• Based on intuition and judgment• No need for a mathematical model• Provides a smooth transition between
members and nonmembers• Relatively simple, fast and adaptive • Less sensitive to system fluctuations• Can implement design objectives,
(difficult to express mathematically), in linguistic or descriptive rules.
Applications DomainApplications Domain• Fuzzy Logic• Fuzzy Control
– Neuro-Fuzzy System– Intelligent Control– Hybrid Control
• Fuzzy Pattern Recognition• Fuzzy Modeling
Some Interesting Some Interesting ApplicationsApplications
• Ride smoothness control• Camcorder auto-focus and jiggle
control• Braking systems• Copier quality control• Rice cooker temperature control• High performance drives• Air-conditioning systems
Conventional or Conventional or crispcrisp sets sets are binary. An element are binary. An element either belongs to the set or either belongs to the set or doesn't. doesn't.
{True, false}{True, false}{1, 0}{1, 0}
Universe (X)
Subset A
Subset B
Subset C
Crisp Set/SubsetCrisp Set/Subset1A
1B1C
0A
9
9 9.5
10e.g. On a scale of
one to 10, how good was the
dive?
• Examples of fuzzy measures include close, heavy, light, big, small, smart, fast, slow, hot, cold, tall and short.
Fuzzy IndicatorsFuzzy Indicators• Can you distinguish between Can you distinguish between
American and French person?American and French person?• Some Rules:Some Rules:
– IfIf speaks English speaks English thenthen American American– IfIf speaks French speaks French thenthen French French– IfIf loves perfume loves perfume thenthen French French– IfIf loves outdoors loves outdoors thenthen American American– IfIf good cook good cook thenthen French French– IfIf plays baseball plays baseball thenthen American American
Fuzzy IndicatorsFuzzy Indicators• Rules may give contradictory indicators{good cook, loves outdoors, speaks French}• The right answer is a question of a degree
of association• Fuzzy logic resolves these conflicting
indicators– Membership of the person in the French set is
0.9– Membership of the person in the American set
is 0.1
• Fuzzy Probability• Probability deals with
uncertainty and likelihood• Fuzzy logic deals with
ambiguity and vagueness
• Fuzzy Probability
• Example #1– Billy has ten toes. The
probability Billy has nine toes is zero. The fuzzy membership of Billy in the set of people with nine toes, however, is nonzero.
Example #2
– A bottle of liquid has a probability of ½ of being rat poison and ½ of being pure water.
– A second bottle’s contents, in the fuzzy set of liquids containing lots of rat poison, is ½.
– The meaning of ½ for the two bottles clearly differs significantly and would impact your choice should you be dying of thirst.
– 50% probability means 50% chance that the water is clean.
– 50% fuzzy membership means that the water has poison.
(cite: Bezdek)
#1
#2
• Crisp membership functions are either one or zero.
• e.g. Numbers greater than 10.
A ={x | x>10}
1
x10
A(x)
• The set, B, of numbers near to 2 can be represented by a membership function
|2|)( xB ex
0 1 2 3
x
B(x)
• A fuzzy set, A, is said to be a subset of B if
• e.g. B = far and A=very far.• For example...
)()( xx BA
)()( 2 xx BA
Fuzzy SetsFuzzy Sets
Tall Short
Fuzzy SetsFuzzy Sets
Tall ShortTall or Short?Tall or Short?
Fuzzy Fuzzy MeasuresMeasures
4’
5’
6’
7’
Very Tall
Tall
Medium
Short
Very Short
Membership FunctionMembership FunctionVery TallTallMediumShortVery Short
4’ 5’ 6’ 7’
1.0
vttmsvs ,,,,
Membership FunctionMembership Function
Short Medium Tall
0,0,0,1,0 0,5.0,5.0,0,0
Very TallTallMediumShortVery Short
4’ 5’ 6’ 7’
1.0
Fuzzy Logic OperationsFuzzy Logic Operations
)](),([max )( A xxx BBA
Fuzzy union operation or fuzzy OR
Fuzzy Logic OperationsFuzzy Logic Operations Fuzzy intersection operation or fuzzy AND
)](),([min )( A xxx BBA
Fuzzy Logic OperationsFuzzy Logic Operations Complement operation
)(1)( xx AA
Fuzzy Logic OperationsFuzzy Logic Operations
)](),([min )( A xxx BBA
)](),([max )( A xxx BBA
Fuzzy union operation or fuzzy OR
Fuzzy intersection operation or fuzzy AND
)(1)( xx AA
Complement operation
0 1 2 3
x
AB(x)
0 1 2 3 x
B(x)
0 1 2 3 x
A(x)1
)](),([min )( A xxx BBA
0 1 2 3 x
B(x)
0 1 2 3 x
A(x)1
A+B (x)
0 1 2 3 x
)](),([max )( A xxx BBA
• Fuzzifier converts a crisp input into a fuzzy variable.
