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ИНЖЕНЕРНЫЕ СИСТЕМЫ В СТРОИТЕЛЬСТВЕ
RESEARCH PAPER / НАУЧНАЯ СТАТЬЯ УДК 536.24:628.8
DOI: 10.22227/1997-0935.2022.2.222-231
Heat transient processes identification of the elements of internal environment system
Rustam Sh. Mansurov1, Yuri E. Voskoboinikov1'2, Vasilisa A. Boeva1
1 Novosibirsk State University of Architecture and Civil Engineering (Sibstrin); Novosibirsk, Russian Federation; 2 Novosibirsk State Technical University; Novosibirsk, Russian Federation
ABSTRACT
Introduction. The study of heat exchange transients in the climate system "Heater-Ventilator-Room", when ventilator capacity varies step-wise, is presented. The construction of functional relations between inputs and outputs of the system is the object of special attention. This allows for a non-parametric identification of impulse responses in the system for simulation and control.
Materials and methods. The climate system is represented by a combination of several different-type elements with step inputs and experimental data as outputs. Mathematical models of the elements are governed by Volterra integral equation of the 2nd kind. Solution of this equation is an ill-posed problem, and specifics of identification experiments do not allow applying computational methods of classical regularization algorithms. A non-parametric identification of impulse responses for the elements is performed by the authors' stable algorithm with due regard for real technical systems specifics. The algorithm is founded on stable differentiation by smoothing cubic splines with optimal smoothing parameter estimation and special type boundary conditions.
Results. Non-parametric identification algorithm is adapted for the investigated climate system. The inverse problems of impulse responses identification and the direct problems of heat flux reactions prediction are solved. A high convergence of theoretical and experimental data is shown.
Conclusions. The behavior of the transients is predictable for the climate system under the particular operation mode. N ^ The algorithm proposed takes proper account of practical problems specifics. The results obtained suggest the efficiency of
jj ® the algorithm for applied identification problems solutions in real complex technical systems.
> in
E — KEYWORDS: climate system, transient process, relative excess heat, non-parametric identification problem, Volterra in-
to ^ tegral equation of the 2nd kind, stable identification algorithm, smoothing cubic splines, combined boundary conditions,
^ optimal smoothing parameter estimation when the measurement noise variance is undefined
2 E Acknowledgements: The reported study was funded by RFBR, project number 20-38-90041.
|2 ,0 FOR CITATION: Mansurov R.Sh., Voskoboinikov Yu.E., Boeva V.A. Heat transient processes identification of the element-
• . sof internal environment system. Vestnik MGSU [Monthly Journal on Construction and Architecture]. 2022; 17(2):222-231.
<u ф DOI: 10.22227/1997-0935.2022.2.222-231 (rus.).
"S Corresponding author: Vasilisa A. Boeva, v.boyeva@sibstrin.ru
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Идентификация переходных процессов теплообмена элементов системы обеспечения микроклимата
Рустам Шамильевич Мансуров1, Юрий Евгеньевич Воскобойников1'2,
" Василиса Андреевна Боева1
^ ° 1 Новосибирский государственный архитектурно-строительный университет (Сибстрин);
о Е г. Новосибирск, Россия;
с5 о 2 Новосибирский государственный технический университет (НГТУ); г. Новосибирск, Россия
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ОТ И АННОТАЦИЯ
от °
— 2 Введение. Выполнено исследование переходных процессов теплообмена в климатическои системе «воздухона-
д,, * греватель-вентилятор-помещение» при скачкообразном изменении объемной производительности на вентиляторе.
О ^ Особое внимание уделяется изучению функциональных связей между входными и выходными характеристиками
О системы. Это позволяет провести непараметрическую идентификацию импульсных переходных функций элементов
^ Е климатической системы для дальнейшего моделирования и управления.
