Intelligent Control Methods

1
Intelligent Control Methods
Slovak University of Technology Faculty of
Material Science and Technology in Trnava

• Lecture 10 Fuzzy Control (1)

2
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)

3
Klassical controller states (calculates) the
control action u(t) according to e(t) For
example PI-controller
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 (?e), we brake stronger (?u)).
4
Control strategy
Rules IF THEN (in a form similar to normal
speech). Derived according to some type of
classical controller.
5
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
linquistic expressions giving the value of
control difference and control action.
6
PD-controller u(t) KP e(t) KD ?e(t)
Fuzzy PD-controller IF e is Ae AND ?e is A?e
THEN u is Bu
7
PI-controller ?u(t) KI e(t) KP ?e(t)
Fuzzy PI-controller IF e is Ae AND ?e is A?e
THEN ?u is B?u Often case, ?u is more native
for people (valve or gas pedal opening or
closing) than the absolute value u (valve open
62 , pedal pressed 16o).
8
PID-controller ?u(t) KI e(t) KP ?e(t)
KD ?2e (t) Fuzzy PID-controller IF e is Ae
AND ?e is A?e AND ?e2 is A?2e THEN ?u is
B?u Assigned for non-linear and unstabil
processes. Problem with great number of
antecedents combinations.
9
Matematical background of fuzzy control (1)
Clasical (crisp) sets A1 ball, cylinder,
cube set of figures given by elements listing A2
x ? Z / 6 lt x lt 10 set of numbers given by
property
10
Matematical background of fuzzy control (2)
11
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.
12
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 ? lt0,1gt F (u,?F(u)/u?U
?F(u1)/u1 ?F(u2)/u2 ... ?F(un)/un
13
Matematical background of fuzzy control (5)

• Fuzzy set example
• Let the temperature in a room is lt0,30gt (oC),
  i.e.
• U lt0,30gt
• Membership functions into sets Cald, Good, Hot
  are
• ?
• 0 15 30
• ?c(25) 0,0 ?g(25) 0,3 ?H(25) 0,7

14
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
15
Matematical background of fuzzy control (7)
Typical membership functions
?(u) 1 0 for
u?? ?(u,?,?,?) (u-?)/(?-?) for ??u??
(?-u)/(?-?) for ??u?? ? ?
? u 1 for u??
?(u) 0 for u?? 1
(u-?)/(?-?) for ??u?? ?(u,?,?,?,?) 1
for ??u?? (?-u)/(?-?) for ??u??
? ? ? ? u 0 for u??
16
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 NS Z PS PM PB
-6 -4 -2 0 2 4
6 u
NB (Negative Big) L(u,-6,-4) NM (Negative
Medium) ?(u,-6,-4,-2) NS (Negative Small)
?(u,-4,-2,0) Z (Zero) ?(u,-2,0,2) PS (Positive
Small) ?(u,0,2,4) PM (Positive Medium)
?(u,2,0,4) PB (Positive Big) ?(u,4,6)
17
Operations with fuzzy sets
? A B
x
? A
Complement (negation) ?A(x) 1 - ?A(x)
x
18
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
19
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
If the definition scopes are continuous, then
20
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
21
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))
22
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 0
Eva 0 1 0 0
  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
Married c. and friends (?M?F(x,y)) M.c. or
friends (?M?F(x,y))
  Peter Ján Milan Igor
Jana 0 0 0 0,3
Iveta 0,5 0 0 0
Eva 0 0,1 0 0
  Peter Ján Milan Igor
Jana 0,8 0,9 0,1 1
Iveta 1 0,6 0,3 0,7
Eva 0,2 1 0,8 0,4
23
Operations with fuzzy relations (2) Projection
Let R is binary relation defined on X x Y. Then
projection R into Y is fuzzy set
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.
Proj R in Y 0,8/Pe 0,9/Já 0,8/Mi 0,7/Ig
  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 X 0,9/Ja 0,7/Iv 0,8/Ev
24
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.
I.e. Cylindric extension means the building of a
table from the function.
  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
F 0,8/Pe 0,7/Já 0,3/Mi 0,6/Ig