Fuzzy Logic Toolbox section. S.D. Shtovba. Introduction to fuzzy set theory and fuzzy logic
Linguistic Variables( LP ) are a way of describing complex systems, the parameters of which are considered not from quantitative positions, but as qualitative ones. At the same time, linguistic variables make it possible to correlate quality characteristics some quantitative interpretation with a given degree of confidence, which makes it possible to process qualitative data on a computer. Another area of application of linguistic variables is fuzzy logical inference, the difference from the usual one is that the truth of logical statements is determined not by two values 0 and 1, but by a set of values in the interval .
The concept of a linguistic variable is based on the concept of an odd variable.
fuzzy variable is a combination of three elements:
< X, U, µ A(u) >,
Where X– name of the fuzzy variable; U– universal set; µ A(u) – fuzzy subset A universal set U. In other words, a fuzzy variable is a named fuzzy set.
Linguistic variable is called a set of five elements:
< L, T(X), U, G, M >,
Where L– name of the linguistic variable;
T(X) – a set of basic terms of a linguistic variable, consisting of a set of names of values of linguistic variables ( T 1 , T 2 , …, Tn), each of which corresponds to a fuzzy variable X universal set U;
U– a universal set on which a linguistic variable is defined;
G– syntactic rule that generates names X variable values;
M– a semantic rule that assigns each fuzzy variable X its meaning M(X), i.e. fuzzy subset of the universal set U.
The terms of a linguistic variable are subject to the ordering requirement: T 1 < T 2 < … < Tn.
The membership functions of fuzzy sets that make up the quantitative meaning of the basic terms of a linguistic variable must satisfy the following conditions:
2. : 
;
4. : 
.
Here n– the number of basic terms of the linguistic variable; umin, umax– boundaries of the universal set U, on which the linguistic variable is defined. If U R (R is the set of real numbers, then U = [umin, umax].
Syntax rule G is a combination of four elements: G = < V T, V N, T, P >,
Where V T– a set of terminal symbols or words; V N– a set of non-terminal symbols or phrases; T– a set of basic terms; R– a set of substitution rules that determine the equivalence of phrases.
Semantic rule M assigns each phrase a new non-
a clear set defined on the basis of membership functions of basic terms and a set of operations with fuzzy sets.
As an example, consider the numerical linguistic variable “person height”. Let the variable values be specified using three basic terms: “low”, “medium”, “high”. The terms are ordered. Universal number set U V in this case is the interval U = .
The membership functions of terms are shown in Fig. 7.6 and satisfy the requirements discussed above.
Rice. 7.6 Linguistic variable “Human height”
As a syntactic rule, we define that the set of non-terminal symbols includes the words “and”, “or”, “more or less”, “not”, “very”, which can be combined with the basic terms “low”, “medium”, “ high”, and the following rules must be followed:
The symbols “and” and “or” can only connect two phrases or basic terms, and the remaining non-terminal symbols are unary, i.e. may precede a phrase or base term; for example, “not high”, “very low”, “low or average”;
The simultaneous negation of two basic terms, for example, “not low and not high,” is equivalent to the remaining basic term, i.e. "average".
By applying these rules, you can construct many phrases and substitution rules. If the syntactic rule cannot be specified algorithmically, then all possible phrases are simply listed.
As a semantic rule, we define the correspondence between non-terminal symbols and operations on fuzzy sets:
“not” – addition;
“and” - intersection;
“or” - union;
“very” - concentration;
"more or less" is an extension.
Using the considered linguistic variable, we can estimate
determine the height of people without resorting to precise measurements.
Thus, with the help of linguistic variables it is possible to describe objects whose precise measurement of characteristics is either extremely labor-intensive or completely impossible.
The formation of a linguistic variable, as a rule, is carried out on the basis of a survey of experts - specialists in the field for which the LP is being built. In this case, special attention is paid to the formation of membership functions of fuzzy sets, which are the basic terms of a linguistic variable, since the definition of syntactic and semantic rules for most linguistic variables is standard and in practice comes down to listing all possible phrases and interpreting non-terminal symbols, as shown above.
The process of forming a linguistic variable includes next steps:
1. Definition of the set of LP terms and its ordering.
2. Construction of the numerical domain of definition of the LP.
3. Determining the scheme for interviewing experts and conducting the survey.
4. Construction of membership functions for each LP term.
Stage 1 involves the expert specifying the number of LP terms and the names of the corresponding fuzzy variables. The number of terms is selected from the range n= 7±2.
At stage 2, the universal set is described U, which can be numeric or non-numeric. The type of universal set depends on the objects being described and determines the method of forming membership functions of LP terms.
Stage 3 is key in the formation of the LP. There are two types
expert survey: direct and indirect. Each of these methods can be individual or group. The simplest from the point of view of organization and
software implementation is an individual way of interviewing experts.
