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Fuzzy Logic and Control
A. Homaifar
Autonomous Control & Information Technology(ACIT) Center
Department of Electrical & Computer Engineering
North Carolina A&T State University
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OUTLINE
Fundamentals
Fuzzy Sets, Operators,
GMP and GMT
Fuzzy Engines
Hybrid Fuzzy-PID Controllers
Generalized Sugeno Controllers
Applications
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1. A B(u) = max { A (u), B(u) }
2. A B(u) = min { A (u), B(u) }
3. A (u) = 1.0 - A (u)
Set Theoretic Operations
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Union Of Fuzzy Sets
0
1
A
X
AB(X)
B
Union of fuzzy sets A & B
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Intersection Of Fuzzy Sets
0
1
A
X
AB(X)
B
Intersection of fuzzy sets A & B
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Complement Of Fuzzy Set
0
1
A
X
A
A(X)
A(X) = 1 - A(X)
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Law of Contradiction
0
1
A
X
A
Fuzzy AA
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Law Of Excluded Middle
0
1
A
X
A
Fuzzy A A X
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Definition oft -norms
t-normsare two-valued functions from [0,1]x[0,1]
that satisfy the following conditions:
vity)(associati(x))(x)),(x),t(t((x))),(x)t((x),t(4.
vity)(commutati(x))(x),t((x))(x),t(3.
ity)(monotonic(x)(x)and(x)(x)if
(x))(x),t((x))(x),t(2.
Xx(x),(x))t(1,(x),1)t(0;t(0,0)1.
C~
B~
A~
C~
B~
A~
A~
B~
B~
A~
D~
B~
C~
A~
D~
C~
B~
A~
A~
A~
A~
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Definition oft -conormst-conormsor s-normsare associative,
commutative, and monotonic two-placed functions
sthat map from [0,1]x[0,1] into [0,1] that satisfy the
following conditions:
vity)(associati(x))(x)),(x),s(s((x))),(x)s((x),s(4.
vity)(commutati(x))(x),s((x))(x),s(3.
ity)(monotonic(x)(x)and(x)(x)if
(x))(x),s((x))(x),s(2.
Xx(x),(x))s(0,(x),0)s(1;s(1,1)1.
C~
B~
A~
C~
B~
A~
A~
B~
B~
A~
D~B~C~A~
D~
C~
B~
A~
A~
A~
A~
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Relationship betweent -norms
andt -conorms
t-normsand t-conorms are related in a sense of
logical duality.
t-conorm
as a two-placed function s mapping from[0,1] x [0,1] in [0,1] such that the function t defined
as
(x))-1(x),-s(1-1(x))(x),t(B
~
A
~
B
~
A
~
Is at-norm.
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R (x,y) defines a relation between x and y:Y1 Y2 Y3
X1 0.2 1 0.4
X2 0 0.6 0.3X3 0 1 0.8
Composition of Rand S
R oS = { [(u,w), sup V (R (u,v) * S (v,w) ], u U,v U, w W}
Fuzzy Relation
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Example:
R = R (x,y) = 0.7 0.5
0.8 0.4
S = S (y,z) = 0.9 0.6 0.2
0.1 0.7 0.5
T = T (x,z) = V y Y(R (x,y) S (y,z) )
= 0.7 0.6 0.5
0.8 0.6 0.4For Example:
T (x,z) = max [ min (0.7,0.9), min (0.5, 0.1) ]
= max [ 0.7, 0.1 ] = 0.7
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Max ProductT = T (x,z) = V y Y(R (x,y) . S (y,z) )
= 0.63 0.42 0.25
0.72 0.48 0.24For Example:
T (x,z) = max [ (0.7)*(0.9) , (0.5)*(0.1) ]
= max [ 0.63, 0.05 ] = 0.63
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Max Average
T = T (x,z) = 0.5V y Y(R (x,y) + S (y,z))
= 0.8 0.65 0.5
0.85 0.7 0.5
For Example:
T (x,z) = 0.5 max [ (0.7)+(0.9) , (0.5)+(0.1) ]
= 0.5 max [ 1.6, 0.6 ] = 0.8
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Typical dual pairs of
nonparameterized t-norms and t-conorms
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))x(),x((B~
A~
otherwise..............................0
1)}x(),x(max{if)}...x(),x(min{ B~A~B~A~
drastic product
drastic sumsw ))x(),x(( B~A~ =
otherwise...............................10)}x(),x(min{if)}...x(),x(max{ B~A~B~A~
}1)()(,0{max))(),((t ~~~~1 xxxxBABA
bounded different
tw =
)}()(,1min{))(),(( ~~~~1 xxxxs BABA bounded sum
