智能控制智能控制智能控制 (17).pdf

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1、Uncertainty&Reality“So far as the laws of mathematics refer to reality,they are not certain.And so far as they are certain,they do not refer to reality.”Albert EinsteinProbability vs FuzzinessI am thinking of a random shape(circle,square,or triangle).What is the probability that I am thinking of a c

2、ircle?Which statement is more accurate?It is probably a circle.It is a fuzzy circle.Probability vs FuzzinessTwo similar but different situations:There is a 50%chance that there is an apple in the fridge.There is half of an apple in the fridge.ParadoxesA heterological word is one that does not descri

3、be itself.For example,“long”；“monosyllabic”Is“heterological”heterological?ParadoxesBertrand Russells barber paradox(1901)The barber shaves a man if and only if he does not shave himself.Who shaves the barber?S:The barber shaves himselfUse t(S)to denote the truth of SS implies not-S,and not-S implies

4、 STherefore,t(S)=t(not-S)=1 t(S)t(S)=0.5Similarly,“heterological”is 50%heterologicalBertrand RussellParadoxesSeries paradox:Premise 1:One million grains of sand is a heap Premise 2:A heap minus one grain is a heap Question:Is one grain of sand a heap?Number of grains“Heap-ness”0100%Fuzzy SetsSets wi

5、th fuzzy boundariesA=Set of tall peopleHeights1701.0Crisp set AMembershipfunctionHeights150170.5.9Fuzzy set A1.0Membership Functions(MFs)Characteristics of MFs:Subjective measures Not probability functionsMembershipHeight160.7.8.6“tall”in Asia“tall”in the US“tall”in NBAFuzzy SetsFormal definition:A

6、fuzzy set A in X is expressed as a set of ordered pairs:AxxxXA(,()|Universe oruniverse of discourseFuzzy setMembershipfunction(MF)A fuzzy set is totally characterized by amembership function(MF).Fuzzy Sets with Discrete UniversesFuzzy set C=“desirable city to live in”X=SF,Boston,LA,XT(discrete and n

7、onordered)C=(SF,0.9),(Boston,0.8),(LA,0.6),(XT,0.95)Fuzzy set A=“sensible number of children”X=0,1,2,3,4,5,6(discrete universe)A=(0,.1),(1,.3),(2,.7),(3,1),(4,.6),(5,.2),(6,.1)Fuzzy Sets with Cont.UniversesFuzzy set B=“about 50 years old”X=Set of positive real numbers(continuous)B=(x,uB(x)|x in XBxx

8、()1150102Alternative NotationA fuzzy set A can be alternatively denoted as follows:AxxAxXiii()/AxxAX()/X is discreteX is continuousNote that S S and integral signs stand for the union of membership grades;“/”stands for a marker and does not imply division.Fuzzy Partition Fuzzy partitions formed by the linguistic values“young”,“middle aged”,and“old”:A linguistic value is a fuzzy set.Summery Uncertainty:Random and fuzzy;Fuzzy set;Expression of fuzzy set:Discrete universes Continuous universes Fuzzy partitions.