contenth. bevrani spring semester, 2010 content introduction basic concepts fctl 2 fuzzy control...
TRANSCRIPT
2010/07/12
1
Graduate School of Science and Technology
H. Bevrani
Spring Semester, 2010
Content
Introduction
Basic Concepts
F C t l
2
Fuzzy Control
Examples
Fuzzy AGC
“The class of tall men” …, or “the class of beautiful women”, do not constitute classes or sets in the usual mathematical sense of these terms. Yet, the fact remains that such imprecisely defined “classes”
3
such imprecisely defined classes play an important role in human thinking, particularly in the domains of pattern recognition, communication of information, and abstraction. Lotfi A. Zadeh, 1965
“Fuzzy” ?
Oxford Dictionary: blurred, indistinct, confused, imprecisely defined
Fuzzy Systems ?
Knowledge-based or rule-based systems: If-Then rules
“As complexity rises, precise statements lose meaningand meaningful statements lose precision.”
Lotfi A. Zadeh, 1965
5
“The theory of fuzzy sets is a theory of graded concepts, a theory in which everything is a matter of degree.” Lotfi Zadeh, 1973
Unlike two-valued Boolean logic fuzzy logic is based on
6
Unlike two valued Boolean logic, fuzzy logic is based on degrees of membership. It deals with degrees of truth.
(a) Boolean Logic. (b) Multi-valued Logic.0 1 10 0.2 0.4 0.6 0.8 100 1 10
2010/07/12
2
7
T a ll M e n
D e g re e o fM e m b e rsh ip1 .0
0 .4
0 .6
0 .8
C risp S e ts
Set of tall men: It covers all men, but their degrees of membership depend on their height.
1 5 0 2 1 01 7 0 1 8 0 1 9 0 2 0 01 6 0
H e ig h t , c mD e g re e o fM e m b e rsh ip
1 5 0 2 1 01 8 0 1 9 0 2 0 0
1 .0
0 .0
0 .2
0 .4
0 .6
0 .8
1 6 0 1 7 0
0 .0
0 .2
H e ig h t , c m
F u z z y S e ts
0 if x < xo
1 if x ≥ xo
(x) =
xo
10
x1 x2
0 if x ≤ x1
(x) = 0 (x) < 1 if x1< x < x2
1 if x ≥ x2
<
Tall men set: consists of three sets: short, average and tall men.
Tall M en
Degree ofM embership
Short Average ShortTall
1.0
0 2
0.4
0.6
0.8
Crisp Sets
150 210170 180 190 200160
Height, cmDegree ofM embership
150 210180 190 200
1.0
0.0
0.2
0.4
0.6
0.8
160 170
0.0
0.2
Fuzzy Sets
Short Average
Tall
Tall
1968, Zadeh: Fuzzy algorithms
1970, Zadeh & Bellman: Fuzzy decision making
1965, Zadeh: “Fuzzy Sets” paper
12
1973, Zadeh: Fuzzy control
1970, Zadeh & Bellman: Fuzzy decision making
1971, Zadeh: Fuzzy ordering
1975, Mamdani: 1st application (Steam engine)
2010/07/12
3
1975-80: Slowly progressed!
1980, Sugeno: Control of a Fuji Water Purification Plant
1983, Sugeno & Nishida: Self-parking Car
13
1983, Sugeno & Nishida: Self parking Car
1984, Togai & Watanabe: 1st VLSI based Fuzzy controller
1987, Hitachi Co.: Sendai Subway System
1988, Hirota & …: Fuzzy Robot arm
1989: Lab. for Int. Fuzzy Eng. Research-Japan
1990: Fuzzy Logic System Institute (FLSI)-Japan
14
1993: 1st Issue of the IEEE Trans. On Fuzzy Sys.
1992: 1st IEEE Int. Conf. on Fuzzy Systems
1996, Hiyama: Fuzzy Power System Stabilizer
15 16
Fuzzy Control
Fuzzy Decision Making
Fuzzy Mathematics
17
Fuzzy Logic and AI
Fuzzy Decision Making
Uncertainty and Information
Flexible
Don’t need precise data
Easy to understand
18
Based on natural language
Don t need precise data
Can model nonlinear functions
2010/07/12
4
Fuzzy-based Automatic Car Speed Control
Example 1:Example 2: Digital Image Stabilizer
If all the points in the picture are moving in the
Same direction, Then the hand is shaking and
20
compensate it.
