Soft Computing by QyNzbnF


									Soft Computing
   Fuzzy logic is part of soft
Congress of Computational Intelligence

   Neural Nets   Evolutionary Algorithms


Fuzzy Logic and Functions
                              Decision Trees
                              and other
Algorithms      Fuzzy logic   Learning

               Neural Nets
    The Definition of Fuzzy Logic
       Membership Function
• A person's height
  membership function
  graph is shown next
  with linguistic values
  of the degree of
  membership as very
  tall, tall, average,
  short and very short
  being replaced by
  0.85, 0.65, 0.50, 0.45
  and 0.15.
•   In traditional logic, statements can be
    either true or false, and sets can
    either contain an element or not.
•   These logic values and set
    memberships are typically
    represented with number 1 and 0.
•   Fuzzy logic generalizes traditional
    logic by allowing statements to be
    somewhat true, partially true, etc.
•   Likewise, sets can have full members,     tall
    partial members, and so on.
•   For example, a person whose height is
    5’ 9” might be assigned a                        medium
    membership of 0.6 in the fuzzy set
    “tall people”.
•   The statement “Joe is tall” is 60%
    true of Joe is 5’9”.
•   Fuzzy logic is a set of “if--then”
    statements based on combining fuzzy
    sets. (Beale & Demuth..Fuzzy Systems
  Fuzzy Sets, Statements, and
• A crisp set is simply a collection of objects taken from
  the universe of objects.
• Fuzzy refers to linguistic uncertainty, like the word
• Fuzzy sets allow objects to have membership in more
  than one set:
   – e.g. 6’ 0” has grade 70% in the set “tall” and grade 40% in the
     set “medium”.
• A fuzzy statement describes the grade of a fuzzy
  variable with an expression:
   – e.g. Pick a real number greater than 3 and less than 8.
 The Definition of Fuzzy
      Logic Rules
• A fuzzy logic system uses fuzzy logic rules, as in an
  expert system where there are many if-then rules.
  – A fuzzy logic rule uses membership functions as
• A fuzzy logic rule is defined as an if variable(s) and
  then output fuzzy variable(s).
• Fuzzy logic variables are connected together like
  binary equations with the variables separated with
  operators of AND, OR, and NOT.
•   Review of classical logic and reasoning systems
•   Fuzzy sets
•   Fuzzy logic
•   Fuzzy logic systems applications
•   Fuzzy Logic Minimization and Synthesis
•   Practical Examples
•   Approaches to fuzzy logic decomposition
•   Our approach to decomposition
•   Combining methods and future research
•be introduced to the topics of:
   – fuzzy sets,
   – fuzzy operators,
   – fuzzy logic
   – and come to terms with the technology
•learn how to represent concepts using fuzzy logic
•understand how fuzzy logic is used to make deductions
•familiarise yourself with the `fuzzy' terminology
Traditional Logic
• One of the main aims of logic is to
  provide rules which can be
  employed to determine whether a
  particular argument is correct or not.
• The language of logic is based on
  mathematics and the reasoning
  process is precise and
 Logical arguments
• Any logical argument consists of
• A statement is a sentence which
  unambiguously either holds true or
  holds false.
 –Example:Today is Sunday
• Example: Seven is an even number
  – This example can be written in a mathematical form
    as follows:
     • 7  {x| x is an even number}
  – or in a more concise way:
     • 7  {x|P(x)}
  – where | is read as such that and P(x) stands for `x
    has property P' and it is known as the predicate.
  – Note that a predicate is not a statement until some
    particular x-value is specified.
  – Once a x value is specified then the predicate
    becomes a statement whose truth or falsity can be
    worked out.
  For All Quantifier
• For all x and y, x2-y2 is the same as (x+y)*(x-
  – This example can be written in a mathematical
    form as well:
     •  x,y ((x,yR) (x2-y2)=(x+y)*(x-y))
• where the is interpreted as 'for all',  is the
  logical operator AND, and R represents what
  is termed as the universe of discourse.
 Universe of Discourse
• Using the universe of discourse one
  assures that a statement is evaluated for
  relevant values.
  – The above predicate is then true only for real
• Similarly for the first example the universe
  of discourse is most likely to be the set of
  natural numbers rather than buildings,
  rivers, or anything else.
  – Hence, using the concept of the universe of
    discourse any logical paradoxes can not arise.
Existential Quantifier
• Another type of quantifier is the
  existential quantifier ().
• The existential quantifier is
  interpreted as 'there exists' or 'for
  some' and describes a statement as
  being true for at least one element of
  the set.
• For example, (x)
Connectives and their truth tables
• A number of connectives exist.
  – Their sole purpose is to allow us to join together
    predicates or statements in order to form more
    complicated ones.
• Such connectives are NOT (~), AND (), OR ().
  – Strictly speaking NOT is not a connective since it only
    applies to a single predicate or statement.
• In traditional logic the main tools of reasoning
  are tautologies, such as the modus ponens
  (A(AB))B ( means implies).
        Truth Tables
                         And        Or            Not
A          B           AÙB         AÚB        ~A
True       True        True        True       False
True       False       False       True       False
False      True        False       True       True
False      False       False       False      True

         This everything will hold true, we will just
         do a small modification to the material on
         logic from the last quarter
      Identities of Fuzzy Logic
• The form of identities used in fuzzy variables are the same as elements in
  fuzzy sets.
• The definition of an element in a fuzzy set, {(x,u a(x))}, is the same as a fuzzy
  variable x and this form will be used in the remainder of the paper.
• Fuzzy functions are made up of fuzzy variables.
The identities for fuzzy algebra are:

Idempotency:    X + X = X, X * X = X
Commutativity:  X + Y = Y + X, X * Y = Y * X
Associativity:  (X + Y) + Z = X + (Y + Z),
                (X * Y) * Z = X * (Y* Z)
Absorption:      X + (X * Y) = X, X * (X + Y) = X
Distributivity:  X + (Y * Z) = (X + Y) * (X + Z),
                 X * (Y + Z) = (X * Y) + (X * Z)
Complement:      X’’ = X
DeMorgan's Laws: (X + Y)’ = X’ * Y’, (X * Y)’ = X’ + Y’
Transformations of Fuzzy
    Logic Formulas
 Some transformations of fuzzy sets with examples follow:
 x’b + xb = (x + x’)b  b
 xb + xx’b = xb(1 + x’) = xb
 x’b + xx’b = x’b(1 + x) = x’b
 a + xa = a(1 + x) = a
 a + x’a = a(1 + x’) = a
 a + xx’a = a
 a + xa + x’b + xx’b = a(1 + x) + x’b(1 + x) = a + x’b
 a + xa + x’a + xx’a = a(1 + x + x’ + xx’) = a
       Differences Between Boolean
          Logic and Fuzzy Logic
•     In Boolean logic the value
  of a variable and its inverse are
  always disjoint (X * X’ = 0)
  and (X + X’ = 1) because the
  values are either zero or one.
•     Fuzzy logic membership
  functions can be either disjoint
  or non-disjoint.
•     Example of a fuzzy non-
  linear and linear membership
  function X is shown (a) with
  its inverse membership
  function shown in (b).
 Fuzzy Intersection and Union
• From the membership
  functions shown in the top in
  (a), and complement X’ (b) the
  intersection of fuzzy variable
  X and its complement X’ is
  shown bottom in (a).

• From the membership
  functions shown in the top in
  (a), and complement X’ (b) the
  union of fuzzy variable X and
  its complement X’ is shown
  bottom in (b).

