Adequacy Of Mathematical Model Research Articles

СИНТЕЗ АЛГОРИТМА ОЦЕНКИ ПАРАМЕТРОВ МНОГОПОЗИЦИОННОЙ СИСТЕМЫ НАБЛЮДЕНИЯ И ИССЛЕДОВАНИЕ ЭФФЕКТА РАСХОДИМОСТИ ПРОЦЕССА ФИЛЬТРАЦИИ

The accuracy of aircraft position measurements directly affects flight safety and is one of the most important tactical characteristics. The introduction of new advanced surveillance tools, such as multi-position surveillance systems (MLAT), can significantly increase the level of flight safety, as well as improve the efficiency of airspace use. The authors consider the task of improving the quality of MLAT functioning and increasing the accuracy of estimating the aircraft coordinates. The accuracy of position-fixing is determined by the error in measuring the time of signal arrival under the influence of noise and interference. Random disturbances must be taken into account to ensure high-quality MLAT operation. This is achieved by applying the methods of Kalman filtration theory. Therefore, to solve the problem of estimating the MLAT state variables, it is proposed to use a Kalman filter (KF). The effectiveness of using the Kalman filter depends on the adequacy of mathematical models and real processes. Model inaccuracies associated with the functioning of navigation systems lead to KF divergence. The paper presents the results of theoretical studies and simulating the MLAT functioning processes based on the implementation of the KF algorithm.

Determination of heat transfer process in vertical cable tunnels of nuclear power plants under real fire conditions

The object of the study was heat and mass transfer processes occurring in vertical cable tunnels. The problem to be solved was the definition of process mechanisms in the inner space of the tunnel. For this, field tests were conducted, mathematical models were created, and computational experiments were conducted to establish specific parameters that affect the temperature regime of a fire in a vertical cable tunnel of a nuclear power plant. The dynamics of temperature changes with known geometric parameters and fire load were determined, and the adequacy of mathematical models built in the Fire Dynamics Simulator software was investigated and computational experiments were carried out. It has been proven that they consist in determining the temperature regime in a vertical cable tunnel of a nuclear power plant with known technical and geometric parameters. Such studies have practical applications in the field of safety of nuclear power plants and the development of new technologies in this field. An important conclusion of these studies is the possibility of determining the fire resistance of building structures of vertical cable tunnels of nuclear power plants with the selection of the most severe temperature regime, according to the conducted field test. This means that research results can be used in practice in designing and evaluating the safety of such objects. The conducted research established that the temperature in the inner space of the tunnel can reach values from 1200 to 1400 ℃. The following factors influence the maximum temperature value and the maximum time to reach the maximum temperature in the fire cell: fire load, height and area of the tunnel. With a lower fire load, the maximum temperature in the vertical cable tunnel of the nuclear power plant was 75 % lower. Therefore, the results of these studies have a direct practical application in the field of safety of nuclear power plants and can be used to improve and develop new technologies in this field

Assessment of Detritus Biomass in Forest Ecosystems

The current method for estimating the biomass of dead organic matter in forest ecosystems is analyzed. The main attention is paid to the study of the adequacy of mathematical models used to estimate the density of decomposing wood. The failure of the traditional approach to estimating the density index, both for an individual decomposing dead tree, and in calculating the average index characterizing the density of the wood of the totality of dead trees is shown. We consider the possibility of applying the concept of effective density to estimate the biomass of dead wood remains as the most corresponding to their heterogeneous structure. We propose a method based on representations of interval analysis, which allows, using standard values of the density of healthy wood, to estimate the density index of rotting wood of the corresponding decomposition class of the debris classification system.

