Week7 Fuzzy Logic Tutorial

8/11/2019 Week7 Fuzzy Logic Tutorial

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October 2005 0

Fuzzy Expert SystemsCS364 Artificial Intelligence


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Recap

Example: Air Conditioner

Example: Cart Pole ProblemCase Study: Building a Fuzzy Expert System

Summary

Fuzzy Expert Systems


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1. Specify the problem; define linguistic variables.

Recap

2. Determine fuzzy sets.

3. Elicit and construct fuzzy rules.

4. Encode the fuzzy sets, fuzzy rules and proceduresto perform fuzzy inference into the expert system.

5. Evaluate and tune the system.

Process of developing a fuzzy expert system:


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Fuzzification: definition of fuzzy sets, anddetermination of the degree of membership of crisp

inputs in appropriate fuzzy sets.

Recap

Inference: evaluation of fuzzy rules to producean output for each rule.

Composition: aggregation or combination ofthe outputs of all rules.

Defuzzification: computation of crisp output

Operation of a fuzzy expert system:


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Recap



Summary



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1a. Specify the problem Air-conditioning involves the delivery of air, which can bewarmed or cooled and have its humidity raised or lowered.

An air-conditioner is an apparatus for controlling, especiallylowering, the temperature and humidity of an enclosedspace. An air-conditioner typically has a fan whichblows/cools/circulates fresh air and has a cooler. The cooleris controlled by a thermostat. Generally, the amount of airbeing compressed is proportional to the ambient

temperature.

1b. Define linguistic variables Ambient Temperature

Air-conditioner Fan Speed


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2. Determine Fuzzy Sets: Temperature 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

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*

Temp(0C).

COLD COOL PLEASANT WARM HOT

0< (T )


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2. Determine Fuzzy Sets: Temperature Temperature Fuzzy Sets

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 5 10 15 20 25 30

Temperature Degrees C

T r u

t h V a

l u e Cold

CoolPleasentWarm

Hot

a


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2. Determine Fuzzy Sets: Fan Speed Rev/sec(RPM)

MINIMAL SLOW MEDIUM FAST BLAST

0 Y* N N N N

10 Y N N N N20 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 N70 N N N Y* N

80 N N N Y Y

90 N N N N Y

100 N N N N Y*


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2. Determine Fuzzy Sets: Fan Speed Speed Fuzzy Sets

0

0.2

0.4

0.60.8

1

0 10 20 30 40 50 60 70 80 90 100

Speed

T r u

t h V a

l u e MINIMALSLOWMEDIUMFASTBLAST


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3. Elicit and construct fuzzy rules

RULE 1: IF temp is cold THEN speed is minimalRULE 2: IF temp is cool THEN speed is slow

RULE 3: IF temp is pleasant THEN speed is mediumRULE 4: IF temp is warm THEN speed is fastRULE 5: IF temp is hot THEN speed is blast


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3. Encode into an Expert System 4. Evaluate and tune the system

Consider a temperature of 16 oC, use the system

to compute the optimal fan speed.

Operation of a Fuzzy Expert System

Fuzzification

Inference Composition Defuzzification


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FuzzificationAffected fuzzy sets: COOL and PLEASANT

COOL (T ) = T / 5 + 3.5

= 16 / 5 + 3.5= 0.3

PLSNT (T ) = T /2.5 - 6

= 16 /2.5 - 6= 0.4

Temp=16 COLD COOL PLEASANT WARM HOT

0 0.3 0.4 0 0


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Inference

RULE 1: IF temp is cold THEN speed is minimalRULE 2: IF

temp is

cool THEN

speed is

slowRULE 3: IF temp is pleasant THEN speed is mediumRULE 4: IF temp is warm THEN speed is fastRULE 5: IF temp is hot THEN speed is blast


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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)

Inference


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Example: Air Conditioner Composition

speed is slow (0.3) speed is medium (0.4)+


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Example: Air Conditioner 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


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Recap



Summary



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Example: Cart Pole Problem

The problem is to balance an upright pole, with a mass m at itshead and mass M at its base. A weightless shaft connects these

two masses. The base can be moved on a horizontal axis. Thetask is to determine the FORCE ( F ) necessary to balance the pole.The calculation of the force F involves the measurement of theangle and the angular velocity, w of the pole .

M

m

g

w


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n b n m n s a z p s p m p b

n b p s p b

n m p m

n s n m n s p s

a z n b n m n s a z p s p m p b

p s n s p s p m

p m n m

p b n b n s


nb : negative big, nm : negative medium, ns : negative small az : approximately zero ps : positive small, pm : positive medium, pb : positive big

IF is negative medium and w is approximately zero THEN F is negative medium

w


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Example: Cart Pole ProblemThe fuzzy sets for , and F are based on the linear equation (x )=ax + b , and are defined based on the following table:

w

w) =1 ifw=

w) =1 ifw=

w) = if

w>=

w


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(b) Consider the case when the input variables are : = 50, = -5.

