ВЕСТНИК ТОМСКОГО ГОСУДАРСТВЕННОГО УНИВЕРСИТЕТА
2024 Управление, вычислительная техника и информатика № 67
Tomsk: State University Journal of Control and Computer Science
Original article UDK 517.9
doi: 10.17223/19988605/67/2
Technology for improving robust quality of control for one-dimensional linear discrete control systems with structural-parametric uncertainty
Andrej N. Parshukov
Industrial University of Tyumen, Tyumen, Russian Federation, [email protected]
Abstract. The article considers one-dimensional linear automatic control systems in discrete time. It is assumed that the operators of the control object contain parametric and/or structural uncertainty. The purpose of the study is to develop a method for the synthesis of modal controllers that provide the maximum robust quality of control of a closed system. The main result is presented in the form of an algorithm for improving the robust quality of control. The efficiency of the algorithm is illustrated by an example.
Keywords: structural perturbations; modal controller; robust control quality.
For citation: Parshukov, A.N. (2024) Technology for improving robust quality of control for one-dimensional linear discrete control systems with structural-parametric uncertainty. Vestnik Tomskogo gosudarstvennogo universiteta. Upravlenie, vychislitelnaja tehnika i informatika - Tomsk State University Journal of Control and Computer Science. 67. pp. 12-21. doi: 10.17223/19988605/67/2
Научная статья
doi: 10.17223/19988605/67/2
Технология повышения свойств робастного качества управления для одномерных линейных дискретных систем управления со структурно-параметрической неопределенностью
Андрей Николаевич Паршуков
Тюменский индустриальный университет, Тюмень, Россия, [email protected]
Аннотация. Рассмотрены одномерные линейные системы автоматического управления в дискретном времени. Предполагается, что операторы объекта управления содержат параметрическую и / или структурную неопределенность. Цель исследования состоит в разработке метода синтеза модальных регуляторов, обеспечивающих максимальное робастное качество управления замкнутой системой. Основной результат оформлен в виде алгоритма повышения робастного качества управления. Эффективность алгоритма проиллюстрирована примером.
Ключевые слова: структурные возмущения; модальный регулятор; робастное качество управления.
Для цитирования: Паршуков А.Н. Технология повышения свойств робастного качества управления для одномерных линейных дискретных систем управления со структурно-параметрической неопределенностью // Вестник Томского государственного университета. Управление, вычислительная техника и информатика. 2024. № 67. С. 12-21. doi: 10.17223/19988605/67/2
Introduction
Currently, in industry, technological processes with different tempo dynamics are increasingly considered as control objects [1, 2]. A feature of these processes is that their mathematical models are restored with
© A.N. Parshukov, 2024
an accuracy of belonging to some families of linear difference equations with different orders (structural-parametric uncertainty). When synthesizing a controller, structural perturbations, as a rule, are not taken into account [3]. Since the properties of stability and quality of control of the system are determined by the location of the zeros of the characteristic polynomial, the question arises: for which operators of structural perturbations in the control object, the closed system will still retain the properties of stability (robust stability) and quality of control (robust quality of control)?
In the modal control scheme [4. P. 8-21] the quality of control is given as an area S on the complex plane; the area S determines the desired location of the zeros of the characteristic polynomial. Consequently, the issues of studying (testing) robust stability and robust quality of control can be considered from a unified standpoint: is it required to check whether the zeros of a given family of polynomials belong to the area S?
The problem of studying robust stability and robust quality of control is widely presented in the literature. There are three main directions in which this problem is solved: 1) the principle of zero exclusion [5-8]; 2) H theory [9, 10]; 3) LMI method [11-13]. At the same time, the main attention is paid to the study of robust stability, and the methods for studying the robust quality of control have not yet been sufficiently developed.
In this paper, we develop the results obtained by us in [14]. The article [14] developed a method for checking the robust quality of control. In the same place, a function was introduced that allows one to compare the robust properties of various dynamic control systems. In this paper, based on the introduced function (measure of robust properties), an algorithm for variation of the modal controller coefficients is developed that consistently increases the property of the robust quality of control.
