Application of fuzzy logic. Fuzzy Logic: Clear Fuzzy Logic Solutions

Fuzzy logic– logic based on the theory of fuzzy sets. Its subject is the construction of models of approximate human reasoning and their use in computer systems. IN fuzzy logic The estimate boundary has been expanded from a two-valued estimate (Either 0 or 1) to an unlimited multi-valued estimate (On the interval ).

A fuzzy set A in a complete space X is defined through the membership function m A (x):

The logic of defining the concept of a fuzzy set does not contain any vagueness. Instead of specifying a specific value (For example, 0.8), it is common to use lower and upper values ​​to set acceptable assessment limits (For example).

In the case of fuzzy logic, you can create an unlimited number of operations, so it does not use basic operations to record the rest. The extensions of NOT, AND, OR to fuzzy operations are especially important. They are called respectively – fuzzy negation, t-norm and s-norm. Since the number of states is unlimited, it is impossible to describe these operations using a truth table. Operations are explained using functions and axioms and represented using graphs.

Axiomatic representation of fuzzy operations:

Fuzzy denial

Axiom N1 preserves the property of double negation, and N2 preserves the rule of double negation. N3 – the most significant: fuzzy negation inverts the sequence of assessments.

A typical fuzzy negation operation is subtraction from 1.

When negated, the value 0.5 is central and usually x and xθ take symmetrical values ​​relative to 0.5.

T-norm.

Axiom T1 is valid, as for clear I. T2 and T3 are the laws of intersection and union. Axiom T4 is a requirement for orderliness.

A typical t-norm is the min operation or logical product:

With a logical product, the graph is constructed symmetrically relative to the plane, formed by the inclined x1 and x2.

S-norm.

A typical s-norm is a logical sum, defined by the max operation.

In addition to it, there are an algebraic sum, a boundary sum and a drastic sum:

As can be seen from the figures, the order is the opposite than in the case of the t-norm.

Examples of fuzzy definition include temperature and valve operation:

Similarities

Fuzzy logic is a generalization of classical crisp logic. Both crisp and fuzzy logic are based on sets and relational operations. Fuzzy operations are an extension of the operations of precise logic.

Differences

In crisp logic, variables are full members of sets, and in fuzzy logic they are only partial members of sets.

In clear logic, a statement is either true or false, and the law of elimination of the middle operates in it. In fuzzy logic, truth or falsity is no longer absolute and statements can be partially true and partially false. In crisp logic the number of possible operations is finite and depends on the number of inputs, whereas in fuzzy logic the number of possible operations is infinite.

3. Example

First we consider the set X of all real numbers between 0 and 10, which we will call the study area. Now, let's define a subset X of all real numbers in range between 5 and 8.

A=

Now let's imagine the set A using a symbolic function, i.e. this function assigns the number 1 or 0 to each element in X, depending on whether the element is in a subset A or not. This results in the following diagram:

We can interpret elements that are assigned the number 1 as as elements that are in set A, and elements that are assigned the number 0, as elements not in the set A.

This concept is sufficient for many application areas. But we can easily find situations where flexibility is lost. To show this, consider the following example, showing the difference between a fuzzy set and a crisp set:

In this example we want to describe a set of young people. More formally we can denote

B = (set of young people)

Since, in general, age starts from 0, bottom line of this set must be zero. The upper limit, on the other hand, must be determined. For the first time, let's define the upper limit of the set, say, 20 years. Therefore we get B as a clear interval, namely:

B=

Now the question arises: why is someone young on his 20th birthday, but not young the next day? Obviously this is - structural problem, because if we move the upper bound from 20 to an arbitrary point, we can pose the same question.

A more natural way to describe a set B is to loosen the strict division between the young and the non-young. We will do this by allowing not only a (clear) solution YES: he/she is in many young, or NO: he/she is not among the young ones, but more flexible phrases like: Okay, he/she belongs a little more to the young crowd or NO, he/she hardly belongs to the multitude of young people.