Use the rule base, execute each of the four tasks to computethe force F necessary to balance the pole using the Centre ofGravity in the Defuzzification task.

(a) Based on the fuzzy sets table draw three graphs showingthe fuzzy sets ( nb , nm , ns , az , ps , pm , pb ) for each , w, and F individually.


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Fuzzification

i) Determine where and the angular velocity fall in the table : pm , pb

w : az

ii) Formulate possible rules from linguistic values obtained

IF is pm AND w is az THEN F is pmIF is pb AND w is az THEN F is pb


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Example: Cart Pole ProblemFuzzification

iii) Compute membership functions

-11.25 -5 0 10

22.5 45 50 67.5 45 50 67.5

1 1 1/22.5 = /17.5

= 0.78 : pm

: pb

67.55045

11/22.5 = /5

= 0.22

1

w: az 1/11.25 = /6.25 = 0.56


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Example: Cart Pole ProblemInference

The two premises in RULE 1 are conjunctive minimum of the two: min{0.78, 0.56}=0.56

1 IF is pm AND w is az THEN F is pm

2 IF is pb AND w is az THEN F is pb

The two premises in RULE 2 are conjunctive minimum of the two: min{0.22, 0.56}=0.22


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Example: Cart Pole ProblemComposition

ps

pb

Defuzzification

3 0.2 (4 5 6) 0.56 (7 8) 0.225.30

0.2 0.56 0.56 0.56 0.22 0.22SoG

= =

C


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Recap



Summary



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A service centre keeps spare parts and repairs failed ones. A customerbrings a failed item and receives a spare of the same type. Failedparts are repaired, placed on the shelf, and thus become spares.

The objective is to advise a manager of the service centre on certain

decision policies to keep the customers satisfied.

Case Study: Building a FuzzyExpert System

Step 1: Specify the problem and definelinguistic variables

There are four main linguistic variables : average waiting time (meandelay) m, repair utilisation factor of the service centre , number ofservers s , and initial number of spare parts n.


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Linguistic variables and their rangesLinguistic Variable: Mean Delay , m

Linguistic Value Notation Numerical Range (normalised)Very Short

ShortMedium

VS

SM

[0, 0.3]

[0.1, 0.5][0.4, 0.7]Linguistic Variable: Number of Servers , s

Linguistic Value Notation Numerical Range (normalised)SmallMediumLarge

SML

[0, 0.35][0.30, 0.70]

[0.60, 1]

Linguistic Variable: Repair Utilisation Factor ,Linguistic Value Notation Numerical Range

LowMediumHigh

LMH

[0, 0.6][0.4, 0.8][0.6, 1]

Linguistic Variable: Number of Spares , nLinguistic Value Notation Numerical Range (normalised)

Very SmallSmallRather SmallMediumRather LargeLargeVery Large

VSS

RSMRLL

VL

[0, 0.30][0, 0.40]

[0.25, 0.45][0.30, 0.70][0.55, 0.75]

[0.60, 1][0.70, 1]

Linguistic Variable: Mean Delay , m

Linguistic Variable: Number of Servers , s

Linguistic Variable: Repair Utilisation Factor ,

Linguistic Variable: Number of Spares , n


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Step 2: Determine fuzzy sets

Fuzzy sets can have a variety of shapes.However, a triangle or a trapezoid can oftenprovide an adequate representation of the expert

knowledge, and at the same time, significantlysimplifies the process of computation.


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Fuzzy sets of M ean Delay m

0.10

1.0

0.0

0.2

0.4

0.6

0.8

0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Mean Delay (normalised)

SVS M

Degree of Membership


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Fuzzy sets of Number of Servers s

0.10

1.0

0.0

0.2

0.40.6

0.8

0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

M LS


Number of Servers (normalised)


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Fuzzy sets of Repair Util isation F actor

0.10

1.0

0.0

0.2

0.40.6

0.8

0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1epair Utilisation Factor

M HL



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Fuzzy sets of Number of Spares n

0.10

1.0

0.0

0.2

0.40.6

0.8

0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

S RSVS M RL L VL


Number of Spares (normalised)


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Step 3: Elicit and construct fuzzy rules

To accomplish this task, we might ask the expertto describe how the problem can be solved usingthe fuzzy linguistic variables defined previously.

Required knowledge also can be collected fromother sources such as books, computerdatabases, flow diagrams and observed human

behaviour.


