|Fuzzy Logic Toolbox|
Fuzzy sets and fuzzy operators are the subjects and verbs of fuzzy logic. These if-then rule statements are used to formulate the conditional statements that comprise fuzzy logic.
A single fuzzy if-then rule assumes the form
where A and B are linguistic values defined by fuzzy sets on the ranges (universes of discourse) X and Y, respectively. The if-part of the rule "x is A" is called the antecedent or premise, while the then-part of the rule "y is B" is called the consequent or conclusion. An example of such a rule might be
Note that good is represented as a number between 0 and 1, and so the antecedent is an interpretation that returns a single number between 0 and 1. On the other hand, average is represented as a fuzzy set, and so the consequent is an assignment that assigns the entire fuzzy set B to the output variable y. In the if-then rule, the word "is" gets used in two entirely different ways depending on whether it appears in the antecedent or the consequent. In MATLAB terms, this is the distinction between a relational test using "==" and a variable assignment using the "=" symbol. A less confusing way of writing the rule would be
In general, the input to an if-then rule is the current value for the input variable (in this case, service) and the output is an entire fuzzy set (in this case, average). This set will later be defuzzified, assigning one value to the output. The concept of defuzzification is described in the next section.
Interpreting an if-then rule involves distinct parts: first evaluating the antecedent (which involves fuzzifying the input and applying any necessary fuzzy operators) and second applying that result to the consequent (known as implication). In the case of two-valued or binary logic, if-then rules don't present much difficulty. If the premise is true, then the conclusion is true. If we relax the restrictions of two-valued logic and let the antecedent be a fuzzy statement, how does this reflect on the conclusion? The answer is a simple one. if the antecedent is true to some degree of membership, then the consequent is also true to that same degree. In other words
p q (p and q are either both true or both false)
The antecedent of a rule can have multiple parts.
in which case all parts of the antecedent are calculated simultaneously and resolved to a single number using the logical operators described in the preceding section. The consequent of a rule can also have multiple parts.
in which case all consequents are affected equally by the result of the antecedent. How is the consequent affected by the antecedent? The consequent specifies a fuzzy set be assigned to the output. The implication function then modifies that fuzzy set to the degree specified by the antecedent. The most common ways to modify the output fuzzy set are truncation using the
min function (where the fuzzy set is "chopped off" as shown below) or scaling using the
prod function (where the output fuzzy set is "squashed"). Both are supported by the Fuzzy Logic Toolbox, but we use truncation for the examples in this section.
Summary of If-Then Rules
Interpreting if-then rules is a three-part process. This process is explained in detail in the next section:
In general, one rule by itself doesn't do much good. What's needed are two or more rules that can play off one another. The output of each rule is a fuzzy set. The output fuzzy sets for each rule are then aggregated into a single output fuzzy set. Finally the resulting set is defuzzified, or resolved to a single number. The next section shows how the whole process works from beginning to end for a particular type of fuzzy inference system called a Mamdani type.
|Logical Operations||Fuzzy Inference Systems|