In our first example, we coded all Study Area elements as 0 or 1. A straightforward way to generalize this concept to a fuzzy set is to define large quantity values ​​between 0 and 1. In fact, we define infinitely many options between 0 and 1, namely the unit interval I =.

The interpretation of the numbers in the fuzzy set assigned to all elements of the Study Area is more difficult. Of course, again the number 1 is assigned to the element as a way to identify the element that is in the set B and 0 - the method in which the element is not defined in the set B. All other values ​​mean gradual belonging to the multitude B.

For greater clarity, we now show the set of young ones, like our first example, graphically using a symbolic function.

With this method, 25-year-old people will still young by 50 percent (0.5). Now you know what a fuzzy set is.

Epimenides of Knossos from the island of Crete is a semi-mythical poet and philosopher who lived in the 6th century. BC, once declared: “All Cretans are liars!” Since he himself was a Cretan, he is remembered as the inventor of the so-called Cretan paradox.

In terms of Aristotelian logic, in which a statement cannot be both true and false, and such self-negations make no sense. If they are true then they are false, but if they are false then they are true.

And here fuzzy logic comes into play, where variables can be partial members of sets. Truth or falsity is no longer absolute - statements can be partly true and partly false. Using this approach allows us to strictly mathematically prove that the Epimenides paradox is exactly 50% true and 50% false.

Thus, fuzzy logic is fundamentally incompatible with Aristotelian logic, especially with regard to the law Tertium non datur (“No third is given” - Latin), which is also called law of exclusion of the average1. To put it briefly, it goes like this: if a statement is not true, then it is false. These postulates are so basic that they are often simply taken for granted.

A more trivial example of the usefulness of fuzzy logic can be given in the context of the concept of cold. Most people are able to answer the question: “Are you cold now?” In most cases (unless you're talking to a physics graduate student), people understand that we're not talking about absolute temperature on the Kelvin scale. Although a temperature of 0 K can, without a doubt, be called cold, many will not consider a temperature of +15 C to be cold.

But machines are not capable of making such fine gradations. If the standard definition of cold is “temperature below +15 C”, then +14.99 C will be regarded as cold, but +15 C will not.

Fuzzy set theory

Let's look at Fig. 1. It shows a graph that helps you understand how a person perceives temperature. A person perceives a temperature of +60 F (+12 C) as cold, and a temperature of +80 F (+27 C) as hot. Temperatures of +65 F (+15 C) may seem cold to some, but quite comfortable to others. We call this group of definitions the function of membership in sets that describe a person’s subjective perception of temperature.

It is just as easy to create additional sets that describe human perception of temperature. For example, you can add sets such as “very cold” and “very hot.” It is possible to describe similar functions for other concepts, such as open and closed states, chiller temperature, or chiller tower temperature.

That is, fuzzy systems can be used as a universal approximator (averager) for a very wide class of linear and nonlinear systems. This not only makes control strategies more reliable in nonlinear cases, but also allows the use of expert assessments to build computer logic circuits.

Fuzzy Operators

To apply algebra to fuzzy values, you need to determine the operators to use. Typically, Boolean logic uses only a limited set of operators, with the help of which other operations are performed: NOT (NOT operator), AND (AND operator) and OR (OR operator).

Many definitions can be given for these three basic operators, three of which are shown in the table. By the way, all definitions are equally valid for Boolean logic (to check, just substitute 0 and 1 in them). In Boolean logic, FALSE is equivalent to 0, and TRUE is equivalent to 1. Similarly, in fuzzy logic, the degree of truth can range from 0 to 1, so the value "Cold" is true to the power of 0.1, and the operation NOT("Cold") will give the value 0.9.

You can go back to Epimenides' paradox and try to solve it (mathematically it is expressed as A = NOT(A), where A is the degree of truth of the corresponding statement). If you want more difficult task, then try to solve the problem of the sound of a clap made by one hand...

Solving problems using fuzzy logic methods

Only a few valves are capable of opening “just a little.” When operating equipment, clear values ​​are usually used (for example, in the case of a bimodal 0-10 V signal), which can be obtained using the so-called “fuzzy logic problem solving”. This approach makes it possible to transform the semantic knowledge contained in the fuzzy system into an implementable control strategy2.

