FUZZY CONTROL DEFINITIONS

1
FUZZY CONTROL DEFINITIONS

• The term control is generally defined as a
  mechanism used to guide or regulate the operation
  of a machine, apparatus or constellations of
  machines and apparatus.
• Often the notion of control is inextricably
  linked with feedback a process of returning to
  the input of a device a fraction of the output
  signal. Feedback can be negative, whereby
  feedback opposes and therefore reduces the input,
  and feedback can be positive whereby feedback
  reinforces the input signal.
• 'Feedback control' is thus a mechanism for
  guiding or regulating the operation of a system
  or subsystems by returning to the input of the
  (sub)system a fraction of the output.
• The machinery or apparatus etc., to be guided or
  regulated is denoted by S, the input by W and the
  output by y, and the feedback controller by C.
  The input to the controller is the so-called
  error signal e and the purpose of the controller
  is to guarantee a desired response of the output
  y.

 
2
FUZZY CONTROL DEFINITIONS
  'Feedback control' is thus a mechanism for
guiding or regulating the operation of a system
or subsystems by returning to the input of the
(sub)system a fraction of the output.   The
machinery or apparatus etc., to be guided or
regulated is denoted by S, the input by W and the
output by y, and the feedback controller by C.
The input to the controller is the so-called
error signal e and the purpose of the controller
is to guarantee a desired response of the output
y.
 
3
FUZZY CONTROL DEFINITIONS Conventional Control
and Fuzzy Control

• 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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FUZZY CONTROL DEFINITIONS Conventional Control
and Fuzzy Control

• Logical rules with vague predicates can be used
  to derive inference from vague formulated data.
  The idea of linguistic control algorithms was a
  brilliant generalisation of the human experience
  to use linguistic rules with vague predicates in
  order to formulate control actions. The main
  paradigm of fuzzy control is that the control
  algorithm is a knowledge-based algorithm,
  described by the methods of fuzzy logic.

 
 
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FUZZY CONTROL DEFINITIONS Conventional Control
and Fuzzy Control
 
 
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FUZZY CONTROL FUZZY CONTROLLERS

• A fuzzy controller is a device that is intended
  to modelise some vaguely known or vaguely
  described process.
• The controller can be used with the process in
  two modes Feedback mode when the fuzzy
  controller will act as a control device and
  feedforward mode where the controller can be used
  as a prediction device. All inputs to, and
  outputs from, the controller are in the form of
  linguistic variables. In many ways, a fuzzy
  controller maps the input variables into a set of
  output linguistic variables.
• A typical fuzzy logic controller is described by
  the relationship between change of control (u(k))
  on the one hand and the error (e(k)) and change
  in the error ?e(k) e(k) -e(k-1). Such a
  control law is formalised as ?u(k) F(e(k),
  ?e(k)).

 
 
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FUZZY CONTROL FUZZY CONTROLLERS
A fuzzy logic controller (FLC) with a rule base
is defined by the matrix
 
 
where the matrix interrelates the error value
e(k) in at a given time k, ?e(k) denotes the
change in error ( e(k) - e(k-1)), and the
control change ?u(k) is defined as the difference
between u(k) and u(k-1). The term-sets of the
input and output variables of the FLC error e,
error change ?e and control change ?u by the
linguistic labels negative (N), approximately
zero (Z) and positive I(P). The above FLC matrix
can equivalent antecedent/consequent rule set
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FUZZY CONTROL FUZZY CONTROLLERS
A fuzzy logic controller (FLC) can be described
as a function of a number of variables - an FLC
helps us to see how the variables are related to
each other.   This relationship uses rules (of
thumb) involving vague predicates like   ?e is
approximately zero ?u is positive e is
negative
 
 
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FUZZY CONTROL FUZZY CONTROLLERS
FORMALLY the rules are expressed as IF U1 is
B11 AND U2 is B12 THEN V is D1 ALSO IF U1 is
B21 AND U2 is B22 THEN V is D2 ALSO  
IF U1 is Bm1 AND U2 is Bm2 THEN V
is Dm   U1, U2 and V are variables   ? U1 and
U2 are input variables ? V is the output
variable.   Bi1, Bi2 and Di are linguistic values
(labels) represented as fuzzy subsets of the
respective universes of discourse X1, X2 and
Y. The membership functions of these linguistic
values are denoted as Bi1(X1) Bi2(X2) and Di(Y)
 
 
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FUZZY CONTROL FUZZY CONTROLLERS
EXAMPLE U1 ? error e(k) U2 ? change of error
?e(k) and V ? change of control ?u B11 ?
positive B32 ? approximately zero Given the
inputs to an FLC are the values U1 x1 and U2
x2, then we are faced with the problem of
determining the appropriate value of the variable
V.
 
