Fuzzy Logic and Fuzzy Modeling - Fuzzy Logic and Fuzzy Modeling Fuzzy Logic: Fuzzy logic deals with

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Transcript of Fuzzy Logic and Fuzzy Modeling - Fuzzy Logic and Fuzzy Modeling Fuzzy Logic: Fuzzy logic deals with

  • Fuzzy Logic and Fuzzy Modeling

    Fuzzy Logic:

    Fuzzy logic deals with fuzzy sets. A fuzzy set or subset is a generalization of an ordinary

    or crisp set. A fuzzy subset can be seen as a predicate whose truth values are drawn from

    the unit interval , I =[0,1] rather than the set {0,1} as in the case of an ordinary set. Thus

    the fuzzy subset has as its underlying logic a multivalued logic. The fuzzy set allows for

    the description of concepts in which the boundary between a property and not having a

    property is not sharp.

    Ex:-

    A set of heights forms a fuzzy set and its subsets include heights that can be categorized

    as tall, medium and short, here the property is height. All the temperatures in a year can

    be clustered into three groups (or fuzzy subsets) that belong to hot, moderate and cold.

    Here the property is temperature. In both these examples, the highest value of the

    property is taken as unity and the rest lie with in the interval [0, 1].

    Membership function:-

    Let X be the universe of discourse (the domain of a property). A subset of A of X is

    associated with a membership function.

    A : x→ [0, 1]

    Where A (x) for each x indicates the degree to which x is a member of the set A. I t is

    also called the degree of association of x in A.

  • Depending on the variation of x in the set A, one can choose a particular shape for the

    membership function. Some of the shapes are described in the following.

    A triangular membership function (MF) is specified by three parameters {a, b, c} as

    follows:

    Triangular (x; a, b, c) =

    0,

    ,

    ,

    0,

    x a

    x a a x b

    b a

    c x b x c

    c b

    c x

      

      

       

       

    

    A trapezoidal MF is specified by four parameters {a, b, c, d} as follows:

  • Trapezoid (a; a, b, c, d) =

    0,

    ,

    ,

    0,

    x a

    x a a x b

    b a

    d x c x d

    d c

    d x

      

      

       

       

    

    A Gaussian MF is specified by two parameters {c, }

    Gaussian (x; c, ) = 1/ 2e 2

    x c

         

  • A generalized bell MF is specified by three parameters {a, b, c}

    Bell (x ; a ,b,c) = 2

    1

    1

    b x c

    a

     

    where the parameter b is usually positive.

  • Fuzzy Set:-

    If X is a collection of objects denoted generally by x, then a fuzzy set A in X is defined as

    a set of ordered pairs:

    A = {[x, A (x) ] | x  X }

    Where A (x) is the MF for the fuzzy set A. Usually x is referred to as the universe of

    discourse or simply the universe.

    Ex :

    Let X=R+ be the set of possible ages for human beings then the fuzzy set B= “about 50

    years old “may be expressed as

  • Support:

    The support of fuzzy set A is the set of all points x in A such that A (x) >0:

    Support (A) = {x │ A (x) >0}

    Core:

    The core of a fuzzy set A is the set of all points x in X such that A (x) =1

    Core (A) = {x │ A (x) =1}

    Crossover point:

    A crossover point of a fuzzy set A is a point x X

    At which A (x) = 0.5

    Crossover (A) = {x │ A (x) =0.5}

    Fuzzy singleton:

    A fuzzy set whose support is a single point in x with A (x) = 1 is called fuzzy singleton.

  • α­cut, strong α­cut:

    The α-cut or α-level set of a fuzzy set A is a crisp set defined by

    Aα = {x │ A (x) ≥α}

    Strong αcut or strong α-level are defined similarly

    Aα’ = {x │ A (x) >α

    Normality:

    A fuzzy set A is normal if its core is nonempty.

    In other words, we can always find a point x X such that A (x) = 1

    Convexity:

    A fuzzy set is convex if and only if for any x1,x2X and any λ [0,1] , A {λx1 + (1-

    λ)x2 } min { A (x1), A (x2) }

    Alternatively, A is convex if all its α –level sets are convex.

    Fuzzy number:

    A fuzzy number A is a fuzzy set in the real line (R) that satisfies the conditions for

    normality and convexity.

