Fuzzy Sets and Control

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Fuzzy Sets and Control
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Fuzzy Logic
The definition of Fuzzy logic is a form of
multi-valued logic derived from fuzzy set theory
to deal with reasoning that is approximate rather
than precise.
Degree of Truths and probabilities range between
0 and 1.
From Wikipedia
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An example of Fuzzy Reasoning
Fuzzy Set Theory defines Fuzzy Operators on Fuzzy
Sets. The problem in applying this is that the
appropriate Fuzzy Operator may not be known. For
this reason, Fuzzy logic usually uses IF-THEN
rules, or constructs that are equivalent, such
as fuzzy associative matrices. Rules are usually
expressed in the formIF variable IS property
THEN action For example, an extremely simple
temperature regulator that uses a fan might look
like thisIF temperature IS very cold THEN stop
fanIF temperature IS cold THEN turn down fanIF
temperature IS normal THEN maintain levelIF
temperature IS hot THEN speed up fan Notice
there is no "ELSE". All of the rules are
evaluated, because the temperature might be
"cold" and "normal" at the same time to different
degrees. The AND, OR, and NOT operators of
boolean logic exist in fuzzy logic, usually
defined as the minimum, maximum, and complement
when they are defined this way, they are called
the Zadeh operators, because they were first
defined as such in Zadeh's original papers. So
for the fuzzy variables x and y NOT x (1 -
truth(x)) x AND y minimum(truth(x), truth(y)) x
OR y maximum(truth(x), truth(y))
From Wikipedia
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Fuzzy Set Definition
The definition of a fuzzy set is given by the
membership function
elements of the universe of discourse U, can
belong to the fuzzy set with any value between 0
and 1. The degree of membership of an element u
when the universe of discourse U, is discrete and
finite, it is given for a fuzzy set A by
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Fuzzy Set Operations
The union of two fuzzy sets
is defined by
The intersection of two fuzzy sets
is defined by
is defined by
The complement of fuzzy set
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Properties of Set Operations
Most of the properties that hold for classical
sets (e.g., commutativity, associativity and
idempotence) hold also for fuzzy sets except for
following two properties
Law of contradiction
the intersection of a fuzzy set and its
complement results in a fuzzy set with membership
values of up to ½ and thus does not equal the
empty set (as in the case of classical sets)
Law of excluded middle
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Intelligent Control

• An intelligent control system is one in which a
  physical system or a mathematical model of it is
  being controlled by a combination of a
  knowledge-base, approximate (humanlike)
  reasoning, and/or a learning process structured
  in a hierarchical fashion.
• Under this simple definition, any control system
  which involves fuzzy logic, neural networks,
  expert learning schemes, genetic algorithms,
  genetic programming or any combination of these
  would be designated as intelligent control.

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Fuzzy Control

• A fuzzy controller consists of three operations
• (1) fuzzification,
• (2) inference engine, and
• (3) defuzzification.
• A common definition of a fuzzy control system is
  that it is a system which emulates a human
  expert. In this situation, the knowledge of the
  human operator would be put in the form of a set
  of fuzzy linguistic rules.
• The human operator observes quantities by
  observing the inputs, i.e., reading a meter or
  measuring a chart, and performs a definite action
  (e.g., pushes a knob, turns on a switch, closes a
  gate, or replaces a fuse) thus leading to a crisp
  action
• The human operator can be replaced by a
  combination of a fuzzy rule-based system (FRBS)
  and a block called defuzzifier. The input sensory
  (crisp or numerical) data are fed into FRBS where
  physical quantities are represented or compressed
  into linguistic variables with appropriate
  membership functions.
• These linguistic variables are then used in the
  antecedents (IF-Part) of a set of fuzzy rules
  within an inference engine to result in a new set
  of fuzzy linguistic variables or consequent
  (THEN-Part). Variables are combined and changed
  to a crisp (numerical) output.

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Fuzzy Control Architecture
System
Defuzzifier
Fuzzifier
Rule Engine
real numbers
real numbers
member ship values
fuzzy sets
Fuzzy Controller
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Simple Rules if antecedent then consequentEx
fuzzyControlForDec
consequent- cuts tts fuzzy set and computes its
area and moment
fuzzifier accepts real number inpuit and
outputs its membership
defuzzifier accumulates the areas and
moments and outputs the centroid
common domain input, x
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Simple Rule Controller
fuzzyControlForDec provides the negative feedback
to stabilize the integrator
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Composite Rules if antecedent1 and antecedent2
then consequentExample fuzzControlFor2To1
AndFn outputs minimum of inputs
x
common domain input, x
v
common domain input, v
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CompositeRule Controller
fuzzyControlFor2To1 provides the control to
settle the spring at zero
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Linear Time Invariant Models
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Inverted Pendulum Fuzzy Control
Linearized inverted pendulum on a cart
fuzzyControlFor2To1 provides the control to keep
the stick stable
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Modeling the Inverted Pendulum
Source http//www.engin.umich.edu/group/ctm/examp
les/pend/invpen.html
Moment of Inertia http//hyperphysics.phy-astr.gs
u.edu/hbase/mi2.htmlrlin
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Inverted Pendulum Swing Up Non-linear Model
source http//www.control.lth.se/publications/ful
ldocs/ast_fur96.pdf
detect when angle reaches turn-off level
detect when pendulum stops rising
reset theta to 0 when reach 2pi
detect when pendulum stops rising
starting from hanging configuration, rod can be
made to reach inverted configuration
with sufficient force acting until horizontal
line is reached