This can be done using various techniques, but to illustrate the process as a whole, let's look at just one example.

In the height defuzzification method, the result is the sum of the fuzzy set peaks, calculated using weights. This method has several disadvantages including bad job with asymmetric set membership functions, but it has one advantage - this method is the easiest to understand.

Let's assume that the set of rules governing the opening of the valve gives us the following result:

"Valve partially closed": 0.2

"Valve partially open": 0.7

"Valve open": 0.3

If we use the height defuzzification method to determine the degree of openness of the valve, we will get the result:

"Valve closed": 0.1

(0,1*0% + 0,2*25% + 0,7*75% + 0,3*100%)/ /(0,1 + 0,2 + 0,7 + 0,3) =

= (0% + 5% + 52,5% + 30%)/(1,3) = = 87,5/1,3 = = 67,3%,

those. the valve must be opened to 67.3%.

Practical application of fuzzy logic

When the theory of fuzzy logic first appeared, one could find articles in scientific journals devoted to its possible areas of application. As developments in this area progress, more practical applications for fuzzy logic began to grow rapidly. This list would be too long at this time, but here are a few examples to help you understand how widely fuzzy logic is used in control systems and expert systems3.

– Devices for automatically maintaining vehicle speed and increasing efficiency/stability automobile engines(Nissan, Subaru companies).

mechanisms of thinking, noticed that in reality there is not just one logic (for example, Boolean), but as many as we wish, because everything is determined by the choice of the appropriate system of axioms. Of course, once the axioms are chosen, all statements built on their basis must be strictly, without contradictions, linked to each other according to the rules established in this system of axioms.

Human thinking is a combination of intuition and rigor, which, on the one hand, considers the world as a whole or by analogy, and on the other hand, logically and consistently and, therefore, represents a fuzzy mechanism. The laws of thought that we would want to include in computer programs must necessarily be formal; the laws of thinking manifested in human-human dialogue are unclear. Can we therefore say that fuzzy logic can be well adapted to human dialogue? Yes - if software, developed taking into account fuzzy logic, will become operational and can be technically implemented, then human-machine communication will become much more convenient, faster and better suited to solving problems.

The term " fuzzy logic" is usually used in two different meanings. In a narrow sense, fuzzy logic is logical calculus, which is an extension multivalued logic. In her in a broad sense, which is the predominant one in use today, fuzzy logic is equivalent to fuzzy set theory. From this point of view, fuzzy logic in the narrow sense is a branch of fuzzy logic in the broad sense.

Definition. Any fuzzy variable characterized by three

Where is the name of the variable, - universal set, is a fuzzy subset of the set, which represents a fuzzy constraint on the value of the variable, conditioned by .

Using the analogy of a traveling bag, fuzzy variable can be likened to a traveling bag with a label that has “soft” walls. Then - the inscription on the label (the name of the bag), - a list of items that, in principle, can be placed in the bag, and - part of this list, where for each item a number is indicated, characterizing the degree of ease with which the item can be placed in the bag.

Let us now consider various approaches to defining the basic operations on fuzzy variables, namely conjunction, disjunction and negation. These operations are fundamental to fuzzy logic in the sense that all its structures are based on these operations. Currently in fuzzy logic as conjunction operations and disjunctions widely use -norms and -conorms, which came to fuzzy logic from the theory of probabilistic metric spaces. They are quite well studied and form the basis of many formal constructions of fuzzy logic. At the same time, the expansion of the scope of applications of fuzzy logic and fuzzy modeling capabilities necessitates the generalization of these operations. One direction is associated with weakening their axiomatics in order to expand the fuzzy modeling tools. Another direction of generalization conjunction operations and disjunction of fuzzy logic is associated with replacing the set of membership values ​​with a linearly or partially ordered set of linguistic credibility assessments. These generalizations of the basic operations of fuzzy logic, on the one hand, are caused by the need to develop expert systems in which the truth values ​​of facts and rules are described by an expert or user directly on a linguistic scale and are of a qualitative nature. On the other hand, such generalizations are caused by a shift in direction active development fuzzy logic from modeling quantitative, measurable processes to modeling human thinking processes, where the perception of the world and decision-making occur on the basis of information granulation and calculation in words.

