Transcript Document

HOW DOES INTELLIGENT
CONTROL WORK ?

Importance
• In the last few years the applications of artificial
intelligence techniques have been opening doors to
convert human experience into a form understandable
by computers. Advanced control based on artificial
intelligence techniques is called intelligent control.

Fuzzy Logic
• Fuzzy logic is a technique to embody human-like
thinking into a control system.
• A fuzzy controller can be designed to emulate human
deductive thinking, that is, the process people use to
infer conclusions from what they know.
USE OF FUZZY CONTROL

Fuzzy control incorporates ambiguous human logic
into computer programs. It suits control problems that
cannot be easily represented by mathematical models :
Weak model
Parameter variation problem
Unavailable or incomplete data
Very complex plants
Good qualitative understanding of plant or process
operation

Because of its unconventional approach, design of
such controllers leads to faster development /
implementation cycles
Traditional control approach requires formal modeling
of the physical reality. Here we show two methods that
may be used to describe a system’s behavior:
 1. Experimental Method

• By experimenting and determining how the process
reacts to various inputs.

2. Mathematical Modeling
• Mathematical model of the controlled process, usually
in the form of differential or difference equations.
Laplace transforms and and z-transforms are
respectively used.
• Problems that can arise: – Model complexity
– Inaccurate values of various parameters

Alternative Approach: Heuristic Method

The heuristic method consists of modeling and understanding in
accordance with previous experience, rules-of-thumb and oftenused strategies.

Heuristic rule: It is a logical implication of the form:
• If <condition> then <consequence>, or in a typical control
situation: If <condition> then <action>

Rules associate conclusions with conditions.
• The heuristic method is actually similar to the experimental method
infused with the control strategies of human operators.

Intelligent control strategies may be implemented by
other means. However, fuzzy implementations are very
efficient for several reasons.
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1965 Introduction of fuzzy sets theory by Lotfi Zadeh (USA)
1972 Toshiro Terano established the first working group on
fuzzy systems in Japan
1973 Paper about fuzzy algorithms by Zadeh (USA)
1974 Steam engine control by Ebrahim Mamdani (UK)
1977 Fuzzy expert system for loan evaluation by Hans
Zimmermann (Germany)
1980 Cement kiln control by F. - L. Smidth & Co. - Lauritz P.
Holmblad (Denmark)
1984 Water treatment (chemical injection) control (Japan)
1984 Subway Sendai Transportation system control (Japan)
1985 First fuzzy chip developed by M. Togai and H. Watanabe
in Bell Labs (USA)
1986 Fuzzy expert system for diagnosing illnesses in Omron
(Japan)
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1987 Container crank control, tunnel excavation, soldering robot,
automated aircraft vehicle landing
• Second IFSA Conference in Tokyo
• Togai Infralogic Inc. first company dedicated to fuzzy control in
Irvine (USA)
1988 Kiln control by Yokogawa
• First dedicated fuzzy controller sold - Omron (Japan)
1989 Creation of Laboratory for International Fuzzy Engineering
Research (LIFE) in Japan
1990 Fuzzy TV set by Sony (Japan)
• Fuzzy electronic eye by Fujitsu (Japan)
• Fuzzy Logic Systems Institute (FLSI) by Takeshi Yamakawa
(Japan)
• Intelligent Systems Control Laboratory in Siemens (Gemiany)
1991 Fuzzy AIl Promotion Centre (Japan)
• USA start to get academic attention
Ten years of hype => We are getting to a ripe technology now !
BASIC PRINCIPLES OF BIVALENT AND
MULTIVALENT LOGIC



Bivalence : something is either true or not true
Logic of Aristotle was the cornerstone of such philosophical
ethics
Multivalence : captures the mismatch between the real world and
our bivalent view of it.
• Medical diagnosis
• Legal decisions


The objective of fuzzy logic is to capture these shades of gray,
these degrees of truth.
Fuzzy logic deals with uncertainty and the partial truth
represented by the various shades of gray in a systematic and
rigorous manner.
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Plato (427-347? Bc) saw degrees of truth everywhere and recoiled from
them. No chair is perfect, it is only a chair to a certain degree.
Charles Sanders Peirce (1839-1914) observed that All that exists is
continuous and such continuums govern knowledge.
Bertrand Russell (1872-1970) Both vagueness and precision are features
of language, not reality. Vagueness clearly is a matter of degree.
Jan Lukasiewicz (1878-1956) proposed a formal model of vagueness, a
logic where 1 stands for TRUE, 0 stands for FALSE, 1/2 stands for
possible. Lukasiewicz was just one step away from Zadeh and his logic
can be considered relative.
Max Black (1909-89) proposed a degree as a measure of vagueness.
Albert Einstein (1879-1955): So far as the laws of mathematics refer to
reality, they are not certain. And so far as they are certain, they do not
refer to reality.
Lotfi Zadeh (1923- ) introduced fuzzy sets logic theory. 'As the
complexity of a system increases, our ability to make precise and
significant statements about its behavior decreases until a threshold is
reached beyond which precision and significance (or relevance) become
almost mutually exclusive characteristics...
PARACONSISTENT LOGIC