If only some points in the picture are moving,
Then the hand is not shaking and leaves it alone.
Open-loopControl
Closed-loopControl
Pure
Takagi-Sugeno-Kang (TSK)
With Fuzzifier and Defuzzifier
U and V are fuzzy sets.In Eng. Systems, inputs/outputs are real values .
U and V are real-valued variables.
Problem: Using mathematical formula against using the human knowledge
2010/07/12
5
x and y are real-valued variables.
Not A
A AA
B
Intersection Union
Complement Containment
BA BAA B
0x
1
( x )
0x
1N o t A
A0
x
1
0x
1
B
A
B
A
( x )
C o m p lem en t x C o n ta in m en t
0x
1
0x
1
A B
In te rsec tio n
0x
1
0x
A B
U n io n0
1
A BA B
x
( x ) ( x )
Containment:
Consider U = {1, 2, 3} and sets A and B
A = 0.3/1 + 0.5/2 + 1/3B = 0.5/1 + 0.55/2 + 1/3
then A is a subset of B, or A B
Complement:Crisp Sets: Who does not belong to the set?Fuzzy Sets: How much do elements not belong to the set?
Union:Crisp Sets: Which element belongs to either set?Fuzzy Sets: How much of the element is in either set?
Fuzzification: crisp to fuzzy
Fuzzification is the process where the crisp quantities are converted to fuzzy.
The conversion of fuzzy values is represented by y p ythe membership functions.
There are various methods to assign membership values or the membership functions to fuzzy variables.
2010/07/12
6
Fuzziness in a fuzzy set is characterized by its membership functions. Membership value is between 0 and 1.
Membership Function
Popular functions:
Triangular Trapezoid Gaussian
Intuition: is based on the human’s own intelligence and understanding to develop the membership functions.
Inference: The membership function is formed from the facts known and knowledge.
Rank Ordering: e. g., best car between
Membership Value Assignments
g gfive cars among 100 people.
Angular: is defined on the universe of angles, hence is repeating shapes every 2Π cycles.
NN: is used to determine the membership values of any input data in the different regions.
GA: The method involved in computing membership functions using GA.
The output of an entire fuzzy process can be union of two or more fuzzy membership functions. To explain this, consider a fuzzy output, which is formed by two parts:
Defuzzification
•It convert fuzzy sets into a crisp output
Defuzzification Methods
(1) Lambda-cut sets,
(2) Centroid method,
(3) Height method,
(4) Weighted average method,
(5) Mean–max method,
(6) Centre of sums,
(7) Centre of largest area,
Centroid method Is the most popular one. It finds a point representing the
centre of gravity (COG)
b
A dxxx
1.0
0.6
0.8
Degree ofMembership
35
b
a
A
a
A
dxx
dxxx
COG
4.675.05.05.05.02.02.02.02.01.01.01.0
5.0)100908070(2.0)60504030(1.0)20100(
COG
0.0
0.2
0.4
0.6
0 20 30 40 5010 70 80 90 10060
Z67.4
In 1973, Lotfi Zadeh published his second most influential paper: Capturing human knowledge in fuzzy rules.
A fuzzy rule can be defined as a conditional statement in the form:
Fuzzy Rules
statement in the form:
IFx is A, THEN y is B
where x and y are linguistic variables; and Aand B are linguistic values determined by fuzzy sets on the universe of discourses X and Y, respectively.
2010/07/12
7
Example: IF height is tallTHEN weight is heavy
In a fuzzy system, all rules fire to some extent. If the antecedent is true to some degree of membership, then the consequent is also true to that same degree.
