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Fuzzy logic

This document discusses the fundamentals of fuzzy logic control systems. It begins by defining fuzzy logic as a problem-solving control system methodology that uses linguistic variables and fuzzy rules to map inputs to outputs. It then outlines the typical elements of a fuzzy logic system, including fuzzy sets, linguistic variables, fuzzy rules, fuzzy inference, and defuzzification. Finally, it provides an example of applying fuzzy logic to control the temperature in a simple heating/cooling system.

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Objectives…Objectives…Objectives…Objectives…Objectives…Objectives…Objectives…Objectives… IntroductionIntroduction.. ApproachApproach.. ApplicationApplication.. 9/12/2013 Mahesh J. vadhavaniya 1 ApplicationApplication.. SummarySummary.. DemoDemo..
IntroductionIntroductionIntroductionIntroductionIntroductionIntroductionIntroductionIntroduction FuzzyFuzzy LogicLogic definesdefines thethe controlcontrol strategystrategy onon aa LinguisticLinguistic levellevel.. ProblemProblem--solvingsolving controlcontrol systemsystem methodologymethodology.. What is Fuzzy Logic ? 9/12/2013 Mahesh J. vadhavaniya 2 ProblemProblem--solvingsolving controlcontrol systemsystem methodologymethodology.. LinguisticLinguistic oror “Fuzzy"“Fuzzy" variablesvariables ExampleExample:: IFIF (process(process isis tootoo hot)hot) ANDAND (process(process isis heatingheating rapidly)rapidly) THENTHEN (cool(cool thethe processprocess quickly)quickly)
IntroductionIntroductionIntroductionIntroductionIntroductionIntroductionIntroductionIntroduction DescribeDescribe thethe valuevalue ofof variablesvariables Apparatus of Fuzzy logic is built on : Fuzzy sets : QualitativelyQualitatively andand quantitativelyquantitativelyLinguistic 9/12/2013 Mahesh J. vadhavaniya 3 QualitativelyQualitatively andand quantitativelyquantitatively describeddescribed byby fuzzyfuzzy setssets Linguistic Variables : AA knowledgeknowledgeFuzzy if - then rules : LinguisticLinguistic valuesvalues toto numericalnumerical valuesvaluesDefuzzification :
IntroductionIntroductionIntroductionIntroductionIntroductionIntroductionIntroductionIntroduction Basic Elements of a Fuzzy logic system : 3.Defuzzyfication Linguistic level Command Variables (Linguistic Values) Measured Variables (Linguistic Values) 1.Fuzzyfication 2. Fuzzy Inference 9/12/2013 Mahesh J. vadhavaniya 4 Defuzzyfication level Numerical level Measured Variables (Numerical Values) Command Variables (Numerical Values) 1.Fuzzyfication
ApproachApproachApproachApproachApproachApproachApproachApproach Preliminary Evaluation : Problem Analysis before Project start ! EvaluationEvaluation CriteriaCriteria : AssessmentAssessment asas toto whetherwhether FuzzyFuzzy logiclogic isis applicableapplicable forfor thethe givengiven applicationapplication.. 9/12/2013 Mahesh J. vadhavaniya 5 HasHas fuzzyfuzzy logiclogic beenbeen previouslypreviously appliedapplied toto aa similarsimilar applicationapplication WithWith success?success? IsIs itit aa multimulti--variablevariable typetype controlcontrol problem?problem? DoDo operatorsoperators andand engineersengineers possesspossess knowledgeknowledge aboutabout anyany relevantrelevant interdependenciesinterdependencies ofof thethe processprocess variables?variables? CanCan furtherfurther knowledgeknowledge aboutabout thethe processprocess behaviorbehavior bebe gainedgained byby observationobservation oror experiments?experiments? IsIs itit difficultdifficult toto obtainobtain aa mathematicalmathematical modelmodel fromfrom thethe process?process?
UsageUsageUsageUsage 1.1. Define the control objectives and criteriaDefine the control objectives and criteria What am I trying to control?What am I trying to control? What do I have to do to control the system?What do I have to do to control the system? What kind of response do I need?What kind of response do I need? What are the possible (probable) system failure modes?What are the possible (probable) system failure modes? ApproachApproachApproachApproachApproachApproachApproachApproach What are the possible (probable) system failure modes?What are the possible (probable) system failure modes? 2.2. Determine the input and output relationshipsDetermine the input and output relationships Choose a minimum number of variables for input to theChoose a minimum number of variables for input to the FL engineFL engine 3.3. Use the ruleUse the rule--based structure of FLbased structure of FL Break the control problem down into a series of rulesBreak the control problem down into a series of rules 4.4. Create FL membership functionsCreate FL membership functions Define the meaning (values) of Input / Output terms usedDefine the meaning (values) of Input / Output terms used in the rulesin the rules 5.5. Test, evaluate, tune and retestTest, evaluate, tune and retest 9/12/2013 Mahesh J. vadhavaniya 6
The Rule MatrixThe Rule Matrix Error (Columns)Error (Columns) ErrorError--dot (Rows)dot (Rows) Input conditionsInput conditions ApproachApproachApproachApproachApproachApproachApproachApproach --veve ErrorError ZeroZero ErrorError +ve+ve ErrorError --veve ErrorError-- dotdotInput conditionsInput conditions (Error and Error(Error and Error-- dot)dot) Output ResponseOutput Response ConclusionConclusion (Intersection of Row(Intersection of Row and Column)and Column) dotdot ZeroZero ErrorError-- dotdot NoNo changechange +ve+ve ErrorError-- dotdot Rule MatrixRule Matrix 9/12/2013 Mahesh J. vadhavaniya 7
ComponentsComponents An electric heating elementAn electric heating element VariableVariable--speed cooling fanspeed cooling fan ApplicationApplicationApplicationApplicationApplicationApplicationApplicationApplication SimpleSimple ProportionalProportional TemperatureTemperature ControllerController : VariableVariable--speed cooling fanspeed cooling fan FunctionalityFunctionality Positive signal output: 0Positive signal output: 0--100% heat100% heat Negative signal output: 0Negative signal output: 0--100% cooling100% cooling 9/12/2013 Mahesh J. vadhavaniya 8
ApplicationApplicationApplicationApplicationApplicationApplicationApplicationApplication Block Diagram of the Control System 9/12/2013 Mahesh J. vadhavaniya 9
