2. Question: Is it warm in here?
Fuzzy Logic
Implication and Inference
Question: Is it warm in here?
Answer
yes
fairly warm
maybe
a little no
not really
IF room is warm THEN set cooling power to 500 watts

3. Fuzzy Inference IF room is warm, THEN set cooling power to 500 watts.
Fuzzy Logic
Implication and Inference
Fuzzy Inference
Premise
Consequence
IF room is warm, THEN set cooling power to 500 watts.
Measurement
The room temperature is 21 °C.
Set cooling at 280 watts.
Action

4. Fuzzy Logic
Fuzzy Logic
Terminologies
Fuzzy Universe: range of all possible values to a chosen variable
Fuzzy Set: set with fuzzy boundaries
Fuzzy Membership Function: used to define fuzzy set
Fuzzy Set Operations: not, or, and
Fuzzy Logic Operators: the realization of set operation
Fuzzy Variable or Fuzzy Linguistic Value: variable assigned to fuzzy sets, such as: tall, high, fast
Fuzzy Linguistic Variable: variable that takes the value of certain fuzzy variable, such as: person, pressure, velocity
Fuzzy Rules: conditional statements that relates fuzzy sets

5. Fuzzy Membership Function
Fuzzy Logic
Membership Function
Fuzzy Membership Function
Single-Valued (Singleton)
Trapezoidal
Triangular
Sigmoid
Gaussian
etc., as can be seen later in Fuzzy Toolbox

6. Fuzzy Membership Function: Tall People
Fuzzy Logic
Membership Function
Fuzzy Membership Function: Tall People
Degree of Membership
Fuzzy Membership Function
Fuzzy Universe

7. Fuzzy Membership Function: Around Noon
Fuzzy Logic
Membership Function
Fuzzy Membership Function: Around Noon
Trapezoid
Triangular
Smooth trapezoid
Smooth triangular

8. Fuzzy Set Operations Min-Max Operators OR AND NOT Fuzzy Logic
Membership Function
Fuzzy Set Operations A B
Min-Max Operators OR
AND
NOT

9. FL-OR must resemble s-Norm FL-AND must resemble t-Norm
Fuzzy Logic
Membership Function
Fuzzy Logic Operators
s-Norm: is monotonous associative operator with
t-Norm: is monotonous associative operator with
FL-OR must resemble s-Norm
FL-AND must resemble t-Norm
Monotony
Associativity

10. Fuzzy Logic Operators NOT Operator OR and AND Operators Max Min
Membership Function
Fuzzy Logic Operators
NOT Operator
OR and AND Operators
Max
Min
Algebraic sum
Algebraic product
Bounded sum
Bounded product

11. Fuzzy Logic Operators Relation between s-Norm and t-Norm:
Membership Function
Fuzzy Logic Operators
Relation between s-Norm and t-Norm:
de Morgan’s Law
Min-Max
Algebraic ? (Prove)
Bounded ? (Prove)

12. Fuzzy Logic
Membership Function
Homework 9
The following membership functions are given: “Temperature is low” μTl(T), “Temperature is middle” μTm(T).
For all three possible realizations of FL-Operators (Min-Max, Algebraic, Bounded), draw the membership functions of: (i) “Temperature is low” AND “Temperature is middle”; (ii) “Temperature is low” OR “Temperature is middle”.

13. Homework 9A Deadline: Monday, 20 March 2017.
Fuzzy Logic
Membership Function
Homework 9A
The following membership functions are given: “Pressure is moderate” μPm(P), “Pressure is high” μPh(P).
For all three possible realizations of FL-Operators (Min-Max, Algebraic, Bounded), draw the membership functions of: (i) “Pressure is moderate” AND “Pressure is not high”; (ii) “Pressure is not moderate” OR “Pressure is high”.
Deadline: Monday, 20 March 2017.