A natural generalization of involutive negation operations of fuzzy logic are non-involutive negations. They are of independent interest and are considered in fuzzy and other non-classical logics. The need to study such negation operations is also caused by the introduction into consideration of generalized conjunction operations and disjunctions connected to each other using negation operations.

Introduction to Fuzzy Logic

Fuzzy logic is logical or control system n-digit logical system, which uses the degrees of state (“degrees of truth”) of the inputs and generates outputs that depend on the states of the inputs and the rate of change of these states. This is not the usual "true or false" (1 or 0), Boolean (binary) logic on which modern computers. It mainly provides the basis for approximate reasoning using imprecise solutions and allows the use of linguistic variables.

Fuzzy logic was developed in 1965 by Professor Lotfi Zadeh at the University of California, Berkeley. The first application was to perform computer data processing based on natural values.

To put it simply, fuzzy logic states can be not only 1 or 0, but also values ​​between them, that is, 0.15, 0.8, etc. For example, in binary logic, we can say that we have a glass of hot water (that is, 1 or logic high) or a glass cold water, that is (0 or low logic level), but in fuzzy logic, we can say that we have a glass of warm water (neither hot nor cold, that is, somewhere in between these two extreme states). Clear logic: yes or no (1, 0). Fuzzy logic: of course, yes; probably no; Hard to say; maybe yes, etc.

Basic architecture of a fuzzy logic system

The fuzzy logic system consists of the following modules:

Fuzzifier (or blur operator). It takes measured variables as input and converts numeric values ​​into linguistic variables. It transforms physical values ​​as well as error signals into a normalized fuzzy subset, which consists of an interval for a range of input values ​​and membership functions that describe the probability of the state of the input variables. Input signal Mainly divided into five states, such as: large positive, medium positive, small, medium negative and large negative.

Controller. It consists of a knowledge base and an inference engine. The knowledge base stores membership functions and fuzzy rules obtained by knowing the operation of the system in the environment. The inference engine processes the resulting membership functions and fuzzy rules. In other words, the inference engine generates output based on linguistic information.

Defuzzifier or clarity restoration operator. It performs the reverse process of phasefire. In other words, it converts fuzzy values ​​into normal numeric or physical signals and sends them to physical system to control the operation of the system.

Operating principle of a fuzzy logic system

Fuzzy operation involves the use of fuzzy sets and membership functions. Each fuzzy set is a representation of a linguistic variable that defines a possible output state. The membership function is a function of the general value in the fuzzy set, so that both the general value and the fuzzy set belong to the universal set.

The degrees of membership in this general value in the fuzzy set determine the output based on the IF-THEN principle. Membership is assigned based on the assumption of the output from the inputs and the rate of change of the inputs. The membership function is basically graphical representation fuzzy set.

Consider a value "x" such that x ∈ X for the entire interval and a fuzzy set A, which is a subset of X. The membership function of "x" in the subset A is given by: fA(x), Note that "x" denotes the membership value . Below is a graphical representation of fuzzy sets.

While the x-axis denotes the universal set, the y-axis denotes degrees of membership. These membership functions can be triangular, trapezoidal, single point, or Gaussian in shape.

Practical example of a fuzzy logic system

Let's develop simple system fuzzy control to control the operation of the washing machine, so that the fuzzy system controls the washing process, water intake, washing time and spin speed. The input parameters here are the volume of clothing, the degree of soiling and the type of dirt. While the volume of clothing would determine the water intake, the degree of contamination would in turn be determined by the clarity of the water, and the type of dirt would be determined by the time the color of the water remained constant.

The first step is to define linguistic variables and terms. For input data, linguistic variables are given below:

• Mud type: (Greasy, Medium, Not Greasy)
• Dirt quality: (Large, Medium, Small)

For output, the linguistic variables are given below:

• Washing time: (Short, Very Short, Long, Medium, Very Long) (short, very short, long, medium, very long).