Effect of extrusion process parameters on pellet crumbliness in fish feed production

The article presents the results of studying the pellets crumbliness index of a pilot batch of extruded compound feed for pike-perch. The regimes of extruded pellets fish compound feed production with minimal crumbling were defined using a mathematical model. A 3-factorial central composite design was implemented to obtain a mathematical model of the process of extrudate resulting in the second-degree polynomial. The process influencing factors include the feed mixture moisture W (%), extrusion temperature T (°C) and steam pressure P (MPa). Experimental studies were conducted according to the experimental plan. The experimental data obtained were entered into the planning matrix. Experimental data was processed in a program prepared in Microsoft Excel. As a result, an adequate second-order mathematical model describing the dependence of the crumbliness index on the feed mixture moisture before extrusion, extrusion temperature and steam pressure was obtained. The mathematical model adequacy was tested based on Fisher's variance ratio. The Fisher criterion is an important statistical tool for verifying the adequacy of the model and analyzing the variance. The analysis of the obtained model and its graphical interpretation are presented. A good extrusion process mathematical model was developed. The fundamental principles of feed mixture processing on a single-screw extruder and the choice of its rational parameters ensure the production of extruded compound feed pellets for pike-perch with minimal crumbling were studied. A minimal crumbling is provided at the following values of the factors - feed mixture moisture before extrusion is 32%, extrusion temperature is 132 °C, and pressure is 0.4 MPa. In the selected levels of factors, the calculated value of extruded pellets crumbling was 1.02%.

Modeling and estimating the model adequacy in muscle tissue electrical stimulator designing

The research object is the mathematical modeling of human skeletal muscle electrical stimulation characteristics during therapy. The subject of research is mathematical models of electrical stimulation characteristics that relate muscle contraction amplitude to the amplitude, the rate of stimulating effects, and other parameters. The research purpose is to study such models, develop an algorithm for their correction and a method for estimating their adequacy. The methods used: mathematical modeling methods, methods of structural and parametric identification of models, optimization methods, methods for estimating the adequacy of models, and interval methods. The results: an algorithm for muscle electrical stimulation characteristics mathematical models correcting during several sessions in case of their change is proposed; a method for estimating the model adequacy area in the external variables space in order to control its adequacy is proposed; using the interval mathematics methods to construct the adequacy actual area is justified; an interval estimating of the error in modeling a certain output characteristic is introduced, that, in the case of characteristic monotonicity, allows checking the adequacy maintaining by checking some inequalities. The results can be used in the design of electrical stimulators and for determining the electrical stimulation effects of individual parameters during one session or a series of sessions. The scientific originality: the interval mathematics methods for approximating the mathematical model adequacy area in a hyperparallepiped and checking for nesting in the target area in the external parameters space in order to control the adequacy during the model correction in electrical stimulation is proposed and justified.

STUDY OF THE INFLUENCE OF REGIONAL DEVELOPMENT FACTORS ON THE FORMATION OF THE STUDENT CONTINGENT

The relevance of the study of the influence of factors of regional development on the formation of the contingent of students has been proven. The purpose of the study is to determine the cause-and-effect relationships between the number of students in higher education institutions and the regional development indicator. To achieve the goal, the following tasks are solved:  characteristics of mathematical tools to ensure the educational process;  development of a mathematical model of the influence of the gross regional product index on the number of students in higher education institutions. Proposed directions Mathematical modeling is implemented through the prism of the relevant stages: formation of information and analytical support regarding the factor of formation of the contingent of students by region and the integral indicator of regional development; application of mathematical modeling methods in relation to establishing the influence of the factor on the integral indicator; development of mathematical models of the influence of the factor on the integral indicator of regional development; determination of criteria for the adequacy of mathematical models; interpretation of the obtained results. The ways of assessing the integral indicator of regional development are determined: determination of local factors of regional development; formation of general indicators of regional development; construction of a multi-level system of integrated assessment indicators; assessment of local factors using quantitative and qualitative methods; development of mathematical models for the assessment of generalizing factors; assessment of generalizing factors using the geometric mean method; construction of an integral model of assessment of regional development; determination of weight coefficients taking into account the corresponding stimulants; assessment of the integral indicator of regional development. As a result of the study, it was determined that the gross regional product index causes a change in the number of students in higher education institutions by 68%. It should be noted that other factors negatively affect the number of applicants. In the study, a mathematical toolkit is proposed, which allows you to form a quantitative basis for making management decisions. It was established that in the future, regional factors and demographic parameters lead to a decrease in the number of students in higher education institutions, which negatively affects the development of the educational system. Keywords: educational components, PhD, educational programs, geodesy and land management, general and professional competences.