During a direct survey of experts, all parameters of the membership functions are directly indicated. The disadvantage here is the manifestation of subjectivity in judgments, as well as the need for an expert to know the basics of fuzzy logic. In an indirect survey, membership functions are formed based on the expert’s answer to “leading” questions. This increases the objectivity of the assessment and does not require knowledge of fuzzy logic, but increases the risk of inconsistency in the expert’s judgments.
With group survey methods, the result is formed based on combining the opinions of several experts. In practice, individual indirect interviews are most often used.
Lecture. Fuzzy calculations
Concept of fuzzy number
One of the areas of application of fuzzy logic is the performance of arithmetic operations with fuzzy sets. To reduce the complexity of such operations, a special type of fuzzy sets is used - fuzzy numbers.
Fuzzy number(NF) is a fuzzy variable that has the following properties: 
; .
In other words, a fuzzy number is a named fuzzy set for which the universal set U represents the real axis interval R.
In real problems, piecewise linear fuzzy numbers are used. To simplify arithmetic operations, piecewise linear membership functions are additionally approximated to obtain a special type of fuzzy numbers - parametric fuzzy numbers or fuzzy numbers
(L–R)-type, which are characterized by compactness of representation and simple-
that implementation of arithmetic operations.
fuzzy number A called fuzzy number (L–R)–type, if its membership function has the following form (Fig. 7.8):
0, 
where are the parameters of the fuzzy number; L(x), R(x) – some functions.
A fuzzy parametric number is denoted by ( a, b, c, d)LR.
Thus, the fuzzy number ( L–R)-type is described by six parameters: four numbers indicating its boundaries, and two functions determining the form of its membership function.
 |
Fig.7.8 Parametric fuzzy numbers
Fuzzy number is called unimodal, if it has only one point at which the membership function is equal to one, i.e. its parameters b And c are equal, otherwise the fuzzy number is called tolerant(see Fig. 7.8). Unimodal fuzzy numbers are denoted by five parameters ( a, b, d)LR.
As LR–functions most often used linear dependencies, given by the following relations:
LR– functions can also be specified by quadratic, exponential and other dependencies.
In case of use linear functions unimodal and tolerant fuzzy numbers are called triangular and trapezoidal, respectively, and denoted by ( a, b, d) And ( a, b, c, d).
For fuzzy numbers, the concept of sign and zero value is defined in a special way.
fuzzy number A called positive, if its base lies in the positive real semi-axis or
fuzzy number A called negative, if its base lies in the negative real semi-axis or
For parametric fuzzy numbers, the sign is determined by the values of the parameters: a positive fuzzy number if a> 0; negative if d < 0; нечеткий ноль, если .
The concept of fuzzy and linguistic variables is used to describe objects and phenomena using fuzzy sets.
Fuzzy variable characterized by three (α, X, A), Where
α — name of the variable;
X— universal set (domain α);
A- fuzzy set on X, describing the restrictions (i.e. μ A(x) ) to the values of the fuzzy variable α.
Linguistic variable (LP) is the set ( β , T, X, G, M), where
β — name of the linguistic variable;
T— a set of its values (term set), which are the names of fuzzy variables, the domain of definition of each of which is a set X. Many T called basic term-set linguistic variable;
G is a syntactic procedure that allows you to operate with elements of the term set T, in particular, to generate new terms (values). The set T∪G(T), where G(T) is the set of generated terms, is called the extended term set of a linguistic variable;
M— a semantic procedure that allows you to turn each new value of a linguistic variable generated by procedure G into a fuzzy variable, i.e. form the corresponding fuzzy set.
Comment. To avoid large quantity characters:
1) symbol β used both for the name of the variable itself and for all its values;
2) use the same symbol to denote a fuzzy set and its name, for example the term “Young”, which is the value of a linguistic variable β = “age”, at the same time there is a fuzzy set M("Young").
Assigning multiple meanings to symbols assumes that context allows possible ambiguities to be resolved.
Example. Let the expert determine the thickness of the manufactured product using the concepts “Small thickness”, “Medium thickness” and “Large thickness”, with the minimum thickness being 10 mm and the maximum being 80 mm.
Formalization of such a description can be carried out using the following linguistic variable ( β , T, X, G, M ), Where
β — thickness of the product;
T— (“Small thickness”, “Medium thickness”, “Large thickness”);
X— ;
G - the procedure for the formation of new terms using connectives “and”, “or” and modifiers such as “very”, “not”, “slightly”, etc. For example: “Small or medium thickness”, “Very small thickness”, etc.;
M- task procedure for X = fuzzy subsets A 1 = "Small thickness", A 2 = "Medium thickness", A 3 = “Large thickness”, as well as fuzzy sets for terms from G (T) in accordance with the rules of translation of fuzzy connectives and modifiers “and”, “or”, “not”, “very”, “slightly” and other operations on fuzzy sets of the form: A⋂IN,A∪INA, CON A =A 2 , DIL A = A 0.5 etc.
