Instructor: Shihua Li School of Automation Southeast university

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1 Course Introduction to Intelligent Control Instructor: Shihua Li School of utomation Southeast university Chap. 2 Fundamentals of Fuzzy Logic Systems(P. 57) 2. Introduction Fuzzy logic a systematic, intuitive and mathematical means of handling vagueness (Linguistic uncertainty) observed in real world (natural and artificial systems). ny comments, please feel free to contact me (oom 68, Central Hall, lsh@seu.edu.cn, Tel.: (o)) 2.2 ackground There are at least three types of uncertainty: Stochastic uncertainty Something is probably true or false (randomness). Informational uncertainty Something is in some interval, but we don t know where. (imprecision) Linguistic uncertainty Something is true to some degree (vagueness). Types of Uncertainty (cont d) e sure you understand the difference. Probability: impersonal uncertainty, occasionality, can be turned to certainty E.g. rolling a dice. imprecision: Whether your bank balance is going up or down is qualitative. credit worthiness, honesty Vagueness: uncertainty in the understanding of concepts, can not be turned to certainty. E.g. How much you like someone. Tall people, young age Vagueness Fuzzy: People think more easily in fuzzy ranges less so with absolute numbers Eample: the temperature in the room? What is warm? What is hot? Humans are fleible (soft) systems.

2 When do we get old? Vagueness re we not old at 39, but old at 4? In crisp logic there would have to be one sharp dividing line (boundary) between the old and the not old. What is Fuzzy Logic? Fuzzy Logic is a superset of conventional (oolean) logic that has been etended to handle the concept of partial truth, i.e. truth values between completely true and completely false. The water is warm. Fuzzy logic might be more intuitive than differential equations for nonlinear problems. It might be easier to decompose the problem using fuzzy logic. Fuzzy allows us to be young and old at the same time to some degree. More-or-Less ather Than Either-Or! What is Fuzzy Logic? Crisp Logic proposition can be true or false only. ob is a student (true) Smoking is healthy (false) The degree of truth is or. Fuzzy Logic The degree of truth is between and. William is young (.3 truth) riel is smart (.9 truth) Differences between crisp and fuzzy oundaries between crisp sets are sharp and clear oundaries between fuzzy sets are gradual, not dramatic ather than think SMLL/IG, think SMLL,SOMEWHT SMLL/MEDIUM/SOMEWHT MEDIUM/IG Don t think EITHE/O, think OTH/ND How strongly does an object/value belong to a set? 2.2. Evolution of Fuzzy Logic The idea behind fuzzy logic dates back to Plato, who recognized not only the logic system of true and false, but also an undetermined area the uncertain. 965, Seminal Paper by Prof. Lotfi Zadeh (U.C. erkeley), Sets the Foundation of the Fuzzy Set Theory Lotfi. Zadeh, Fuzzy sets, Information and Control, Vol. 8, , 965 Evolution of Fuzzy Logic 972, Seminar paper rational for fuzzy control, Zadeh, 974, Mamdani (England), Fuzzy control steam engine control 975, Introduction of Fuzzy Logic in Japan Fuzzy Logic has been around since the mid 6 s but was not readily accepted until the 8 s and 9 s. lthough now prevalent throughout much of the world, China, Japan and Korea were the early adopters 2

3 Evolution of Fuzzy Logic 98, Holmblad and Ostergaard, first industrial application of FL, control of cement kiln (Denmark) 985, road pplication of Fuzzy Logic in Japan(P. 6-63) 99, road pplication of Fuzzy Logic in Europe 995, road pplication of Fuzzy Logic in the U.S. 2, Fuzzy Logic ecomes a Standard Technology in many application field: Control, Data and Sensor Signal nalysis, etc. Popular pplications Quality ssurance Error Diagnostics Control Theory Pattern ecognition Decision Support Otis Elevators Vacuum Cleaners Hair Dryers Specific Fuzzified pplications ir Control in Soft Drink Production Noise Detection on Compact Disks Cranes Electric azors Camcorders Television Sets Showers Sendai subway system Impact of Lotfi Zadeh's work on fuzzy logic The number of papers in the literature which contain the word "fuzzy" in title (data drawn from the INSPEC and Mathematical eviews databases) INSPEC/fuzzy : : 2, : 23,753 total : 26,68 Math.Sci.Net/fuzzy : : 2, : 8,428 total :,357 Number of citations in the Citation Inde: over,. Chap. 2 Fundamentals of Fuzzy Logic Systems Crisp set vs. Fuzzy set 2.3 Fuzzy sets 2.3. Membership function. Definition X : universe of discourse. oolean/crisp set (Cantor, end of 9 th century) is a mapping for the elements of X to the set {, }, i.e., : X {, } Characteristic { function: () = if is an element of set if is not an element of set Fuzzy set F is a mapping for the elements of X to the interval [, ], i.e., F: X [, ] Membership function: F () means full membership, means no membership and anything in between, e.g.,.5 is called graded membership 3

