1. Introduction to Fuzzy Logic
Fuzzy Inference
Shadi T. Kalat
2/2
05/27/2016

2. Fuzzy Sets
Fuzzy sets
Crisp sets:

3. Membership Functions Triangular membership function
Trapezoidal membership function
Membership Grade
Membership Grade
Gaussian membership function
Generalized bell membership function
Membership Grade
Membership Grade

4. Fuzzy Rules Assume A is a fuzzy member of X 𝑦=𝑓(𝑥) 𝑥=𝐴 𝑦=𝑓 𝐴 =𝐵
𝜇 𝐵 𝑦 = 𝜇 𝐴 (𝑥)
𝜇 𝐵 𝑦 = 𝜇 𝐴 ( 𝑓 −1 (𝑦))
𝜇 𝐵 𝑦 ≜𝑚𝑎𝑥 𝜇 𝐴 (𝑥)
𝑥= 𝑓 −1 (𝑦)
𝑦=𝑓(𝑥) 𝑥=𝐴 𝑦=𝑓 𝐴 =𝐵
𝑆𝑚𝑎𝑙𝑙=
𝑦= 𝑥 2
𝑆𝑚𝑎𝑙 𝑙 2 =

5. Fuzzy Relations Max-Min Composition Max-Dot Product
𝑣∈𝑉, 𝑥∈𝑋, 𝑦∈𝑌
𝜇 𝑅 𝑥,𝑦 =𝑚𝑎𝑥 𝜇 𝑅1 𝑥,𝑣 , 𝜇 𝑅2 𝑣,𝑦
𝑣∈𝑉, 𝑥∈𝑋, 𝑦∈𝑌

6. IF THEN rules If 𝑒 is 𝑒 1 and 𝑒 is 𝑒 1 Then 𝑢 is 𝑢 1 𝜇( 𝑢 1 ) 𝜇( 𝑒 1 )
𝜇( 𝑒 1 )
𝜇 1
𝜇 2 𝑒 𝑒 𝑢
𝑒 1
𝑒 1
𝑢 1 =?
T-norm

7. Fuzzy and Approximate Reasoning
Inference of a (possible) conclusion from a set of premises
Fuzzy Reasoning
Approximate Reasoning
This tomato is red
If a tomato is red, then it is ripe
This tomato is ripe
This tomato is very red
If a tomato is red, then it is ripe
This tomato is very ripe

8. Fuzzy Inference Zadeh/Mamdani Inference: min⁡( 𝜇 1 , 𝜇 2 )
Larsen Product: 𝜇 1 × 𝜇 2
Bounded Product: max⁡( 𝜇 1 + 𝜇 2 −1,0)
Drastic Product: 𝜇 1 𝑓𝑜𝑟 𝜇 2 =1 𝜇 2 𝑓𝑜𝑟 𝜇 1 =1 0 𝑓𝑜𝑟 𝜇 1,2 <1

9. Approximate Inference (SISR)
𝜇 𝐵 ′ 𝑦 =max⁡( 𝜇 𝐴 ′ 𝑥 , 𝜇 𝐴 𝑥 )∧ 𝜇 𝐵 (𝑦) 𝑤

10. Approximate Inference (MISR)
𝜇 𝐶 ′ 𝑧 =max⁡( 𝜇 𝐴 ′ 𝑥 , 𝜇 𝐴 𝑥 )∧ max⁡( 𝜇 𝐵 ′ 𝑦 , 𝜇 𝐵 𝑦 )𝜇 𝐶 (𝑧)
𝑤 1
𝑤 2
𝑤= 𝑤 1 ∧ 𝑤 2
Firing Strength

11. Fuzzy Inference System
A Fuzzy Inference System (FIS) is a way of mapping an input space to an output space using fuzzy logic
FIS uses a collection of fuzzy membership functions and rules, instead of Boolean logic, to reason about data.
The rules in FIS (sometimes may be called as fuzzy expert system) are fuzzy production rules of the form:
if p then q, where p and q are fuzzy statements.
For example, in a fuzzy rule
if x is low and y is high then z is medium.
Here x is low; y is high; z is medium are fuzzy statements; x and y are input variables; z is an output variable, low, high, and medium are fuzzy sets.

12. Fuzzy Control

13. Mamdani Fuzzy Inference System

14. Sugeno FIS
Output∈𝑍

15. Tsukamoto FIS

16. Example Automotive Speed Controller 3 inputs: speed (5 levels)
acceleration (3 levels)
distance to destination (3 levels)
1 output:
power (fuel flow to engine)
Set of rules to determine output based on input values

17. Example

18. Example
Example Rules
IF speed is TOO SLOW and acceleration is DECELERATING,
THEN INCREASE POWER GREATLY
IF speed is SLOW and acceleration is DECREASING,
THEN INCREASE POWER SLIGHTLY
IF distance is CLOSE,
THEN DECREASE POWER SLIGHTLY

19. Example Output Determination
Degree of membership in an output fuzzy set now represents each fuzzy action.
Fuzzy actions are combined to form a system output.

20. Example
Steps
Fuzzification: determines an input's degree of membership in overlapping sets.
Rules: determine outputs based on inputs and rules.
Combination/Defuzzification: combine all fuzzy actions into a single fuzzy action and transform the single fuzzy action into a crisp, executable system output. May use centroid of weighted sets.

21. Example Defuzzification Max Membership Weighted Average Centroid
a b z  .9 .5
z* z  1
Max Membership
Weighted Average
z* z  1
a z* b z  1
Centroid
Mean max

22. Summary
Note there would be a total of 95 different rules for all combinations of inputs of 1, 2, or 3 at a time.
In practice, a system won't require all of the rules.
System could be improved by adding or changing rules and by adjusting set boundaries.
Doesn't require an understanding of process but any knowledge will help formulate rules.
Complicated systems may require several iterations to find a set of rules resulting in a stable system.