                                   Fuzzy          Fuzzy union
Validation of Fuzzy Functions

       • Two fuzzy functions are valid iff the function outputs are  0.5 under all
         possible assignments.
       • This is like doing EXOR of two binary functions shown in (b) which is the
         same as union.
       • Two fuzzy functions are inconsistent iff the function output is  0.5 under all
         possible assignments. Thus, if the output of the two fuzzy functions is < 0.5
         then the two fuzzy functions are inconsistent.
       • This is like exnor of two binary functions of shown in (a) which is the same
         as intersection.
      Fuzzy Logic
• The concept of fuzzy logic was introduced
  by L.A Zadeh in a 1965 paper.
• Aristotelian concepts have been useful and
  applicable for many years.
• But these traditional approaches present us
  with certain problems:
  – Cannot express ambiguity
  – Lack of quantifiers
  – Cannot handle exceptions
Traditional Logic Problems
– Cannot express ambiguity:
  • Consider the predicate `X is tall'.
  • Providing a specific person we can turn the predicate into
    a statement.
  • But what is the exact meaning of the word `tall'?
  • What is `tall' to some people is not tall to others.
– Lack of quantifiers:
  • Another problem is the lack of being able to express
    statements such as `Most of the goals came in the first
    half '.
  • The `most' quantifier cannot be expressed in terms of the
    universal and/or existential quantifiers.
Traditional Logic Problems
– Cannot handle exceptions:
  • Another limitation of traditional predicate logic is
    expressing things that are sometimes, but not
    always true.
  Traditional sets
• In order to represent a set we use curly brackets {}.
• Within the curly brackets we enclose the names of the
  items, separating them from each other by commas.
• The items within the curly brackets are referred to as
  the elements of the set.
   – Example: Set of vowels in the English alphabet = {a,e,i,o,u}
• When dealing with numerical elements we may replace
  any number of elements using 3 dots.
   – Example: Set of numbers from 1 to 100 = {1,2,3,...,100}
   – Set of numbers from 23 to infinity = {23,24,25,...}
        Traditional sets
• Rather than writing the description of a set
  all the time we can give names to the set.
• The general convention is to give sets names
  in capital letters.
  – Example:
    • V = set of vowels in the English alphabet.
    • Hence any time we encounter V implies the set {a, e, i, o, u}.

  – For finite size sets a diagrammatic representation
    can be employed which can be used to assist in
    their understanding.
    • These are called the cloud diagrams
  Cloud Diagrams
                             A cloud diagram


           Set order
• The order in which the elements are
  written down is not important.
  – Example: V = {a,e,i,o,u} = {u,o,i,e,a} = {a,o,e,u,i}
• The names of the elements in a set must
  be unique.
  – Example:
     • V = {a,a,e,i,o,u}
     • If two elements are the same then there is no point
       writing them down twice (waste of effort)
     • but if different then we must introduce a way to tell
       them apart.
 Set membership
• Given any set, we can test if a certain
  thing is an element of the set or not.
• The Greek symbol, , indicates an
  element is a member of a set.
• For example, xA means that x is an
  element of the set A.
• If an element is not a member of a set, the
  symbol  is used, as in A.
  Set equality & subsets
• Two sets A and B are equal, (A= B) if every
  element of A is an element of B and every
  element of B is an element of A.

• A set A is a subset of set B, (A  B) if every
  element of A is an element of B.

• A set A is a proper subset of set B, (A  B) if
  A is a subset of B and the two sets are not
 Set equality & subsets
• Two sets A and B are disjoint, (A  b) if and only
  if their intersection is the empty set.
• There are a number of special sets. For instance:
  –   Boolean B={True, False}
  –   Natural numbers N={0,1,2,3,...}
  –   Integer numbers Z={...,-3,-2,-1,0,1,2,3,...}
  –   Real numbers R
  –   Characters Char
  –   Empty set  or {}
  –   The empty set is not to be confused with {0} which is a
      set which contains the number zero as its only
      Set operations
• We have a number of possible operators acting
  on sets.
• The intersection ( ), the union ( ), the difference
  (/), the complement (').
  – Intersection results in a set with the common elements
    of two sets.
  – Union results in a set which contains the elements of
    both sets.
  – The difference results in a set which contains all the
    elements of the first set which do not appear in the
    second set.
  – The complement of a set is the set of all element not in
    that set.
  Set operations example
• Using as an example the two following sets A and
  B the mathematical representation of the
  operations will be given.
  – A = {cat, dog, ferret, monkey, stoat}
  – B = {dog, elephant, weasel, monkey}
• C = A  B = {x e u | (x e A) (x e B)}={dog, monkey}
• C = A  B = {x e u | (x e A) (x e B)}={cat, ferret,
  stoat, dog, elephant, weasel, monkey}
• C = A / B = {x e u | (x e A)  ~(x e B)}={cat, ferret,
• C = A‘ = {x e u | ~(x e A)}
Set operations example using
       Venn diagrams
Intersection        Union

 A        B           A           B

 Difference         om ent
                   C plem

 A    B
                     A       A'
Computing and
 Fuzzy Theory
What is fuzziness
• The concept of fuzzy logic was introduced in a 1965 paper by
  Lotfi Zadeh.
• Professor Zadeh was motivated by his realization of the fact
  that people base their decisions on imprecise, non-numerical
• Fuzzification should not be regarded as a single theory but as
  a methodology
   – It generalizes any specific theory from a discrete to a continuous
• For instance:
   –    from Boolean logic to fuzzy logic,
   –   from calculus to fuzzy calculus,
   –    from differential equations to fuzzy differential equations,
   –   and so on.
What is fuzziness
• Fuzzy logic is then a superset of
  conventional Boolean logic.
• In Boolean logic propositions take a
  value of either completely true or
  completely false
• Fuzzy logic handles the concept of
  partial truth, i.e., values between the
  two extremes.
  What is fuzziness
• For example, if pressure takes values
  between 0 and 50 (the universe of
  discourse) one might label the range 20 to
  30 as medium pressure (the subset).
• Medium is known as a linguistic variable.
• Therefore, with Boolean logic 15.0 (or
  even 19.99) is not a member of the
  medium pressure range.
• As soon as the pressure equals 20, then
  it becomes a member.
Boolean Medium Pressure
  Membership Grade
     following Figure shows the membership function
using Boolean logic.
   0.4   Not-Member   Member    Not-Member
    0                                             Pressure
               10     20       30     40     50
      What is fuzziness
• Contrast with the Figure of the next page
  which shows the membership function using
  fuzzy logic.
• Here, a value of 15 is a member of the medium
  pressure range with a membership grade of
  about 0.3.
• Measurements of 20, 25, 30, 40 have grade of
  memberships of 0.5, 1.0, 0.8, and 0.0
• Therefore, a membership grade progresses
  from non-membership to full membership and
  again to non-membership.
           Fuzzy Medium Pressure
Membership Grade
          Not-Member   Member        Not-Member
     0                                                 Pressure
                10     20       30     40         50
               Fuzzy Sets
• Fuzzy logic is based upon the notion of fuzzy sets.
  – Recall from the previous section that an item is an
    element of a set or not.
  – With traditional sets the boundaries are clear cut.
  – With fuzzy sets partial membership is allowed.
  – Fuzzy logic involves 3 primary processes :
     • Fuzzification
     • Rule evaluation
     • Defuzzification
  – With fuzzy logic the generalised modus ponens is
    employed which allows A and B to be characterised by
    fuzzy sets.
Fuzzy Set Theory
           Fuzzy Sets
•   Definition
•   Operations
•   Identities
•   Transformations
• Traditional sets, influenced from the
  Aristotelian view of two-valued logic, have only
  two possible truth values, namely TRUE or
  FALSE, 1 or 0, yes or no etc.
• Something either belongs to a particular set or
  does not.
• The characteristic function or alternatively
  referred to as the discrimination function is
  defined below in terms of a functional
• In fuzzy sets, something may belong
  partially to a set.
• Therefore it might be very true or
  somewhat true, 0.2 or 0.9 in numerical
• The membership function using fuzzy
  sets defined in terms of a functional
  mapping is as shown below:
• Fuzzy logic allows you to violate the laws of
  noncontradiction since an element can be a
  member of more than one set.
• More set operations are available
• The excluded middle is not applicable, i.e.,
  the intersection of a set with its complement
  does not necessarily result to an empty set.
• Rule based systems using fuzzy logic in
  some cases might increase the amount of
  computation required in comparison with
  systems using classical binary logic.

• If fuzzy membership grades are restricted to
  {0,1} then Boolean sets are recovered.
• For instance, consider the Set Union
  operator which states that the truth value of
  two arguments x and y is their maximum:
  – truth(x or y) = max(truth(x), truth(y)).

• If truth grades are either 0 or 1 then
  following table is found:
  – x y truth
     • which is the same truth table as in the Boolean logic.
• So, every crisp set is fuzzy, but not
Definition of Fuzzy Set
• A fuzzy set, defined as A, is a subset of a universe of discourse
  U, where A is characterized by a membership function uA(x).

• The membership function uA(x) is associated with each point in
  U and is the “grade of membership” in A.

• The membership function uA(x) is assumed to range in the
  interval [0,1], with value of 0 corresponding to the non-
  membership, and 1 corresponding to the full membership.

• The ordered pairs form the set {(u,uA(x))} to represent the
  fuzzy set member and the grade of membership.
         Operations on Fuzzy Sets
• The fuzzy set operations are defined as follows.
    – Intersection operation of two fuzzy sets uses the symbols: , *, , AND, or min.
    – Union operation of two fuzzy sets uses the symbols: , , +, OR, or max.

• Equality of two sets is defined as A = B  u a(x) = u b(x) for all x  X.