ApplicationApplicationApplicationApplicationApplicationApplicationApplicationApplication WorkingWorking Establish a meaningful system for representing theEstablish a meaningful system for representing the linguistic variables in the Rule Matrixlinguistic variables in the Rule Matrix "N" = "negative" error/ error"N" = "negative" error/ error--dot input leveldot input level"N" = "negative" error/ error"N" = "negative" error/ error--dot input leveldot input level "Z" = "zero" error/ error"Z" = "zero" error/ error--dot input leveldot input level "P" = "positive" error/ error"P" = "positive" error/ error--dot input leveldot input level "H" = "Heat" output response"H" = "Heat" output response ""--" = "No Change" to current output" = "No Change" to current output "C" = "Cool" output response"C" = "Cool" output response 9/12/2013 Mahesh J. vadhavaniya 10
ApproachApproachApproachApproachApproachApproachApproachApproach UsageUsageUsageUsage 1.1. Define the control objectives and criteriaDefine the control objectives and criteria What am I trying to control?What am I trying to control? What do I have to do to control the system?What do I have to do to control the system? What kind of response do I need?What kind of response do I need? What are the possible (probable) system failure modes?What are the possible (probable) system failure modes?What are the possible (probable) system failure modes?What are the possible (probable) system failure modes? 2.2. Determine the input and output relationshipsDetermine the input and output relationships Choose a minimum number of variables for input to theChoose a minimum number of variables for input to the FL engineFL engine 3.3. Use the ruleUse the rule--based structure of FLbased structure of FL Break the control problem down into a series of rulesBreak the control problem down into a series of rules 4.4. Create FL membership functionsCreate FL membership functions Define the meaning (values) of Input / Output terms usedDefine the meaning (values) of Input / Output terms used in the rulesin the rules 5.5. Test, evaluate, tune and retestTest, evaluate, tune and retest 9/12/2013 Mahesh J. vadhavaniya 11
1.1. Define the control objectives and criteriaDefine the control objectives and criteria What am I trying to control?What am I trying to control? System temperatureSystem temperature What do I have to do to control the system?What do I have to do to control the system? Proper balance and control of the functional devicesProper balance and control of the functional devices ApplicationApplicationApplicationApplicationApplicationApplicationApplicationApplication Proper balance and control of the functional devicesProper balance and control of the functional devices What kind of response do I need?What kind of response do I need? Stable Environment temperatureStable Environment temperature What are the possible (probable) system failureWhat are the possible (probable) system failure modes?modes? The lack of the “No change” regionThe lack of the “No change” region 9/12/2013 Mahesh J. vadhavaniya 12
What is being controlled and how?What is being controlled and how? ApplicationApplicationApplicationApplicationApplicationApplicationApplicationApplication Typical control system responseTypical control system response 9/12/2013 Mahesh J. vadhavaniya 13
ApproachApproachApproachApproachApproachApproachApproachApproach UsageUsageUsageUsage 1.1. Define the control objectives and criteriaDefine the control objectives and criteria What am I trying to control?What am I trying to control? What do I have to do to control the system?What do I have to do to control the system? What kind of response do I need?What kind of response do I need? What are the possible (probable) system failure modes?What are the possible (probable) system failure modes?What are the possible (probable) system failure modes?What are the possible (probable) system failure modes? 2.2. Determine the input and output relationshipsDetermine the input and output relationships Choose a minimum number of variables for input to theChoose a minimum number of variables for input to the FL engineFL engine 3.3. Use the ruleUse the rule--based structure of FLbased structure of FL Break the control problem down into a series of rulesBreak the control problem down into a series of rules 4.4. Create FL membership functionsCreate FL membership functions Define the meaning (values) of Input / Output terms usedDefine the meaning (values) of Input / Output terms used in the rulesin the rules 5.5. Test, evaluate, tune and retestTest, evaluate, tune and retest 9/12/2013 Mahesh J. vadhavaniya 14
ApplicationApplicationApplicationApplicationApplicationApplicationApplicationApplication 2.2. Determine the input and output relationshipsDetermine the input and output relationships2.2. Determine the input and output relationshipsDetermine the input and output relationships Define the minimum number of possible input productDefine the minimum number of possible input product combinations and corresponding output responsecombinations and corresponding output response conclusionsconclusions INPUT#1: ("Error", positive (P), zero (Z), negative (N)) INPUT#2: ("Error-dot", positive (P), zero (Z), negative (N)) CONCLUSION: ("Output", Heat (H), No Change (-), Cool (C))CONCLUSION: ("Output", Heat (H), No Change (-), Cool (C)) INPUT#1 System Status Error = Command-Feedback P=Too cold, Z=Just right, N=Too hot INPUT#2 System Status Error-dot = d(Error)/dt P=Getting hotter Z=Not changing N=Getting colder OUTPUT Conclusion & System Response Output H = Call for heating - = Don't change anything C = Call for cooling 9/12/2013 Mahesh J. vadhavaniya 15
ApproachApproachApproachApproachApproachApproachApproachApproach UsageUsageUsageUsage 1.1. Define the control objectives and criteriaDefine the control objectives and criteria What am I trying to control?What am I trying to control? What do I have to do to control the system?What do I have to do to control the system? What kind of response do I need?What kind of response do I need? What are the possible (probable) system failure modes?What are the possible (probable) system failure modes?What are the possible (probable) system failure modes?What are the possible (probable) system failure modes? 2.2. Determine the input and output relationshipsDetermine the input and output relationships Choose a minimum number of variables for input to theChoose a minimum number of variables for input to the FL engineFL engine 3.3. Use the ruleUse the rule--based structure of FLbased structure of FL Break the control problem down into a series of rulesBreak the control problem down into a series of rules 4.4. Create FL membership functionsCreate FL membership functions Define the meaning (values) of Input / Output terms usedDefine the meaning (values) of Input / Output terms used in the rulesin the rules 5.5. Test, evaluate, tune and retestTest, evaluate, tune and retest 9/12/2013 Mahesh J. vadhavaniya 16