X Материалы и методы. Изучаемая климатическая система рассматривается в виде комбинации нескольких разно-
н £ типных элементов со скачкообразными входными и выходными сигналами, представленными зашумленными экс-
¡^ ¡^ периментальными данными. Математические модели элементов системы описываются интегральным уравнением
Ш > Вольтерры второго рода. Решение этого уравнения является некорректно поставленной задачей. Специфика эксперимента по идентификации не дает возможность использовать для решения вычислительные схемы классических
© Rustam Sh. Mansurov, Yuri E. Voskoboinikov, Vasilisa A. Boeva, 2022 Распространяется на основании Creative Commons Attribution Non-Commercial (CC BY-NC)
регуляризирующих алгоритмов. Непараметрическая идентификация импульсных переходных функций элементов климатической системы выполняется разработанным авторами устойчивым алгоритмом, способным учитывать специфические особенности реальных технических систем. Построение алгоритма основано на вычислении устойчивых производных и интеграла свертки сглаживающими кубическими сплайнами с подбором оптимального параметра сглаживания и специальными краевыми условиями.
Результаты. Приведен авторский алгоритм непараметрической идентификации, адаптированный для работы с исследуемой климатической системой. Решены обратные задачи идентификации импульсных переходных функций и прямые задачи прогнозирования реакций теплового потока. Показано высокое соответствие теоретических и экспериментальных характеристик.
Выводы. Анализ полученных результатов показал, что поведение рассматриваемых переходных процессов предсказуемо и характерно для данного режима работы климатической системы. Предложенный алгоритм непараметрической идентификации способен учитывать специфические особенности практических задач и экспериментальных данных и может быть применен при решении прикладных задач идентификации в реальных технических системах.
КЛЮЧЕВЫЕ СЛОВА: климатическая система, переходный процесс, относительная избыточная теплота, задача непараметрической идентификации, интегральное уравнение Вольтерры второго рода, устойчивый алгоритм идентификации, сглаживающие кубические сплайны, комбинированные краевые условия, оценивание оптимального параметра сглаживания при неизвестной дисперсии шумов измерений
Благодарности: Исследование выполнено при финансовой поддержке РФФИ в рамках научного проекта 20-3890041.
ДЛЯ ЦИТИРОВАНИЯ: Мансуров Р.Ш., Воскобойников Ю.Е., Боева В.А. Heat transient processes identification of the elements of internal environment system // Вестник МГСУ. 2022. Т. 17. Вып. 2. С. 222-231. DOI: 10.22227/19970935.2022.2.222-231
Автор, ответственный за переписку: Василиса Андреевна Боева, v.boyeva@sibstrin.ru
INTRODUCTION
When building operation and maintenance, a number of problems of great importance are solved to provide comfortable conditions in rooms. Among the problems are development, control and optimization of internal environment systems and microclimate stabilization in rooms between the preset values on exposure to internal or external thermal agitations. The quality of climate parameters and stability of a system are defined by system response on the agitations arising named a transient process, or simply a transient [1]. Non-parametric identification of dynamic transients in the elements of the researchable system is an initial crucial phase for all of these problems.
Transients study in the climate system "Heater -Ventilator - Room" appears as response prediction of the one of climate system element on agitation caused by another element [1]. Construction of functional relations between the input and output values of the element allows for a non-parametric identification of the impulse responses, but it requires to formulate the reliable mathematical model in various dynamic states [2].
Conventionally, mathematical models for various technical processes and states of an element are presented as differential equations. In recent times, energy specialists tend to reject this approach because it either does not respond for some properties of the element or simply is unrealizable [3]. At present, more universal integral models are actively used which successfully serve as applications in experimental data processing, automation control theory, filtration and restoration of signals and images, computed tomography and optimization [4-10]. The advantages of the integral equations include compactness in the description of dynamic systems, the natural practical content of the kernels of the equation as reactions to typical input agitations,
and high stability of the numerical implementation. But the main problem that arises when solving integral equations is related to the conventional correctness of their solutions. Classical regularizing algorithms, such as methods of Runge-Kutta type, h-regularization methods (by Tichonov, Apartsyn, Bakushinsky, Deni-sov etc.), different quadrature methods, Wiener-Kalman method etc., do not take into account the features of real objects and practical problems.
Thus, the development of stable identification algorithms for the practical problems with experimental initial data is not only relevant for applications, but also has a theoretical significance in the development of computational methods of ill-posed problems under incomplete a priori information.