4 Membership function of fuzzy logic Symbolic representation DOM Degree of Membership.5 Fuzzy values Young Middle Old ge {(, ( )), ( ) [,]} = X { ( )} = ( ) Zadeh notation = Continuous Universe X Discrete Universe = ( )/ + ( )/ + L+ ( )/ + L 2 2 i i ordered pairs( 序偶 ) ( i) = i X i Eample (Continuous Universe) Eample (Discrete Universe) U : the set of positive real numbers {(, ( )) } = X ( ) = lternative epresentation: = ( )/ = + 5 ( ) 4 5 X + 4 possible ages The common method of representing fuzzy set is : about 5 years old ( ) :age X = {,2,3,4,5,6,7,8} (,.), (2,.3), (3,.8), (4,), = (5,.9), (6,.5), (7,.2), (8,.) ( ) : # courses # courses a student may take in a semester. appropriate # courses taken Eample (Discrete Universe) X = {,2,3,4,5,6,7,8} epresentation : (,.), (2,.3), (3,.8), (4,), = (5,.9), (6,.5), (7,.2), (8,.) # courses a student may take in a semester. appropriate # courses taken epresentation 2: =./ +.3/ 2+.8/ 3+. / 4+.9 / 5+.5/ 6+.2 / 7+./ Fuzzy logic operations(p.68) Equality = () = () for all X Complement(negation, NOT) () = - () for all X Containment () () for all X P.7 hot and NOT hot 4

5 Fuzzy set operators Eample fuzzy set operations Zadeh Operator for any a,b in [,] a v b=ma{a,b}, a b = min{a,b} Union(disjunction, O, 并 ) () = () v () =ma( (), ()) for all X Intersection(conjunction,ND, 交 ) () = () () =min( (), ()) for all X asic laws of fuzzy logic Eample fuzzy set operations properties Involution Commutativity ssociativity Distributivity Idempotence bsorption = = ( ) = = = ( ) C = ( C) ( ) C = ( C) ( C) = ( ) ( C) ( C) = ( ) ( C) = = ( ) = ( ) = De Morgan s laws = = = u u2 u3 u4 u = u u u u u U = I = u u u u u u u u u u U = { u, u, u, u, u } U = I = = u u u u u u u u u u u u u u u Eample fuzzy set operations Properties The following properties are invalid for fuzzy sets: The laws of Eclusion = = U 5

6 2.5 Triangular norms f :[,] [,] [,] Generalized Intersection T-norm Generalized Union T-conorm or S-norm T-Norm T :[,] [,] [,] a= ( ), b= ( y) Tab (, ). Monotonicity 2, y y2 T(, y) T( 2, y2) 2. Symmetry(Commutativity) T(, y) = T( y, ) 3. ssociativity 4. oundary Conditions or TTy ( (, ), z) = TT (, ( yz, )) T(,) = T(,) = atb. Monotonicity S-norm S :[,] [,] [,] 2. Symmetry(Commutativity) 3. ssociativity, y y S(, y ) S(, y ) Sy (, ) = Sy (, ) SSy ( (, ), z) = SSyz (, (, )) 4. oundary Conditions S (,) = S (,) = Q: MonotonicityDe Morgan s laws true or not? Eamples: T-Norm & T-Conorm Minimum/Maimum(Zadeh): Lukasiewicz(boundary): Probabilistic: Tab (, ) = min( ab, ) = a b Sab (, ) = ma( ab, ) = a b Tab (, ) = ma( a+ b,) = LNDab (, ) Sab (, ) = min( a+ b,) = LOab (, ) T ( a, b) = ab = PND( a, b) S( a, b) = a+ b ab= PO( a, b) 2. Fuzzy set operators Minimum operator is the largest T-norm and maimum operator is the smallest S-norm. (P. 79-8) Ty min( y, ) ma( y, ) Sy Zadeh T,S-norms : simple and efficient, but not smooth enough Principle of selecting T,S-norms operator (4 conditions) min{ ab, } Zadeh a b probabilistic a b= ma{, a + b } boundary user defined M a b? ( a+ b a b) (2)Other Definitions for Set Operations Probabilistic Operator Probabilistic Intersection (lgebraic Product) ( ) = ( ) ( ) Probabilistic Union (lgebraic Sum) ( ) = ( ) + ( ) ( ) ( ) laws of Idempotence, Distributivity, bsorption, Eclusion 6