• Containment of two sets is defined as A subset B,
     A  B  u a(x)  u b(x) for all x  X.

• Complement of a set A is defined as A’, where u a’(x) = 1 – u a(x) for all x  X.

• Intersection of two sets is defined as A  B
  where u a  b (x) = min{(u a(x),u b(x))} for all x  X.
  Where C  A, C  B then C  A  B.

• Union of two sets is defined as A  B where u a  b(x) = max{(u a(x), u b (x))}
  for all x  X where D  A, D  B then D  A  B.
Fuzzy sets, logic, inference, control
– This is the appropriate place to clarify not what
  the terms mean but their relationship.
– This is necessary because different authors and
  researchers use the same term either for the
  same thing or for different things.
– The following have become widely accepted:
  • Fuzzy logic system
     – anything that uses fuzzy set theory
  • Fuzzy control
     – any control system that employs fuzzy logic
  • Fuzzy associative memory
     – any system that evaluates a set of fuzzy if-then rules uses fuzzy
       inference. Also known as fuzzy rule base or fuzzy expert
  • Fuzzy inference control
     – a system that uses fuzzy control and fuzzy inference
           Fuzzy sets
• A traditional set can be considered as a special
  case of fuzzy sets.
• A fuzzy set has 3 principal properties:

  – the range of values over which the set is mapped

  – the degree of membership axis that measures a domain
    value's membership in the set

  – the surface of the fuzzy set - the points that connect the
    degree of membership with the underlying domain
                Fuzzy sets
• Therefore, a fuzzy set in a universe of discourse U is
  characterised by the membership function µx, which takes
  values in the interval [0,1] namely µx:U[0,1].
  – A fuzzy set X in U may be represented as a set of ordered pairs of a
    generic element u and its grade of membership µx as
     X={u,µX(u)/u _ U},
     • i.e., the fuzzy variables u take on fuzzy values µx(u).
  – When a fuzzy set, say X, is discrete and finite it may expressed as
     • where `+' is not the summation symbol but the union operator, the `/’
       does not denote division but a particular membership function to a
       value on the universe of discourse.
Fuzzy set



            Fuzzy sets
• As an example consider:
  – the universe of discourse U={0,1,2,...,9}
  – and a fuzzy set X1, `young generation decade'.
• A possible presentation now follows:
  – X1={1.0/0+1.0/1+0.85/2+0.7/3+0.5/4+0.3/5+0.15/6+
  – The set is also shown in a graphical form




      Fuzzy set
• Another set, X2 might be `mid-age
  generation decade. In discrete form this can
  be depicted as
  +0.0/7+0.0/8+0.0/9. 0.9
                 Grade 0.5
                             0   1   2   3   4   5    6   7   8   9
– Support of a fuzzy set:                 Support, Crossover,
   • The support of a fuzzy set is the
     set of all elements of the               Singleton
     universe of discourse that their
     grade of membership is greater
     than zero.
   • For X2 the support is {2,3,4,5,6}.
   • Additionally, a fuzzy set has
     compact support if its support
     is finite.                                 0.9
– Crossover point:                              0.7
   • The element of a fuzzy set that      Grade 0.5
     has a grade of membership                  0.3
     equal to 0.5 is known as the               0.2
     crossover point.                             0
                                                      0   1   2   3   4   5    6   7   8   9
   • For X2 the crossover point is 2.                                 Decade
– Fuzzy singleton:
                                             Support, Crossover,
    • The fuzzy set whose support is a
      single point in the universe of
      discourse with grade of
      membership equal to one is
      known as the fuzzy singleton.
–   -Level sets:
    • The fuzzy set that contains the
      elements which have a grade of             1
      membership greater than the -            0.8
      level set is known as the -Level         0.6
                                         Grade 0.5
      set.                                     0.4
    • For X2 the -Level set when               0.2
       =0.6 is {3,4,5}.                          0
                                                     0   1   2   3   4   5    6   7   8   9

    • Whereas for X2 the -Level set                                  Decade

      when =0.4 is {2,3,4,5}.
   Popular Membership Functions
• Membership Functions are used in order to return
  the degree of membership of a numerical value for
  a particular set.
  – Fuzzy membership functions can have different shapes,
    depending on someone's experience or even
  – Here we review some of the membership functions used
    in order to capture the modeler's sense of fuzzy
  – Membership functions can be drawn using:
     • Subjective evaluation and elicitation
         – (Experts specify at the end of an elicitation phase the appropriate
           membership functions) or
     • Ad-hoc forms
         – (One can draw from a set of given different curves.
Popular Membership Functions
• This simplifies the problem, for example to
  – 1. choosing just the central value and the slope on
    either side)
  – 2. Converted frequencies (Information from a frequency
    histogram can be used as the basis to construct a
    membership function
  – 3. Learning and adaptation.
• For example, let us consider the fuzzy
  membership function of the linguistic variable
• The following function can be one presentation:
  Popular Membership Functions
                    0 if height(x) < 5 feet
          height(x) - 5
Tall(x) =               if 5 feet  height(x)  7 feet
                   1 if height(x) > 7 feet

• Given the above definition the membership grade for an
  expression like `Dimitris is Tall' can be evaluated.
  Assuming a height of 6' 11'' the membership grade is 0.54
• Other popular shapes used are triangles and trapezoidals.
The S-Function
                              0 for x  
                                     2
                       x  
                      2           for  x  
                       
 S ( x; ,  ,  )              
                                     2
                     1  2 x    for   x  
                               
                                   
                               1 for x
     The S-Function
   0 .9

   0 .8

   0 .7

   0 .6

   0 .5

   0 .4

   0 .3

   0 .2

   0 .1

     0                 x
           a   b   g
    The S-Function
• As one can see the S-function is flat at a value of 0
  for xa and at 1 for xg. In between a and g the S-
  function is a quadratic function of x.
  – To illustrate the S-function we shall use the fuzzy
    proposition Dimitris is tall.
  – We assume that:
     • Dimitris is an adult
     • The universe of discource are normal people
        – (i.e., excluding the extremes of basketball players etc.)
  – then we may assume that anyone less than 5 feet is not
    tall (i.e., a=5) and anyone more than 7 feet is tall (i.e.,
     • Hence, b=6.
     • Anyone between 5 and 7 feet has a membership function which
       increases monotonically with his height.
                     0 for x  5
                 x 5 

                 2  for5 x  6
S ( x;5,6,7)           
                 x  7 2
               1   2  for6 x  7
                        
                      1 for x7
           Hence the membership of 6 feet tall people is 0.5,
            whereas for 6.5 feet tall people increases to 0.9.
   0 .9

   0 .8

   0 .7

   0 .6

   0 .5

   0 .4

   0 .3

   0 .2

   0 .1


                 5             6            7          Heigh

                                    
                  S ( x;   ,   ,  ) for x  
( x;  ,  )                      2
                1  S ( x; ,   ,    ) for x  
                                  2





0.5                         b




      0                                         x
            g-b   g-(b/2)   g   g+(b/2)   g+b
• The P-function goes to zero at  < , and the
  0.5 point is at  = (/2).
• Notice that the  parameter represents the
  bandwidth of the 0.5 points.





0.5                       6




      0                                            x
            1    4         7         10       13

                              S ( x;1,4,7) for x  7
                 ( x;6,7)  
                             1  S ( x;7,10,13) for x  7
Many argument
• An example of fuzzy operations: X = { 1, 2, 3, 4, 5} and
  fuzzy sets A and B.
• A = {(3,0.8), (5,1), (2,0.6)} and
  B = {(3,0.7), (4,1), (2,0.5)} then
• A  B = {(3, 0.7), (2, 0.5)}
• A  B = {(3, 0.8), (4, 1), (5, 1), (2, 0.6)}
• A’ = {(1, 1), (2, 0.4), (3, 0.2), (4, 1), (5,0)}
Fuzzy operators
• What follows is a summary of some fuzzy set
  operators in a domain X.
• For illustration purposes we shall use the following
  membership sets:
   – A= 0.8/2 + 0.6/3 + 0.2/4, and B = 0.8/3 + 0.2/5
   – as well as X1 and X2 from above.
• Set equality:
   – A=B if A(x)=B(x) for all xX
• Set complement:
   – A' A' (x)=1-A (x) for all xX.
   – This corresponds to the logic `NOT' function.
   – A' (x) = 0.2/2 + 0.4/3 + 0.8/4
      Fuzzy operators
• Subset: AB if and only if A(x)B(x) for all xX