 
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FUZZY CONTROL FUZZY CONTROLLERS
A CONTROL PROCEDURE   FIND the firing level of
each of the rules ?FUZZIFICATION FIND the output
of each of the rules ?INFERENCE AGGREGATE the
individual rule outputs to obtain the overall
system output ?COMPOSITION OBTAIN a crisp
value to be input to the controlled system
? DEFUZZIFICATION
 
 
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FUZZY CONTROL FUZZY CONTROLLERS
The level of matching between the linguistic
label Bi1 and the input value x1 is determined by
the membership grade of x1 in the fuzzy set
representing Bi1 Bi1(x1) as the level of
matching for the first antecedant. Similarly
Bi2(x2) is the level of matching for the second
antecedant. 1. The level or degree of firing for
a rule set of the type Rule If U1 is Bi1 AND
U2 is Bi2 THEN V is Di   which has a conjunctive
connection.  The degree of firing (DOF) of the
ith rule with respect to input values U1 is x1
and U2 is x2 is given as ?i ?i Bi1(x1) ?
Bi2(x2) ? min(Bi1(x1), Bi2(x2))
 
 
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FUZZY CONTROL FUZZY CONTROLLERS
2. The output of individual rule (Fi(y)) depends
on the interaction between the DOF) (?i) and the
consequent of the rule (Di). The MAMDANI method
suggests that   Fi(y) ?i ? Di ?
min (?i, Di)
 
 
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FUZZY CONTROL FUZZY CONTROLLERS
3. The aggregation of the individual rule
outputs to obtain the overall control system
output (Fy). The rules are chained through
ALSO thus individual rule outputs are aggregated
using the disjunctive connective ALSO. The
overall system output is (Fy) ViFi(y)
Vi(?i?Di(y)) OR connective   Recall
IF U1 is B11 AND U2 is B12 then U is D1
ALSO IF U1 is Bm1 AND V2 is Bm2 then V is Dm
 
 
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FUZZY CONTROL FUZZY CONTROLLERS
4. The extraction of a crisp output value The
output fuzzy set cannot be used directly as input
to the controlled system. We need to select one
element y from the universe Y (e.g., ?u from
all possible values of ?u - the change of
control). So we should defuzzify typically used
methods of defuzzification include the so-called
centre of area method. Let the output Y be a
finite universe of discourse and F(y) be a
DISCRETE membership function
 
 
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FUZZY CONTROL FUZZY CONTROLLERS- An example
A worked example Consider an FLC of Mamdani type
 
 
which expresses rules like   Rule 1 If e(k) is
negative AND ?e(k) is negative then ?u(k) is
negative ALSO Rule 9 If e(k) is positive AND
?e(k) is positive then ?u(k) is positive
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FUZZY CONTROL FUZZY CONTROLLERS- An example
The membership functions for the three elements
of the term set for the error e are given as
 
 
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FUZZY CONTROL FUZZY CONTROLLERS
The reference fuzzy set for the error for a
specific case is given as
 
 
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FUZZY CONTROL FUZZY CONTROLLERS- An example
The membership functions for the three elements
of the term set for the change in error ?e are
given as
 
 
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FUZZY CONTROL FUZZY CONTROLLERS
The reference fuzzy set for the change in error
?e is given as
 
 
21
FUZZY CONTROL FUZZY CONTROLLERS- An example
The membership functions for the three elements
of the term set for the change in control ?u are
given as
 
 
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FUZZY CONTROL FUZZY CONTROLLERS
The reference fuzzy set for the change in control
?u is given as
 
 
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FUZZY CONTROL FUZZY CONTROLLERS An example

