Computational Intelligence Lecture 6:Fuzzy Rule Base and Fuzzy Inference Engine

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1 Computational Intelligence Lecture 6:Fuzzy Rule Base and Fuzzy Inference Engine Farzaneh Abdollahi Department of Electrical Engineering Amirkabir University of Technology Fall 200 arzaneh Abdollahi Computational Intelligence Lecture 6 /24

2 Fuzzy Rule Base Properties of Set of Rules Fuzzy Inference Composition Based Inference Individual-Rule Based Inference Farzaneh Abdollahi Computational Intelligence Lecture 6 2/24

3 Fuzzy Rule Base Fuzzy Rule Base X in U Fuzzifier Defuzzifier y in V Fuzzy Sets in U Fuzzy Inference Engine Fuzzy sets in V Farzaneh Abdollahi Computational Intelligence Lecture 6 3/24

4 Fuzzy Rule Base We study only the multi-input-single-output case a multioutput system can be decomposed into a collection of single-output systems. A fuzzy rule base : a set of fuzzy IF-THEN rules. It is of the fuzzy system: all other components are used to implement these rules in a reasonable and efficient manner. The Canonical Fuzzy IF-THEN Rule Ru (l) : IF x is A l and... and x n l is A l n, THEN y is B l A l i : fuzzy set in U i R Bl : Fuzzy set in V R X = (x,..., x n ) T U: Input linguistic variable y V : Output linguistic variable l =,..., M, M: # of rules in the fuzzy rule base Farzaneh Abdollahi Computational Intelligence Lecture 6 4/24

5 Most of Fuzzy Rules can be expressed by Canonical Fuzzy Rule Partial Rules: IF x is A l and... and x m is A l m, THEN y is B l, m < n It is equivalent to: IF x is A l and... and x m is A l m and x m+ is I and... and x n is I THEN y is B l I is a fuzzy set in R, µ I (x) =, x R Or Rules: IF x is A l and... and x m is A l m or x m+ is A l m+ and... and x n is A l n THEN y is B l, It is equivalent to: IF x is A l and... and x m is A l m THEN y is B l IF x m+ is A l m+ and... and x n is A l n THEN y is B l Farzaneh Abdollahi Computational Intelligence Lecture 6 5/24

6 Single fuzzy statement: y is B l, It is equivalent to: IF x is I and... and x n is I THEN y is B l Gradual Rules: The smaller the x, the bigger the y, Define S : a fuzzy set for smaller : µs (x) = +exp(5(x+2)) Define B : a fuzzy set for bigger : µ B (y) = +exp( 5(y 2)) The rule is equivalent to: IF x is S THEN y is B Non-fuzzy Rules: When µa l i and µ B l can only take values or 0 Farzaneh Abdollahi Computational Intelligence Lecture 6 6/24

7 Properties of Set of Rules Complete Fuzzy Rules x U, at least one rule (k) in fuzzy rule base s.t. µ A k i 0, for all i =,..., n Example: Consider a 2-input -output system The completer Fuzzy rule set: IF x is S and x 2 is S 2, THEN y is B IF x is S and x 2 is L 2, THEN y is B 2 IF x is M and x 2 is S 2, THEN y is B 3 IF x is M and x 2 is L 2, THEN y is B 4 IF x is L and x 2 is S 2, THEN y is B 5 IF x is L and x 2 is L 2, THEN y is B 6 If each of them is missing is will loose the completeness Farzaneh Abdollahi Computational Intelligence Lecture 6 7/24

8 Properties of Set of Rules Consistent Fuzzy Rules: there are no rules with the same IF parts but different THEN parts. Consistency is very important for nonfuzzy production rules, it causes conflicting rules For fuzzy rules: the proper inference engine and defuzzifier use average solution to resolve the conflicting results Continuous Fuzzy Rules: there is no neighboring rules whose THEN part fuzzy sets have empty intersection. I/O behavior of fuzzy system should be smooth Farzaneh Abdollahi Computational Intelligence Lecture 6 8/24

9 Fuzzy Inference Fuzzy Rule Base X in U Fuzzifier Defuzzifier y in V Fuzzy Sets in U Fuzzy Inference Engine Fuzzy sets in V Farzaneh Abdollahi Computational Intelligence Lecture 6 9/24

10 Fuzzy Inference Using fuzzy logic principles, the fuzzy IF-THEN rules in the fuzzy rule base are combined to map a fuzzy set A in U to a fuzzy set B in V. It is the brain of fuzzy system There are two methods to infer the rules. Composition based inference 2. Individual-rule based inference Composition Based Inference all rules in the fuzzy rule base are combined into a single fuzzy relation in U V It is viewed as a single fuzzy IF-THEN rule. Farzaneh Abdollahi Computational Intelligence Lecture 6 0/24