Tall menHeavy men
180
Degree ofMembership1.0
0.0
0.2
0.4
0.6
0.8
Height, cm
190 200 70 80 100160
Weight, kg
120
Degree ofMembership1.0
0.0
0.2
0.4
0.6
0.8
38
Design Steps
• Fuzzification (of the input variables)
• Rule evaluation (Inference)
• Aggregation (of the rule outputs)
• Defuzzification
Consider five temperature control switches:
Example: Air conditioner
1. Specify the problem and define linguistic variables •Temperature, Fan Speed
COLD, COOL, PLEASANT, WARM, HOT.
The corresponding speeds of the motor controlling the fan on the air-conditioner has the graduations:
MINIMAL, SLOW, MEDIUM, FAST, BLAST.
The rules governing the air-conditioner are as follows:
RULE 1: IF TEMP is COLD THENSPEED is MINIMAL
RULE 2: IF TEMP is COOL THENSPEED is SLOW
2. Construct fuzzy rules
RULE 3: IF TEMP is PLEASANT THEN SPEED is MEDIUM
RULE 4: IF TEMP is WARM THENSPEED is FAST
RULE 5: IF TEMP is HOT THENSPEED is BLAST
Temperature fuzzy set
Temp (0C).
COLD COOL PLEASANT WARM HOT
0 Y* N N N N
5 Y Y N N N
10 N Y N N N
12.5 N Y* N N N
15 N Y N N N
17.5 N N Y* N N
Temp Temp ((00C).C).
COLDCOLD COOLCOOL PLEASANTPLEASANT WARMWARM HOTHOT
0< (T)<1
(T)=1
Y : temp value belongs to the set (0<A(x)<1)
Y* : temp value is the ideal member to the set (A(x)=1)
N : temp value is not a member of the set (A(x)=0)
20 N N N Y N
22.5 N N N Y* N
25 N N N Y N
27.5 N N N N Y
30 N N N N Y* (T)=0
2010/07/12
8
The analytically expressed membership for the temperatureare:
COLD: for 0 ≤ t ≤ 10 COLD(t) = – t / 10 + 1
Cool: for 0 ≤ t ≤ 12.5 COOL(t) = t / 12.5
Pleasant: for 12.5 ≤ t ≤ 17.5 PLEASENT(t) = – t / 5 + 3.5
etc… all based on the linear equation: y = ax + b
Temperature Fuzzy Sets
0 6
0.7
0.8
0.9
1
alu
e Cold
Cool
0
0.1
0.2
0.3
0.4
0.5
0.6
0 5 10 15 20 25 30
Temperature Degrees C
Tru
th V
a
Pleasent
Warm
Hot
Rev/sec
(RPM)
MINIMAL SLOW MEDIUM FAST BLAST
0 Y* N N N N
10 Y N N N N
20 Y Y N N N
30 N Y* N N N
40 N Y N N N
50 N N Y* N N
60 N N N Y N
Speed fuzzy set
60 N N N Y N
70 N N N Y* N
80 N N N Y Y
90 N N N N Y
100 N N N N Y*
Y : temp value belongs to the set (0<A(x)<1)
Y* : temp value is the ideal member to the set (A(x)=1)
N : temp value is not a member of the set (A(x)=0)
The analytically expressed membership for the speedare:
MINIMAL: for 0 ≤ v ≤ 30 COLD(t) = – v / 30 + 1COLD
SLOW: for 10 ≤ v ≤ 30 SLOW(t) = v / 20 – 0.5
for 30 ≤ v ≤ 50 SLOW(t) = – v / 20 + 2.5
etc… all based on the linear equation: y = ax + b
Speed Fuzzy Sets
0.6
0.8
1
Val
ue MINIMAL
SLOW
0
0.2
0.4
0.6
0 10 20 30 40 50 60 70 80 90 100
Speed
Tru
th V MEDIUM
FAST
BLAST
3. Evaluate fuzzy rules
Example: Use the system to compute the optimal fan speed, for temperature of 16oC.
