ict619 intelligent systems topic 3: fuzzy systems

ICT619 Intelligent SystemsICT619 Intelligent Systems

Topic 3: Fuzzy SystemsTopic 3: Fuzzy Systems

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Fuzzy SystemsFuzzy SystemsPART APART A IntroductionIntroduction ApplicationsApplications Fuzzy sets and fuzzy logicFuzzy sets and fuzzy logic Probability and fuzzy logicProbability and fuzzy logic Fuzzy reasoningFuzzy reasoning Design of a fuzzy controllerDesign of a fuzzy controller

PART BPART B Building fuzzy systemsBuilding fuzzy systems Advantages and limitations of fuzzy systemsAdvantages and limitations of fuzzy systems Case StudiesCase Studies

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Why fuzzy systemsWhy fuzzy systems

Vagueness or imprecision is inherent in many real life objects or Vagueness or imprecision is inherent in many real life objects or propertiespropertiesWhat is the definition of “warm” or "tall"? What is the definition of “warm” or "tall"? – – There are many such imprecise conceptsThere are many such imprecise concepts

Application of hard boundaries for categorisation gives Application of hard boundaries for categorisation gives unsatisfactory resultsunsatisfactory results

Ability to handle imprecision is an attribute of intelligenceAbility to handle imprecision is an attribute of intelligence

Fuzzy logic provides a methodology for reasoning using imprecise Fuzzy logic provides a methodology for reasoning using imprecise rules and assertions rules and assertions

Intelligent control and decision support systems based on fuzzy Intelligent control and decision support systems based on fuzzy logic have proved their superiority over conventional hard logic logic have proved their superiority over conventional hard logic based systemsbased systems

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Fuzzy system applicationsFuzzy system applicationsFuzzy control systems Fuzzy control systems – controlling machinery– controlling machinery

Most renowned fuzzy Most renowned fuzzy control system in use - control system in use - Sendai subway (since 1987)Sendai subway (since 1987)

Japanese appliances Japanese appliances - vacuum cleaners, - vacuum cleaners, washing machines, washing machines, camcorderscamcorders

Fuzzy auto transmission & Fuzzy auto transmission & ABS in carsABS in cars

Fuzzy lift control systemFuzzy lift control system Fuzzy TV!Fuzzy TV!

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Fuzzy intelligent systems in business - Fuzzy intelligent systems in business - making decisionsmaking decisions

Fuzzy expert systems are proving to be a powerful tool Fuzzy expert systems are proving to be a powerful tool in business knowledge decision supportin business knowledge decision support

Successfully applied inSuccessfully applied in TransportationTransportation Managed health careManaged health care Financial services such as insurance risk assessment and Financial services such as insurance risk assessment and

company stability analysiscompany stability analysis Product marketing and sales analysisProduct marketing and sales analysis Extraction of information from databases (data mining)Extraction of information from databases (data mining) Resource and project managementResource and project management

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FS applicationsFS applications

Table: Approximate estimated numbers of commercial and industrial Table: Approximate estimated numbers of commercial and industrial applications of fuzzy systems (Munakata 1994)applications of fuzzy systems (Munakata 1994)

Fuzzy systems are suitable for complex problems or applications Fuzzy systems are suitable for complex problems or applications that involve intuitive thinkingthat involve intuitive thinking

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Fuzzy sets – the basis of fuzzy logicFuzzy sets – the basis of fuzzy logic

In classical logic, the boundary of a set is In classical logic, the boundary of a set is sharp: sharp: eg, all people earning $75,000 or higher eg, all people earning $75,000 or higher are members of set are members of set high-income earnerhigh-income earnerAnyone earning less than $75,000 is notAnyone earning less than $75,000 is not

Because of the sharpness of the set Because of the sharpness of the set boundary, classical logic sets are known boundary, classical logic sets are known as as crisp setscrisp sets

As the domain value (in this case, As the domain value (in this case, income) increases, the degree of income) increases, the degree of membership in the set high-income membership in the set high-income earner remains zero, but jumps to 1 earner remains zero, but jumps to 1 (true) as income reaches $75000 (true) as income reaches $75000

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Fuzzy setsFuzzy sets For a For a fuzzy setfuzzy set, membership , membership

values lie within the range values lie within the range zero (no membership) to 1 zero (no membership) to 1 (complete membership)(complete membership)

eg. the membership graph eg. the membership graph of the fuzzy set of the fuzzy set high-income high-income earnerearner may have the shape may have the shape shown belowshown below