Comment. Along with the basic values of the linguistic variable “Thickness” discussed above (T =(“Small thickness”, “Medium thickness”, “Large thickness”)) possible values depend on the domain of definition X. In this case, the values of the linguistic variable “Product thickness” can be defined as “about 20 mm”, “about 50 mm”, “about 70 mm”, i.e. in the form of fuzzy numbers.
The term set and extended term set in the example conditions can be characterized by the membership functions shown in Fig. 1.5 and 1.6.
Rice. 1.5. Fuzzy set membership functions: “Small thickness” = A 1,"Medium thickness" = A 2, "Large thickness" = A 3
Rice. 1.6. Fuzzy set membership function “Small or medium thickness” = A 1 ∪ A 2
Fuzzy numbers
Fuzzy numbers- fuzzy variables defined on the number axis, i.e. a fuzzy number is defined as a fuzzy set A on the set of real numbers ℝwith membership function μ A(X) ϵ , where X- real number, i.e. X ϵ ℝ.
fuzzy number It's okay if tah μ A(x) = 1; convex, if for any X ≤ at ≤ z running
μ A (x)≥ μ A(at) ˄ μ A(z).
Many α -fuzzy number level A defined as
Aα= {x/μ α (x) ≥ α } .
Subset S A⊂ ℝ is called the support of the fuzzy number A, If
S A= { x/μ A(x) > 0 }.
fuzzy number And unimodally, if condition μ A(X) = 1 is valid only for one point of the real axis.
Convex fuzzy number A called fuzzy zero, If
μ A(0) = sup ( μ A(x)).
fuzzy number And positively, if ∀ xϵ S A, X> 0 and negative, if ∀ X ϵ S A, X< 0.
Operations on fuzzy numbers
Extended binary arithmetic operations(addition, multiplication, etc.) for fuzzy numbers are determined through the corresponding operations for clear numbers using the generalization principle as follows.
Let A And IN- fuzzy numbers, and - fuzzy operation corresponding to an arbitrary algebraic operation * on ordinary numbers. Then (using here and henceforth the notation instead instead of ) we can write
Fuzzy Numbers (L-R)-Type
Fuzzy numbers (L-R)-type are a type of fuzzy numbers of a special type, i.e. specified according to certain rules in order to reduce the amount of calculations when performing operations on them.
Membership functions of (L-R)-type fuzzy numbers are specified using functions of the real variable L(, non-increasing on the set of non-negative real numbers x) and R( x), satisfying the following properties:
a) L(- x) = L( x), R(- x) = R( x);
b) L(0) = R(0).
Obviously, the class of (L-R)-functions includes functions whose graphs look like those shown in Fig. 1.7.
Rice. 1.7. Possible form of (L-R) functions
Examples of analytical tasks of (L-R) functions can be
Let L( at) and R( at)—(L-R)-type (concrete) functions. Unimodal fuzzy number A With fashion a(i.e. μ A(A) = 1) using L( at) and R( at) is given as follows:
where a is the mode; α > 0, β > 0 — left and right fuzziness coefficients.
Thus, for given L( at) and R( at) fuzzy number (uni-modal) is given by a triple A = (A, α, β ).
The tolerant fuzzy number is specified, respectively, by four parameters A = (a 1 , A 2 , α, β ), Where A 1 and A 2 - limits of tolerance, i.e. in between [ a 1 , A 2 ] the value of the membership function is 1.
Examples of graphs of membership functions of (L-R)-type fuzzy numbers are shown in Fig. 1.8.
Rice. 1.8. Examples of graphs of membership functions of fuzzy numbers (L-R)-type
Note that in specific situations the functions L (y), R (y), as well as parameters A, β fuzzy numbers (A, α, β ) And ( a 1 , A 2 , α, β ) must be selected in such a way that the result of the operation (addition, subtraction, division, etc.) is exactly or approximately equal to a fuzzy number with the same L (y) and R (y), and the parameters α" And β" the results did not go beyond the restrictions on these parameters for the original fuzzy numbers, especially if the result will subsequently participate in operations.
Comment. Solving problems of mathematical modeling of complex systems using the apparatus of fuzzy sets requires performing a large volume of operations on various kinds of linguistic and other fuzzy variables. For ease of execution of operations, as well as for input-output and data storage, it is advisable to work with standard-type membership functions.
Fuzzy sets, which have to be operated in most problems, are, as a rule, unimodal and normal. One of possible methods approximation of unimodal fuzzy sets is approximation using (L-R)-type functions.
Examples of (L-R)-representations of some linguistic variables are given in Table. 1.2.
Table 1.2. Possible (L- R)-representation of some linguistic variables
2.9.1. Definition. Using the methods of fuzzy set theory, semantic concepts are described, for example, for the concept of “reliability of a node” it is possible to define such components as “small value of node reliability”, “average value of node reliability”, “large value of node reliability”, which are specified as fuzzy sets on a basic set defined by all possible values of reliability values.