APPLICATION (Contd.) 3.3. Use the ruleUse the rule--based structure of FLbased structure of FL ApplicationApplicationApplicationApplicationApplicationApplicationApplicationApplication 9/12/2013 Mahesh J. vadhavaniya 17
ApproachApproachApproachApproachApproachApproachApproachApproach UsageUsageUsageUsage 1.1. Define the control objectives and criteriaDefine the control objectives and criteria What am I trying to control?What am I trying to control? What do I have to do to control the system?What do I have to do to control the system? What kind of response do I need?What kind of response do I need? What are the possible (probable) system failure modes?What are the possible (probable) system failure modes?What are the possible (probable) system failure modes?What are the possible (probable) system failure modes? 2.2. Determine the input and output relationshipsDetermine the input and output relationships Choose a minimum number of variables for input to theChoose a minimum number of variables for input to the FL engineFL engine 3.3. Use the ruleUse the rule--based structure of FLbased structure of FL Break the control problem down into a series of rulesBreak the control problem down into a series of rules 4.4. Create FL membership functionsCreate FL membership functions Define the meaning (values) of Input / Output terms usedDefine the meaning (values) of Input / Output terms used in the rulesin the rules 5.5. Test, evaluate, tune and retestTest, evaluate, tune and retest 9/12/2013 Mahesh J. vadhavaniya 18
4.4. CreateCreate FLFL membershipmembership functionsfunctions thatthat definedefine thethe meaningmeaning (values)(values) ofof Input/OutputInput/Output termsterms usedused inin thethe rulesrules ApplicationApplicationApplicationApplicationApplicationApplicationApplicationApplication 9/12/2013 Mahesh J. vadhavaniya 19
A sample caseA sample case ApplicationApplicationApplicationApplicationApplicationApplicationApplicationApplication 9/12/2013 Mahesh J. vadhavaniya 20
ApproachApproachApproachApproachApproachApproachApproachApproach UsageUsageUsageUsage 1.1. Define the control objectives and criteriaDefine the control objectives and criteria What am I trying to control?What am I trying to control? What do I have to do to control the system?What do I have to do to control the system? What kind of response do I need?What kind of response do I need? What are the possible (probable) system failure modes?What are the possible (probable) system failure modes?What are the possible (probable) system failure modes?What are the possible (probable) system failure modes? 2.2. Determine the input and output relationshipsDetermine the input and output relationships Choose a minimum number of variables for input to theChoose a minimum number of variables for input to the FL engineFL engine 3.3. Use the ruleUse the rule--based structure of FLbased structure of FL Break the control problem down into a series of rulesBreak the control problem down into a series of rules 4.4. Create FL membership functionsCreate FL membership functions Define the meaning (values) of Input / Output terms usedDefine the meaning (values) of Input / Output terms used in the rulesin the rules 5.5. Test, evaluate, tune and retestTest, evaluate, tune and retest 9/12/2013 Mahesh J. vadhavaniya 21
Backing up a Truck…Backing up a Truck…Backing up a Truck…Backing up a Truck…Backing up a Truck…Backing up a Truck…Backing up a Truck…Backing up a Truck… Loading dock (Loading dock ( XXff ,, YYff )) RearRear ( X( X , Y, Y )) 9/12/2013 Mahesh J. vadhavaniya 22 FrontFront ɸɸ ϴϴ Diagram of Truck and Loading zoneDiagram of Truck and Loading zone
Backing up a Truck…Backing up a Truck…Backing up a Truck…Backing up a Truck…Backing up a Truck…Backing up a Truck…Backing up a Truck…Backing up a Truck… TheThe goalgoal isis toto makemake thethe trucktruck arrivearrive atat thethe loadingloading dockdock atat aa rightright angleangle ((ɸɸff==9090)) andand toto alignalign thethe positionposition (x,(x, y)y) ofof thethe trucktruck withwith thethe desireddesired loadingloading dockdock (( XXff ,, YYff )).. TheThe trucktruck movesmoves backwardbackward byby somesome fixedfixed distancedistance atat Goal ? 9/12/2013 Mahesh J. vadhavaniya 23 TheThe trucktruck movesmoves backwardbackward byby somesome fixedfixed distancedistance atat everyevery stagestage.. TheThe loadingloading zonezone correspondscorresponds toto planeplane [[00,,100100]] XX [[100100,,00]] andand (( XfXf ,, YfYf )) equaledequaled ((5050,,100100)) AtAt everyevery stagestage thethe FuzzyFuzzy controllercontroller shouldshould produceproduce thethe steeringsteering angleangle ϴϴ thatthat backsbacks upup thethe trucktruck toto thethe loadingloading dockdock fromfrom anyany initialinitial positionposition andand fromfrom anyany angleangle inin thethe loadingloading zonezone..
Backing up a Truck…Backing up a Truck…Backing up a Truck…Backing up a Truck…Backing up a Truck…Backing up a Truck…Backing up a Truck…Backing up a Truck… TheThe threethree statestate variablesvariables ɸɸ,, xx andand yy exactlyexactly determinedetermine thethe trucktruck positionposition.. TheThe coordinatecoordinate pairpair (x,(x, y)y) specifiesspecifies thethe positionposition ofof thethe rearrear centercenter ofof thethe trucktruck inin thethe planeplane.. ɸɸ specifiesspecifies thethe angleangle ofof thethe trucktruck withwith thethe horizontalhorizontal.. 9/12/2013 Mahesh J. vadhavaniya 24 rearrear centercenter ofof thethe trucktruck inin thethe planeplane.. First specify controller’s input and output variables. TheThe inputinput variablesvariables areare thethe trucktruck angleangle ɸɸ andand xx –– positionposition coordinatecoordinate xx.. TheThe outputoutput isis steeringsteering angleangle signalsignal ϴϴ.. WeWe assumeassume enoughenough clearanceclearance betweenbetween thethe trucktruck andand thethe loadingloading dockdock soso wewe cancan ignoreignore thethe yy –– positionposition coordinatecoordinate..
