Another fuzzy membership function that is often used to represent vague, linguistic terms is the Gaussian which is given by:

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1 Gaussian Membership Functions Another fuzzy membership function that is often used to represent vague, linguistic terms is the Gaussian which is given by: µ A i(x) = exp( (c i x) 2 ), (1) 2σ 2 i where c i and σ i are the centre and width of the ith fuzzy set A i, respectively. 1 C i Gaussian σ i Centres 1

2 local although not strictly compact the output is very smooth not probability Multivariate Gaussian functions can be formed from the product of the univariate sets. Gaussian fuzzy membership functions are quite popular in the fuzzy logic literature, as they are the basis for the connection between fuzzy systems and radial basis function (RBF) neural networks. 2

3 Fuzzy operators Fuzzy Intersection: AND The fuzzy intersection of two sets A and B refers to linguistic statement (x is A) AND (y is B), where x and y could potentially refer to the same variable. For fuzzy intersection, the family of operators is known as the set of triangular or T-norms and the new membership function is generated by: µ A B (x, y) = µ A (x)ˆ µ B (y), (2) where ˆ is the T-operator. For ease of notation, let a, b, c, d [0,1] denote the value of the fuzzy membership functions. 3

4 Definition(T-norm) T-norms is the class of functions obeying the following relationships: aˆ b = bˆ a (aˆ b)ˆ c = aˆ (bˆ c) If a c and b d then aˆ b cˆ d aˆ 1 = a If a T-norm is Archimedean then aˆ a < a, a (0,1). e.g., algebraic product and min functions: µ A B (x, y) = µ A (x) µ B (y), (3) µ A B (x, y) = min{µ A (x), µ B (y)}, (4) for which the former is an Archimedian T- norm. 4

5 algebraic product / min operator the min operator acts as a truncation operator losing information contained in the membership functions {µ A (x), µ B (y)} the product operator allows error information to be back propagated through the network as the first derivative is well defined. generally gives a smoother output surface when univariate B-spline and Gaussian fuzzy membership functions are used. min operator forms an upper bound on the space of fuzzy intersection operators. For any T-norm, we have µ A (x)ˆ µ B (x) min{µ A (x), µ B (x)}, (5) 5

6 In a fuzzy algorithm, the antecedent of a fuzzy production rule is formed from the fuzzy intersection of n-univariate fuzzy sets: (x 1 is A i 1 ) AND... AND (x n is A i n), which produces a new multivariate membership function µ A i 1... A i n (x 1,..., x n ) or µ A i(x), defined on the original n-dimensional input space whose output is given by µ A i(x) = ˆ (µ A i 1 (x 1 ),..., µ A i n (x n )), (6) where ˆ is the multivariate T-norm operator. 6

7 Fuzzy Union: OR The fuzzy union of two sets A and B refers to a linguistic statement of the form: (x is A) OR (y is B), where x and y could potentially refer to the same variable. A new fuzzy membership function is generated by this operation defined on the X Y space, and is denoted by µ A B (x, y). This family of operators is known as S-norms and new membership functions are obtained from: µ A B (x, y) = µ A (x)ˆ+µ B (y), (7) where ˆ+ is the binary S-norm operator. 7

8 Definition 4.9 (S-norm) The set of triangular w-norms or S-norms is the class of functions which satisfy the following: aˆ+b = bˆ+a, (aˆ+b)ˆ+c = aˆ+(bˆ+c), If a c, b d, then aˆ+b cˆ+d, aˆ+0 = a. e.g., algebraic sum and max functions: µ A B (x, y) = µ A (x) + µ B (y), (8) µ A B (x, y) = max{µ A (x), µ B (y)}. (9) 8

9 If the fuzzy membership functions do not form a partition of unity, the sum operator is sometimes replaced by the bounded sum: µ A B (x, y) = µ A (x) + µ B (y) µ A (x) µ B (y), (10) which always has a membership value that lies in the unit interval. The max operator can be shown to be the most pessimistic S-norm as max{µ A (x), µ B (y)} µ A (x)ˆ+µ B (y). (11) Unlike T-norms, when the arguments of an S-norm are unimodal, the resulting membership function is unlikely to retain the unimodal property. µ (y) µ (y) 1 1 B1 B2 B1 B2 y y (a) (b) Comparison of max (a) and sum (b) operators where the shaded area represents the membership function µ B 1 B 2 ( ) 9

10 Fuzzy Implication: IF (.) THEN (.) A general fuzzy algorithm is composed of a set of production rules of the form: r ij : IF (x 1 is A i 1 AND... AND x n is A i n) THEN (y is B j ) c ij, (12) where r ij rule is the ijth fuzzy production rule which relates the ith input fuzzy set A i to the jth output fuzzy set B j. The degree of confidence, with which the input fuzzy set A i (which is composed of fuzzy intersection (AND) of several univariate fuzzy sets) is related to the output fuzzy set B j, is given by a rule confidence c ij [0,1]. When c ij is zero, the rule is inactive and does not contribute to the output. Otherwise, it partially fires whenever its antecedent is activated to a degree greater than zero. 10

11 The rule base is characterized by a set of rule confidence {c ij } (i = 1,2,..., p; j = 1,2,..., q) which can be stored in a rule confidence matrix C = {c ij }. Once fuzzy membership functions have been defined, the rule confidences encapsulate the experts knowledge about a particular process and also form a convenient set of parameters to train the fuzzy system from training data. An example: Some cars have a separate antilock brake system using fuzzy control, with sensors that monitor the rotational speeds of wheels when brakes are applied. When one of these wheels approaches lockup, the controller reduces brake torque to that wheel (or set of wheels) just enough to allow rotation again. Discuss how a fuzzy intersection operator can be used to describe the condition of a wheel lockup, relating 11

12 to its rotational speed and vehicle speed, and how this information can be used to control the brake. Sample answer: Intersection operator AND define two situations happen simultaneously. Wheel lockup happens when the rotational speed approaches zero (small), but the speed of the bus is still large. A fuzzy rule can be formed such as IF (Wheel rotational speed is small) AND (speed of the bus is not small) The fuzzified value of the above rule can be transformed into a control signal as the input to antilock system, e.g. turn on/off the antilock that reduces the brake torque.

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