Klassical controller states (calculates) the control action
u(t) according to e(t): For example PI-controller:
u (t )  Pe(t )  I  e(t )dt
u (t )  K i e(t )  K p e(t )
Fuzzy controller states the actuating signal according to
control strategy based on rules IF – THEN. F.E:
IF difference is big and difference of difference is small
THEN difference of control action is big.
Usual by people decision and performance:
(If a car rides faster than we want (e) and it reduced
gently (Δe), we brake stronger (Δu)).
3
Control strategy:
Rules IF – THEN (in a form similar to normal speech).
Derived according to some type of classical controller.
4
P-controller:
u(t) = KP e(t)
u(t) – control action
e(t) – control difference
Fuzzy P-controller:
IF e is Ae THEN u is Bu
Ae, Bu –
and
linquistic expressions giving the
value of control difference
control action.
5
PD-controller:
u(t) = KP e(t) + KD Δe(t)
Fuzzy PD-controller:
IF e is Ae AND Δe is AΔe THEN u is Bu
6
PI-controller:
Δu(t) = KI e(t) + KP Δe(t)
Fuzzy PI-controller:
IF e is Ae AND Δe is AΔe THEN Δu is BΔu
Often case, Δu is more native for people (valve
or gas pedal opening or closing) than the
absolute value u (valve open 62 %, pedal
pressed 16o).
7
PID-controller:
Δu(t) = KI e(t) + KP Δe(t) + KD Δ2e (t)
Fuzzy PID-controller:
IF e is Ae AND Δe is AΔe AND Δe2 is AΔ2e
THEN Δu is BΔu
Assigned for non-linear and unstabil processes.
Problem with great number of antecedents
combinations.
8
Matematical background of fuzzy control (1):
Clasical (crisp) sets:
A1 = {ball, cylinder, cube}
set of figures given by elements listing
A2 = {x  Z / 6 < x < 10}
set of numbers given by property
xA
T
F
A(x) =
1 for x  A
charakteristic function of a
set A
0 for x  A
(gives membership of
elements to the
set A)
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Matematical background of fuzzy control (2):
U
A
negation
U
A
B
intersection
U
A
B
union
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Matematical background of fuzzy control (3):
Pojem relácie (v prípade ostrých množín):
Let X and Y are definition scopes and let their cartesian
product is U = X x Y. Then a binary relation R is each subset
R  U.
(the same definition is valid for n-dimensional relations)
Example:
X = {Jana, Iveta, Eva} and Y = {Peter, Ján, Milan, Igor} are
definition scopes (universes)
R = {(Jana, Igor), (Iveta, Peter), (Eva, Ján)} is relation
„married couples“ defined on X x Y.
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Matematical background of fuzzy control (4):
Fuzzy set definition:
Fuzzy set is the set of elements, which can belong into the set
partially.
The membership of element into the set is given by
membership function (what is generalised characteristic
function of the set).
F:
U  <0,1>
F = {(u,F(u)/uU} = F(u1)/u1 + F(u2)/u2 + ... +F(un)/un
n
   F (ui ) / ui
i 1


F
(u ) / u
uU
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Matematical background of fuzzy control (5):
Fuzzy set example:
Let the temperature in a room is <0,30> (oC), i.e.
U = <0,30>
Membership functions into sets Cald, Good, Hot are:
1 
0
c(25) = 0,0
15
g(25) = 0,3
30
H(25) = 0,7
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Matematical background of fuzzy control (6):
Typical membership functions (linear, therefore simple):
(u)
1


0
for u
(u,,) = (u-)/(-) for u
1
for u
u
(u)
1


1
for u
L(u,,) = (-u)/(-) for u
0
for u
u
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Matematical background of fuzzy control (7):
Typical membership functions:
(u)
1


(u)
1

 