Backing up a Truck…Backing up a Truck…Backing up a Truck…Backing up a Truck…Backing up a Truck…Backing up a Truck…Backing up a Truck…Backing up a Truck… PositivePositive valuesvalues ofof ϴϴ representsrepresents clockwiseclockwise rotationsrotations.. Next specify Fuzzy set values of input and output variables. ɸɸ specifiesspecifies thethe angleangle ofof thethe trucktruck withwith thethe horizontalhorizontal.. NegativeNegative valuesvalues ofof ϴϴ representsrepresents countercounter clockwiseclockwise rotationsrotations.. 9/12/2013 Mahesh J. vadhavaniya 25 AngleAngleAngleAngleAngleAngleAngleAngle ɸɸ RB Right Below RU Right Upper RV Right Vertical VE Vertical LV Left Vertical LU Left Upper LB Left Below XXXXXXXX--------position xposition xposition xposition xposition xposition xposition xposition x LE Left LC Left Center CE Center RC Right Center RI Right Steering AngleSteering AngleSteering AngleSteering AngleSteering AngleSteering AngleSteering AngleSteering Angle ϴϴϴϴϴϴϴϴ NB Negative Big NM Negative Medium NS Negative Small ZE Zero PS Positive Small PM Positive Medium PB Positive Big
Backing up a Truck…Backing up a Truck…Backing up a Truck…Backing up a Truck…Backing up a Truck…Backing up a Truck…Backing up a Truck…Backing up a Truck… Next specify Fuzzy Membership Functions. 9/12/2013 Mahesh J. vadhavaniya 26 0 ≤ x ≤ 1000 ≤ x ≤ 100
Backing up a Truck…Backing up a Truck…Backing up a Truck…Backing up a Truck…Backing up a Truck…Backing up a Truck…Backing up a Truck…Backing up a Truck… Next specify Fuzzy Membership Functions. 9/12/2013 Mahesh J. vadhavaniya 27 --90 ≤90 ≤ ɸɸ ≤ 270≤ 270
Backing up a Truck…Backing up a Truck…Backing up a Truck…Backing up a Truck…Backing up a Truck…Backing up a Truck…Backing up a Truck…Backing up a Truck… Next specify Fuzzy Membership Functions. 9/12/2013 Mahesh J. vadhavaniya 28 --30 ≤30 ≤ ϴϴ ≤ 30≤ 30
Backing up a Truck…Backing up a Truck…Backing up a Truck…Backing up a Truck…Backing up a Truck…Backing up a Truck…Backing up a Truck…Backing up a Truck… Next specify Fuzzy “rulebase” or bank of FAM rules. xxxx LE LC CE RC RI RB PS PM PM PB PB 9/12/2013 Mahesh J. vadhavaniya 29 ɸɸɸɸɸɸɸɸ RB PS PM PM PB PB RU NS PS PM PB PB RV NM NS PS PM PB VE NM NM ZE PM PM LV NB NM NS PS PM LU NB NB NM NS PS LB NB NB NM NM NS
Backing up a Truck…Backing up a Truck…Backing up a Truck…Backing up a Truck…Backing up a Truck…Backing up a Truck…Backing up a Truck…Backing up a Truck… Next Defuzzification. 9/12/2013 Mahesh J. vadhavaniya 30 CorrelationCorrelation –– minimum inference withminimum inference with centroidcentroid defuzzificationdefuzzification methodmethod
Summary …Summary …Summary …Summary …Summary …Summary …Summary …Summary … 1.1. TheThe plantplant isis observableobservable andand controllablecontrollable :: state,state, inputinput andand outputoutput variablesvariables areare usuallyusually availableavailable forfor observationobservation andand measurementmeasurement oror computationcomputation.. Assumptions in a Fuzzy control system design : 2.2. ThereThere existsexists aa bodybody ofof knowledgeknowledge comprisedcomprised ofof aa setset ofof 9/12/2013 Mahesh J. vadhavaniya 31 2.2. ThereThere existsexists aa bodybody ofof knowledgeknowledge comprisedcomprised ofof aa setset ofof linguisticlinguistic rules,rules, engineeringengineering commoncommon sense,sense, intuition,intuition, oror aa setset ofof inputinput –– outputoutput measurementsmeasurements datadata fromfrom whichwhich rulesrules cancan bebe extractedextracted.. 3.3. AA solutionsolution existsexists.. 4.4. AA controlcontrol engineerengineer isis lookinglooking forfor aa “good“good enough”enough” solution,solution, notnot necessarilynecessarily thethe optimumoptimum oneone..
Summary …Summary …Summary …Summary …Summary …Summary …Summary …Summary … 5.5. TheThe controllercontroller willwill bebe designeddesigned withinwithin anan acceptableacceptable rangerange ofof precisionprecision.. Assumptions in a Fuzzy control system design : 9/12/2013 Mahesh J. vadhavaniya 32 6.6. TheThe problemsproblems ofof stabilitystability andand optimalityoptimality areare notnot addressedaddressed explicitlyexplicitly ,, suchsuch issuesissues areare stillstill openopen problemsproblems inin FuzzyFuzzy controllercontroller designdesign..
Summary …Summary …Summary …Summary …Summary …Summary …Summary …Summary … 1.1. IdentifyIdentify thethe variablesvariables (inputs,(inputs, statesstates andand outputs)outputs) .. The Steps in designing a simple Fuzzy control system : 2.2. PartitionPartition thethe universeuniverse ofof disclosuredisclosure oror thethe intervalinterval spannedspanned byby eacheach variablevariable intointo aa numbernumber ofof FuzzyFuzzy subsets,subsets, assigningassigning eacheach aa linguisticlinguistic labellabel (subset(subset includeinclude allall thethe 9/12/2013 Mahesh J. vadhavaniya 33 assigningassigning eacheach aa linguisticlinguistic labellabel (subset(subset includeinclude allall thethe elementselements inin thethe universe)universe).. 3.3. AssignAssign oror determinedetermine aa membershipmembership functionfunction forfor eacheach FuzzyFuzzy subsetsubset.. 4.4. AssignAssign thethe FuzzyFuzzy relationshipsrelationships betweenbetween thethe inputs’inputs’ oror states’states’ FuzzyFuzzy subsetssubsets onon thethe oneone handhand andand thethe outputs’outputs’ FuzzyFuzzy subsetssubsets onon thethe otherother hand,hand, thusthus formingforming thethe rulerule–– basebase..
Summary …Summary …Summary …Summary …Summary …Summary …Summary …Summary … 5.5. ChooseChoose appropriateappropriate scalingscaling factorsfactors forfor thethe inputinput andand outputoutput variablesvariables inin orderorder toto normalizenormalize thethe variablesvariables toto thethe [[00,, 11]] oror thethe [[--11,, 11]] intervalinterval.. The Steps in designing a simple Fuzzy control system : 6.6. FuzzifyFuzzify thethe inputsinputs toto thethe controllercontroller..6.6. FuzzifyFuzzify thethe inputsinputs toto thethe controllercontroller.. 7.7. UseUse FuzzyFuzzy approximateapproximate reasoningreasoning toto inferinfer thethe outputoutput contributedcontributed fromfrom eacheach rulerule.. 8.8. AggregateAggregate thethe FuzzyFuzzy outputsoutputs recommendedrecommended byby eacheach rulerule.. 9.9. ApplyApply defuzzificationdefuzzification toto formform aa crispcrisp outputoutput.. 9/12/2013 Mahesh J. vadhavaniya 34
9/12/2013 Mahesh J. Vadhavaniya 35
9/12/2013 Mahesh J. Vadhavaniya 37

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Fuzzy logic

  • 2.