• Proper Subset:
  – AB if A(x)B(x) and A(x) B(x) for at least one xX

• Set Union:
  – AB AB(x)=(A(x),B(x)) for all xX where  is the
    join operator and means the maximum of the
     • This corresponds to the logic `OR' function.
  – AB(x) = 0.8/2 + 0.8/3 + 0.2/4 + 0.2/5
Fuzzy Union Diagram
         1                              X2
       0.8                              Union
 Grade 0.6
             0 1   2 3    4 5     6 7   8 9

  – AB(x) = 0.8/2 + 0.8/3 + 0.2/4 + 0.2/5
         Fuzzy operators
• Set Intersection:
  – AB  AB(x)=( A(x),  B(x)) for all xX where  is
    the meet operator and means the minimum of the
     • This corresponds to the logic `AND' function.
  –  AB(x) = 0.6/3
• Set product:
  – AB AB(x)=A(x)B(x)

• Power of a set:
  – AN AN (x)=(A(x))N
Fuzzy Intersection diagram

          0.8                              Inter.
    Grade 0.6
                0 1   2 3    4 5     6 7   8 9
             Fuzzy operators
• Bounded sum or bold union: AB
  – AB(x)=(1,(A(x)+B(x))) where is minimum and + is
    the arithmetic add operator.
• Bounded product or bold intersection: AB
  – AB(x)=(0,(A(x)+B(x)-1)) where  is maximum and + is
    the arithmetic add operator.
• Bounded difference: A - B
  – A- B(x)=(0,(A(x)-B(x)))
     • where  is maximum and - is the arithmetic minus operator.
  – This operation represents those elements that are more
    in A than B.
Single argument
 Concentration set operator
• CON(A) CON(A)=(A(x))2
  – This operation reduces the membership grade
    of elements that have small membership
• If TALL=-.125/5+0.5/6+0.875/6.5+1/7+1/7.5+1/8
  since VERY TALL=TALL2.
Concentration set operator
    0.8           Original
          0   2         4    6        8      10   12   14
Dilation set operator
• DIL(A) DIL(A)=(A(x))0.5
   – This operation increases the membership grade of
     elements that have small membership grades.
   – It is the inverse of the concentration operation.
• If TALL=-.125/5+0.5/6+0.875/6.5+1/7+1/7.5+1/8
  0.354/5+0.707/6+0.935/6.5+1/7+1/7.5+1/8 since
Dilation set operator
        0     2        4      6       8   10   12   14
   Intensification set operator
• This operation raises the membership grade of
  those elements within the 0.5 points and
• This operation reduces the membership grade of
  those elements outside the crossover (0.5) point.
• Hence, intensification amplifies the signal within
  the bandwidth while reducing the `noise'.
  – If TALL = -.125/5+0.5/6+0.875/6.5+1/7+1/7.5+1/8 then
  – INT(TALL) = 0.031/5+0.5/6+0.969/6.5+1/7+1/7.5+1/8.

                      2(  A ( x)) 2 for 0   A ( x)  0.5
    INT ( A) ( x)  
                     1  2(1   A ( x)) for 0.5   A ( x)  1
Intensification set operator
   0.9       intensification
   0.5                             Original
         0         2           4     6        8   10   12   14
 Normalization set operator
• NORM(A)(x)=A(x)/max{A(x)} where the
  max function returns the maximum
  membership grade for all elements of x.
  – If the maximum grade is <1, then all
    membership grades will be increased.
  – If the maximum is 1, then the membership
    grades remain unchanged.
• NORM(TALL) = TALL since the
  maximum is 1
   Hedges –
language related
• The above diagram shows the relationship between
  linguistic variables, term sets and fuzzy representations.
   – Cold, cool, warm and hot are the linguistic values of the linguistic
     variable temperature.
   – In general a value of a linguistic variable is a composite term u = u1,
     u2,...,un where each un is an atomic term.

  linguistic            Temperature

    Cold            Cool          Warm       Hot

       term set

                 fuzzy set
• From one atomic term by employing
  hedges we can create more terms.
   • Hedges     such as very, most, rather,
     slightly, more or less etc.
   • Therefore, the purpose of the hedge is to
     create a larger set of values for a
     linguistic variable from a small collection
     of primary atomic terms.
– This is achieved using the processes of:
  • normalisation,
  • intensifier,
  • concentration, and
  • dilation.
– For example, using concentration very u
   is defined by :
  very u = u2 and
  very very u = u4.
– Let us assume the following definition for linguistic
  variable slow (first is membership function, second
– u= 1.0/0 + 0.7/20 + 0.3/40 + 0.0/60 + 0.0/80 + 0.0/100.
– Then,
   • Very slow = u2 = 1.0/0 + 0.49/20 + 0.09/40 + 0.0/60 +
     0.0/80 + 0.0/100
   • Very Very slow = u4 = 1.0/0 + 0.24/0 + 0.008/40 +
     0.0/60 + 0.0/80 + 0.0/100
   • More or less slow = u0.5 = 1.0/0 + 0.837/20 + 0.548/40
     + 0.0/60 + 0.0/80 + 0.0/100


                                 more or less

        0.6                              slow


        0.3                                 very slow
        0.2                                                  Speed


             0       very very
                 0     slow20    40         60          80     100
        Hedge rather
– The hedge rather is a linguistic modifier that moves each
  membership by an appropriate amount C.
   • Setting C to unity we get.
   • Rather slow = 0.7/0 + 0.3/20 + 0.0/40 + 0.0/60 + 0.0/80
– The slow but not very slow is a modification which is using the
  connective but, which in turn is an intersection operator.
– The membership function in its discrete form was found as follows:
   •   slow = 1.0/0 + 0.7/20 + 0.3/40 + 0.0/60 + 0.0/80 + 0.0/100
   •   very slow = 1.0/0 + 0.49/20 + 0.09/40 + 0.0/60 + 0.0/80 + 0.0/100
   •   not very slow = 0.0/0 + 0.51/20 + 0.91/40 + 1.0/60 + 1.0/80 + 1.0/100
   •   slow but not very slow = min(slow, not very slow) = 0.0/0 + 0.51/20 +
       0.3/40 + 0.0/60 + 0.0/80 + 0.0/100
Hedges slow but not very slow and rather




                                   slow but not very slow




0.1           rather slow

          0           20    40                60            80   100
       Hedge slightly
– The slightly hedge is the fuzzy set operator for
  intersection acting on the fuzzy sets Plus slow
  and Not (Very slow).
– Slightly slow = INT(NORM(PLUS slow and NOT
  VERY slow) where Plus slow is slow to the power
  of 1.25, and is the intersection operator.
   • slow = 1.0/0 + 0.7/20 + 0.3/40 + 0.0/60 + 0.0/80 + 0.0/100
   • plus slow = 1.0/0 + 0.64/20 + 0.222/40 + 0.0/60 + 0.0/80 +
   • not very slow = 0.0/0 + 0.51/20 + 0.91/40 + 1.0/60 + 1.0/80
     + 1.0/100
   • plus slow and not very slow = min(plus slow, not very
     slow) = 0.0/0 + 0.51/20 + 0.222/40 + 0.0/60 + 0.0/80 +
  • norm (plus slow and not very slow) = (plus
    slow and not very slow/max) = 0.0/0 + 1.0/20 +
    0.435/40 + 0.0/60 + 0.0/80 + 0.0/100
– slightly slow = int (norm) = 0.0/0 + 1.0/20 +
  0.87/40 + 0.0/60 + 0.0/80 +0.0/100.
                                    not very
                                     slightly slow
              plus slow
          0   20          40   60              80    100
• Now we are in a better position to understand the
  meaning of the syntactic and semantic rule.
  – A syntactic rule defines, in a recursive fashion, more
    term sets by using a hedge.
  – For instance T(slow)={slow, very slow, very very
  – The semantic rule defines the meaning of terms such as
    very slow which can be defined as very slow = (slow)2.
  – One is obviously allowed either to generate new hedges
    or to modify the meaning of existing ones
    Linguistic variables
• Looking at the production rules of either a expert system
  or a fuzzy expert system one can not see any differences
   – except that the fuzzy system is employing linguistic descriptors
     rather than absolute numerical values.
   – However, both parts of fuzzy rules have associated `levels of belief'
     something lacking in traditional production rules.
• Secondly, with traditional production rules even when
  more than one rule applies only one executes.
   – With fuzzy rules all applicable rules contribute in calculating the
     resulting output.
       • All in all, fuzzy expert systems require fewer production rules
         since fuzzy rules embody more information.
   Linguistic variables
• A major reason behind using fuzzy logic is
  the use of linguistic expressions.
• A linguistic variable consists of:
  – the name of the variable (u),
  – the term set of the variable (T(u)),
  – its universe of discourse (U) in which the fuzzy
    sets are defined,
  – a syntactic rule for generating the names of
    values of u, and
  – a semantic rule for associating with each value
    its meaning.
   Linguistic variables