11 Composition Based Inference What is appropriate logic to combine? There are two diff. views:. Each rule is independent conditional statement operator: union For M rules, Ru (l) = A l... A l n B l Interpreted as a single fuzzy relation called Mamdani combination Q M = M l= Ru(l) µ QM (x, y) = µ Ru ()(x, y) µ Ru ()(x, y) +. is s-norm 2. The rules are strongly coupled conditional statements; all the rules must be satisfied to have impact appropriate operator is intersection Combined as a fuzzy relation called Godel combination µ QG (x, y) = M l= Ru(l) µ QG (x, y) = µ Ru ()(x, y)... µ Ru ()(x, y) is t-norm Farzaneh Abdollahi Computational Intelligence Lecture 6 /24

12 Composition Based Inference Output of fuzzy inference engine is obtained using the generalized modus ponens: µ B (y) = sup x U t[µ A (x), µ QM/G (x, y)] The computational procedure of the composition based inference:. Consider M canonical IF-THEN rules (Ru (l) : IF x is A l and... and x l n is A l n, THEN y is B l ), determine membership fcn: µ A l... µ A l n (x,..., x n ) = µ A l (x )... µ A l n (x ) for l =,..., M Farzaneh Abdollahi Computational Intelligence Lecture 6 2/24

13 Composition Based Inference Output of fuzzy inference engine is obtained using the generalized modus ponens: µ B (y) = sup x U t[µ A (x), µ QM/G (x, y)] The computational procedure of the composition based inference:. Consider M canonical IF-THEN rules (Ru (l) : IF x is A l and... and x l n is A l n, THEN y is B l ), determine membership fcn: µ A l... µ A l n (x,..., x n ) = µ A l (x )... µ A l n (x ) for l =,..., M 2. Considering A l... Al n as FP, and B l as FP2, determine µ Ru (l)(x,..., x n, y) = µ A l... A l n Bl (x,..., x n, y) for l =,..., M using Dienes, Lukasiewicz, Zadeh, Godel, or Mamdani implications Farzaneh Abdollahi Computational Intelligence Lecture 6 2/24

14 Composition Based Inference Output of fuzzy inference engine is obtained using the generalized modus ponens: µ B (y) = sup x U t[µ A (x), µ QM/G (x, y)] The computational procedure of the composition based inference:. Consider M canonical IF-THEN rules (Ru (l) : IF x is A l and... and x l n is A l n, THEN y is B l ), determine membership fcn: µ A l... µ A l n (x,..., x n ) = µ A l (x )... µ A l n (x ) for l =,..., M 2. Considering A l... Al n as FP, and B l as FP2, determine µ Ru (l)(x,..., x n, y) = µ A l... A l n Bl (x,..., x n, y) for l =,..., M using Dienes, Lukasiewicz, Zadeh, Godel, or Mamdani implications 3. Determine µ QG (x, y) = µ Ru ()(x, y)... µ Ru ()(x, y) or µ QM (x, y) = µ Ru ()(x, y) µ Ru ()(x, y) Farzaneh Abdollahi Computational Intelligence Lecture 6 2/24

15 Composition Based Inference Output of fuzzy inference engine is obtained using the generalized modus ponens: µ B (y) = sup x U t[µ A (x), µ QM/G (x, y)] The computational procedure of the composition based inference:. Consider M canonical IF-THEN rules (Ru (l) : IF x is A l and... and x l n is A l n, THEN y is B l ), determine membership fcn: µ A l... µ A l n (x,..., x n ) = µ A l (x )... µ A l n (x ) for l =,..., M 2. Considering A l... Al n as FP, and B l as FP2, determine µ Ru (l)(x,..., x n, y) = µ A l... A l n Bl (x,..., x n, y) for l =,..., M using Dienes, Lukasiewicz, Zadeh, Godel, or Mamdani implications 3. Determine µ QG (x, y) = µ Ru ()(x, y)... µ Ru ()(x, y) or µ QM (x, y) = µ Ru ()(x, y) µ Ru ()(x, y) 4. Determine output B, give input A and µ B (y) = sup x U t[µ A (x), µ QM/G (x, y)] Farzaneh Abdollahi Computational Intelligence Lecture 6 2/24

16 Individual-Rule Based Inference Each rule in the fuzzy rule base determines an output fuzzy set The output of the whole fuzzy inference engine is the combination of the M individual fuzzy sets using intersection or union for combination. The computational procedure of the individual-rule based inference:. Follow Step and 2 of composition based inference. Farzaneh Abdollahi Computational Intelligence Lecture 6 3/24