• Recall that there are nine rules in all
• IF e(k) is NEGATIVE De(k) is NEGATIVE THEN
  Du(k) is NEGATIVE
• IF e(k) is NEGATIVE De(k) is ZERO THEN Du(k)
  is NEGATIVE
• IF e(k) is NEGATIVE De(k) is POSITIVE THEN
  Du(k) is ZERO
• IF e(k) is ZERO De(k) is NEGATIVE THEN Du(k)
  is NEGATIVE
• IF e(k) is ZERO De(k) is ZERO THEN Du(k) is
  ZERO
• IF e(k) is ZERO De(k) is POSITIVE THEN Du(k)
  is POSITIVE
• IF e(k) is POSITIVE De(k) is NEGATIVE THEN
  Du(k) is ZERO
• IF e(k) is POSITIVE De(k) is ZERO THEN Du(k)
  is POSITIVE
• IF e(k) is POSITIVE De(k) is POSITIVE THEN
  Du(k) is POSITIVE

 
 
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FUZZY CONTROL FUZZY CONTROLLERS An example
We need the the membership functions for the
output in a discrete form. By equidistant
discretization of the universe Y
Y-6,-4.5,-3,-1.5,0,1.5,3,4.5,6) we have   
 
 
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FUZZY CONTROL FUZZY CONTROLLERS An example
Consider the case where e(k) -2.1 and De(k)
0.5. The fuzzification of the input leads to two
observations
 
 
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FUZZY CONTROL FUZZY CONTROLLERS An example
For the case where e(k) -2.1 and De(k) 0.5, the
level or degree of firing for the 9-rule rule set
 
 
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FUZZY CONTROL FUZZY CONTROLLERS An example
The output of individual rule (Fi(y)) depends on
the interaction between the degree (level) of
firing (DOF or ?i) and the consequent of the rule
(Di). We are considering the case where e(k)
-2.1 and De(k) 0.5.
 
 
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FUZZY CONTROL FUZZY CONTROLLERS An example
The MAMDANI method suggests that Fi(y) ?i ? Di
? min (?i, Di) We are considering the case where
e(k) -2.1 and De(k) 0.5.
 
 
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FUZZY CONTROL FUZZY CONTROLLERS An example
The aggregation of the individual rule outputs to
obtain the overall control system output (Fy).
The rules are chained through ALSO thus
individual rule outputs are aggregated using the
disjunctive connective ALSO.
 
 
The overall system output for the inputs e(k)
-2.1 and De(k) 0.5 is
30
FUZZY CONTROL FUZZY CONTROLLERS An example
The extraction of a crisp output value The
output fuzzy set cannot be used directly as input
to the controlled system. We need to select one
element y from the universe Y (e.g., ?u from
all possible values of ?u - the change of
control). So we should defuzzify typically used
methods of defuzzification include the so-called
centre of area method. Let the output Y be a
finite universe of discourse and F(y) be a
DISCRETE membership function
 
31
FUZZY CONTROL FUZZY CONTROLLERS An example
The extraction of a crisp output value Centre of
Area Computation The weighted average of the
output fuzzy set with the corresponding value of
the membership function for each of the change in
control Du for the inputs e(k) -2.1 and De(k)
0.5 is
 
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FUZZY CONTROL FUZZY CONTROLLERS Another example
For the case where e(k) -0.9 and De(k) 0.2, the
level or degree of firing for the 9-rule rule set
 
 
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FUZZY CONTROL FUZZY CONTROLLERS Another example
The output of individual rule (Fi(y)) depends on
the interaction between the degree (level) of
firing (DOF or ?i) and the consequent of the rule
(Di). We are considering the case where e(k)
-0.9 and De(k) 0.2
 
 
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FUZZY CONTROL FUZZY CONTROLLERS Another example
The MAMDANI method suggests that Fi(y) ?i ? Di
? min (?i, Di) We are considering the case where
e(k) -0.9 and De(k) 0.2.
 
 
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FUZZY CONTROL FUZZY CONTROLLERS Another example
The aggregation of the individual rule outputs to
obtain the overall control system output (Fy).
The rules are chained through ALSO thus
individual rule outputs are aggregated using the
disjunctive connective ALSO.
 
