Volume 2(51)
CONTENT
1. Busalov A. A. Nonlinear stationary problem of transport theory in the diffusion approximation
5. Savelyev V. P., Sutyagina N. I. Mathematical models of regional business dynamics
6. Sorokina M. Application of optimal evaluation of linear time-varying systems using reachable sets
Lobachevsky State University of Nizhny Novgorod, 603022, Nizhny Novgorod, Russia
NONLINEAR STATIONARY PROBLEM OF TRANSPORT THEORY IN THE DIFFUSION APPROXIMATION
UDK 519.633.2
DOI:10.24412/2073-0667-2021-2-6-14
One of the most effective mathematical approximations for describing the processes of transfer of X-ray radiation is the multi-group diffusion approximation. The equations of the one-group diffusion approximation are a system of two differential equations for two unknown functions: density and radiation flux.
The article considers a nonlinear problem of the theory of radiation transfer in the diffusion approximation; formulates a linearizing iterative algorithm for solving the nonlinear problem. In addition, the possibility of applicability of the corresponding linearizing iterative algorithm for a nonlinear system of integro-differential equations of radiation transfer and statistical equilibrium for a two-level atom model are investigated. It should be noted that the model of a two-level atom reflects the important meaningful problems. Also it can be considered as an integral part of the study of more complex problems of a multi-level atom. A linearizing iterative solution algorithm is proposed and numerically investigated for solving the arising system of equations. The results of a numerical analysis of the proposed iterative algorithm are presented. They confirm the possibility of its application. The properties of the used difference scheme are discussed.
Key words: radiation transfer, diffusion approximation, stationarity equations, nonlinear system of equations of the theory of transfer, iterative methods.
Bibliographic reference: Busalov A. A. Nonlinear stationary problem of transport theory in the diffusion approximation //journal “Problems of informatics”. 2021, № 2. P. 6-14. DOI:10.24412/2073-0667-2021-2-6-14
Samara Federal Research Scientific Center RAS, Institute for the Control of Complex Systems RAS, 443020, Samara, Russia
ONLINE DSS ALGORITHMS FOR SELECTING THE DISTRIBUTION LAW OF A POSITIVELY DEFINED RANDOM VARIABLE
UDK: 51-37
DOI: 10.24412/2073-0667-2021-2-15-25
The problem of making online DSS for selecting a two-parametric distribution law of a continuous positively defined random variable with a finite second moment is considered. These RVs are very important for describing such parameters of real systems as distances, time intervals, value deviations, reliability characteristics, economic indices etc. Algorithms and scenarios of the system operation are given. The main priority is a practical application-oriented approach to selection of the most appropriate type of distribution law out of a finite set of models. The approach is based on usable criteria and is applicable in case of small datasets.
The proposed algorithm for selecting the distribution law of random variables is based on a natural factorization of the space of empirical characteristics -- the average and mean standard deviation. In exponential mode, if the average and mean standard deviation are close, the exponential distribution is suggested. In hyperexponential mode (mean standard deviation is greater than the average), the following distributions are proposed as models of a random variable: hyperexponential distribution of a special type, Weibull distribution, gamma distribution, lognormal distribution, inverse Gaussian distribution. For hypoexponential mode (the average is greater than mean standard deviation), the following distributions are offered: hypoexponential distribution, Erlang distribution, Weibull distribution, gamma distribution, lognormal distribution, left truncated normal distribution, inverse Gaussian distribution. If the average is divisible by mean standard deviation, Erlang distribution is used instead of hypo-exponential distribution, and gamma distribution, which degenerates into a special case of a discrete parameter, coincides with the Erlang distribution. Special importance and attention are given to two-phase hyperexponential distribution of a special type, proposed by one of the co-authors of this article, and hypoexponential distribution with two types of phases. These are convenient models for approximating two-parameter distributions of positively defined random variables. The advantages of these distributions are: requirement of two parameters only; existence and uniqueness of the solution of the system of moment equations for hyperexponential distribution of a special type; explicit solution of the system of moment equations for hypoexponential distribution; the physical meaning of the RV distributed according to hyperexponential distribution is the sojourn time in two parallel connected states with exponential distribution laws; the physical meaning of the RV distributed according to hypoexponential distribution is the sojourn time in a series-coupled states with exponential distribution laws; explicit analytical view of the restoration function. Distribution parameters are determined by numerical solution of systems of equations using functions of python libraries. To effectively search for solutions of systems of equations, the initial parameter values are selected adaptively using the equation root localization procedure.