The Model of Socio-Economic Stratification of Society Based on An Agent Approach, Taking Into Account Rank Exchange

The purpose of the study is to demonstrate the possibility of adequate mathematical modeling of property inequality in a stratified society based on an agent approach, taking into account the unequal (rank) exchange between agents with different socio-economic status. To achieve this goal, it was necessary to solve the following tasks:1. To develop minimum criteria for the adequacy of mathematical models of socio-economic inequality.2. To substantiate the advantages of mathematical models based on an agent approach using rank exchange.3. To present the author’s version of the mathematical model of the goods’ movement in society based on an agent approach, taking into account rank exchange.4. To check the author’s mathematical model for compliance with the minimum qualitative criteria of adequacy and to create a quantitatively coinciding with the real distribution of the country’s population of the Russian Federation by income.5. Determine the limitations of the developed mathematical model by the authors.Materials and methods. The paper used statistical data from Rosstat, the Federal Tax Service of the Russian Federation, the World Bank, the US Bureau of the Census, the Edelman Trust Barometer, as well as data published by domestic and foreign researchers of inequality. These data were compared with the results of inequality calculations obtained using the mathematical model of the goods’ movement in society based on an agent approach, taking into account rank (nonequivalent) exchange.Results. The minimum criteria that any adequate model of stratification of society must meet are defined: 1. reveals a lognormal distribution of the population by income with a heavy tail; 2. reflects in long-term historical retrospect the trend of inequality growth; 3. demonstrates a short-term reduction of inequality during periods of socio-economic crises.The proposed model meets these requirements, it demonstrates not only qualitative, but also quantitative adequacy – reproduces by calculation the curve of the actual distribution of Russian society by income.Common concepts of inequality, in particular, the theory of superstars or skill-based technological change do not allow achieving such a result. The limitations of the created mathematical model are shown:– the impossibility of creating an adequate Lorentz curve with insufficient computing power;– inability to describe changes in cross-country inequality, since countries are not subjects of rank exchange, although interstate agreements certainly have an impact on the exchange between economic subjects of social interaction;– inability to describe situations of absolute poverty, i.e. long-term decline in income, although in the real economy such situations are sometimes observed;– the endless growth of inequality over time, whereas in reality economic processes of inequality growth are always opposed by social processes of counteraction to this growth.Conclusion. The mechanism of spontaneous emergence and growth of income (property) inequality as an inevitable consequence of market relations is demonstrated.The agent-based approach, taking into account the rank exchange, can allow identifying critical moments in time, after which extraordinary economic and socio-political consequences will have an irreversible character, and can also help in studying the influence of individual behavior of economic agents of different levels on the evolution of the entire economic, and as a consequence, socio-political system.The results obtained will be useful for strategic planning, when developing target indexes for the socio-economic development of regions and the country as a whole, and improving the methodology for calculating key performance indexes for civil servants

КОНЦЕПЦИЯ УПРАВЛЕНИЯ ПРОМЫШЛЕННОЙ ПОЛИТИКОЙ В ЧАСТИ ГОСОБОРОНЗАКАЗА В УСЛОВИЯХ СПЕЦИАЛЬНОЙ ВОЕННОЙ ОПЕРАЦИИ

With the beginning of a special military operation, the defense industrial complex of the Russian Federation became more active. Work with the state defense order is carried out under special conditions and is regulated by a number of regulations. The regulation on the procedure for the formation of prices for state defense order products divides pricing methods into two large groups: methods based on the analysis of the external environment and costly methods. Methods based on the analysis of the external environment are based on the study of prices for similar products. The mechanism of such pricing and cost planning correlates with the Target-costing method, but this method does not take into account the specifics of production processes at the enterprise. In the group of cost methods, the price is set in the form of a planned surcharge, the most accurate method of calculating the cost to date is the TBC-costing method. The article proposes a new concept for the implementation of the mechanism of industrial policy of the Russian Federation, based on the author’s method of calculating the cost of production – «claster-costing», which is based on the methodology of TVS- costing, as the most accurate method of calculating the cost of industrial products to date. The concept suggests taking into account the principles of behaviorism, in order to achieve maximum adequacy of mathematical models, as well as the principles of target-costing, since in the implementation of the state defense order, clear planning of budget expenditures is necessary. The implementation of all the stated elements of the industrial policy management concept of the Russian Federation should be based, according to the authors, on the principle of fairness. The proposed concept based on the claster-costing method will contribute to a more effective implementation of the industrial policy of the Russian Federation in accordance with the tasks set by the Government of the Russian Federation in this area.

