Informatics, computer technology and management. Рубрика в журнале - Siberian Aerospace Journal
N-version programming for nanosatellite telemetry processing
Статья научная
Software is a key element that ensures the functioning of any modern complex technical system. One such system is the constellation of spacecraft and associated ground control complexes that provide reception, transmission, and processing of collected telemetry. The process of data acquisition and its subsequent processing is critical to the flight control of the spacecraft and its onboard scientific equipment. Furthermore, telemetry data processed by ground control systems involves large volumes of raw data, the processing of which is a complicated and time-consuming task. In order to solve this problem, various methods of automatic data processing are used. Improving them is a key factor in ensuring the fault tolerance of onboard software and hardware, improving its reliability. Of all the existing widely-used methods of data processing, we shall focus on N-version programming (NVP) approach. N-version programming has firmly established itself as an effective method for increasing software reliability and designing fault-tolerant systems. Since its inception in the 1970s, this approach has been deeply connected with the development of aerospace software systems, including, among others, satellite ground control stations. In light of the aforementioned, this paper discusses the application of NVP for processing telemetry data gathered from nanosatellites (CubeSats). Due to the fact that there exists a skeptical view on the NVP approach in terms of its efficiency, the author covers this issue in existing literature in terms of the approach’s applicability for processing satellite telemetry.
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Статья научная
In the presented work, an assessment of the noise immunity of the parallel algorithm of the broadband signals with frequency manipulation with minimal shift time search is given. The threshold value of the Signal-to-Noise ratio per one L signal is used as a criterion, which ensures the required accuracy of code synchronization with a given error probability. In this case, the task of temporary search is formed as a task of L signals recognition which modulating code sequences { } kl d differ by a time cyclic shift τk (k 1) τэ (k 1,L,L the length of the modulating code sequence, э the duration of the signal element or the element of the modulating code sequence) equal to energy. The L -dimensional probability density of the modules 1,..., LVV ( i V the correlation module of the analyzed and comparison signals at the output of the quadrature correlator) is preliminarily determined. The probability of error is determined by L -multiple integration of the L -dimensional probability density of modules 1,..., L VV. The problem of determining the energy losses in the analyzed signal under the influence of structural interference is solved. In this case, a quadrature recognition circuit of one of two discrete signals is used and the Signal-to- Noise ratio at the output of the recognition circuit is determined, taking into account the effect of structural interference, and a comparison is made with the Signal-to-Noise ratio for the analyzed signal acting at the input of the quadrature recognition circuit. An assessment of the structural interference impact on the temporary search is given, taking into account the normalized periodic autocorrelation function (NPACF) of the analyzed signal and structural interference (at L 16383). It is shown that it is most rational to work with NPACF in its ascending section (cycle interval [ 361τ , 540τ э э]), while the reciprocal correlation coefficient will not exceed 4 · 10–3 and the equivalent energy losses of the analyzed signal will not exceed 3 dB. At an intensity of 34 dB of structural interference, the use of an ascending section of the NPACF allows minimizing the energy losses of the analyzed signal and temporary searching for a time of 0.35 s (at τэ = 2.5 microseconds).
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Non-parametric identification and control algorithms for T-processes
Статья научная
In this paper, nonparametric identification and control methods are considered for multidimensional discrete-continuous processes with a delay inherent in many real productions. Such systems are typical for practice, including in the rocket and space industry, as well as in the technological processes of space technology production. Considering multidimensional processes, it is necessary to take into account the relationships between input and output variables, as well as their relationships with each other. Moreover, these connections are not always known to the researcher. When taking into account unknown connections of input variables, the researcher will deal with tubular processes or H-models, and when taking into account unknown connections of output variables, the model for one or another channel of the object will be analogs of implicit functions. In general, the model of a multidimensional object will be represented as a system of nonlinear implicit equations. In this case, the solution of the identification problem will be reduced to finding the prediction of the vector of output variables from the known values of the vector of input variables and can be obtained only as a result of solving the corresponding system of equations, which were called T-models. The solution of a system of nonlinear implicit equations by parametric identification methods will not lead to the desired result due to the lack of sufficient a priori information, and here there is a need for the use of nonparametric identification methods, as well as the use of system analysis methods. A priori information in the tasks of nonparametric statistics is insufficient, which conventional identification methods cannot cope with. When controlling multidimensional processes, it is necessary to take into account the dependencies of output variables, in connection with which another important feature arises, namely: random values from the domain of determining output variables cannot be used as setting influences; they must be selected from their common intersection.