  • 3.
    IntroductionIntroductionIntroductionIntroductionIntroductionIntroductionIntroductionIntroduction FuzzyFuzzy LogicLogic definesdefinesthethe controlcontrol strategystrategy onon aa LinguisticLinguistic levellevel.. ProblemProblem--solvingsolving controlcontrol systemsystem methodologymethodology.. What is Fuzzy Logic ? 9/12/2013 Mahesh J. vadhavaniya 2 ProblemProblem--solvingsolving controlcontrol systemsystem methodologymethodology.. LinguisticLinguistic oror “Fuzzy"“Fuzzy" variablesvariables ExampleExample:: IFIF (process(process isis tootoo hot)hot) ANDAND (process(process isis heatingheating rapidly)rapidly) THENTHEN (cool(cool thethe processprocess quickly)quickly)
  • 4.
    IntroductionIntroductionIntroductionIntroductionIntroductionIntroductionIntroductionIntroduction DescribeDescribe thethe valuevalueofof variablesvariables Apparatus of Fuzzy logic is built on : Fuzzy sets : QualitativelyQualitatively andand quantitativelyquantitativelyLinguistic 9/12/2013 Mahesh J. vadhavaniya 3 QualitativelyQualitatively andand quantitativelyquantitatively describeddescribed byby fuzzyfuzzy setssets Linguistic Variables : AA knowledgeknowledgeFuzzy if - then rules : LinguisticLinguistic valuesvalues toto numericalnumerical valuesvaluesDefuzzification :
  • 5.
    IntroductionIntroductionIntroductionIntroductionIntroductionIntroductionIntroductionIntroduction Basic Elements ofa Fuzzy logic system : 3.Defuzzyfication Linguistic level Command Variables (Linguistic Values) Measured Variables (Linguistic Values) 1.Fuzzyfication 2. Fuzzy Inference 9/12/2013 Mahesh J. vadhavaniya 4 Defuzzyfication level Numerical level Measured Variables (Numerical Values) Command Variables (Numerical Values) 1.Fuzzyfication
  • 6.
    ApproachApproachApproachApproachApproachApproachApproachApproach Preliminary Evaluation : ProblemAnalysis before Project start ! EvaluationEvaluation CriteriaCriteria : AssessmentAssessment asas toto whetherwhether FuzzyFuzzy logiclogic isis applicableapplicable forfor thethe givengiven applicationapplication.. 9/12/2013 Mahesh J. vadhavaniya 5 HasHas fuzzyfuzzy logiclogic beenbeen previouslypreviously appliedapplied toto aa similarsimilar applicationapplication WithWith success?success? IsIs itit aa multimulti--variablevariable typetype controlcontrol problem?problem? DoDo operatorsoperators andand engineersengineers possesspossess knowledgeknowledge aboutabout anyany relevantrelevant interdependenciesinterdependencies ofof thethe processprocess variables?variables? CanCan furtherfurther knowledgeknowledge aboutabout thethe processprocess behaviorbehavior bebe gainedgained byby observationobservation oror experiments?experiments? IsIs itit difficultdifficult toto obtainobtain aa mathematicalmathematical modelmodel fromfrom thethe process?process?
  • 7.
    UsageUsageUsageUsage 1.1. Define thecontrol objectives and criteriaDefine the control objectives and criteria What am I trying to control?What am I trying to control? What do I have to do to control the system?What do I have to do to control the system? What kind of response do I need?What kind of response do I need? What are the possible (probable) system failure modes?What are the possible (probable) system failure modes? ApproachApproachApproachApproachApproachApproachApproachApproach What are the possible (probable) system failure modes?What are the possible (probable) system failure modes? 2.2. Determine the input and output relationshipsDetermine the input and output relationships Choose a minimum number of variables for input to theChoose a minimum number of variables for input to the FL engineFL engine 3.3. Use the ruleUse the rule--based structure of FLbased structure of FL Break the control problem down into a series of rulesBreak the control problem down into a series of rules 4.4. Create FL membership functionsCreate FL membership functions Define the meaning (values) of Input / Output terms usedDefine the meaning (values) of Input / Output terms used in the rulesin the rules 5.5. Test, evaluate, tune and retestTest, evaluate, tune and retest 9/12/2013 Mahesh J. vadhavaniya 6
  • 8.
    The Rule MatrixTheRule Matrix Error (Columns)Error (Columns) ErrorError--dot (Rows)dot (Rows) Input conditionsInput conditions ApproachApproachApproachApproachApproachApproachApproachApproach --veve ErrorError ZeroZero ErrorError +ve+ve ErrorError --veve ErrorError-- dotdotInput conditionsInput conditions (Error and Error(Error and Error-- dot)dot) Output ResponseOutput Response ConclusionConclusion (Intersection of Row(Intersection of Row and Column)and Column) dotdot ZeroZero ErrorError-- dotdot NoNo changechange +ve+ve ErrorError-- dotdot Rule MatrixRule Matrix 9/12/2013 Mahesh J. vadhavaniya 7
  • 9.
    ComponentsComponents An electric heatingelementAn electric heating element VariableVariable--speed cooling fanspeed cooling fan ApplicationApplicationApplicationApplicationApplicationApplicationApplicationApplication SimpleSimple ProportionalProportional TemperatureTemperature ControllerController : VariableVariable--speed cooling fanspeed cooling fan FunctionalityFunctionality Positive signal output: 0Positive signal output: 0--100% heat100% heat Negative signal output: 0Negative signal output: 0--100% cooling100% cooling 9/12/2013 Mahesh J. vadhavaniya 8
  • 10.
  • 11.
    ApplicationApplicationApplicationApplicationApplicationApplicationApplicationApplication WorkingWorking Establish a meaningfulsystem for representing theEstablish a meaningful system for representing the linguistic variables in the Rule Matrixlinguistic variables in the Rule Matrix "N" = "negative" error/ error"N" = "negative" error/ error--dot input leveldot input level"N" = "negative" error/ error"N" = "negative" error/ error--dot input leveldot input level "Z" = "zero" error/ error"Z" = "zero" error/ error--dot input leveldot input level "P" = "positive" error/ error"P" = "positive" error/ error--dot input leveldot input level "H" = "Heat" output response"H" = "Heat" output response ""--" = "No Change" to current output" = "No Change" to current output "C" = "Cool" output response"C" = "Cool" output response 9/12/2013 Mahesh J. vadhavaniya 10
  • 12.
