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ECTE441/841/941Intelligent ControlAutumn 2020Lecture 02 2Coronavirus COVID-19Be mindful of othersUOW is committed to creating a safe and respectfulenvironment for all members of our community. Racism willnot be tolerated. For more info:• Visit• Download UOW Safezone to your phoneFeeling Unwell?If you develop a fever, cough, sore throat or shortness ofbreath within 14 days of travel to an affected area, seekmedical attention Phone HealthDirect 1800 022 222 Phone your local GP Phone your local emergency departmentSupport for studentsTake care of each other and if you do need support,reach out to your Student Support Advisor or StudentCounsellingFor more info, you can’t attend classesContact Student Central• Phone: 1300 275 869• Email: [email protected]• Visit: COVID-19Practice good hygiene• Wash your hands thoroughly• Sneeze/cough into your elbow• Stay at home if you feel unwellFurther Information:••• UOW will communicate with students via email and updates on its website and social mediaplatformsStudent SupportAdvisers (SSAs)SSA for Engineering and Information Sciences FacultyMitz PerezEmail: [email protected] Building 4, room 105Available Monday to Friday51. We provide confidential appointments for students either face to face, over the phone and viaemail.2. We can help with general welfare or personal issues that may impact your study, and arrangesupport.3. We give information about UOW services and refer on when needed. We also link you withexternal services.4. If you are confused about how things work at UOW, or need information about policies,processes and your rights we can help you.5. Support for International students who are experiencing issues with: visas, accommodation,loneliness, study, fees, adjustment to their new country and English language skills.6. Provide information about where to start with a problem or question.What can Student Support Advisers dofor you?6Subject Outline Introduction to intelligent control and fuzzy sets Fuzzy operations and rules Fuzzy inference and control Fuzzy controller design and tuning Fuzzy extension TSK fuzzy control7Objectives of the Lecture Be able to use the basic fuzzy operators. Be able to use linguistic variables Be able to use fuzzy if-then rules Be able to use Matlab Fuzzy Inference System forapplication of fuzzy sets and rulesFuzzy Operation and rules8Lecture Structure1. Basic fuzzy set operators2. Linguistic variables, values and hedges3. Fuzzy T norm and S norm4. Fuzzy If … Then … rules5. Introduction to Matlab Fuzzy Inference Systems91 Basic Fuzzy Set OperationsSimilar to crisp set, various operations can be applied to fuzzy sets.The most basic operations are: Fuzzy set Identity Fuzzy set Subsethood Fuzzy set Union Fuzzy set Intersection Fuzzy set ComplementA fuzzy set is totally characterised by its membership function.Hence, the fuzzy set operations take place on the membershipfunctions of the sets.10IdentityA fuzzy set A in the universe U is identical to another fuzzy set B if,for every element x in U, its membership degree in A is equal to itsmembership degree in B.A  B xU A(x)  B(x)Crisp Sets1 BA Fuzzy Sets TemperaturexBA11SubsethoodA fuzzy set A in the universe U is a subset of another fuzzy set B if,for every element x in U, its membership degree in A is less than orequal to its membership degree in B.A B xU A(x)  B(x)BCrisp Sets1BFuzzy Sets TemperaturexAA12low mediumFuzzy Set UnionThe fuzzy set OR operation is also called the fuzzy set unionoperation and can be defined as the maximum operation. For twofuzzy membership functions low and medium, the union will belowmedium(x)  maxlow(x), medium(x)A BCrisp Sets1Low MediumFuzzy Sets Temperaturex13Fuzzy Set IntersectionThe fuzzy AND operation is also called fuzzy intersection operationand can be defined as the minimum operation. This is shown asfollows for two membership functions low and medium:low  MediumLow Medium1TemperatureA BCrisp Sets Fuzzy Setslowmedium(x)  minlow(x),medium(x)x14A BFuzzy Set ComplementThe fuzzy complement operation is also called NOT operation.When the not operation is applied to a fuzzy set it becomes thecomplement of that fuzzy set. MF of the complement is obtainedusing the following equationCrisp Sets Fuzzy Sets Temperaturemedium(x) 1 medium(x)Medium Medium1×15 Set-Theoretic Operations Subset: Complement: Union: Intersection: Example: not tall, tall or not tall, tall and not tallA  B  A  BC  A B  c (x)  max( A(x),B (x))  A(x)  B (x)C  A B  c (x)  min( A(x),B (x))  A(x)  B (x)A  X  A  A(x) 1 A(x)Degree of MembershipFuzzyMarkJohnTomBobBill1 1 1 0 0 0 0 Height, cm20519818116715515215817217920816Example1 2 3 4 51 2 3 4 50.5 0.3 0 0.1 0.70.2 0.7 1 0 0.5x x x x xBx x x x xA        ? ? ?