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Статья научная
The paper discusses new classes of models of multidimensional inertia-free systems with a delay in the condi-tions of a lack of a priori information. The subject is multidimensional discrete-continuous processes, the com-ponents of the vector of output variables of which are stochastically dependent in an unknown way. There are also processes, through some channels of which aprior information corresponds simultaneously to both the par-ametric and nonparametric type of source data about the studied process. The mathematical description of such processes leads to a system of implicit nonlinear equations, some of which will be unknown, while others will be known with accuracy to the parameter vector. The main purpose of a model of an object having stochastic de-pendencies of output variables is to find a forecast of output variables with known input variables. To find the predicted values of the output variables from known inputs, it is necessary to solve a system of im-plicit nonlinear equations. The problem is to solve a system that is actually unknown, when only equations for some channels of a multidimensional system are known. Thus, a rather nontrivial situation arises when solving a system of implicit nonlinear equations under conditions when, in one channel of a multidimensional system, the equations themselves are not in the usual sense, and in others they are known accurate to parameters. Therefore, an object model cannot be constructed using the methods of the existing identification theory because of a lack of aprior information. The purpose of this work is the solution of the identification problem in the presence of a partially-parameterized discrete-continuous process, and despite the fact that the parameterization stage cannot be overcome without additional priori information about the process under study. The control algorithm for multidimensional processes with dependencies of output variables is a sequential multi-step algorithmic chain that allows finding the control action and bring the object to the desired state. Computational experiments to study the proposed models and to control multidimensional discrete-continuous processes have shown quite satisfactory results. The article presents the results of computational experiments illustrating the effectiveness of the proposed technology for predicting the values of output varia-bles from known input variables, as well as for managing these processes.
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Nonparametric identification of dynamic systems under normal operation
Статья научная
The research gives nonparametric identification algorithms under the conditions of incomplete a priory information. The identification case differs from the previously known ones due to the fact that, besides the control action, an uncontrollable variable, but a measurable one, impacts on the object input. In contrast to parametric identification, the research considers the situation when the equations describing dynamic objects are not given with accuracy to the parameters. In this case, there are some features to study while getting the recovery characteristics of various object channels. The main characteristic is that the transition response of a channel is taken when the other channel is in a stable position. Moreover, the identification problem is analyzed under normal object operation, opposite to the previously known nonparametric approach based on Heaviside function input to the object and further Duhamel integral application. An arbitrary signal is input to the object during normal operation as a result we have a corresponding response of the object output. It should be noted that the measurements of the input and output variables are carried out with random noise. As a result, we have a sample of input-output variables. As linear dynamical system can be described by the Duhamel integral, with known input and output object variables, corresponding values of the weight function can be found. This is achieved by discrete representation of the latter. Having such realization, nonparametric estimate of the weight function in the form of the nonparametric Nadaraya-Watson estimate is used later. Substituting this with the Duhamel integral, we obtain a nonparametric model of a linear dynamical system of unknown order. The article also describes the case of constructing nonparametric model when a delta-shaped function is input to the object. It is interesting to find out how delta-shaped function might differ from the delta function. The weight function is determined in the class of nonparametric Nadaraya-Watson estimates. Previously proposed nonparametric algorithms consider the case when Heaviside function is applied to the object; this narrows the scope of nonparametric identification practical use. It is important to construct nonparametric model of the dynamic object under conditions of normal operation.