7 (2)Other Definitions for Set Operations oundary Operator oundary Intersection (oundary Product) oundary Union (oundary Sum) ( ) ( ) = ma, ( ) + ( ) ( ) ( ) = min, ( ) + ( ) Intersection Operator largest T-norm laws of Idempotence, Distributivity, bsorption, Eclusion Union Operator 2.6 Implication 蕴涵 (if -then) fuzzy rule is a fuzzy relation. smallest S-norm Use fuzzy sets and fuzzy operators as the subjects and verbs of fuzzy logic to form rules. if is then y is. (Fuzzy elations [,]) antecedent consequence where and are linguistic terms defined by fuzzy sets on the sets X and Y respectively. What is the difference between classical and fuzzy rules? classical IF-THEN rule uses binary logic, for eample, ule: IF speed is > THEN stopping_distance is long ule: 2 IF speed is < 4 THEN stopping_distance is short The variable speed can have any numerical value between and 22 km/h, but the linguistic variable stopping_distance can take either value long or short. In other words, classical rules are epressed in the black-and-white language of oolean logic. We can also represent the stopping distance rules in a fuzzy form: ule: IF speed is fast ule: 2 IF speed is slow THEN stopping_distance is long THEN stopping_distance is short In fuzzy rules, the linguistic variable speed also has the range (the universe of discourse) between and 22 km/h, but this range includes fuzzy sets, such as slow, medium and fast. The universe of discourse of the linguistic variable stopping_distance can be between and 3 m and may include such fuzzy sets as short, medium and long. 7

8 Eamples Fuzzy ules as elations If is then y is. If the road is slippery, then driving is dangerous. If a tomato is red, then it is ripe. If the speed is high, then apply the brake a little. fuzzy rule can be defined as a binary relation with MF If is then y is. ( y, ) = (, y) Depends on how to interpret Fuzzy Implication Fuzzy If-Then ules Method : Mamdani Implication [ ] (, y) = min ( ), ( y), X, y Y Not intuitively satisfying, but provides a good, robust result (more commonly used) Method 2 : Lukasiewicz Implication ( y, ) = min[,{ ( ) + ( y)} ], X, y Y n intuitive appeal [(NOT ) O ] Two ways to interpret If is then y is : is coupled with entails y y Local global 46 Table 2.3 (P. 85) Eam. 2.6 (P ) epresentati Meaning on of -> (Not ) Or ( ) ( nd ) Or (Not ) nd Comments Same as in crisp binary logic; a weaker(global) implication, meaning entails Same as above stronger(local) implication, meaning is coupled with T-norms Eamples D implication ma{ ( ), ( y)} Luka implication min{, ( ) + ( y)} Zadeh implication ma[ min{ ( ), ( y)}, ( ) ] Mamdani implication min{ ( ), ( y)} Larsen implication ( ) ( y) Probabilistic T-norm Larson Mamdani Zadeh D Luka T-norm? Most local! Global! 8