The purpose of the research is to study responses behavior of the elements of the climate system "Heater-Ventilator-Room" and calculate the responses mathematically by a priori information about the experimental observational data. The accompanying objectives are:
• to construct the functional relations in the climate system in input-output terms and formulate the reliable mathematical model;
• to detect behavior specifics of climate system signals under the definite operation mode;
• to perform a non-parametric identification of the impulse responses of the elements by the authors' algorithm;
• to solve the direct prediction problems of the theoretical responses on internal or external agitations;
• to study the behavior of the theoretical responses in comparison with the experimental data.
To solve the objectives, the authors use methods and algorithms self-developed reported in [11-13]. This paper represents the application of these methods and algorithms in terms of experiments reported in [1, 2, 14].
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MATERIALS AND METHODS
Heat exchange transients are investigated in the climate system "Heater-Ventilator-Room", considered in detail in [14], when ventilator capacity varies step-wise from 0.6 to 0.7 (from 42 to 49 Hz) in fractions of maximum controller frequency, and the heater is inactive and characterizes by zero constant capacity. In this operation mode, the ventilator is an active element, the heater is a reactive element, and the transients in the system are actuated by ventilator capacity step.
On Fig. 1 [1] functional block diagram of the investigated climate system is shown. Heater capacity is operated by the triac capacity controller AW, and ventilator capacity is operated by frequency current converter AF. The parameters registered from sensors in the system are:
• dust heat flow rate u;
• pressure differential across the ventilator AP;
• dust relative humidity
• heater input temperature Tv
• heater output temperature T2;
• room input temperature T3;
• room output temperature T4.
The parameters of heat flux state were registered at the dust inspection points at 1 s intervals.
Fig. 2 depicts the circulation of heat fluxes generated/absorbed by a moist air flux though the elements
of the climate system [14]. The air mass flow, while experimental data processing, is defined as G = p(T3)uF, where F, m2, is the sectional area of an air duck at the location points of velocity u and temperature T3 sensors; p(T3), kg/m3, is dry air density at T3 temperature. The heat absorbed by the heat flux from the heater is defined as Q. = c GAT, AT. = T. - T., from the ven-
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tilator as Q = c GAT, AT = T. - T., from the control
^v a v v 3 2'
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thermal capacity. The heat assimilated and accumulated by the heat flux at room air space is defined as Qr = = c GAT, where AT = T, - T,. Heat injection to the en-
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vironment from the room is defined as Q = c GAT,
^ e a e'
where AT = T - T .
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According to [14], a similarity parameter that describes transitions in the climate system is relative excess heat Q r. This characteristic involves contribution of all heat flux parameters and defined by
q;
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= 1...4,
(1)
where Qr. for i = 1 is the relative excess heat Q rh generated/absorbed by the moist air flux though the heater; for i = 2 is the relative excess heat Q rv generated/absorbed by the moist air flux though the ventilator; for i = = 3 is the relative excess heat Q rr assimilated by the heat
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Fig. 1. Functional block diagram of the climate system "Heater-Ventilator-Room" (source: Mansurov R.Sh., Rudyak V.Ya. Transients processes in the system the heater-fan when changing the operating mode of the fan. News of Higher Educational Institutions. Construction. 2019; 3:50-63)
Fig. 2. Scheme of heat fluxes ventilation in the climate system "Heater-Ventilator-Room:
flux though the room; for i = 4 is the relative excess heat Q rs generated/absorbed by the moist air flux though the control subsystem; Q.(t) is the heat generated at the current time t; Q(0) is the heat generated at the initial time t = 0; Q/t*) is the heat generated at the finite time t* = 500 s.
Fig. 3 shows the scheme of non-parametric identification experiment in the climate system "Heater-Ventilator-Room" under the considered operation mode, with regard to heat fluxes circulation and thermal agitations. The system can be represented as a combination of two simpler subsystems characterized by the different proceeding physical processes. The subsystems are the subsystem of climate control in room, also called the climate control subsystem "Heater-Ventilator", and the controlled subsystem "Room". Alternatively, the climate control subsystem is composed of two objects — the heater and the ventilator. When performing experiments, stationarity and linearity of the climate control subsystem and its objects had been observed [2]. The room is not included in the general identification scheme because of unpredictability of the proceeding stochastic physical processes [13].
