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Fuzzy Logic and Fuzzy Inference

- Why use fuzzy logic?
- Tipping example
- Fuzzy set theory
- Fuzzy inference

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What is fuzzy logic?

- A super set of Boolean logic
- Builds upon fuzzy set theory
- Graded truth. Truth values between True and False. Not everything is either/or, true/false, black/white, on/off etc.
- Grades of membership. Class of tall men, class of far cities, class of expensive things, etc.
- Lotfi Zadeh, UC/Berkely 1965. Introduced FL to model uncertainty in natural language. Tall, far, nice, large, hot, …
- Reasoning using linguistic terms. Natural to express expert knowledge. If the weather is cold then wear warm clothing

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Fuzzy logic – A Definition

- Fuzzy logic provides a method to formalize reasoning when dealing with vague terms.
- Traditional computing requires finite precision which is not always possible in real world scenarios. Not every decision is either true or false, or as with Boolean logic either 0 or 1.
- Fuzzy logic allows for membership functions, or degrees of truthfulness and falsehoods. Or as with Boolean logic, not only 0 and 1 but all the numbers that fall in between.

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Why use fuzzy logic?

Pros:

- Conceptually easy to understand w/ “natural” maths
- Tolerant of imprecise data
- Universal approximation: can model arbitrary nonlinear functions
- Intuitive
- Based on linguistic terms
- Convenient way to express expert and common sense knowledge

Cons:

- Not a cure-all
- Crisp/precise models can be more efficient and even convenient
- Other approaches might be formally verified to work

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Tipping example

- The Basic Tipping Problem: Given a number between 0 and 10 that represents the quality of service at a restaurant what should the tip be?Cultural footnote: An average tip for a meal in the U.S. is 15%, which may vary depending on the quality of the service provided.

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Tipping example: The non-fuzzy approach

- Tip = linearly proportional to service from 5% to 25%tip = 0.20/10*service+0.05

- What about quality of the food?

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Tipping example: Extended

- The Extended Tipping Problem: Given a number between 0 and 10 that represents the quality of service and the quality of the food, at a restaurant, what should the tip be?How will this affect our tipping formula?

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Tipping example: The non-fuzzy approach

- Tip = 0.20/20*(service+food)+0.05

- We want service to be more important than food quality. E.g., 80% for service and 20% for food.

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Tipping example: The non-fuzzy approach

- Tip = servRatio*(.2/10*(service)+0.05) + servRatio = 80% (1-servRatio)*(.2/10*(food)+0.05);

- Seems too linear. Want 15% tip in general and deviation only for exceptionally good or bad service.

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Tipping example: The non-fuzzy approach

if service < 3,

tip(f+1,s+1) = servRatio*(.1/3*(s)+.05) + ... (1-servRatio)*(.2/10*(f)+0.05);

elseif s < 7,

tip(f+1,s+1) = servRatio*(.15) + ...

(1-servRatio)*(.2/10*(f)+0.05);

else,

tip(f+1,s+1) = servRatio*(.1/3*(s-7)+.15) + ...

(1-servRatio)*(.2/10*(f)+0.05);

end;

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Tipping example: The non-fuzzy approach

- Nice plot but
- ‘Complicated’ function
- Not easy to modify
- Not intuitive
- Many hard-coded parameters
- Not easy to understand

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Tipping problem: the fuzzy approach

What we want to express is:

- If service is poor then tip is cheap
- If service is good the tip is average
- If service is excellent then tip is generous
- If food is rancid then tip is cheap
- If food is delicious then tip is generous

or

- If service is poor or the food is rancid then tip is cheap
- If service is good then tip is average
- If service is excellent or food is delicious then tip is generous

We have just defined the rules for a fuzzy logic system.

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Tipping problem: fuzzy solution

- Before we have a fuzzy solution we need to find out
- how to define terms such as poor, delicious, cheap, generous etc.
- how to combine terms using AND, OR and other connectives
- how to combine all the rules into one final output

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Fuzzy sets

- Boolean/Crisp set A is a mapping for the elements of S to the set {0, 1}, i.e., A: S {0, 1}
- Characteristic function:

A(x) =

{

1 if x is an element of set A

0 if x is not an element of set A

- Fuzzy set F is a mapping for the elements of S to the interval [0, 1], i.e., F: S [0, 1]
- Characteristic function: 0 F(x) 1
- 1 means full membership, 0 means no membership and anything in between, e.g., 0.5 is called graded membership

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Fuzzy Sets (contd.)