    ApproachApproachApproachApproachApproachApproachApproachApproach UsageUsageUsageUsage 1.1. Define thecontrol objectives and criteriaDefine the control objectives and criteria What am I trying to control?What am I trying to control? What do I have to do to control the system?What do I have to do to control the system? What kind of response do I need?What kind of response do I need? What are the possible (probable) system failure modes?What are the possible (probable) system failure modes?What are the possible (probable) system failure modes?What are the possible (probable) system failure modes? 2.2. Determine the input and output relationshipsDetermine the input and output relationships Choose a minimum number of variables for input to theChoose a minimum number of variables for input to the FL engineFL engine 3.3. Use the ruleUse the rule--based structure of FLbased structure of FL Break the control problem down into a series of rulesBreak the control problem down into a series of rules 4.4. Create FL membership functionsCreate FL membership functions Define the meaning (values) of Input / Output terms usedDefine the meaning (values) of Input / Output terms used in the rulesin the rules 5.5. Test, evaluate, tune and retestTest, evaluate, tune and retest 9/12/2013 Mahesh J. vadhavaniya 11
  • 13.
    1.1. Define thecontrol objectives and criteriaDefine the control objectives and criteria What am I trying to control?What am I trying to control? System temperatureSystem temperature What do I have to do to control the system?What do I have to do to control the system? Proper balance and control of the functional devicesProper balance and control of the functional devices ApplicationApplicationApplicationApplicationApplicationApplicationApplicationApplication Proper balance and control of the functional devicesProper balance and control of the functional devices What kind of response do I need?What kind of response do I need? Stable Environment temperatureStable Environment temperature What are the possible (probable) system failureWhat are the possible (probable) system failure modes?modes? The lack of the “No change” regionThe lack of the “No change” region 9/12/2013 Mahesh J. vadhavaniya 12
  • 14.
    What is beingcontrolled and how?What is being controlled and how? ApplicationApplicationApplicationApplicationApplicationApplicationApplicationApplication Typical control system responseTypical control system response 9/12/2013 Mahesh J. vadhavaniya 13
  • 15.
    ApproachApproachApproachApproachApproachApproachApproachApproach UsageUsageUsageUsage 1.1. Define thecontrol objectives and criteriaDefine the control objectives and criteria What am I trying to control?What am I trying to control? What do I have to do to control the system?What do I have to do to control the system? What kind of response do I need?What kind of response do I need? What are the possible (probable) system failure modes?What are the possible (probable) system failure modes?What are the possible (probable) system failure modes?What are the possible (probable) system failure modes? 2.2. Determine the input and output relationshipsDetermine the input and output relationships Choose a minimum number of variables for input to theChoose a minimum number of variables for input to the FL engineFL engine 3.3. Use the ruleUse the rule--based structure of FLbased structure of FL Break the control problem down into a series of rulesBreak the control problem down into a series of rules 4.4. Create FL membership functionsCreate FL membership functions Define the meaning (values) of Input / Output terms usedDefine the meaning (values) of Input / Output terms used in the rulesin the rules 5.5. Test, evaluate, tune and retestTest, evaluate, tune and retest 9/12/2013 Mahesh J. vadhavaniya 14
  • 16.
    ApplicationApplicationApplicationApplicationApplicationApplicationApplicationApplication 2.2. Determine theinput and output relationshipsDetermine the input and output relationships2.2. Determine the input and output relationshipsDetermine the input and output relationships Define the minimum number of possible input productDefine the minimum number of possible input product combinations and corresponding output responsecombinations and corresponding output response conclusionsconclusions INPUT#1: ("Error", positive (P), zero (Z), negative (N)) INPUT#2: ("Error-dot", positive (P), zero (Z), negative (N)) CONCLUSION: ("Output", Heat (H), No Change (-), Cool (C))CONCLUSION: ("Output", Heat (H), No Change (-), Cool (C)) INPUT#1 System Status Error = Command-Feedback P=Too cold, Z=Just right, N=Too hot INPUT#2 System Status Error-dot = d(Error)/dt P=Getting hotter Z=Not changing N=Getting colder OUTPUT Conclusion & System Response Output H = Call for heating - = Don't change anything C = Call for cooling 9/12/2013 Mahesh J. vadhavaniya 15
  • 17.
    ApproachApproachApproachApproachApproachApproachApproachApproach UsageUsageUsageUsage 1.1. Define thecontrol objectives and criteriaDefine the control objectives and criteria What am I trying to control?What am I trying to control? What do I have to do to control the system?What do I have to do to control the system? What kind of response do I need?What kind of response do I need? What are the possible (probable) system failure modes?What are the possible (probable) system failure modes?What are the possible (probable) system failure modes?What are the possible (probable) system failure modes? 2.2. Determine the input and output relationshipsDetermine the input and output relationships Choose a minimum number of variables for input to theChoose a minimum number of variables for input to the FL engineFL engine 3.3. Use the ruleUse the rule--based structure of FLbased structure of FL Break the control problem down into a series of rulesBreak the control problem down into a series of rules 4.4. Create FL membership functionsCreate FL membership functions Define the meaning (values) of Input / Output terms usedDefine the meaning (values) of Input / Output terms used in the rulesin the rules 5.5. Test, evaluate, tune and retestTest, evaluate, tune and retest 9/12/2013 Mahesh J. vadhavaniya 16
  • 18.
    APPLICATION (Contd.) 3.3. Usethe ruleUse the rule--based structure of FLbased structure of FL ApplicationApplicationApplicationApplicationApplicationApplicationApplicationApplication 9/12/2013 Mahesh J. vadhavaniya 17
  • 19.
    ApproachApproachApproachApproachApproachApproachApproachApproach UsageUsageUsageUsage 1.1. Define thecontrol objectives and criteriaDefine the control objectives and criteria What am I trying to control?What am I trying to control? What do I have to do to control the system?What do I have to do to control the system? What kind of response do I need?What kind of response do I need? What are the possible (probable) system failure modes?What are the possible (probable) system failure modes?What are the possible (probable) system failure modes?What are the possible (probable) system failure modes? 2.2. Determine the input and output relationshipsDetermine the input and output relationships Choose a minimum number of variables for input to theChoose a minimum number of variables for input to the FL engineFL engine 3.3. Use the ruleUse the rule--based structure of FLbased structure of FL Break the control problem down into a series of rulesBreak the control problem down into a series of rules 4.4. Create FL membership functionsCreate FL membership functions Define the meaning (values) of Input / Output terms usedDefine the meaning (values) of Input / Output terms used in the rulesin the rules 5.5. Test, evaluate, tune and retestTest, evaluate, tune and retest 9/12/2013 Mahesh J. vadhavaniya 18
  • 20.
