TO THE QUESTION OF RESEARCH OF NONLINEAR IDENTIFICATIONS OF COMPLEX OBJECTS

АННОТАЦИЯ

В данной статье рассмотрены существующие проблемы в железнодорожной автоматике и телемеханике. Он направлен на расчет протяженности железнодорожных линий новыми методами. Объяснены автоматические и телемеханические устройства управления на железных дорогах и определение полных интервалов времени с использованием нового метода, т.е. математического метода. При этом сокращается время ожидания поездов на железной дороге, она имеет значительную экономическую эффективность и автоматическую надежность.

ABSTRACT

This article considered the existing problems in railway automation and telemechanics. It is aimed at calculating the length of railway lines using new methods. Automatic and telemechanical control devices on railways and determination of total time intervals using a new method, i.e. mathematical method, are explained. In this case, railway waiting times for trains are reduced, it has significant economic efficiency and automatic reliability.

Keywords: Rele (R), automation and telemechanics, non-combustible polyvinyl chloride (NPCH), electricity centralization (EC), transformer box (TB)

Ключевые слова: Реле (Р), автоматика и телемеханика, негорючий поливинилхлорид (НПХ), электроцентрализация (ЭЦ), трансформаторная будка (ТБ)

Experimental and analytical research methods are used in many areas of railway automation and telemechanics. In some cases, carrying out sufficiently complete experimental studies requires significant expenditures of material resources and time [12]. Consider a product (component, element, device), in which the relationship between the output y and input x parameters is expressed by a continuous dependence. Elements, devices with a discrete (relay) characteristic are characterized by the fact that up to a certain critical value of the input parameter X_i = X_ave ( called the pickup parameter) they are out of service and the output parameter has the value y_i=y_min (y_min=0) in the case of a repeater element or y_i = y_min in the case of an inverter element (Fig. 1)

Figure 1. Changing the output parameter depending on the value of the input parameter

With an increase in the value of the input parameter X_i over X_i = X_ave, the output parameter y will remain constant or change slightly [11]. Analytical methods are devoid of these shortcomings, however, it becomes possible to use them only when it is possible to describe the factors influencing the process under study with exact mathematical equations. In the case of a decrease in the input parameter x from the value at some other critical value X_i = X_out, (called the return or release parameter), the output parameter y abruptly decreases from У_мах to y_min (at the repeater element), or from У_min to У_мах (for the element-inverters ). With a further increase in the input parameter х_i to the value х_i - х_ave  the output parameter y changes abruptly to У_мах (for the repeater element) or to У_min (for the inverter element). With a further decrease in the input parameter x to zero, the output parameter y will retain its new value or change slightly (Fig. 2).

Figure 2. Changing the output parameter depending on the decrease in the input parameter to zero

The critical values х_ave and х_out are not strictly constant. With each cycle of changing the input parameter from х=0 tо х=х_шах and back, there will be observed their own values of х_ave and х_out. They will be about some average values х_ave and х_out, and the deviations of the values  and   will, as experience shows, obey, as a rule, the normal distribution law [3, 10].

The probability of transition from one state to another - from out of working state to working state is possible when the input parameter х=х_р exceeds the value Х_ave.

                            (1)

where x_ave – is the average value of the parameter that determines the transition of the element from the non-working state to the working one (the average current of the relay operation);

σ_ave – is the standard deviation of this parameter.

Here 

The probability of transition from the working state to the out-of-work state with a decrease in the input parameter to the value х₀ = х_out will be

                       (2)

where х_out – is the average value of the parameter that determines the transition of the element from the working state to the out-of-work state; σ_out – standard deviation; х₀ – value of the input parameter in the out-of-work state of the element

                                                     (3)

In order to obtain a mathematical model of control, two-position relays with hysteresis describe a complex function

                                       (4)

where B – is  the amplitude describing the functions. A fairly universal and unified approach to determining the class of an unknown system, synthesizing optimal structure models, determining identifiability conditions, and developing methods for parametric identification of linear dynamic systems and nonlinear systems consisting of compounds of linear dynamic and nonlinear inertialess parts has been developed [1]. The applied identification methods form the basis of the theory of constructing discrete models "input-output" of the optimal structure of dynamic systems with lumped parameters, operating in open and closed loops under disturbing influences [4,9].

Models "input-output" of the optimal structure of classes of linear and nonlinear dynamical systems are, in essence, optimal predictors, one step ahead of the output signal.

By constructing such a model, the problem of synthesis of control actions that minimize the dispersion of deviations of the system output signal from the required ones is also practically solved [5,8]. A nonlinear system is often modeled as various combinations of linear dynamic links and non-linear inertialess elements. In the Hammerstein model, the non-linear elements come before the linear dynamic part. In this case, the relationship between the input x(t) and the output y(t) with the signals of the stationary system is described by the Hammerstein operator

                                   (5)

where ω(t) – is the impulse transition function; f(х) – is a non-linear function.

In the discrete case

                                          (6)

where ω[t] is the impulse transition function of the discrete system.

The obvious physical interpretation and convenience in practical applications can explain the attention of many researchers to methods for modeling and identifying nonlinear systems based on the use of Hammerstein models of various structures and parametrizations [2, 6].

