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8/12/2019 Chapt 3 - Fuzzy Logic-2
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INTELLIGENT CONTROL SYSTEM (ICS) Semester2 session 20132014
8/12/2019 Chapt 3 - Fuzzy Logic-2
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Contents
Fuzzy Inference
Fuzzification of the input variables
Rule evaluation
Aggregation of the rule outputs
Defuzzification
Mamdani
Sugeno
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Fuzzy Inference
The most commonly used fuzzy inference technique is the so-
called Mamdanimethod.
In 1975, Professor Ebrahim Mamdani of London University built
one of the first fuzzy systems to control a steam engine and
boiler combination. He applied a set of fuzzy rules supplied by
experienced human operators.
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Mamdani Fuzzy Inference
The Mamdani-style fuzzy inference process is performed in
four steps:
1. Fuzzification of the input variables
2. Rule evaluation (inference)
3. Aggregation of the rule outputs (composition)
4. Defuzzification.
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Mamdani Fuzzy Inference
We examine a simple two-input one-output problem that includes
three rules:
Rule: 1 Rule: 1
IF x is A3 IF project_funding is adequate
OR y is B1 OR project_staffing is small
THEN z is C1 THEN risk is low
Rule: 2 Rule: 2
IF x is A2 IF project_funding is marginal
AND y is B2 AND project_staffing is large
THEN z is C2 THEN risk is normal
Rule: 3 Rule: 3
IF x is A1 IF project_funding is inadequate
THEN z is C3 THEN risk is high
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TABLE 2.1Mathematical Characterization of
Triangular Membership Functions
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TABLE 2.2Mathematical Characterization of
Gaussian Membership Functions
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Step 1: Fuzzification
The first step is to take the crisp inputs, x1 and y1 (project
fundingandproject staffing), and determine the degree to which
these inputs belong to each of the appropriate fuzzy sets.
Crisp Input
y1
0.1
0.7
1
0 y1
B1 B2
Y
Crisp Input
0.2
0.5
1
0
A1 A2 A3
x1
x1 X
(x=A1)= 0.5(x=A2)= 0.2
(y=B1)= 0.1(y=B2)= 0.7
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Step 2: Rule Evaluation
The second step is to take the fuzzified inputs, (x=A1)=0.5, (x=A2)= 0.2, (y=B1)= 0.1 and (y=B2)= 0.7, and apply them to
the antecedents of the fuzzy rules.
If a given fuzzy rule has multiple antecedents, the fuzzy operator
(AND or OR) is used to obtain a single number that representsthe result of the antecedent evaluation.
This number (the truth value) is then applied to the consequent
membership function.
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Step 2: Rule Evaluation (cont)
RECAL:
To evaluate the disjunction of the rule antecedents, we use the OR
fuzzy operation. Typically, fuzzy expert systems make use of the
classical fuzzy operation union:
AB(x) = max [A(x), B(x)]
Similarly, in order to evaluate the conjunction of the rule
antecedents, we apply the AND fuzzy operation intersection:
AB(x) = min [A(x), B(x)]
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A3
1
0 X
1
y10 Y
0.0
x1 0
0.1C1
1
C2
Z
1
0 X
0.2
0
0.2C1
1
C2
Z
A2
x1
Rule3:
A11
0 X 0
1
Zx1
THEN
C1 C2
1
y1
B2
0 Y
0.7
B10.1
C3
C3
C30.5 0.5
OR(max)
AND(min)
OR THENRule1:
AND THENRule2:
IFxisA3 (0.0) isB1 (0.1) z isC1 (0.1)
IFxisA2 (0.2) isB2 (0.7) z isC2 (0.2)
IFxisA1 (0.5) z isC3 (0.5)
Step 2: Rule Evaluation (cont)
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Now the result of the antecedent evaluation can be applied to
the membership function of the consequent.
There are two main methods for doing so:
Clipping Scaling
Step 2: Rule Evaluation (cont)
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The most common method of correlating the rule consequentwith the truth value of the rule antecedent is to cut theconsequent membership function at the level of the antecedenttruth. This method is called clipping(alpha-cut).
Since the top of the membership function is sliced, the clippedfuzzy set loses some information.
However, clipping is still often preferred because it involves lesscomplex and faster mathematics, and generates an aggregatedoutput surface that is easier to defuzzify.
Step 2: Rule Evaluation (cont)
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While clipping is a frequently used method, scalingoffers a
better approach for preserving the original shape of the fuzzy
set.