9 2.7 Some definitions Height of a fuzzy set of is the largest membership grade obtained by any element in. hgt( ) = sup ( ) X Support set ( 支集 ) of a fuzzy set within a universal set X is the crisp set that contains elements of X that have non zero membership in. { ( ) } S = X > Modal point α Cut α-cut of a fuzzy set defined on X for a given α is the crisp set: ( ), [,] α = X α α Note: { } Strong α cut of a fuzzy set defined on X for a given α is the crisp set: { } ( ), [,] α = X > α α if α < then α, 2 α α2 Some definitions Support set of = strong cut of The -cut of (all the elements that have membership = ) is called the core of. fuzzy set is called normal if its height is. Otherwise it is called subnormal. Eample Fuzzy sets : young, middle aged and old Membership Young Mid ge 2 Old ge Discrete approimation of middle age concept D2: {,2,4,6,,8} [,] Eample Not in {22,24,,58} {22,58} {24,56} {26,54} {28,52} {3,5} {32,48} {34,46} {36,38,,44} D2() cut of D2 =?.3-cut of D2 =?.53-cut of D2=? Height of D2=? Modal points of D2=? S(D2)=? The core of D2=? -cut of D2 = X.3-cut of D2 = {22,24,,58}.53-cut of D2={28,3,,52} Height of D2= Normal set Modal points of D2=36,38,,44 S(2)={22,24,,58} The core of D2={36,38,,44} 9

10 2.9 Fuzzy relations Crisp elations {,} & Fuzzy elations [,] Crisp elations Ordered pairs showing connection between two sets: (a,b): a is related to b (2,3) are related with the relation < elations are set themselves < = {(,2), (2, 3), (2, 4),.} < elations can be epressed as matrices 2 2 M inary elation () a a 2 a 3 a 4 = b b 2 b 3 b 4 b 5 a b a b a 3 2b5 ( a, b),( a, b3),( a2, b5) = ( a3, b),( a3, b4),( a4, b2) a 3b a 3b a 4 4 b2 The eal-life elation is close to y and y are numbers depends on y and y are events and y look alike and y are persons or objects If is large, then y is small is an observed reading and y is a corresponding action Fuzzy elations Triples showing connection between two sets: (a,b,#): a is related to b with degree # Fuzzy relations are set themselves Fuzzy relations can be epressed as matrices Fuzzy elations Matrices Eample: Color-ipeness relation for tomatoes (, y) unripe semi ripe ripe green.5 yellow.3.4 ed.2

11 2.9. nalytical representation of a fuzzy relation {(, ), } = a b a b fuzzy relation is a 2-Dimension(2D) DOM function: {(( ab, ), ( ab), ) ( ab, ) } = Fuzzy matri (discrete case): = { u, L, un}, = { v, M = { rij} n m = { ( ui L, vm}, v )} j n m 2- n- Fuzzy elations {(( a, L, an), ( a, L, an) ) ( a, L, an) L n} = Eample (pproimate Equal) {(( ab, ), ( ab), ) ( ab, ) } = = = U = {,2,3,4,5} M = Cartesian Product of fuzzy sets Fuzzy rule: If (u)-error Then (v)- output of control Fuzzy elation: Given DOM Functions of and, how to conclude DOM Functions of? Cartesian Product Cartesian Product( 笛卡儿积 ) : Minimum Operator ( 直积 ) = = U V U V ( u, v) /( u, v) min( ( u), ( v)) /( u, v) Other T-norm Operator (Probabilistic Operator etc.) = ( u, v)/( u, v) = U V U V ( ut ) ( v)/( uv, ) ule: if C is slow, then is fast. C = slow= /+.7/2+.3/4+ / 6 + /8 + /; = fast= /+ /2+.3/4+.7 / 6 + / 8 + /; = fast= C= slow? C =? Eample C = min( ( u), ( v)) uv, C.7.3 = min(.3.7 ) =

12 Eample 2. Composition and inference C.7 uv,.3 = t( = C( u) ( v) uv, =? = More smooth C = ( u) T ( v).3.7 ) pplication of a fuzzy input to a fuzzy relation in order to infer a fuzzy output is the basis of decision-making in a fuzzy knowledge-based system (KS). fuzzy inference, approimate reasoning Inference: evaluation of fuzzy rules to produce an output for each rule. Composition: aggregation or combination of the outputs of all rules. elation Composition (Sup-T Operator) elation Composition (Sup-T Operator) If Then, if Then C obtain relations between and C I/O relation X Y Z : fuzzy relation defined on X and Y. S: fuzzy relation defined on Y and Z. S: the composition of and S. fuzzy relation defined on X an Z. (, z) = sup{ (, y) T ( y, z)} S o S y Y Supremum (, z) = sup{ (, y) T ( y, z)} Sup-Min Sup-Dot S o S y Y ( ( y S y z )) = sup min (, ), (, ) y Y ( y y z ) = sup (, ) (, ) y Y S Eample Eample a b c d min sup o S α β γ S o (, y) = sup{ (, v) S(, v y)} v S α β γ a.9..3 b.2..8 c d dot sup o S α β γ fat mot son.2.8 S Gfat Gmot dau.6. fat.5.7 mot. min sup (, y) = sup{ (, v) T (, v y)} S o S v o S Gfat Gmot son.2.2 dau.5.6 o S Gfat Gmot son..4 dau

