Fuzzy Logic

1
Fuzzy Logic

• An approach to uncertainty that combines real
  values 0,1 and logic operations
• Fuzzy logic is based on the ideas of fuzzy set
  theory and fuzzy set membership often found in
  language
• You are tall what is tall?
• You are very tall how does this differ from
  tall?

2
Fuzzy Sets

• In normal sets, membership is binary
• An item is either in the set or not in the set
• In fuzzy sets, membership is based on a degree
  between 0 and 1
• 0 item not in set
• 1 item is in set
• If degree is between 0 and 1, then this degree is
  the degree to which the item is thought to be in
  the set
• In a way, this is like the statistic approaches
  covered in chapter 8, but it differs

3
Example Young

• Ann is 28, .8 in set Young
• Bob is 40, .1 in set Young
• Cathy is 23, 1.0 in set Young
• Unlike statistics and probabilities, the degree
  is not describing a probability that the item is
  in the set, but instead describes to what extent
  the item is in the set
• Also note that the degrees do not add up to 1

4
Distinctions to Probabilities

• Why both fuzzy sets and probabilities use real
  numbers to describe a degree of membership they
  differ
• membership functions are not necessarily based on
  statistic distributions
• fuzzy logic deals with deterministic
  plausibilities
• probabilities deal more with non-deterministic
  but stochastic events and their likelihoods

5
Fuzzy Sets

• A is a fuzzy set described by the items, u, where
  the set A u/a(u) u in U
• That is, A is a set of items combined with their
  degrees of being in the universe U
• a(u) is a membership function that maps onto the
  set 0,1 which derives the degree that each u is
  or is not in U
• Example The set of young people might be
  Ann/0.8, Bob/0.1, Cathy/1.0

6
Set Operations

• Fuzzy sets, like sets, have operations for
• is-a-member-of a(u)
• union A u B u/max(a(u), b(u) u in U
• intersection A n B u/min(a(u), b(u) u in
  U
• complement A u/1-a(u) u in U
• With these rules, other set properties can be
  used such as the commutative law, DeMorgans
  laws, etc...

7
Fuzzy Relations

• A is a fuzzy set in the universe U with
  membership function a(u)
• B is a fuzzy set in the universe V with
  membership function b(v)
• AxB(u,v)/min(a(u),b(v)) u in U, v in V
• R is a fuzzy relation such that RU ? V where
  UxV(u,v)/m(u,v) u in U, v in V where m(u,v)
  is a membership function

8
Ex Temperature Regulator

• Given two input values
• E the difference between the current
  temperature and the target temperature
• dEthe time derivative
• Determine the output value Wthe change in the
  heating or cooling source
• W can take on one of 5 values, negative big,
  negative small, zero, positive small, positive big

9
Fuzzy Rules for Example

• Given E and dE, we have the following
  rules dE NB NS ZO PS PB NB PB NS
  PS E ZO PB PS ZO NS NB PS NS
  PB NB
• For instance, if E is ZO and dE is NB, then W is
  PB

10
Computing dE and E

• See figure 1 from the handout (p.73)
• Suppose E0.75 and dE0 (that is, at time zero,
  the temperature is .75 from where it should be)
• Use the membership function in figure 1
• E comes in at the horizontal .75 point which is
  midway between PS and PB. PS has a value of 0.5
  and PB has a value of 0.5. dE 0 which has ZO
  with a value of 1.0.

11
Selecting Rules

• We have two applicable rules, 8 and 9
• dEZO (membership of 1.0)
• EPB (membership of 0.5) and PS (membership of
  0.5)
• The weight of each rule is determined by the
  minimum of the two memberships (0.5 and 1.0) so
  both rules have equal weight (0.5)
• However, we only want one output, so we must
  select a single rule. This is done through a
  difuzzification procedure finding the center of
  gravity in the output function

12
Fuzzy Logic

• Just as fuzzy sets are an extension to sets,
  fuzzy logic is an extension to logic
• Union Or, Intersection And, Complement Not
• A gt B or If A then B is a fuzzy implication and
  can be derived as AxB

13
Two possible fuzzy logics

• There have been two fuzzy logics developed that
  might be applicable to inference
• Consider two items A and B with degrees of
  membership (or truth) of a and b
• Not A 1-a (the same in either fuzzy logic)
• A and B min(a,b) or ab
• A or B max(a,b) or ab-ab
• A?B max (1-a, b) or 1-aab
• A xor B max(min(a,1-b), min(1-a,b)) or
  ab-3abaababb-aabb

14
Example using the fuzzy logics

• A domain has hypotheses H1, H2, H3, H4, H5
• H1, H2, and H3 are primitive hypotheses, H4 and
  H5 are derived by the following rules
• H4H1 or H2
• H5not(H1 or H2) and H3
• Given values for H1, H2 and H3, values for H4 and
  H5 can be derived using the prior fuzzy logic
  rules for Not, And, Or
• We can now compute the likelihood of each
  hypothesis and select all that are probable

15
Fuzzy System Applications

• Fuzzy logic is most commonly used as controllers
  of systems, applying fuzzy rules to determine
  changes in output
• Cement Kiln - first expert system to use fuzzy
  logic, incorporates experience of operators in
  cement production facilities to mix ingredients
• Sendai Subway - most celebrated fuzzy logic
  system, controller uses fuzzy rules to control
  speed of cruising, braking and switching
• Yamaichi Fuzzy Fund - fuzzy rules determined
  monthly by analysts and then used to predict
  trends and suggest trades

16
New Applications

• Home appliances fuzzy logic used in
  refrigerators, vacuum cleaners, washers, dryers,
  air conditioners to control temperature,
  pressure, etc...
• Video Cameras fuzzy automatic focusing and
  exposure rules
• Automotive fuzzy controlled fuel injection,
  transmission and brakes systems
• Robotics fuzzy controlled robots
• Aerospace fuzzy logic temperature control for
  space shuttle

17
Future Development - Hybrids

• Fuzzy neural net the two areas can complement
  each weakness (fuzzy logic - does not learn,
  neural networks - little control)
• Fuzzy genetic algorithms GAs can be used to
  tune membership functions

18
Limitations of Fuzzy Logic

• Stability since fuzzy logic is not formal,
  there is no guarantee that a fuzzy system will
  function correctly
• Learning no ability to learn membership
  functions and no memory to learn during problem
  solving
• Determining good membership functions and fuzzy
  rules is not always easy or straightforward
• Verification and Validation requires extensive
  testing (as in any expert system). This is
  especially important of controllers where safety
  becomes a key factor (e.g. subway system, space
  shuttle)

19
Fuzzier Methods

• Fuzzy Logic is (roughly) based on a mathematical
  formalism
• Fuzzy methods do not require this mathematical
  backing
• Certainty factors are a compromise between using
  fuzzy logic and a looser approach
• Qualitative fuzzy rules go further and completely
  divorce the rules from mathematics