 
The overall system output for the inputs e(k)
-0.9 and De(k) 0.2 is
36
FUZZY CONTROL FUZZY CONTROLLERS Another example
The extraction of a crisp output value Centre of
Area Computation The weighted average of the
output fuzzy set with the corresponding value of
the membership function for each of the change in
control Du for the inputs e(k) -0.9 and De(k)
0.2 is
 
The defuzzified value of Du is
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FUZZY CONTROL FUZZY CONTROLLERS Takagi-Sugeno
Controllers
According to Yager and Filev, a known
disadvantage of the linguistic modules is that
they do not contain in an explicit form the
objective knowledge about the system if such
knowledge cannot be expressed and/or incorporated
into fuzzy set framework' (1994192).   Typically,
such knowledge is available often for example
in physical systems this kind of knowledge is
available in the form of general conditions
imposed on the system through conservation laws,
including energy mass or momentum balance, or
through limitations imposed on the values of
physical constants.
 
 
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FUZZY CONTROL FUZZY CONTROLLERS Takagi-Sugeno
Controllers

• Tomohiro Takagi and Michio Sugeno recognised two
  important points
• Complex technological processes may be described
  in terms of interacting, yet simpler sub
  processes. This is the mathematical equivalent of
  fitting a piece-wise linear equation to a complex
  curve.
• The output variable(s) of a complex physical
  system, e.g. complex in the sense it can take a
  number of input variables to produce one or more
  output variable, can be related to the system's
  input variable in a linear manner provided the
  output space can be subdivided into a number of
  distinct regions.

 
 
Takagi, T., Sugeno, M. (1985). Fuzzy
Identification of Systems and its Applications to
Modeling and Control. IEEE Transactions on
Systems, Man and Cybernetics. Volume No. SMC-15
(No.1) pp 116-132.
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FUZZY CONTROL FUZZY CONTROLLERS Takagi-Sugeno
Controllers
Literature on conventional control systems has
suggested that a complex non-linear system can be
described as a collection of subsystems that were
combined based on a logical (Boolean) switching
system function.   In realistic situations such
disjoint (crisp) decomposition is impossible, due
to the inherent lack of natural region boundaries
in the system, and also due to the fragmentary
nature of available knowledge about the system.
 
 
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FUZZY CONTROL FUZZY CONTROLLERS Takagi-Sugeno
Controllers
Takagi and Sugeno (1985) have argued that in
order to develop a generic and simple
mathematical tool for computing fuzzy
implications one needs to look at a fuzzy
partition of fuzzy input space. In each fuzzy
subspace a linear input-output relation is
formed. The output of fuzzy reasoning is given
by the values inferred by some implications that
were applied to an input.
 
 
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FUZZY CONTROL FUZZY CONTROLLERS Takagi-Sugeno
Controllers
Consider a domain where all fuzzy sets are
associated with linear membership functions.
  Let us denote the membership function of a
fuzzy set A as mA(x), x ?X. All the fuzzy sets
are associated with linear membership functions.
Thus, a membership function is characterised by
two parameters giving the greatest grade 1 and
the least grade 0. The truth value of a
proposition x is mA and y is mB is expressed as
 
 
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FUZZY CONTROL FUZZY CONTROLLERS Takagi-Sugeno
Controllers
Takagi and Sugeno have described a fuzzy
implication R is of the format  R if (x1 is
mA(x1), xk is mA(xk)) then y g(x1, ,
xk) Where
 
 
43
FUZZY CONTROL FUZZY CONTROLLERS Takagi-Sugeno
Controllers
In the premise if mA(xi) is equal to Xi for some
i where Xi is the universe of discourse of xi,
this term is omitted xi is unconditioned. The
following example will help in clarifying the
argumentation related to 'conditioned' and
'unconditioned' terms in a given
implication   R if x1 is small and x2 is big
then y x1 x2 2x3.   The above implication
comprises two conditioned premises, x1 and x2,
and one unconditioned premise, x3. The
implication suggests that if x1 is small and x2
is big, then the value of y would depend upon and
be equal to the sum of x1, x2, and 2x3., where x3
is unconditioned in the premise.
 
