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1
Fuzzy Logic and Fuzzy Systems –X: Relationship and Membership
1
Khurshid Ahmad, Professor of Computer Science,Department of Computer Science
Trinity College,Dublin-2, IRELANDNovember 19th, 2008.
https://www.cs.tcd.ie/Khurshid.Ahmad/Teaching.html
2
Fuzzy SetsMembership Functions
Triangular MF: trimf x a b c x ab a
c xc b( ; , , ) max min , ,=
−−
−−
0
Trapezoidal MF: trapmf x a b c d x ab a
d xd c( ; , , , ) max min , , ,=
−−
−−
1 0
Generalized bell MF: gbel lmf x a b cx c
b
b( ; , , ) =+
−1
12
Gaussian MF: gaussmf x a b c ex c
( ; , , ) =−
−
12
2
σ
3
Fuzzy SetsSigmoid Membership Function
Membership function for "being well-off (or earning around 3K Euros per month)"
00.10.20.30.40.50.60.70.80.9
1
0 5 10 15 20
Monthly Income in Euros
Tru
th V
alue
Series1
))((1
1)( cxawealthy
ex −×−+
=µ
4
Fuzzy SetsGaussian Membership Function
Membership function for the proposition 'about 50 years old'
00.10.20.30.40.50.60.70.80.9
1
0 25 50 75 100
Age
Tru
th V
alue
Series1
250
1050
1
1)(
−+=
xxoldsoorµ
5
Fuzzy SetsCartesian Products and Patches
The cartesian or cross product of fuzzy subsets A and B, of sets X and Y respectively is denoted as
A ×××× BThis cross product relationship T on the set X ×××× Y is denoted as
T = A ×××× B
EXAMPLEA = {1/a1, 0.6/a2,0.3/a3},
B = {0.6/b1, 0.9/b2,0.1/b3}.
A ×××× B = { 0.6/(a1,b1), 0.9/(a1,b2), 0.1/(a1,b3),0.6/(a2,b1), 0.6/(a2,b2), 0.1/(a2,b3),0.3/(a3,b1), 0.3/(a3,b2), 0.1/(a3,b3)}
))](),([(),( yxMINyx BAT µµµ =
6
Fuzzy SetsCartesian Products and Patches
More generally, if A1, A2, ……An, are fuzzy subsets of X1, X2, ……Xn, then their cross product
A1× A2× A3 × … × An,is a fuzzy subset of
X1× X2× X3 × … × Xn, and
‘Cross products’ facilitate the mapping of fuzzy subsets that belong to disparate quantities or observations. This mapping is crucial for fuzzy rule based systems in general
and fuzzy control systems in particular.
)]([),.....,,( 321 iiAi
nT xMINxxxx µµ =
7
Fuzzy SetsFuzzy Relationships
•Electric motors are used in a number of devices; indeed, it is impossible to think of a device in everyday use that does not convert electrical energy into mechanical energy – air conditioners, elevators or lifts, central heating systems, …..•Electric motors are also examples of good control systems that run on simple heuristics relating to the speed of the (inside) rotor in the motor: change the strength of the magnetic field to adjust the speed at which the rotor is moving.
Electric motors can be electromagnetic and electrostatic; most electric motors are rotary but there are linear motors as well.
8
Fuzzy SetsFuzzy Relationships
•Electric motors are also examples of good control systems that run on simple heuristics relating to the speed of the (inside) rotor in the motor:
If the motor is running too slow, then speed it up.If motor speed is about right, then not much change is needed.If motor speed is too fast, then slow it down.
INPUT: Note the use of reference fuzzy sets representing linguistic values TOO SLOW, ABOUT RIGHT, and, TOO FAST. The three linguistic values form the term set SPEED.
9
Fuzzy SetsFuzzy Relationships
If the motor is running too slow, then speed it up.If motor speed is about right, then not much change is needed.If motor speed is too fast, then slow it down.
OUTPUT: In order to change speed, an operator of a control plant will have to apply more or less voltage: there are three reference fuzzy sets representing the linguistic values:
increase voltage (speed up); no change (do nothing); and, decrease voltage (slow down).
The three linguistic values for the term set VOLTAGE.