Although not very known in engineering applications,
Paraconsistent Logic has been under interest of Philosophers and
Mathematicians before the foundations of fuzzy logic
• Roughly speaking, a paraconsistent logic is a logic rejecting the
principle of non contradiction (PNC).
• The philosophical focus was for a long time whether a negation not
obeying (PNC) is still a negation.

Newton da Costa assembled in 1963 the foundations based on
the work of Stanislaw Jaskowski. Francisco Miró Quesada
suggested this name (paraconsistent) in 1976.
• An example could be a decision support system which is able to
support contradictions like someone is 0.7 tall and 0.5 short.

Latter you will observe that despite mathematical
formalities, in engineering applications one can shift the
membership functions to get a similar effect.
CONCEPT OF FUZZY DESCRIPTIONS
A linguistic variable has values expressed in words,
representing qualitative concepts or impressions about
the state of a system
 The idea is to capture vague, imprecise, inexact, fuzzy
verbal descriptions

• Large, medium, big, not very large, etc…
To convert linguistic terms into numeric values one
needs the fundamentals of set theory
 In crisp set theory, an element is
 either a member (membership grade = 1)
 or not a member (membership grade =0)

Concept of a Fuzzy Number



Zero
Almost Zero
Near Zero
-3
-2
-1
0
1
2
3
-3
-2
-1
0
1
2
3
-3
-2
-1
0
1
2
3
Fuzzy Sets

In fuzzy set theory an element’s membership may
have a value in the interval from 0 to1:
x2
x1
Set A
Universe of Discourse

Example : define speed over the range of 50 km/h and
100 km/h
1.0
grade 
Membership
• What is the set of speeds equal or greater that 70 km/h ?
Set of
speeders
0.5
Km/h
50

70
100
There is a sharp change from 0 to 1
Fuzzy Membership
Fuzzy sets allow gradual transitions between membership and
non-membership;  = any value in the interval [0,1]

Example :”a set of speeds greater than 70 km/h”
1.0
grade 
Membership

Set of
speeders
0.5
Km/h
50

70
Concept of Universe of Discourse
100
FUZZIFICATION

Process of decomposing a system input into one or more fuzzy
sets. Triangular or trapezoidal shaped are the most common.

With 50% of overlapping, any input will “fire” two fuzzy sets
COLD
COOL GOOD WARM
HOT
1.0
8o
11o 14o
17o 20o 23o
26o 29o 32o
o
Temperature ( C)
Translating Semantics into
Numerical Values

“The temperature is cool.”
• It the temperature was 13OC we would assign such
statement the truth value of 0.8. However, we would
also assign a truth value of 0.2 for the following
statement as well :
“The temperature is cold.”
 The set terminology is as follows:

• The temperature is a member of the set of the cool
feeling with COOL(x) = 0.8
• The temperature is a member of the set of the cold
feeling with COLD(x) = 0.2
Establishing a Logic Kernel
To establish a complete system of fuzzy logic, one
needs to define some logic operations:
 EMPTY
 EQUAL
 COMPLEMENT (NOT)
 CONTAINMENT
 ALPHA-CUT
 INTERSECTION (AND)
 UNION (OR)


Measure of fuzziness : it is the closeness of its
elements to the membership grade of 0.5
f A    A  x    A 1  x dx
2
where A1/2 is the ½ cut of A(x)

a-cut of a fuzzy set is the crisp set formed by
elements A whose membership grade is greater than
or equal to a given value a. This is denoted by:
Aa  x  X  A x a 
in which X is the universe to which A belongs.
Intersection (AND)

Given two sets A and B , A E, B  E, where E is their common
universe of discourse, he intersection A  B is a set of all elements x
which are members of both A and B. This is illustrated in the Venn
diagram.
E
B


U
A
A B
The overlapping portion of sets A and B and, as a result, it is always
smaller than any of the individual sets A and B.
Membership function AB(u) , u  U, of the intersection A  B is
defined point-by-point by :
AB(u) = A(u) t B(u)  min [A(u), B(u)]
Considering the fundamental definition of AND operator
as min [A(u), B(u)], the figure depicts the resulting
membership function for A AND B
A
A
B
B
U