Development of new mathematical methods and algorithms for verifying the adequacy of mathematical models of objects based on data from a natural experiment to determine the area of functional stability

Development of new mathematical methods and algorithms for verifying the adequacy of mathematical models of objects based on data from a natural experiment to determine the area of functional stability

Principles and stages of creation of automatic control systems with a model of complex technological processes

The object of this study is an effective approach to designing automatic control systems with a model of complex technological processes, which include interrelated and complementary stages from the formulation of the task to the implementation of the control system at the facility. It includes a set of measures, starting with the analysis of the process in its hardware design with the construction of an information and logical scheme to the development of all types of security and commissioning. The main problem in the implementation of a control system with a technological process model is the limited ranges of adequacy of mathematical models. Therefore, when changing the load on the unit, changing external and internal perturbations, it is necessary to constantly ensure the necessary level of adequacy of the models. It is proposed to use a combined model as a mathematical model of the control object, combining the advantages of analytical and experimental-statistical models. This makes it possible to significantly expand the information base of the resulting model. A simple and effective iterative algorithm for calculating this model is also proposed. It includes sequential steps to determine the parameters of the model by basic dependences (the deterministic part of the model), followed by clarifying them according to the current data from the object (experimental statistical part of the model). The effectiveness of the approach is confirmed by the example of ASC TP of the ammonia synthesis column. By improving the accuracy of determining the control parameters and narrowing the range of their change around the optimal value, the volume of ammonia release increases by 5–8 %. The application of the described approach on the example of the development of an automatic control system for the technological process of ammonia synthesis confirmed the economic feasibility of implementing the proposed solutions

An approach to estimating the range of wild fire firebrands transportation by wind

Forest wild fire prevention that is carried out to minimize of wild fires number and possible damage from them, is inconceivable without the development of fire-prevention procedures, which are based on data from wild fire monitoring. One of the areas of wild fires monitoring is the evaluation of forest wild fires spreading ability to transportation of firebrands by the wind. This way observed at crown-type of forest wild fires, which are formed from ground ones by transferring the flame to the trees crowns when the wind reaches a certain speed. A mathematical model has been developed for the firebrands transportation by the wind. Mathematical model adequacy was evaluated by inclusion of transfer coefficient as a ratio of firebrand speed to wind speed. The modeling of the horizontal firebrands transportation (sphere, cylinder and disc) has been carried out. Experimental conditions are firebrands have various sizes, fall off from a height of 5-20 m, at a wind speed of 1-5 m⋅s-1. Spherical and discshaped particles have the best transportation ability. The adequacy of the model of horizontal firebrands transportation has been experimentally proved by conducting our own experiments and analyzing information obtained from the literature on similar studies. The results of the experiments showed that with an increase in the air flow speed up to 10 m⋅s-1, the speed of the burning particle does not exceed the speed of the air flow that stimulates it. Thus, it is shown that within the air flow velocity of 1-10 m⋅s-1, the value of the transfer coefficient does not exceed one.

Evaluation of the adequacy of phase equilibria modeling based on various sets of experimental data

Objectives. The purpose of the paper is to compare the adequacy of mathematical models of vapor–liquid equilibrium (VLE) and their ability to reproduce the phase behavior of the ternary system benzene–cyclohexane–chlorobenzene using different experimental data sets to evaluate binary interaction parameters.Methods. The research methodologies were mathematical modeling of VLE in the Aspen Plus V.10.0 software package using activity coefficient models (Non-Random Two-Liquid (NRTL), Wilson) and the Universal quasichemical Functional-group Activity Coefficients (UNIFAC) group model, which allows for independent information. For the benzene–cyclohexane–chlorobenzene ternary system, the use of the NRTL equation is warranted because it provides a better description of the VLE experimental data.Results. The diagram construction of the constant volatility of cyclohexane relative to benzene lines revealed three topological structures. Only one of them can be considered reliable because it corresponds to the experimental data and coincides with the UNIFAC model diagram constructed based on independent UNIFAC model data. The results indicate that to study systems containing components with similar properties, it is necessary to improve the description quality of the available data sets (the relative error should not exceed 1.5%).Conclusions. The reproduction of the thermodynamic features of various manifolds in the composition simplex obtained by processing direct VLE data can be used to supplement the adequacy of the model. For the cyclohexane–benzene–chlorobenzene system, the best NRTL equation parameters are those regressed from the extensive experimental VLE data available in the literature for the ternary system as a whole.