The online calculation system is being developed in Python using the Dash framework and is a convenient tool for modeling random variables, their distributions, and characteristics for engineers and economists. The advantages of the developed automated online system are interactive intuitive interface; availability without software installation; ability to save graphic elements as files; minimal set of statistically reliable and intuitive input data; ability to visually compare different models of random variables by means of various criteria.
The input data block enables the user to upload a file with statistical data. Output forms are: RV distributions and characteristics (densities, parameters, entropies); graphs of densities, distribution functions, and failure rates; scattering diagrams; the table of metrics with quantitative difference estimation for pairs of distributions. These characteristics are applied to visually compare the distribution laws. The block of multi-criteria ranking of distribution laws is applied to rank the distribution laws. The following simple and intuitive criteria, focused on solving practical problems, are proposed for a user (engineer, economist, biologist, etc.): maximization of the differential entropy, fitting empirical quantiles, fitting empirical moments of higher orders, the presence or absence of analytically defined restoration function, the behaviour of the intensity function, the possibility of decomposition into exponential phases. The significance of these six criteria is determined by the user by assigning them to one of the four groups of significance: very important, important, slightly important, ignore. Quantification of these fuzzy assessments into criteria weight coefficients can be done by the method of Piyavsky S. A. Then the multi-criteria problem is to be solved by reducing it to a linear convolution with the criteria weight coefficients obtained. Herewith, each distribution is assigned the number of points (from 1 to N, where N is the size of distribution set according to the natural factorization of the space of empirical characteristics) due to predefined rules. Some additional data are to be specified by user. They are the number of higher-order moments, the number of quantiles, the behavior of the failure rate function (monotonic; non-monotonic (break-in character); other) if respective criteria are taken into account.
The DSS can be applied in combination with other existing methods for selecting the distribution law of a random variable.
Key words: DSS, multi-criteria selection, random variable, distribution law, statistical data processing.
References
Bibliographic reference: Ivleva A., Smirnov S. Online DSS algorithms for selecting the distribution law of a positively defined random variable //journal “Problems of informatics”. 2021, № 2. P. 15-25. DOI:10.24412/2073-0667-2021-2-15-25
Lobachevsky State University, 603950, Nizhny Novgorod, Russia
OPTIMAL ROTOR STABILIZATION IN AN ELECTROMAGNETIC SUSPENSION SYSTEM USING TAKAGI-SUGENO FUZZY MODELS
UDK: 517.977
DOI: 10.24412/2073-0667-2021-2-26-37
An electromagnetic suspension is a rotor located in the field of gravity and the magnetic attraction force acting on the side of the electromagnet. When the two forces are equal, the rotor is stationary. The operation principle of an electromagnetic suspension is based on the phenomenon of magnetic levitation, which is applied in active magnetic bearings. Thanks to this, it becomes possible to overcome gravity without contact and to provide the rotor hanging in active magnetic bearings. The obvious advantage of such systems is, first of all, the absence of mechanical contact and, as a consequence, the absence of friction. This advantage allows significantly increase the service life and efficiency compared to traditional mechanical counterparts.
The magnetic forces acting on the electromagnet are controlled by a control system that ensures a stable equilibrium position of the rotor. It is necessary to note that an electromagnetic suspension is inherently unstable system and it is describes nonlinear differential equations which present additional difficulties for the control low realizing. The standard scheme of the electromagnetic suspension assumes the presence of a sensor of the position of the suspended body as the main element for forming feedback in the control loop. Also there is another scheme that involves measuring only the current in the circuit of the electromagnet without measuring the position of the body and its speed. The advantages of such a scheme are the compactness and reliability of the design, as well as the lower cost compared to the traditional scheme.
The control of the rotor in an electromagnetic suspension is an important and actual task associated with the wide practical application of electromagnetic bearings. The practical application of electromagnetic bearings covers a wide range of different areas of industry and technology, as well as some areas of medicine. For the construction of regulators, the most widely used approach is based on the use of linearized models. Despite its simplicity and convenience, the obvious disadvantage of linearized models is their limited applicability. Linearized models correctly describe the dynamics of an object only in the neighborhood of the equilibrium position, with small initial deviations. In fact, the initial perturbations of an object can go far beyond the applicability of linearized models. As a result, the regulators calculated on the basis of linearized systems are only operable for small initial perturbations. One of the ways to take into account the nonlinearities of the object and build corresponding controllers can be the use of fuzzy logic and fuzzy systems based on it.