A B
Union (OR)

Given two sets A and B , A  E, B  E, where E is their common
universe of discourse, the union A  B is a set of all elements x
which are members of set A or set B or both A and B. This is
illustrated in the Venn diagram.
E
AU B
B
A


Union is the smallest subset of the universe of discourse E, which
includes both sets A and B.
Membership function A  B(u) , u  U, of the union A  B is defined
point-by-point by :
A B(u) = A(u) s B(u)  max [A(u), B(u)]

Considering the fundamental definition of OR operator
as max [A(u), B(u)], the figure depicts the resulting
membership function for A OR B
A
A
B
B
AU B
There are several T-norm and
S-norm operators
MIN / MAX
ALGEBRAIC
SCHWEIZER &
SKLAR
HAMACHER
UNION
INTERSECTION
max [ A x ,  B x ]
min [ A x ,  B x ]
 A x  B x 
 A  x  +  B  x    A  x  B  x 
[
]
1  max 0, 1 -  A  x  + 1 +  B  x   1
-P
-P
1/ P



DUBOIS &
PRADE
DOMBI
1
YAGER


l
l
éæ 1
ö
æ 1
ö ù
1 + êçç
 1÷÷ + çç
 1÷÷ ú
êëè  A  x  ø
è  B  x  ø úû
-P
+  B x 
None
-P
1
None
]
1 / P
 A  x  B  x 
g + 1  g  A  x  +  B  x    A  x  B  x 
 A  x  +  B  x   2  g  A  x  B  x 
1  1  g  A  x  B  x 
é
s1  A  x   1 s1  B  x  ù
1  log S ê1 +
ú
s 1
ë
û
1
W
W Wù
é
min ê1,  A  x  +  B  x 
úû
ë
 A  x  +  B  x    A  x  B  x   min  A  x ,  B  x ,1  a 
max 1   A  x ,1   B  x ,a 
FRANK
[
max 0,  A  x 
PARAMETER



é
s  A x  1 s  B x  1 ù
log S ê1 +
ú
s 1
ë
û
W
W
ì
1  min í1, 1   A  x  + 1   B  x 
î
 A  x  B  x 
max  A  x ,  B  x ,a 
[
]
1
W
1
l
g  0,+¥ 
s  0,+¥ 
w  0,+¥ 
a  0,1
l  0,+¥ 
1

ü
ý
þ
p   ¥,+¥ 
1
l
l l
éæ 1
ö æ 1
ö ù
1 + êçç
 1÷÷ + çç
 1÷÷ ú
êëè  A  x  ø è  B  x  ø úû
Properties
Given universe of discourse E, and three
fuzzy sets A  E, B E, C  E , then the
following applies:
Commutativity properties:
A B = B A
A B = B A
Fuzzy set and its complement(*):
A  A’  0
A  A’  E
Fuzzy set and the null set:
A= 0
Associativity properties:
(A  B)  C = A  (B  C)
(A  B)  C = A  (B  C)
A= A
Idempotence:
AA=A
AA=A
AE = E
Distributivity with respect to intersection:
A  (B  C) = (A  B)  (A  C)
Distributivity with respect to union:
A  (B  C) = (A  B)  (A  C)
Fuzzy set and the universal set
AE =A
Involution property:
( A’)’ = A
De Morgan’s theorem:
(A  B)’= A’ B’
(A  B)’= A’ B’
HEDGES
Sometimes we need to refine descriptions, making them more
meaningful and accurate. Fuzzy sets can be modified to reflect
this kind of linguistic refinement by applying hedges. Once a
hedge has been applied to a fuzzy set, the degrees of membership
of the members of the set are altered. The membership of Fast
can be altered by VERY fast by an exponential operator like the
one indicated in the figure :
1
Degree of
Membership

Fast
VERY Fast
0
80 Km/h
120 Km/h
Speed
Some examples of hedges :
 VERY, A(x) = [A(x)]b
 SOMEWHAT, A(x) = [A(x)]1/b
 The hedge “Approximately” uses the concept of subsethood. It
requires an auxiliary fuzzy set (distribution of weights) to
evaluate assertions like
“How Much Approximately Cold is Hot ?

distribution
of weights
Degree of
Membership
1
APPROXIMATELY Cold
Cold
Temperature
0
0 oC
o
2.5 C
o
5 C
Operations on the Same
Universe of Discourse

Keeping simple : the operation of fuzzification
requires a generic element and its membership degree

