Fuzzy Logic Outside resources ì Fuzzy Sets § Professor Lo/i Zadeh, UC Berkeley, 1965 “People do not require precise, numerical informaBon input, and yet they are capable of highly adapBve control.” § Accepts noisy, imprecise input! 3 Fuzzy Logic Introduction Fuzzy Inference System Fuzzy Sets ì superset of convenBonal (Boolean) logic that has been extended to handle the concept of parBal truth ì central noBon of fuzzy systems is that truth values (in fuzzy logic) or membership values (in fuzzy sets) are indicated by a value on the range [0.0, 1.0], with 0.0 represenBng absolute Falseness and 1.0 represenBng absolute Truth. ì deals with real world vagueness Linguistic variable, linguistic term ì Linguis'c variable: A linguis(c variable is a variable whose values are sentences in a natural or arBﬁcial language. ì For example, the values of the fuzzy variable height could be tall, very tall, very very tall, somewhat tall, not very tall, tall but not very tall, quite tall, more or less tall. ì Tall is a linguis(c value or primary term ì Hedges are very, more or less so on ì If age is a linguisBc variable then its term set is ì T(age) ì young, not young, very young, not very young ì middle aged, not middle aged ì old, not old, very old, more or less old, not very old young middle aged old 1.0 µ 0.0 Age Operations A B A ∧ B A ∨ B ¬A Fuzzy Rules ì Fuzzy rules are useful for modeling human thinking, percepBon and judgment. ì A fuzzy if-­‐then rule is of the form “If x is A then y is B” where A and B are linguisBc values deﬁned by fuzzy sets on universes of discourse X and Y, respecBvely. ì “x is A” is called antecedent and “y is B” is called consequent. Examples, for such a rule are ì If pressure is high, then volume is small. ì If the road is slippery, then driving is dangerous. ì If the fruit is ripe, then it is soY. Example Air CondiBoning Controller Example: ì IF Cold then Stop ì If Cool then Slow ì If OK then Medium ì If Warm then Fast ì IF Hot then Blast Fuzzy Air Conditioner 0 100 90 80 Bla Fa If Hot then Blast st st If Warm then Fast 70 60 Med 50 40 If Just Right then Medium ium Slo IF Cool then Slow w 30 if Cold then Stop 20 o St 10 p 0 Ho t W ar m Jus Rig t ht ld Co Co ol 1 0 45 50 55 60 65 70 75 80 85 90 Mapping Inputs to Outputs 1 0 100 90 80 s Bla Fa t t st 70 60 Med 50 40 iu m Sl ow 30 20 o St 10 p 0 ld Ho t W ar m Ju s Rig t ht Co Co ol 1 0 45 50 55 60 65 70 75 80 85 90 14 Fuzzy Logic Introduction Fuzzy Inference System 15 Fuzzy Logic Introduction • Fuzzy Inference System... Mamdani Method • In 1975, Professor Ebrahim Mamdani of London University built one of the first fuzzy systems to control a steam engine and boiler combination. He applied a set of fuzzy rules supplied by experienced human operators. 16 Fuzzy Logic Introduction Fuzzy Inference System… 17 Fuzzy Logic Introduction • Fuzzy Inference System… o An example ì Two inputs (x, y) ì One output (z) ì Rules: Rule1: C1 Rule2: C2 Rule3: C3 If x is A3 or y is B1 Then z is If x is A2 and y is B2 Then z is If x is A1 Then z is 18 Fuzzy Logic Introduction • Fuzzy Inference System… o Input x: research_funding o Input y: project_staﬃng o Output z: risk ì Rules: Rule1: If research_funding is adequate or project_staﬃng is small Then risk is low Rule2: If research_funding is marginal and project_staﬃng is large Then risk is normal Rule3: If research_funding is inadequate Then risk is high 19 Step 1: Fuzzification 20 Step 2: Rule Evaluation 21 Applying to the membership function The result of the antecedent evalua(on can be applied to the membership func(on of the consequent in two diﬀerent ways: 22 Step 3: Rule Evaluation 23 Step 4: Defuzzification Using Center of Gravity method, but other methods can also be used Why Fuzzy Logic? § Advantages § Mimicks human control logic § Uses imprecise language § Inherently robust § Fails safely § Modified and tweaked easily Why Fuzzy Logic? § Disadvantages § Operator's experience required § System complexity Game using Fuzzy Logic – Battle City What are advantages of this approach?