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Статья научная
Testing the hypothesis of independence of random variables is one of the main stages of sys-tem analysis of statistical data. Based on its results, the synthesis of effective decision-making algorithms is carried out. The traditional method of testing the hypothesis of independence of random variables is based on the use of the Pearson criterion, which contains a difficult to formalize stage of dividing the range of the values of random variables into multidimensional intervals. A method for testing the hypothesis of independence of random variables is proposed, which uses a nonparametric pattern recognition algorithm corresponding to the maximum likelihood criterion. Its application makes it possible to circumvent the problem of decomposing the range of the values of random variables into intervals. The idea of the approach is to form a training sample based on the initial statistical data to solve a two-alternative pattern recognition problem. Each class is defined under the assumption of independence or dependence of random variables, which is manifested in the difference in their distribution laws in the classes. Under these conditions, it becomes possible to replace the initial hypothesis with the task of checking the reliability of the difference in the probabilities of pattern recognition errors in classes. Using the apparatus of graph theory, the proposed method is developed in the formation of sets of independent random variables. The obtained results are generalized when testing the hypothesis of independence of random variables for large volumes of statistical data based on compression of the original information. This allows increasing the computational efficiency of the problem being solved. The article substantiates the method for testing the hypothesis of independence of random variables, based on the use of a nonparametric pattern recognition algorithm in conditions of large volumes of statistical data. The results of comparing the technique with the generally recognized Pearson consensus criterion in the study of ambiguous dependences between random variables of varying complexity are presented. The effectiveness of the proposed method is confirmed by the results of its application in processing remote sensing information from anthropogenic territories in the vicinity of the city of Krasnoyarsk.
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Object tracking with deep learning
Статья научная
Tracking objects is a key task of video analytics and computer vision, which has many applications in various fields. A lot of tracking systems include two stages: detecting objects and tracking changes in the position of objects. At the first stage, objects of interest are detected in each frame of the video sequence, and at the second, the correspondence of the detected objects in neighboring frames is assessed. Nevertheless, in difficult conditions of video surveillance, this task has a number of difficulties associated with changing the illumination of the frame, changing the shape of objects, for example, when a person is walking, and the task is also complicated in the case of camera movement. The aim of the work is to develop a method for tracking objects on the basis of deep learning, which allows to track several objects in the frame, including those in the rough conditions of video surveillance. The paper provides an overview of modern methods for solving objects tracking tasks, among which the most promising one is deep learning neural networks application. The main approach used in this paper is neural networks for detecting regions (R-CNN), which has proven to be an effective method for solving problems of detection and recognition of objects in images. The proposed algorithm uses an ensemble containing two deep neural networks to detect objects and to refine the results of classification and highlight the boundaries of the object. The article evaluates the effectiveness of the developed system using the classical in the field MOT(Multi-Object tracking) metric for objects tracking based on the known databases available in open sources. The effectiveness of the proposed system is compared to other well-known works.
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On incorrect representation of the shock process on shock polars in a viscous heat-conducting gas
Статья научная
The shock gas-dynamic processes that have found wide application in rocket and space technology in the design and optimization of devices and power plants are considered. The analysis of known exact and asymptotic relations/conditions on a shock wave, in particular, generalized differential relations (GDRs) on a curvilinear oblique shock wave (SW) (COSW) for a model of a viscous heat-conducting gas at high Reynolds numbers is made. The advantages of using the discrete-analytical approach are shown, for exam-ple: 1) the ability to make maximal use of the smoothness of the shock gas-dynamic formation (shock wave) in the tangential direction; 2) to build efficient computational algorithms devoid of the negative effect of approximation/artificial viscosity on the schematized discontinuity. In parallel, a very common graphical method for mapping the results of gas-dynamic calculations on the plane of shock polars, proposed by Busemann, and a volumetric (3D) polaroid, proposed by V. N. Uskov, are considered. The mathematical apparatus of shock polars itself is based on exact relations of the Rankine-Hugoniot type and has proven itself well even in modeling the flows of a viscous heat-conducting gas. However, in numerous literature sources there are results (shock solutions) of both physical and computational experiments, which are not mapped strictly on shock polars. In this paper, we show that in rare cases this very common way of such mapping may be incorrect. It has been proven that the main causes of such a defect are the combined ac-tion of three main factors: the nonuniform flow in front of the shock formation, the edge/boudary effect be-hind it, the action of the external factor of viscosity and the heat conduction mechanism.
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On remote sensing of the Earth by spacecraft
Статья научная
Remote sensing is a process which implies collecting information about an object. Due to their properties, satellite images are widely used in both practical and scientific fields. Satellite imagery is used in research aimed at the comprehensive study of natural resources, the dynamics of natural phenomena, and in the tasks of environmental protection. Special attention is paid to the use of space information for daily operational monitoring of the state of the environment in the implementation of geo-ecological monitoring of regions. In particular, this poses the problem to find the regions of the earth's surface with the characteristics determined by the considered parameters using the values of established parameters at certain points of the earth's surface. In this paper, we consider the special case of this problem when the given four points of the earth's surface determine the regions of the earth's surface (the so-called kernels of generalized squares) that have a specified configuration (square).