Mathematical Modelling to Predict Angular Distortion in Butt-welded Aluminium 1200 Plates

MIG welding is an inert gas shielded arc welding process using a consumable electrode. The process finds huge applications in various engineering industries due to its versatility and its ability to get automated. In the present work, angular distortion (AD) occurring in the butt welds has been analysed using mathematical modelling. The important input parameters selected are wire feed rate (WFR), welding speed, and voltage. These variables are taken at three levels to analyse their effect on AD. The investigation was carried out using a statistical design of experiments technique. The ANOVA technique is employed to evaluate the mathematical model's adequacy. The graphical results were evaluated using the response surface methodology (RSM). The optimal combination of input parameters was determined to have minimum possible AD.The present study is expected to provide useful results which may prove beneficial for the aluminium fabrication practices.

THE GROUNDS OF MATHEMATICAL MODEL OF PROGNOSTICATION OF MEDIUM-SIZED CITY SUSTAINABLE DEVELOPMENT

The analysis having been done gives evidence of the fact that every city is socially, economically, ecologically, geographically unique and inimitable as to its substance. Accordingly, the model of its sustainable development is based on different outer and inner factors which determine their development and which are and will always be specific and individual. That`s why the elaboration of the model of sustainable development prognostication of medium-sized cities including Rivne has become urgent. The basis of sustainable development prognostication of the city is the achieved level of the development of social, economic, ecological spheres of its vital activity, proportionality and balance of these spheres functioning, taking into account the preconditions of their perspective development. The research and prognostication of the sustainable development of medium-sized cities were carried out in the following three stages: primary, analytical and calculation, final. The algorithm of the structure of mathematical models for determination the indices of social, economic and ecological spheres development of the cities assumes the realization of the following structural elements: making regressive and corrective analyses, control of the adequacy of mathematical models, making prognosis of the development of these spheres and integral index of the city sustainable development. It is ascertained that the values of the indices of the city social, economic and ecological spheres development are stipulated by the indices stimulants and destimulants and are deseribed by mathematical models which have the form of multifactor linear regressins with the coefficient of determination from 0,841 to 0,921. Aimed at the improvement of the level of city development within the period of the year 2025, it is projected to improve 2 indices of social sphere, 3 – economic, 2 – ecological in Ivano-Frankivsk, 3 indices of social sphere, 2 – economic, 2- ecological in Lutsk, 5 indices of social sphere, 4 – economic, 3 – ecological in Melitopol, 4 indices of social sphere, 3 – economic, 4 – ecological in Rivne. All these will allow Ivano-Frankivsk to reach the high level of development (0,75), for Lutsk – 0,619, for Melitopol – 0,594, for Rivne – 0,679 average level of their socio-economic-ecological systems development.

Confirmation of the mathematical model adequacy describing the dynamics of implementing a computer attack possibility in time

One of the urgent problems of information security at the moment is the lack of a unified approach for predicting computer attacks that is confirmed by both international and Russian experts as well as by practice. The paper proposes a mathematical model describing the dynamics of the computer attack in time built on the basis of the Diffusion of Innovations Theory. The substantiation of the possibility of using the Diffusion of Innovations Theory is given; the specifics of cybercrimes from other types of crimes are described. The adequacy of the proposed model is confirmed, in particular, by full-scale modeling of cyber-attacks on the organization's resources via the Internet site. The data of practical experiments confirm the mathematical model; in particular, the probability function of the cyber-attack implementation in time, i.e. the change in the number of infected nodes in time can be approximated using an s-shaped Pearl curve or a cascade model. The optimality of this approximation is proved by the method of least squares.