10
Fuzzy SetsFuzzy Relationships
http://www.fuzzy-logic.com/ch3.htm
A fuzzy patch between the terms SPEED & VOLTAGE.
11
Fuzzy SetsFuzzy Relationships
Slow down
Not much change
Spee
d up
Too sl
ow
About rightToo fast
2.36
2.40
2.44
2362 2420 2478
12
Fuzzy Systems:Fuzzy Sets and RelationshipsEXAMPLE:In order to understand how two fuzzy subsets are mapped onto each other to obtain a cross product, consider the example of an air-conditioning system. Air-conditioning involves the delivery of air which can be warmed or cooled and have its humidity raised or lowered.
An air-conditioner is an apparatus for controlling, especially lowering, the temperature and humidity of an enclosed space. An air-conditioner typically has a fanwhich blows/cools/circulates fresh air and has cooler and the cooler is under thermostatic control. Generally, the amount of air being compressed is proportional to the ambient temperature.
Consider Johnny’s air-conditioner which has five control switches: COLD, COOL, PLEASANT, WARM and HOT. The corresponding speeds of the motorcontrolling the fan on the air-conditioner has the graduations: MINIMAL, SLOW, MEDIUM, FAST and BLAST.
13
Fuzzy Systems:Fuzzy Sets and RelationshipsEXAMPLE:The rules governing the air-conditioner are as follows:RULE#1: IF TEMPis COLD THEN SPEEDis MINIMAL
RULE#2: IF TEMPis COOL THEN SPEEDis SLOW
RULE#3: IF TEMPis PLEASENT THEN SPEEDis MEDIUM
RULE#4: IF TEMPis WARM THEN SPEEDis FAST
RULE#5: IF TEMPis HOT THEN SPEEDis BLAST
The rules can be expressed as a cross product:CONTROL = TEMP × SPEED
14
Fuzzy Systems:Fuzzy Sets and RelationshipsEXAMPLE:
The rules can be expressed as a cross product:CONTROL = TEMP × SPEED
WHERE:TEMP = {COLD, COOL, PLEASANT, WARM, HOT}
SPEED = {MINIMAL, SLOW, MEDIUM, FAST, BLAST}
)](),([(),(
300&100:1#
)](),([(),(
VTMINVT
RPMVCTIFRULE
VTMINVT
SPEEDTEMPCONTROL
SPEEDTEMPCONTROL
µµµ
µµµ
=≤≤≤≤
=
o
15
Fuzzy Systems:Fuzzy Sets and RelationshipsEXAMPLE (CONTD.):The temperature graduations are related to Johnny’s perception of ambient temperatures:
Y*NNNN30
YNNNN27.5
NYNNN25
NY*NNN22.5
NYNNN20
NNY*YN17.5
NNNY*N12.5
NNNYN10
NNNYY5
NNNNY*0
HOTWARMPLEASANTCOOLCOLDTemp (0C).
16
Fuzzy Systems:Fuzzy Sets and RelationshipsEXAMPLE (CONTD.):Johnny’s perception of the speed of the motor is as follows:
YNNNN90
Y*NNNN100
YYNNN80
NY*NNN70
NYYNN60
NNY*YN50
NNYYN40
NNNY*Y30
NNNYY20
NNNYY10
NNNNY*0
BLASTFASTMEDIUMSLOWMINIMALRev/second (RPM)
17
Fuzzy Systems:Fuzzy Sets and RelationshipsEXAMPLE (CONTD.):The analytically expressed membership for the reference fuzzy subsets for the temperatureare:
301)(
3025115.2
)(''
5.275.225.55
)(
5.225.175.35
)('''
;205.1785.2
)(
5.171565.2
)(''
;5.175.125.35
)(
5.1205.12
)(''
;100110
)(''
)2(
)1(
)2(
)1(
)2(
)1(
)2(
)1(
≥=
≤≤−=
≤≤−−=
≤≤−=
≤≤+−=
≤≤−=
≤≤+−=
≤≤=
≤≤+−=
TT
TT
THOT
TT
T
TT
TWARM
TT
T
TT
TPLEASENT
TT
T
TT
TCOOL
TT
TCOLD
HOT
HOT
WARM
WARM
PLEA
PLEA
SLOW
SLOW
COLD
µ
µ
µ
µ
µ
µ
µ
µ
µ
18
