Fuzzy Logic Control

1. Fuzzy Logic Control

2. Feedback Control System Plant, u, y Controller, e, u Reference input, r Sensor

3. Classical Controller Design • Given mathematical model of plant (i.e. transfer function) • Given objectives of Feedback Control System • Use standard design techniques to determine controller to meet objectives (root locus, ITAE, Bode, etc.)

4. Fuzzy Logic Controllers • Use Experienced operators • Construct IF-THEN rules that describe how the operator uses his knowledge of the objectives and the measured outputs to determine inputs to the plant • Convert the IF-THEN rules into mathematics (i.e. convert human knowledge into computer knowledge)

5. Elements of FL Controller • Fuzzifier • Takes measurement data (crisp numbers) produces fuzzy sets (singleton fuzzifier, membership functions) • Rule Base or Knowledge Base • IF-THEN rules from expert operator • Inference engine • Choice of AND, OR, NOT. Determines how to interpret (implement) IF-THEN statements • De-fuzzifier • Produces single crisp output to be sent to plant

6. IF-THEN rules and Math • IF antecedent THEN conclusion • Logical IF-THEN statement • IF condition THEN action • Programming IF-THEN statement • Programming IF-THEN in FL • To the extent that the condition is true, implement the action (part of inference engine) • Collection of rules = Rule Base = Knowledge Base • Multiple rules are joined by inference engine and de-fuzzifier

7. FL implementation of functions • Any mathematical function can be interpreted as IF-THEN statements • Crisp logic interpretation of y = x2 • IF x = 1 THEN y = 1 • IF x = 2 THEN y = 4 • Etc • One rule active at a time • FL interpretation of y = x2 • IF x is close to 1 THEN y is close to 1 • IF x is close to 2 THEN y is close to 4 • Etc • Many rules active at once. How to blend all active rules?

8. FL implementation of functions • FL interpretation of y = x2 • IF x is close to 1 THEN y is close to 1 • IF x is close to 2 THEN y is close to 4 • Many rules active at once. How to blend all active rules? • Consider a number between x=1 and x=2. • The extent to which a number, x, is close to 1 is • The extent to which a number, x, is close to 2 is • The actions of the two rules needs to be blended together to get the value of the function for numbers other than 1 and 2

9. Single Antecedent Formula • Consider N fuzzy rules • IF x is in A1 THEN y is y1 • more similar rules • IF x is in AN THEN y is yn • The extent to which x is in Ak is measured by the kth membership function • The rules are blended together by

10. Examples/Assignments • FL version of y = x2, y = 3x/(x+1) • FL version of y = sin(x), y = cos(x), y = tan(x) • FL version of (you pick a function) • FL version of {(x,y)}  experimental data (thermocouple data) • Proportional controller with saturation • Choices • Membership functions

11. Compound Antecedents • Choose implementation of AND • Min or * or etc • IF x1 is A1,k AND x2 is A2,k THEN y is yk • Two input linguistic variables • Each LV has its term set

12. Compound Antecedent formula Interpret x as a vector with components

13. Examples/Assignments • Heat index formula • Chill factor index formula • PI controller • PD controller • PID controller • You choose a function

14. More complex consequents • Constant consequents • Linear consequents • Quadratic consequents • Generic function consequents

15. Re-Discussion of FL controller components • Fuzzy logic controllers convert an expert’s collection of IF-THEN rules into a mathematical function that operates similar to the expert • Fuzzifier • Rule Base • Inference Engine • Defuzzifier

16. Coding Rules in an Array • Each row in array is a rule • First N columns • N LV input variables • Entries determine term in term set of LV, hence membership function • Last few columns • Determine consequent • Programming in Sysquake/Matlab • Persistent variables

17. Examples/Assignments • Code the FL functions using arrays • Array • Super Membership function for each LV • Membership function for each term • Function • Sum • Prod • Random assignment of functions to students

18. Sysquake Code function y = muz(x,left,right) i=find(x<left); y=ones(size(i)); x = x(length(i)+1:length(x)); i=find(x<right); y =[y (right-x(i))/(right-left)]; x = x(length(i)+1:length(x)); y = [y zeros(size(x))]; return;

19. function y = mus(x,left,right) i=find(x<left); y=zeros(size(i)); x = x(length(i)+1:length(x)); i=find(x<right); y =[y (x(i)-left)/(right-left)]; x = x(length(i)+1:length(x)); y = [y ones(size(x))]; Return;

20. function y = mutri(x,left,center,right) i=find(x<left); y=zeros(size(i)); x = x(length(i)+1:length(x)); i=find(x<center); y =[y (x(i)-left)/(center-left)]; x = x(length(i)+1:length(x)); i=find(x<right); y =[y (right-x(i))/(right-center)]; x=x(length(i)+1:length(x)); y = [y zeros(size(x))]; Return;

21. function y = mu(LV,T,x) // change LV to name of LV temp = 1; humidity= 2; //For each LV, something like: If (LV == temp) N = 1; Z = 2; P = 3 //term set with 3 terms left1 = -1; right1 = 0; // term N left2 = -1 ; center2 = 0; right2 = 1; //term Z left3 = 0; right3 = 1; //term P if (T==N) y = muz(x,left1,right1);//muN if (T==Z) y = mutri(x,left2,center2,right2);//muZ if (T==P) y = mus(x,left3,right3);//muP end //For 2nd LV (similar to first) if (LV==humidity) end return;

22. Rules as Arrays function y = fuzzyfn(x) N1 = 1; Z1 = 2; P1 = 3; // term set for LV 1 N2 = 1; Z2 = 2; P2 = 3; // term set for LV 2 nLV = 2; // number of linguistic variables A = [N1 N2 10 // rule 1 N1 Z2 6 // rule 2 N1 P2 3]; // rule 3 // more rules num = 0; den = 0 for i = 1:length(A(:,1)); mu = 1; for LV = 1:nLV; mu=mu*mu(LV,A(i,LV),x(LV)); end num = num+mu*A(i,nLV+1); sum = sum+mu; //sum =1? end y = num/sum; // not needed if sum =1 return

23. Assignment • Add rules • Modify to accommodate linear consequents

24. P Controller • u = KPe • Rules like • If e = ek then u = uk

25. D controller • u =KD (d/dt)e • Rules like • If du = duk Then u = uk

26. I Controller If I = Ik then u = uk or If e = ek then u = u+uk (or du=duk)

27. PID Controller