• For example:
• if u is temperature,
• then its term set T(temperature) could be:
  – T(temperature)={cold, cool, warm, hot} over a
    universe of discourse U=[0,300].
Linguistic variables
 linguistic            Temperature

   Cold            Cool          Warm   Hot

      term set

                fuzzy set
Fuzzy Set Definitions

         We can have either continuous or
         discrete definition of a fuzzy set
  Linguistic Variable
• …. Terms, Degree of Membership, Membership
  Function, Base Variable…..
Recap AI and Expert
 Fuzzy logic in this
            Overview of AI
– The realization by the Artificial Intelligence
  community during the 1960's of the weakness
  of general purpose problem solvers led to the
  development of expert systems.
– Expert systems held the greatest promise for
  capturing intelligence and have received more
  attention than any other sub-discipline of
  Artificial Intelligence.
  • The term knowledge-based systems is used
    interchangeably to avoid the mis-understandings
    and mis-interpretations of the word 'expert'.
       Expert Systems
– Irrespective of the adjective, each such system is
  designed to operate in one of a variety of narrow areas.
– The design involves attempts to model and codify the
  knowledge of human experts.
– One might wonder what makes expert systems different
  from conventional ones. One might remark that in some
  sense, any computer program is expert at something.
   • A payroll program incorporates knowledge about accountancy,
     but it is not included in the expert class.
      – The differences originate from the type of programming language
      – Additionally, expert systems can reason using incomplete data and
        can generate explanations and justifications, even during
        execution of their actions.
Components of ES
– Knowledge-base module:
  • this is the essential component of any
  • It contains a representation in a variety of
    forms of knowledge elicited from a human
– Inference engine module:
  • the inference engine utilizes the contents
    of the knowledge base in conjunction with
    the data given by the user in order to
    achieve a conclusion.
Components of ES
– Working memory module:
  • this is where the user's responses and the system's
    conclusions for each session are temporarily stored.
– Explanation module:
  • this is an important aspect of an expert system.
  • Answers from a computer are rarely accepted
  • This is particularly true for responses from an expert
  • Any system must be able to explain how it reached its
    conclusions and why it has not reached a particular
 Components of ES
– Justification module:
  • using this module the system provides the user
    with justification(s) of why some piece of
    information is required.
– User interface module:
  • the user of an expert system asks questions,
    enters data, examines the reasoning etc.
  • The input-output interface, using menus or
    restricted language, enables the user to
    communicate with the system in a simple and
    uncomplicated way.
  Methods of inference
• Much of the power of an expert system comes from the
  knowledge embedded in it.
• In addition, the way the system infers conclusions is of
  equal importance.
   – Most expert systems apply forward and/or backward chaining.
      • The mode of chaining describes the way in which the production
        rules are activated.
      • With forward chaining the user of the expert system asks what
        conclusions can be made when this data is true.
      • The expert system might or might not ask for further data.
      • With backward chaining the user of the expert system asks what
        conclusions can be made.
      • The expert system will ask the user for data.
 Methods of inference
– To illustrate the two modes consider the
  following situation.
– As you are driving you notice that behind you is
  a police car with its lights and siren on.
– So the data is `light is on' and `siren is on'.
  • The expert system will come to a conclusion such as
    `stop the car' and `someone else to stop the car'.
  • Obviously, the system can not make a hard decision
    and asks for more data.
      – You suddenly realise that the policeman in the car
        is waving at you.
      – This third piece of data `policeman is waving at
        me' suggests to the system that they want you to
        stop the car rather than someone else.
Methods of inference
– The previous scenario describes the forward chaining of
  your expert system which in this case happens to be
  your brain.
– Now, the system starts applying backward chaining.
   • There are numerous conclusions of why the police want you to
   • For instance, 50 miles in a 30-mile zone, not-working brake light,
     stolen plate number, turning to a one-way street etc.
   • Therefore, your system starts collecting data to support any of
     the hypothesised reasons.
   • Since you just passed your MOT, know that this is your car, it is
     not a one-way street the system deduces that you were
     Control Strategies
• This refers to how the expert system comes
  to a conclusion, i.e., the mode of reasoning
  describes the way in which the system as a
  whole is organised.
  – For instance, the order of looking at the rules;
  – how to use meta-rules in order to check for
    outstanding queries, of a completed goal and the
    initiation of the evaluation of rules.
  – The order of looking at the rules usually is in
    lexical order viz. when scanning rules it will first
    look at rule 1, and then rule 2 etc.
     • When it searches, it inspects each rule to see if the left
       hand conditions are true.
  Control Strategies
• This is achieved by either reading the
  working memory or by asking questions or
  by generating further subgoals.
• In most cases the system continues to the
  next rule until all rules have inspected.
  – All rules that can execute are placed in a conflict
    set and one of the rules is selected.
  – The selected rule then executes. This is what is
    known as the match, select and execute cycle.
 Fuzzy Logic
Principles and
 Fuzzy Logic Principles
• Fuzzy control produces actions using a set
  of fuzzy rules based on fuzzy logic
       This involves:
          fuzzifying: mapping sensor readings into a set
           of fuzzy inputs
          fuzzy rule base: a set of IF-THEN rules
          fuzzy inference: maps fuzzy sets onto other
           fuzzy sets using membership fncts.
          defuzzifying: mapping a set of fuzzy outputs
           onto a set of crisp output commands
  Fuzzy Control
• Fuzzy logic allows for specifying
  behaviors as fuzzy rules
       Such behaviors can be smoothly blended
        together (e.g., Flakey robot)
       Fuzzy rules can be learned
Industrial Application of
  Fuzzy Logic Control
History, State of the Art, and Future
Types of Uncertainty and the Modeling
           of Uncertainty
 • Stochastic Uncertainty:
   – The Probability of Hitting the Target is 0.8

Lexical Uncertainty:
     Methods of inference under
• This is very important to consider when using
  expert systems since sometimes data is uncertain
  (i.e., ambiguous, incomplete, noisy etc.).
• A number of theories have been devised to deal
  with uncertainty.
  – These include classical probability, Bayesian
    probability, Shannon theory, Dempster-Shafer theory
    among others.
  – A popular method of dealing with uncertainty uses
    certainty factors
     Methods of inference under
• The certainty factor indicates the net belief in the
  conclusion and premises of a rule based on some
• Certainty factors are hand-crafted by asking
  potential users questions such as `How much do
  you believe that opening valve x will start a
  flooding' and `How much do you disbelieve that
  opening valve x will start a flooding'.
• The degree of certainty is the difference between
  the two responses.
         Production Rules
• Assuming that the knowledge-base
  module contains knowledge represented
  in the the format of production rules the
  following    sections    introduce    the
    • the concept of a production rule
    • the concept of linguistic variables
    • the fuzzy inference concept
    • the concept of fuzzification and how to
      accomplish the crisp to fuzzy transformation
    • the concept of defuzzification and how to
      accomplish the fuzzy to crisp transformation
 Knowledge presentation using
      production rules
• From a philosophical point the concept of knowledge is highly
  ambiguous and debatable
   – knowledge-base builders treat knowledge from a narrower point of
• This way the knowledge is easier to model and understand.
• It remains diverse including:
   –    rules,
   –   facts,
   –   truths,
   –   reasons,
   –   defaults and
   –    heuristics.
• The knowledge engineer needs some technique for capturing
  what is known about the application.
Knowledge presentation using
     production rules
• The technique should provide expressive adequacy
  and notational efficacy.
• Knowledge representation is very much under
  constant research.
• Several schemes have been suggested in the
  literature, namely:
  – semantic nets,
  – frames and
  – logic.
• Production rules have also been suggested and are
  the most popular way of representing knowledge.
 Knowledge presentation using
      production rules
• Production rules are small chunks of knowledge
  expressed in the form of if..then statements.
  – The left hand side (IF) represents the antecedent or conditional
  – The right hand side (THEN) represents the conclusion or
    action part.
  – A number of rules collectively define a modularized know-how
  – The principal use of production rules is in the encoding of
    empirical associations between incoming patterns of data and
    actions that the system should perform as a consequence.
  – The production rules are either expressed by an expert of the
    field, or derived using induction.
Fuzzy Logic Control
• Fuzzy controller design consist of turning
  intuitions, and any other information about how
  to control a system, into set of rules.
• These rules can then be applied to the system.
• If the rules adequately control the system, the
  design work is done.
• If the rules are inadequate, the way they fail
  provides information to change the rules.
 Control a Plant
                                                                 A valve in an
    pressure            steam turbine                               engine that
temperature              system                         throttle    regulates the
                                                                    amount of
                                                                    fuel entering
                                                                    the cylindres
                        Rules presentation of ...
                  An Expert system       A fuzzy system
               If temperature > 680      if temperature is hot
                and pressure < 25        and pressure is low
               then throttle is 165     then throttle is positive
         Using Fuzzy Logic for
       Autonomous Vehicle Motion
• Findings of Stanford Research
  Institute (SRI)
• Based on the performance of the robot
  “Flakey” circa 1993
• Discussion of autonomous navigation
  and path planning in an uncertain
• Paper: “Using Fuzzy Logic for
  Autonomous Vehicle Motion
Difficulties of this problem