• Definition of the membership functions must– reflect the designer's knowledge– provide smooth transition between
member and nonmembers of a fuzzy set– be simple to calculate
• Typical shapes of the membership function are Gaussian, trapezoidal and triangular.
• Assume we want to evaluate the health of a person based on his height and weight.
• The input variables are the crisp numbers of the person’s heightheight and weight.
• Fuzzification is a process by which the numbers are changed into linguistic words
Fuzzification of Height
4’ 5’ 6’ 7’
Very Short Short Medium Tall Very Tall
VS = very shortS = ShortM = Mediumetc.
1.0
Fuzzification of Weight
Very HeavyHeavyMediumSlimVery Slim
150lb 200lb 250lb 300lb
1.0
VS = very slimS = SlimM = Mediumetc.
• Rules reflect expert’s decisions.• Rules are tabulated as fuzzy words• Rules can be grouped in subsets• Rules can be redundant• Rules can be adjusted to match desired
results
• Rules are tabulated as fuzzy words– Healthy (H)– Somewhat healthy (SH)– Less Healthy (LH)– Unhealthy (U)
• Rule function f f{ U, LH, SH, H}
f
f{ U, LH, SH, H}
Weight
Height
VerySlim Slim Mediu
m Heavy VeryHeavy
VeryShort H SH LH U U
Short SH H SH LH UMediu
m LH H H LH U
Tall U SH H SH UVeryTall U LH H SH LH
• For a given person, compute the membership of his/her weight and height
• Example:– Assume that a person’s height is 6’ 1”– Assume that the person’s weight is 140 lbs
Very Short Short Medium Tall Very Tall
4’ 5’ 6’ 7’
1.0
0.7
0.3
height ={ VS, S, M, T , VT}
height={ 0 0 0.7 0.3 0 }
Very Slim Slim Medium Heavy Very Heavy
weight ={ VS, S, M, H , VH}
Weight={ 0.8 0.2 0 0 0 }
150lb 200lb 250lb 300lb
1.0
0.8
0.2
Weight
Height
Very Slim Slim Mediu
m Heavy VeryHeavy
VeryShort H SH LH U U
Short SH H SH LH UMediu
m H LH U
Tall H SH UVeryTall U LH H SH LH
LHSHHLHUVeryTall
USHH0.3
ULHH0.7
ULHSHHSHShort
UULHSHHVeryShort
VeryHeavyHeavyMediu
m0.20.8
Height
Weight
SHU
HLH
Weight
Height
0.8 0.2Mediu
m(0)
Heavy(0)
V.Heavy(0)
V.Short(0) 0 0 0 0 0
Short(0) 0 0 0 0 0
0.7 0.7 0.2 0 0 0
0.3 0.3 0.2 0 0 0
V.Tall(0) 0 0 0 0 0
f = {U,LH,SH,H}f = {0.3,0.7,0.2,0.2}
Weight
Height
0.8 0.2V.Short(
0) 0 0
Short(0) 0 0
0.7 0.7 0.20.3 0.3 0.2
V.Tall(0) 0 0
Weight
Height
0.8 0.2V.Short(
0) H SH
Short(0) SH H0.7 LH H0.3 U SH
V.Tall(0) U LH
f = { U, LH, SH, H}f = { 0.3, 0.7, 0.2, 0.2}
• Use the fuzzified rules to compute the final decision.
• Two methods are often used. - Maximum Method(not often
used) - Centroid
• Fuzzy set with the largest membership value is selected.
• Fuzzy decision:f={U, LH, SH, H}f={0.3, 0.7, 0.2, 0.2}
• Final Decision(FD)=Less Healthy• If two decisions have same membership
max, use the average of the two.
