Fuzzy logic
Fuzzy logic ( engl. fuzzy, blurred ',' blurred ',' undefined ', fuzzy logic, fuzzy theory, fuzzy logic ' or ' fuzzy theory ') is a theory which especially for the modeling of uncertainty and vagueness of colloquial descriptions has been developed. It is a generalization of two-valued Boolean logic. For example, the so -called fuzziness of data are "very" captured in mathematical models as "a bit ", " pretty ", " strong" or. The fuzzy logic is based on the fuzzy sets ( fuzzy sets ) and so-called membership functions that map objects on fuzzy sets, as well as appropriate logical operations on these sets and their inference. In industrial applications in addition to the fuzzification and defuzzification methods need to be considered, that is, methods for the conversion of information and relationships in the fuzzy logic and back again, for example, as a set value for a heater, as a result.
- 4.1 Example of a non-linear fuzzy logic function
Historical development
The reflections on a logic of blur goes back to ancient Greece. Even the philosopher Plato postulated that true and false lies a third region between the terms. This stood in stark contrast to his contemporaries, Aristotle, who founded the precision of mathematics is that a statement can be true or false only either.
The fuzzy set theory, ie the fuzzy set theory, in 1965, developed by Lotfi Zadeh at the University of California, Berkeley. The fuzzy technology adopted in the 1980s, especially in Japan its rise to the so-called Japanese fuzzy wave. A historical example is the regulation of fully automated subway Sendai, the first successful large-scale application of fuzzy logic in practice. Later, the fuzzy logic has been widely used in consumer electronics devices. The European fuzzy wave came only in the mid -1990s, when the policy discussions on the fuzzy logic subsided.
Fuzzy Set Theory
The fuzzy set theory is distinguished from the multivalued logic that described in the 1920s, the Polish logician Lukasiewicz January. In a narrower sense, the so-called fuzzy logic can be interpreted as a multi-valued logic though, and so far there is a certain proximity to the multi-valued logic, for whose truth value of a logical statement numbers from the real unit interval [0, 1] ( the real numbers from 0 to 1) are used. However Zadeh summarizes the fuzzy set theory as a formalization of indefinite term volumes in terms of referential semantics, which allows him to specify the fuzziness of the membership of objects as elements to be defined quantities gradually over numerical values between 0 and 1. Thus, a more extensive, linguistic interpretation of the fuzzy set theory opened as the basis of a logic of blur. The concept of fuzzy logic was initially not in use by Zadeh but later also of the teaching in Berkeley linguist George Lakoff, after Joseph Goguen, a graduate student Zadeh, had introduced a logic of fuzzy concepts. Comparisons in this context, the term coined by Hegel the doubled middle.
Basis of the fuzzy logic are the so-called fuzzy sets (English: fuzzy sets). Unlike traditional quantities (called in the context of fuzzy logic and sharp quantities ) in which contain an element of a predetermined base amount of either or not, is only approximate ( fuzzy) quantity is not defined by the objects that elements of this set are (or are not ), but the degree of their belonging to this set. This is done by membership functions uA: X → [ 0,1] that each element of the definition set X = { x} assign a number from the real-valued interval [ 0,1] of the target amount, which the membership degree uA (x ) for each element x as defined fuzzy set A indicates. Thus, each element is the element of each fuzzy set, but with different, a certain subset defining membership degrees. Zadeh said this new set operations that constitute the multi-valued fuzzy logic operations as a new logical calculus and prove them to be a generalization of two-valued classical logic, which is included as a special case in it. These operations on fuzzy sets are defined as in crisp sets, such as the formation of intersection (AND), union set ( OR ) and Komplementmengen (NOT ). To model the logical operators of conjunction ( AND ), disjunction ( OR ), and negation (NOT ) involves the functional classes of T- norm and T- conorm.
