The Principle of Incompatibility | My Assignment Tutor

ECTE441/841/941Intelligent ControlPart AZheng LiAutumn 2021Lecture 01 2An Example of Cost-Usefulness Relationship Usually it takes a driver less than half a minute to parallel park.However, if we are asked to park a car in a parking space suchthat the outside of the wheels are precisely 0.01 mm from the sidelines of the curb, and the wheels are within 0.01 degree from aspecified angle, the amount of time required to park the care willbe much more.3The Principle of Incompatibility The underpinning of a fuzzylogic to achieve the twoobjectives described before isbased on an observation aboutthe trade off between precisionand cost, which is referred byZadeh as the principle ofincompatibility. This implies thatthere is a cost for high precision,and the cost for precisemodelling and analysis of acomplex system can be too highto be practical.CostUsefulness Domain ofFuzzy Logi Systems that areboth useful andcost effective.4Intelligent Control The development of Fuzzy logic was motivated in pursuit oftwo objectives: It aims to alleviate difficulties inmodeling, developing and analysingcomplex systems encountered byconventional mathematical tools. Observing that human reasoning canutilise concepts and knowledge that donot have well-defined, sharp boundaries(i.e. vague concepts.) These two motivations together not only make fuzzy logicunique and different from other technologies that focus on onlyone of these goals but also enable fuzzy logic to be a naturalbridge between the quantitative world and the qualitativeworld.5v6Panasonic SR-DE103 Rice Cookerhttp://www.amazon.com/exec/obidos/ASIN/b004sktthi/mitrasites-207Intelligent Control A control methodology that uses human, animalor biologically motived techniques andprocedures to develop or implement a controllerfor dynamic systems The techniques or procedures includes forms ofrepresentation or decision making8Conventional Controller vs Fuzzy Controller What happens when we build a classical controller? How about fuzzy controller?MathematicalModelingPerformanceObjectivesControllerDesignPerformanceEvaluation•Modeling is not necessary•Linguistic statement based IF-THEN inference•Heuristic knowledge is crucial in control design•Fuzzy controller is non-linear, many tools forLTI system could not be usedIntelligent controllers use empirical models that form the framework on how andnot why the controlled plant behaves in a particular manner, instead of relying onexplicit mathematical models of the plant. The fundamental problem in developingan intelligent controller is elicitation and representation of the knowledge andexperience of human operators in a manner that is amenable to computationalprocessing.9Advantages of Fuzzy Control It can be used for controlling process that is toononlinear or too ill-understood to use conventionalcontrollers It enables control engineers to easily implement controlstrategies used by human operators (heuristicknowledge)10Fuzzy Control Structure11Intelligent Control Methodologies Fuzzy Control Expert control – where a rule-based expert system is used. Learning control – where learning theories are used Neural control – which is motivated by low-level biological neuralnetworks for representation and decision making. Biologically motivated genetic algorithms to solve control problems. A combination of the above approaches such as Neuro-fuzzycontrol.12AI History