 
44
FUZZY CONTROL FUZZY CONTROLLERS Takagi-Sugeno
Controllers
Typically, for a Takagi-Sugeno controller, an
implication is written as R if x1 is m1 and
and xk is mk then y p0 p1x1 pkxk. The
assumption here is that only and connectives
are used in the antecedants or premises of the
rules. And, that the relationship between the
output and inputs is strictly a LINEAR (weighted
average) relationship. (The weights here are p0
,p1.. pk).
 
 
45
FUZZY CONTROL FUZZY CONTROLLERS Takagi-Sugeno
Controllers
Reasoning Algorithm Recall the arguments related
to the membership functions of the union and
intersection of fuzzy sets. The union of sets A
and B is given as  mAÇB min (mA, mB)  We
started this discussion by noting that we will
explore the problems of multivariable control
(MultipleInputSingleOutput). Usually, the rule
base in a fuzzy control system comprises a number
of rules in the case of multivariable control
the relevant rules have to be tested for what
they imply.
 
 
46
FUZZY CONTROL FUZZY CONTROLLERS Takagi-Sugeno
Controllers
Consider a system with n implications (rules)
the variable of consequence, y, will have to be
notated for each of these implications, leading
to yi variables of consequence. There are three
stages of computations in Takagi-Sugeno
controllers FUZZIFICATION Fuzzify the input.
For all input variables compute the implication
for each of the rules INFERENCE or CONSEQUENCES
For each implication compute the consequence for
a rule which fires. Compute the output y for the
rule by using the linear relationship between the
inputs and the output (y p0 p1x1
pkxk.). AGGREGATE ( DEFUZZIFICATION) The
final output y is inferred from n-implications
and given as an average of all individual
implications yi with weights y yi y (S
y yi yi )/ S y yi
 
 
47
FUZZY CONTROL FUZZY CONTROLLERS Takagi-Sugeno
Controllers
Consider the following fuzzy implications (or
rules) R1,R2, R3 used in the design of a
Takagi-Sugeno controller R1 ? If x1 is small1
x2 is small2 then y(1) x1x2 R2? If x1 is big1
then y(2) 2x1 R3? If x2 is big2 then y(3)
3x2 where y (i) refers to the consequent
variable for each rule labelled Ri and x1 and x2
refer to the input variables that appear in
premise of the rules.
 
 
48
FUZZY CONTROL FUZZY CONTROLLERS Takagi-Sugeno
Controllers
The membership function for small1, small2, big1
and big2 are given as follows
 
 
49
FUZZY CONTROL FUZZY CONTROLLERS Takagi-Sugeno
Controllers
The membership function for small1, small2, big1
and big2 are given as follows
 
 
50
FUZZY CONTROL FUZZY CONTROLLERS Takagi-Sugeno
Controllers An example
Let us compute the FINAL OUTPUT y for the
following values x1 12 x2 5 using Takagi
and Sugenos formula y (S y yi yi )/ S
y yi where y yi stands for the truth
value of a given proposition.
 
 
51
FUZZY CONTROL FUZZY CONTROLLERS Takagi-Sugeno
Controllers An example
FUZZUFICATION We have the following values of
the membership functions for the two values x1
12 x2 5
 
 
52
FUZZY CONTROL FUZZY CONTROLLERS Takagi-Sugeno
Controllers An example
INFERENCE CONSEQUENCE x1 12 x2 5
 
53
FUZZY CONTROL FUZZY CONTROLLERS Takagi-Sugeno
Controllers An example
AGGREGATION (DEFUZZIFICATION) x1 12 x2 5
y (S y yi yi )/ S y yi Using a
Centre of Area computation for y we get
 
54
FUZZY CONTROL FUZZY CONTROLLERS
Key difference between a Mamdani-type fuzzy
system and the Takagi-Sugeno-Kang System? A
zero-order Sugeno fuzzy model can be viewed as a
special case of the Mandani fuzzy inference
system in which each rule is specified by fuzzy
singleton or a pre-defuzzified consequent. In
Sugenos model, each rule has a crisp output, the
overall input is obtained by a weighted average
this avoids the time-consuming process of
defuzzification required in a Mandani model. The
weighted average operator is replaced by a
weighted sum to reduce computation further.
(Jang, Sun, Mizutani (199782)).