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On the application of the Bayesian search method for detecting underwater objects
Статья научная
Underwater archaeology and, in particular, the study of shipwrecked vessels, is one of the most advanced fields in the discipline, covering a range of theoretical and applied problems. In the practice of searching for sunken ships, it is quite rare to find a ship on the basis of previously (a priori) known data. In this regard, as well as in connection with the use of definitions from the field of probability theory and mathematical statistics, Bayesian statistics, namely, a search method that has found application in a number of well-known international search projects, has become the natural direction for the development of search systems. The Bayesian search method for determining the location of sunken ships, as well as their identification, has been almost never used in domestic underwater archeology practice. However, there is a need for its use, as it was demonstrated by the 2024 expedition to search for the transport ship Tbilisi, sunk during the Great Patriotic War in the Yenisei Gulf: despite the relatively small area of the search zone, determining the ship’s location became a very labor-intensive process. At the same time, the use of Bayesian search could significantly simplify this task. In this regard, the paper examines the methodology of using Bayesian search to detect sunken ships (an example of constructing a probability distribution in the search zone of the ship Tbilisi is given). In addition, the author considered the issue of using the Bayesian method for identifying objects (a database model is proposed with the inclusion of various search parameters). As a real example, the results of the 2024 expedition are presented with a description of the historical object, search conditions, and problems that arose during this work.
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On the choice of orbits for spacecrafts
Статья научная
The problem of distribution of a given number of spacecrafts over a certain structured set of orbits consisting of k np orbits is considered. The solution to this problem is given under the condition that the set of possible orbits for spacecraft coincides with the number of spacecrafts. In addition, it is assumed that the given set is divided into disjoint subsets of orbits, and the number of orbits in the indicated subset is the same. In the situation under consideration, it is equal to some prime number p. Currently, several orbits are used to place satellites on them, depending on the tasks they solve. Geostationary orbit is used for live TV broadcasting. Low satellite orbits are used for communication between satellite phones. Their own orbits exist for satellites of navigation systems GPS, Navstar, GLONASS, military satellites, satellites for various scientific research. Naturally, under these conditions, the problem of structuring a set of orbits with some restrictions on the location of the spacecraft in given orbits, depending on the purpose of the spacecraft arises. The problem of the complexity of calculating the number of orbits under these constraints is considered.
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On the construction of stress discontinuity lines for a two-dimensional plastic region
Статья научная
The paper considers the plasticity equations in case of two dimensions and the construction of stress discontinuity lines. The construction of stress discontinuity lines is based on the fact that they are located at the intersection point of lines of the same family (characteristics) and are directed along the bisector of the angle formed by these characteristics. Therefore, to find these lines, we constructed characteristics. Such a problem is easier to solve in case of plastic torsion, then there is only one characteristic, it is directed along the normal to the outer contour, and finding the sliding lines and their intersection points is quite simple. Consequently, most of the works devoted to the construction of stress discontinuity lines solve the problem of plastic torsion for isotropic and anisotropic media. For problems of plane strain of plastic ma-terial, this method is not sufficiently developed. This is due to the complexity of constructing sliding lines for such problems and the presence of two families of sliding lines. In this paper, we construct a homotopy of two known exact solutions: that of Prandtl and of Nadai, that is, a continuous transformation of one solution into another. In this case, one can observe the evolution of characteristics that depend on the group parameter a: for a = 1, the characteristics of the Prandtl solution are obtained; at a = 0, the characteristics of the Nadai solution, at a = 0.5, the characteristics of one fami-ly begin to intersect and lines of stress discontinuity appear. These lines are constructed in this paper.