The horizontal axis of the The horizontal axis of the graphs that represent these graphs that represent these fuzzy sets is called the fuzzy sets is called the universe of discourse universe of discourse over a over a variable of interest xvariable of interest x

The vertical axis is a degree The vertical axis is a degree of membership in the set of membership in the set mm(x) and is always in the (x) and is always in the range [0,1]range [0,1]

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FuzzificationFuzzification

According to this According to this membership membership functionfunction, someone earning , someone earning $30,000 will have a membership $30,000 will have a membership value of 0.1 value of 0.1

Someone earning $74,900 will Someone earning $74,900 will have a membership value of have a membership value of 0.9980.998

This is called This is called fuzzificationfuzzification

All incomes at or below $25,000 All incomes at or below $25,000 have membership value 0 have membership value 0

All those at or above $75,000 All those at or above $75,000 have membership value 1have membership value 1

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Fuzzy set examplesFuzzy set examples Depending on the application, Depending on the application,

fuzzy set membership functions fuzzy set membership functions can have different shapes can have different shapes including S-shape, triangle, including S-shape, triangle, trapezoid trapezoid

eg, membership functions of fuzzy eg, membership functions of fuzzy sets sets warmwarm and and hothot are bell- are bell-shapedshaped

Continuous valued degrees of Continuous valued degrees of membership in fuzzy sets enable membership in fuzzy sets enable handling of imprecise concepts handling of imprecise concepts such as such as highhigh, , weakweak, , warmwarm, which , which are commonly encountered in are commonly encountered in real life problemsreal life problems

In practice, these curves are In practice, these curves are often replaced by simpler often replaced by simpler triangular and trapezoidal triangular and trapezoidal functions, which are much faster functions, which are much faster to compute to compute

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Fuzzy logic is not just probabilityFuzzy logic is not just probability

A lot of discussion about the nature of fuzzy logic since its A lot of discussion about the nature of fuzzy logic since its appearance in the 1970sappearance in the 1970s

Many regard it as just a form of probability and question the Many regard it as just a form of probability and question the soundness of its basis and its reliability – the name “fuzzy” soundness of its basis and its reliability – the name “fuzzy” has not helpedhas not helped

Both fuzzy logic and probability deal with the issue of uncertaintyBoth fuzzy logic and probability deal with the issue of uncertainty

Both use a continuous 0 to 1 scale for measuring uncertaintyBoth use a continuous 0 to 1 scale for measuring uncertainty

But despite their apparent similarity, there is an important But despite their apparent similarity, there is an important difference between the two paradigms...difference between the two paradigms...

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Fuzzy logic and probability - the Fuzzy logic and probability - the differencedifference

Probability deals with Probability deals with likelihoodlikelihood – the chance of something happening or – the chance of something happening or something having a certain propertysomething having a certain property

Fuzzy logic deals not with likelihood of something having a certain Fuzzy logic deals not with likelihood of something having a certain property, but the property, but the degree to which it has that propertydegree to which it has that property

The "high card" drawing example: The "high card" drawing example: P(high_card) = 16/52 (picture cards) versus P(high_card) = 16/52 (picture cards) versus m m9Hearts9Hearts(high_cards) = 0 (picture cards) or 9/13 (linear scale)(high_cards) = 0 (picture cards) or 9/13 (linear scale)

Fuzzy set theory and fuzzy logic provide a mathematical tool for handling Fuzzy set theory and fuzzy logic provide a mathematical tool for handling this second kind of uncertainty this second kind of uncertainty

Despite the associated debate, its usefulness as a powerful tool for Despite the associated debate, its usefulness as a powerful tool for solving problems is well established. solving problems is well established.

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Fuzzy reasoningFuzzy reasoning The fuzzy model of a problem consists of a series of unconditional The fuzzy model of a problem consists of a series of unconditional

and conditional fuzzy propositionsand conditional fuzzy propositions

A unconditional fuzzy proposition A unconditional fuzzy proposition has the form has the form x x isis Y Y

where where x x is a is a linguistic variablelinguistic variable , , YY is the name of a fuzzy set. is the name of a fuzzy set.

xx is called a linguistic variable because its value in the proposition is called a linguistic variable because its value in the proposition is expressed by a human expert using a word (linguistic is expressed by a human expert using a word (linguistic expression) rather than a numberexpression) rather than a number