Negation
The negation in fuzzy logic is carried out by subtracting the input values of 1 So
NOT ( A) = 1-A Non-exclusive - OR gate
The adjunction is done by selecting the higher of the value of the input values. so
OR ( A, B ) = A if A> B B if A < = B AND circuit
The conjunction is effected by the choice of the lesser value of the input values. so
AND ( A, B ) = A if A = B Negative - OR circuit
For the disjunction of one complements the smaller of two values and selects the smaller of the two. For more than two input values is given to the result of the last operation recursively with the next input value. Simple: take the difference of less extremes of opposing him extreme value. so
XOR ( A, B ) = A if A> B and A <= ( 1-B) B if A> B and A > = ( 1-B) B where B > A, and B < = ( 1-A) A if B> A and B> = ( 1-A) fuzzy functions
Summaries of individual membership functions yield the fuzzy functions. An example is a fuzzy function of the age of a person. This could consist of several roof-shaped triangles, which in turn are available for different age types and represent the membership functions of these different age types. Each triangle covers an area of several years of human age. A man of 35 years had the characteristic features: still young with an initial rating 0.75 (which is still quite a lot ), mean age with an initial rating of 0.25 ( which is a bit ) and nothing from the other functions. In other words, at 35 you are pretty much still young and a bit of medium. The fuzzy function assigns each age value to a membership function that characterizes him.
In many cases, fuzzy functions are generated using tables from statistical surveys. These can also be raised by the application itself as far as feedback is given, as in the elevator control. Practically important is to have the experience and intuition of an expert be included in the relevant field in a fuzzy function, especially when no statistical statements are present, for example if it is a complete re -describing system.
This triangular shape is, however, not mandatory, in general, the values of fuzzy functions can have any shape as long as their function values are in the interval [ 0,1]. In practice, such delta functions are, however, often used due to their ease of predictability. Relatively widespread are still Trapeze (not necessarily mirror symmetry ), but also semi-circles can be found in some applications. Also, in principle, more than two portions of a fuzzy function overlap ( in the example considered here, but that does not seem to make sense ).
For example for a non- linear fuzzy logic function
An example of a non-linear membership function is the following sigmoidal function:
The curve suppressed by the shape of the letter S an increasing belonging to the amount each described by a value in the range [0,1] from. Depending on the application, can be a decreasing membership by a corresponding Z - curve expressed as:
The α parameter here indicates the inflection point of the S- curve, the value of δ determines the slope of the curve. The larger is chosen δ, the flatter the shape of the resulting function.
The age of a person can be using this curve as follows as a fuzzy function represent:
The colloquial modifiers can be represented very, more or less, and not by simple modification of a given function:
- The colloquial enhancing modifier very can be represented in terms of increased exponent ( in the example). The result is a steeper curve compared to the initial function.
- The colloquial modifier more or less can be expressed by using a lower or the square root of the exponent to a given function (). The result is a flatter curve in comparison to the initial function.
- The negation of a colloquial expression can represent by a simple subtraction ().
According to the applications, it is in this form of representation to linguistic variables. Ultimately, a single numerical value is calculated from the individual weighted statements, which can express the age in mathematical form. This value can then be precisely continue working. In this so-called Defuzzyfikation many methods are possible, the most famous (but by no means always best) is certainly the method center -of -gravity, in which the numerical value weighted by the mass of the geometric shape of the individual sections of the membership function is formed. Another possibility is to simply form a weighted average of the function values .
Application Examples
Fuzzy logic is used today in different areas: Application finds in areas of automation technology, business administration, medical technology, consumer electronics, automotive engineering, control engineering, artificial intelligence, inference systems, speech recognition and other areas. Useful use of fuzzy logic is often, if not a mathematical description of an issue or problem is present, but only a verbal description. Even if - as almost always - is the existing knowledge has gaps or partially obsolete, the use of fuzzy logic offers to still arrive at a valid opinion about a current or future system state. Then a mathematical description language is derived from the formulated principles and rules by means of fuzzy logic, can be used in computer systems. It is interesting that with the fuzzy logic and then controlled systems useful (or controlled) can be when a mathematical relationship between the input and output variables of a system can not be represented - or could be done only with great effort, so that automation to expensive or not be realized in real time.
Other applications include the control of subways, the prognosis of the future load in routers, gateways or mobile phone base stations, control of automatic transmissions in cars, alarm systems for anesthesia, intermediate frequency filter in radios, ABS for automobiles, fire alarm systems, the forecast of energy consumption energy providers, AF - coupled multi - automatic exposure and AF predict in SLR cameras, etc.
Also in business applications fuzzy logic has successfully taken hold. A successful example is the Intelligent Claim Evaluation (ISP ), protect against insurance fraud with the worldwide insurance company.
Distinction between
Not to be confused with the fuzzy logic is the fuzzy search, which enables fuzzy search in databases, for example, if the exact spelling of a name or term is not known. Even if the belongingness values from the interval [ 0,1] look formally as probability values , so blur is something fundamentally different probability. In particular it should be noted that the sum of the values of two functions, the overlap does not need to be 1. It can be equal to 1, but also be higher or lower.