Further, the following notation is adopted: = is equal by definition; Rn is the n-dimensional space of real numbers; Cn is the n-dimensional space of complex numbers; C1 is the complex plane; T is the transposition operation; E is the identity matrix of the corresponding dimension; 0 is a column vector consisting entirely of zeros; j is the imaginary unit; s is a variable (generally complex); s* is the number complex conjugate of s; t is discrete time (t = 0, ±1, ±2, ...); z is advance operator by one cycle; S is the area on C1; dS is the boundary of the area S; int S is the interior of the area S.
Let fx) be a function of the vector argument x defined on the domain X c Rn; denote
|/| =max|/(x)|, |/|_ =min|/(x)|.
xsJL xsJi
Operator of the form
n
f (n, z) = £ f . z', (1)
i=0
is called a polynomial operator of degree n. Replacing z with s in (1), where s is a variable, we obtain an algebraic polynomial
n
f (n,s) = X f • s'.
i=0
The set of zeros of the polynomial f n, s) is denoted by Af):
A (/) = {A,I.eC1:/(»5XI.) = 05 iel^i}.
Family of operators of the form
f(n,F,z) = ^f(n,f,z) = fjJrz': f = (f0, /B)ejp|, (2)
F = {f g Rn+1: f G[f0-Af; f0 + Af],
fV0, Af. >0, i = 0n},
is called an interval polynomial operator of degree n.
Replacing z with s in (2), we obtain an interval polynomial:
f (n,F,s) = jf (n,f,s) = ¿f. • s' : f g Fj. (3)
Interval polynomial (3) can be represented as
f (n, F, s ) = f 0( n, f0, s) + f (n, AF, s), (4)
where
n
f(n,f»=(/0°, ..., r),
1=0
ordinary polynomial, and
/(«,AF,s) = j/(«,8f,.s,) = ^]8/] -s' :8f = (5/0,...,5/B)e Af|
l 1=0
AF = {sf GRn+1 : Sf e[-Af ; Af ], i = 0^},
interval polynomial with symmetric coefficient uncertainty bounds.
Let A be a matrix whose coefficients are complex numbers, then denote by ARe and Aim the matrices whose coefficients are, respectively, the real and imaginary parts of the coefficients of the matrix A; in this sense, we denote
ARe=Re(A), Ata=Im(A). Let's call the generalized gradient of the function fx) a vector that coincides with the ordinary gradient Vx f ( x ) at those points where the ordinary gradient exists, and is redefined as a zero vector at those points where the ordinary gradient does not exist, that is:
хД H 0, 3Vx/(x),
here Vg is the generalized gradient.
1. Method of synthesis of a modal controller for a control object under conditions of structural-parametric disturbances
Let a linear one-dimensional discrete control object is given by a model of the form:
v (l,V, z )• a (n, A, z) y (t) = w (h,W, z )• b (m, B, z) u (t), n > m, l > h, (5)
here u is the input (control) signal, y is the output (control) signal, a, b, v, w are interval operators of the form (4) such that
a° = 1, Aan = 0, v00 = 1, Av0 =0, w° = 1, Aw0 = 0.
Model
a0 (n, a0, z)y (t) = b0 (m,b0, z)u (t), (6)
belonging to the family of models (5), we will call "nominal"; interval operators a, b are "basic dynamics"; operators v, w are "structural perturbations".
The quality of control will be assigned in the form of an area S, which determines the admissible location of zeros of the characteristic polynomial on Cl. We assume that the area S satisfies the following conditions: it is located inside a circle of unit radius; simply connected; also holds s* e S for any point s e S . In addition, based on the meaning of the problem, we require that the operators of structural perturbations v, w satisfy the expressions:
A( v )c int S, A( w )c int S. (7)
Note that the fulfillment of expressions (7) is easy to check using the methods described in [8. P. 7-10].