_A B  A B  A 17Example1 2 3 4 51 2 3 4 50.5 0.3 0 0.1 0.70.2 0.7 1 0 0.5x x x x xBx x x x xA        1 2 3 4 5_1 2 3 4 5 1 2 51 2 3 4 50.8 0.3 0 1 0.50.2 0.3 0 0 0.5 0.2 0.3 0.50.5 0.7 1 0.1 0.7x x x x xAx x x x x x x xA Bx x x x xA B                18Properties of Set Theory19Difference: Crisp Set vs. Fuzzy SetA A U  A A Law of ContradictionLaw of Excluded Middle Crisp Set Fuzzy SetY YN NTemperature Medium MediumxFuzzy Set  xMedium Medium0011A A A A U  A A  A A U  20A numerical variable takes numerical values:Age = 65A linguistic variable takes linguistic values:Age is oldA linguistic value is a fuzzy set.All linguistic values form a term set:T(age) = {young, not young, very young, …middle aged, not middle aged, …old, not old, very old, more or less old, …not very yound and not very old, …}2 Linguistic Variables, Values and Hedges21Linguistic Values (Terms)22Example: Linguistic Variables10.90.70.823Concentration and Dilation Let A be a linguistic value characterized by a fuzzy setwith membership function A(.). Then Ak is interpretedas a modified version of the original linguistic valueexpressed as Concentration is defined asCON(A)=A2 Dilation is defined asDIL(A)=A0.5 Conventionally CON(A) and DIL(A) are taken to be theresults of applying the hedges very and more or less,respectively. XkAkA [ (x)] / x24Concentration and DilationNumber close to 5More or lessVeryExtremely25Exampleand .Determine , ,3010011( ) ( ,30,3,100)2011( ) ( ,20,2,0)Given64young but not very young extremly oldmore or less old not young and not oldxx bell xxx bell xoldyoung  26       XXoldyoungxx xnot young and not old young oldxxmore or less old DIL old oldxbell xxx bell x4 660.56430100111201113010011( )We have3010011( ,30,3,100)2011( ) ( ,20,2,0)Given27                         XXxxextremely old CON CON CON old oldxx xyoung but not very young young young862 2 224 423010011( ( ( ))) (( ) )201112011( )28Linguistic Values (Terms)292( ) , 0.5 ( ) 12 , 0 ( ) 0.5( ) 22    A xA xINT AAA Intensification to Reduce FuzzinessINT increases the values of A(x) which are greaterthan 0.5 & decreases those which are less or equalthat 0.5Contrast intensification has effect of reducing thefuzziness of the linguistic value A30Linguistic Variables and HedgesHedge MathematicalExpressionA littleSlightlyVeryExtremelyGraphical Representation[A(x)]1.3[A(x)]1.7[A(x)]2[A(x)]8Hedge MathematicalExpression Graphical RepresentationVery veryMore or lessIndeedSomewhat2 [A(x)]2A(x)A(x)if 0  A  0.5if 0.5 < A  11  2 [1  A(x)]2[A(x)]4Typical hedges:31Linguistic Values (Terms)323 Fuzzy T-norm operators T-norm, [0, 1] × [0, 1] → [0, 1], is the general operatorfor the intersection/conjunction operation of twofuzzy sets A and B.minminThey have the relation0 , 11 1Drastic product : ( , )Bounded product : ( , ) 0 ( 1)Algebraic product : ( , )Minimum : ( , ) min( , )Examples are( ) ( ( ), ( ))T T T Tif a bb if aa if bT a bT a b a bT a b abT a b a b a bx T x xdp bp apdpbpapA B A B            33Minimum:Tm(a, b)Algebraicproduct:Ta(a, b)tnorm.mT-norm Operator34T norm cont.Boundedproduct:Tb(a, b)Drasticproduct:Td(a, b)35T-norm operator A T-norm operator is a two-place function statisfying( , ( , )) ( ( , ), ) (associati vity)( , ) ( , ) (commutati vity)( , ) ( , ) if and (monotonic ity)(0,0) 0, ( ,1) (1, ) (boundary)T a T b c T T a b cT a b T b aT a b T c d a c b dT T a T a a    36S-norm (T-conorm)T-conorm, [0, 1] × [0, 1] → [0, 1], is the general operatorfor the union operation of two fuzzy sets A and Bas bs dsdsbsasA B A BS S S Sif a bb if aa if bS a bS a b a bS a b a b abS a b a b a bx S x x  >       maxmaxTheyhavethe relation1 , 00 0Drasticsum: ( , )Bounded sum ( , ) 1 ( )Algebraic sum ( , )Maximum: ( , ) max( , )Examples are ( ) ( ( ),  ( ))37 T-conorm operator is a two-place function satisfying( , ( , )) ( ( , ), ) (associati vity)( , ) ( , ) (commutati vity)( , ) ( , ) if and (monotonic ity)(1,1) 1, ( ,0) (0, ) (boundary)S a S b c S S a b cS a b S b aS a b S c d a c b dS S a S a a    T-conorm operator38tconorm.mMaximum:Sm(a, b)Algebraicsum:Sa(a, b)T-conorm or S-norm39S norm cont.Boundedsum:Sb(a, b)Drasticsum:Sd(a, b)40T-Norm S-Norm414 Fuzzy If-Then Rules (Mamdani)Examples:if service is good then tip is averageIf x is A then y is BThe if-then rule statements are used to formulate the conditionalstatements that comprise fuzzy logic. A single fuzzy if-then rule assumes the form bellow, where A and Bare linguistic values defined by fuzzy sets on the ranges (universesof discourse) X and Y, respectively.antecedentor premise consequentor conclusion42Example – Tipping Problem in US Problem: Given two numbers between 0 and 10 thatrepresents the quality of service and food at a restaurant(when 10 is excellent for service and delicious for food)what should the tip be for a specific visit? This problem is based on a tipping custom in UnitedStates which is on average 