Fuzzy logic
Fuzzy logic is a form of many-valued logic in which the truth value of variables may be any real number between 0 and 1. It is employed to handle the concept of partial truth, where the truth value may range between completely true and completely false. By contrast, in Boolean logic, the truth values of variables may only be the integer values 0 or 1.
The term fuzzy logic was introduced with the 1965 proposal of fuzzy set theory by mathematician Lotfi Zadeh. Basic fuzzy logic had, however, been studied since the 1920s, as infinite-valued logic—notably by Łukasiewicz and Tarski. The works of Zadeh and Joseph Goguen in the 1960's and 1970's went further by considering issues such as linguistic variables and lattices.
Fuzzy logic is based on the observation that people make decisions based on imprecise and non-numerical information. Fuzzy models or fuzzy sets are mathematical means of representing vagueness and imprecise information. These models have the capability of recognising, representing, manipulating, interpreting, and using data and information that are vague and lack certainty.
Fuzzy logic has been applied to many fields, from control theory to artificial intelligence.
Overview
only permits conclusions that are either true or false. However, there are also propositions with variable answers, which one might find when asking a group of people to identify a color. In such instances, the truth appears as the result of reasoning from inexact or partial knowledge in which the sampled answers are mapped on a spectrum.Both degrees of truth and probabilities range between 0 and 1 and hence may seem identical at first, but fuzzy logic uses degrees of truth as a mathematical model of vagueness, while probability is a mathematical model of ignorance.
Applying truth values
A basic application might characterize various sub-ranges of a continuous variable. For instance, a temperature measurement for anti-lock brakes might have several separate membership functions defining particular temperature ranges needed to control the brakes properly. Each function maps the same temperature value to a truth value in the 0 to 1 range. These truth values can then be used to determine how the brakes should be controlled. Fuzzy set theory provides a means for representing uncertainty.Linguistic variables
In fuzzy logic applications, non-numeric values are often used to facilitate the expression of rules and facts.A linguistic variable such as age may accept values such as young and its antonym old. Because natural languages do not always contain enough value terms to express a fuzzy value scale, it is common practice to modify linguistic values with adjectives or adverbs. For example, we can use the hedges rather and somewhat to construct the additional values rather old or somewhat young.
Fuzzy systems
Mamdani
The most well-known system is the Mamdani rule-based one. It uses the following rules:- Fuzzify all input values into fuzzy membership functions.
- Execute all applicable rules in the rulebase to compute the fuzzy output functions.
- De-fuzzify the fuzzy output functions to get "crisp" output values.
Fuzzification
For example, in the image below, the meanings of the expressions cold, warm, and hot are represented by functions mapping a temperature scale. A point on that scale has three "truth values"—one for each of the three functions. The vertical line in the image represents a particular temperature that the three arrows gauge. Since the red arrow points to zero, this temperature may be interpreted as "not hot"; i.e. this temperature has zero membership in the fuzzy set "hot". The orange arrow may describe it as "slightly warm" and the blue arrow "fairly cold". Therefore, this temperature has 0.2 membership in the fuzzy set "warm" and 0.8 membership in the fuzzy set "cold". The degree of membership assigned for each fuzzy set is the result of fuzzification.
Image:Fuzzy logic temperature en.svg|thumb|center|upright=1.5|Fuzzy logic temperature
Fuzzy sets are often defined as triangle or trapezoid-shaped curves, as each value will have a slope where the value is increasing, a peak where the value is equal to 1 and a slope where the value is decreasing. They can also be defined using a sigmoid function. One common case is the standard logistic function defined as
which has the following symmetry property
From this it follows that
Fuzzy logic operators
Fuzzy logic works with membership values in a way that mimics Boolean logic. To this end, replacements for basic operators AND, OR, NOT must be available. There are several ways to accomplish this. A common replacement is called the s:| Boolean | Fuzzy |
| AND | MIN |
| OR | MAX |
| NOT | 1 – x |
For TRUE/1 and FALSE/0, the fuzzy expressions produce the same result as the Boolean expressions.
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.
However, an arbitrary choice table does not always define a fuzzy logic function. In the paper, a criterion has been formulated to recognize whether a given choice table defines a fuzzy logic function and a simple algorithm of fuzzy logic function synthesis has been proposed based on introduced concepts of constituents of minimum and maximum. A fuzzy logic function represents a disjunction of constituents of minimum, where a constituent of minimum is a conjunction of variables of the current area greater than or equal to the function value in this area.
Another set of AND/OR operators is based on multiplication, where
x AND y = x * y
NOT x = 1 - x
Hence,
x OR y = NOT, NOT )
x OR y = NOT
x OR y = NOT *
x OR y = 1 - *
x OR y = x + y - xy
Given any two of AND/OR/NOT, it is possible to derive the third. The generalization of AND is an instance of a t-norm.
IF-THEN rules
IF-THEN rules map input or computed truth values to desired output truth values. Example:IF temperature IS very cold THEN fan_speed is stopped
IF temperature IS cold THEN fan_speed is slow
IF temperature IS warm THEN fan_speed is moderate
IF temperature IS hot THEN fan_speed is high
Given a certain temperature, the fuzzy variable hot has a certain truth value, which is copied to the high variable.
Should an output variable occur in several THEN parts, then the values from the respective IF parts are combined using the OR operator.
Defuzzification
The goal is to get a continuous variable from fuzzy truth values.This would be easy if the output truth values were exactly those obtained from fuzzification of a given number.
Since, however, all output truth values are computed independently, in most cases they do not represent such a set of numbers.
One has then to decide for a number that matches best the "intention" encoded in the truth value.
For example, for several truth values of fan_speed, an actual speed must be found that best fits the computed truth values of the variables 'slow', 'moderate' and so on.
There is no single algorithm for this purpose.
A common algorithm is
- For each truth value, cut the membership function at this value
- Combine the resulting curves using the OR operator
- Find the center-of-weight of the area under the curve
- The x position of this center is then the final output.
Takagi–Sugeno–Kang (TSK)
IF temperature IS very cold = 2
An example of a rule with a linear output would be instead:
IF temperature IS very cold AND humidity IS high = 2 * temperature + 1 * humidity
The main advantage of using TSK over Mamdani is that it is computationally efficient and works well within other algorithms, such as PID control and with optimization algorithms. It can also guarantee the continuity of the output surface. However, Mamdani is more intuitive and easier to work with by people. Hence, TSK is usually used within other complex methods, such as in adaptive neuro fuzzy inference systems.