A generalization of the description of linguistic variables from a formal point of view is the introduction of fuzzy and linguistic variables.
N clear variable is called a triple of sets, where a- name of the fuzzy variable, X- domain of definition, - fuzzy subset in the set X, describing restrictions on the possible values of the variable a.
Linguistic variable is called a collection of sets
From the definition it follows that a linguistic variable is a variable specified on a quantitative (measurable) scale and taking values that are words or phrases of the natural language of communication. Fuzzy variables describe the values of a linguistic variable. In Fig. Figure 2.20 shows the relationship between the basic concepts.
Thus, linguistic variables can be used to describe difficult-to-formalize concepts in the form of a qualitative, verbal description. When describing a linguistic variable and all its values, it is associated with a specific quantitative scale, which, by analogy with the basic set, is sometimes called the basic scale.
Using linguistic variables, it is possible to formalize qualitative information in management systems, which is formulated in verbal form by specialists (experts). This allows you to build fuzzy models of control systems (fuzzy controllers).
2.9.2. Type of membership functions. Let us consider the requirements that are put forward for the type of membership functions of fuzzy sets that describe the terms of linguistic variables.
Let the linguistic variable
Many T arrange according to the expression
"T i ,T j ÎT i>j«($xÎC i)("yÎC j)(x>y). (2.5)
Expression (2.5) requires that a term that has a support located to the left receives a lower number. Then the term set of any linguistic variable must satisfy the conditions:
("T i ÎT)($xÎX)( ); (2.8)
("b)($x 1 ОR 1)($x 2 ОR 2)("xОX)(x 1
Condition (2.6) requires that the values of the membership functions of the extreme terms (T 1 And T 2) at points x 1 And x 2 accordingly, equal to one and so that the appearance of bell-shaped curves is not allowed, as shown in Fig. 2.21.
Fig.2.21
Condition (2.7) prohibits in the base set X pairs of terms of type T 1 And T 2, T 2 And T 3. For a couple T 1 And T 2 there is no natural differentiation of concepts. For a couple T 2 And T 3 segment no concept matches. Condition (2.7) prohibits the existence of terms of the type T 4, since every concept has at least one typical object. Condition (2.8) determines the physical limitation (within the problem) on the numerical values of the parameters.
In Fig. Figure 2.22 shows an example of specifying the membership functions of the terms “small price value”, “small price value”, “average price value”, “sufficiently large price value”, “large price value” of the linguistic variable “product price”.
2.9.3. Universal scales. Membership functions are constructed based on the results of expert surveys. However, the procedure for using fuzzy sets constructed based on the results of a survey of experts has the disadvantage that a change in the operating conditions of the model (object) requires adjustment of the fuzzy sets. Adjustments can be made based on the results of a repeated survey of experts.
One of the ways to overcome this shortcoming is the transition to universal scales for measuring the values of the estimated parameters. The well-known methodology for constructing universal scales involves describing the frequency of phenomena and processes, which at a qualitative level in natural language is determined by the following words and phrases: “never”, “extremely rarely”, “rarely”, “neither rarely nor often”, “often”, “ very often”, “almost always” (or similar). A person uses these concepts to assess the subjective frequencies of events (the ratio of the number of events characterized by the concept to the total number of events).
The universal scale is built on a segment and represents a series of intersecting bell-shaped curves corresponding to the scaled frequency estimates. A universal scale of a linguistic variable for a given estimated parameter of a control object is constructed according to the following procedure.
1. According to the expert survey, the minimum xmin and maximum xmax variable scale values X.
2. Based on the results of an expert survey, membership functions of fuzzy sets describing the values of a linguistic variable defined on a scale are constructed X. In Fig. Figure 2.23 shows an example of constructing membership functions, where a 1 , a 2 , a 3- some names of fuzzy variables.
3. Points ( xmin,0) and ( xmax,1) are connected by a straight line p 0, which is the mapping function p 0:X®.
4. The transition from a scale of relative frequencies of occurrence of events to frequency estimates, called quantifiers, occurs as follows.
For an arbitrary point z on the universal scale its prototype is built on the scale X. Then, using the membership functions of fuzzy sets corresponding to the terms a 1 , a 2 , a 3, the values are determined which are taken as the values of the corresponding membership functions at point z on the universal scale. Function p (p=p 0 in the example considered) is determined by an expert survey, because its choice affects the adequacy of the model to the object under study.
2.9.4. Multiple display functions. Unambiguous definition of the mapping function p limit the possibility of simultaneous consideration of different criteria in the control system, which may even be in antagonism with respect to each other, as well as the possibility of simultaneous consideration of various control conditions determined by the properties of the controlled object.