15% of the bill. This,however, can change depending on the quality of theservice provided. The if-then rule is If the service is excellent or the food is delicious then tip isgenerous43Step 1. Fuzzify Inputs44Step 2. Apply Fuzzy Operator45Step 3. Apply Implication Method46Fuzzy If-Then Rules1. Fuzzification of the input variables using fuzzy sets,2. Application of the fuzzy operator (AND or OR) in theantecedent,3. Implication from the antecedent to the consequent,47Fuzzy Air ConditionerStopSlowMediumFastBlast01020304050607080901000145 50 55 60 65 70 75 800ColdCool85 90JustRightWarmHotif Coldthen StopIF CoolthenSlowIf Just RightthenMediumIf WarmthenFastIf HotthenBlastAir Temperature in FFan Speed in %48Mapping Inputs to Outputs1StopSlowMediumFastBlast01020304050607080901000145 50 55 60 65 70 75 800ColdCool85 90JustRightWarmHott495 Fuzzy Logic Toolbox >> fuzzyMatlab FuzzyInferenceSystem (FIS)50FIS Editor51FIS for Tipper Select Edit > Add variable > Input. A second yellow box labeled input2 appears. Click the yellow box input1. This box is highlighted with a redoutline. Edit the Name field from input1 to service, and press Enter. Click the yellow box input2. This box is highlighted with a redoutline. Edit the Name field from input2 to food, and press Enter. Click the blue box output1. Edit the Name field from output1 to tip, and press Enter. Select File > Export > To Workspace Enter the Workspace variable name tipper and click OK52Membership Function Editor53Membership Function EditorYou can open the Membership Function Editor in one of three ways:Within the Fuzzy Logic Designer window, select Edit > MembershipFunctions.Within the Fuzzy Logic Designer window, double-click the blue icon calledtip.At the command line, type mfedit.The Membership Function Editor is the tool that lets you display and edit all ofthe membership functions.Double-click the input variable service to open the Membership FunctionEditor.In the Membership Function Editor, enter [0 10] in the Range and theDisplay Range fields.Create membership functions for the input variable service.Select Edit > Remove All MFs to remove the default membershipfunctions for the input variable service.Select Edit > Add MFs to open the Membership Functions dialog box.54Membership FunctionsIn the Membership Functions dialog box, select gaussmfas the MF Type.Verify that 3 is selected as the Number of MFs.Click OK to add three Gaussian curves to the inputvariable service.Rename the membership functions for the input variableservice, and specify their parameters.Click on the curve named mf1 to select it, and specifythe following fields in the Current Membership Function(click on MF to select) area:In the Name field, enter poor.In the Params field, enter [1.5 0].55The two inputs of Params represent the standard deviation and center for theGaussian curve.TipTo adjust the shape of the membership function, type in the desiredparameters or use the mouse, as described previously.Click on the curve named mf2 to select it, and specify the following fieldsin the Current Membership Function (click on MF to select) area:In the Name field, enter good.In the Params field, enter [1.5 5].Click on the curve named mf3, and specify the following fields in theCurrent Membership Function (click on MF to select) area:In the Name field, enter excellent.In the Params field, enter [1.5 10].The Membership Function Editor window looks similar to the followingfigure.In the FIS Variables area, click the input variable food to select it.Enter [0 10] in the Range and the Display Range fields.56MF of Service57MF for Food Select Edit > Remove All MFs to remove the defaultMembership Functions for the input variable food ifneeded. Select Edit > Add MFs to open the MembershipFunctions dialog box. In the Membership Functions dialog box, selecttrapmf as the MF Type. Select 2 in the Number of MFs drop-down list. Click OK to add two trapezoidal curves to the inputvariable food.58MF of Tip59Rule Editor60Fuzzy Rules Create rules by selecting an item in each input andoutput variable box, selecting one Connection item, andclicking Add Rule. You can choose none as one of thevariable qualities to exclude that variable from a givenrule and choose not under any variable name to negatethe associated quality. Delete a rule by selecting the rule and clicking DeleteRule. Edit a rule by changing the selection in the variablebox and clicking Change Rule.61If (service is poor) or (food is rancid) then (tipis cheap) To insert a rule in the Rule Editor, select the following: poor under the variable service rancid under the variable food The or radio button, in the Connection block cheap, under the output variable, tip. Then, click Add rule. The resulting rule isRule for Tipping


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