МАТЕМАТИЧНЕ МОДЕЛЮВАННЯ РОБОТИ СУДНОВИХ ГВИНТІВ ПРИ РІЗНИХ РЕЖИМАХ МАНЕВРУВАННЯ

In the study of ship maneuvers, in the development of ship's autopilots and simulators, the availability of adequate mathematical models of the ship’s propulsion complex, which would describe with high accuracy the dynamics of the ship for the widest ranges of phase coordinates, plays a decisive role. Mathematical models of ship dynamics consist of mathematical models of inertial and non-inertial forces and moments on the ship's hull. The latter include hydrodynamic and aerodynamic forces and moments on the ship's hull, forces and moments caused by the operation of propellers, rigging, ship rudders. For the adequacy of the general mathematical model of the ship's dynamics, the adequacy of the mathematical models of all components of non-inertial forces and moments on the ship's hull is necessary. Mathematical models of non-inertial forces and moments on a ship's hull are usually empirical and made by processing data from experimental tests in ship model basin or in full-scale sea trials. In particular, this applies to the forces and moments caused by the operation of the propeller. In the mathematical modeling of the indicated forces and moments, the rectilinear movement of the vessel, or movement at small values of the drift angles, is usually considered. In this case, only the longitudinal action of the propeller is usually considered, and the transverse component of the force and the moment on the ship's propeller are unreasonably neglected. In addition, well-known mathematical models usually contain coefficients and functions that are difficult to determine using tables and graphs, which is inconvenient for numerical modeling. This work is intended to somewhat eliminate the indicated disadvantages. In particular, it was investigated how the curvilinear movement of the ship affects the operation of the propeller, analyzed the existing mathematical models, and new effective mathematical models of the longitudinal and transverse components of the force and moment caused by the operation of the ship's propeller in dimensional and dimensionless forms were proposed. In particular, representations are obtained for the dimensionless coefficients of the propeller stop and the moment on the propeller shaft, the wake fraction coefficient, the thrust-deduction factor, and skew of the flow on the propeller. It is shown how these parameters change for all possible values of the drift angle and angular velocity. For a number of commercial vessels of various types, technical characteristics are given, and parameters for constructing mathematical models of the dynamics of the propeller are calculated.

Problem of Mathematical Model Adequacy in Assessing the Seismic Risk

The problem of mathematical models' adequacy in assessing seismic risk is considered. It is demonstrated that the currently used methods of testing such models make it possible to assess only the consistency of simulation and real data by counting the number of earthquake epicenters that appear in the areas with increased values of the fields with various indicators. The paper proposes a fundamentally new approach to testing adequacy of the seismic risk assessment models based on examining statistical hypotheses. Application of this approach is considered in a seismic risk assessment model for the territory of Armenia and the adjacent regions. Practical implementation of the proposed approach and the results obtained convincingly confirm that the tested mathematical model is adequate. Normality of the seismic risk values general set distribution calculated by the probabilistic model for Armenia and the adjacent territories is presented. Correlation coefficients of theoretical and empirical frequencies distribution are 0.75--0.99. It is shown that adequacy of the seismic risk assessment probabilistic model should be checked taking into account the earthquake abyssal levels. Conclusions are provided on operability and possibility of further use of the considered method in checking adequacy of assessing the seismic risk mathematical models

NONLINEAR PROBLEM OF FRACTURE MECHANICS OF AN INTERFACE CRACK SUBJECTED TO SHEAR WAVE

Solution for the problem for an interface crack under the action of a harmonic shear wave in normal direction is presented. The contact of the crack faces is put into consideration. The problem is solved by the boundary integral equations method, the vector components in the boundary integral equations are presented by Fourier series. The unilateral Signorini boundary conditions are involved to prevent the interpenetration of the crack faces and the emergence of tensile forces in the contact zone. Amonton-Coulomb Friction Law included allows to put into consideration relative resting of the crack opposite faces or their relative motion when one crack face is slipping or sliding across another face. The contact boundary restrictions are implemented using the iterative correction algorithm. The mathematical model adequacy is checked by comparing with classical model solution for statics problems that takes into account the crack faces contact. Numerical researches of friction influence on the displacement and contact forces distribution, size of contact zone are carried out. Influence of the faces contact and value of the friction coefficient on the distribution of stress intensity coefficients of normal rupture and transverse shear, which are the parameters of the biomaterial fracture, are presented and analyzed. It is shown that the nature of change in the distribution of the stress intensity coefficients for the conditions of tensile and shear waves is fundamentally different. It is concluded that it is possible to extend the approach proposed to the problems of three-dimensional fracture mechanics for composites with interfacial cracks at arbitrary dynamic loading.