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On the function of time distribution of a complex computing system uptime
Статья научная
Any space computing complex is a complicated system. A complicated system is understood as a set of functionally related heterogeneous devices designed to perform certain functions and solve problems facing the system. One of the important characteristics of a system is its uptime. This characteristic is often considered to be a random variable. However, such a mathematical model is quite limited, since the uptime depends on many characteristics (parameters) that describe a system. Therefore, the uptime can be assumed to be a continuous random field (that is, a random function of many variables). It is this approach that is used in this work. If there are certain restrictions on the uptime of a computing system, upper estimates are found for the distributions of a random number of system failures. Therefore, the problem of estimating Gaussian field distribution in Hilbert space arises. Two theorems that allow calculating the probability of a Gaussian vector falling into a sphere of a given radius are proved in the paper. The paper is devoted to the reliability of a computing system. The random number of a computing system failures v(r) is a characteristic of its reliability. The v(r) distribution is the distribution of the sum of a computing system random uptime. It is impossible to write down the distribution v (r) explicitly. Therefore, one has to look for an estimate of these distributions from above. Assuming that the uptime of a computing system is the sum of many variables, the authors of the paper obtained the following results: it is shown that the problem of estimating the distributions of a random number of system failures can be considered as the problem of estimating the convergence rate in the central limit theorem in Banach spaces; if there are certain restrictions on the uptime of a computing system, upper estimates are found for the distributions of a random number of system failures. The estimates obtained can be used for further research in the theory of computing systems reliability. Knowing these upper estimates, it is possible to predict the level of average costs for computer systems restoration, as well as for the development of special mathematical and algorithmic support for analysis systems, for management, decision-making and information processing tasks.
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On the location of spacecraft in a given number of orbits
Статья научная
Space vehicles are an expensive product. For example, just putting such a device into orbit costs at least one hundred million dollars plus the cost of the satellite itself and scientific equipment it carries. However, the cur-rent state of human civilization does not allow us to do without the presence of satellites in orbit. There were 2,062 active satellites in the international database as of March 2019. Compared to 2018, the number of new devices increased by 15 %. Experts warn that in the coming years, the world is expecting a «satellite boom» with a projected increase in the number of devices of about 15–30 % annually. All these satellites are rather different. Currently, several orbits are used for placing satellites on them, depending on the tasks they solve. A geostationary orbit is used for live television broadcasting. Low satellite orbits are used for communication between satellite phones. There are some orbits for navigation systems (GPS, Navstar, GLONASS). Naturally, under these conditions, there is a prob-lem of placing spacecraft over a given number of orbits, with some restrictions on the location of the spacecraft in certain orbits, depending on the purpose of the spacecraft. The solution to this problem is considered on the condition that the number of spacecraft coincides with the number of possible orbits in which they can be placed with some additional re-strictions on the possibility of their placement in orbit. Several solutions to this problem are obtained that allow us to calculate the number of possible combinations for such placement of spacecraft over a given number of orbits.
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One class of solutions to the equations of ideal plasticity
Статья научная
Much attention is given to the study and solution of nonlinear differential equations in the modern mathematical literature. Despite this, there are not many methods for researching and solving such equations. These are point and contact transformations of equations, various methods of separating variables, the method of differential connections, the search for various symmetries and their use to construct solutions, as well as conservation laws. The paper considers a nonlinear differential equation describing the plastic flow of a prismatic rod. A group of point symmetries is found for this equation. The optimal system of onedimensional subalgebras is calculated. Conservation laws corresponding to Noetherian symmetries are given, and it is also shown that there are infinitely many non-Noetherian conservation laws. Several new invariant solutions of rank one, i. e. depending on one independent variable, are constructed. It is shown how classes of new solutions can be constructed from two exact solutions, passing to a linear equation. Thus, in this short article, almost all methods of modern research of nonlinear differential equations are involved.
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Статья научная
Currently, the development of large-sized space structures and, in particular, transformable reflectors is actively developing. A feature of these devices is a small volume during transportation and large dimensions in the expanded working condition. Therefore, it is important to carry out a reliable and smooth deployment, adjust the shape of the active radio-reflecting surface with a given accuracy, and adjust the orbital position. In outer space, the system is constantly exposed to radiation, there is a large temperature difference in near-Earth orbit, there is a solar wind, which mainly affects the radiation pattern. In this paper, the process of deployment of the reflector spokes in the presence of disturbances and measurement errors is considered. The solution to the problem is presented using the separation theorem. To estimate the parameters of the system in the presence of measurement noise, the Kalman filter is applied. Its performance is shown at various values of the noise intensity. A random process such as white noise was selected as external disturbances and measurement noises. The control problem is solved using the optimal control algorithm according to the hierarchy of target criteria. The possibility of minimizing energy costs by means of interval switching on of measuring sensors is shown. The results of numerical simulation are presented.