Taking into account various conditions and criteria is determined by a subjective approach to solving the problem. If we accept the mapping function of an unambiguous form, then different points of view will be reduced to a “common denominator” or actually rejected. Practice shows that when managing processes that are difficult to formalize, taking into account all variants of subjective views improves the quality of management, increasing resistance to various kinds of disturbances. However, it should be noted that it is almost never possible to take into account in people all the conditions that influence the choice of control and all the characteristics of the object. Let's consider how formalized accounting of control conditions is carried out when interviewing experts in the form of multiple mapping functions.
Let the composition of the states of the object under study be quantitatively and qualitatively determined from expert surveys. An object's states are assessed based on the values of the attributes y i ОY=(y 1 ,y 2 ,…,y p ).
It is impossible to take into account everything, therefore, when assessing states, it is better to use fuzzy categories, and fuzzy definitions of parameter values should be made with a certain degree of uncertainty about the correctness of the definitions. Indeed, one can always assume that there are some set of signs 
, not indicated by experts for various reasons: they were forgotten; experts believe that these features do not affect accuracy; These parameters cannot be assessed, a consequence of technical difficulties.
Display functions p i ОP=(p 1 ,p 2 ,…,p b ) degrees of confidence are compared b(p i)О, which are asked by experts. Also each display function p i weight is compared a(pi), which corresponds to the expert’s level of competence. Weight values a(pi) are determined by the numbers of the segment. So the multiple mapping function P=(p 1 ,p 2 ,…,p b ) consists of a set of mapping functions p i, each of which is associated with a degree g(pi), defined as the conjunction of degrees of competence and confidence in the correct definition of mapping functions p i, i.e. g(pi)=a(p i)&b(p i).
The practical use of multiple functions has shown that, within the limits of a certain competence of experts, the constructed multiple mapping function is in good agreement with their individual opinions about the most plausible correspondence of fuzzy concepts to points on the subject scale X.
FUZZY LOGIC
Fuzzy AND operation
Defining fuzzy sets allows one to generalize clear logical operations into their fuzzy analogues. A fuzzy extension of the AND operation is the triangular norm T, Another name T– the norms are S–conorma. In Fig. 3.1 shows a schematic representation T-norms.
The fuzzy AND operation in general form is defined as the mapping:
for which the axioms hold:
Axioms of boundary conditions T– norms:
Axiom of orderliness:
In the theory of fuzzy sets, there are an innumerable number of fuzzy “AND” operations, which are determined by the ways of specifying the operation (T) when conditions (3.1) - (3.2) are met. In the theory of fuzzy control, the following methods for specifying an operation (T), listed below, are applicable.
Logical product[Zadeh, 1973]:
, "xÎ R. (3.6)
Algebraic product[Bandler, Kohout, 1980]:
, "xÎ R, (3.7)
Where "." - a product accepted in classical algebra.
Boundary product[Lukashevich, Giles, 1976]:
, (3.8)
where is the boundary product symbol.
Strong, or drastic (drastic), work[Weber, 1983]:
(3.9)
where D is the strong product symbol.
In Fig. Figure 3.2 shows the membership function for logical, algebraic, boundary and strong products of fuzzy sets.
Fuzzy OR operation
A fuzzy extension of the OR operation is S-norm. Sometimes the name is used T–conorma. In Fig. 3.3 shows a schematic representation S-norms.
The fuzzy OR operation is defined as the mapping
for which mappings are performed:
Axioms of boundary conditions T– norms:
, ; (3.10)
Axioms of unification (recombination):
Axiom of orderliness:
From infinite number fuzzy operations satisfying axioms (3.10) – (3.14), the following operations listed below have found application in control theory.
Logical sum[Zadeh, 1973]:
, "xÎ R. (3.15)
Algebraic sum[Bandler and Kohout, 1980]:
, "xÎ R, (3.16)
Limit amount[Lukashevich, Giles, 1976]:
, (3.17)
Strong, or drastic (drastic), amount[Weber, 1983]:
(3.18)
Comparison of axioms T–norms with axioms S-norms shows that the difference between them lies only in the axioms of the boundary conditions.
In Fig. Figure 3.4 shows the membership function for logical, algebraic, boundary and strong sum of fuzzy sets.
Fuzzy operation "NOT"
The fuzzy “NOT” operation is defined as a mapping for which the following axioms hold:
The set of mappings that satisfy axioms (3.19) – (3.21) is a fuzzy negation. The operation of fuzzy negation in the form of a diagram is shown in Fig. 3.5.
From the infinite number of fuzzy “NOT” operations that satisfy axioms (3.19) – (3.21), the following operations listed below have found application in control theory.
Fuzzy "NOT" according to Zadeh(1973) is defined as subtracting from one:
. (3.22)
Fuzzy "NOT" according to Sugeno(1977) or l-complement is defined as
. (3.23)
At l=0 equation (3.23) coincides with equation (3.22).
Fuzzy "NOT" according to Yager(1980) is defined as:
, (3.24)
Where p>0- parameter. At p=1 equation (3.24) coincides with equation (3.22).
