Fuzzy Systems i i

8/9/2019 Fuzzy Systems i i

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Fuzzy Inference: Fuzzy Associative Matrix

FUZZY SYSTEMSFUZZY SYSTEMS


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FUZZINESS IS NOT PROBABILITYFUZZINESS IS NOT PROBABILITYFUZZINESS IS NOT PROBABILITYFUZZINESS IS NOT PROBABILITY

• Probability is used, for example, in weather forecasting

• Probability is a number between 0 and 1 that is thecertainty that an event will occur

• Fuzziness is more than probability; probability is asubset of fuzziness

• Probability is only valid for future/unknown events

• Fuzzy set membership continues after the event


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Fuzzy Relations


The fuzzy set operators allow rudimentary reasoning aboutfacts

For example, consider the three fuzzy sets tall, good_athleteand good-basketballplayer. Now assume

If we know that a good basketball player is tall and is agood athlete, then which one of Peter or Carl will be thebetter basketball player?


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Fuzzy Relations


Through application of the intersection operator, we get

Using the standard set operators, it is possible todetermine that Peter will be better at the sport than Carl

The example above is a very simplistic situation. Formost real-world problems, the sought outcome is afunction of a number of complex events, or scenarios


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Fuzzy Relations


For example, actions made by a controller are determinedby a set of if-then rules. The if-then rules describe situations

that can occur, with a corresponding action that thecontroller should execute

It is, however, possible that more than one situation, asdescribed by if-then rules, are simultaneously active, with

different actions. The problem is to determine the bestaction to take

A mechanism is therefore needed to infer an action from aset of activated situations


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Fuzzy rule based reasoning system


For fuzzy systems in general, the dynamic behavior of thatsystem is characterized by a set of linguistic fuzzy rules

These rules are based on the knowledge and experience of ahuman expert within that domain. Fuzzy rules are of thegeneral form

if antecedent(s) then consequent(s)

The antecedents of a rule form a combination of fuzzy setsthrough application of the logic operators (i.e. complement,intersection, union)

The consequent part of a rule is usually a single fuzzy set,

with a corresponding membership function


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Fuzzification


The antecedents of the fuzzy rules form the fuzzy “inputspace,” while the consequents form the fuzzy “outputspace”

The input space is defined by the combination of inputfuzzy sets, while the output space is defined by thecombination of output sets

The fuzzification process is concerned with finding a fuzzyrepresentation of non-fuzzy input values. In which, inputvalues from the universe of discourse are assignedmembership values to fuzzy sets


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Inferencing


The task of the inferencing process is to map the fuzzifiedinputs to the rule base, and to produce a fuzzified output

For the consequents in the rule output space, a degree of membership to the output sets are determined based on thedegrees of membership in the input sets and the relationshipsbetween the input sets

The output fuzzy sets in the consequent are then combined toform one overall membership function for the output of therule


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Fuzzy Relations

Classical relation between two universesU = {1, 2} and V = {a, b, c} is defined as:

a b c

R = U x V = 1 1 1 12 1 1 1



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Fuzzy Relations

Example:

Determine fuzzy relation between A 1 and A 2

A1 = 0.2/x 1 + 0.9/x 2A2 = 0.3/y 1 + 0.5/y 2 + 1/y 3



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Fuzzy Relations

R = R (A 1, A 2)

= 0.2 0.2 0.20.3 0.5 0.9

A1

A2

a11(0.2)

a12(0.9)

a22(0.5)

a21(0.3)

0.2 0.3

0.2 0.5

a23(1.0)

0.20.9



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Fuzzy Associative Matrix

So for the fuzzy rule:If X is A then Y is B

We can define a matrix M (nxp) which relates A to B

M = A x B

It maps fuzzy set A to fuzzy set B and is used in the fuzzyinference process



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• Fuzzification :In the fuzzification subprocess, the membership functions definedon the input variables are applied to their actual values, to

determine the degree of truth for each rule premise

• Inference :The truth value for the premise of each rule is computed, andapplied to the conclusion part of each rule. This results in onefuzzy subset to be assigned to each output variable for each rule



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• Composition :All of the fuzzy subsets assigned to each output variable arecombined together to form a single fuzzy subset for each output

variable.