Following the method of modal control [14. P. 6-7], the controller for the nominal model (6) is sought in the form of a difference equation of the n - 1 degree:
P( n-1, z) u (t) = a(n-1, z) y (t ) + x(n-1, z) g (t), p„_i=1, (8)
here g is a given program signal. In the modal control scheme, the coefficients of the operators P and a of the controller (8) are calculated from the condition of conversion to the identity of the equation
ast (2n -1,s) = a0 (n,a0,s)p(n -1,s)-b0 (m,b0,s)a(n-1,s),
<-i=l, (9)
where ast is the given characteristic polynomial of the standard control system (hereinafter referred to as the "standard"); the choice of standard is limited by the condition
A( ast )c int S. (10)
It is obvious that the choice of the polynomial x does not affect the properties of robust stability and the robust quality of control, so the issue of calculating the polynomial x in the modal control method is not considered. In this sense, the vector of coefficients of the polynomials P and a is called the vector of coefficients of the controller and is denoted
r = co/(r1;r2),
»"i =co/(Po,Pi,---,P„_2)5 r2=col(a 0,au---,an_,); After closing the original object (5) by the controller synthesized according to the scheme (8)-(10), it is easy to obtain the following expression for the characteristic polynomial of the closed system
a° (2n +1 -1, A, B,V ,W, s) = {ac (2n +1 -1, a, b, v, w, s):
Va g A, b g B, v gV, w gW }, (11)
ac(2ra + /-l,a,b,v,w,s) = v(/, v,s)-a(«,a,5,)-p(«-l,s)--w(h, w,s)• b(m,b,s)• a(n -1,s). A closed system with a characteristic polynomial of the form (11) has a robust control quality if the set
A( ac ) = {X,: 3a g A, 3b g B, 3 v gV,
3w gW, ac (2n +1 -1, a, b, v,w, X,.) = 0, i =1,2n +1-1}
lies inside the area S, i.e.
A( ac )c int S. (12)
In the presence of structural disturbances in the control object, it is impossible to guarantee in advance that the modal controller calculated by formulas (8)-(10) will ensure the fulfillment of condition (12). Thus, the problem of synthesizing a modal controller in the presence of structural disturbances in the control object consists of the following steps:
1) synthesis of a modal controller for a nominal object (6) (according to formulas (8)-(10);
2) subsequent verification of the fulfillment of condition (12) for a given family of characteristic polynomials (11) (the problem of testing the robust quality of control).
Theorem 1 [14. P. 8]. Let the family ofpolynomials (11) satisfy conditions (7), (10) and
p(s) = Pj (s)-p2 (s) > 0, VscdS, (13)
where
Pl (s) = |v° (/, Y°,s) + v(/,5v,s)| • |ast (2n - l,s) +
+a (n -1,5a, s) • p (n -1, s) - b (m,5b, s) • a (n -1, s)| ,
P2 (s) = |v° (/, v°,s) + v(/,5v,s) - w° (h,w°,s) - w(/2,5w,s)| •
• |b0 (m,b0,s) + b (m,5b,s)| •|a(n-1,s)|. (14)
Then the family of polynomials (11) satisfies (12).
An algorithm for checking the robust quality of control based on Theorem 1 is described in [14. P. 9]. The function p, defined by formulas (13)-(14), makes it possible to compare various closed systems with each other, thus, the function p is a measure of robust properties. The introduction of the measure p makes it possible to pose the problem of increasing the robust properties of a closed system.
2. Method for improving the properties of robust control quality
2.1. Statement of the problem of improving the properties of the robust quality of control
Let us assume that the boundary dS is replaced by a finite number of points [8. P. 10]:
Q = {s, edS, i e1~N}, (N <w), here the points Si included in the set Q are chosen on the boundary dS based on the condition:
max A arg (ra - ss+1 <-1--— radian.
eS v ^ 2« +1 -1 2
Suppose that for a given control object (5)-(7) a controller (8) is synthesized that satisfies conditions (9)-(10). Note that in the synthesis scheme of the modal controller (5)-(10), freedom is allowed in choosing the standard ast within the limits of condition (10). The new chosen standard can be considered as a result of the variation of the vector of controller coefficients (8):
r(fc + l) = r(fc) + 8r(fc), k = §,\,2,...,K <<x>. (15)
The present work is devoted to the search for the vector r* providing a local maximum of the function p in the space of the vector of the controller coefficients. The local maximum is sought in the vicinity of the vector r(0).
Consider the procedure for searching for the vector r* as an iterative process, let us denote the number of iteration as k (see expression (15)). Further, for convenience of presentation, we will indicate the iteration number as the first of the arguments of functions, vectors and matrices: for example, the vector r, the polynomial ast(2n - 1, s), the functions p(s), p, and so on will be written, respectively , r(k), ast(k, 2n - 1, s), p(k, s), p(k).
