1. Intelligent Control Methods Lecture 11: Fuzzy control 2 Slovak University of Technology Faculty of Material Science and Technology in Trnava

2. 2 Fuzzy system uses rules of the type: rule ::= If then antecedent ::= {and|or } consequent ::= ::= is Example: if (v is small) and (d is medium) then (F is small)

3. 3 Linguistic variable: Variable, which has a verbal value. Example: (speed – small, medium, big) Definition: Linguistic variable is a tetrad (x, L x, U x, M x ), where: x – the name of the variable (speed, v, e) L x – set of verbal values (small, medium, big) U x – definition scope (possible values of the variable (physical interval of the speed expressed by numbers, e.g. 0 – 120)) M x – function, which expresses the verbal values by fuzzy sets in definition scope

4. 4 Linguistic variable example: v – the name of linguistic variable „speed“ L x – the set of verbal values (N – nearly null, M - mini, S - medium, V - big) U x – universe (definition scope) M x – function, which maps the verbal values in universe by fuzzy sets, e.g.:

5. 5 Combination of assertions: ::= is Konjunction: Let p and q are atomic fuzzy assertions p: „x is A“ and q: „y is B“, where A and B are defined in the same universe. The value of assertion (x is A) and (y is B) is given by konjunction of fuzzy sets A and B, i.e. by value  A  B = min (  A,  B ). Disjunction: The value of assertion (x is A) or (y is B) is given by disjunction of fuzzy sets A and B, i.e. by value  A  B = max (  A,  B ). Negation: The negation of the assertion „x is A“ (i.e. „x is not A“) is given by complement A’ of fuzzy set A, i.e. by value  A’ = 1-  A.

6. 6 Combination of assertions in different universes (1): Let a and b are linguistic variables defined in universes U a, U b. Let p and q are assertions „a is F 1 “ and „b is F 2 “. The cylindric extension of sets F 1 a F 2 into cartesian product U a x U b is necessary before the assertion combination.

7. 7 Combination of assertions in different universes (2): Let a and b are linguistic variables defined in definition scopes U a, U b. Let p and q are assertions „a is F 1 “ and „b is F 2 “. Conjunction: Combined assertion (a is F 1 ) and (b is F 2 ) is given by fuzzy relation defined on cartesian product U a x U b

8. 8 Combination of assertions in different universes (3): Let a and b are linguistic variables defined in universes U a, U b. Let p and q are assertions „a is F 1 “ and „b is F 2 “. Disjunction: Combined assertion (a is F 1 ) or (b is F 2 ) is given by fuzzy relation defined on cartesian product U a x U b

9. 9 Fuzzy implication: Fuzzy implication is a construction if (fuzzy assertion) then (fuzzy assertion), where (fuzzy assertion) is atomic or combined one. „if (a is F 1 ) then (b is F 2 )“ Boolean implication:p  q = (not p) or q ((not p) is given as 1-p, or is calculated as max after cylindric extension)

10. 10 Fuzzy implication (2): Boolean implication is not used. It is replaced by implication by Lukasziewicz, Zadeh, Larsen and by others. The most used implication is expression, which is not an implication (it is only called implication) – implication by Mandami. p  q = p and q (!!!)

11. 11 Implication example „if (a is F 1 ) then (b is F 2 )“ F 1 = 0,1/a 1 + 0,4/a 2 +0,7/a 3 + 1,0/a 4 F 2 = 0,2/b 1 + 0,5/b 2 +0,9/b 3 b1b1 b2b2 b3b3 a1a1 0,1 a2a2 0,4 a3a3 0,7 a4a4 1,0 b1b1 b2b2 b3b3 a1a1 0,9 a2a2 0,6 a3a3 0,3 a4a4 000 b1b1 b2b2 b3b3 a1a1 0,20,50,9 a2a2 0,20,50,9 a3a3 0,20,50,9 a4a4 0,20,50,9 ce(F 1 ) ce(F 1 ’) ce(F 2 ) Boolean implication: ce(F 1 ’)  ce(F 2 ): b1b1 b2b2 b3b3 a1a1 0,9 a2a2 0,6 0,9 a3a3 0,30,50,9 a4a4 0,20,50,9 Mandami implication: ce(F 1 )  ce(F 2 ): b1b1 b2b2 b3b3 a1a1 0,1 a2a2 0,20,4 a3a3 0,20,50,7 a4a4 0,20,50,9

12. 12 Valuation of list of rules: k. (one) rule: Value of all rules: kk kk

13. 13 Fuzzy systems Systems, which variables (input, state, output) are defined by linguistic values (by fuzzy sets) Structure of (technical) fuzzy system: Fuzzificati- on modul Inference engine and rules base Defuzzica- tion modul Database Crisp Fuzzy Fuzzy Crisp Input valuesinput values output values output values

14. 14 Database and base of rules: Database consists of data about fuzzy sets of all fuzzy variables (the definition scope, the form given by membership function). Base of rules consists of inference rules (in the form of Mandami fuzzy implications).