For T- norms and S- norms, there may be various versions of negations due to the infinite number of possible fuzzy “NOT” operations. However, it is advisable to choose negation options that satisfy the following conditions:
These conditions, by analogy with clear logic, are called De Morgan’s fuzzy laws. Operations (3.25) and (3.26) are called mutually dual, because in the theory of fuzzy sets it is proven that from (3.25) follows (3.26) and, conversely, from (3.26) follows (3.25).
The following fuzzy operations are also mutually dual:
; (3.29)
Fuzzy inference algebra
3.4.1. Base of fuzzy rules. In fuzzy logic there is the concept of a fuzzy proposition. A fuzzy sentence is defined as the statement " ". Symbol " x" denotes a physical quantity (current, voltage, pressure, speed, etc.), the symbol " " denotes a linguistic variable (LP), and the symbol " p" - abbreviation proposition - proposal. For example, in the statement “the magnitude of the current is large” of the physical variable x is the "magnitude of current" that can be measured by a current sensor. The fuzzy set is defined by the “big” LP and formalized by the membership function m A (x). The connective “is” corresponds to an ordering operation in the form of equality, which is denoted by the symbol “=”. Receives a formalized form of the sentence " » .
A fuzzy sentence can consist of several separate fuzzy sentences connected to each other by connectives “AND” and “OR”. The choice of logical connectives “AND”, “OR” depends on the meaning and context of sentences, on the relationship between them. Note that the operations of fuzzy “AND” and “OR” according to Zadeh (formulas (3.6) and (3.15)) in control theory are preferable to the others, because they have no redundancy. When fuzzy sentences are not equivalent, but are correlated and interconnected, then it is possible to use T- norms and S- norms according to Lukashevich (formulas (3.8) and (3.17)).
Offer p can be represented as a fuzzy relation R with membership function: . To compose a fuzzy sentence consisting of several separate fuzzy sentences connected by “AND” connectives, use the “if” indicator. As a result, we obtain a system of conditional fuzzy statements:
.
Fuzzy sentences are called conditions or prerequisites.
A set of conditions allows one to construct a set conclusions or conclusions. In this case, the “then” indicator is used.
Production fuzzy rule(fuzzy rule) is a set of conditions and conclusions:
R 1: if x 1 = and x 2 = and..., then y 1 = and y 2 = And …
……………………………………………………………,
where is the symbol R 1– abbreviation “rule” - rule.
For example, the rule for controlling water temperature is formulated as follows: “ R 1: if the water temperature is cold and the air temperature is cold, then turn the hot water valve to the left to a large angle and the cold water valve to the right to a large angle.”
Fuzzy conditions for solving the problem:
-x 1- water temperature (measured by sensor); - cold;
-x 2- air temperature (measured by sensor); - cold;
Fuzzy inference conditions:
-y 1- the angle of rotation of the valve to the left is large;
-y 2- the angle of rotation of the valve to the right is large.
This linguistic fuzzy rule corresponds to a formalized notation:
R 1: if x 1 = and x 2 = , then y 1 = and y 2 = , (3.31)
Where , , and – fuzzy sets defined by membership functions.
The set of fuzzy production rules forms a base of fuzzy rules, where R i: if..., then...;. The following properties are valid for the base of fuzzy rules: continuity, consistency, completeness.
Continuity is defined by the following concepts: an ordered collection of fuzzy sets; adjacent fuzzy sets.
Collection of fuzzy sets (Ai) called orderly, if the order relation is specified for them: «<»:A 1 <…
If a collection of fuzzy sets { } is ordered, then the sets and , and are called adjacent provided that these fuzzy sets are overlapping.
The base of fuzzy rules is called continuous, if for rules
R k: if x 1 = and x 2 = , then y= and k’¹k
conditions are met:
‑ Ù and are adjacent;
‑ Ù and are adjacent;
- and are adjacent.
Let us consider the consistency of the fuzzy rules base using an example. The base of fuzzy rules for controlling the robot is given in the form:
………………………………….
R i: if there is an obstacle ahead, then move to the left,
R i +1: if there is an obstacle ahead, then move to the right,
……………………………………
The rule base is inconsistent.
An example of a consistent fuzzy rule base is the following:
R 1: if x 1 = or x 2 = , then y= ;
R 2: if x 1 = or x 2 = , then y= ;
R 3: if x 1 = or x 2 = , then y= .
If the rules contain two conditions and one output, then these rules represent a system with two inputs x 1 And x 2 and one exit y. This system can be presented in matrix form:
| x 2 | x 1 | ||
| y= | |||
| y= | |||
| y= |
The base of fuzzy rules is consistent.
A fundamental mathematical concept is the concept of a variable. In practical applications of fuzzy set theory, fuzzy and linguistic variables are usually used.
Fuzzy and linguistic variables are used in the natural language description of various objects and phenomena, in formalizing processes and making decisions in difficult-to-formalize situations.