According to the known vector r(k), the polynomial ast(k, 2n - 1, s) is determined by the formula
ast (k,2n -1,s) = a0 (n,a0,s)p(k,n -1,s)-
-b0 (m, b0, s )a( k, n-1, s), (16)
the polynomials P(k, n - 1, s) and a(k, n - 1, s) can be expressed in terms of the components of the vector r(k):
P(k,n -1,s) = sn-1 + • r (k), a(k, n -1,s) = o2 • r2 (k),
here we use the notation
Oj o2
Suppose that at the current k iteration r(k), ast(k, 2n - 1, s), p(k) are defined, and for the polynomial ast(k, 2n - 1, s) done
a( ast (k ))c int S. (17)
It should be noted that for k = 0 condition (17) is satisfied.
At the next k + 1 iteration, we will look for a vector Sr(k) such that:
1) after substituting the vector Sr(k) into (15) for the polynomial ast(k + 1, 2n - 1, s) done
A( ast (k + 1))c int S. (18)
2) done
p(k + 1)>p(k). (19)
2.2. Algorithm for improving the property of robust control quality
Denote by © the set of points s at which the function p(k, s) takes the minimum value, the total number of minimum points at the k iteration will be denoted by N(k), meaning N(k) < N. The set © can be written:
© = |s = argminp(k,s), i = 1, N(k)}.
For each point s,. e0, we find the nominated generalized gradient of the function p(k, si) with respect to r(k), we denote such a gradient by p(k, i):
p(*,z>V?p(M,)/|V?p(M,)|. (20)
It's not hard to get
Vg p( k, s) = Vg p1 (k, s) - Vg P2 (k, s), (21)
Vg p1( k, s ) = |v°(l, v0, s) + v (l, Sv, s )|^Vg k, s),
Vgp2(k, s) = ( 0 e Rn-1, Vgp2(k, s)), Vg p( k, s ) = |v0(l, v0, s) + v (l, Sv, s)- w0( h, w0, s )-
-w(h,Sw,s)|+ • |b0(m,b0,s) + b(m,Sb,s)| •Vg^ (k,s),
here the designations are introduced
9 = ast (k,2n -l,s) + a(n -l,Sa,s)- sn~l,
H = (a(n -l,Sa,s) • o1, -¿(wi,Sb,s) -o2), = + r\2(k,s) = |o2 -r2(£)|.
The gradients of the functions n and n are
V, n \ (9Re + ^Re • r)^Re +(9]m + ^Ln • r)^L
V ^ k,s )=-MM
at those points © where | ^1(k, s) | ^ 0, and
/ \ 1 V^k, s ) = -
Im
•r.'fT. -t- ffT _ _ • r_ • I
K(k, s)| '
at those points © where | ^2(k, s) | ^ 0. Thus,
Vgp1(k, s ) = |v0(l, v0, s ) + v (l, Sv, s )| •
Vg k, s), (k, s )|*0, 0 e R2"-1, k, s )| = 0,
Vgp (k, s) = |v0 (l, v0, s) + v(l, Sv, s) - w0 (h, w0, s) -w(h,Sw,s)|+ • |b0(m,b0,s) + b(m,Sb,s)| •
i o e Rn-1 ^
, (k, s)|*0,
(22)
(23)
k, s),
0 e R2n-1, k, s)| = 0.
The normalized generalized gradient p(k, i) is calculated by formulas (20)-(23). It follows from these formulas that the vector p(k, i) is not defined only when
M k, si )|=0, U 1^2 ( k, si )|=°
Denote by I the index set of points s{ e© at which the vector p(k, i) exists:
I = {i e 1,N(k): 3 p(k,i)}.
The set of vectors p(k, i) calculated for all points of the set © will be denoted by P:
P = {p(k,i): i e I}. Consider the case when the set I is empty
I = O. (24)
This case means that at all points of the set © the gradient of the function p(k, si) is either equal to zero or does not exist; therefore, at the k iteration, the local maximum of the function is reached and the problem of improving the robust control quality is solved.