In the Wiener model, the linear dynamic part comes before the non-linear inertialess element. In this case, the dependence of the output signal of the nonlinear stationary system on the input signal is represented as

                       (7)

In the case of using a discrete model

                                 (8)

The Wiener-Hammerstein model is obtained from a series connection of three links of two linear dynamic ones and one non-linear inertialess one located between the linear ones. In this case, the output signal

              (9)

or moreover, the deviations of the values x_ave and x_out

                                             (10)

                                                       (11)

where ω₁ (t) ω₂ (t), ω₁ [t] ω₂ [t] – are the impulse transition functions of the corresponding continuous or discrete linear links.

Figure 3. Hammerstein models
NS - non-linear link realizing the non-linearity of on-off relays with hysteresis; LD - linear dynamic link

Modeling procedure

From the foregoing, any complex dynamic system can be represented as a set of linear inertial and inertial non-linearities.

Widely used Hammerstein models were chosen as the object under study

Figure 4. Exact point

For the object model, the identification problem is solved by correlation, dispersion methods of identification. When using correlation identification to determine the optimal operator of a one-dimensional dynamic object by the criterion of the minimum mean square error, it is necessary to know the correlation function of the variable and the mutual correlation of the input and output variables. According to the results of measuring input and output variables of the object, estimates of the autocorrelation function of the input K_xx (t) and the cross-correlation function of the input u are determined. exit К_ух (t). Obtaining an estimate is approximated by the corresponding functions К_хх (t)u К_ух (t), according to which the weight function of the object is determined analytically along with other necessary characteristics.

                          (12)

The practical determination of the characteristics by realizations y(t) and x(t) in the construction of the dispersion model is carried out in the following order.

Based on the implementation of x(t), an estimate of the conditional expectation   of the estimate of the generalized auto (θ_хх)and mutual (θ_ух) dispersion functions is determined. Then, according to the smoothed characteristics of these functions, by solving the equation

                                         (13)

determine the dispersion weight function. Comparison of the dispersion equation (2) with the Wienor-Honf correlation equation (1) shows their formal coincidence, but instead of the correlation functions in (2), generalized dispersion functions are used. This is a significant advantage of equation (2), since it makes it possible to use the computational algorithms used in the correlation identification theory [7] for the identification of nonlinear objects by dispersion methods.

Used in modeling for correlation identification

                                        (14)

for dispersion identification.

                     (15)

and for them by the formula

                                 (16)

the values of the residual dispersions were calculated, and they were compared with each other.

Two variants of the signal-to-noise dispersion ratio are considered:

D_x = D_e , i.e. equal to 1;

, i.e. equal to 2.

normal distribution laws. As the NS link, the non-linearity of a two-position relay with hysteresis is realized, and as the LD link, typical links were considered. For the first-order link, the output value y at each step is determined by the recursive formula

                                  (17)

for quadratic interpolation

+                      (18)

For a specific when modeling objects, the transfer function ω(р) is chosen equal for the first order

                                                (19)

for the second order

                              (20)

Based on the results of the experiments, we can conclude that the exact estimate ,  was obtained by dispersion identification, which describes the model more accurately than  by the correlation identification method. In this paper, attempts are made to obtain mathematical models for the control of a complex nonlinear object.

References

1. Б.С. Сотсков. Основы теории и расчета элементов и устройств автоматики и вычислительной техники. М.: Изд. В.Ш. 1970.
2. Дисперсионная идентификация // Райбман Н.С., Капитопенко В.В., Карлака П.М. // М.: Наука, 1981.
3. Каменское В. А. Оценивание параметров дискретных систем класса Гаммерштейна. Автоматика и телемеханика. 1975. №7.
4. Каминское В. Шудлоуское К. Управление одномерными экстремальными динамическими объектами класса Винера // Статическая проблема управления // Вильнюс, 1984. С.бЗ.
5. Растригин Л.Н., Маджаров Н.Е. Введение в идентификацию объектов управления. М.: Энергия. 1972.
6. Электронные устройства железнодорожной автоматики, телемеханики и связь. // Под.ред. Шилейко А.В. // М.: Транспорт, 1989.
7. Теория автоматического управления// Под. ред. Натушила АВ. // Изд. 2-е М.: 1976.
8. Baltaev S T., Rakhmonov B. B., Mohiddinov A. O., Saitov A. A.. Toshboyev Z. B. development of a block model for intelligent control of the position of operating devices in the electrical interlocking system. 2021.
9. H.H., Butakova M.A. Queuing systems: development of theory, methodology of modeling and synthesis: monograph. - Rostov-on-Don, 2004. - 200 p.
10. Zohid Toshboyev., Aziz Saitov., Janibek Kurbanov., Sunnatillo Boltayev. (IMPROVEMENT OF CONTROL DEVICES FOR ROAD SECTIONS OF RAILWAY AUTOMATION AND TELEMECHANICS). № 267.
11. I.V. Dergacheva, The study of the model of an intra-company organization // Bulletin of the. - 2004. №1.
12. L.A. Zadeh, Mathematics today. Fundamentals of a new approach to the analysis of complex systems and decision-making processes: Znanie, 2010. - pp.5-49.