In identification scheme (Fig. 3), the input ^(t) of the climate control subsystem is capacity step on the active ventilator. The output f1(t) is heat flux active agitation by ventilator activity represented by the relative heat Q[. Being the reactive object of the climate control subsystem, the heater not only affects the heat flux by its own heat release but also reacts on ventilator capacity step. At the initial moment, the heater is inactive, or operates under zero capacity, so heater activity includes its reaction on ventilator capacity step only. The output f2(t) is the reaction of the heat flux through the heater on ventilator capacity step ^(t)
9(x)
Heater mode "0"
V V
Climate control subsystem k(t)
\t
Heater K(t) Ventilator K®
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Fig. 3. Scheme of identification in the climate system "Heater-Ventilator-Room"
represented by the relative heat Q rK The output f(t) is the reaction of the climate control subsystem on the input step ^(t) represented by the relative heat Q r
Physical processes proceeding in the climate control subsystem and its objects are complicate for an adequate mathematical description. In this case, let us assume a model of the subsystem is a non-detailed "black-box" involving only an input and an output. In input-output terms, this mathematical model can be formulated by Volterra integral equation of the 1st kind with a difference kernel [15, 16]. However, there is shown in the literature [6] that solution of the integral equation is an ill-posed problem by Hadamard due to possible violation of the solution stability under noisy experimental data on the right side of the equation [7, 8]. Let us employ the approach from [14] involving the conversion of Vol-terra integral equation of the 1st kind to Volterra integral equation of the 2nd kind:
where k(t) is an impulse response, ^(t) is an input, f(t) is an output. In the investigated subsystem, the input 9(t) is the step signal from 0.6 to 0.7 with the constant amplitude . The outputs fl(f), f2(t), f(t) are a priori inexact noisy observations from the laboratory facility described in [2, 13] contaminated by measurement errors. All the signals of the subsystem are equally spaced, plotted by N = 500 samples with the time step At = 1 s.
For simulation and operation the climate control subsystem described by the model (2), it is required to solve the problem of non-parametric identification consisting in evaluation of climate control subsystem impulse response k(t), ventilator impulse response kl(1t) and heater impulse response k2(t) by measured noisy outputs when ventilator capacity varies step-wise. In terms of causal relationships, the problem is inverse because it requires to regenerate the impulse response on the known reaction of the subsystem or object.
Note that the problem formulated in the form (2) remains ill-posed and means calculation of the first derivatives from noisy outputs so the solution obtained may be unstable under measurement noises in initial data. A stable solution of the problem posed is suggested to calculate by authors' non-parametric identification algorithm for technical systems reported in [11, 12, 14] which evaluates impulse responses by smoothing cubic splines.
RESULTS OF THE RESEARCH
Stable non-parametric identification algorithm for dynamic systems adapted for the practical identification problems in the climate control subsystem and the objects involves the following stages.
Stage 1. Analysis of information about measurement noise characteristics in noisy initial data
In this stage it is concluded whether pre-processing (or filtration) of noisy initial data would be advisable.
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As is shown in [17], in case of initial data contamination by impulse noise or Gaussian noise more than 10 % it is advisable to pre-filtrate the initial data to reduce the identification error.
The initial experimental noisy outputs fl(t), f2(t), f(t) are represented on Fig. 4-6. There is no a priori information available about measurement noise characteristics in the outputs. Measurement noise level in each output has been evaluated by methods [10], and for the ventilator it was 5n1 = 3.4 %, for the heater — 5l2 = = 6.7 %, for the climate control subsystem — = 3.1 %, so no pre-filtration of noisy outputs is required.
Stage 2. Calculation of smoothing cubic splines with their stable first derivatives S9,a (t), Sfa (t) from the values of inputs and outputs
Smoothing cubic splines construction is fully considered in [18, 19] but essentially depends on the two following factors not properly appreciated in literature.
The first factor is boundary conditions. When solving practical engineering problems, boundary conditions assumed to be set on account of problem specifics and distinctive features of the field observations. In the investigated problems, it would be incor-
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Fig. 5. Output of the heater f(t)
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rect to set the left boundaries of the impulse responses in zero values for a variety of reasons undermentioned. At the same time, the reactions f1(t), f2(t), f(t) in the end of the time interval converge to steady state, and the according right boundaries of the impulse responses are expected to be zero [20, 21]. The classical natural boundary conditions or boundary conditions of the 1st kind seem to be inappropriate for the investigated problem. It would make sense using the combined boundary conditions, when at the left boundary the first derivative value is set, and the second zero derivative value is at the right boundary [22].