Let us consider the case when the set I is not empty, then the set P is the direction of growth of the function p(k). A common direction for all elements of the set P (of course, if it exists at all) can be found, for example, using the "linear classifier" algorithm [15. P. 113-115]. We will not give the entire algorithm of the linear classifier here, but we will introduce the concepts necessary for the functioning of the algorithm. The algorithm assumes that the set I is not empty; admissible error value 80 (80 > 0). If there is a common vector t g R^-1 such that
{t • p(k,i) > s0, Vi g /}, (25)
then such a vector can be found in no more than e-2 iterations of the algorithm (Novikov's theorem [15. P. 28]). If for the specified number of iterations of the linear classification algorithm it is not possible to obtain a vector t that satisfies (25), then, according to the Novikov theorem, such a vector does not exist. As applied to the problem we are considering, this means that the function p(k) has reached a local maximum, and our algorithm must be stopped. Therefore, it is further assumed that the vector t has been found. The vector Sr(k) in expression (15) is sought in the form:
5r (k) = y • t, y>0, (26)
here t is the general direction of increase of the function p(k, si) at the points of the set © if the number of such points is more than one, and
t = p (k, i),
if the set © consists of one point si.
The coefficient of proportionality y in (26) is chosen based on conditions (18)-(19). At the first stage, for each point s, gq , one should determine the value of the parameter y in (26), at which the zeros of the polynomial ast(k + 1, 2n - 1, s) reach the point si for the first time. We denote such a value by y(s).
For further presentation, it is necessary to obtain an explicit expression for the polynomial ast(k + 1, 2n - 1, s). By analogy with formula (16), we can write
ast (k + 1,2n -1, s ) = a0( n, a , s )p( k + 1, n-1, s )-
-b0( m, b0, s )a( k + 1, n-1, s),
here
P(k + 1,n-1,s) = sn-1+ Oj • r (k + 1),
a( k + 1, n -1, s) = o2 • r2 (k + 1),
col (r (k + 1), r2 (k + 1)) = r (k + 1),
the vector r(k + 1) is determined by formulas (15) and (26)
r (k + 1) = r (k) + y • t.
After reducing similar ones, we get the following expression for the polynomial ast(k + 1, 2n - 1, s):
ast (k + 1,2n -1, s) = ast (k, 2n -1, s) + y • Aast (k, 2n - 2,s), (27)
Aast (k, 2n - 2, s ) = (a0 (n, a0, s ) • o , -b0 ( m, b0, s ) • o2 ) • t.
The desired value y(sO can be found from the condition that the polynomial ast(k + 1, 2n - 1, s) vanishes at the point si, which corresponds to the system of equations:
< (k, 2n -1, s, ) + y • AaR (k, 2n - 2, s, ) = 0,
(28)
asm (k, 2n-1, si + y-AaSm (k, 2n-2, s'0.
The compatibility condition for system (28) is established by the Kronecker-Capelli theorem. If the system of equations (28) is consistent, then its solution has the form:
Y( s ) =
-as (k,2n-1,s.) t ,
Re( ,-, Aast (k,2n-2,s.W0,
Aast (k,2n -2,s.), Re (, ,i)
Re ( , , i ) (29) -aS (k, 2n -1, s.)
Im( , , i) Aa^(k,2n-2,st)*0.
[Aa^ (k, 2n -2, s,
It is easy to verify that the simultaneous execution
AaRe (k, 2n -2, st) = 0, and Aaj* (k, 2n -2, s, ) = 0,
is equivalent to a violation of condition (17) and is thus excluded.
Note that when searching for y(s), the following situations are possible:
1) the number y(s) found by formula (29) can be negative, such a solution in the framework of the problem we are considering does not make sense;
2) the zeros of the polynomial ast(k + 1, 2n - 1, s) do not intersect the boundaries of the area S at the point si for any y.
In these cases, the desired value y(sO will be taken equal to +». Define the function
Y1 = min y+( st),
fy(s,), 3Y(s,)ay(S,)>0,
\ +oo 3y(s;.)vy(s,)<0.
Based on the above, we can conclude that condition (18) is satisfied over the entire interval r = [(), y,).
The proof of this statement for y = 0 is obvious, and for y = Y1 the zeros of the polynomial (27) for the first time come to the boundary of the area S.
On the interval r, one should find such a value of the parameter y (we denote it by y*) for which condition (19) is satisfied for p(k + 1). The existence of y* follows from the existence of the vector t, the general direction of increase of the function p(k, s). The search for y* can be performed using the segment dichotomy method.
The above reasoning is the rationale for the algorithm given below for improving the property of robust control quality. The stopping condition of the algorithm is the achievement of a local maximum of the function p in the space of vectors r. The achievement of the local maximum of the function p at the k + 1 iteration corresponds to the fulfillment of at least one of the conditions:
1) the set I is empty, that (24) is satisfied;
2) there is no vector t such that it satisfies inequalities (25). As a result, we can formulate the following algorithm.