• Autonomous operation of a mobile robot in a real-
  world unstructured environment poses a series of
  – knowledge about the environment is usually:
     • incomplete
     • uncertain, and
     • approximate
  – Perceptually acquired information is not reliable:
     • noise introduces uncertainty and imprecision
     • limited range and visibility introduces incompleteness
     • errors in interpretation
More Difficulties with this
        Problem                                           Flakey

• Real world environments have complex and
  largely unpredictable dynamics
  – objects can move
  – the environment may be modified
  – features may change
• – Vehicle action execution is not reliable:
     • the results produced by sending a given command to an
       effector can only be approximately estimated
     • action execution may fail entirely
Robot Architecture using Fuzzy
         Controller                           Flakey
                        of the
 LPS                    rooms

Key is “Local

LPS is data structure
picture around
vehicle                          Camera,etc
   The Fuzzy Controller                               Flakey

• Physical motion based on complex fuzzy
  – Provides a layer of robust high-level motor skills.
• Basic building block of controller is a
     • A behavior is defined as implementing an atomic
       motor skill aimed at achieving or maintaining a give
       goal situation
        – – e.g. follow a wall.
Implementing Behaviors                            Flakey

• Each behavioral skill is represented by means
  of a “desirability function” that expresses
  preferences over possible actions with
  reference to the goal:
  – e.g. a behavior aimed at following a given wall
    prefers actions that keep the agent parallel to the
    wall at a safe distance
Behavior through Fuzzy Rules
• Each behavior was implemented by a set of fuzzy rules of
  the form
   – IF A THEN C
      • A is composed of fuzzy predicates and connectives, and
      • C is a fuzzy set of control vectors
   – An example of a “keep off” behavior rule is:
 IF obstacle-close-in-front AND NOT obstacle-
 close-on-left THEN turn-sharp-left

                                Last slide used
           Fast Reactive

• Purely reactive behaviors, intended to provide quick
  simple reactions to potential dangers typically use
  sensor data that has undergone little or no
• Since quick response is necessary to avoid disaster,
  little processing can be done.
  Control Structures                                                          Flakey

• Purposeful behavior like attempting to reach a certain
  location must take explicit goals into consideration.
• Goals represented in the LPS by means of control
• Control structure is a triple
   – S = (A,B,C)
      • A is a virtual object (artifact) in the LPS
      • B is a behavior that specifies the way to react to the presence of this object,
      • C is a fuzzy predicate expressing the context where the control structure is
Control Structure Example
     S = (A,B,C)                                                                      Flakey
          A is a virtual object (artifact) in the LPS
          B is a behavior that specifies the way to react to the presence of this object, and
          C is a fuzzy predicate expressing the context where the control structure is relevant

• An example control structure is
   – S1=(CP1, go-to-CP, near(CP1)
      • CP1 is a control-point (marker for a location), together with a heading and a
      • go-to-CP reacts to the presence of S1 in the LPS by generating the
        commands to reach the location, heading and velocity specified by CP1.
• go-to-CP includes rules like:
   – IF facing(CP1) AND too-slow-for(CP1) THEN
Blending of Behaviors                               Flakey

• Many behaviors can be simultaneously active
• Fuzzy controller selects the controls that best
  satisfy the active behaviors
• Satisfaction is weighted by each behavior’s
  relevance to the current situation.
   – e.g. can’t follow a wall if there isn’t one
• Context dependent blending of behaviors is
  implemented by combining the output of all the
  behaviors using context rules
             Generating a plan:
     S = (A,B,C)
          A is a virtual object (artifact) in the LPS
          B is a behavior that specifies the way to react to the presence of this object, and
          C is a fuzzy predicate expressing the context where the control structure is relevant
• simple goal-regressing planner used:
   – based on a topological map annotated with approximate measurements (no
     obstacles) working backwards from goal.
• An example plan might be:
    S1 = (Obstacle, keep-off, near(Obstacle))
    S2 = (Corr1, follow,~near(obstacle) AND at(Corr2) AND
    S3 = (Corr2, follow,~near(obstacle) AND at(Corr2) AND
    S4 = (Door5, cross,~near(Obstacle) AND near(Door5))
  Control structure
        Executing the Plan                                  Flakey

•   S1 = (Obstacle, keep-off, near(Obstacle))
•   S2 = (Corr1, follow,~near(obstacle) AND at(Corr2) AND
•       ~near(Corr2))
•   S3 = (Corr2, follow,~near(obstacle) AND at(Corr2) AND
•      ~near(Door5))
•   S4 = (Door5, cross,~near(Obstacle) AND near(Door5))
• 1. Discuss a Fuzzy Logic Control System
• 2. Steps in Designing a Fuzzy Logic Control System
• 3. Design of a Fuzzy Logic Control System:
  – Input membership function,
  – Fuzzy logic rules table,
  – Output membership function.
Components of Fuzzy system:
 • The components of a conventional
   expert system and a fuzzy system are
   the same.
 • Fuzzy      systems though    contain
   – Fuzzifiers convert crisp numbers into
     fuzzy numbers,
 • Fuzzy systems contain `defuzzifiers',
   – Defuzzifiers convert fuzzy numbers into
     crisp numbers.
• The function of the fuzzification component
  is to convert crisp numbers to equivalent
  fuzzy sets.
• Please notice that the inputs might require
  some pre-processing in order to fit the
  range of the fuzzy system.
• The output of the combined
  operation is defuzzified before being
  broadcast to the external world.
• This implies the conversion of a
  fuzzy set to a crisp number.
• There are several techniques of
Conventional vs Fuzzy system
                Components of a ...
     conventional expert    fuzzy
     system                    system
                                                             precise value
              physical                       physical
             device                          device
                                             fuzzifier     value
   precise               precise
   value                 value
Sections of a Fuzzy Logic Control

 • Fuzzifier section
    – System inputs with range of values
    – Mapped to membership function can be non-linear in correspondence
 • Fuzzy control section
    – Input fuzzy values
    – Through fuzzy rules
    – Produce the fuzzy output values
 • De-fuzzifier section
    – Fuzzy output values are combined together in values needed for output
 Steps in Designing a Fuzzy Logic Control System
         1.     Identify the system input variables, their ranges, and
                membership functions.
         2.     Identify the output variables, their ranges, and membership
         3.     Identify the rules that describe the relations of the inputs to
                the outputs.
         4.     Determine the de-fuzzifier method of combining fuzzy rules
                into system outputs.
               Calculate                               Fuzzy     Combine
Inputs                      Fuzzy Inputs   Rule-Base                       Crisp        Output
              Memberships                              Outputs   Outputs

    Fuzzification step                                           Defuzzification step
     Fuzzification step
Example design of a Fuzzy Logic
Control System – Cruise Control
 The block diagram of the intelligent cruise
  control system.
Input membership functions

The three input membership functions
 Fuzzy Logic Rules Table
cruise    slow / distance from car ahead     fast / distance from car ahead
speed    15    30      45     60     75    15       30     45      60     75
 +10     0.15 0.15    0.15   0.15   0.15   0.15    0.15   0.15    0.15   0.15