For example, For example, salary salary isis high highThe truth value of this proposition is given by the degree of The truth value of this proposition is given by the degree of membership of membership of salarysalary in the fuzzy set in the fuzzy set highhigh

This membership value is computed from the actual case-specific This membership value is computed from the actual case-specific numeric value with which numeric value with which salarysalary is instantiated, and the fuzzy is instantiated, and the fuzzy membership function membership function highhigh

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Fuzzy reasoning (cont'd)Fuzzy reasoning (cont'd)

A conditional fuzzy proposition, or rule,A conditional fuzzy proposition, or rule, has the form has the form

IF IF ww is is Z Z THENTHEN x x isis Y Y

This should be interpreted as: This should be interpreted as: x is a member of Y to the degree that w is a member of Zx is a member of Y to the degree that w is a member of Z

The consequent (RHS) of the rule is applied or executed only to the The consequent (RHS) of the rule is applied or executed only to the extent that the antecedent (LHS) is trueextent that the antecedent (LHS) is true

In the example fuzzy ruleIn the example fuzzy rule

IF IF years_in_job years_in_job is is high high THENTHEN salary salary isis high, high,

The membership value of The membership value of salarysalary in the fuzzy set in the fuzzy set highhigh is determined by the is determined by the membership value of membership value of years_in_job years_in_job in set in set high high

The fuzzy region for the set The fuzzy region for the set highhigh for salary will be truncated to a level for salary will be truncated to a level determined by the truth value of the proposition “determined by the truth value of the proposition “salary salary isis high” high”

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Inferencing through fuzzy reasoningInferencing through fuzzy reasoning

A number of fuzzy propositions is evaluated for their degrees of A number of fuzzy propositions is evaluated for their degrees of truthtruth

All propositions having some truth contribute to the final output All propositions having some truth contribute to the final output state of the solution variablestate of the solution variable

Unlike conventional expert systems, fuzzy reasoning is based on Unlike conventional expert systems, fuzzy reasoning is based on the the parallel processingparallel processing principle principle

All rules are fired even if not all of them contribute to the final All rules are fired even if not all of them contribute to the final outcome and some may contribute only partiallyoutcome and some may contribute only partially

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Fuzzy reasoning exampleFuzzy reasoning example

A fuzzy rule based system for determining salaryA fuzzy rule based system for determining salary

Rule base may consist of the rules:Rule base may consist of the rules:IF IF years_in_job years_in_job is is high high THENTHEN salary salary isis high high

IF IF years_in_job years_in_job is is medium medium THENTHEN salary salary isis medium medium

IF IF years_in_job years_in_job is is low low THENTHEN salary salary isis low low

IF IF products_sold products_sold is is high high THENTHEN salary salary isis high high

IF IF products_sold products_sold is is medium medium THENTHEN salary salary isis medium medium

IF IF products_sold products_sold is is low low THENTHEN salary salary isis low low

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Fuzzy reasoning example (cont’d)Fuzzy reasoning example (cont’d)

Inferencing value of solution variableInferencing value of solution variable salary salary

Given membership of Given membership of years_in_job years_in_job in set in set high high = 0.5, = 0.5,the contribution of the rule the contribution of the rule

IF IF years_in_job years_in_job is is high high THENTHEN salary salary isis high highto making salary to making salary highhigh will be to a degree of 0.5 will be to a degree of 0.5

Truth values of all rules contributing to the membership Truth values of all rules contributing to the membership of of salary salary in in highhigh, are combined using the , are combined using the min-max rulemin-max rule to give the aggregate truth value for high salary to give the aggregate truth value for high salary

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Fuzzy reasoning example (cont’d)Fuzzy reasoning example (cont’d)

Other rules give truth values for propositions Other rules give truth values for propositions salary salary isis mediummedium and and salary salary isis low low

The ultimate solution value of the variable The ultimate solution value of the variable salarysalary is also is also determined through a combination process determined through a combination process

Combination of the fuzzy spaces for high, medium and Combination of the fuzzy spaces for high, medium and low salary creates an low salary creates an aggregated fuzzy regionaggregated fuzzy region

A A defuzzification processdefuzzification process computes the numerical computes the numerical output value for salary from the aggregated fuzzy output value for salary from the aggregated fuzzy output regionoutput region

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The Min-max ruleThe Min-max rule Fuzzy rules of inference are used to combine the fuzzy regions Fuzzy rules of inference are used to combine the fuzzy regions

produced by the application of many rules run in parallel produced by the application of many rules run in parallel

