2. Homework 8 Min Max “Temperature is low” AND “Temperature is middle”
Fuzzy Logic
Membership Function
Homework 8
Min
“Temperature is low” AND “Temperature is middle”
Max
“Temperature is low” OR “Temperature is middle”

3. Homework 8 Algebraic product Algebraic sum
Fuzzy Logic
Membership Function
Homework 8
Algebraic product
“Temperature is low” AND “Temperature is middle”
Algebraic sum
“Temperature is low” OR “Temperature is middle”

4. Homework 8 Bounded product Bounded sum
Fuzzy Logic
Membership Function
Homework 8
Bounded product
“Temperature is low” AND “Temperature is middle”
Bounded sum
“Temperature is low” OR “Temperature is middle”

5. Further Fuzzy Set Operations
Fuzzy Logic
Fuzzy Control
Further Fuzzy Set Operations
Dilation
Concentration

6. Fuzzy Logic
Fuzzy Control
Fuzzy Control Loop

7. Fuzzy Logic
Fuzzy Control
Fuzzy Inference
Prior to fuzzy control, the followings must be defined:
Fuzzy membership functions
Fuzzy logic operators
Fuzzy rules, including fuzzy linguistic value and linguistic variable
The processing steps in a fuzzy control include:
Fuzzification
Implication / Inference Core
Accumulation
Defuzzification

8. Fuzzy Rules Example of a fuzzy rule while “Driving a Car”:
Fuzzy Logic
Fuzzy Control
Fuzzy Rules
Example of a fuzzy rule while “Driving a Car”:
“IF the distance to the car in front is small, AND the distance is decreasing slowly, THEN decelerate quite big”
The question that arises:
Given a certain distance and a certain change of distance, what (crisp) value of acceleration should we select?

9. Definition of Fuzzy Membership Functions
Fuzzy Logic
Fuzzy Control
Definition of Fuzzy Membership Functions
v. small
Distance
small
perfect
big
v. big
moderate
Distance decrease
slow
fast
very fast
v. slow
Acceleration
–small
zero
+small
+big
–big

10. Observation/ measurement
Fuzzy Logic
Fuzzy Control
Fuzzification
Observation/ measurement
Observation/
measurement
v. small
Distance
small
perfect
big
v. big
moderate
Distance decrease
slow
fast
very fast
v. slow
Acceleration
–small
zero
+small
+big
–big
Distance between small and perfect
Distance decrease can be moderate or fast
What acceleration should be applied?

11. Observation/ measurement
Fuzzy Logic
Fuzzy Control
Implication of Rules
Observation/ measurement
v. small
Distance
small
perfect
big
v. big
Acceleration
–small
zero
+small
+big
–big
0.55
RULE 1: IF distance is small THEN decelerate small
Inference core: Clipping
Clip the fuzzy membership function of “–small” at the height given by the premises (0.55). Later, the clipped area will be considered in the final decision

12. Observation/ measurement
Fuzzy Logic
Fuzzy Control
Implication of Rules
Observation/ measurement
moderate
Distance decrease
slow
fast
very fast
v. slow
Acceleration
–small
zero
+small
+big
–big
0.7
RULE 2: IF distance decrease is moderate THEN keep the speed
Inference core: Clipping
Clip the fuzzy membership function of “zero” at the height given by the premises (0.7). Later, the clipped area will be considered in the final decision

13. Fuzzy Logic
Fuzzy Control
Accumulation
From each rule, a clipped area is obtained. But, in the end only one single output is wanted. How do we make a final decision?
–small
zero
+small
+big
–big
Acceleration
Rule 1
Rule 2
In the accumulation (aggregation) step, all clipped areas are merged into one merged area (taking the union).
Rules with high premises will contribute large clipped area to the merged area. These rules will “pull” that merged area towards their own central value.

14. Fuzzy Logic
Fuzzy Control
Defuzzification
–small
zero
+small
+big
–big
Acceleration
Center of gravity
Crisp value
In this last step, the returned value is the wanted acceleration.
Out of many possible ways, the center of gravity is the commonly used method in defuzzification.

15. Fuzzy Logic
Fuzzy Control
Inference Core
There are two approaches that can be used for inference core:
acceleration
1. Clipping approach:
0.55
Min-Operator
Fuzzification value
Membership function
acceleration
2. Scaling approach:
0.55
Algebraic Product

16. Review on Center of Gravity
Fuzzy Logic
Fuzzy Control
Review on Center of Gravity
Rectangle
Triangle

17. Review on Center of Gravity
Fuzzy Logic
Fuzzy Control
Review on Center of Gravity
Isosceles Trapezoid
Trapezoid

18. Summary of Fuzzy Control
Fuzzy Logic
Fuzzy Control
Summary of Fuzzy Control
Fuzzify inputs, determine the degree of membership for all terms in the premise.
Apply fuzzy logic operators, if there are multiple terms in the premise (min-max, algebraic, bounded).
Apply inference core (clipping, scaling, etc.)
Accumulate all outputs (union operation i.e. max, sum, etc.)
Defuzzify (center of gravity of the merged outputs, max-method, modified center of gravity, height method, etc)

19. Limitations of Fuzzy Control
Fuzzy Logic
Fuzzy Control
Limitations of Fuzzy Control
Definition and fine-tuning of membership functions need experience (covered range, number of MFs, shape).
Defuzzification may produce undesired results (needs redefinition of membership functions).

20. Acceleration adj. [m/s2]
Fuzzy Logic
Fuzzy Control
Homework 9
A fuzzy controller is to be used in driving a car. The fuzzy membership functions for the two inputs and one output are defined as below.
v. small
Distance to next car [m]
small
perfect
big
v. big 1
Speed change [m/s2]
constant
growing
declining
–small
zero
+small
+big
–big
Acceleration adj. [m/s2]
–2 –
–10 –

21. Fuzzy Logic
Fuzzy Control
Homework 9 (Cont.)
A fuzzy controller is to be used in driving a car. The fuzzy rules are given as follows.
Rule 1: IF distance is small AND speed is declining, THEN maintain acceleration.
Rule 2: IF distance is small AND speed is constant, THEN acceleration adjustment negative small.
Rule 3: IF distance is perfect AND speed is declining, THEN acceleration adjustment positive small.
Rule 4: IF distance is perfect AND speed is constant, THEN maintain acceleration.

22. Fuzzy Logic
Fuzzy Control
Homework 9 (Cont.)
Using Min-Max as fuzzy operators, clipping as inference core, union operator as accumulator, and center of gravity method as defuzzifier, find the output of the controller if the measurements confirms that distance to next car is 13 m and the speed is increasing by 2.5 m/s2.

23. Fuzzy Logic
Fuzzy Control
Homework 9A
A driver of an open-air car determine how fast he drives based on the air temperature and the sky conditions. The corresponding fuzzy membership functions can be seen here.

24. Fuzzy Logic
Fuzzy Control
Homework 9A (Cont.)
After years of experience, he summarizes his personal driving rules as follows:
Rule 1: IF it is sunny AND warm, THEN drive fast.
Rule 2: IF it is partly cloudy AND hot, THEN drive slow.
Rule 3: IF it is partly cloudy, THEN drive fast.
You are now assigned to design a fuzzy control with the following requirements:
Fuzzy logic operators: algebraic sum / product
Inference core: scaling
Accumulator: union operator
Defuzzification: center of gravity method
The speed limit is 120 km/h. How fast will the driver go if in one day the temperature is 65 °F and the cloud cover is 25 %?