    Fuzzy set operations:

    Before introducing the fuzzy set operations, first we shall define the notation of

    containment, which plays a central role in fuzzy sets.

    Containment or subset:

    A fuzzy set A is contained in set B (or, equivalent A is α subset of B) if and only if A

    (x) ≤ B (x) for all. In symbols,

    A B  A (x) ≤ B (x)

    Union (disjunction):

  • The union of two fuzzy sets A and B is a fuzzy set C , written as C=A  B or C=a or B,

    whose MF is related to those of A and B by

    C (x)= max{ A (x), B (x)}= A (x)  B (x) it is the “smallest” fuzzy set

    containing both A and B.

    Intersection (conjunction):

    The interconnection of two fuzzy sets A and B is a fuzzy set C written as C=A  B or C=

    A and B whose MF is related to those of A and B by

    C (x) = min { A (x), B (x)}= A (x)  B (x)

    It is the “largest” fuzzy set, which is contained in both A and B.

    Complement (negation):

    The complement of a fuzzy set A is denoted by A

    1 AA  

    MFs of two Dimensions:

    Some times it is necessary to use MFs with two inputs, each on a different universe of

    discourse. MFs of this kind are generally referred to as two-dimensional MFs, where as

    ordinary MFs are referred to as one-dimensional MFs. One natural way to extend one-

    dimensional MF to two-dimensional ones is via cylindrical extension.

    Cylindrical extension of one-dimensional fuzzy sets:

    If A is a fuzzy set in X, then its cylindrical extension in X x Y is a fuzzy set c(A) defined

    by

    C (A) = A XxY

     /(x,y)

  • Usually, A is referred to as a base set. The operation of projection on the other hand

    decreases the dimension of a given (multidimensional) MF.

    Projection of fuzzy sets:

    Let R be a two dimensional fuzzy set on XxY. Then the projection of R onto X and Y are

    defined as

    xR = [max y R x

     (x,y)]/x

    yR = [max x R y

     (x,y)]/y

    respectively.

    T and S operators:

    The four conditions: commutativity, associativity, monotonicity and respective identities

    have been used to characterize the T and S operators which in turn define the general

    class of intersection and union.

    An operator T: [ 0,1] x [ 0,1]  [ 0,1 ]

    Is called a t-norm operator if

    (1) T (a, b) = T (b, a) Commutativity

    (2) T( a,T( b,c)) = T(T ( a,b ),c) Associativity

    (3) T (a, b) ≥ T (c,d) if a ≥ c and b ≥ d Monotonicity

    (4) T(a,1) = a One identity

    To see that the T reduces to the crisp intersection we note that (4) implies T(0,1) =0 and

    T (1,1) =1.

    Condition (1) implies T(0,1) = T(1,0) = 0. Finally, this fact along with condition (3)

    implies that T (0,0) = 0.

  • We note that the Min and product operators are examples of these t-norm operators. Part

    of the uniqueness of the Min operator as a choice for the implementation of the

    intersection operator is based on the fact that the Min operator is the largest of possible t-

    norms, T(a,b) ≤ Min (a,b).

    t-co norm or s operator:

    An operator s: [0,1] x [0,1]  [ 0,1]

    Is called a t-conorm operator if

    (1) s(a,b)= s(b,a) Commutativity

    (2) s(a, s(b,c)) = s(s(a,b),c) Associativity

    (3) s(a,b) ≥ s(c,d) if a ≥ c and b ≥ d Monotonicity

    (4) s(a,0) = a Zero identity

    It can be seen that the above conditions imply

    s(1,1) = s(1,0) = s(0,1) = 1

    s(0,0) = 0

    We note that the Max and the a + b – ab operators are examples of these t- conorm

    operators. Another example of the t- conorm is the bounded sum Min [1, a+b]. part of the

    uniqueness of the max operator as a choice for the implementation of the union operator

    is based on the fact that the Max operator is the smallest of all the t-co norms, for all s:

    Max(a,b) ≤ s(a,b).

    It should be noted that the only distinction between the T and S operators is in conditions

    (4) and (4’). These conditions can be seen as the defining characteristics of the respective

    operators. Essentially condition (4) implies that the smallest argument is the most

    influential in the formula