• Defuzzification :Sometimes it is useful to just examine the fuzzy subsets that are theresult of the composition process, but more often, this fuzzy valueneeds to be converted to a single number - a crisp value. This iswhat the defuzzification subprocess does



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Fuzzy Inference: Composition Operator, Max-min Inference



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Composition of Fuzzy Relations

Let there be three universes U, V and W

Let R be the relation that relates elements from U to V

e.g. R = 0.6 0.80.7 0.9

And let S be the relation between V and W

e.g. S = 0.3 0.10.2 0.8



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Composition of Fuzzy Relations

With the help of an operation called “composition” we can findthe relation T that maps elements of U to W

By max-min rule T = R S = max v V { min( R (u, v), S(v, w)) }

0.6 0.8 0.3 0.1 = 0.3 0.80.7 0.9 0.2 0.8 0.3 0.8

Whereelement (1,1) is obtained by max{min(0.6, 0.3), min(0.8, 0.2)}

= 0.3Note that S R = 0.3 0.3 R S

0.7 0.8



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Example:

If Temperature is normal then Speed is medium

Let A = [0/100, 0.5/125, 1/150, 0.5/175, 0/200]B = [0/10, 0.6/20, 1/30, 0.6/40, 0/50]

M = 0 0 0 0 00 0.5 0.5 0.5 00 0.6 1 0.6 00 0.5 0.5 0.5 00 0 0 0 0



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Max-Min Inference

Let Temperature = 125 o F = A currentThen it may be thought of as a new fuzzy set defined over theuniverse of discourse of variable Temperature

Acurrent = [0/100, 0.5/125, 0/150, 0/175, 0/200]

We can find the relationship (i.e. mapping or FAM) betweenAcurrent and A as

A’=A current x A = 0 x [0 0.5 1 0.5 0]0.50 = [0 0.5 0.5 0.5 0]0 (if we eliminate all 0 rows)0



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Max-Min Inference

Now the composition of A’ and M will produce a newrelationship, which we call B’

A’ M = B’

0 0 0 0 00 0.5 0.5 0.5 0

[0 0.5 0.5 0.5 0] 0 0.6 1.0 0.6 0 = [0 0.5 0.5 0.5 0]0 0.5 0.5 0.5 00 0 0 0 0



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Multi-Premises Rules

If A B then C

C A’ = A’ M ACC B’ = B’ M BC

C’ = C A’ C B’ = min( CA’ , CB’ )

C A’C B’

C A’C B’



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Multi-Premises Rules

If A B then CC’ = C A’ C B’

= max( CA’ , CB’ )

C A’C B’ C A’

C B’



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Multiple Fuzzy Rules

In a regular rule-based system, if two rules aresimultaneously satisfied, a conflict resolution policy decides

the precedence

The system proceeds sequentially, with one rule firing at atime

In fuzzy rule based systems, all rules are executed duringeach pass through the system



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Fuzzy Inference: Combining Multiple Rules



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EXAMPLEEXAMPLE – – DINNERDINNER

Rule 2 If service is good,then tip is average

Rule 3 If service is excellent orfood is delicious, then

tip is generous

The inputs are crisp (non-fuzzy) numbers limited toa specific range

All rules are evaluatedin parallel using fuzzyreasoning

The results of the rulesare combined anddistilled (de-fuzzyfied)

The result is a crisp (non-fuzzy) number

OutputTip (5-25%)

Dinner for two: this is a 2 input, 1 output, 3 rulesystem

Input 1

Service (0-10)

Input 2

Food (0-10)

Rule 1 If service is poor orfood is rancid, thentip is cheap


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Multiple Fuzzy Rules



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Fuzzy Inference: Defuzzification



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Defuzzification

In most applications we need to obtain a crisp value afterinferring a fuzzy set B’

The most popular defuzzification technique used is the fuzzy centroid method



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References

Engelbrecht Chapter 18 & 20


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