13 elation Composition (Sup-Min Operator) elation Composition (Sup-Min Operator) Properties: o I = I o = o = o = m+ m+ n = m = m o o n Laws of Distributivity ssociativity Containment Inversion ( U T ) o S = ( o S) U ( T o S) o( T U S) = ( ot ) U ( o S) ( I T ) o S = ( o S) I ( T o S) o( T I S) = ( ot ) I ( o S) o( S ot ) = ( o S) ot IF S T, THEN S o T o ( o S) T T T = S o Commutativity o S = S o Evaluation of fuzzy rules In oolean logic: p q if p is true then q is true In fuzzy logic: p q if p is true to some degree then q is true to some degree..5p =>.5q (partial premise implies partially) How? Fuzzy inference Conventional inference If then ; Now, If tomato is red, then it is ripe. red ripe Fuzzy inference If then ; Now,? If tomato is red, then it is ripe. sort of red sort of ripe Inference methods: Mamdani, Larson, et al. approimate reasoning Fuzzy inference If then ; Now, = = ( u) T( v)/( u, v) U V Method :Composition through matri multiplication ' = ' o Eample 2.24(P. 8-9) How to Calculate? approimate reasoning Fuzzy inference If then ; Now, = = ( u) T( v)/( u, v) U V Method :Composition through matri multiplication ' = ' o Method 2: Clipping method ( y) = sup ( ) (, ) ' ' How to Calculate? 3

14 approimate reasoning If then ; Now, ( ) = sup min( ( ), ( )) ( y) = α ( y) ' ' Sup-min + Mamdani implication ( y ) = sup ( ) (, ) ' ' ( y ) = sup min( ( ), (, )) ' ( y ) = sup min( ( ), ( ), ( ))) ' ( ) = sup min( ( ), ( )) ( y) = α ( y) ' multiple input reasoning If and then C; Now and C ealization of fuzzy Inference (discrete case: two methods) Matri method: C' = ( ' ') o C (a)d=, (b)d DT d L d n (c)=dt C D = M L M (d)d = dm L dmn (e)d DT (f) C =DT DT = [ d L d ] T n d2 L d mn Clipping method Eample = + = + + C = + 2 y y2 y3 z z = + = + + C' = ( ' ') o C=? 2 y y2 y D = = = DT C = [.2 ] = D = = [.5.2 ]. = C' = o =.2.2 [ ] [ ] Q: How about using clipping method? Fuzzy inference multiple input reasoning If and then C; Now and C C' = ( ' ') o C = sup (, ) (,, ) ( y yz) C' ' ' C y, y, y, ( y y z ) = sup min( ( ), ( )) min( ( ), ( ), ( )) ' ' C ( y y z ) = sup min( ( ), ( ), ( ), ( ), ( )) ' ' C ( ) ( ) = sup min( ( ), ( )) sup min( ( y), ( y)) ( z) = ( α α ) ( z) C ' ' C y multiple input reasoning α = min{ α, α } C 4

15 Eample.5..5 = + = y y2 y = + = + + y y y C = + z z 2 ( ) ( ) ' C' = ( ' ') o C=? α = sup min( ( ), ( )) = sup.8,. =.8 ( y y ) ( ) α = sup min( ( ), ( )) = sup.,.2, =.2 ' y [ ] C ' = ( α α ) =.2 =.2.2 C C y Fuzzy inference multiple input-multiple rule reasoning If and then C; (if e=p and de=ze then u=p) If n and n then Cn; Now and C C' = ( ' ') o = ( ' ') ou( C) i i i i C' = U(( ' ') o ( ) = U(( α α ) ) i i i Ci i i C i i multiple input-multiple rule reasoning Larsen implication method C α u v w multiple input-multiple rule reasoning If and then C; If n and n then Cn; Now and C 2 2 C 2 α 2 C' = U(( ' ') o ( C) = U(( α α) C) i i i i i i i i u v min w Larsen implication method α u v w α 2 u v min w Graphical representation of Larsen method with fuzzy set input 5