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))x(),x(( B~A~ t1.5 =)]x().x()x()x([2
)x().x(
B~
A~
B~
A~
B~
A~
Einsteinproduct
s1.5 ))x(),x(( B~A~ =
Einstein
sum)x().x(1
)x()x(
B~
A~
B~
A~
t2 ))x(),x(( B~A~ =
s2 ))x(),x(( B~A~ =
)x().x(B~
A~
)x().x()x()x(B~
A~
B~
A~
Algebraic product
Algebraic sum
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t2.5 ))x(),x(( B~A~ =)x().x()x()x(
)x().x(
B~
A~
B~
A~
B~
A~
Hamacher
product
s2.5 ))x(),x(( B~A~ =)x().x(1
)x().x(2)x()x(
B~
A~
B~A~B~A~
Hamacher
sum
t3 ))x(),x(( B~A~ =
s3 ))x(),x(( B~A~ =
min
max
)}x(),x({B~
A~
)}x(),x({B~
A~
minimum
maximum
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Theorem 1:
All t-norm operators, are bounded below by drastic product tw,
and bounded above by t3
twt1t1.5t2t2.5t3
Theorem 2:
Alls-norm operators, are bounded below bys3, and bounded
above by drastic sumsw
sws1s1.5s2s2.5s3
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Fuzzy Logic & Approximate Reasoning
1- Generalized Modus Ponens (GMP):
Premise 1: x is A (meaning not exactly A),Premise 2: if x is A then y is B,
consequence: y is B (i.e. not exactly B)
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Fuzzy Logic & Approximate Reasoning
1- Generalized Modus Tollens (GMT):
Premise 1: y is B (meaning not exactly B),Premise 2: if x is A then y is B,
consequence: x is A (i.e. not exactly A)
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Fuzzy Logic Controller
Fuzzification Interface
Measures the values of input variables,
Performs a scale mapping (if necessary) that
transfers the range of values of input variables intocorresponding universes of discourse
Performs the function of fuzzification that
converts the crisp real world input data into suitable
linguistic values, which may be viewed as labels of
fuzzy sets.
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Knowledge Base
Fuzzy rules consist of a premise with one or
more antecedents, and a conclusion with one
or more consequences. The individual rules in
the rule base are connected through the
operator also.
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Decision-making Logic
The two most common types are Min of
Mamdani and product of Larson [lee,1990].
The Min operator takes the minimum of allfuzzy membership values in the "if-side" for the
rule being evaluated, and clips the corresponding
output membership at this level.
The product operator scales the output
membership as opposed to clipping it.
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Defuzzification Interface
The defuzzification interface performs the
following functions:
a scale mapping, which converts the range ofvalues of output variables into corresponding
universes of discourse
defuzzification which yields a nonfuzzy controlaction from an inferred fuzzy control action.
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Defuzzification Interface
Two methods of defuzzification are most often
used:
maximum membership: chooses the output value
corresponding to the maximum degree of
membership in the output fuzzy set.
centroid or center of gravity: is the most
commonly used defuzzification method. In thecase of a discrete universe, the centroid method
yields a weighted sum of the output values [Lee,
1990].
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KB
Fuzzification
Interface
Defuzzificatio
Interface
Decision
Making
LogicFuzzy Fuzzy
Process Output & State Actual Control
Non-Fuzzy
Controled
System
(Process)
Block Diagram
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Hybrid Fuzzy PID
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HYBRID FUZZY-PID CONTROLLER
A Hybrid Fuzzy-PID controller is a type of fuzzycontroller in which the fuzzy engine is placed above a
conventional PID controller in the control hierarchy. This
is shown by the picture below.