 +5      0.15 0.15    0.35   0.35   0.35   0.15    0.15   0.15    0.35   0.35

  0      0.15 0.50    0.50   0.50   0.50   0.15    0.15   0.35    0.50   0.50

 -5      0.35 0.65    0.65   0.85   0.85   0.15    0.35   0.50    0.65   0.85

 -10     0.35 0.65    0.85   0.85   0.85   0.15    0.50   0.65    0.85   0.85
Output membership function
 The output is the
 accelerator percentage
 change needed to keep
 the car a safe distance
 behind the car ahead
 and to keep the car at
 cruising speed.
Fuzzy Inference: creating fuzzy
• At the end of the fuzzification step the working
  memory module contains the values of the
  fuzzified input.
  – Each production rule is examined and all the rules that
    have their premises satisfied `fire'.
  – Hence, the only rules which do not fire are those that at
    least one of their premises has a membership degree of
  – In the case that more than one rule fires, this is common
    and desirable, the system generates a single fuzzy
      • This is achieved by combining all fuzzy outputs.
  – The single fuzzy output is then passed to the
    defuzzification module which generates a crisp value.
     • Then the system is ready to start the entire process all over
SIMPLE EXAMPLE: a four-rule
• Assumptions
  – Let X,Y and Z be the linguistic variables.
  – Let the membership functions be low and
  – Let the membership functions be the
    same for all linguistic variables.
  – Define the membership functions as:
    • low(linguistic) = 1 - t
    • high(linguistic) = t
       – where t is a value in the interval [0,1].
A Simple Example (Membership

A Simple Example (Rules)

• The rule base contains the following four
  – if X is low and Y is low then Z is high (rule-1)
  – if X is low and Y is high then Z is low (rule-2)
  – if X is high and Y is low then Z is low (rule-3)
  – If X is high and Y is high then Z is high (rule-4)
A Simple Example (Calculations)

• Next assume that the inputs are 0 and 0.32 for
  the linguistic variables X and Y respectively.
  – The problem then is to find if Z is high or low and
    its crisp value.
• The next steps are followed:
  – Step-1: Find the membership grade for the
    premises of each rule. We have 8 premises but
    due to replication only 4 are needed.
     •   Low(X) = 1- t = 1-0 = 1
     •   Low(Y) = 1- t = 1-0.32 = 0.68
     •   High(X) = t = 0
     •   High(Y) = t = 0.32
A Simple Example (Calculations)
 – Step-2: Identify the rules
   that can fire. In our case
   only the first two rules can
   fire since the last two
   rules have a premise with
   a     zero      degree    of                        1

 – Step-3: For each rule that           X

   can fire find the strength
   of the firing.
    • Rule-1 = min(1,0.68) = 0.68
    • Rule-2 = min(1,0.32) = 0.32   Y

    • Note that min is used
      because the premises are
      connected with a logical
A Simple Example





                          •Rule-2 = min(1,0.32) = 0.32

A Simple Example (Calculations)

  – Step-4: All the fuzzy outputs are
    combined together to form a single
    fuzzy subset.
  – One way is to take the pointwise
    maximum value over all fuzzy outputs.
  – In our case we have two fuzzy outputs,
    so for each point we take the maximum
A Simple Example                  •   The rule base contains the following four
                                        –   if X is low and Y is low then Z is high (rule-1)
                                        –   if X is low and Y is high then Z is low (rule-2)
                                        –   if X is high and Y is low then Z is low (rule-3)
                                        –   If X is high and Y is high then Z is high (rule-4)


 In our case we have two fuzzy outputs, so for each point we take the maximum value.
A Simple Example (Calculations)
 – Step-5: The fuzzy combined output needs to
   be converted to a single crisp value.
 – Using one method which takes the Average-
   of-Maxima. With this method one finds the
   maximum peak of the fuzzy combined output -
   in our case this is 0.68.
 – Then one collects all the t values for which the
   maximum value occurs - in our case there are
   42 cases.
 – Finally, the crisp value is the average of such
   variables - in our case 0.84.
 Do not discuss in the class – this is home reading
        Development Cycle
   Define model's functional                      Define post-model
and operational characteristics                   normalisation and data
  Define fuzzy sets

 Define rules

                                   Define acceptable limits
Define method of
defuzzification               for performance

Run simulation
of system

 Tune and validate the                              Connect to production
system                                             systems
            Development Cycle
– Step 1: The goal is to establish the characteristics
 of the system, and also to define the specific
 operating properties of the proposed fuzzy model.
   • Traditional systems analysis and knowledge engineering
     techniques can be employed at this stage. The designer
     must identify the relevant and appropriate inputs to the
     system; the basic transformations, if any, that are performed
     on the inputs,; and what output is expected from the system.
   • The designer must also decide if the fuzzy system is a
     subsystem of a `global' system and if so to define where the
     fuzzy subsystem fits into the `global architecture', or if it is
     the `global' system itself.
   • The numerical ranges of inputs and outputs must also be
     specified. This step applies to the development of all
     systems. Be prepared to throw away original versions at this
 Development Cycle
– Step 2:     The goal is to decompose each control
 variable (i.e., inputs and outputs) into fuzzy sets and
 to give unique names to them.
  • It has been reported in the literature that the number of
    labels associated with a control variable should generally be
    odd and between five and nine.
  • Also, in order to obtain a smooth transition from a state to
    another each label should overlap somewhat with its
    neighbours. Overlapping of 10 to 50 percent is advised.
  • Finally, the density of the fuzzy sets should be highest
    around the optimal control point of the system and should
    thin out as the distance from that point increases.
Development Cycle
• If possible begin with an exhaustive list of
  production rules
• Deal with redundant, impossible and
  implausible rules and/or conditions later.
  – When the number of rules increases
    dramatically, may be several rule-bases
    might be constructed.
  – Each rule-base to deal with a particular
    situation/condition of the system to be
Development Cycle
– Step 3: The goal is to obtain the production
  rules that tie the input values to the output
– Since each production rule `declares' a small
  chunk of knowledge, the order in the
  knowledge base is unimportant.
– Nevertheless, in order to maintain the
  knowledge base one should group the
  production rules by their premise variable.
– How many rules is obviously dependent on
  the application and is related to the number
  of control variables.
Development Cycle
– Step 4: The goal is to decide on the way that is going to
  be used in order to convert a an output fuzzy set into a
  crisp solution variable.
– As already indicated in another section there are many
  ways to perform the conversion but by and large,
  process control applications use the centroid technique.
– The rest of the steps are similar to any modelling
  exercise. For instance, at the end of the fuzzy system
  construction the process of simulation commences. The
  model is compared against known test cases and the
  results are validated. When the results are not as
  desired changes are made until the desired performance
  is achieved.
Popular Fuzzy
Popular Fuzzy Inference Methods
• Under the fuzzy inference process the grade of
  each premise is found and applied to the
  conclusion part of each rule.
  – MIN or PRODUCT are two popular inference
  – With MIN inferencing the output of the conclusion
    part is clipped off at a height equal to the rule's
    degree of firing.
     • This was used in the simple example before and the MIN
       inference for the first rule is re-shown below
     • (Recall that the rule's degree of firing was 0.68):
 1                         Original High
0.7                         MIN
      0          0.5   1
• With PRODUCT inferencing the output
  membership function is scaled by the rule's
  degree of firing.
• Using the first rule of the simple example
  and the PRODUCT inference we get the
 1                      Original High
      0    0.5      1
methods for
     Popular methods for
      combining outputs
• Under the combining outputs process all
  outputs are combined together to produce a
  single fuzzy output.
  – MAX and SUM are two popular techniques for
    combining outputs.
  – With MAX the pointwise maximum for all fuzzy
    outputs is taken.
  – For the first rule of the simple example we used
    the MAX techniques which resulted in the
    following diagram:
0.7                                   MAX
0.3                      Rule 2
0.1       Rule 1

      0            0.5            1
• With SUM the pointwise sum for all fuzzy outputs
  is taken.
• Again for the first rule of the simple example we
  get:             1
                 0.9                                SUM
                 0.5                       Rule 1
                 0.1              Rule 2

                       0          0.5                 1
Popular defuzzification methods
• Under the defuzzification process a fuzzy output is
  converted to a crisp number.
   – Two of the more popular techniques are the MAXIMUM and the
   – With MAXIMUM one selects the maximum value of the fuzzy output
     as the crisp value.
   – There are several variations to the MAXIMUM theme.
      • One such variation is the AVERAGE-OF-MAXIMA which was used in
        the simple example and gave us a crisp value of 0.84.