This paper presents the results of solving the problem of constructing fuzzy output regulators for an electromagnetic suspension system based on the use of Takagi-Sugeno fuzzy models. Two control problems are considered: the construction of stabilizing regulators and the construction of optimal regulators according to a given quadratic quality criterion.
To solve these problems, the original nonlinear mathematical model was transformed to a special form, and then replaced with an equivalent fuzzy model consisting of a set of linear subsystems. To construct a fuzzy mathematical model of the system, triangular distribution functions were used. Ultimately, the fuzzy model was represented as a weighted sum of all linear subsystems. To synthesize the control laws, we used the apparatus of linear matrix inequalities, extended to the case of fuzzy systems. In this case, for each linear subsystem is consistent with its own system of linear matrix inequalities.
As a result of numerical calculations, fuzzy controllers of both types were obtained, which were then alternately substituted into the original nonlinear object closed by the fuzzy controller. To check the operability of the regulators, mathematical modeling of the rotor dynamics was performed. Transient processes in a closed system are presented as simulation results.
The results of numerical calculations and mathematical modeling showed that using the Takagi-Sugeno fuzzy models, it is possible to construct both a stabilizing regulator and an optimal regulator according to a given quadratic quality criterion for controlling the rotor in an electromagnetic suspension. The found regulators provided stabilization of the rotor in a fairly wide range of initial perturbations, up to the maximum possible values. Based on the results obtained, it can be concluded that the presented approach, based on the use of fuzzy Takagi-Sugeno models, allows to stabilize the rotor in an electromagnetic suspension system in a wide range of initial perturbations.
Key words: electromagnetic suspension, magnetic levitation, rotor, stabilization, fuzzy Takagi-Sugeno models, linear matrix inequalities.
References
Bibliographic reference: Mukhin A. V. Optimal rotor stabilization in an electromagnetic suspension system using Takagi-Sugeno fuzzy models //journal “Problems of informatics”. 2021, № 2. P. 26-37. DOI:10.24412/2073-0667-2021-2-26-37
E. V. Proydakova, M. A. Fedotkin
Lobachevsky State University of Nizhni Novgorod (UNN), 603950, Nizhnij Novgorod, Russia
OPTIMIZING THE DYNAMICS OF THE HOSPITAL NETWORK TAKING INTO ACCOUNT THE ERRORS OF OBSERVATION RESULTS
UDK: 330.46; 519.71; 519.213
DOI: 10.24412/2073-0667-2021-2-38-48
In Russia, over the past few years, there has been a tendency for an increase in the number of elderly citizens and the incidence of this category of persons. The situation that has arisen requires an increase in the efficiency of the existing system of medical institutions for each specific subject in the context of limited financial and other resources in health care. This task cannot be solved without an analytical study aimed at economic optimization of costs in each medical institution and in the network of medical institutions of the subject as a whole.
In this paper, we use a representation of the process of functioning of a network of medical institutions in the form of a control cybernetic system. The authors, using the cybernetic approach, have synthesized a mathematical model of the functioning of a typical medical institution of the network in the form of a control system. The created mathematical model allows for a single measurement with a given accuracy to generate the values of the main indicators of the work of a medical institution during each reporting period and, thus, to obtain additional statistics of any final volume for these indicators.
The data obtained using the constructed model can be used to study the quality and dynamics of the functioning of a medical institution. And also to study the influence of erroneous information on the performance indicators of the functioning of the network of medical institutions of the subject. In this work, on the basis of the additional statistics obtained, a solution to the problem of determining the mechanism for the optimal distribution of resources between medical institutions of a network of a particular subject is proposed, using the example of Nizhny Novgorod.
Key words: cybernetic control system, functioning of a medical institution, basic indicators, measurement accuracy, realization of a random variable, sample values, optimization.
Reference
Bibliographic reference: Proydakova E. V., Fedotkin M. A. Optimizing the dynamics of the hospital network taking into account the errors of observation results//journal “Problems of informatics”. 2021, № 2. P. 38-48. DOI:10.24412/2073-0667-2021-2-38-48
V. P. Savelyev, N. I. Sutyagina
Nizhny Novgorod State University of Engineering and Economics, 606340, Knyaginino, Russia
MATHEMATICAL MODELS OF REGIONAL BUSINESS DYNAMICS
UDK: 519.8
DOI: 10.24412/2073-0667-2021-2-49-58
Key words: region, small business enterprises, dynamic system, equilibrium state, linearization, phase portrait.