The second step involves constructing membership functions. Below are plots that define the membership functions for the two inputs. Accessory functions for mud quality:

Accessory functions for mud type:

The third step involves developing a set of rules for the knowledge base. Below is a set of rules using IF-THEN logic:

IF Dirt Quality Small AND Dirt Type Greasy, THEN Washing Time Long.
IF dirt quality Medium AND Dirt type Greasy, THEN Washing time Long.
IF dirt quality Large and dirt type Greasy, THEN Washing time Very Long.
IF dirt quality Small AND Dirt type Medium, THEN Washing time Medium.
IF dirt quality Medium AND Dirt type Medium, THEN Washing time Medium.
IF dirt quality Large and dirt type Medium, THEN Washing time Medium.
IF dirt quality Small and dirt type Non-Greasy, THEN Washing time Very Short.
IF dirt quality Medium AND Dirt type Non-Greasy, THEN Washing time Medium.
IF dirt quality Large and dirt type Greasy, THEN Washing time Very Short.

Fazifire, which initially converted the sensor inputs into these linguistic variables, now applies the above rules to perform fuzzy set operations (e.g., MIN and MAX) to determine the output fuzzy functions. A membership function is developed based on the output fuzzy sets. The last step is the defasification step, in which the defasifier uses the output membership functions to determine the wash time.

Application areas of fuzzy logic

Fuzzy logic systems can be used in automotive systems such as automatic transmissions. Applications in the area household appliances include microwaves, air conditioners, washing machines, TVs, refrigerators, vacuum cleaners, etc.

Advantages of Fuzzy Logic

• Fuzzy logic systems are flexible and allow rules to be changed.
• Such systems also accept even inaccurate, distorted and erroneous information.
• Fuzzy logic systems can be easily designed.
• Because these systems are associated with human reasoning and decision making, they are useful in forming decisions in complex situations in various types applications.

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The mathematical theory of fuzzy sets and fuzzy logic are generalizations of classical set theory and classical formal logic. These concepts were first proposed by the American scientist Lotfi Zadeh in 1965. The main reason for the emergence of the new theory was the presence of fuzzy and approximate reasoning when humans describe processes, systems, and objects.

Before fuzzy modeling approach complex systems received recognition all over the world, more than a decade has passed since the birth of the theory of fuzzy sets. And along this path of development of fuzzy systems, it is customary to distinguish three periods.

The first period (late 60s–early 70s) is characterized by the development of the theoretical apparatus of fuzzy sets (L. Zadeh, E. Mamdani, Bellman). In the second period (70–80s), the first practical results appeared in the field of fuzzy control of complex technical systems(steam generator with fuzzy control). At the same time, attention began to be paid to the issues of building expert systems based on fuzzy logic and the development of fuzzy controllers. Fuzzy expert systems for decision support are widely used in medicine and economics. Finally, in the third period, which lasts from the late 80s and continues today, software packages for building fuzzy expert systems appear, and the areas of application of fuzzy logic are noticeably expanding. It is used in the automotive, aerospace and transportation industries, in the field of household appliances, in finance, analysis and management decision-making, and many others.

The triumphal march of fuzzy logic around the world began after Bartholomew Cosco proved the famous FAT theorem (Fuzzy Approximation Theorem) in the late 80s. In business and finance, fuzzy logic gained recognition after expert system based on fuzzy rules for predicting financial indicators, the only one predicted a stock market crash. And the number of successful fuzzy applications now numbers in the thousands.

Mathematical apparatus

A characteristic of a fuzzy set is the Membership Function. Let us denote by MF c (x) the degree of membership in the fuzzy set C, which is a generalization of the concept of the characteristic function of an ordinary set. Then a fuzzy set C is a set of ordered pairs of the form C=(MF c (x)/x), MF c (x) . The value MF c (x)=0 means no membership in the set, 1 means complete membership.

