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Fuzzy Inference and Reasoning Proposition 2 Logic variable 3 Basic connectives for logic variables (1)Negation (2)Conjunction 4 Basic connectives for logic variables (3) Disjunction (4)Implication 5 Logical function 6 Logic Formula 7 Tautology 9 Tautology 10 Predicate logic 11 Fuzzy Propositions • Assuming that truthand falsity are expressed by values 1 and 0, respectively, the degree of truth of each fuzzy proposition is expressed by a number in the unit interval [0, 1]. Fuzzy Propositions p : temperature (V) is high (F). Fuzzy Propositions p : V is F is S • • • • V is a variable that takes values v from some universal set V F is a fuzzy set onV that represents a fuzzy predicate S is a fuzzy truth qualifier In general, the degree of truth, T(p), of any truth-qualified proposition p is given for each v e V by the equation T(p) = S(F(v)). p : Age (V) is very(S) young (F). Representation of Fuzzy Rule 17 Representation of Fuzzy Rule 18 Fuzzy rule as a relation If x is A, then y is B x is A, y is B fuzzy predicatesA(x), B( y ) If A(x), thenB( y ) can be represented by relation R( x, y ) : A(x) B( y ) R( x, y ) can be considereda fuzzy set with 2 - dim membershipfunction R ( x, y ) f ( A ( x), B ( y )) wheref is " fuzzy implication function", performsthe task of transforming themembershipdegrees of x in A and y in B into thoseof ( x, y ) in A B 19 Fuzzy implications 20 Example of Fuzzy implications Let T and H be universeof temperatu re and humidity, and define variablest T and h H. A " high", A T B " fairly high", B H t hen t herule can be rewrit t enas R(t ,h) : If t is A, thenh is B R(t ): t is A, R(h) : h is B R(t ,h) : R(t ) R(h) R(t ,h) A B A (t ) B (h) /(t , h) 21 Example of Fuzzy implications R(t, h) A B A (t ) B (h) /(t , h) h t 20 30 40 20 50 70 90 0.1 0.2 0.2 0.1 0.5 0.6 0.1 0.5 0.7 0.1 0.5 0.9 22 Example of Fuzzy implications When" temperature is fairlyhigh"or t is A' , A' T We can use composition of fuzzy relations to find R(h) R( h) R(t ' ) R C (t , h) h t 20 30 40 20 50 70 90 0.1 0.2 0.2 0.1 0.5 0.6 0.1 0.5 0.7 0.1 0.5 0.9 23 Representation of Fuzzy Rule Single input and single output Fact: u is A ' : R(u) Rule: If u is A then w is C : R(u, w) Result: w is C ' : R( w) R(u) R(u, w) Multiple inputs and single output Fact: u1 is A1' ' and u2 is A2' ' and ... and un is An' ' Rule: If u1 is A1 and u2 is A2 and ... and un is An then w is C Result: w is C ' Multiple inputs and Multiple outputs Fact: u1 is A1' and u2 is A2' and ... and un is An' Rule: If u1 is A1 and u2 is A2 and ... and un is An then w1 is C1 , w2 is C2 ,..., wm is Cm Result: w1 is C1' , w2 is C2' ,..., wm is Cm' 24 Representation of Fuzzy Rule Multiple rules Fact : u1 is A' 1 and u2 is A' 2 and ... and un is A' n Rule j : If u1 is A' 1 j and u2 is A' 2 j and ... and un is A' nj , then w1j is C'1j , w 2j is C' 2j , ..., w mj is C' mj Result : w1 is C'1 , w 2 is C' 2 , ..., w m is C' m 25 Compositional rule of inference The inference procedure is called as the “compositional rule of inference”. The inference is determined by two factors : “implication operator” and “composition operator”. For the implication, the two operators are often used: For the composition, the two operators are often used: 26 Representation of Fuzzy Rule Fact: u is A ' : R(u) Rule: If u is A then w is C : R(u, w) Result: w is C ' : R( w) R(u) R(u, w) Max-min composition operator R(u, w) : A C Mamdani: min operator for the implication Larsen: product operator for the implication 27 One singleton input and one fuzzy output Mamdani Fact: u is A ' : R(u) Rule: If u is A then w is C : R(u, w) Result: w is C ' : R( w) R(u) R(u, w) 28 One singleton input and one fuzzy output Mamdani 29 One singleton input and one fuzzy output Larsen Fact: u is A ' : R(u) Rule: If u is A then w is C : R(u, w) Result: w is C ' : R( w) R(u) R(u, w) 30 One singleton input and one fuzzy output Larsen 31 One fuzzy input and one fuzzy output Mamdani Fact: u is A ' : R(u) Rule: If u is A then w is C : R(u, w) Result: w is C ' : R( w) R(u) R(u, w) 32 One fuzzy input and one fuzzy output Mamdani 33 Ri consists of R1 and R2 Rule i : If u is Ai and v is Bi , then wis Ci C 'i (A' , B' ) (Ai and Bi Ci ) μ C'i (μ A ' , μ B' ) (μ A i Bi μ C ) (μ A ' , μ B' ) (min(μ A i , μ Bi ) μ C ) (μ A ' , μ B' ) min[(μ A i μ C ), (μ Bi μ C )] max min{(μ A ' , μ B' ), min[(μ A i μ C ), (μ Bi μ C )]} u ,v max min{min[μ A ' , (μ A i μ C )],min[μ B' , (μ Bi μ C )]} u ,v min{[μ A ' (μ A i μ C )],[μ B' (μ Bi μ C )]} C 'i [A' (Ai Ci )] [B' (Bi Ci )] [A' R 1i ] [A' R i2 ] C1i Ci2 34 Example R : if x is A and y is B, thenz is C where A (0,1,2), B (1,2,3),C (5,6,7)are triangular fuzzy sets. If input x0 1 and y0 1.5 (Singleton) , thenoutput? 35 Two singleton inputs and one fuzzy output Mamdani Fact: u is A ' and v is B ' : R(u, v) Rule: If u is A and v is B then w is C : R(u, v, w) Result: w is C ' : R(w) R(u, v) R(u, v, w) 36 Two singleton inputs and one fuzzy output Mamdani 37 Example R : if x is A and y is B, thenz is C where A (0,1,2), B (1,2,3),C (5,6,7)are triangular fuzzy sets. If input x0 1 and y0 1.5 (Singleton) , thenoutput? 38 Two fuzzy inputs and one fuzzy output Mamdani Fact: u is A ' and v is B ' : R(u, v) Rule: If u is A and v is B then w is C : R(u, v, w) Result: w is C ' : R(w) R(u, v) R(u, v, w) 39 Two fuzzy inputs and one fuzzy output Mamdani 40 Two fuzzy inputs and one fuzzy output Mamdani 41 Example R : if x is A and y is B, thenz is C where A (0,1,2), B (1,2,3),C (5,6,7)are triangular fuzzy sets. If input A' (1,2,3)and B' (1.5,2.5,3.5)(Fuzzy set) , thenoutput? 42 Multiple rules 43 Multiple rules 44 Multiple rules 45 Example R 1 : if x is A1 , thenz is C1 R 2 : if x is A 2 , thenz is C 2 where A1 (0,1,2), C1 (1,2,3),A 2 (0.5,1.5,2.5),C 2 (2,3,4) are triangular fuzzy sets. If input x0 1 (Singleton) , thenoutput ? 46 Mamdani method 47 Mamdani method 48 Mamdani method 49 Mamdani method 50 Larsen method 51 Larsen method 52 Larsen method 53 Larsen method 54 Fuzzy Logic Controller 55 Inference 56 Inference 57 Inference 58 Inference 59 Defuzzification • Mean of Maximum Method (MOM) 60 Defuzzification • Center of Area Method (COA) 61 Defuzzification • Bisector of Area (BOA) 62