CC BY
0
UDC 004.896
MODELING IN AUTOMATED DESIGN OF COMPLEX SYSTEMS
N. V. Shaposhnikova, Y. S. Ganzha Scientific supervisor - L. V. Lipinskiy
Reshetnev Siberian State University of Science and Technology 31, Krasnoyarskii rabochii prospekt, Krasnoyarsk, 660037, Russian Federation E-mail: shapninel@yandex.ru, yanavaio@yandex.ru
This article discusses modeling in computer-aided design of complex systems. Models of multidimensional multiply connected objects are presented, the a priori information about which is a combination of different levels. Nonparametric control algorithms were considered, modifications of the algorithms were proposed, and their effectiveness was investigated.
Keywords: modeling, control algorithms, combined models, model building.
МОДЕЛИРОВАНИЕ В АВТОМАТИЗИРОВАННОМ ПРОЕКТИРОВАНИИ
СЛОЖНЫХ СИСТЕМ
Н. В. Шапошникова, Ю. С. Ганжа Научный руководитель - Л. В. Липинский
Сибирский государственный университет науки и технологий имени академика М. Ф. Решетнева Российская Федерация, 660037, г. Красноярск, просп. им. газеты «Красноярский рабочий», 31 E-mail: shapninel@yandex.ru, yanavaio@yandex.ru
Рассматривается моделирование в автоматизированном проектировании сложных систем. Представлены модели многомерных многосвязных объектов, априорная информация о которых представляет собой комбинацию различных уровней. Рассмотрены непараметрические алгоритмы управления, предложены модификации алгоритмов и исследована их эффективность.
Ключевые слова: моделирование, алгоритмы управления, комбинированные модели, построение моделей.
Introduction. Currently, there is an intensive complication and increase in the scale of industrial production, and consequently, a greater complication of management objects. This is accompanied by a decrease in a priori information. The problem of designing control systems is acquiring an increasingly important role in modern technological systems.
Many methods have been obtained for constructing models of certain processes, objects, and phenomena. A large number of models have been accumulated that are constantly used to solve specific problems. But nevertheless, classical theories of identification and control presuppose the existence of such a volume of a priori information, which is very rare in practice. Often there are situations in which it is not possible to use standard methods.
A situation arises of a combination of different levels of a priori information. Especially often, such problems arise when building models of complex production complexes. Models built in such conditions are called combined. In the general case, the task of constructing a combined model is solved in two stages. At the first stage, a system of equations is constructed that describes the object, and at the second stage, this system is solved with fixed values of the input variables to obtain an output forecast [2].
Let there be some object described by the equation
x(t) = A{u(t), Ji(t),m(t), g(t),t) (1)
where x(t) = (xj(i), 5 , xP ( t)) - is the output vector, U (t) = {u1(t), 5 , UK (t)) - is the
vector of controlled input variables, ju(t) = {ju1(t), 5 , juM (t)) - is the vector of uncontrolled
input variables, (o(t) = {p1(t),...,(0N(t)) - is the vector of uncontrolled input variables (the
researcher may not know about their existence), g(t) = (^1(t), 5 , gL (t)) - are random
perturbations with zero mathematical expectation and limited dispersion, t - is the time, AQ - is unknown operator that converts the input variables of the object to the output. The nonparametric model of the object (1) is written as
/u -u pA^ Svn -M.pf
is an
М(хк | u,p)= ~ =
1=1
1=1
V CSu
П«
n=1
с
SVn
S K
да
J=1 l=1
u -Щ [j]
V CSu
M
к=1, P
n=1
с
SMn
(2)
In the general case, the problem of choosing the blur parameter is reduced to the problem of minimizing some optimality criterion characterizing the quality of the constructed estimate.
Most often, the standard error between the output of the object and the output of the model is selected as the criterion
w (c) =
1 S
-Z(x[i] - xs и)2
S
(3)
The modified algorithm (2) is written as follows
X (N) =-
i=i
p=i
uv ~up[]
C
Sup
П*
/=1
0~£ [i]
v )
a
X (N -1) - Xh[i]
C
Sxh
S к
J=1 p=1
r~ г -Л up ~uplJ]
с
Sup
П*
/=1
0-^/ [J]
с
v s$
X (N-1) - xJ
(N),
с
SXh
(4)
The choice of the criterion for stopping the iterative process is of great importance.
It is proposed to stop the iteration process when the condition is satisfied
X(N)-x,(N-1)| <S, (5)
where 8 - is a given small number.
Algorithms are studied by statistical modeling methods, therefore, systems of equations describing objects were selected in a special way, and samples of input and output variables were generated by the Monte Carlo method.
The construction of the model is carried out in two stages. The first stage consists in training the model, and the second in quality control based on the examination sample. The training consists in adjusting the model parameters of each local object and in setting the blur parameters corresponding to the discrepancies in the output prediction calculation formulas. Checking the quality of tuning the model consists in finding the output vector of the resulting model for a given input and comparing it with the true output vector.
The parameters for variation in model research can be: the type of bell-shaped function in nonparametric statistics, initial approximations of parameters in parametric equations, the volume
1
of training and exam samples, the optimization method of blur parameters, various levels of additive noise in the measurement channels.
Graphs of the dependence of the value of the control error on the level of interference for various methods for choosing the blur parameters with parameters optimized by the deformable polyhedron method for each of the methods.
Fig. 1. Graph of control error value versus interference level
Fig. 2. Graph of control error value versus sample size
A significant impact on the quality of the algorithm is provided by the choice of parameters Px
and /3M corresponding to the values of the blur parameters, and the choice of a parameter a related
to the calculation of a learning additive. The possibility of optimizing these parameters was considered.
In the absence of interference, the values of the parameters J3x and (3a little more than unity showed the best result, in the case of interference, it is better to take the parameter /3slightly larger.
An increase in the sample size leads to a decrease in the control error when using various methods for choosing the blur parameters at any interference levels.
The nonparametric control algorithm with active accumulation of information gives a fairly small control error even in the presence of interference, which can be considered a good result.
In this paper, we consider the place of modeling in the automated design of complex systems. Models of multidimensional multiply connected objects are presented, the a priori information about which is a combination of different levels, such as Bayes, parametric and non-parametric. Such models are systems of stochastic nonlinear equations that are solved using nonparametric procedures. Such models were considered in the presence of delay. An algorithm was also proposed to search for the roots of the system, in the case when their number is more than one. Nonparametric control algorithms were considered, modifications of the algorithms were proposed, and their effectiveness was investigated.
The considered algorithms were studied by the method of statistical modeling, the results were obtained, according to which it can be said that the constructed combined models describe the initial objects well enough, and nonparametric control algorithms successfully cope with the task.
References
1. Norenkov I P. Fundamentals of Computer Aided Design. - M.: MSTU. N.E.Bauman, 2002.
2. Krasnoshtanov A.P. Combined multiply connected systems. - Novosibirsk: Science, 2001.
3. Granichin O.N., Polyak B.T. Randomized estimation and optimization algorithms with almost arbitrary interference. - M.: Science, 2003.
4. Medvedev A.V. Nonparametric Adaptation Systems. - Novosibirsk: Nauka., 1983.
5. Medvedev A.V. Elements of the theory of nonparametric control systems. Actual problems of computer science, applied mathematics and mechanics. Novosibirsk-Krasnoyarsk: Publ. SB RAS, 1996.
© Shaposhnikova N. V., Ganzha Y. S., 2020