The original membership function of the rule consequent is
adjusted by multiplying all its membership degrees by the truth
value of the rule antecedent.
This method, which generally loses less information, can be
very useful in fuzzy expert systems.
Step 2: Rule Evaluation (cont)
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Step 2: Rule Evaluation (cont)
Degree ofembership
1.0
0.0
0.2
Z
Degree ofembership
Z
C2
1.0
0.0
0.2
C2
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Step 3: Aggregation of the rule outputs
Aggregation is the process of unification of the outputs of all
rules.
We take the membership functions of all rule consequents
previously clipped or scaled and combine them into a single
fuzzy set.
The input of the aggregation process is the list of clipped or
scaled consequent membership functions, and the output is one
fuzzy set for each output variable.
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0
0.1
1C1
Cis 1 (0.1)
C2
0
0.2
1
Cis 2 (0.2)0
0.5
1
Cis 3 (0.5)ZZZ
0.2
Z0
C30.5
0.1
Step 3: Aggregation of the rule outputs (cont)
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Step 4: Defuzzification
The last step in the fuzzy inference process is defuzzification.
Fuzziness helps us to evaluate the rules, but the final output of a
fuzzy system has to be a crisp number.
The input for the defuzzification process is the aggregate output
fuzzy set and the output is a single number.
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There are several defuzzification methods, but probably themost popular one is the centroid technique. It finds the point
where a vertical line would slice the aggregate set into two equal
masses. Mathematically this centre of gravity(COG) can be
expressed as:
Step 4: Defuzzification (cont)
b
a
A
b
a
A
dxx
dxxx
COG
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Step 4: Defuzzification (cont)
Centroid defuzzification method finds a point representing thecentre of gravity of the fuzzy set,A, on the interval, ab.
A reasonable estimate can be obtained by calculating it over a
sample of points.
(x)1.0
0.0
0.2
0.4
0.6
0.8
160 170 180 190 200
a b
210
A
150
X
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Step 4: Defuzzification (cont)
1.0
0.0
0.2
0.4
0.6
0.8
0 20 30 40 5010 70 80 90 10060
Z
Degree ofembership
67.4
4.675.05.05.05.02.02.02.02.01.01.01.0
5.0)100908070(2.0)60504030(1.0)20100(
COG
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Sugeno Fuzzy Inference
Mamdani-style inference, as we have just seen, requires us tofind the centroid of a two-dimensional shape by integrating
across a continuously varying function. In general, this process
is not computationally efficient.
Michio Sugeno suggested to use a single spike, a singleton, as
the membership function of the rule consequent.
A singleton, or more precisely a fuzzy singleton, is a fuzzy set
with a membership function that is unity at a single particularpoint on the universe of discourse and zero everywhere else.
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Sugeno-style fuzzy inference is very similar to the Mamdanimethod. Sugeno changed only a rule consequent. Instead of afuzzy set, he used a mathematical function of the input variable.The format of the Sugeno-style fuzzy ruleis
IF xisAAND yis B
THEN zis f(x, y)
wherex, yand zare linguistic variables;Aand Bare fuzzy sets
on universe of discoursesXand Y, respectively; and f(x, y) is amathematical function.
Sugeno Fuzzy Inference (cont)
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The most commonly used zero-order Sugeno fuzzy modelapplies fuzzy rules in the following form:
IF xisA
AND yis B
THEN zis k
where kis a constant.
In this case, the output of each fuzzy rule is constant. All
consequent membership functions are represented by singletonspikes.
Sugeno Fuzzy Inference (cont)
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Sugeno Rule Evaluation
A3
1
0 X
1
10 Y
0.0
x1 0
0.1
1
Z
1
0 X
0.2
0
0.2
1
Z
A2
x1
IFxisA1 (0.5) zis k3 (0.5)Rule3:
A11
0 X 0
1
Zx1
THEN
1
y1
B2
0 Y
0.7
B1
0.1
0.5 0.5
OR(max)
AND(min)
OR isB1 (0.1) THEN zis k1 (0.1)ule1:
IFxisA2 (0.2) AND isB2 (0.7) THEN zis k2 (0.2)Rule2:
k1
k2
k3
IFxisA3 (0.0)
A3
1
0 X
1
10 Y
0.0
x1 0
0.1
1
Z
1
0 X
0.2
0
0.2
1
Z
A2
x1
IFxisA1 (0.5) zis k3 (0.5)Rule3:
A11
0 X 0
1
Zx1
THEN
1
y1
B2
0 Y
0.7
B1
0.1
0.5 0.5
OR(max)
AND(min)
OR isB1 (0.1) THEN zis k1 (0.1)ule1:
IFxisA2 (0.2) AND isB2 (0.7) THEN zis k2 (0.2)Rule2:
k1
k2
k3
IFxisA3 (0.0)
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Sugeno Aggregation of the Rule Outputs
zis k1 (0.1) zis k2 (0.2) zis k3 (0.5)
0
1
0.1
Z 0
0.5
1
Z0
0.2
1
Zk1 k2 k3 0
1
0.1
Zk1 k2 k3
0.20.5
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Sugeno Defuzzification
Weighted Average (WA)
655.02.01.0
805.0502.0201.0
)3()2()1(
3)3(2)2(1)1(
kkk
kkkkkkWA
0 Z
Crisp Output
z1
z1
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Mamdani or Sugeno?