- fuzzy set A
- A = {(x,µA(x))| x X} where µA(x) is called the membership function for the fuzzy set A. X is referred to as the universe of discourse.
- The membership function associates each element x X with a value in the interval [0,1].

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Example: Crisp set Tall

- Fuzzy sets and concepts are commonly used in natural languageJohn is tallDan is smartAlex is happyThe class is hot
- E.g., the crisp set Tall can be defined as {x | height x > 1.8 meters}But what about a person with a height = 1.79 meters?What about 1.78 meters?…What about 1.52 meters?

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Example: Fuzzy set Tall

- In a fuzzy set a person with a height of 1.8 meters would be considered tall to a high degreeA person with a height of 1.7 meters would be considered tall to a lesser degree etc.
- The function can changefor basketball players,Danes, women, children etc.

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Membership functions: S-function

- The S-function can be used to define fuzzy sets
- S(x, a,b, c) =
- 0 for xa
- 2(x-a/c-a)2 for axb
- 1 – 2(x-c/c-a)2 for bxc
- 1 for x c

a

b

c

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Membership functions of one dimension

- These membership functions are some of the commonly used membership functions in the fuzzy inference systems.
- Triangle(x; a, b, c) = 0 if x a;

= (x-a)/(b-a) if a x b;

= (c-x)/(c-b) if b x c;

= 0 if c x.

- Trapezoid(x; a, b, c, d) = 0 if x a;

= (x-a)/(b-a) if a x b;

= 1 if b x c;

= (d-x)/(d-c) 0 if c x d;

= 0, if d x.

- Sigmoid(x; a, c) = 1/(1 + exp[-a(x-c)]) where a controls slope at the crossover point x = c.

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Membership functions of two dimensions

- One dimensional fuzzy set can be extended to form its cylindrical extension on second dimension
- Fuzzy set A = “(x,y) is near (3,4)” is
- µA(x,y) = exp[- ((x-3)/2)2 -(y-4)2 ]

= exp[- ((x-3)/2)2 ]exp-(y-4)2 ]

=gaussian(x;3,2)gaussian(y;4,1)

- This is a composite MF since it can be decomposed into two gaussian MFs

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b-a

b-a/2

b+a/2

b+a

a

Membership functions: P-Function- P(x, a, b) =
- S(x, b-a, b-a/2, b) for xb
- 1 – S(x,b, b+a/2, a+b) for x b

E.g., close (to a)

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Simple membership functions

- Piecewise linear: triangular etc.
- Easier to represent and calculate saves computation

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Other representations of fuzzy sets

- A finite set of elements:F = 1/x1 + 2/x2 + … n/xn+ means (Boolean) set union
- For example:TALL = {0/1.0, 0/1.2, 0/1.4, 0.2/1.6, 0.8/1.7, 1.0/1.8}

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Fuzzy sets with a discrete universe

- Let X = {0, 1, 2, 3, 4, 5, 6} be a set of numbers of children a family may possibly have.
- fuzzy set A with “sensible number of children in a family” may be described by
- A = {(0, 0.1), (1, 0.3), (2, 0.7), (3, 1), (4, 0.7), (5, 0.3), (6, 0.1)}

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Fuzzy sets with a continuous universe

- X = R+ be the set of possible ages for human beings.
- fuzzy set B = “about 50 years old” may be expressed as
- B = {(x, µB(x)) | x ЄX}, where
- µB(x) = 1/(1 + ((x-50)/10)4)

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We use the following notation to describe fuzzy sets.

- A = ΣxiЄ XµA(xi)/ xi, if X is a collection of discrete objects,
- A = ∫X µA(x)/ x, if X is a continuous space.