Let's illustrate this with a simple example. Let's formalize the imprecise definition of "hot tea". The x (discussion area) will be the temperature scale in degrees Celsius. Obviously, it will vary from 0 to 100 degrees. A fuzzy set for the concept “hot tea” might look like this:

C=(0/0; 0/10; 0/20; 0.15/30; 0.30/40; 0.60/50; 0.80/60; 0.90/70; 1/80; 1 /90; 1/100).

Thus, tea with a temperature of 60 C belongs to the “Hot” set with a degree of membership of 0.80. For one person, tea at a temperature of 60 C may be hot, for another it may not be too hot. This is precisely where the vagueness of specifying the corresponding set manifests itself.

For fuzzy sets, as for ordinary sets, the basic logical operations are defined. The most basic ones needed for calculations are intersection and union.

Intersection of two fuzzy sets (fuzzy “AND”): A B: MF AB (x)=min(MF A (x), MF B (x)).
Union of two fuzzy sets (fuzzy "OR"): A B: MF AB (x)=max(MF A (x), MF B (x)).

Developed in fuzzy set theory general approach to the execution of intersection, union and complement operators, implemented in the so-called triangular norms and conorms. The above implementations of the intersection and union operations are the most common cases of t-norm and t-conorm.

To describe fuzzy sets, the concepts of fuzzy and linguistic variables are introduced.

A fuzzy variable is described by a set (N,X,A), where N is the name of the variable, X is a universal set (domain of reasoning), A is a fuzzy set on X.
The values ​​of a linguistic variable can be fuzzy variables, i.e. linguistic variable is at more high level than a fuzzy variable. Each linguistic variable consists of:

• titles;
• set of its values, which is also called the basic term set T. The elements of the basic term set are the names of fuzzy variables;
• universal set X;
• syntactic rule G, according to which new terms are generated using words of a natural or formal language;
• semantic rule P, which associates each value of a linguistic variable with a fuzzy subset of the set X.

Let's consider such a fuzzy concept as “Share price”. This is the name of the linguistic variable. Let’s form a basic term set for it, which will consist of three fuzzy variables: “Low”, “Moderate”, “High” and set the scope of reasoning in the form X= (units). The last thing left to do is to construct membership functions for each linguistic term from the base term set T.

There are over a dozen standard curve shapes for specifying membership functions. The most widely used are: triangular, trapezoidal and Gaussian membership functions.

The triangular membership function is defined by a triple of numbers (a,b,c), and its value at point x is calculated according to the expression:

$$MF\,(x) = \,\begin(cases) \;1\,-\,\frac(b\,-\,x)(b\,-\,a),\,a\leq \,x\leq \,b &\ \\ 1\,-\,\frac(x\,-\,b)(c\,-\,b),\,b\leq \,x\leq \ ,c &\ \\ 0, \;x\,\not \in\,(a;\,c)\ \end(cases)$$

When (b-a)=(c-b) we have the case of a symmetric triangular membership function, which can be uniquely specified by two parameters from the triple (a,b,c).

Similarly, to specify a trapezoidal membership function, you need four numbers (a,b,c,d):

$$MF\,(x)\,=\, \begin(cases) \;1\,-\,\frac(b\,-\,x)(b\,-\,a),\,a \leq \,x\leq \,b & \\ 1,\,b\leq \,x\leq \,c & \\ 1\,-\,\frac(x\,-\,c)(d \,-\,c),\,c\leq \,x\leq \,d &\\ 0, x\,\not \in\,(a;\,d) \ \end(cases)$$

When (b-a)=(d-c) the trapezoidal membership function takes on a symmetrical form.

The membership function of Gaussian type is described by the formula

$$MF\,(x) = \exp\biggl[ -\,(\Bigl(\frac(x\,-\,c)(\sigma)\Bigr))^2\biggr]$$

and operates with two parameters. Parameter c denotes the center of the fuzzy set, and the parameter is responsible for the slope of the function.

The collection of membership functions for each term in the underlying term set T is usually plotted together on a single graph. Figure 3 shows an example of the linguistic variable “Share Price” described above; Figure 4 shows a formalization of the imprecise concept “Person’s Age”. Thus, for a 48-year-old person, the degree of membership in the set “Young” is 0, “Average” – 0.47, “Above average” – 0.20.

