24-07-2012, 02:31 PM
FUZZY LOGIC
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4.1 INTRODUCTION
Normally in logic we have a series of statement which are either true or false, yes or no, 0 or 1.In this true or false. In this context, the statement „the temperature is 25 degree Celsius‟ is an objective one and is either true or false. However, for many situations the answer is more like Em, not sure, may be, that depends and so on. For examples, on a pleasant summer’s day the statement „the temperature is too high‟ is neither true nor false. The statement is a qualitative one-it represent an opinion rather than an objective fact. For example, it needs to be a bright sunny day on the beach for me to feel warm. On the other hand, I could mention some visiting scientists at control systems principles who feel comfortable in a snow storm on top of a mountain. Do you see what I mean? There is no certainty to the situation-it depends upon the context.
Fuzzy logic deals with uncertainty in engineering by attaching degrees of certainty to the answer to a logical question. Why should this be useful? The answer is commercial and practical. Commercially, fuzzy logic has been used with great success to control machines and consumer product. In the right applications fuzzy logic systems are simple to design, and can be understood and implemented by non-specialists in control theory. In most cases some with an intermediate technical background can design a fuzzy logic controller. The control system will not be optimal but it can be acceptable. Control engineers also use it in applications where the on board computing is very limited and adequate control is enough.
4.2 CRISP SET VS FUZZY SET
In short, for a crisp set (subset) elements of the set definitely do belong to the set, while in a fuzzy set (subset) elements of the set have a degree of membership in the set. To make things clearer:
Suppose we have a reference set X= {x_1,} and a subset Y= {y_1,} of X. If Y represents a crisp subset of X, then for all x_and belonging to X, either belongs to X or Y or does not belong to Y. We can write this by assigning a function C which takes each member of X to 1 if it belongs to Y, and 0 if it does not belong to Y. E. G. Suppose we have the set {1, 2, 3, 4, 5}. For the crisp subset {1, 2, 4} we could write this in terms of a function C which takes 1 to 1, 2 to 1, 3 to 0, 4 to 1, and 5 to 0, or we can write {(1, 1), (2, 1), (3, 0), (4, 1), (5, 1)}.
For a fuzzy subset F of a reference set X the elements of F may belong to F to a degree in between 0 and 1 (as well as may belong to F to degree 0 or 1). We can write this by assigning a function M which takes each member of X to a number in the interval of real numbers from 0 to 1, [0, and 1] to represent its degree of membership. Here "larger" numbers represent a greater degree of membership in the fuzzy subset F. For example, for the reference set {1, 2, 3, 4, 5} we could have a function M which takes 1to .4, 2 to 1, 3to .6, 4 to .2, and 5 to 0, or {(1, .4), (2, 1), (3, .6), (4, .2), (5, 0)}, with 3 having a greater degree of membership in F than 4 does, since .6>.2.
4.3 REAL TIME EXAMPLE
Fuzzy set theory defines fuzzy operators on fuzzy sets. The problem in applying this is that the appropriate fuzzy operator may not be known. For this reason, fuzzy logic usually uses IF-THEN rules, or constructs that are equivalent, such as fuzzy associative matrices.
There is no "ELSE" – all of the rules are evaluated, because the temperature might be "cold" and "normal" at the same time to different degrees.
The AND, OR, and NOT operators of Boolean logic exist in fuzzy logic, usually defined as the minimum, maximum, and complement; when they are defined this way, they are called the Zadeh operators. So for the fuzzy variables x and y:
There are also other operators, more linguistic in nature, called hedges that can be applied. These are generally adverbs such as "very", or "somewhat", which modify the meaning of a set using a mathematical formula.
In this context, FL is a problem-solving control system methodology that lends itself to implementation in systems ranging from simple, small, embedded micro-controllers to large, networked, multi-channel PC or workstation-based data acquisition and control systems. It can be implemented in hardware, software, or a combination of both. FL provides a simple way to arrive at a definite conclusion based upon vague, ambiguous, imprecise, noisy, or missing input information. FL's approach to control problems mimics how a person would make decisions, only much faster.
4.4 FUZZY CONCEPT
Fuzzy logic is the comprehensive form of classical logic. In this chapter classical logic and fuzzy logic are discussed and between them analyzed. Fuzzy logic is the superset of classical logic with the introduction of "degree of membership." The introduction of degree of membership allows the input to interpolate between the crisp set. The operators in both logic are similar except that their interpretation differs.
4.5 FUZZIFICATION
Fuzzification is the process of changing a real scalar value into a fuzzy value. This is achieved with the different types of fuzzifiers. There are generally three types of fuzzifiers, which are used for the fuzzification process; theyare
1. Singleton fuzzifier,
2. Gaussian fuzzifier, and
3. Trapezoidal or triangular fuzzifier.
4.6 DE- FUZZIFICATION
Fuzzy logic is a rule-based system written in the form of horn clauses (i.e., if- then rules). These rules are stored in the knowledge base of the system. The input to the fuzzy system is a scalar value that is fuzzified. The set of rules is applied to the fuzzified input. The output of each rule is fuzzy. These fuzzy outputs need to be converted into a scalar output quantity so that the nature of the action to be performed can be determined byte system.
The process of converting the fuzzy output is called defuzzification. Before an output is defuzzified allthe fuzzyoutputsof the system are aggregated with an union operator
4.7 FUZZY RULE
Fuzzy systems are built to replace the human expert with a machine using the logic a human would use to perform the tasks. Suppose we ask someone how hot it is today. He maytellus that it is hot, moderately hot or cold. He cannot tellus the exact temperature. Unlike classical logic which can only interpret the crisp set such as hot or cold, fuzzy logic has the capability to interpret the natural language. Thus, fuzzy logic can make human-like interpretations and is a veryusefultool in artificial intelligence, machine learning and automation.
The concept of linguistic variable was introduced to process the natural language. The linguistic variable discussed later in Example 2.8 is temperature. The linguistic variable can take the verbal values such as hot, moderately hot or cold. The terms temperature is hot and temperature is cold and temperature is moderate are known as fuzzy propositions.
FUZZY PROPOSITION
Afuzzyproposition can be an atomic or compound sentence. For example
"Temperature is hot" is an atomic fuzzy proposition.
"Temperature is hot and humidity is low" is a compound fuzzy proposition.
Compound fuzzy relations are expressed with fuzzy connectives such as and, or and complement
The fuzzy rules are written as
If <fuzzy proposition> then <fuzzy proposition> the fuzzy proposition can be atomic or compound.
Method of Implication
The If-Then rules can be interpreted in classical logic byte implication operators. Suppose there is a statement such as "If aThen b", then the classical set represents this by a b. The truth table for this rule can be given as
The above equivalence can easily be shown with the above truth table.
As discussed earlier, the If-Then rules for fuzzy logic can be written as If <fuzzy proposition> Then <fuzzy proposition>. The propositional variables a and b are replaced by fuzzy propositions, and the implication can be replaced by fuzzy union, fuzzy intersection and fuzzy complement. There are many fuzzy implications. Only the two most important fuzzy implications are discussed here.