The control inference of the HFPID is of the following:IF e is bigand de/dt issmalland edt issmallTHEN P is bigand D is
smalland I is medium
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Sugeno Engines
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GENERALIZED SUGENO CONTROLLER
A Generalized Sugeno Controller (GSC) is a fuzzy
controller which maps the input space to the output
space by the following fuzzy control
inference:
Ri: IF x1is A1iand ... xnis An
iTHEN y = Pi(x); 1
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PROBLEM STATEMENT
The Generalized Sugeno Controller (GSC) requires thedetermination of several unknown parameters for its design
and implementation. The search for these parameters can
be quite exhaustive.
Therefore, the purpose of this research is to develop asimple approach to determine the unknown parameters of a
Generalized Sugeno Controller.
Our approach is applied to the following systems:
A Hybrid Fuzzy-PID controlled robot manipulator arm
An approximation of an optimal feedback control law
for a ship tracking problem
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GSC vs. HFPID
The GSC has more desirable properties than the HFPID
The GSC is easier to design since the consequence
of each rule is caluclated automatically.
Rule evaluation and defuzzification is easier for theGSC than the HFPID.
It is easier to analyze the GSC in a qualitative
manner for stablility, controllability, observability,
and other issues of control systems.
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APPLICATIONS
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Applicability of Using A Fuzzy Controller
for DC-DC Converters
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Dynamic Control of Paralleled
DC-DC Converters(Current Sharing)
Desirable Characteristic:
Steady State Load Sharing
Transient Load Sharing
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BASIC CELL STRUCTURE OF DC-DC POWER MODULE
Vref+
-
VinIL
d
VC+-Hi Hv
Vo
Basic Cell
I V
+
-
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DIAGRAM OF TWO PARALLELED MODULES WITH MSC
WITH A DEDICATED MASTER
Module I
Module n
Cell #1
Cell #2
LO
AD
VoIo1
Io2
Vref1+
-Hv
+
+
+
-Hv
+
-Hi
Vref2
Module #1
Module #2
CS_Bus
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BLOCK DIAGRAM OF #N PARALLELED
CONVERTERS WITH MASTER-SLAVE CONTROL
Cell #1
Cell #n
Module #1
Module #n
L
OA
D
VoVinIo1
Ion
Vref1
+
++-Hv +
-Hi
++
+-Hv
+-Hi
Vrefn
CS Bus
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Classical Control
Complexity of analysis:
Huge number of loops and transfer
functions
Stability analysis is almost impossible
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Fuzzy Control
Linguistic Rules:
Ease of analysis
Ease of implementation
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Fuzzy Implementation
Input and output selection for the FLC
Membership function definitionFuzzy rules definition
(If x is A and y is B, then z is C)
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INPUT
N NS Z PS P
OUTPUT
N NS Z PS P
ATTRIBUTES OF THE MEMBERSHIP FUNCTION
IN DESIGNING FLC OF LOAD SHARING
NB: Negative Big NS: Negative SmallZ: Zero
PS: Positive Small PB: Positive Big
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BASIC MEMBERSHIP FUNCTION OF INPUT AND
OUTPUT FOR THE CSC FUZZY LOGIC CONTROLLER
(err) NB NS Z PS PB
-2 -1 0 1 2 err
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The centroid method is determined from the
fuzzy set as follow:
i
iic
i
iici
z
zz
z
)(
)(
0
Defuzzification
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If the error (difference of output currents)is PB,
then the duty cycle change of slave should be PB
If the error is NS, then the duty cycle change ofslave should be NS
If the error is PS and its derivative is NB, then the
duty cycle change of slave should be Z
Example of Fuzzy Control Rules
master-slave problem
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Block Diagram
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DEFFERENCE BETWEEN TWO OUTPUT CURRENTS
OUTPUT VOLTAGE VO
SYSTEM USING A CLASSICAL CSC SYSTEM USING A FUZZY LOGIC CSC
two paralleled buck converters output waveforms
Results
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DEFFERENCE BETWEEN TWO OUTPUT CURRENTS
OUTPUT VOLTAGE VO
SYSTEM USING A CLASSICAL CSC SYSTEM USING A FUZZY LOGIC CSC
STEP LOAD RESPONSES FOR 50% OF THE LOAD
Results
1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8 3
x 10-3
-1.5
-1
-0.5
0
0.5
1
T
OutputCurrent1-OutputCurrent2
1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8 3
x 10-3
13
14
15
16
17
18
19
T
Outp
utVoltage