    4.4. CreateCreate FLFLmembershipmembership functionsfunctions thatthat definedefine thethe meaningmeaning (values)(values) ofof Input/OutputInput/Output termsterms usedused inin thethe rulesrules ApplicationApplicationApplicationApplicationApplicationApplicationApplicationApplication 9/12/2013 Mahesh J. vadhavaniya 19
  • 21.
    A sample caseAsample case ApplicationApplicationApplicationApplicationApplicationApplicationApplicationApplication 9/12/2013 Mahesh J. vadhavaniya 20
  • 22.
    ApproachApproachApproachApproachApproachApproachApproachApproach UsageUsageUsageUsage 1.1. Define thecontrol objectives and criteriaDefine the control objectives and criteria What am I trying to control?What am I trying to control? What do I have to do to control the system?What do I have to do to control the system? What kind of response do I need?What kind of response do I need? What are the possible (probable) system failure modes?What are the possible (probable) system failure modes?What are the possible (probable) system failure modes?What are the possible (probable) system failure modes? 2.2. Determine the input and output relationshipsDetermine the input and output relationships Choose a minimum number of variables for input to theChoose a minimum number of variables for input to the FL engineFL engine 3.3. Use the ruleUse the rule--based structure of FLbased structure of FL Break the control problem down into a series of rulesBreak the control problem down into a series of rules 4.4. Create FL membership functionsCreate FL membership functions Define the meaning (values) of Input / Output terms usedDefine the meaning (values) of Input / Output terms used in the rulesin the rules 5.5. Test, evaluate, tune and retestTest, evaluate, tune and retest 9/12/2013 Mahesh J. vadhavaniya 21
  • 23.
    Backing up aTruck…Backing up a Truck…Backing up a Truck…Backing up a Truck…Backing up a Truck…Backing up a Truck…Backing up a Truck…Backing up a Truck… Loading dock (Loading dock ( XXff ,, YYff )) RearRear ( X( X , Y, Y )) 9/12/2013 Mahesh J. vadhavaniya 22 FrontFront ɸɸ ϴϴ Diagram of Truck and Loading zoneDiagram of Truck and Loading zone
  • 24.
    Backing up aTruck…Backing up a Truck…Backing up a Truck…Backing up a Truck…Backing up a Truck…Backing up a Truck…Backing up a Truck…Backing up a Truck… TheThe goalgoal isis toto makemake thethe trucktruck arrivearrive atat thethe loadingloading dockdock atat aa rightright angleangle ((ɸɸff==9090)) andand toto alignalign thethe positionposition (x,(x, y)y) ofof thethe trucktruck withwith thethe desireddesired loadingloading dockdock (( XXff ,, YYff )).. TheThe trucktruck movesmoves backwardbackward byby somesome fixedfixed distancedistance atat Goal ? 9/12/2013 Mahesh J. vadhavaniya 23 TheThe trucktruck movesmoves backwardbackward byby somesome fixedfixed distancedistance atat everyevery stagestage.. TheThe loadingloading zonezone correspondscorresponds toto planeplane [[00,,100100]] XX [[100100,,00]] andand (( XfXf ,, YfYf )) equaledequaled ((5050,,100100)) AtAt everyevery stagestage thethe FuzzyFuzzy controllercontroller shouldshould produceproduce thethe steeringsteering angleangle ϴϴ thatthat backsbacks upup thethe trucktruck toto thethe loadingloading dockdock fromfrom anyany initialinitial positionposition andand fromfrom anyany angleangle inin thethe loadingloading zonezone..
  • 25.
    Backing up aTruck…Backing up a Truck…Backing up a Truck…Backing up a Truck…Backing up a Truck…Backing up a Truck…Backing up a Truck…Backing up a Truck… TheThe threethree statestate variablesvariables ɸɸ,, xx andand yy exactlyexactly determinedetermine thethe trucktruck positionposition.. TheThe coordinatecoordinate pairpair (x,(x, y)y) specifiesspecifies thethe positionposition ofof thethe rearrear centercenter ofof thethe trucktruck inin thethe planeplane.. ɸɸ specifiesspecifies thethe angleangle ofof thethe trucktruck withwith thethe horizontalhorizontal.. 9/12/2013 Mahesh J. vadhavaniya 24 rearrear centercenter ofof thethe trucktruck inin thethe planeplane.. First specify controller’s input and output variables. TheThe inputinput variablesvariables areare thethe trucktruck angleangle ɸɸ andand xx –– positionposition coordinatecoordinate xx.. TheThe outputoutput isis steeringsteering angleangle signalsignal ϴϴ.. WeWe assumeassume enoughenough clearanceclearance betweenbetween thethe trucktruck andand thethe loadingloading dockdock soso wewe cancan ignoreignore thethe yy –– positionposition coordinatecoordinate..
  • 26.
    Backing up aTruck…Backing up a Truck…Backing up a Truck…Backing up a Truck…Backing up a Truck…Backing up a Truck…Backing up a Truck…Backing up a Truck… PositivePositive valuesvalues ofof ϴϴ representsrepresents clockwiseclockwise rotationsrotations.. Next specify Fuzzy set values of input and output variables. ɸɸ specifiesspecifies thethe angleangle ofof thethe trucktruck withwith thethe horizontalhorizontal.. NegativeNegative valuesvalues ofof ϴϴ representsrepresents countercounter clockwiseclockwise rotationsrotations.. 9/12/2013 Mahesh J. vadhavaniya 25 AngleAngleAngleAngleAngleAngleAngleAngle ɸɸ RB Right Below RU Right Upper RV Right Vertical VE Vertical LV Left Vertical LU Left Upper LB Left Below XXXXXXXX--------position xposition xposition xposition xposition xposition xposition xposition x LE Left LC Left Center CE Center RC Right Center RI Right Steering AngleSteering AngleSteering AngleSteering AngleSteering AngleSteering AngleSteering AngleSteering Angle ϴϴϴϴϴϴϴϴ NB Negative Big NM Negative Medium NS Negative Small ZE Zero PS Positive Small PM Positive Medium PB Positive Big
  • 27.