The second significant factor is selection of the best estimate aL of the optimal smoothing parameter a that minimizes the smoothing mean-square error [8, 20]. When solving practical problems with experimental data, there is no information about the numerical characteristics of measurement noises, and the estimate aL is calculated by L-curve method [11]. In this experiment the value of estimate was aL = 6,183-103.
By substitution of the calculated spline derivatives S' aL(t), Sfjtt) into equation (2), equation (3) can be obtained:
V'l )
SfJt)
(3)
Stage 3. Calculation of the convolution integral i0S^ a (t - T)k(T)dT from equation (3)
Approximating the convolution integral we obtain:
. (4)
i = l...N, j = 2...N
Then matrix ®'(N - 1)(N - 1) is formed with i = 1 ... N - 1, j = 1 ... N - 1 elements calculated by the rule:
0 for i< j
for * J
(5)
Stage 4. Construction of the impulse response estimate k(t)
Approximating equation (3) by a system of linear equations we obtain:
1 +
^.J'l) KJO
f
(6)
Fig. 6. Output of the climate control subsystem ft)
where I is an identity matrix (N- 1)(N- 1); k is solution vector, and its components are the estimates for identified impulse response values.
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s a,
400 500
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Fig. 7. Impulse response estimate h1(t) of the ventilator 0.04
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0
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Fig. 8. Impulse response estimate k2(t) of the heater
400 500
0.4
0.3
0.2
o a
0.1
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0
100
400
500
200 300 Time t, s
Fig. 9. Impulse response estimate k(t) of the climate controlled subsystem
Fig. 7-9 shows the calculated estimates ^(t), k2(t), k(t) for the climate control subsystem and its objects. The reason of nonzero initial values of the impulse responses is the testing environment of the field observations for the experimental series under the investigation. Heater capacity control is provided by the triac capacity controller generating control signal as the sinewave with average period factory setting in 57 s [23]. The experiments are conducted serially from the eighteen consecutive single experiments lasting 500 s including serial heater capacity increase in the even experiments and decrease in the odd experiments. After that, the received observational data were averaged over the nine even heating experiments and the nine odd cooling experiments. This compensates, fully or partially, the sinewave feature of the experimental data [1].
When the heater operates under zero capacity and nominally inactive, the applied power to the heater is not more than 0.01 from the maximum value of the electric capacity anyway. This fact is due to the operation of safety control system of heater fuel elements. If there is not the minimum required heating, safety control system states the heater in the emergency operation [1]. Summing up, the reason of the sinewave experimental signals is the operation of heater safety control system and the triac capacity controller. The initial values of the impulse responses associated with heater switching torque from capacity increase/decrease which mistimes with the initial zero value. The sinewaves in signals are theoretically irrelevant for construction of mathematical models so they must be reduced. In subsequent experiments the triac capacity controller will be replaced with the thyristor one, linearly controlling heater capacity.
The solutions of the direct problems shown on Fig. 10-12 are the predictions of heat flux reactions fj(t), f2(t), f(t) on the agitation ^(t) by calculated impulse response estimates kj(t), k2(t), k(t). The values of the solution relative error are: for the ventilator — 5fi = = 4.6 %, for the heater — Sf = 15.6 %, for the climate control subsystem — Sf = 4.4 %. There is the evidence that the theoretical reactions fj(t), f2(t), f(t) corresponds highly to the experimental dataf1(t),f2(t),ft). Note that the theoretical characteristics are sufficiently smooth. Measurement noises and instrumental errors introducing non-informative sinewaves are reduced that affects the solution relative errors.
CONCLUSION AND DISCUSSION
Let us analyze the behavior of the heat transients of the climate system elements.