Fuzzy Systems:Fuzzy Sets and RelationshipsTriangular membership functions can be described through the equations:
100 90 80 70 60 50 40 30 20 10 0
1 0
BLASTFAST
MEDIUM
SLOWSTOP
AIR MOTOR SPEED
45 50 55 60 65 70 75 80 85 90
1 0
CO
LD CO
OL
JUST
R
IGH
T
Temperature in Degrees in F
WA
RM
HO
T
≥
≤≤−−
≤≤−−
≤
=
cx
cxbbc
xc
bxaab
ax
ax
cbaxf
0
,0
),,;(
19
Fuzzy Systems:Fuzzy Sets and RelationshipsTriangular membership functions can be more elegantly and compactly expressed as
100 90 80 70 60 50 40 30 20 10 0
1 0
BLASTFAST
MEDIUM
SLOWSTOP
AIR MOTOR SPEED
45 50 55 60 65 70 75 80 85 90
1 0
CO
LD CO
OL
JUST
R
IGH
T
Temperature in Degrees in F
WA
RM
HO
T
)0),,max(min(),,;(bc
xc
ab
axcbaxf
−−
−−=
20
Fuzzy Systems:Fuzzy Sets and Relationships
A graphical representation of the two linguistic variables Speed and Temperature.
100 90 80 70 60 50 40 30 20 10 0
1 0
BLASTFAST
MEDIUM
SLOWSTOP
AIR MOTOR SPEED
45 50 55 60 65 70 75 80 85 90
1 0
CO
LD CO
OL
JUST
R
IGH
T
Temperature in Degrees in FW
AR
M
HO
T
21
7030BLAST
907050FAST
605040MEDIUM
503010SLOW
130MINIMAL
cbaMembership functionTerm
Fuzzy Systems:Fuzzy Sets and RelationshipsEXAMPLE (CONTD.):The analytically expressed membership for the reference fuzzy subsets for speed are:
ca
VVMINIMAL +−=)(µ
−−
−−= 0,,minmax)(
bc
Vc
ab
aVVSLOWµ
−−
−−= 0,,minmax)(
bc
Vc
ab
aVVMEDIUMµ
−−
−−= 0,,minmax)(
bc
Vc
ab
aVVFASTµ
−= 1,min)(a
cVVBLASTµ
22
Fuzzy Systems:Fuzzy Sets and Relationships
0-0.252.2555
00250
0.250.251.7545
0.50.51.540
0.750.751.2535
11130
0.751.250.7525
0.51.50.520
0.251.750.2515
02010
Speed (V)
−−
ab
aV
−−
bc
Vc
EXAMPLE (CONTD.):A sample computation of the SLOW membership function as a triangular membership function:
−−
−−= 0,,minmax)(
bc
Vc
ab
aVVSLOWµ
23
Fuzzy Systems:Fuzzy Patches and Rules
A fuzzy patch is defined by a fuzzy rule: a patch is a mapping of two membership functions, it is a product of two geometrical objects, line segments, triangles, squares etc.
100 90 80 70 60 50 40 30 20 10 0
0
BLAST
FAST
MED
IUM
SLOW
STOP
AIR
MO
TO
R S
PE
ED
45 50 55 60 65 70 75 80 85 90
1 0
CO
LD CO
OL
JUST
R
IGH
T
Temperature in Degrees in F
WA
RM
HO
T
IF WARM THEN FAST
24
Fuzzy Systems:Fuzzy Patches and Rules
In a fuzzy controller, a rule in the rule set of the controller can be visualized as a ‘device’for generating the product of the input/output fuzzy sets.
Geometrically a patch is an area that represents the causal association between the cause (the inputs) and the effect (the outputs).
The size of the patch indicates the vagueness implicit in the rule as expressed through the membership functions of the inputs and outputs.
25
Fuzzy Systems:Fuzzy Patches and Rules
The total area occupied by a patch is an indication of the vagueness of a given rule that can be used to generate the patch.
Consider a one-input-one output rule: if the input is crisp and the output is fuzzy then the patch becomes a line. And, if both are crisp sets then the patch is vanishingly small – a point.