The most common method for this combination process is the The most common method for this combination process is the min-max rulemin-max rule::

The composite membership value of the LHS is the minimum of The composite membership value of the LHS is the minimum of the memberships of all of the conditions on the LHSthe memberships of all of the conditions on the LHS

Example: Given the ruleExample: Given the rule

IF IF aa is is X ANDX AND bb is is Y Y THEN THEN c c isis Z Z

If the membership value of If the membership value of aa in in X X is 0.5, and that of is 0.5, and that of b b in in YY is 0.2, is 0.2, the degree of truth of the consequent (membership value of the degree of truth of the consequent (membership value of cc in in ZZ) ) will be min(0.5,0.2) = 0.2will be min(0.5,0.2) = 0.2

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The Min-max rule (cont’d)The Min-max rule (cont’d)

If a number of rules lead to different membership values for an If a number of rules lead to different membership values for an output variable, the maximum of these values is taken as the output variable, the maximum of these values is taken as the membership value.membership value.

Given a number of rules producing different truth values T1, T2, .., Given a number of rules producing different truth values T1, T2, .., Tn for the membership of Tn for the membership of c c inin Z Z, the aggregated truth value is , the aggregated truth value is maximum(T1, T2, .., Tn ) maximum(T1, T2, .., Tn )

The following rules lead to differing membership values (shown in The following rules lead to differing membership values (shown in parentheses) for the output variable parentheses) for the output variable riskrisk in the fuzzy set in the fuzzy set highhigh,,IF IF ageage is is middlemiddle THEN THEN riskrisk is is mediummedium (0.3) (0.3)IF IF assetasset is is mediummedium THEN THEN riskrisk is is mediummedium (0.2) (0.2)IF IF credit_historycredit_history is is reasonablereasonable THEN THEN riskrisk is is mediummedium (0.8) (0.8)

Variable Variable riskrisk will have a membership value of max(0.3, 0.2, 0.8) = will have a membership value of max(0.3, 0.2, 0.8) = 0.8 in 0.8 in mediummedium. .

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DefuzzificationDefuzzification With the application of a number of rules for the person in the With the application of a number of rules for the person in the

above example, the values for his/her membership in the above example, the values for his/her membership in the smallsmall and and highhigh sets will also be similarly evaluated using the min-max sets will also be similarly evaluated using the min-max rulesrules

Suppose these values are 0.4 for small, and 0.2 for highSuppose these values are 0.4 for small, and 0.2 for high

These membership values will truncate the fuzzy spaces for the These membership values will truncate the fuzzy spaces for the sets sets smallsmall, , mediummedium and and highhigh as shown below as shown below

%

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Defuzzification (cont’d)Defuzzification (cont’d)

Fuzzy spaces truncated by membership values Fuzzy spaces truncated by membership values for the sets small, medium and highfor the sets small, medium and high

These fuzzy regions are combined to give the aggregated fuzzy space for These fuzzy regions are combined to give the aggregated fuzzy space for the output variable the output variable riskrisk

The numerical value for The numerical value for riskrisk is computed from the aggregated fuzzy space is computed from the aggregated fuzzy space by by defuzzificationdefuzzification

%

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Defuzzification assigns an exact numerical value to the aggregated fuzzy Defuzzification assigns an exact numerical value to the aggregated fuzzy region for the output variableregion for the output variable

The most common defuzzification method is the The most common defuzzification method is the centroid centroid oror centre of centre of gravitygravity method method

It is a weighted average It is a weighted average R R of the output membership function:of the output membership function:

Were Were ddii is the is the iith value along the horizontal axis, th value along the horizontal axis, nn is the maximum value is the maximum value of the range on the horizontal axis and of the range on the horizontal axis and m(dm(dii)) is the membership value for is the membership value for that pointthat point

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The centroid method for calculating a fuzzy systems output value.The centroid method for calculating a fuzzy systems output value.

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Fuzzy system operation - an overall Fuzzy system operation - an overall viewview

The operation of a fuzzy system is shown in the schematic diagram The operation of a fuzzy system is shown in the schematic diagram below.below.