ConventionalAIFuzzy LogicNeural SystemGenetic Algorithm1940s1947Cybernetics1943 McCullochPitts neuron model1950s1956 ArtificialIntelligence1957 Perceptron1960s1960 Lisplanguage1965 Fuzzy setslofti Zadeh1960s AdalineMadaline1970sMid-1970sexpertsystems1974 fuzzycontroller1974 Birth of Backpropagationalgorithm1970s Geneticalgorithm1980s1985 Fuzzymodelling (TSKmodel)1980 Selforganizing map1982 Hopfield NetMid- 1980s ArtificiallifeImmune modelling1990s1990s Neuro-fuzzymodellingANFIS (Adaptive NeuroFuzzy Inference System)1990 Geneticprogramming 13Control Theory ClassicalControl (TF)PIDPole PlacementModernControl (SS)DigitalControlECTE344 ECTE442/942 OptimalControlNonlinearControlAdaptiveControlRobustControlIntelligentControl•LQG•Kalman filter•MPC•H2•H∞•Lyapunov’stheory•Fuzzy•Neural•Genetic•…ECTE441/941ECTE94414Subject Outline Fuzzy Control — Weeks 1-6 Introduction to intelligent control and fuzzy sets Fuzzy inference systems Extension principle Mamdani fuzzy control systems Fuzzy control design issues TSK fuzzy inference systems and fuzzy controller Neuro System — Weeks 8-12 An overview of Artificial Neural Networks (ANN), Supervised Learning,Perceptrons; non-linear separability; multilayered Perceptrons. Specialized ANN – radial basis functions, Unsupervised Learning –Competitive learning, Recurrent NN, applications in control, FuzzyModel Reference Learning Control (FMRLC) Neuro Fuzzy Systems; ANFIS Neuro-fuzzy modeling Evolutionary Algorithms Reinforcement Learning Review and further considerations15Course Information Lecturers Dr. Zheng Li (coordinator) 35.G29 zli@uow.edu.au Prof Haiping Du 35.G24A hdu@uow.edu.au Reference book Soft computing and intelligent systems design F. O. Karray and C. D. Silva Neuro-Fuzzy and Soft Computing J.-S. R. Jang, C. –T. Sun, E. Mizutani Fuzzy Logic, Control and Information, John Yen and Reza Lagari Fuzzy Control, Addison-Wesley K.M. Passina and S. Yurkovich, Free online book athttp://www.ece.osu.edu/~passino/FCbook.pdf A printable version of the Fuzzy Logic Toolbox User’s Guide is also available in PDF formatat this site www.mathworks.com/help/pdf_doc/fuzzy/fuzzy.pdf16 Final Exam 60% Project 1 (on moodle) 20%Submission: Refer to subject outline. Reports (Submitted in subjectwebsite in Elearning)11.30am Friday, Week 8. Project 2 (on moodle) 20%Submission: Refer to subject outline. Reports (Submitted in subjectwebsite in Elearning)11.30am Friday, Week 13.Assessment for ECTE 441/84117 Final Exam 70% Project 1 (on moodle) 15%Submission: Refer to subject outline. Reports (Submitted in subjectwebsite in Elearning)11.30am Friday, Week 8. Project 2 (on moodle) 15%Submission: Refer to subject outline. Reports (Submitted in subjectwebsite in Elearning)11.30am Friday, Week 13.Assessment for ECTE 94118Marking Criteria for Report 1 System modelling and controller design 25%Accuracy of results Matlab simulation 25%Using correct excitation and take appropriate measurements to exposecharacteristics of control systems in both transient and steady phase Critical analysis and evaluation of the simulation resultsClear and logical presentation, layout, graphs, diagrams and tables25%  25%19Lecture & WorkshopWed. 