Mamdani method is widely accepted for capturing expertknowledge. It allows us to describe the expertise in more
intuitive, more human-like manner. However, Mamdani-type
fuzzy inference entails a substantial computational burden.
On the other hand, Sugeno method is computationally effective
and works well with optimisation and adaptive techniques, which
makes it very attractive in control problems, particularly for
dynamic nonlinear systems.
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Takagi-Sugeno Fuzzy Systems
The fuzzy system defined in the previous sections will be referred to as a
standard fuzzy system.
In this section we will define a functional fuzzy system, of which the Takagi-
Sugeno fuzzy system is a special case.
For the functional fuzzy system, that uses singleton fuzzification, and the ith
MISO rule has the form
where simply represents the argument of the function giand the bi are
not output membership function centers. The premise of this rule is defined
the same as it is for the MISO rule for the standard fuzzy system
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chooseomay want tinstance,forpossible,areothersmanybutfunctionsaffineandlineardiscusswillweBelow,
.consideredbeingnapplicatioon thedependsfunctiontheofchoiceThe
used.bealsomayvariables
otherbut21,termsthecontainsofargumentoften thethatNotice
function.membership
associatedanhavenotdoesthatsystem)fuzzylfunctionanamethe(hence
()functionauseweconsequentin thefunction,membershipassociatedanwith
termlinguisticaofInsteadhowever.different,arerulestheofsconsequentThe
,, . . ., n,iug
gb
ii
ii
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For the functional fuzzy system, we can use an appropriate operation for
representing the premise (e.g., minimum or product), and defuzzification
may be obtained using
whereiis defined in Equation bellow:
It is assumed that the functional fuzzy system is defined so that no matter
what its inputs are, we have
One way to view the functional fuzzy system is as a nonlinear interpolatorbetween the mappings that are defined by the functions in the
consequents of the rules.
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An Interpolator Between Linear Mappings
In the case where
mappings.linearbetweenioninterpolatnonlineara
performsyessentiallsystemfuzzySugeno-Takagithat theseeweOverall,
e.conveniencformapping
linearaasmappingaffinetherefer towillwestandard,isashowever,Often,
affine.calledismapping
then the,0ifandmappinglinearaismapping()then the,0If
system.fuzzySugeno-Takagia
astoreferredissystemfuzzyfunctionalthenumbersrealarethewhere
00 i,ii,
i,j
aga
a
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As an example, suppose that n = 1, R = 2, and that we have rules
havewe,~
representsand~
representsthatso2.5Figureingivenfordiscourseofuniversewith the
2
12
1
1
11
AA
u
For u1> 1,1= 0, so y = 1+u1, which is a line. If u1< 1,2= 0, so y = 2+u1,
which is a different line.
Between 1 u1 1, the output y is an interpolation between the two lines.
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Figure 2.5Membership functions for Takagi-Sugeno fuzzy system example.
Finally, it is interesting to note that if we pick
(i.e., ai,j = 0 forj > 0), then the Takagi-Sugeno fuzzy system is equivalent to
a standard fuzzy system that uses center-average defuzzification with
singleton output membership functions at ai,0.
It is in this sense that the Takagi-Sugeno fuzzy systemor, more generally,
the functional fuzzy systemis sometimes referred to as a general fuzzy
system.
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An Interpolator Between Linear Systems
It is important to note that a Takagi-Sugeno fuzzy system may have any linear
mapping (affine mapping) as its output function, which also contributes to its
generality.
