Ideas of the fuzzy control methodology
Before coming to the details, the main ideas of fuzzy control methodology on the example of a situation will be illustrated in which everyone feels himself an expert. One of the most widely used control systems is the simplest rule-based system imaginable, a thermostatic temperature controller. This rule-based system operates with two rules:
(1) IF temperature is below set point THEN heat is on
(2) IF temperature is above set point THEN heat is off
The success of this controller is due to the combination of the properties, that it is simple, robust and does not require a complex process model. The model is: when the heat is on, the temperature rises slowly, and when the heat is off, the temperature falls slowly.
The two IF-THEN clauses above can also be formally rewritten as
IF is THEN is for
In the thermostat example there is only one input variable (linguistic variable), the temperature . In the general case, there are several input variables , so, in addition to the logical connective IF-THEN, another logical connective is needed, AND. Then the IF-THEN clauses are
IF is AND is ... AND is THEN is .
Here, is the rule number, and and are words from natural language (linguistic terms), like ``below set point'', ``on'', ``small'', ``large'', ``approximately 1.5'', etc. If the standard mathematical notation for IF-THEN and AND is used, the above rules can be re-formulated as follows:
where . The set of rules is usually called a rule base. The left-hand side of a rule is called premise, the right-hand side conclusion and the rule itself implication.
The general idea is to represent the rule base in a computer. It has a clear structure. A rule base consists of rules and each rule, in its turn, is obtained from properties expressed by linguistic variables and terms and using logical connectives. In view of this structure, it is reasonable to represent the rule base by first representing the basic elements of the rule base, premises and conclusions, and then by extending this representation to the rule base as a whole. It makes sense to use the following steps in the methodology:
Representation of the basic properties and .
Representation of the logical connectives.
The representations of the basic properties and of the logical connectives is used to get the representations of all the rules.
Combination of the representations of different rules into a representation of a rule base.
As a result of these four steps, one obtains an expert system that can give advise for a specialist, who has to make a decision. In control, a system to automatically make a decision based on its own conclusions is wanted. Therefore, for control situations, a fifth follow-up step is needed:
Based on facts for and on the rule base a reasoning procedure makes a decision.
In the following section from the very beginning it is described how these five steps are implemented.
A typical fuzzy logic system has four components: a fuzzifier, a fuzzy rule base, an inference engine, and a defuzzifier. The function of the fuzzifier is to determine the degree of membership of a crisp input in a fuzzy set. The fuzzy rule base is used to represent the fuzzy relationships between input-output fuzzy variables. The output of the fuzzy rule base is determined based on the degree of membership specified by the fuzzifier. The inference engine calculates the rule’s conclusion based on its membership degree. Optionally, if needed, a defuzzifier is used to convert outputs of the fuzzy rule base into crisp values. Figure 1 illustrates a block diagram of a fuzzy system.
Figure 1: A Block Diagram of a Fuzzy System
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