15:30-17:30 (online)Tutorial & Laboratory Tue.Tue.11:30-12:30 (online)12:30-13:30 (6.227) ConsultationTue.Fri.8:30-10:30 (zoom meeting)8:30-10:30 (zoom meeting) Email to make appointmentsTeaching & Learning Methods201. Explain concept of fuzzy sets2. Characterize a fuzzy set using a membership functionObjective of This Lecture21 Sets with fuzzy boundariesSets with sharp boundariesA = Set of tall people Membershipfunction1.8m 1.9m Heights.5.9Fuzzy set A1.0Fuzzy Sets vs. Crisp Sets1.8m Heights1.0Crisp set ADegree of MembershipFuzzyMarkJohnTomBobBill1 1 1 0 01.001.000.980.820.78PeterStevenMikeDavidChrisCrisp1 0 0 0 00.240.150.060.010.00Name Height, cm205198181167155152158172179208Everything is a matter of degree in fuzzy sets22Characteristics of MFs: Subjective measures Context sensitive Not probability functionsMFsHeights.5.8.1“tall” in Asia“tall” in the US“tall” in NBAMembership Functions (MFs)1.8m 2.0m23A Classical (Crisp) Set To understand the fuzzy set moreclearly, let’s consider the classicalset of the days of a week. It certainlyincludes Monday, Thursday andSaturday but does not include shoepolish, butter, liberty, or matches. This is called a classical setbecause it was Aristotle who firstformulated the Law of ExcludedMiddle which says x must eitherbe in set A or in Set not A.Monday…Thursday…SaturdayButterLibertyShoePolishMatches24Days in a WeekendThu Fri Sat Sun Mon0.01.0Crisp LogicThu Fri Sat Sun Mon0.01.0Fuzzy LogicSaturdayButterLibertyShoePolishMatchesFridaySundayThursdayMonday Now let’s look at set ofdays which make aweekend. Most wouldagree that Saturdayand Sunday belong,but what about Friday.It “feels” like a part ofthe weekend. AlsoSunday is not part ofthe weekend as muchas Saturday.25Continuous-scale Weekend-ness The two bar charts for the crisp and fuzzy logics showing theweekend-ness of a day can be converted to continuous scale. Thediagram associated with the fuzzy logic shows a smoothly varyingcurve that accounts for the fact that part of Friday and even part ofThursday to a small degree partake of the quality of weekend-ness.The smooth curve is known in fuzzy control as a membershipfunction.Thu Fri Sat Sun Mon0.01.0Thu Fri Sat Sun Mon0.01.0Crisp Logic Fuzzy Logic26A Simple Membership Function A Membership function is a curve that defines how each point inthe input or output space is mapped to a membership value (ordegree of membership). As an example the membership functionillustrated below implies the membership definition:-2 0 x(x)1  0 otherwiseif 2 0221 if 0( ) x xx x-1 0.5 For x = -1, the degree of membership is o.5.27Formal definition:A fuzzy set A in X is expressed as a set of ordered pairs:A  {(x, A(x)) | x X }Universe oruniverse of discourseFuzzy set Membership function(MF)A fuzzy set is totally characterized by amembership function (MF).Fuzzy Sets: Definition28 A fuzzy set is a set with smooth boundary and it is an expansion of crisp settheory. In a fuzzy set partial membership is possible. Mathematically, the membership to a crisp set C is described by Whereas for a fuzzy set F the membership is defined as Where U is the universal set defined for a specific problem.Fuzzy Sets x Cx CC x0 if1 if ( )F :U  [0,1]29 A fuzzy set A can be alternatively denoted as follows:A x xAx Xi ii ( ) /A x xAX   ( ) /X is discreteX is continuousNote that S and integral signs stand for the union ofmembership grades; “/” stands for a marker and doesnot imply division.Alternative Notation30 Fuzzy set C = “desirable city to live in”X = {Wollongong, Sydney, Perth} (discrete and non-ordered)C = {(Wollongong, 0.9), (Sydney, 0.8), (Perth, 0.1)} F= integers close to 10,X={7,8,9,10,11,12,13}F=0.1/7+ 0.5/8 +0.8/9 +1/10 +0.8/11+ 