0
for u
(u,,,) = (u-)/(-) for u
(-u)/(-) for u
u
1
for u
0
for u
(u-)/(-) for u
(u,,,,) = 1
for u
(-u)/(-) for u
u
0
for u
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Matematical background of fuzzy control (8):
Often (general) case of description of definition scope
by fuzzy sets without considering the physical
parameters:
 1 NB NM
-6
-4
NS
Z
PS
PM
PB
-2
0
2
4
6
NB (Negative Big): L(u,-6,-4)
NS (Negative Small): (u,-4,-2,0)
PS (Positive Small): (u,0,2,4)
PB (Positive Big): (u,4,6)
u
NM (Negative Medium): (u,-6,-4,-2)
Z (Zero): (u,-2,0,2)
PM (Positive Medium): (u,2,0,4)
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Operations with fuzzy sets:

A
B
x

A’
Complement (negation):
A’(x) = 1 - A(x)
x
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Operations with fuzzy sets:

A
B
Intersection:
AB(x) = min (A(x), B(x))
x

A
B
Union:
AB(x) = max (A(x), B(x))
x
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Fuzzy relation:
Let U and V are definition scopes and let it is given
the function R: UxV  0,1. Binary fuzzy relation
R is fuzzy set of ordered couples
R    R (u, v) /(u, v)
UxV
If the definition scopes are continuous, then:
R

R
(u, v) /(u, v)
UxV
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Fuzzy relation (example):
X = {Jana, Iveta, Eva} and Y = {Peter, Ján, Milan, Igor}
are definition scopes.
Relation „Friends“ defined on X x Y:
Peter
Ján
Milan
Igor
Jana
0,8
0,9
0,1
0,3
Iveta
0,5
0,6
0,3
0,7
Eva
0,2
0,1
0,8
0,4
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Operations with fuzzy relations: (intersection and
union)
Let R and S are binary relations defined on X x Y. Then
membership functions for intersection and union of
relations R and S are defined for all x,y as follow:
Intersection:
RS(x,y) = min (R(x,y), S(x,y))
Union:
RS(x,y) = max (R(x,y), S(x,y))
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Operations with fuzzy relations
(example for intersection and
union):
X = {Jana, Iveta, Eva} and Y = {Peter, Ján, Milan, Igor} are definition
scopes.
Relations „Married couples“ and „Friends“ defined on X x Y:
Married couples (M):
Friends (F):
Peter
Ján
Milan
Igor
Jana
0
0
0
1
Iveta
1
0
0
Eva
0
1
0
Peter
Ján
Milan
Igor
Jana
0,8
0,9
0,1
0,3
0
Iveta
0,5
0,6
0,3
0,7
0
Eva
0,2
0,1
0,8
0,4
Married c. and friends (MF(x,y))
Peter
Ján
Milan
Igor
Jana
0
0
0
0,3
Iveta
0,5
0
0
Eva
0
0,1
0
M.c. or friends (MF(x,y))
Peter
Ján
Milan
Igor
Jana
0,8
0,9
0,1
1
0
Iveta
1
0,6
0,3
0,7
0
Eva
0,2
1
0,8
0,4
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Operations with fuzzy relations (2):
Projection
Let R is binary relation defined on X x Y. Then projection R into Y is
fuzzy set
projRnaY   max  R ( x, y ) / y
x
Y
I.e.: Projection R into Y means the finding of maximal value R in
each column y1, y2, ... yn in the table and assignment of this value to
element yj.
Peter
Ján
Milan
Igor
Jana
0,8
0,9
0,1
0,3
Iveta
0,5
0,6
0,3
0,7
Eva
0,2
0,1
0,8
0,4
Proj R in Y = 0,8/Pe + 0,9/Já + 0,8/Mi + 0,7/Ig
Proj R in X = 0,9/Ja + 0,7/Iv + 0,8/Ev
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Operations with fuzzy relations (3):
Extension
Opposit operation for projection:
Let F is a fuzzy set defined on Y. Then cylindric extension F to X x Y is
the set of all couples (x,y)  X x Y with membership function CE(F)(x,y),
i.e.:
ce( F ) 

F
( y) /( x, y)
XxY
I.e.: Cylindric extension means the building of a table from the
function.
F = 0,8/Pe + 0,7/Já + 0,3/Mi + 0,6/Ig
Peter
Ján
Milan
Igor
Jana
0,8
0,7
0,3
0,6
Iveta
0,8
0,7
0,3
0,6
Eva
0,8
0,7
0,3
0,6
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