DIAGNOSTICS OF ACTUATORS OF MACHINE TOOLS DRIVES ACCORDING TO THE IDENTIFIABILITY CRITERION BY THE STATE SPACE

Currently, in view of the fact that almost all the tasks of the practice of controlling CNC machine and robot drives cannot be accurately represented by linear models, and there is no solution to non-linear models in the general case, a very important task is to develop control algorithms based on discrete models. Discrete models of nonlinear systems assume variable state, control, and measurement matrices that determine an infinite number of variants of this model. Therefore, some tool is needed to calculate the degree of adequacy of mathematical models and real objects. The paper considers theoretical statements related to the main directions of research in the field of theoretical issues - the study of the dynamics of CNC machine and robot drives and their modeling. The paper studies the drives of CNC machine and robots by the criterion of identifiability based on a discrete digital control model. Criteria of observability, controllability and identifiability of drives are considered as a function of the rank of an extended state matrix with a measurement matrix, in which the relative errors of the information-measuring system are analytically taken into account. An algorithm for calculating the identifiability criterion for a nonlinear control system in a discrete linearization version is proposed. It is proposed to use identification in terms of the correspondence of the mathematical model to the results of the operation of the object. Drive control by means of a discrete vector-matrix algorithm involves the calculation of the state matrix at each step. Therefore, at each step, the determinant of the extended matrix is calculated, which is compared with a constant that numerically divides the space of the state matrices. Thus, the operation of the drives itself makes it possible to determine its identifiability. As a criterion for the optimality of the identification algorithm, a decision-making optimality criterion is chosen in combination with an identifiability criterion for an optimal control algorithm by the criterion of minimum quadratic form. The vector-matrix model of drives in the state space is presented taking into account the relative accuracy of measuring the state of the information-measuring subsystem of drives. It is proposed for practical problems to determine the identifiability criterion by modeling the state matrix for cases when the state matrix parameters exit the space of realizable parameters of serviceable drives. The linearized model of CNC machine and robot drives has limitations in accordance with technical characteristic, for example, restrictions on the strength of electric current and voltage. The obtained research results can be used to build diagnostic systems for CNC machine and robot drives.

Development of mathematical models of strip heat transfer in the inter-stand spacings of the finishing group during wide-strip hot rolling

The analysis of regularities of heat transfer in the finishing group of a wide-strip hot rolling mill is carried out and mathematical models are proposed to describe thermal processes in various zones of the inter-stand spacings. A significant difference in conditions of heat transfer in the middle, most extended zones of the inter-stand spacing of the finishing group was revealed, which consists in the presence of a large amount of water vapor between the stands and absorption of thermal radiation from the strip by steam. A mathematical model of absorption of radiation from a strip by water vapor is proposed, which complements the models of convective and radiant heat transfer of the strip in the spacing. The difference lies in the determination of the reduced emissivity of the medium/strip system, taking into account the absorption capacity of the vapor, depending on the temperature and partial pressure. Using an extended model of strip cooling in the spacing, a numerical simulation of the rolling process was carried out for the conditions of a 2000 wide-strip mill with various strip cooling schemes. It is shown that the prediction of strip temperatures at the exit of the finishing group without taking into account the absorption of radiation can have an error of up to 25 °C or more, which will negatively affect the prediction accuracy of the structure and mechanical properties of rolled products. It is noted that the detailing of the description of thermal processes in the finishing group makes it possible to reduce uncertainty and increase the adequacy of mathematical models when controlling the mill. The proposed model is included in the integrated model of the 2000 mill finishing group and is used when adjusting the strip cooling modes, which provided a significant reduction in the yield of inappropriate products by the “ripple” defect. The numerical implementation of the model was carried out by M. N. Oreshina, Cand. Phys.-Math. and M. R. Dabas, Master (Lipetsk State Technical University).