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Fuzzy set operators

- EqualityA = BA (x)= B (x) for all x X
- ComplementA’A’ (x)= 1 - A(x)for all x X
- ContainmentA BA (x)B (x) for all x X
- UnionA BA B(x) = max(A (x),B (x)) for all x X
- IntersectionA BA B(x) = min(A (x),B (x)) for all x X

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Crossover point of a fuzzy set A is a point x in X such that

- {(x, µA(x)) |µA(x) = 0.5 }
- α-cut of a fuzzy set A is set of all points x in X such that
- {(x, µA(x)) |µA(x) ≥ α }

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Support(A) is set of all points x in X such that

- {(x, µA(x)) |µA(x) > 0 }
- core(A) is set of all points x in X such that
- {(x, µA(x)) |µA(x) =1 }
- Fuzzy set whose support is a single point in X with µA(x) =1 is called fuzzy singleton

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Fuzzy intersection and Union

- AB = T(A(x), B(x)) where T is T-norm operator. There are some possible T-Norm operators.
- Minimum: min(a,b)=a ٨b
- Algebraic product: ab
- Bounded product: 0 ٧(a+b-1)

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C(x) = AB = S(A(x), B(x)) where S is called S-norm operator.

- It is also called T-conorm
- Some of the T-conorm operators
- Maximum: S(a,b) = max(a,b)
- Algebraic sum: a+b-ab
- Bounded sum: = 1 ٨(a+b)

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More or less tall

Very tall

Linguistic Hedges- Modifying the meaning of a fuzzy set using hedges such as very, more or less, slightly, etc.
- Concentration or Con operator
- Very F = F2
- Dilation or Dil operator
- More or less F = F1/2
- more or less tall

= DIL(tall);

- extremely tall

= CON(CON(CON(tall)))

- etc.

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Fuzzy relations

- A fuzzy relation for N sets is defined as an extension of the crisp relation to include the membership grade.R = {R(x1, x2, … xN)/(x1, x2, … xN) | xi X, i=1, … N}

which associates the membership grade, R , of each tuple.

- E.g. Friend = {0.9/(Manos, Nacho), 0.1/(Manos, Dan), 0.8/(Alex, Mike), 0.3/(Alex, John)}

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Binary fuzzy relation

- A binary fuzzy relation is a fuzzy set in X × Y which maps each element in X × Y to a membership value between 0 and 1. If X and Y are two universes of discourse, then
- R = {((x,y), R(x, y)) | (x,y) ЄX × Y } is a binary fuzzy relation in X × Y.
- X × Y indicates cartesian product of X and Y

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Fuzzy Rules

- Fuzzy rules are useful for modeling human thinking, perception and judgment.
- A fuzzy if-then rule is of the form “If x is A then y is B” where A and B are linguistic values defined by fuzzy sets on universes of discourse X and Y, respectively.
- “x is A” is called antecedent and “y is B” is called consequent.

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Examples, for such a rule are

- If pressure is high, then volume is small.
- If the road is slippery, then driving is dangerous.
- If the fruit is ripe, then it is soft.

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The fuzzy rule “If x is A then y is B” may be abbreviated as A→ B and is interpreted as A × B.

- A fuzzy if then rule may be defined (Mamdani) as a binary fuzzy relation R on the product space X × Y.
- R = A→ B = A × B =∫X×YA(x) T-normB(y)/ (x,y).

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Fuzzy inference

- Fuzzy logical operations
- Fuzzy rules
- Fuzzification
- Implication
- Aggregation
- Defuzzification

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- 0 0 0
- 0 1 0
- 0 0
- 1 1 1

- A B A or B
- 0 0 0
- 0 1 1
- 0 1
- 1 1 1

- AND, OR, NOT, etc.
- NOT A = A’ = 1 - A(x)
- A AND B = A B = min(A (x),B (x))
- A OR B = A B = max(A (x),B (x))

From the following truth tables it is seen that fuzzy logic is a superset of Boolean logic.

min(A,B)

max(A,B)

1-A

A not A

0 1

1 0

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If-Then Rules

- Use fuzzy sets and fuzzy operators as the subjects and verbs of fuzzy logic to form rules.

if x is A then y is B

where A and B are linguistic terms defined by fuzzy sets on the sets X and Y respectively.

This reads

if x == A then y = B

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Evaluation of fuzzy rules

- In Boolean logic: p qif p is true then q is true
- In fuzzy logic: p qif p is true to some degree then q is true to some degree.0.5p => 0.5q (partial premise implies partially)
- How?

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Evaluation of fuzzy rules (cont’d)

- Apply implication function to the rule
- Most common way is to use min to “chop-off” the consequent(prod can be used to scale the consequent)

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Summary: If-Then rules

- Fuzzify inputsDetermine the degree of membership for all terms in the premise.If there is one term then this is the degree of support for the consequent.
- Apply fuzzy operatorIf there are multiple parts, apply logical operators to determine the degree of support for the rule.
- Apply implication methodUse degree of support for rule to shape output fuzzy set of the consequent.