AND, OR and NOT can be performed like:
 (X and Y) = MIN ((X), Y))
 (X or Y) = MAX ((X), (Y))
 (not X) = 1.0 -  (X)


Let us assume an air-conditioner vent. Suppose it has blades that
control the openings and can be controllable so as the inclination angle
of the vent might be directed downward or upward. Such angle control
sends the air-flow towards the floor or to the ceiling.
Fuzzy sets DOWNWARD and UPWARD describe the position of the
vent blades. If the blades are totally rotated to –45 degrees in respect to
the horizontal then they are completely downward. If the blades are
totally rotated to +45 degrees, they are completely upward.
Downward
1.0
-45 0
1.0
45
Upward
AND
Downward
-45 0
45
Upward
OR
Downward
1.0
1.0
-45 0
Upward
45
-45 0
45
Statements


Statements assert facts or states of affairs; they give descriptions
that can be organized in several rules of reasoning. Application
of mathematical descriptions and the use of logical rules for
formulating hypotheses was developed by several philosophers.
They have been influenced by the earlier syllogistic logic, in
which premises were manipulated to produce true conclusions.
A typical syllogistic rule of inference  that goes by the Latin
name modus ponens (affirmative mode)  can be given as
• If A is true
• And A implies B
• Then B is true
where the connectives and, or, and not are essential to derive
the truth of above rules.
Operations in Different
Universes of Discourse




What are the operations between sets belonging to
different universes of discourse ?
In control systems, mappings between input and output
are our main concern. These mappings are between input
variable sets A( u) U and output variable sets B(v)  V
through the conditional statement of inference:
AB
or: IF A(u) THEN B(v)
Such map links the antecedent (condition) set A (defined
by its membership vector A(u), u  U with the
consequent (result or action) set B (defined by its
membership vector B(v), v  V).

Suppose A and B are crisp sets. Defining P(u, v) = A(u) X B(v),
where the symbol ”X” stands for the Cartesian product
operator. Since both sets A and B are characterized by their
respective membership vectors , their Cartesian product will be
a matrix of crisp numbers.

Cartesian product of fuzzy sets => In this case, the membership
grades, , are in M = [0,1]. If the spaces are, for example, U1 =
{x} and U2 = {y}, then the Cartesian product is P{x, y} = U1 X
U2 with membership function c(x, y) where each ordered pair
is in [0,1]. In fuzzy set theory, the component sets of a Cartesian
product are always universes of discourse, hence they are
always crisp.

In practice, the t-norms min and algebraic product are mostly
used .

A fuzzy set A E1 will induce another fuzzy set B  E2 whose
membership function will be B(y/x). Then one can write:
B(y) = A(x) ° R(x,y)
where “°” is the compositional operator which indicates a
compositional rule of inference. For the purpose of practical
computation, it can also be written in terms of the membership
functions of the respective sets:
Max-min composition:
B(y) = MAX[MIN(A(x), R(x,y))] x E1
Max-product composition:
B(y) = MAX[A(x) . R(x,y))] x E1

A multi-input multi-output system can be characterized
by a set of rules :
IF var1 = A <connective> var2 = B <connective> … THEN Out1 = C <connective>…
<connective’>
IF var1 = D <connective> var2 = E <connective> … THEN Out2 = E <connective>…
<connective’>
……
……
where A, B, C, D and E are crisp or fuzzy sets, and
<connective> represents the particular fuzzy operator
chosen to express the fuzzy inference or fuzzy
implication desired.
Fuzzy Rules

Rules of inference : IF <conditions> THEN <consequences>
Membership function
curves for inputs
Membership function
curves for outputs
MEDIUM
MEDIUM

SMALL
1
The lower of the two
IF condition outputs
are selected

Distance (D)


1
Speed (S)
MEDIUM
1
MEDIUM
Defuzzification
1

HARD

Center of
gravity
Brake action for
speed reduction
Rule 1 If the distance between two cars is medium and the speed
of the car is medium Then brake medium for speed reduction
Rule 2 If the distance between two cars is small and the speed of
the car is medium Then brake hard for speed reduction
Rule-based fuzzy controllers
have several advantages





Fuzzy control rules are easy to understand by maintenance
personnel
Control functions associated with a rule can be tested
individually. This improves maintainability because the
simplicity of the rules allows the use of less skilled personnel.
Individual rules combine to form a structured complex control;
parallel processing allows fuzzy logic to control complex
systems using simple expressions.
Rules can be added for alarm conditions, both linear and
nonlinear control functions can be implemented by a rule-based
system, using expert knowledge formulated in linguistic terms.
Fuzzy controllers are inherently reliable and robust. A partial
system failure may not significantly degrade the controller’s
performance.