17 Individual-Rule Based Inference Each rule in the fuzzy rule base determines an output fuzzy set The output of the whole fuzzy inference engine is the combination of the M individual fuzzy sets using intersection or union for combination. The computational procedure of the individual-rule based inference:. Follow Step and 2 of composition based inference. 2. For given input fuzzy set A U, find output fuzzy set B V for each individual rule RU (l) according to the generalized modus ponens µ B l = sup x U t[µ A (x), µ Ru (l)(x, y)] for l =,..., M Farzaneh Abdollahi Computational Intelligence Lecture 6 3/24

18 Individual-Rule Based Inference Each rule in the fuzzy rule base determines an output fuzzy set The output of the whole fuzzy inference engine is the combination of the M individual fuzzy sets using intersection or union for combination. The computational procedure of the individual-rule based inference:. Follow Step and 2 of composition based inference. 2. For given input fuzzy set A U, find output fuzzy set B V for each individual rule RU (l) according to the generalized modus ponens µ B l = sup x U t[µ A (x), µ Ru (l)(x, y)] for l =,..., M 3. The output of the fuzzy inference engine is the combination of the M fuzzy sets B l,..., B M either by union µ B (y) = µ B (y) µ B M (y) or by intersection µ B (y) = µ B (y)... µ B M (y) Farzaneh Abdollahi Computational Intelligence Lecture 6 3/24

19 There are a variety of choices in the fuzzy inference engine based on Composition based inference or individual-rule based inference, and within the composition based inference, Mamdani inference or Godel inference Type of implication: Dienes-Rescher implication, Lukasiewicz implication, Zadeh implication, Godel implication, or Mamdani implications Different operations for the t-norms and s-norms Three main criteria to choose these alternatives: Intuitive appeal: The choice should make sense intuitively. For example, if an expert who believes that the rules are independent of each other, use engine with combined by union. Computational efficiency Special properties: Some choice may result in an inference engine that has special properties. Farzaneh Abdollahi Computational Intelligence Lecture 6 4/24

20 A Number of Popular Fuzzy Inference Engine Product Inference Engine: Includes: Individual rule based inference with union combination Mamdani s product implication Algebraic product for t-norms and max for all the s-norms µ B (y) = max M l= [sup x U(µ A (x) n i= µ A l (x i)µ i B l (y))] for given fuzzy set A U az input Minimum Inference Engine: Includes: Individual-rule based inference with union combination Mamdani s minimum implication min for t-norms and max for s-norms µ B (y) = max M l= [sup x U min(µ A (x), µ A l (x i ),..., µ A l n (x i ), µ B l (y))] for given fuzzy set A U az input Product and Minimum inference engines are computationally simple; they are intuitively appealing for many practical problems, especially for fuzzy control. Farzaneh Abdollahi Computational Intelligence Lecture 6 5/24

21 A Number of Popular Fuzzy Inference Engine A disadvantage of the product and minimum inference engines: if at some x U the µ A l (x i ) are very small, the obtained µ B (y) is very i small. (due to local implication) Lukasiewicz Inference Engine: Includes: Individual-rule based inference with intersection combination Lukasiewicz implication min for t-norms µ B (y) = min M l= [sup x U min(µ A (x), µ Ru (l)(x, y))] µ Ru (l)(x, y) = min(, min n i= (µ A l (x i)) + µ B (y))) i µ B (y) = min M l= [sup x U min(µ A (x), min n i= (µ A l i (x i)) + µ B (y))] Farzaneh Abdollahi Computational Intelligence Lecture 6 6/24

22 Zadeh Inference Engine: Includes: Individual rule based inference with intersection combination Zadeh implication min for t-norms. µ B (y) = min M l={sup x U min[µ A (x), max(min(µ A l i (x ),..., µ A l n (x n ), µ l B (y)), min n i=(µ A l i (x i )))]} Dienes-Rescher Inference Engine: Includes: Individual rule based inference with intersection combination Dienes-Rescher implication min for t-norms. µ B (y) = min M l={sup x U min[µ A (x), max( min n i=(µ A l i (x i )), µ l B (y))]} Farzaneh Abdollahi Computational Intelligence Lecture 6 7/24