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Design of a fuzzy controllerDesign of a fuzzy controller

Actions of a fuzzy controller are defined by a rule baseActions of a fuzzy controller are defined by a rule base

Five steps in the construction of this rule base:Five steps in the construction of this rule base:

1.1. Identify and list the input variables and their ranges,Identify and list the input variables and their ranges,2.2. Identify and list the output variables and their ranges,Identify and list the output variables and their ranges,3.3. Define a fuzzy membership function for each of the input and Define a fuzzy membership function for each of the input and

output variables,output variables,4.4. Construct the rule base that will govern the controller’s Construct the rule base that will govern the controller’s

operation,operation,5.5. Determine how the control actions will be combined to form Determine how the control actions will be combined to form

the executed action.the executed action.

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Fuzzy controller design - a simplified Fuzzy controller design - a simplified exampleexample

Controller to be used to smoothly slow and Controller to be used to smoothly slow and stop a train travelling at any speed and at any stop a train travelling at any speed and at any distance from stationdistance from station

Step 1: Identify and list linguistic input variables Step 1: Identify and list linguistic input variables and their rangesand their ranges Two input variables: train speed and distance to Two input variables: train speed and distance to

stationstation Five ranges each of speed (km/hr) and distance (m)Five ranges each of speed (km/hr) and distance (m)

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Fuzzy controller design - a simplified Fuzzy controller design - a simplified exampleexample

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Fuzzy controller design - a simplified Fuzzy controller design - a simplified example (cont’d)example (cont’d)

Step 2: Identify and list linguistic output variables and their Step 2: Identify and list linguistic output variables and their numeric rangesnumeric ranges Two input variables: train throttle and train brakeTwo input variables: train throttle and train brake Five ranges each of train throttle (%) and brake (%):Five ranges each of train throttle (%) and brake (%):

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Step 3: Define a set of fuzzy membership functions for each Step 3: Define a set of fuzzy membership functions for each of the input and output variablesof the input and output variables Low and high values are used to define trapezoidal membership Low and high values are used to define trapezoidal membership

functions for each of the input rangesfunctions for each of the input ranges Height of each function is 1.0 and function bounds do not exceed high Height of each function is 1.0 and function bounds do not exceed high

and low ranges listed for each rangeand low ranges listed for each range

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Step 4: Construct rule base that will govern Step 4: Construct rule base that will govern controller’s operationcontroller’s operation

Rule base is represented as a matrix of combinations of each Rule base is represented as a matrix of combinations of each of the input range variablesof the input range variables

Each matrix entry contains each of the two output range Each matrix entry contains each of the two output range variables related to the input variablesvariables related to the input variables

Rule base matrix for example problem has only 12 rules that Rule base matrix for example problem has only 12 rules that describe the interaction between input and output variablesdescribe the interaction between input and output variables

Each entry in rule base is defined by AND-ing together the Each entry in rule base is defined by AND-ing together the inputs to produce each individual output response. inputs to produce each individual output response.

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In the example diagram below, the shaded matrix entry In the example diagram below, the shaded matrix entry meansmeans IF IF speedspeed is stopped AND IF is stopped AND IF distancedistance is at THEN full brake is at THEN full brake IF IF speedspeed is stopped AND IF is stopped AND IF distancedistance is at THEN no throttle is at THEN no throttle

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Step 5: Determine how control actions will be combined to Step 5: Determine how control actions will be combined to form the executed action at the action interfaceform the executed action at the action interface Centroid defuzzification used for rule combination procedureCentroid defuzzification used for rule combination procedure

Consider the inputs:Consider the inputs: speed = 2 km/hr and distance = 1 mspeed = 2 km/hr and distance = 1 m

What is the correct % brake and % throttle?What is the correct % brake and % throttle? First task: Determine which membership functions are activated and First task: Determine which membership functions are activated and

to what degreeto what degree

Four membership functions are activated: Four membership functions are activated: the speed functions for the speed functions for Very SlowVery Slow and and SlowSlowthe distance functions for the distance functions for AtAt and and Very NearVery Near and and

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Membership of the speed = 2 km/hr in fuzzy set for Membership of the speed = 2 km/hr in fuzzy set for Very SlowVery Slow is is 1.01.0

Membership of the speed = 2 km/hr in the fuzzy set for Membership of the speed = 2 km/hr in the fuzzy set for SlowSlow is is 0.2. Mathematically, they are denoted as0.2. Mathematically, they are denoted as

MMVery SlowVery Slow(2) = 1.0,(2) = 1.0,

MMSlowSlow(2) = 0.2.(2) = 0.2.

