A powerful representation technique using fuzzy rules and sets is similar in nature to rule-based systems with the difference that the rules include statements with fuzzy variables that are assigned fuzzy values. Fuzzy values are represented mathematically in fuzzy sets. Fuzzy logic is then applied to these rules and sets to process the reasoning.
Fuzzy logic basics:
- Fuzzy set: a generalization of the crisp set, defined as an order set of pairs: A = {(x, μA(x)) ⎜x∈X}, where X is the universe (e.g. numbers, measurement results, …), A is a name of the fuzzy set, μA(x) is a degree of membership of the element x into the set X (μA(x)∈[0,1]). Basic characteristics of the fuzzy set: support, α-cut, α-level, kernel, height. Fuzzy sets operations: fuzzy intersection, fuzzy union, fuzzy complement
- Fuzzy expression (formula): a mapping function f: [0,1]→[0,1], or in n-dimension f: [0,1]n→[0,1]). The operations are the same as used in the classical logic (∧,∨,¬)
- Fuzzy logic: a logic represented by the fuzzy expression, satisfying certain conditions
- Linguistic variable: quintuple (x, T(x), U, G,, M), where x: name of variable, T(x): a set of linguistic terms (fuzzy set) which can be a value of the variable, U: a set of universe of discourse which defines the characteristics of the variable, G: syntactic grammar which produces terms in T(x), M: semantic rules which map terms in T(x) to fuzzy sets in U
- Linguistic values: values of linguistic variable, e. g. {Large negative, Small negative, Zero, Small Positive, Large Positive}
- Popular types of membership functions: discrete or continuous; linear, nonlinear or singleton; S+, S-, Π, Z+, Z-, triangular, trapezoidal
- Fuzzy rule: the general form: If x is A, then y is B.
An example: standard Boolean and fuzzy logic representation of the concept “height is tall”
Fuzzy Reasoning: also called approximate reasoning, the process of drawing conclusions from fuzzy sets and fuzzy rules
Fuzzy Logic System (Fuzzy Inference System, FIS): a framework which is based on fuzzy sets, fuzzy rules and fuzzy reasoning. It has four main modules:
- The fuzzification interface (Fuzzifier) – transforms input crisp values into fuzzy values and involves the following functions: receives the input values, transforms the range of values of input variable into corresponding universe of discourse, and converts input data into suitable linguistic values (fuzzy sets).
- The fuzzification interface (Fuzzifier) – transforms input crisp values into fuzzy values and it involves the following functions: receives the input values, transforms the range of values of input variable into corresponding universe of discourse, and converts input data into suitable linguistic values (fuzzy sets).
- The knowledge base (Rules) – contains a knowledge of the application domain and the control goals. It consists of a data base and a linguistic rule base. The data base contains necessary definitions which are used in control rules and data manipulation. The linguistic rule base defines the control strategy and goals by means of linguistic control rules.
- The decision-making logic (Inference) – performs the following functions: simulates the human decision-making procedure based on fuzzy concepts, and infers fuzzy control actions employing fuzzy implication and linguistic rules. Examples of inference methods: Mamdani, Larsen, Sharp, Gőedel implications, etc.
- The defuzzification interface (Defuzzifier) – performs the functions: a scale mapping which converts the range of output values into corresponding universe of discourse, and defuzzification which yields a nonfuzzy control action from an inferred fuzzy control action. There are multiple defuzzification methods, e.g. Center of Gravity, Height Method (Max-Membership Principle), First of Maxima, Last of Maxima, Mean of Maxima, Center of Sums, Bisector of Area, Weighted Average, Center of Largest Area, etc.
Fuzzy Logic System
The fuzzy system performs the 4 basic operations (see 4 steps in the example below).
Advantages and disadvantages of Fuzzy Logic systems are given in the table below:
| Advantages | Disadvantages |
| Have easy and understandable structureA solution to complex problems in all fields of lifeAnswers uncertainties and ambiguities created by human language where everything cannot be described in precise and discrete termsMay not offer accurate reasoning, but the only acceptable reasoningMostly robust as no precise inputs requiredCan easily be modified to improve or alter system performance | Setting exact, fuzzy rules and, membership functions is a difficult taskFuzzy outputs can be interpreted in a number of ways making analysis difficultDon’t have the capability of machine learning as-well-as neural network type pattern recognitionTheir validation and verification needs extensive testing with hardwareSome fuzzy time logic is confused with probability theory and the terms |
An example: Mamdani’s fuzzy inference method [2]
Literature
- Mendel, J. M.: Fuzzy logic systems for engineering: A tutorial. Proc. of the IEEE, vol. 83, no. 3, pp. 345-377, 1995
- eMathTeacher: Mamdani’s Fuzzy Inference Method. http://www.dma.fi.upm.es/recursos/aplicaciones/logica_borrosa/web/fuzzy_inferencia/main_en.htm