23 Lemma: The product inference engine is unchanged if individual rule based inference with union combination is replaced by composition based inference with Mamdani combination Proof: composition based inference with Mamdani combination: µ B (y) = sup x U t[µ A (x), µ QM/G (x, y)] µ QM (x, y) for max as s-norm, product as t-norm: µ QM/G (x, y) = max M l= (µ Ru (l)(x, y)) µb (y) = sup x U [µ A (x) max M l= (µ Ru (l)(x, y))] Use Mamdani product and µ A l... µ A l n (x,..., x n ) = µ A l (x )... µ A l n (x ) µ B (y) = sup x U max M l= [µ A (x) n l= µ A (x i)µ l i B l (y)] Since max M l= and sup x U are interchangeable the above equation is equal to individual rule based inference with union combination Farzaneh Abdollahi Computational Intelligence Lecture 6 8/24

24 { if x = x If the fuzzy set A is a fuzzy singleton: µ A (x) = 0 otherwise where x is some point in U The product inference engine is µ B (y) = max M i= [ n i= µ A (x l i i )µ B l (y)] The minimum inference engine is µ B (y) = max M i= [min(µ A (x l ),..., µ A (x l n n ), µ B l (y))] The Lukasiewicz inference engine is µ B (y) = min M l=[, min n i=(µ A l i (xi )) + µ B l (y))] The Zadeh inference engine is µb (y) = min M l={max[min(µ A l i (xi ),..., µ A l n (xn ), µ l B (y)), minn i=(µ A l i (xi ))]} The Dienes-Rescher inference engine is µ B (y) = min M l={max( min n i=(µ A l i (xi )), µ l B (y))} Farzaneh Abdollahi Computational Intelligence Lecture 6 9/24

25 Graphical Minimum inference Engine Farzaneh Abdollahi Computational Intelligence Lecture 6 20/24

26 Graphical Product inference Engine Farzaneh Abdollahi Computational Intelligence Lecture 6 2/24

27 Example A fuzzy rule base: IF x is A and... and x n is A n, THEN y is B { y if y µ B (y) = 0 otherwise { if x = x A is a fuzzy singleton: µ A (x) = 0 otherwise Find µ B (y) using B P, B M, B L, B Z, B D Let min[µ A (x ),..., µ A n (xn)] = µ Ap (xp) n i= µ A i (xi ) = µ A(x ) Farzaneh Abdollahi Computational Intelligence Lecture 6 22/24

28 µ B P (y) = µ A (x )µ B (y) µ Ap (x ) > 0.5, µ Ap (x ) µ B p µ B p y y Farzaneh Abdollahi Computational Intelligence Lecture 6 23/24

29 µ B P (y) = µ A (x )µ B (y) µ B M (y) = min(µ A p (x p), µ B (y)) µ Ap (x ) > 0.5, µ Ap (x ) µ B M µ B p y µ B p µ B;M y Farzaneh Abdollahi Computational Intelligence Lecture 6 23/24

30 µ B P (y) = µ A (x )µ B (y) µ B M (y) = min(µ A p (xp), µ B (y)) µ B L(y) = min[, µ A p (xp) + µ B (y)] µ Ap (x ) > 0.5, µ Ap (x ) µ B L µ B L µ B M µ B p µ B M µ B p y y Farzaneh Abdollahi Computational Intelligence Lecture 6 23/24

31 µ B P (y) = µ A (x )µ B (y) µ B M (y) = min(µ A p (xp), µ B (y)) µ B L(y) = min[, µ A p (xp) + µ B (y)] µ B Z (y) = max{min[µ A p (xp), µ B (y), µ A p (xp)]} µ Ap (x ) > 0.5, µ Ap (x ) µ B L µ B Z µ B L µ B Z µ B M µ B p µ B M µ B p y y Farzaneh Abdollahi Computational Intelligence Lecture 6 23/24

32 µ B P (y) = µ A (x )µ B (y) µ B M (y) = min(µ A p (xp), µ B (y)) µ B L(y) = min[, µ A p (xp) + µ B (y)] µ B Z (y) = max{min[µ A p (xp), µ B (y), µ A p (xp)]} µ B D (y) = max( µ A p (xp), µ B (y)) µ Ap (x ) > 0.5, µ Ap (x ) µ B D µ µ µ B Z B L B D µ B L µ B Z µ B M µ B p µ B M µ B p y y Farzaneh Abdollahi Computational Intelligence Lecture 6 23/24

33 Example Cont d µ Ap (x ) < 0.5 µ B P, µ B M is very small; µ B L(y),µ B Z (y),µ B D (y) is very large Product and Minimum inf. eng. are similar; Zadeh, Dienes, Lukasiewicz inf. eng. are similar Lukasiewicz inf. eng.: Largest output; Product inf. eng.: Smallest output Farzaneh Abdollahi Computational Intelligence Lecture 6 24/24