One mapping that has proven to be particularly useful is to have a linear
dynamic system as the output function so that the ithrule has the form
system.fuzzytheinput toldimensiona
-theis)](.,..),(),([)(anddimension,eappropriatofmatricesinputandstatetheare21,andinput,modelldimensiona
-theis)](.,..),(),([)(inputs,ofnumberthey)necessaril
notis(nowstateldimensiona-theis)](.,..),(),([)(Here,
T
21
T
21
T
21
ptztztztz, . . ., R,iBA
mtutututu
nntxtxtxtx
p
ii
m
n
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This fuzzy system can be thought of as a nonlinear interpolator between
R linear systems, it takes the input z(t) and has an output
If R = 1, we get a standard linear system.
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ruleswith2and,1),()(thatsupposeexample,anAs
)].(),([)(sometimesor),()(chooseinstance,for),(forpossiblearechoicesMany
output.thetocontributeand
onturnwillrulescertainonly),(ofegiven valuaand1forGenerally,
Rmnptxtz
tutxtztxtztz
tzR
).with
2.5Figureofaxishorizontaltherelabelwe(i.e.,lyrespective,~and~for
functionsmembershiptheas2.5Figurefromandusethat weSuppose
1
2
1
1
1
21
x
AA
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Takagi-Sugeno fuzzy system interpolates between the two linear systems.
We see that for changing values ofx1(t), the two linear systems that are in the
consequents of the rules contribute different amounts.
We think of one linear system being valid on a region of the state space that
is quantified via1and another on the region quantified by2(with a fuzzy
boundary in between).
For the higher-dimensional case, we have premise membership functions for
each rule quantify whether the linear system in the consequent is valid for aspecific region on the state space.
As the state evolves, different rules turn on, indicating that other combinations
of linear models should be used.
Overall, we find that the Takagi-Sugeno fuzzy system provides a very intuitiverepresentation of a nonlinear system as a nonlinear interpolation between R
linear models.
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Figure 2.6: Commonly used fuzzy if-then rules and fuzzy mechanism.
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Lets consider the nonlinear system below:
The goal is to derive a T-S fuzzy model from the above given nonlinear system
equations by the sector nonlinearity approach as if the response of the T-S
fuzzy model in the specified domain exactly match with the response of the
original system with the same input u.
The following steps should be taken to derive the T-S fuzzy model of aboveequation. For simplicity, we assume thatx1 [0.5, 3.5] and x2 [-1, 4]. Here x1
and x2 are nonlinear terms in the equations in the last equations so we make
them as our fuzzy variables.
Generally they are denoted as z1, z2 and are known as premise variables that
may be functions of state variables, input variables, external disturbances
and/or time. Therefore z1 = x1 and z2 = x2. Equation above can be written as
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wherex(t) = [x1(t) x2(t)]T. The first step for any kind of fuzzy modeling is to
determine the fuzzy variables and fuzzy sets or so-called membership
functions. Although there is no general procedure for this step and it can be
done by various methods predominantly trial and error, in exact fuzzy modeling
using sector nonlinearity, it is quite routine. It is assumed that the premise
variables are just functions of the state variables for the sake of simplicity. Thisassumption is needed to avoid a complicated defuzzification process of the
fuzzy controllers [9].
To acquire membership functions, we should calculate the minimum and
maximum values of z1(t) and z2(t) which under x1 [0.5, 3.5] and x2 [-1. 4],
they are obviously obtained as follows:
max z1(t) = 3.5; min z1(t) = 0.5;max z2(t) = 4; min z2(t) = -1:
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Figure 2.7: Membership functions M1(z1(t)), M2(z2(t)), N1(z2(t)) and N2(z2(t)).
We name the membership functions "Positive", "Negative," "Big," and "Small,"
respectively. Figure 2.7 shows these membership functions.
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Here, we can generalize that the ithrule of the continuous T-S fuzzy models
are of the following forms:
Model Rule i:
IF z1(t) is Mi1 and and zp(t) is Mip,
Here, Mij is the fuzzy set and r is the number of model rules; x(t) is the state
vector, u(t) is the input vector, y(t) is the output vector, Ai is the square matrix
with real elements and z1(t),..,zp(t) are known premise variables asmentioned before. Each linear consequent equation represented byAix(t) +
Biu(t) is called a subsystem.
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Therefore, the nonlinear system equation previous is modeled by the
following fuzzy rules (we don't consider input u(t) in this stage):
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Therefore, if z1 = x1 = 2.75 and z2 = x2 = 0.25, the T-S fuzzy modeling
implication can be derived as:
Figure 2.8: Given the value of z1 = 2.75 and z2 = 0.25 to the membership
functions
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