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Similarly, Similarly, membership values for the distance = 1 m in the fuzzy membership values for the distance = 1 m in the fuzzy

set for set for AtAt and and Very NearVery Near are: are:

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This results in four rules firing in the rule base matrixThis results in four rules firing in the rule base matrix

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Next, membership values are combined using the AND (min) Next, membership values are combined using the AND (min) operator for each rule combination:operator for each rule combination:

Rule 1:Rule 1: MMVerySlowVerySlow AND AND MMAtAt = min(1.0, 0.8) = 0.8, = min(1.0, 0.8) = 0.8, Rule 2:Rule 2: MMSlowSlow AND AND MMAtAt = min(0.2, 0.8) = 0.2, = min(0.2, 0.8) = 0.2, Rule 3:Rule 3: MMVerySlowVerySlow AND AND MMVeryNearVeryNear = min(1.0, 0.4) = 0.4, = min(1.0, 0.4) = 0.4, Rule 4:Rule 4: MMSlowSlow AND AND MMVeryNearVeryNear = min(0.2, 0.4) = 0.2. = min(0.2, 0.4) = 0.2.

The values 0.8, 0.2, 0.4 and 0.2 are the firing strengths of rules 1 The values 0.8, 0.2, 0.4 and 0.2 are the firing strengths of rules 1 to 4, respectively, for the input (2,1).to 4, respectively, for the input (2,1).

Next, output value for each rule is determined by truncating the Next, output value for each rule is determined by truncating the corresponding output membership function using its firing strengthcorresponding output membership function using its firing strength

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The resulting aggregated fuzzy output region for the The resulting aggregated fuzzy output region for the rules for variable rules for variable brakebrake : :

Rule 1

Rule 2Rule 3

Rule 4

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Finally, defuzzification using centroid method yields Finally, defuzzification using centroid method yields output value of 78 percent application of the brakeoutput value of 78 percent application of the brake

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Fuzzy Controller OperationFuzzy Controller Operation

During operation, input values are continually sampled and During operation, input values are continually sampled and presented to the fuzzy controllerpresented to the fuzzy controller

The fuzzy controller then repeats the process described above in The fuzzy controller then repeats the process described above in Step 5:Step 5:

Determine the fuzzy membership values activated by the inputsDetermine the fuzzy membership values activated by the inputs Determine which rules are activated (fired) in the rule base matrixDetermine which rules are activated (fired) in the rule base matrix Combine the membership values for the activated rules using the Combine the membership values for the activated rules using the

AND operatorAND operator

Determine the aggregated fuzzy region for each output variableDetermine the aggregated fuzzy region for each output variable Use defuzzification to compute the values for each output variableUse defuzzification to compute the values for each output variable

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REFERENCESREFERENCES Cox, E., Cox, E., The Fuzzy Systems HandbookThe Fuzzy Systems Handbook, AP Professional, San , AP Professional, San

Diego 1999.Diego 1999. Dhar, V., & Stein, RDhar, V., & Stein, R., Seven Methods for Transforming ., Seven Methods for Transforming

Corporate Data into Business IntelligenceCorporate Data into Business Intelligence., Prentice Hall ., Prentice Hall 1997, pp. 126-148, 203-2101997, pp. 126-148, 203-210..

Mcneill, F., & Thro, E., Mcneill, F., & Thro, E., Fuzzy Logic a Practical ApproachFuzzy Logic a Practical Approach, AP , AP Professional, Boston 1994.Professional, Boston 1994.

Munakata, T., & Jani, Y., Munakata, T., & Jani, Y., Fuzzy Systems: an OverviewFuzzy Systems: an Overview, , Communications of the ACM, Vol.37, No.3, 1994, pp.69-76.Communications of the ACM, Vol.37, No.3, 1994, pp.69-76.

Medsker,L., Medsker,L., Hybrid Intelligent SystemsHybrid Intelligent Systems, Kluwer Academic Press, , Kluwer Academic Press, Boston 1995.Boston 1995.

Negnevitsky, M. Negnevitsky, M. Artificial Intelligence A Guide to Intelligent Artificial Intelligence A Guide to Intelligent Systems, Systems, Addison-Wesley 2005. Chapter Addison-Wesley 2005. Chapter Sangalli, A., Sangalli, A., The Importance of being FuzzyThe Importance of being Fuzzy, Princeton University , Princeton University Press, 1998.Press, 1998.

Zahedi, F., Zahedi, F., Intelligent Systems for BusinessIntelligent Systems for Business, Wadsworth , Wadsworth Publishing, Belmont, California, 1993.Publishing, Belmont, California, 1993.

ict619 intelligent systems topic 3: fuzzy systems

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