    Backing up aTruck…Backing up a Truck…Backing up a Truck…Backing up a Truck…Backing up a Truck…Backing up a Truck…Backing up a Truck…Backing up a Truck… Next specify Fuzzy Membership Functions. 9/12/2013 Mahesh J. vadhavaniya 26 0 ≤ x ≤ 1000 ≤ x ≤ 100
  • 28.
    Backing up aTruck…Backing up a Truck…Backing up a Truck…Backing up a Truck…Backing up a Truck…Backing up a Truck…Backing up a Truck…Backing up a Truck… Next specify Fuzzy Membership Functions. 9/12/2013 Mahesh J. vadhavaniya 27 --90 ≤90 ≤ ɸɸ ≤ 270≤ 270
  • 29.
    Backing up aTruck…Backing up a Truck…Backing up a Truck…Backing up a Truck…Backing up a Truck…Backing up a Truck…Backing up a Truck…Backing up a Truck… Next specify Fuzzy Membership Functions. 9/12/2013 Mahesh J. vadhavaniya 28 --30 ≤30 ≤ ϴϴ ≤ 30≤ 30
  • 30.
    Backing up aTruck…Backing up a Truck…Backing up a Truck…Backing up a Truck…Backing up a Truck…Backing up a Truck…Backing up a Truck…Backing up a Truck… Next specify Fuzzy “rulebase” or bank of FAM rules. xxxx LE LC CE RC RI RB PS PM PM PB PB 9/12/2013 Mahesh J. vadhavaniya 29 ɸɸɸɸɸɸɸɸ RB PS PM PM PB PB RU NS PS PM PB PB RV NM NS PS PM PB VE NM NM ZE PM PM LV NB NM NS PS PM LU NB NB NM NS PS LB NB NB NM NM NS
  • 31.
    Backing up aTruck…Backing up a Truck…Backing up a Truck…Backing up a Truck…Backing up a Truck…Backing up a Truck…Backing up a Truck…Backing up a Truck… Next Defuzzification. 9/12/2013 Mahesh J. vadhavaniya 30 CorrelationCorrelation –– minimum inference withminimum inference with centroidcentroid defuzzificationdefuzzification methodmethod
  • 32.
    Summary …Summary …Summary…Summary …Summary …Summary …Summary …Summary … 1.1. TheThe plantplant isis observableobservable andand controllablecontrollable :: state,state, inputinput andand outputoutput variablesvariables areare usuallyusually availableavailable forfor observationobservation andand measurementmeasurement oror computationcomputation.. Assumptions in a Fuzzy control system design : 2.2. ThereThere existsexists aa bodybody ofof knowledgeknowledge comprisedcomprised ofof aa setset ofof 9/12/2013 Mahesh J. vadhavaniya 31 2.2. ThereThere existsexists aa bodybody ofof knowledgeknowledge comprisedcomprised ofof aa setset ofof linguisticlinguistic rules,rules, engineeringengineering commoncommon sense,sense, intuition,intuition, oror aa setset ofof inputinput –– outputoutput measurementsmeasurements datadata fromfrom whichwhich rulesrules cancan bebe extractedextracted.. 3.3. AA solutionsolution existsexists.. 4.4. AA controlcontrol engineerengineer isis lookinglooking forfor aa “good“good enough”enough” solution,solution, notnot necessarilynecessarily thethe optimumoptimum oneone..
  • 33.
    Summary …Summary …Summary…Summary …Summary …Summary …Summary …Summary … 5.5. TheThe controllercontroller willwill bebe designeddesigned withinwithin anan acceptableacceptable rangerange ofof precisionprecision.. Assumptions in a Fuzzy control system design : 9/12/2013 Mahesh J. vadhavaniya 32 6.6. TheThe problemsproblems ofof stabilitystability andand optimalityoptimality areare notnot addressedaddressed explicitlyexplicitly ,, suchsuch issuesissues areare stillstill openopen problemsproblems inin FuzzyFuzzy controllercontroller designdesign..
  • 34.
    Summary …Summary …Summary…Summary …Summary …Summary …Summary …Summary … 1.1. IdentifyIdentify thethe variablesvariables (inputs,(inputs, statesstates andand outputs)outputs) .. The Steps in designing a simple Fuzzy control system : 2.2. PartitionPartition thethe universeuniverse ofof disclosuredisclosure oror thethe intervalinterval spannedspanned byby eacheach variablevariable intointo aa numbernumber ofof FuzzyFuzzy subsets,subsets, assigningassigning eacheach aa linguisticlinguistic labellabel (subset(subset includeinclude allall thethe 9/12/2013 Mahesh J. vadhavaniya 33 assigningassigning eacheach aa linguisticlinguistic labellabel (subset(subset includeinclude allall thethe elementselements inin thethe universe)universe).. 3.3. AssignAssign oror determinedetermine aa membershipmembership functionfunction forfor eacheach FuzzyFuzzy subsetsubset.. 4.4. AssignAssign thethe FuzzyFuzzy relationshipsrelationships betweenbetween thethe inputs’inputs’ oror states’states’ FuzzyFuzzy subsetssubsets onon thethe oneone handhand andand thethe outputs’outputs’ FuzzyFuzzy subsetssubsets onon thethe otherother hand,hand, thusthus formingforming thethe rulerule–– basebase..
  • 35.
    Summary …Summary …Summary…Summary …Summary …Summary …Summary …Summary … 5.5. ChooseChoose appropriateappropriate scalingscaling factorsfactors forfor thethe inputinput andand outputoutput variablesvariables inin orderorder toto normalizenormalize thethe variablesvariables toto thethe [[00,, 11]] oror thethe [[--11,, 11]] intervalinterval.. The Steps in designing a simple Fuzzy control system : 6.6. FuzzifyFuzzify thethe inputsinputs toto thethe controllercontroller..6.6. FuzzifyFuzzify thethe inputsinputs toto thethe controllercontroller.. 7.7. UseUse FuzzyFuzzy approximateapproximate reasoningreasoning toto inferinfer thethe outputoutput contributedcontributed fromfrom eacheach rulerule.. 8.8. AggregateAggregate thethe FuzzyFuzzy outputsoutputs recommendedrecommended byby eacheach rulerule.. 9.9. ApplyApply defuzzificationdefuzzification toto formform aa crispcrisp outputoutput.. 9/12/2013 Mahesh J. vadhavaniya 34
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