0.5/12 +0.1/13Fuzzy Sets with Discrete Universe31 Fuzzy set B = “about 50 years old”X = Set of positive real numbers (continuous)B = {(x, B(x)) | x in X}2105011( )xxBFuzzy Sets with Continuos Universemembership function32Effective Membership Functions1TriangularFunction 1 TrapezoidFunction Matlab function:y = trimf(x,[a b c])Matlab function:y = trapmf(x,[a b c d])a b c a b c d The best membership functions are the ones which can be easily defined with a smallnumber of parameters. This will not only help the design of the fuzzy controller but willalso facilitate the automatic tuning of the fuzzy controller by adjusting the parameters. The most popular membership functions which fit into such requirements have beenTriangular and Trapezoid functions.33Membership Functions cont.GaussianFunctionGeneralized bellFunctionMatlab function:y = gaussmf(x,[σ c])Matlab function:y = gbellmf(x,[a b c])c34 Triangular MF:   trimf (x;a,b,c)  max   min   bx  aa , cc bx   ,0 Trapezoidal MF:   trapmf (x;a,b,c,d)  max   min   bx  aa ,1, dd  cx   ,0Generalized bell MF: bax cgbellmf x a b c 211( ; , , )Gaussian MF:21 2( ; , )x cgaussmf x c eMF Formulation35 Sigmoidal MF:( )1( ; , )1 a x csigmf x a ce Extensions:Abs. differenceof two sig. MFProductof two sig. MFdisp_sig.mMF Formulationy = dsigmf(x,[a1 c1 a2 c2])y = psigmf(x,[a1 c1 a2 c2])3600.20.40.60.81trapmf gbellmf trimf gaussmf gauss2mf smf00.20.40.60.81zmf psigmf dsigmf pimf sigmf37 MFMF Terminology XCoreCrossover pointsSupporta – cut .51 0a‘ { | ( ) }{ | ( ) }( ) { | ( ) 0.5}( ) { | ( ) 1}( ) { | ( ) 0}a  aa  a aa       Strong cut A x xcut A x xCrossover A x xCore A x xSuport A x xAAAAA38In Class Exesice ExampleForA = {0.2/a, 0.4/b, 1/c, 0.8/d, 0/e}B = {0/a, 0.9/b, 0.3/c, 0.2/d, 0.1/e}calculate the Support, Core and alpha cut A0.5Solution:SupportSupp(A) = {a, b, c, d}Supp(B) = {b, c, d, e}CoreCore(A) = {c}Core(B) = {}Alpha cutA0.5= {c, d}B0.5= {b}‘ { | ( ) }{ | ( ) }( ) { | ( ) 0.5}( ) { | ( ) 1}( ) { | ( ) 0}a  aa  a aa       Strong cut A x xcut A x xCrossover A x xCore A x xSuport A x xAAAAA39MF Terminology in Class ExciseGiven fuzzy set ADetermineTriangular MF:   trimf (x;1,0,1)  max min   0x  ((11)) ,11 0x   ,0‘ { | ( ) }{ | ( ) }( ) { | ( ) 0.5}( ) { | ( ) 1}( ) { | ( ) 0}a  aa  a aa       Strong cut A x xcut A x xCrossover A x xCore A x xSuport A x xAAAAA40MF Terminology in Class ExciseTriangular MF:   trimf (x;1,0,1)  max min    0x  ((11)) ,11 0x    ,0 -11x1A ‘ { | ( ) }{ | ( ) }( ) { | ( ) 0.5}( ) { | ( ) 1}( ) { | ( ) 0}a  aa  a aa       Strong cut A x xcut A x xCrossover A x xCore A x xSuport A x xAAAAA41MF Terminology in Class ExciseTriangular MF:   trimf (x;1,0,1)  max min    0x  ((11)) ,11 0x    ,0 -11x1A ‘ { | ( ) }{ | ( ) }( ) { | ( ) 0.5}( ) { | ( ) 1}( ) { | ( ) 0}a  aa  a aa       Strong cut A x xcut A x xCrossover A x xCore A x xSuport A x xAAAAA42MF Terminology in Class ExciseTriangular MF:   trimf (x;1,0,1)  max min    0x  ((11)) ,11 0x    ,0 -11x1A ‘ { | ( ) }{ | ( ) }( ) { | ( ) 0.5}( ) { | ( ) 1}( ) { | ( ) 0}a  aa  a aa       Strong cut A x xcut A x xCrossover A x xCore A x xSuport A x xAAAAA43MF Terminology ExampleTriangular MF:   trimf (x;1,0,1)  max min    0x  ((11)) ,11 0x    ,0 -1 1x1A ‘ { | ( ) 0.1} { | 0.9 0.9}{ | ( ) 0.1} { | 0.9 0.9}( ) { | ( ) 0.5} { 0.5,0.5}( ) { | ( ) 1} {0}( ) { | ( ) 0} { | 1 1}0.10.1                      Strong cut A x x x xcut A x x x xCrossover A x xCore A x xSuport A x x x xAAAAAa a  44