Misconceptions about Fuzzy concepts Fuzziness is not Probability!!! How to make a decision on which method to apply – Mamdani or Sugeno?

2. Fuzziness is not Vague • we shall have a look at some propositions. • Dimitris is six feet tall – The first proposition (traditional) has a crisp truth value of either TRUE or FALSE. • He is tall – The second proposition is vague. – It does not provide sufficient information for us to make a decision, either fuzzy or crisp. – We do not know the value of the pronoun. – Is it Dimitris, John or someone else?

3. Fuzziness is not Vague • Andrei is tall – This proposition is a fuzzy proposition. – It is true to some degree depending in the context, i.e., the universe of discourse. – It might be SomeWhat True if we are referring to basketball players or it might be Very True if we are referring to horse-jockeys.

4.Fuzziness is not Multi-valued logic – The limitations of two-valued logic were recognised very early. – A number of different logic theories based on multiple values of truth have been formulated through the years. – For example, in three-valued logic three truth values have been employed. – These are TRUTH, FALSE, and UNKNOWN represented by 1, 0 and 0.5 respectively. – In 1921 the first N-valued logic was introduced. – The set of truth values Tn were assumed to be evenly divided over the closed interval [0,1]. – Fuzzy logic may be considered as an extension of multi- valued logic but they are somewhat different. – Multi-valued logic is still based on exact reasoning whereas fuzzy logic is approximate reasoning.

5.Fuzziness is not Probability!! !

6. Fuzziness is not Probability • Let X be the set of all liquids (i.e., the universe of discourse) . • Let L be a subset of X which includes all suitable for drinking liquids. A Bottle A label is marked as B The label of bottle B is marked probability of L is 0.9. membership of L is 0.9. Which one would you drink?

7. Fuzziness is not Probability • This is better explained using an example. • Let X be the set of all liquids (i.e., the universe of discourse) . • Let L be a subset of X which includes all suitable for drinking liquids. • Suppose now that you find two bottles, A and B. • The labels do not provide any clues about the contents. • Bottle A label is marked as membership of L is 0.9. • The label of bottle B is marked as probability of L is 0.9. • Given that you have to drink from the one you choose, the problem is of how to interpret the labels.

8. Fuzziness is not Probability • Well, membership of 0.9 means that the contents of A are fairly similar to perfectly potable liquids. • If, for example, a perfectly liquid is pure water then bottle A might contain, say, tonic water. • Probability of 0.9 means something completely different. • You have a 90% chance that the contents are potable and 10% chance that the contents will be unsavoury, some kind of acid maybe. • Hence, with bottle A you might drink something that is not pure but with bottle B you might drink something deadly. So choose bottle A.

9. Fuzziness is not Probability • Opening both bottles you observe beer (bottle A) and hydrochloric acid (bottle B). • The outcome of this observation is that the membership stays the same whereas the probability drops to zero. • All in all: – probability measures the likelihood that a future event will occur, – fuzzy logic measures the ambiguity of events that have already occurred. • In fact, fuzzy sets and probability exist as parts of a greater Generalized Information Theory. • This theory also includes: – Dempster-Shafer evidence theory, – possibility theory, – and so on.

10.Applications of Fuzzy concepts

11. Fuzzy inference for practical control by Mamdani The most commonly used fuzzy inference technique is the so-called Mamdani method. In 1975, Professor Ebrahim Mamdani of London University built one of the first fuzzy systems to control a steam engine and boiler combination. He applied a set of fuzzy rules supplied by experienced human operators.

12. Mamdani versus Sugeno Models • Most of our examples were for Mamdani Model. • Another famous model comes from Sugeno. • We will discuss and compare both models.Mo

13.3/18 and Soft ng 9 Sugeno fuzzy inference  Mamdani-style inference, as we have just seen, requires us to find the centroid of a two-dimensional shape by integrating across a continuously varying function. In general, this process is not computationally efficient.  Michio Sugeno suggested to use a single spike, a singleton, as the membership function of the rule consequent. A singleton, or more precisely a fuzzy singleton, is a fuzzy set with a membership function that is unity at a single particular point on the universe of discourse and zero everywhere else.