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The rule table

Rule m s n Rule m s n Rule m s n1 VS S L VS 10 VS S M S 19 VS S H VL

2 S S L VS 11 S S M VS 20 S S H L

3 M S L VS 12 M S M VS 21 M S H M

4 VS M L VS 13 VS M M RS 22 VS M H M5 S M L VS 14 S M M S 23 S M H M

6 M M L VS 15 M M M VS 24 M M H S

7 VS L L S 16 VS L M M 25 VS L H RL

8 S L L S 17 S L M RS 26 S L H M9 M L L VS 18 M L M S 27 M L H RS


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Rule Base 1 1. If (utilisation_factor is L) then (number_of_spares is S)2. If (utilisation_factor is M) then (number_of_spares is M)3. If (utilisation_factor is H) then (number_of_spares is L)

4. If (mean_delay is VS) and (number_of_servers is S) then (number_of_spares is VL)5. If (mean_delay is S) and (number_of_servers is S) then (number_of_spares is L)6. If (mean_delay is M) and (number_of_servers is S) then (number_of_spares is M)

7. If (mean_delay is VS) and (number_of_servers is M) then (number_of_spares is RL)8. If (mean_delay is S) and (number_of_servers is M) then (number_of_spares is RS)9. If (mean_delay is M) and (number_of_servers is M) then (number_of_spares is S)

10.If (mean_delay is VS) and (number_of_servers is L) then (number_of_spares is M)11.If (mean_delay is S) and (number_of_servers is L) then (number_of_spares is S)12.If (mean_delay is M) and (number_of_servers is L) then (number_of_spares is VS)


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Step 4: Encode the fuzzy sets, fuzzy rulesand procedures to perform fuzzyinference into the expert system

To accomplish this task, we may choose one oftwo options: to build our system using aprogramming language such as C/C++ or Pascal,or to apply a fuzzy logic development tool suchas MATLAB Fuzzy Logic Toolbox or FuzzyKnowledge Builder.


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Step 5: Evaluate and tune the system

The last, and the most laborious, task is toevaluate and tune the system. We want to seewhether our fuzzy system meets therequirements specified at the beginning.

Several test situations depend on the meandelay, number of servers and repair utilisationfactor.

The Fuzzy Logic Toolbox can generate surface tohelp us analyse the systems performance.


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Three-dimensional plots for Rule Base 1

00.2

0.40.6

0.81

00.2

0.4

0.6

0.2

0.3

0.4

0.5

0.6

number_of_serversmean_delay

n u m b e r

_ o f_ s p a r e s


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00.2

0.40.6

0.81

0

0.2

0.4

0.6

0.2

0.3

0.4

0.5

0.6

utilisation_factormean_delay

n u m b e r_ o f_ s p a r e s


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00.2

0.40.6

0.81

0

0.2

0.4

0.6

0.15

0.2

0.25

0.3

0.35


n u m b e r

_ o f_ s p a r e s


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00.2

0.40.6

0.81

0

0.2

0.4

0.6

0.2

0.3

0.4



0.5


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However, even now, the expert might not be

satisfied with the system performance.

To improve the system performance, we mayuse additional sets Rather Small and Rather Large on the universe of discourse Number ofServers , and then extend the rule base.

Tune the system.


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Th di i l l f R l B 3


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00.2

0.40.6

0.81

0

0.2

0.4

0.6

0.15

0.2

0.25

0.3

0.35



Th di i l l f R l B 3


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00.2

0.40.6

0.81

0

0.2

0.4

0.6

0.2

0.3

0.4



0.5


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Tuning fuzzy systems

1. Review model input and output variables, and ifrequired redefine their ranges.

2. Review the fuzzy sets, and if required defineadditional sets on the universe of discourse.

The use of wide fuzzy sets may cause the fuzzysystem to perform roughly.

3. Provide sufficient overlap between neighbouring

sets. It is suggested that triangle-to-triangle andtrapezoid-to-triangle fuzzy sets should overlapbetween 25% to 50% of their bases.


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4. Review the existing rules, and if required add newrules to the rule base.

5. Examine the rule base for opportunities to writehedge rules to capture the pathological behaviourof the system.

6. Adjust the rule execution weights. Most fuzzylogic tools allow control of the importance of rulesby changing a weight multiplier.

7. Revise shapes of the fuzzy sets. In most cases,fuzzy systems are highly tolerant of a shapeapproximation.


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Recap



Case Study: Building a Fuzzy Expert System

Summary



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Summary

Process of developing a fuzzy expert system

Operation of a fuzzy expert system

Examples: Air Conditioner; Cart Pole Problem

Case Study: Building a Fuzzy Expert System

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Week7 Fuzzy Logic Tutorial

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