• With the CENTROID method, the center of gravity of the
  fuzzy output gives the crisp value.
• Using PRODUCT inferencing and the SUM combination the
  CENTROID results in a crisp value of 0.56.
Example of calculation
    of Maximum
  fuzzy point * strength of point
  i 1
               fuzzy point
              i 1

where n is the total number of fuzzy points (strips).
the output of a
 Ways of Combining Fuzzy Logic
         Output Values
 There are four different
techniques to combine the
   fuzzy logic rules output

They are:
• Maximizer
• Average
• Centroid
• Singleton
  Calculating the output of a

• Rules have promises which are usually
  combined together by the connective and.
• The output is calculated by taking the
  degree of membership of the lesser of all
  premises as the value of the combination
  and truncating the output fuzzy set at that
Calculating the output of a rule

Premise-1   0.45

                                   Truncated at
                   0.45            0.2

 Combining rule outputs
• When all rules have been evaluated a
  single fuzzy set is calculated by
  combining all outputs.
• The combination involves the
  connective or.
• The single output is calculated by taking
  the maximum of their respective output
  fuzzy sets grades at each point along
  the horizontal axis.
           Combining rule outputs
Output-1                                   Combined Output


                                           Think about the set of fuzzy rules as a
                                           network with fuzzy literals and operators
                                           MIN, MAX, etc. In primary inputs you have
                            Shifted        fuzzification, in primary outputs you have
                            horizontally   defuzzification. The network analogy is very
                                           useful in your thinking about fuzzy logic, mv
                                           logic, or any other kind of data structure to
                                           store knowledge.

             Do in class a complete example of designing a fuzzy logic network
             for a simple robot control with about 6 rules. Discuss various
             operators and fuzzifiers, defuzzifiers.
• Up to now we have seen that fuzzy logic supports
  the task of designing a control system.
  – What we examined in previous sections is known as the
    `static' fuzzy system.
  – In such a system, conventional techniques are
    employed in order to elicit or induce production rules.
  – Such a system, receives inputs which are then
    normalised and converted to fuzzy representations, the
    knowledge base is executed, an output fuzzy set is
    generated which is then converted to a crisp value.
static' fuzzy system.

       Normalisation and

       Execution of rules   Output fuzzy
        Inference engine

        Production rules

        Fuzzy sets


   – The rules do not change, except if modified by the hand of the
     designer, for all the knowledge base lifetime.
   – Static systems are fine for applications in which the environment is
     known and predictable.
        • But because they can not adapt to gradual changes in their
          environments they can lead to disaster when the assumptions upon
          which they are built are violated.
• An adaptive system adjusts to time or process conditions.
• This means that such a system modifies:
   –    the charactersitics of the rules,
   –   the topology of the fuzzy sets, and
   –    the method of defuzzification
   –    among others.
An adaptive system
                          Buffer       Performance
     Normalisation and

                              Output fuzzy
     Execution of rules
      Inference engine

      Production rules
      Fuzzy sets


•  A performance metric, often another expert system or
  an algorithm, compares the current and the stored array
  of past solutions and the comparison result is passed
  to an adaptation machine.
• The adaptation machine, often another expert system,
  decides what changes to make in the underlying fuzzy
    – For instance:
       • the contribution weights connected with each rule can be modified
       • the membership functions re-drawn.
 about Fuzzy
      Fuzziness is not Vague
• we shall have a look at some propositions.

• Dimitris is six feet tall
   – The first proposition (traditional) has a crisp truth value
     of either TRUE or FALSE.

• He is tall
   – The second proposition is vague.
   – It does not provide sufficient information for us to
     make a decision, either fuzzy or crisp.
   – We do not know the value of the pronoun.
   – Is it Dimitris, John or someone else?
  Fuzziness is not Vague
• Andrei is tall
  – The third proposition is a fuzzy
  – It is true to some degree depending in
    the context, i.e., the universe of
  – It might be SomeWhat True if we are
    referring to basketball players or it
    might be Very True if we are referring to
 Fuzziness is not Multi-valued
– The limitations of two-valued logic were recognised very early.
– A number of different logic theories based on multiple values of
  truth have been formulated through the years.
– For example, in three-valued logic three truth values have been
– These are TRUTH, FALSE, and UNKNOWN represented by 1, 0 and
  0.5 respectively.

– In 1921 the first N-valued logic was introduced.
– The set of truth values Tn were assumed to be evenly divided over
  the closed interval [0,1].
– Fuzzy logic may be considered as an extension of multi-valued
  logic but they are somewhat different.
– Multi-valued logic is still based on exact reasoning whereas fuzzy
  logic is approximate reasoning.
            Fuzziness is not
• This is better explained using an example.
• Let X be the set of all liquids (i.e., the universe of
  discourse) .
• Let L be a subset of X which includes all suitable for
  drinking liquids.
• Suppose now that you find two bottles, A and B.
• The labels do not provide any clues about the contents.
• Bottle A label is marked as membership of L is 0.9.
• The label of bottle B is marked as probability of L is 0.9.
• Given that you have to drink from the one you choose,
  the problem is of how to interpret the labels.
         Fuzziness is not
• Well, membership of 0.9 means that the contents of A
  are fairly similar to perfectly potable liquids.
• If, for example, a perfectly liquid is pure water then
  bottle A might contain, say, tonic water.
• Probability of 0.9 means something completely
• You have a 90% chance that the contents are potable
  and 10% chance that the contents will be unsavoury,
  some kind of acid maybe.
• Hence, with bottle A you might drink something that
  is not pure but with bottle B you might drink
  something deadly. So choose bottle A.
               Fuzziness is not
• Opening both bottles you observe beer (bottle A) and
  hydrochloric acid (bottle B).
• The outcome of this observation is that the membership stays the
  same whereas the probability drops to zero.
• All in all:
   – probability measures the likelihood that a future event will occur,
   – fuzzy logic measures the ambiguity of events that have already occurred.
• In fact, fuzzy sets and probability exist as parts of a greater
  Generalized Information Theory.
• This theory also includes:
   – Dempster-Shafer evidence theory,
   – possibility theory,
   – and so on.
Applications of
Fuzzy concepts
   Where is fuzzy logic
– Fuzzy logic is a powerful problem-solving methodology.
   • It is used directly and indirectly in a number of applications.
– Fuzzy logic is now being applied all over Japan, Europe
  and more recently in the United States of America.
– It is true though that all we ever hear about is Japanese fuzzy logic.
– Products such as:
   •   the Panasonic rice cooker,
   •   Hitachi's vacuum cleaner,
   •   Minolta's cameras,
   •   Sony's PalmTop computer,
   •   and so on.
– This is not unexpected since Japan adopted the
  technology first.
Where is fuzzy logic
– whereas in the West companies might keep
  their fuzzy development secret because of
  the implication of the word `fuzzy',
– or because companies want to preserve
  competitive advantage,
– or because fuzzy logic is embedded in
  products without advertisement.
– Most applications of fuzzy logic use it as
  the concealed logic system for expert
      Where is fuzzy logic
• The areas of potential fuzzy implementation are
  numerous and not just for control:
  –   Speech recognition,
  –   fault analysis,
  –   decision making,
  –   image analysis,
  –   scheduling
  –   and many more are areas where fuzzy thinking can
• Hence, fuzzy logic is not just control but can be
  utilized for other problems.
Where is fuzzy logic used?
• One business problem, namely that of fraud detection,
  was recently addressed using fuzzy logic.
• The system detects probable fraudulent behavior:
   – by evaluating all the characteristics of a provider's claim data in
   – against the normal behavior of a small ( in demographic terms )
• An all-American success story is the use of fuzziness on keeping a
  commercial refrigerator thermally controlled (0.1 C).
   – The excitement comes due to the fact that this refrigerator has
     flown on several space shuttle flights.
Where is fuzzy logic used?
• Another interesting application has been reported by Aptronix
• Fuzzy logic was there used as a means of determining correct
  focus distance for cameras with automatic focusing system.
• Traditionally, such a camera focuses at the middle of the view
• This can be inaccurate though when the object of interest is not
  at the center.
• Using fuzzy logic, three distances are measured from the view
  finder; left, center and right.
• For each measurement a plausibility value is calculated and the
  measurement with the highest plausibility is deemed as the
  place where the object of interest is located.
When to use Fuzzy Logic?
• If the system to be modelled is a
  linear   system     which   can    be
  represented by a mathematical
  equation or by a series of rules then
  straightforward techniques should be
• Alternatively, if the system is
  complex, fuzzy logic may be the
  technique to follow.
      When to use Fuzzy Logic?
• We define a complex system :
  – when it is nonlinear, time-variant, ill-
  – when variables are continuous;
  – when a mathematical model is either too
    difficult to encode or does not exist or is
    too complicated and expensive to be
  – when noisy inputs;
  – and when an expert is available who can
    specify the rules underlying the system
• Fuzzy logic captures intuitive, human
• Fuzzy sets, statements, and rules are the
  basis of control.
• The technique is extremely powerful, and
  appears in mills at a growing rate.
• It is combined with other methods and is
  the base of soft computing
• Used much in Intelligent Robotics
• Paul and Mildred Burkey
• Weilin Pan
• Xuekun Kou
• A. Ferworn
• Kevin Morris
• Dr Dimitris Tsaptsinos
   Kingston University, Mathematics

To top