1. Fuzzy systems

2. Calculate the degree of matching Fuzzy inference engine Defuzzification module Fuzzy rule base General scheme of a fuzzy system

3. Linguistic rules: IF x = A THEN y = B A is the rule antecedent, B is the rule consequent Example: „IF traffic is heavy in this direction THEN keep the green light longer” Fuzzy rule base relation R containing two fuzzy rules A 1 → B 1, A 2 → B 2 (R 1, R 2 ) If x = A then y = B "fuzzy point" A×B If x = A i then y = B i i = 1,...,r "fuzzy graph" Fuzzy rule = fuzzy relation (R i ) Fuzzy rule base = fuzzy relation (R), is the union (s-norm) of the fuzzy rule relations R i :

4. Fuzzy rule base relation: more dimensional case:

5. Linguistic variables and their representations Linguistic variable (linguistic term) defined by Zadeh: "By a linguistic variable we mean a variable whose values are words or sentences in a natural or artificial language. For example, Age is a linguistic variable if its values are linguistic rather than numerical, i.e., young, not young, very young, quite young, old, not very old and not very young, etc., rather than 20, 21, 22, 23,..."

6. Frame of Cognition (fuzzy partition) Partition A={A i } "covers" the universe X; i.e. each element of this universe is assigned to at least one granule with a nonzero degree of membership. Thus:  > 0 denotes the level of "coverage" of X. The fuzzy partition (frame of cognition)  -covers the universe of discourse X

7. Ruspini-partition sup(supp(A i (x)))=inf(core(A i+1 (x))) sup(core(A i (x)))= inf(supp(A i+1 (x)))

8. Boolean partition A induced by the fuzzy partition:

9. Specificity of fuzzy partitions Fuzzy partition A* is more specific than A if all the elements of A* are more specific (e.g. in terms of their specificity measure) than the elements of A. Then, the number of elements of A* is greater than the number of linguistic terms in A. For instance, the fuzzy partition: A = { Negative, Zero, Positive} is less specific than the fuzzy partition A* containing seven items:

10. Specificity of fuzzy partitions Fuzzy Partition A containing three linguistic terms Fuzzy Partition A* containing seven linguistic terms

11. Fuzzy inference mechanism (Mamdani) If x 1 = A 1,i and x 2 = A 2,i and...and x n = A n,i then y = B i The weighting factor w ji characterizes, how far the input x j corresponds to the rule antecedent fuzzy set A j,i in one dimension The weighting factor w i characterizes, how far the input x fulfils to the antecedents of the rule R i.

12. Conclusion The conclusion of rule R i for a given x observation is y i

13. The whole inference

14. Compositional Rule of Inference

15. Takagi-Sugeno method If x 1 = A 1,i and x 2 = A 2,i and...and x n = A n,i then y i = f i (x 1,x 2,...,x n ) where w i is the weighting factor, the level of the firing of the rule R i, similarly to the Mamdani method

16. Defuzzification methods Center of Gravity Method (COG)

17. Defuzzification methods Center of Sums Method (COS)

18. Defuzzification methods Mean of Maxima Method (MOM)

19. Fuzzy systems: an example Fuzzy systems operate on fuzzy rules: IF temperature is COLD THEN motor_speed is LOW IF temperature is WARM THEN motor_speed is MEDIUM IF temperature is HOT THEN motor_speed is HIGH TEMPERATUREMOTOR_SPEED

20. Inference mechanism (Mamdani) Temperature = 55Motor Speed Motor Speed = 43.6 RULE 1 RULE 2 RULE 3