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 Fuzzification is the process of assigning the numerical input of a system to fuzzy sets with some degree of membership. This degree of membership may be anywhere within the interval [0,1]. If it is 0 then the value does not belong to the given fuzzy set, and if it is 1 then the value completely belongs within the fuzzy set. Any value between 0 and 1 represents the degree of uncertainty that the value belongs in the set. These fuzzy sets are typically described by words, and so by assigning the system input to fuzzy sets, we can reason with it in a linguistically natural manner. 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 (truth values) 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 (pointing at 0.2) may describe it as "slightly warm" and the blue arrow (pointing at 0.8) "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.
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 (which can have a length of 0 or greater) 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 : S(x) = \frac{1}{1 + e^{-x}} which has the following symmetry property : S(x) + S(-x) = 1. From this it follows that (S(x) + S(-x)) \cdot (S(y) + S(-y)) \cdot (S(z) + S(-z)) = 1
Fuzzy logic operators Fuzzy logic works with membership values in a way that mimics
Boolean logic. To this end, replacements for basic
operators ("gates") AND, OR, NOT must be available. There are several ways to accomplish this. A common replacement is called the
s: 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 (Zaitsev, et al), 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 (to the right of the function value in the inequality, including the function value). 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( AND( NOT(x), NOT(y) ) ) x OR y = NOT( AND( 1 - x, 1 - y) ) x OR y = NOT( (1 - x) * (1 - y) ) x OR y = 1 - (1 - x) * (1 - y) 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, 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) The
Takagi–Sugeno or
Takagi–Sugeno–Kang (
TSK) system was introduced by Tomohiro Takagi and
Michio Sugeno for fuzzy identification of systems and applications to modeling and control. Sugeno and Kang later developed methods for structure identification of such fuzzy models from input-output data. The TSK system is similar to Mamdani, but the defuzzification process is included in the execution of the fuzzy rules. These are also adapted, so that instead the consequent of the rule is represented through a polynomial function, usually constant in a zero-order model or linear in a first-order model. An example of a rule with a constant output would be: IF temperature IS very cold THEN fan_speed = 2 In this case, the output will be equal to the constant of the consequent (e.g. 2). In most scenarios we would have an entire rule base, with 2 or more rules. If this is the case, the output of the entire rule base will be the average of the consequent of each rule i (Yi), weighted according to the membership value of its antecedent (hi): \frac{\sum_i (h_i \cdot Y_i)}{\sum_i h_i} An example of a rule with a linear output would be instead: IF temperature IS very cold AND humidity IS high THEN fan_speed = 2 * temperature + 1 * humidity In this case, the output of the rule will be the result of the function in the consequent. The variables in the consequent are the system's input variables evaluated at the current input, while the antecedent membership values determine the weights of the rules. Same as before, in case we have an entire rule base with 2 or more rules, the total output will be the weighted average between the output of each rule. 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. ==Forming a consensus of inputs and fuzzy rules==