References
Bibliographic reference: Savelyev V. P., Sutyagina N. I. Mathematical models of regional business dynamics//journal “Problems of informatics”. 2021, № 2. P. 49-58. DOI:10.24412/2073-0667-2021-2-49-58
Lobachevsky University, 603950, Nizhniy Novgorod, Russia
APPLICATION OF OPTIMAL EVALUATION OF LINEAR TIME-VARYING SYSTEMS USING REACHABLE SETS
UDK: 517.9
DOI: 10.24412/2073-0667-2021-2-59-68
Supported by Ministry of Science and Higher Education of the Russian Federation (project 0729-2020-0055)
The paper is devoted to reachable sets of linear time-varying systems under uncertain initial states and disturbances with a bounded uncertainty measure. The uncertainty measure is the sum of a quadratic form of the initial state and the integral over the finite-time interval from a quadratic form of the disturbance. One of the main problems of dynamic system control theory is researching the opportunity of reaching this or that state under control. Reachable sets studying allows us to solve it. Reachable sets play a large role in different parts of control theory. The main are optimal control problems, disturbance estimation, etc. It makes them applicable to practically all spheres of activity: technical field (preservation of given trajectory by pilotless aircraft, taking into account the speed and direction of the wind~[1] and building manipulator path~[2]), economics~[3], medicine~[4], chemistry~[5], etc. Reachable set for system is class of all trajectory ends emerging from closed set in state space defining class of possible initial system states. These trajectories correspond to different values of disturbance. For the system under disturbance reachable set describes area in which the system comes under disturbance and allows us to evaluate accuracy of system hitting a finite state~[6]. Reachable sets allow us to realise whether it is possible to put the system into the given state provided we add a control to the system. There are two problems in this article: reacable sets finding problem and reachable sets estimation problem.
We solve some special matrix differential Riсcati equations for finding reachable sets. We demonstrate it using the Mathieu equation as the example of uncertain initial states and disturbances problem and linear oscillator with floating stiffness coefficient equation as the example of parametric uncertainty problem. Method of estimation of ellipsoidal reachable sets has been cosidered for such systems also using matrix differential Riсcati equation. Applying this method allows to find minimal ellipsoidal set that is defined by optimal observer. Besides, linear time-varying system with parametric time-varying uncertainty is being examined. Evaluation of ellipsoidal reachable sets is also given in the article. Applying the method is demonstrated with numerical modeling with the Mathieu-Hill dying-away equation for parametric vibrations and resonance. Euler iterative method is applied to compute required estimations.
Key words: reachable sets, ellipsoidal sets, optimal observer, parametric uncerainty.
References
Bibliographic reference: Sorokina M. Application of optimal evaluation of linear time-varying systems using reachable sets //journal “Problems of informatics”. 2021, № 2. P. 59-68. DOI:10.24412/2073-0667-2021-2-59-68.
Lobachevsky State University of Nizhni Novgorod 603950, Nizhnij Novgorod, Russia
NUMERICAL INVESTIGATION AND OPTIMIZATION OF OUTPUT PROCESSES IN CYCLIC CONTROL OF CONFLICTING FLOWS
UDK : 519.21
DOI : 10.24412/2073-0667-2021-2-69-80
In this article, we consider a non-classical queuing system with waiting, in which m conflicting flows are managed in a class of cyclic algorithms. Conflicting threads mean that they cannot be summed up and this does not allow you to reduce the problem to a simpler case with a single thread. Requirements from different conflicting threads are served at non-overlapping intervals. In addition, there are additional time intervals ― readjustments, due to which the problem of conflicting threads is resolved. Such systems are adequate models of real-world systems for processing and transmitting information, technological systems, transport systems, etc.
Unlike most well-known works, the so-called non-local description of the requirements flow proposed in [1-6] is used to construct a mathematical model of output flows. The description of output streams includes the state of the service device and the values of queues for conflicting streams. Note that the functioning of the system under consideration for servicing heterogeneous requirements and managing conflict flows in continuous time is a complex non-Markov process. Therefore, studying the characteristics of the system and the properties of the output streams in continuous time is a difficult task. Using the theoretical results of works [7-12], this article substantiates the method of numerical investigation of the system by simulation methods using computer and information technologies. The results of studies of the dynamics of output processes for servicing requirements on a simulation model are interpreted on the problem of managing conflict heterogeneous traffic flows at isolated intersections.
Key words: conflict flow, homogeneous Markov sequence, conditional distribution, Markov process.
References
Bibliographic reference: Fedotkin A. M. Numerical investigation and optimization of output processes in cyclic control of conflicting flows //journal “Problems of informatics”. 2021, № 2. P. 69-80. DOI:10.24412/2073-0667-2021-2-69-80
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