The ventilator has an internal heat source and affects the heat flux by the heat production of its own and by the heat exchange with the flux. On Fig. 10 it can be observed that the heat Q V generated by the ventilator increases monotonically and stabilizes then when the heat flux is healed. Objectively, at the beginning of the transient, the difference ATv = T3 - T2 is incremental due to
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the small decrease of pre-ventilator temperature T2 and constant post-ventilator temperature T3, and Q rv evolution is governed by the increasing air mass flow G. After that, Q rv increase changes qualitatively and keeps growing because of active ventilator heat production by its own [11]. Finally, heat generation intensity by the ventilator and the heat exchange with the heat flux stabilizes and sets Q rv to 1. The evolution of climate control subsystem heat Qrs on Fig. 12 changes similarly. As is noted in [1], the characteristics plotted on Fig. 10 are the typical Q rv evolution for blasting when the ventilator is active. In this case, the heat Q\ changes at most in accordance with heater capacity operation mode, and the value of ventilator capacity step is insufficient.
On Fig. 11 it can be observed that at the beginning of the transient the heat Q rh generated by the heater is attended with the increase of G. Transient time depends on the factory setting of the triac capacity control-
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ler [23]. Then heat exchange process between the heat flux and heater surface stabilizes, Q rh decreases slightly and sets to 1. The characteristics plotted on Fig. 11 are the typical Q h evolution for blasting when the heater is reactive. The literature [1] states that Q rh evolution is sufficiently defined by heater capacity operation mode, and the more capacity is, the more the influence of ventilator capacity step.
For the theoretical characteristics on Fig. 10-12 it is simpler to predict their behavior under the certain conditions in comparison with the experimental data contaminated with measurement noises. The theoretical characteristics are qualitatively more accurate and informative and can be used further for modeling and controlling the room.
A non-parametric identification of real dynamic systems is a rather complicated and time-consuming problem as a matter of actual practice. On the one hand, it is due to ill-posedness of equation (2) solution, and on the other hand, it caused by different types of inputs and outputs of identified systems. The algorithm proposed in paper enables to perform a stable non-parametric identification whether for a whole system or for its elements individually with completely regard to the specifics of a problem posed. Smoothing cubic splines allow constructing the efficient algorithm for filtration of various statistical measurement noises in inputs and outputs of a system. Because of smoothing cubic splines application, identification error value depends on the differentiation errors of inputs involving matrix ® elements (5) and of outputs involving vector f ' components (6). Optimal smoothing parameter evaluation meets the minimum of the differentiation errors for noisy inputs and outputs of an identified system and the minimum of the identification error as a consequence [24].
Successful solution of the real engineering problem of the non-parametric identification applied to the cli-
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mate system "Heater-Ventilator-Room", when the observational data are noise-contaminated, proves the efficiency of the algorithm proposed for applied problems solution. The non-parametric identification of the impulse responses for the elements of the climate system allows predicting heat flux reaction on the agitation with
the high accuracy. The results of the investigation represented permit advising the algorithm for identification of impulse responses in other real dynamic systems. Particularly, the algorithm can be used for modeling and studying of transients in the investigated climate system under the more complex operation modes.
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19. Li K., Huang T.Z., Li L., Lanteri S. Non-intrusive reduced-order modeling of parameterized electromagnetic scattering problems using cubic spline interpolation. Journal of Scientific Computing. 2021; 87(2):52. DOI: 10.1007/s10915-021-01467-2
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22. Voskoboinikov Yu.E., Boeva V.A. Descriptive signal smoothing in a single algorithm nonparametric identification of technical systems. Modern High Technologies. 2020; 7:24-28. DOI: 10.17513/snt.38128 (rus.).
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Received December 3, 2021.
Adopted in revised form on February 23, 2022.
Approved for publication on February 23, 2022.
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24. Toshniwal D., DiPasquale M. Counting the dimension of splines of mixed smoothness. Advances in Computational Mathematics. 2021; 47:1-29. DOI: 10.1007/s10444-020-09830-x
Bionotes: Rustam Sh. Mansurov — Candidate of Technical Sciences, Associate Professor; Head of the Department of Heat and Gas Supply and Ventilation; Novosibirsk State University of Architecture and Civil Engineering (Sibstrin); 113 Leningradskaya st. Novosibirsk, 630008, Russian Federation; 377487; rmansurov@inbox.ru;
Yuri E. Voskoboinikov — Doctor of Physics and Mathematics, Professor, Head of the Department of Applied Mathematics; Novosibirsk State University of Architecture and Civil Engineering (Sibstrin); 113 Leningradskaya st., Novosibirsk, 630008, Russian Federation; Professor of the Department of Automation; Novosibirsk State Technical University; 20 Karla Marksa Avenue, Novosibirsk, 630073, Russian Federation; 13547, Scopus: 6602170597, ORCID: 0000-0002-9731-8618; voscob@mail.ru;
Vasilisa A. Boeva — postgraduate student, Assistant of the Department of Applied Mathematics; Novosibirsk State University of Architecture and Civil Engineering (Sibstrin); 113 Leningradskaya st., Novosibirsk, 630008, Russian Federation; 1027172; v.boyeva@sibstrin.ru.
