Lecture 4 The Fuzzy Controller design

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Lecture 4 The Fuzzy Controller design
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• By a fuzzy logic controller (FLC) we mean a
  control law that is described by a
  knowledge-based system consisting of IF...THEN
  rules with vague predicates and a fuzzy logic
  inference mechanism.
• The rule base is the main part of the FLC. It is
  formed by a family of logical rules that
  describes the relationship between the input e
  and the output u of the controller.
• The main difference between conventional control
  system and fuzzy logic controlled system is not
  only in the type of logic (Boolean or fuzzy) but
  in the inspiration. The former attempted to
  increase the efficiency of conventional control
  algorithms the latter were based on the
  implementation of human understanding and human
  thinking in control algorithms.

 
 
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4.1 fuzzy logic controller structure
Fuzzy controller architecture.
The fuzzy controller has four main components
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• 1. A rule-base (a set of If-Then rules), which
  contains a fuzzy logic quantification of the
  expert's linguistic description of how to achieve
  good control.
• 2. An inference mechanism (also called an
  "inference engine" or "fuzzy inference" module),
  which emulates the expert's decision making in
  interpreting and applying knowledge about how
  best to control the plant.
• 3. A fuzzification interface, which converts
  controller inputs into information that the
  inference mechanism can easily use to activate
  and apply rules.
• 4. A defuzzification interface, which converts
  the conclusions of the inference mechanism into
  actual inputs for the process.

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4.2 Fuzzy control algorithm
consider a fuzzy controller with three inputs and
a single output only
Given a MISO controller with inputs x1, x2, x3
and output y and assuming that the linguistic
control rules are of the form
then the membership function of the output of the
controller is given by
operator -- implies max-min or max-product
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4.2 Fuzzy control algorithm
the degree of fulfillment of the j-th rule
is defined by
Mamdani max-min implication
Mamdani max-min implication
max-product implication
max-product implication
the degree of fulfillment a measure of how
closely the inputs to the controller match the
control rules. They can be viewed conveniently as
weights that are assigned to every rule.
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4.2 Fuzzy control algorithm
any instant k the membership function of the
output of the controller is
Mamdani max-min implication
max-product implication
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4.2 Fuzzy control algorithm
Steps in the fuzzification algorithm
1?determination of the minimum intercepts for
each input, i.e., their membership value.
2?determination of the degrees of fulfillment of
every rule. 3?determination of the composite
membership function of the output of the
controller.
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4.2 Fuzzy control algorithm
Example Graphical interpretation of
fuzzification
assume that the controller has two inputs and a
single output. Assume that the first input
to t he controller Input_1 ( x1) is specified
by 5 fuzzy sets, while Input_2 ( x2) is
specified by 3 fuzzy sets. The linguistic
variables are assumed to be VLVery_Low,
LOLOw, ZOZerO, LHLittle_H ig h , MHMediu m_
High, and VHVery _ High. Assume that the
following 15 control rules constitute the rule
base
Assume that the following 15 control rules
constitute the rule base
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4.2 Fuzzy control algorithm
rule matrix
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4.2 Fuzzy control algorithm
For simplicity, assume, furthermore that the
fuzzy sets of the inputs and outputs are
triangular
universes of discourse percentages of their
maximum permissible values
pressure deviation
temperature deviation
the opening of a servo-valve
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4.2 Fuzzy control algorithm
The first five rules
the instantaneous inputs to the controller
are - 20 and -50 respectively.
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4.2 Fuzzy control algorithm
membership value
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4.2 Fuzzy control algorithm
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4.2 Fuzzy control algorithm
Larsen implication,
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4.2 Fuzzy control algorithm
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4.2.2 Defuzzification
yield a single crisp value that uniquely
specifies the desired control action.
there is no theoretical basis for deciding
Simplicity and speed of computation are
invariably the primary requirements in industrial
controllers
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4.2.2 Defuzzification
1. Center of gravity (COG) defuzzification
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4.2.2 Defuzzification
2. Center-average
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4.2.2 Defuzzification

• 3. Max criterion (maximum method)
• one of the variable values at which the fuzzy
  subset has its maximum truth value is chosen as
  the crisp value for the output variable.

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4.2.2 Defuzzification
4. Mean of maximum
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4.2.2 Defuzzification
5. Center of area (COA) defuzzification
Weighted average
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4.2.2 Defuzzification
Examples ( water temperature middle)
0.0/00.0/100.33/200.67/301.0/40
1.0/500.75/600.5/700.25/800.0/900.0/100
COA
(00.0100.0200.33300.67401.0501.0
600.75700.5800.25900.01000.0)
/(0.00.00.330.671.01.00.750.50.250.00.0)
48.2
maximum
(40 50)/245
Mean of maximum
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4.4 Design Considerations
Shape of the fuzzy sets
basic criteria in selecting the shapes of
membership functions

• Experience in similar controllers
• computational ease

In practice triangular and trapezoidal
functions are generally used
there is no theory which can guide the designer
on the best shape to use for a specific
application.
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4.4 Design Considerations
Coarseness of the fuzzy sets
The number of fuzzy sets that are required to
specify a variable is termed the coarseness of
the controller and determines the accuracy of the
controller.
fuzzy sets for coarse-fine control
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4.4 Design Considerations
Completeness of the fuzzy sets
The fuzzy control algorithm must lead to a unique
control action for any set of inputs. This
property is termed completeness and depends on
the contents of the knowledge-base as well as the
number and shape of the fuzzy sets used to
describe the inputs and outputs of the
controller.
overlap