14.4/18 and Soft ng 9 Sugeno-style fuzzy inference is very similar to the Mamdani method. Sugeno changed only a rule consequent. Instead of a fuzzy set, he used a mathematical function of the input variable. The format of the Sugeno-style fuzzy rule is IF x is A AND y is B THEN z is f (x, y) where x, y and z are linguistic variables; A and B are fuzzy sets on universe of discourses X and Y, respectively; and f (x, y) is a mathematical function.

15.5/18 and Soft ng 9 The most commonly used zero-order Sugeno fuzzy model applies fuzzy rules in the following form: IF x is A AND y is B THEN z is k where k is a constant. In this case, the output of each fuzzy rule is constant. All consequent membership functions are represented by singleton spikes.

16.Sugeno-style rule evaluation 1 1 1 A3 B1 0.1 OR 0.1 0.0 (max) 0 x1 X 0 y1 Y 0 k1 Z Rule 1: IF x is A3 (0.0) OR yis B1 (0.1) THEN z is k1 (0.1) 1 1 1 0.7 A2 0.2 B2 AND 0.2 (min) 0 x1 X 0 y1 Y 0 k2 Z Rule 2: IF xis A2 (0.2) AND yis B2 (0.7) THEN z is k2 (0.2) 1 1 A1 0.5 0.5 0 x1 X 0 k3 Z Rule 3: IF xis A1 (0.5) THEN z is k3 (0.5)

17.7/18 and Soft ng 9 Sugeno-style aggregation of the rule outputs 1 1 1 1 0.5 0.5 0.2 0.1 0.2 0.1 0 k1 Z 0 k2 Z 0 k3 Z 0 k1 k2 k3 Z z isk1 (0.1) z isk2 (0.2) z isk3 (0.5) 

18.8/18 and Soft ng 9 Weighted average (WA): (k1) k1  ( k2) k2  (k3) k3 0.120  0.2 50  0.5 80 WA  65 ( k1)  ( k2)  ( k3) 0.1  0.2  0.5 Sugeno-style defuzzification 0 z1 Z Crisp Output z1

19.9/18 and Soft ng 9 How to make a decision on which method to apply – Mamdani or Sugeno?  Mamdani method is widely accepted for capturing expert knowledge. It allows us to describe the expertise in more intuitive, more human-like manner. However, Mamdani-type fuzzy inference entails a substantial computational burden.  On the other hand, Sugeno method is computationally effective and works well with optimisation and adaptive techniques, which makes it very attractive in control problems, particularly for dynamic nonlinear systems.

20.Example 4 (Mamdani Fuzzy Model) Single input single output Mamdani fuzzy model with 3 rules: If X is small then Y is small  R1 If X is medium then Y is medium  R2 Is X is large then Y is large  R3 X = input [-10, 10] Y = output [0,10]

21. Using centroid defuzzification, we obtain the following overall input- output curve 21 Overall input-output curve Single input single output antecedent & consequent MFs

22./18 Example 5 and Soft ng 9 (Mamdani Fuzzy model) Two input single-output Mamdani fuzzy model with 4 rules: If X is small & Y is small then Z is negative large If X is small & Y is large then Z is negative small If X is large & Y is small then Z is positive small If X is large & Y is large then Z is positive large

23. X = [-5, 5]; Y = [-5, 5]; Z = [-5, 5] with max- min composition & centroid defuzzification, we can determine the overall input output surface Two-input single output antecedent & consequent MFs

24./18 and Soft X = [-5, 5]; Y = [-5, 5]; Z = [-5, 5] with max- ng 9 min composition & centroid defuzzification, we can determine the overall input output surface Z Y X Overall input-output surface

25./18 and Soft ng 9 Overall input- output surface X = [-5, 5]; Y = [- 5, 5]; Z = [-5, 5] with max- min composition & centroid defuzzification, we can determine the overall input output surface

26.EXAMPLE 6 Example of Mamdani: Cement Kiln Example

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