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1. Мансуров Р.Ш., Рудяк В.Я. Переходные процессы в системе нагреватель-вентилятор при изменении режима работы вентилятора // Известия вузов. Строительство. 2019. № 3. С. 50-63. DOI: 10.32683/0536-1052-2019-723-3-50-63
2. Мансуров Р.Ш. Термодинамические процессы в элементах систем обеспечения микроклимата // Сантехника, отопление, кондиционирование. 2014. № 1 (145). С. 90-93.
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8. Gabbasov N.S. On numerical solution of one class of integro-differential equations in a special case // Computational Mathematics and Mathematical Physics. 2020. Vol. 60. Pp. 1666-1678. DOI: 10.1134/ s0965542520090092
9. VoytishekA.V. Classification and applications of randomized functional numerical algorithms for the solution of second-kind Fredholm integral equations // Journal of Mathematical Sciences. 2021. Vol. 254. Issue 7. Pp. 589-605. DOI: 10.1007/s10958-021-05328-z
10. Solodusha S.V., Mokry I.V. A numerical solution of one class of Volterra integral equations of the first kind in terms of the machine arithmetic features // Bulletin of the South Ural State University. Series Mathematical Modelling, Programming & Computer Software. 2016. Vol. 9. Issue 3. Pp. 119-129. DOI: 10.14529/mmpl60310
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Поступила в редакцию 3 декабря 2021 г. Принята в доработанном виде 23 февраля 2022 г. Одобрена для публикации 23 февраля 2022 г.
Об авторах: Рустам Шамильевич Мансуров — кандидат технических наук, доцент, заведующий кафедрой теплогазоснабжения и вентиляции; Новосибирский государственный архитектурно-строительный университет (Сибстрин); 630008, г. Новосибирск, ул. Ленинградская, д. 113; РИНЦ ID: 377487; rmansurov@inbox.ru;
Юрий Евгеньевич Воскобойников — доктор физико-математических наук, профессор, заведующий кафедрой прикладной математики; Новосибирский государственный архитектурно-строительный университет (Сибстрин); 630008, г. Новосибирск, ул. Ленинградская, д. 113; профессор кафедры автоматики; Новосибирский государственный технический университет (НГТУ); 630073, г. Новосибирск, пр-т Карла Маркса, д. 20; РИНЦ ID: 13547, Scopus: 6602170597, ORCID: 0000-0002-9731-8618; voscob@mail.ru;
Василиса Андреевна Боева — аспирант, ассистент кафедры прикладной математики; Новосибирский государственный архитектурно-строительный университет (Сибстрин); 630008, г. Новосибирск, ул. Ленинградская, д. 113; РИНЦ ID: 1027172; v.boyeva@sibstrin.ru.
electromagnetic scattering problems using cubic spline interpolation // Journal of Scientific Computing. 2021. Vol. 87. Issue 2. P. 52. DOI: 10.1007/s10915-021-01467-2
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21. Wang J., Ricardo A.R.M, MengH., RubenM.M., Jorge L.S. Introducing system identification strategy into model predictive control // Journal of Systems Science and Complexity. 2020. Vol. 33. Issue 5. Pp. 1402-1421. DOI: 10.1007/s11424-020-9058-3
22. Воскобойников Ю.Е., Боева В.А. Дескриптивное сглаживание сигнала в одном алгоритме непараметрической идентификации технических систем // Современные наукоемкие технологии. 2020. № 7. С. 24-28. DOI: 10.17513/snt.38128
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24. Toshniwal D., DiPasquale M. Counting the dimension of splines of mixed smoothness // Advances in Computational Mathematics. 2021. Vol. 47. Pp. 1-29. DOI: 10.1007/s10444-020-09830-x
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