The number of terms in a linguistic variable rarely exceeds 7.

Fuzzy inference

The basis for carrying out the fuzzy logical inference operation is a rule base containing fuzzy statements in the form of “If-then” and membership functions for the corresponding linguistic terms. In this case, the following conditions must be met:

1. There is at least one rule for each linguistic term of the output variable.
2. For any term of the input variable there is at least one rule in which this term is used as a prerequisite ( left side rules).

Otherwise, there is an incomplete base of fuzzy rules.

Let the rule base have m rules of the form:
R 1: IF x 1 is A 11... AND... x n is A 1n, THEN y is B 1
…
R i: IF x 1 is A i1 ... AND ... x n is A in , THEN y is B i
…
R m: IF x 1 is A i1 ... AND ... x n is A mn, THEN y is B m,
where x k, k=1..n – input variables; y – output variable; A ik – given fuzzy sets with membership functions.

The result of fuzzy inference is a clear value of the variable y * based on the given clear values ​​x k , k=1..n.

In general, the inference mechanism includes four stages: introduction of fuzziness (phasification), fuzzy inference, composition and reduction to clarity, or defuzzification (see Figure 5).

Fuzzy inference algorithms differ mainly in the type of rules used, logical operations and a type of defasification method. Mamdani, Sugeno, Larsen, Tsukamoto fuzzy inference models have been developed.

Let's take a closer look at fuzzy inference using the Mamdani mechanism as an example. This is the most common method of inference in fuzzy systems. It uses minimax composition of fuzzy sets. This mechanism includes the following sequence of actions.

1. Phasification procedure: degrees of truth are determined, i.e. values ​​of membership functions for the left sides of each rule (prerequisites). For a rule base with m rules, we denote the degrees of truth as A ik (x k), i=1..m, k=1..n.

Fuzzy output. First, the cutoff levels for the left side of each rule are determined:

$$alfa_i\,=\,\min_i \,(A_(ik)\,(x_k))$$

$$B_i^*(y)= \min_i \,(alfa_i,\,B_i\,(y))$$

Composition, or combination of the resulting truncated functions, for which the maximum composition of fuzzy sets is used:

$$MF\,(y)= \max_i \,(B_i^*\,(y))$$

where MF(y) is the membership function of the final fuzzy set.

Dephasification, or bringing to clarity. There are several defuzzification methods. For example, the mean center method, or centroid method:
$$MF\,(y)= \max_i \,(B_i^*\,(y))$$

The geometric meaning of this value is the center of gravity for the MF(y) curve. Figure 6 graphically shows the Mamdani fuzzy inference process for two input variables and two fuzzy rules R1 and R2.

Integration with intelligent paradigms

Hybridization of methods of intellectual information processing is the motto under which the 90s passed among Western and American researchers. As a result of combining several technologies artificial intelligence a special term appeared - “soft computing”, which was introduced by L. Zadeh in 1994. Currently, soft computing combines such areas as: fuzzy logic, artificial neural networks, probabilistic reasoning and evolutionary algorithms. They complement each other and are used in various combinations to create hybrid intelligent systems.

The influence of fuzzy logic turned out to be perhaps the most extensive. Just as fuzzy sets expanded the scope of classical mathematical set theory, fuzzy logic has “invaded” almost the majority of Data methods Mining, giving them new functionality. Below are the most interesting examples such associations.

Fuzzy neural networks

Fuzzy neural networks carry out inferences based on fuzzy logic, but the parameters of membership functions are adjusted using NN learning algorithms. Therefore, to select the parameters of such networks, we apply the error backpropagation method, originally proposed for training a multilayer perceptron. For this purpose, the fuzzy control module is represented in the form multilayer network. A fuzzy neural network usually consists of four layers: a layer of phasification of input variables, a layer of aggregation of condition activation values, a layer of aggregation of fuzzy rules and an output layer.

The most widely used fuzzy neural networks architectures are ANFIS and TSK. It has been proven that such networks are universal approximators.