• Fuzzification• Inference• Composition • Defuzzification
2010/07/12
9
• Fuzzification
Affected fuzzy sets: COOL and PLEASANT
COOL(T) = – T / 5 + 3.5 PLSNT(T) = T /2.5 - 6COOL( )
= – 16 / 5 + 3.5
= 0.3
PLSNT( )
= 16 /2.5 - 6
= 0.4
Temp=16 COLD COOL PLEASANT WARM HOT
0 0.3 0.4 0 0
• Inference
RULE 1: IF temp is cold THEN speed is minimal
RULE 2:RULE 2: IF temp is cool THEN speed is slow
RULE 3: IF temp is pleasant THEN speed is medium
RULE 4: IF temp is warm THEN speed is fast
RULE 5: IF temp is hot THEN speed is blast
• Inference
RULE 2: IF temp is cool (0.3) THEN speed is slow (0.3)
RULE 3: IF temp is pleasant (0.4) THEN speed is medium (0.4)
• Composition
speed is slow (0.3) speed is medium (0.4) +
• Defuzzification
COG = 0.125(12.5) + 0.25(15) + 0.3(17.5+20+…+40+42.5) + 0.4(45+47.5+…+52.5+55) + 0.25(57.5)
0.125 + 0.25 + 0.3(11) + 0.4(5) + 0.25
= 45.54rpm
Fuzzy Modeling
+ fuzzy logic Static fuzzy system+ fuzzy logic
+ fuzzy logic
Static fuzzy system
Dynamic fuzzy system
2010/07/12
10
Modeling Example
The input-output relationship of the model: smoothed fuzzy description (circles) and piecewise linear description (dashed lines).
Fuzzy Control
FLC Internal
Structuree(t)
e(t)u(t) u(t)
C S ye uw
A typical fuzzy logic controller is described by the relationship between change of control u(t), at a given time t, on the one hand
u(t) = f(e(t), e(t))
and the change in the error e(t): e(t) = e(t) – e (t – 1)
Z-1
Z-1
e(t)
Fuzzy Control Design Steps
Define control problem andsystem characteristics
Define variables andFuzzy sets
Define Defuzzification Method
Define Inference Rules
Design Example
Inverted Pendulum Problem
X ref X
State variables: Angle of the Pendulum, Rate of change of the angle,
Position of the cart or cart speed
Control problem: Keep pendulum upright by moving cart left or right.
Fuzzy Control System
Input variables: Angle of the Pendulum, Rate of change of the angle
Output variables: Position of the cart or cart speed (produced by a PWM signal)
60
2010/07/12
11
MN SN ZE SP MP
03 +3d
Input fuzzy sets for angle of the pole and rate of change of angle
Membership functions
LN MN SN ZE SP MP LP
-100 0 100 Speed of vehicle
0-3 +3
-4 +4
d
Output fuzzy set
Pendulum Angle
Inputs
Cart Speed
Pendulum AngularVelocity
Output
Fuzzy Rules
IF angle is zero and angular velocity is zero THEN speed shall be zero.
The fuzzy rules are
If is MN and d is MN then output is LNIf is ZE and d is ZE then output is ZEIf is ZE and d is ZE then output is ZEIf is SP and d is SP then output is MPetc.
The rules are best summaries by the Fuzzy Associative Memory (FAM) table.
Fill in the blanks in the table.
\ d MP SP ZE SN MNMP LPSP MP ?SPSP MPZE ZESNMN LN
?
?
MN
SP
if angle is zero and angular velocity is zero then
Defuzzification
2010/07/12
12
if angle is zero and angular velocity is zero then
if angle is zero and angular g gvelocity is negative low then
if angle is positive low andangular velocity is zero then
if angle is positive low and angular velocity is negative low then
Defuzzificationmust now be
done on fused output.
Fuzzy Logic AGC
1. Fuzzy logic controller
A general scheme for fuzzy logic based AGC
A general scheme for adaptive fuzzy logic AGC
2. Fuzzy based PI (PID) controller
t
IP τdτACEktACEktu0
)()()(
n
iIP TiACETkkTACEkkTu
1
)()()(
)1()( TkACEkTACE )(
)1()()1()()( kTACEk
T
TkACEkTACEkTkukTukTu IP
Fuzzy PI control scheme
3. Fuzzy Tuner
Fuzzy logic for tuning of PI based AGC system