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Likewise, we derive formulas for some other operations on fuzzy numbers. Rather than going over detailed calculations, we summarize the final results

  division

where G1 and G2 are obtained from the original membership functions and computed in the form

and

for ω in [0, 1].

  inverse

4.3.3. Accumulation of fuzziness in computing with fuzzy numbers

The problem of practical relevance that needs to be mentioned here concerns a phenomenon of accumulation of fuzziness - in manifestation quite similar to that encountered in error accumulation in numerical computations, e.g., when performing some iterative calculations. To illustrate this effect, consider an iterative process of adding a fuzzy number with triangular membership function, A = (0, 1, 2) to a single numerical quantity (1, 1, 1). Following the addition formula for triangular fuzzy numbers, we get

We note that the supports of the successive fuzzy numbers expand very quickly; while the lower bound becomes fixed, the upper gets bigger from step to step and the resulting membership function gets skewed. This result indicates clearly that one should proceed with caution when computing with fuzzy numbers: a length of the chain of iterative computations should be eventually reduced to retain the obtained meaningful results.

4.4. Fuzzy rule-based computing

Rules provide a formal way of representing strategies, directives, hints and are often appropriate when a domain knowledge about a problem under discussion comes in the form of empirical associations or experimental findings (Yager and Filev, 1994). Rule-based systems are built upon a set of rules and use a collection of facts to derive conclusions. The spectrum of rule-based computing is amazingly broad. They appear in knowledge-based systems as a generic form of knowledge representation fully supported by numerous implementations with a significant variation of two basic inference mechanisms known as backward and forward chaining. As far as diversity goes, we can encounter rule-based models, rule-based classifiers, rule-based controllers, etc. Rules are used as an important part of many topologies such as e.g., subsumption architectures encountered in robotics. Logic-oriented languages (such as PROLOG) heavily rely on rules.

The role of fuzzy sets in this computational paradigm is essential: they help alleviate a significant deficiency of brittleness. Owing to the fact that both antecedents and consequents are linguistic terms, the levels of firing (activation) of rules are continuous (rather than exclusively 0 or 1) and this contributes to a significant robustness. Implicitly, this helps design systems in a more efficient way. The benefits manifest when it comes to learning procedures. That is why the reader can encounter a number of fuzzy rule-based systems (including fuzzy models, approximate reasoning, fuzzy controllers). Especially, the fuzzy controller is often referred to as an example of a commonly exploited knowledge-based and model-free control paradigm.

In what follows, we start with a brief summary of several classes of rules to expose the reader to their evident versatility and ability to capture all necessary facets of domain knowledge. In the sequel, we will look into key design aspects and, finally, conclude with an analysis and synthesis of fuzzy controllers.

4.4.1. Rules with fuzzy sets

The rules in their standard form are conditional statements

- if condition is A then conclusion is B

(equivalently, conditions are referred to as antecedents while conclusions are also called consequents). Both A and B are fuzzy sets (linguistic terms). Depending upon the specific form of fuzzy sets the rules can be made either more general or more specific. Let the conclusion B is fixed. Then if A becomes more specific (in the sense of the specificity measure), the rule becomes more specific (less general). On the other hand, if A is just unknown (modeled by a fuzzy set with a constant membership function equal identically 1), then the rule becomes simply an unconditional statement

conclusion is B

that holds independently of the current condition - the rule becomes absolutely general (as it always holds meaning that the conclusion is unconditionally valid).

The rules come also with some quantifiers. This adds an extra certainty quantification

- if condition is A then conclusion is B with certainty λ

λ ∈[0, 1] is a certainty factor. If λ = 1 this rule converts to the form previously discussed.

The rules can represent knowledge that deals with gradual relationships between the antecedents and the condition; these come in the form

- the more A the more B

or more generally

- the τ is A the τ is B

where τ stands for a linguistic modifier.

The dimensional generalization of the rules involves a number of subconditions

- if condition1 and condition1 and ... and condition1 then conclusion

The rules can assume a tremendous number of formats, Fig. 4.3, especially when it comes to the type of the conclusion.


Figure 4.3  Several selected classes of fuzzy rules

The one we have just discussed are the simplest set-to-set mappings completely defined by the linguistic granulae (A and B) occurring therein. The rules can assume a form of the set-to - function mappings. Take the rule

- if condition is A then y = f(x, param)

where f is a function from X to Y coming with a vector of parameters (param). The function viewed in the context of the rule is very much local as it holds for the condition being activated; this correspondence does not hold (or holds partially) for x whose membership in A is negligible.

The form of the function itself (f) varies from case to case. For instance, one can consider

  linear functions

  quadratic functions

  polynomials


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