How do we then combine several rules?

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Multiple rules

- We aggregate the outputs into a single fuzzy set which combines their decisions.
- The input to aggregation is the list of truncated fuzzy sets and the output is a single fuzzy set for each variable.
- Aggregation rules: max, sum, etc.
- As long as it is commutative then the order of rule exec is irrelevant.

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max-min rule of composition

- Given N observations Ei over X and hypothesis Hi over Y we have N rules: if E1 then H1if E2 then H2if EN then HN
- H = max[min(E1), min(E2), … min(EN)]

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Center of largest area

Defuzzify the output- Take a fuzzy set and produce a single crisp number that represents the set.
- Practical when making a decision, taking an action etc.

I x

I

I=

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Limitations of fuzzy logic

- How to determine the membership functions? Usually requires fine-tuning of parameters
- Defuzzification can produce undesired results

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General Fuzzified Applications

- Quality Assurance
- Error Diagnostics
- Control Theory
- Pattern Recognition

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Specific Fuzzified Applications

- Otis Elevators
- Vacuum Cleaners
- Hair Dryers
- Air Control in Soft Drink Production
- Noise Detection on Compact Disks
- Cranes
- Electric Razors
- Camcorders
- Television Sets
- Showers

Japan, Korea, China were the early adapters of

Fuzzy Logic into industrial applications

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Expert Fuzzified Systems

- Medical Diagnosis
- Legal
- Stock Market Analysis
- Mineral Prospecting
- Weather Forecasting
- Economics
- Politics

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References

- Slides from CS 561 (Intro to AI) course, Sessions 20-21 of Paolo Pirjanian
- Slides from OperMgt 345 course of Mitch Pence, Boise State University
- Slides from CS623-Lec11-4Sept06, S.G.Sanjeevi of IIT Bombay.
- Fuzzy Logic. Fuzzy Logic - a powerful new technology. http://www.austinlinks.com/Fuzzy/
- FuzzyNet On-line. Automatic Transmission http://www.aptronix.com/fuzzynet/applnote/transmis.htm
- Garner, Martin. Weird Water and Fuzzy Logic: More notes of a Fringe Watcher.
- Generation 5. An Introduction to Fuzzy Logic. http://www.generation5.org/fuzzyintro.shtml
- Sowell, Thomas . FUZzy Logic For “Just Plain Folks” .
- You Fuzzin’ with me? http://www.doc.ic.ac.uk/~nd/surprise_96/journal/vol1/sbaa/article1.html
- Zadeh, Lotfi A. http://http.cs.berkeley.edu/People/Faculty/Homepages/zadeh.html
- Zadeh, L. (1965), "Fuzzy sets", Information and Control, 8: 338-353
- Jang J.S.R., (1997): ANFIS architecture. In: Neuro-fuzzy and Soft Computing (J.S. Jang, C.-T. Sun, E. Mizutani, Eds.), Prentice Hall, New Jersey

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The following, illustrates a basic “fuzzy” automatic transmission system. The transmission uses four fuzzy sensor inference inputs to control the best gear selection for the given conditions. The inputs are throttle, vehicle speed, engine speed and engine load.An Example

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An Example (cont.)

Using the labels as defined in the previous slides, rules can be written for the fuzzy interface unit. The rules provide a tangible knowledge base required for the decision process and are represented as English like if-then statements.

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An Example (cont.)

To create the fuzzy interface unit, rules such as the following would be developed to facilitate the automatic shifting of the vehicle.

If vehicle speed is low, variation of vehicle speed is small, slope resistance is positive large and accelerator is medium then mode is steep uphill mode.

If vehicle speed is medium, variation of vehicle speed is small, slope resistance is negative large and accelerator is small then mode is gentle downhill mode.

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An Example (cont.)

If Mode is Steep uphill mode, the Shift is No. 2

If Mode is Gentle downhill mode, then Shift is No. 3

The previous slides illustrate how fuzzy logic can provide a powerful tool when addressing complex situations that are not feasible using conventional approaches. By employing fuzzy logic, we have the ability to include additional variables and rules to take into account factors that might improve the behavior of the control system.

* See reference (see notes): http://www.aptronix.com/fuzzynet/applnote/transmis.htm

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