Algorithm for improving the property of robust control quality.
For the algorithm to function, it is necessary to define r(0), ast(0, 2n - 1, s), Q, p(0, si), p(0), 80.
Accept k = 0.
Step 1. Form sets © h I.
Step 2. Check condition (24). If this condition is met, then go to stop, otherwise go to the next step. Step 3. Using the linear classifier algorithm, find the vector t that satisfies the system of inequalities (26). If direction t is found, go to the next step, if direction t does not exist, go to stop. Step 4. Calculate the value of y* for which conditions (18)-(19) are satisfied. Step 5. Calculate vector
r (k + 1) = r (k ) + y * •t. Accept r* = r (k + 1).
Step 6. Accept k = k + 1. Go to step 1. Stop.
The convergence of this algorithm to the local maximum point of the function p was justified above. The number of iterations of the algorithm is finite, since the value of the function p is bounded.
3. An example of improving the properties of robust control quality
Let the control object be given by
(v° (2,z) + Av(2, z))•(a0 (2, z) + Aa(1,z))y (t) =
= ( w0 (2, z ) + Aw (2, z)) • (b0 (1, z ) + Ab (1, z )) u (t), (30)
here
a0 (2, z) = z2 -1,918z + 0,923, b0 (1, z) = 0,232z - 0,179, Aa(1, z) = 10-4 [-1; 1]z + 10-4 [-2; 2], Ab (1, z) = 10-2 [-1; 1] z + 10-2 [-1; 1]. v0 (2, z) = 1,830z2 -2,707z + 1, w° (2,z) = 1,848z2-2,717z + 1, Av(2,z) = 10-4 [-5; 5]z2 +10-3 [-1; 1]z, Aw(2, z) = 10-4 [-5; 5] z2 +10-3 [-1,5; 1,5] z. Zeros of polynomials v0(2, z) and w0(2, z):
A(v0) = {0,715, 0,764} c C1,
A (w0) = {0,735 + 0,027j, 0,735 - 0,027j } c C1. Requirements for the quality of control are set by the area
S = {s: < Isl |arg(s)<?}»
where n = 0,607, n = 0,961, 9 = n/4 radian. Sets A(v0) and A(w0) lie inside the area S.
In accordance with the above synthesis scheme, the modal controller has the structure of a first-order difference equation:
(z + p0 )u (t) = (a1z + a0 ) y (t) + X0 g (t), r = COl(po, a0, a1 ). (31)
In [14], for the control object (30) and the standard
ast (3, z ) = z3 -2,715 z2 +2,456 z-0,741. we have calculated a modal controller of the form (31) with a vector of coefficients [14. P. 10]:
r (0) = col (-0,833, 0,157, -0,158).
Let us set the problem: it is required to find a vector r* such that it provides a local maximum of the function p(r) in the vicinity of the vector r(0).
Solution. As a result of applying the above algorithm, after 12 iterations, we obtain the vector
r* = col (-0,838, 0,135, -0,134).
For the vector r(0) the value of p is equal to 6J36-10-4, for the vector r* the value of p is equal to 7,894-10-4. Note that p(r*) > p(r(0)) holds. The required vector r* is found.
Conclusions
In this article, we have developed an algorithm for improving the robust quality of control for a closed control system consisting of a linear one-dimensional control object with structural-parametric uncertainty and a modal controller. The efficiency of the algorithm is illustrated by an example. The application of this algorithm will make it possible to obtain closed automatic control systems with higher indicators of robust stability margins and robust control quality. The design and implementation of such systems will lead to significant savings in the operation of control systems under conditions of uncertainty.
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Information about the author:
Parshukov Andrej N. (Candidate of Technical Sciences, Associate Professor, Industrial University of Tyumen, Tyumen, Russian Federation). E-mail: [email protected]
The author declares no conflicts of interests.
Информация об авторе:
Паршуков Андрей Николаевич - доцент, кандидат технических наук, доцент кафедры «Электроэнергетика» Тюменского индустриального университета (Тюмень, Россия). E-mail: [email protected]
Автор заявляет об отсутствии конфликта интересов.
Received 29.11.2023; accepted for publication 03.06.2024 Поступила в редакцию 29.11.2023; принята к публикации 03.06.2024