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Optimization control actions for the electrolytic method of aluminium production
Статья научная
The most common indicator of the aluminium production process managing efficiency is the cost of the metal production, but this concept includes a lot of components. First, this is the cost of raw materials and electricity in this region, as well as the labour cost per ton of products, consumption coefficients of raw materials and energy, capital costs for construction and repairs, waste disposal cost, environmental payments, etc. At the same time, there is no single functional of the process quality, depending on technological parameters, that is, the problem of complete and relatively strict mathematical process optimization as a whole is currently not solvable, not only because of its volume, but because of the lack of a complete efficiency model. In this study, particular efficiency criteria are considered, the improvement of which is aimed at the optimization model of control actions developed by the authors, which are selected based on the possible levers of the current automated process control system (APCS) for aluminium electrolysis. All tests were carried out on Virtual cell software without transfer to a real control object.
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Optimizing the readability of tests generated by symbolic execution
Статья научная
Taking up about half of the development time, testing remains the most common method of software quality control and its disadvantage can lead to financial losses. With a systematic approach, the test suite is considered to be complete if it provides a certain amount of code coverage. At the moment there are a large number of systematic test generators aimed at finding standard errors. Such tools generate a huge number of difficult-to-read tests that require human verification which is very expensive. The method presented in this paper allows improving the readability of tests that are automatically generated using symbolic execution, providing a qualitative reduction in the cost of verification. Experimental studies of the test generator, including this method as the final phase of the work, were conducted on 12 string functions from the Linux repository. The assessment of the readability of the lines contained in the optimized tests is comparable to the case of using words of a natural language, which has a positive effect on the process of verification of test results by humans.
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Статья научная
The article proposes an approach to developing the architecture of a service-oriented information processing system, modeling and process control. The system, which is being developed, is a tool for identifying, predicting and controlling discrete-continuous processes. Its mathematical apparatus is based on nonparametric algorithms of identification and control. The software architecture includes the following main modules: the module for processing data, modeling and forecasting output process variables and the process control module. The first module includes data preprocessing algorithms: normalization, centering and analysis of outliers and omissions. The modeling module is an algorithm for research and recovery dependencies between process variables, process identification using nonparametric estimation of the regression function from observations. The last module is an implementation of nonparametric dual control algorithms. Control devices built on the basis of these algorithms perform functions of both object control and its study. The article discusses the application of architectural solutions based on two proven approaches in the field of software development: the composite approach and the service- oriented approach.. The main principles of composite architecture as a set of software systems with many characteristics that perform a specific task and service-oriented architecture as a modular approach to software development are described. The advantages of the applied composite service-oriented architecture over other variants of software architecture for control systems are shown, in particular, monolithic software architecture is compared with composite service-oriented architecture. This means that a researcher can use a single operation, which is a logically isolated, repeated task related to the production process of the enterprise. At the same time, it is necessary to ensure positive results when integrating with existing software products of enterprises which greatly complicates and requires the development of new components, as well as support for the "inherited" parts of the system.
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Piecewise approximation based on nonparametric modeling algorithms
Статья научная
In this research the issue of inertialess processes modeling is under study. The main modeling algorithm is the non-parametric recovery algorithm of the regression function. The algorithm allows to build a process model under conditions of low a priori information. This feature may be particularly important in modeling processes of large dimensions prevailing in the space industry. One important feature of the algorithm for nonparametric estimation of the regression function is that the accuracy of modeling using this algorithm highly depends on the quality of the observations sample. Due to the fact that in processes with large dimensions of input and output variable vectors observation sampling elements are in most cases unevenly distributed, the development of modifications to improve the quality of mod-eling is relevant. The modification of the nonparametric dual algorithm based on piecewise approximations has been devel-oped. According to the proposed modification, the process area is divided into sub-areas and a non-parametric esti-mate of the regression function for each of these sub-areas is recovered. The proposed modification reduces the impact of some observation sampling features, such as sparseness or voids in observation samples on the quality of the built model. The computational experiments were carried out, during which a comparison was made between the classical algorithm of non-parametric estimation of regression function and the developed modification. As the computa-tional experiments have shown, with uniform distribution of the sample elements of observations, the developed modification does not lead to the improvement of the quality of modeling. With a substantial uneven distribution of the observations sample elements, the developed modification resulted in a 2-fold improvement in the quality of the simulation. The results suggest that the proposed modification can be used to model complex technologi-cal processes, including those in the space industry.
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