A feature of human thinking is the ability to analyze and select information relevant to the problem being analyzed, that is, the ability to evaluate heterogeneous information. This ability plays an important role in describing complex phenomena and processes.
Consider a person’s ability to evaluate the concept “Temperature”. In many cases, when estimating temperature values, people do not operate with a numerical characteristic, but with vaguely expressed concepts, such as “low”, “medium”, “normal”, “high”, etc. Moreover, if we are talking about estimating temperature, for example , in certain types of ovens, the human operator is more easily guided by qualitative information, such as “normal temperature,” than by a specific numerical value.
In such a qualitative assessment of information reflecting the nature of a phenomenon or process, natural language plays an important role, which makes it possible to express basic concepts.
Let us introduce the concepts of fuzzy and linguistic variables, which, like an ordinary variable, can change their values.
So, fuzzy variable characterized by three:
< ,X,WITH >,
where is the name of the fuzzy variable;
X– a universal set (finite or infinite), that is, the domain of definition of a fuzzy variable; X= {X};
WITH = { X (X) ) – fuzzy subset of the set X, which is a fuzzy constraint on the values of a variable X.
Example 3.19. Let the universal set X= describes the scope of definition of the parameter – “Temperature in the reactor”. This parameter characterizes the quality of the ongoing technological process. A fuzzy set describing the fuzzy variable “Normal” ( = “Normal”) can be represented by a human operator as follows:
WITH = ((4800), (4810.3), (4820.4), (4830.5), (4841), (4851), (4861) , (4870.5), (4880.4), (4890.3), (4900)).
Obviously, with this definition of a fuzzy set WITH for a human operator who controls the temperature in the reactor, the concept of “Normal temperature” is fully consistent with temperature values from 484 to 486, to a lesser extent - temperature values from 481 to 483 and from 487 to 489. Temperature values in the reactor that are less than 481 and more than 489, cannot be characterized by the concept “Normal”, that is, they are not elements of the carrier of this fuzzy set.
Let's move on to consider the linguistic variable, which is a higher order variable.
Linguistic variable is a variable whose values are words or sentences in a natural or artificial language.
A linguistic variable is characterized by a set of:
< ,T ,X,G, M >,
where is the name of the linguistic variable;
Tβ is the term set of the variable, i.e. the set of its values, which are the names of fuzzy variables, the domain of definition of each of which is the set X with base variable X;
X– universal set;
G – syntactic rule generating set terms T β ();
M is a semantic rule that assigns to each fuzzy variable Tβ fuzzy set WITH , and WITH denotes a fuzzy subset of the set X.
In a more simplified form, a linguistic variable is described by a tuple:< ,T β , X>.
Example 3.20. The values of the linguistic variable “Quality” (β = “Quality”) can be: “Low”, “Average”, “Low”, “High”, “Very High”, etc. Each of these values is the name of a fuzzy variable. This is why the linguistic variable is considered a higher order variable.
Let us discuss all the components of the concept of “linguistic variable”.
Let's look at example 3.20. The adjectives “Low”, “Average”, etc., which define the linguistic variable “Quality”, reflect a certain set of quality characteristics. Each of these values represents a constraint determined by the name and method of specifying the corresponding fuzzy set. From this point of view, the definitions of quality are “Very high”, “Extremely high”, “Not very high”, etc. – names of fuzzy sets formed by action modifiers “very”, “extremely”, “not very” to the fuzzy set “High”.
The set of values of a linguistic variable constitutes a term set of this variable. This set can, generally speaking, be an infinite number of elements.
Example 3.21. Let's consider ways to describe the term set of the linguistic variable “Quality”:
Tβ (Quality) = (“Very low”, “Low”, “Not low”, “Average”, “Rather high than average”, “High”, “Very high”);
Tβ (Quality) = “Very low”“Low”“Not low”…“Very high”.
A term whose name consists of one word or several words that always appear together with each other is called atomic term . Terms consisting of more than one atomic terms are called compound terms. The formation of a compound term by assigning component chains to each other is called concatenation , and the attributed components are subterms and compound term.
If it is necessary to explicitly indicate that the term was generated by the grammar G (syntactic rule G), we will write:
T β * = Tβ G( T β),
Where Tβ * – compound term.
As for the semantic rule M, it can be performed using one of the standard operations on fuzzy sets discussed in Chapter 3.2. The most frequently used modifiers and their corresponding operations on fuzzy sets are:
“not” – addition;
“very” – concentration;
“more or less” – stretching;
“and” – intersection;
“or” is a union.
Linguistic variables play an important role in the construction of fuzzy models: with their help, qualitative information about the object of decision-making, presented in verbal form by specialist experts, is formalized. It is fundamentally important that any linguistic variable, like all its values, is determined by a specific quantitative scale called base scale . It follows from this another definition linguistic variable:
Linguistic variable is a variable defined on a certain scale (base scale) and taking values that are words and phrases of natural language. The values of a linguistic variable are described by fuzzy variables.