..........
DDD
FDu
u
u LH LH
LH
0.44292.02.07.03.0
0.82.00.62.00.47.00.20.3
FD
•Crisp Decision Index (D) = 0.4429
• Assume that we need to evaluate student applicants based on their GPA and GRE scores.
• Let us assume that the decision should be Excellent (E), Very Good(VG), Good(G), Fair(F) or Poor(P)
• An expert will associate the decision to the GPA and GRE score. They are then Tabulated.
• Assume that we need to evaluate student applicants based on their GPA and GRE scores.
• For simplicity, let us have three categories for each score [High (H), Medium (M), and Low(L)]
• Let us assume that the decision should be Excellent (E), Very Good (VG), Good (G),
Fair (F) or Poor (P)• An expert will associate the decisions to
the GPA and GRE score. They are then Tabulated.
Excellent = 95-100%Very Good = 90 - 94% Good = 80 - 89% Fair = 70 - 79% Poor = 0 - 69%
Issue94 is VG, but is also very close to Excellent
• Fuzzifier converts a crisp input into a fuzzy variable.
• Definition of the membership functions must – Reflects the designer’s knowledge– Provides smooth transition between member
and nonmembers of a fuzzy set– Simple to calculate• Typical shapes of the membership function are
Gaussian, trapezoidal and triangular.
GRE = {L , M ,
H }
GRE
GPA = {L , M ,
H }
GPA
Fn
• Assume a student with GRE=900 and GPA=3.6
• The decisions on the classification of the applicant are – Excellent– Very good– Etc.
GRE=900
GRE = {L = 0.8 , M = 0.2 , H = 0}
GRE
GPA=3.6
GPA = {L = 0 , M = 0.4 , H = 0.6}
GPA
GRE = {L = 0.8 , M = 0.2 , H = 0}GPA = {L = 0 , M = 0.4 , H = 0.6}
Fn
0.6
0.4
0.2
•Converting the output fuzzy variable into a unique number
•Two defuzzifier methods are often used. –Maximum Method (not often used)
–Centroid
• Fuzzy set with the largest membership value is selected.
• Fuzzy decision: Fn = {P, F, G,VG, E}Fn = {0.6, 0.4, 0.2, 0.2, 0}
• Final Decision (FD) = Poor Student• If two decisions have same
membership max, use the average of the two.
Fn
..........
VGEfn
FDVGE
VGE
if
if
Final Decision (FD) = Fair Student
706.04.02.02.0
606.0704.0802.0902.01000
FD
F
Feedback (error, change in error)
Reference
FuzzyFuzzyControllerController
SystemSystemu
Input
Output
CELN MN SN ZE SP MP LP
LN LN LN LN LN MN SN SNMN LN LN LN MN SN ZE ZESN LN LN MN SN ZE ZE SP
E ZE LN MN SN ZE SP MP LPSP SN ZE ZE SP MP LP LPMP ZE ZE SP MP LP LP LPLP SP SP MP LP LP LP LP
• Requires mathematical models• Nonlinear processes are linearized• Poor Performance when model
deviates• Difficult to tune for unknown
dynamics• Poor performance for widely varying
operation
• Poor performance in noisy environments
• Inclusion of some design objectives can be challenging (e.g. comfort of ride and safety)
• Engineers are comfortable with the classical control design.
– Well-established technologies.– Verifiable overall system stability. – System’s reliability can be
evaluated.
• Stability and reliability studies are based on linearized models
• Most systems do not behave linearly
• Most systems do drift
• Neural and Fuzzy Control.• Based on intuitions and judgments.• Relatively simple, fast and
adaptable.• Can implement design objectives.
Difficult to express mathematically; in linguistic or descriptive rules.
• No need for mathematical model .• Less sensitive to system fluctuations. • Design objectives difficult to express
mathematically can be incorporated in a fuzzy controller by linguistic rules.
• Implementation is simple and straight forward.
System Inputs
Execution Layer ….FLC FLC FLC
Supervisor Layer
MLFC Outputs
-3 -2 -1 0 1 2 3
LN MN SN ZE SP MP LP
0
1
0 1 3 6-1-3-60
1ZE SP MP LPSNMNLNE
CE