Fast learning algorithms and interpretability of accumulated knowledge - these factors have made fuzzy neural networks today one of the most promising and effective tools soft computing.

Adaptive fuzzy systems

Classic fuzzy systems have the disadvantage that to formulate rules and membership functions it is necessary to involve experts of one or another subject area, which is not always possible to ensure. Adaptive fuzzy systems solve this problem. In such systems, the selection of fuzzy system parameters is carried out in the process of training on experimental data. Algorithms for training adaptive fuzzy systems are relatively labor-intensive and complex compared to algorithms for training neural networks, and, as a rule, consist of two stages: 1. Generation of linguistic rules; 2. Correction of membership functions. The first problem is an exhaustive search type problem, the second is an optimization problem in continuous spaces. In this case, a certain contradiction arises: to generate fuzzy rules, membership functions are needed, and to carry out fuzzy inference, rules are needed. In addition, when automatically generating fuzzy rules, it is necessary to ensure their completeness and consistency.

A significant part of the methods for teaching fuzzy systems uses genetic algorithms. In the English-language literature, this corresponds to a special term – Genetic Fuzzy Systems.

A significant contribution to the development of the theory and practice of fuzzy systems with evolutionary adaptation was made by a group of Spanish researchers led by F. Herrera.

Fuzzy queries

Fuzzy queries to databases are a promising direction in modern systems information processing. This tool makes it possible to formulate queries in natural language, for example: “Display a list of inexpensive housing offers close to the city center,” which is impossible when using the standard query mechanism. For this purpose, a fuzzy relational algebra and special SQL language extensions for fuzzy queries. Most of the research in this area belongs to Western European scientists D. Dubois and G. Prade.

Fuzzy association rules

Fuzzy association rules(fuzzy associative rules) – a tool for extracting patterns from databases that are formulated in the form of linguistic statements. Here, special concepts of fuzzy transaction, support and reliability of a fuzzy association rule are introduced.

Fuzzy cognitive maps

Fuzzy cognitive maps were proposed by B. Kosko in 1986 and are used to model the causal relationships identified between the concepts of a certain area. Unlike simple cognitive maps, fuzzy cognitive maps are a fuzzy directed graph whose nodes are fuzzy sets. The directed edges of the graph not only reflect the cause-and-effect relationships between concepts, but also determine the degree of influence (weight) of the connected concepts. Active use fuzzy cognitive maps as a means of modeling systems is due to the possibility visual representation the analyzed system and ease of interpretation of cause-and-effect relationships between concepts. The main problems are related to the process of constructing a cognitive map, which cannot be formalized. In addition, it is necessary to prove that the constructed cognitive map is adequate to the real system being modeled. To solve these problems, algorithms for automatically constructing cognitive maps based on data sampling have been developed.

Fuzzy Clustering

Fuzzy clustering methods, in contrast to clear methods (for example, Kohonen neural networks), allow the same object to belong to several clusters simultaneously, but with varying degrees. Fuzzy clustering in many situations is more “natural” than clear clustering, for example, for objects located on the border of clusters. The most common are the c-means fuzzy self-organization algorithm and its generalization in the form of the Gustafson-Kessel algorithm.

Literature

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• Kruglov V.V., Dli M.I. Intelligent information systems: computer support for fuzzy logic and fuzzy inference systems. – M.: Fizmatlit, 2002.
• Leolenkov A.V. Fuzzy modeling in MATLAB and fuzzyTECH. – St. Petersburg, 2003.
• Rutkowska D., Pilinski M., Rutkowski L. Neural networks, genetic algorithms and fuzzy systems. – M., 2004.
• Masalovich A. Fuzzy logic in business and finance. www.tora-centre.ru/library/fuzzy/fuzzy-.htm
• Kosko B. Fuzzy systems as universal approximators // IEEE Transactions on Computers, vol. 43, No. 11, November 1994. – P. 1329-1333.
• Cordon O., Herrera F., A General study on genetic fuzzy systems // Genetic Algorithms in engineering and computer science, 1995. – P. 33-57.