15. 15 Fuzzification: It transforms the crisp value of variable into fuzzy one.  F (x) 1 F x x = 1.614  F (x) = 0.6 NB NM ZO PMPB v = -3.67 NB0.7 NM0.3 ZO0 PM0 PB0

16. 16 Inference engine: It evaluates the set of rules. The result is a fuzzy set.

17. 17 Inference engine on example of fuzzy PI-controller: IF e is A e AND  e is A  e THEN  u is B  u E.g.: IF e is PS AND  e is PM THEN  u is PB NB NMNSZO NB NMNSZ0PS NB NMNSZ0PSPM NBNMNSZ0PSPMPB NMNSZ0PSPMPB NSZ0PSPMPB Z0PSPMPB  e NB NM NS ZO PS PM PB NB NM NS e ZO PS PM PB

18. 18 Inference engine on example of fuzzy PI- controller (2): e = 0.024  e = 0.016 e  e NB00 NM00 NS00 ZO0,40.3 PS0.60.7 PM00 PB00 NB NM NS ZO PS PM PB e = 0.024 -0,20,2 e Let U e = U  e = U  u = NB NM NS ZO PS PM PB  e = 0.016 -0,1 0,1  e NB NM NS ZO PS PM PB -5 5  u

19. 19 Inference engine on example of fuzzy PI- controller (3): e = 0.024  e = 0.016 e  e NB00 NM00 NS00 ZO0,40.3 PS0.60.7 PM00 PB00 NB NMNSZO NB NMNSZ0PS NB NMNSZ0PSPM NBNMNSZ0PSPMPB NMNSZ0PSPMPB NSZ0PSPMPB Z0PSPMPB  e NB NM NS ZO PS PM PB NB NM NS e ZO PS PM PB 4 active rules (from 49): IF e is ZO AND  e is ZO THEN  u is ZO (1) IF e is ZO AND  e is PS THEN  u is PS (2) IF e is PS AND  e is ZO THEN  u is PS (3) IF e is PS AND  e is PS THEN  u is PM (4)

20. 20 Inference engine on example of fuzzy PI- controller (4): Before the inference: The crisp values of input variables (e,  e) are fuzzificated. Inference = Evaluation of active rules: 1.Each rule is evaluated independently (there are allways two fuzzy assertions connected by AND in the antecedent, the Mandami implication (min) is used) 2.Partiall results from active rules are aggregated by operation or (i.e. max is used, see slade 12). Fuzzy set is a result. After the inference: The obtained fuzzy set is defuzzificated to crisp number.

21. 21 Inference engine on example of fuzzy PI- controller (5): NB NM NS ZO PS PM PB -0,2 e=0.024 0.2 e NB NM NS ZO PS PM PB -0,1  e=0.016 0.1  e NB NM NS ZO PS PM PB -5 5  u IF e is ZO AND  e is ZO THEN  u is ZO 0.4 0.3 MIN 0.3 NB NM NS ZO PS PM PB -0,2 e=0.024 0.2 e NB NM NS ZO PS PM PB -0,1  e=0.016 0.1  e NB NM NS ZO PS PM PB -5 5  u IF e is ZO AND  e is PS THEN  u is PS 0.4 0.7 MIN 0.4

22. 22 Inference engine on example of fuzzy PI- controller (6): NB NM NS ZO PS PM PB -0,2 e=0.024 0.2 e NB NM NS ZO PS PM PB -0,1  e=0.016 0.1  e NB NM NS ZO PS PM PB -5 5  u IF e is PS AND  e is ZO THEN  u is PS 0.6 0.3 MIN 0.3 NB NM NS ZO PS PM PB -0,2 e=0.024 0.2 e NB NM NS ZO PS PM PB -0,1  e=0.016 0.1  e NB NM NS ZO PS PM PB -5 5  u IF e is PS AND  e is PS THEN  u is PM 0.6 0.7 MIN 0.6

23. 23 Inference engine on example of fuzzy PI- controller (7): NB NM NS ZO PS PM PB -5 5  u Resultant fuzzy set: 1 0.7 0.6 0.4 0.3 0

24. 24 Defuzzification: The result of inference is a fuzzy set. The goal of defuzzification is to obtain a crisp value from this fuzzy set. There are more methods of defuzzification. Centre of Area (centre of Gravity) - COA (COG): y y*

25. 25 Defuzzification (2): Center of Sum (COS), reflects the overlapped areas: y y*y*

26. 26 Defuzzification (3): FoM, SoM (First of Maximum, Smallest of Maximum): y* = inf {y  Y/  F (y) = hgt(F)} y y* hgt(F) LoM (Last of Maximum, Largest of Maximum): y* = sup {y  Y/  F (y) = hgt(F)} MoM (Middle of Maxima): y* = (y 1 +y 2 )/2 FoMMoMLoM