There are no special requirements for the name of a linguistic variable and the names of its terms. An expert who describes the system with qualitative or fuzzy concepts directly works with these quantities, behind which the mathematical apparatus of fuzzy sets is hidden. However, certain requirements are imposed on the functions that approximate these fuzzy concepts, as well as on their relative position.
Let us highlight a number of restrictions that the terms of linguistic variables must satisfy. Let Tβ – basic term set of a linguistic variable<,T β , X>,T β = { T i },i= 1, 2, …,m. To each term T i Tβ corresponds to a fuzzy variable<T i ,X,WITH i >.
First of all, the basic term set Tβ must be ordered according to the expression:
(T i T β)( T j T β)( i j)(XS WITH i)(yS WITH j)(x y), (3.36)
Where S WITH i– carrier of a fuzzy set:
S WITH i = {xX Sc i (x) 0 },
that is, this is a set of strict level = 0
Expression (3.36) means that a term that has a carrier located to the left receives a lower number.
The restriction imposed on type of membership functions , corresponding to the basic terms, looks like this:
T 1 (X min) = 1. Tn (x max) = 1, (3.37)
Where n– the number of terms in the base term set, X min and x max – boundaries of the universal set X, on which the linguistic variable is defined.
In accordance with expression (3.37), the membership functions of the terms T 1 and T n must be ammodal.
The following condition can be defined as completeness and consistency :
(T i Tβ)(0 sup C i C (i +1) (x) 1). (3.38)
This expression means that the natural distinction of concepts must be observed when the same point of the universal set X cannot simultaneously belong (with degree of confidence 1) to two or more terms. On the other hand, each value from the domain of definition of a linguistic variable must be described by at least one term.
The next condition is normality – is determined by the following expression:
(T i T β)( XX: C i (x) = 1). (3.39)
Each concept in a linguistic variable must have at least one reference or typical object.
The last condition is limitation :
(β)( X 1 R)(x 2 R)((xX)(x 1 x x 2)), (3.40)
Where R– real axis.
Domain of definition X must be limited to a finite set of points, since in any analysis and decision-making problem there are real restrictions on the numerical values of object parameters.
In Fig. Figure 3.14 presents the linguistic variable β with the number of terms equal to 5, and illustrates the non-fulfillment of the listed conditions and restrictions.
Rice. 3.14. Restrictions imposed on the basic terms of a linguistic variable
So, when forming the basic term set of the linguistic variable β, the following errors were made:
On the boundaries of the universal set X the values of the membership functions of terms denoting the minimum and maximum values of the linguistic variable β must be unity. In Fig. 3.14 therm T 1 has an irregular form (unimodal), and the term T 6 – correct (ammodal).
Existence in the base term set is prohibited Tβ pairs of terms of type T 2 and T 3, since there is no natural limitation of the concepts approximated by terms. These terms illustrate the failure of the consistency condition.
The completeness condition is violated by a pair of terms T 3 and T 4, since the area X no concept matches.
The presence of terms in the base term set is prohibited T 5 having sup C i (x) 1. Since terms must be described by normalized membership functions, in Fig. 3.14 the normality condition is violated.
The use of linguistic variables to describe complexly formalized systems in practice inevitably poses the preliminary task of forming linguistic variables, that is, determining all its components. This, as a rule, is implemented on the basis of surveys of experts - highly qualified specialists in the field for which a fuzzy model is being built using a linguistic variable. Particular attention is paid to the formation of membership functions of fuzzy sets, which are terms of the base term set.
The process of forming a linguistic variable includes the following stages :
Definition of a set of terms of a linguistic variable and its ordering.
Construction of the numerical domain of definition of a linguistic variable.
Determining the scheme for interviewing experts and conducting the survey.
Construction of membership functions for each term of a linguistic variable.
At stage 1, the expert forming the linguistic variable sets the number of terms in the set Tβ and the names of the corresponding fuzzy variables.
At stage 2, the universal set is described X. The implementation of this stage may be accompanied by a number of difficulties caused by the type of linguistic variable. So, for example, the type of universal set for the linguistic variable “Temperature in the reactor” is obvious - it will be a certain interval of temperature values, specified on a certain temperature scale, and the temperature values defining the boundaries of the interval will also not cause any difficulties for the expert. However, if formalization of the concept of “Quality” is required, which is defined as “High”, “Medium” or “Low”, then there is a need to artificially introduce a numerical universal set XR=(–; +), on which the approximable fuzzy concepts will be defined. This procedure will make it possible in the future to use unified approaches for working with linguistic variables of various types.
Stage 3 is key in the formation of a linguistic variable. The scheme for conducting a survey of an expert (or experts) chosen at this stage already assumes that the method for constructing the membership functions of interest to us has also been selected.