Method for determining and refining the interval of data collection of gas-dynamic processes by safety criterion

To solve the problems of controlling the gas-dynamic processes of production areas and air distribution in the ventilation network, their mathematical description is required, which can be obtained only on the basis of a detailed study and determination of the parameters and characteristics of gas-dynamic processes and regularities of air distribution. The mathematical description of the production ventilation system is complicated by the fact that the variables that determine the state of the control system are random functions of time. Therefore, the main attention should be paid to determining the probabilities of the characteristics of aerogasdynamic processes. The mathematical description of the ventilation of the production area into static and dynamic characteristics was the basis for the development of an optimal control algorithm for the production area. The control algorithm for the ventilation of the mining area consists of subalgorithms for controlling the parameters of the mine atmosphere, primary processing of the received information, and generating control actions. Their development is possible on the basis of research and mathematical description of gas-dynamic processes and methods for controlling the parameters of the mine atmosphere. Effective control of gas-dynamic processes in production areas is possible only when considering the entire ventilation system, due to the interconnectedness of aerodynamic parameters. The task of optimal air distribution control in the network is to ensure the required airflow rates for ventilation, determined as a result of solving the first problem with minimal energy costs for ventilation.


Introduction
In accordance with the requirements of safety rules, deviations of the methane concentration from the limits of the norm, even of a single, short-term nature, are completely unacceptable, since they can lead to an emergency mode [23]. When monitoring such an object, it is important not to miss any, even rather rare, excess of the maximum permissible level [1][2][3][4]24]. Therefore, the problem of determining the measurement interval ∆tβ by the safety criterion will be solved in the following setting. Let in the process of normal operation of the object on the observation interval (0, T) a nonstationary discrete sequence of methane concentration values is obtained С(tk), tk = tk-1+∆t, ∆t = const (K = 1, 2, 3, …) [17][18][19][20]. It is required to find such a functional IC, which for the time interval between two adjacent measurements of methane concentration should not exceed the value of the safety criterion β.
When conducting research, as functional I, it is proposed to choose the absolute difference in the values of the controlled process Х(t) in neighboring points tk and tk+1: or rms measurement function: In the general case [5][6][7], the functional I depends on the values of the controlled process X(tk), time t ,and measuring interval ∆t.

Materials and methods
During the experiments, it was shown that with a limited input action on an inertial object, 0˂ U(t) ≤ M, М>0 the functional (1) and (2) constructed from the realizations of the output signal significantly depend on both the M number and the dynamic characteristics of the object. This influence is reflected, for example, in the fact that the slew rate of the output signal is always limited for any law of variation of U(t), including a stepwise one. Therefore, with a limited effect on an inertial object, the output signal increment will always be a finite value for a fixed measurement interval ∆t.
Since the mining area is an inertial object with a limited input impact, a slightly modified functional can be used to solve the problem posed (1). The required IC functional must correspond in its structure and be always positive, i.e. IC > 0. Taking this requirement into account, we choose as the functional IC the positive difference between the methane concentration values at adjacent measurement points and tk и tk+1: For the selected functional, the dependence can be plotted according to the discrete sequence of the methane concentration process as follows.
Let us choose the discreteness interval equal to tn =∆t and determine the values of a function (4) for K = 1,2. Let the values of IC obtained in this case the value as not satisfying condition (5) and determine the next value of the function ) the found positive values we will choose the largest one. Let, then, having fixed the largest value, we take the next step ≥ the largest value, we take the next step, determine the value of the function ) ( 5 t I C for K = 5, compare it with, etc [5][6][7][8]. Having thus passed the entire discrete sequence C(tk) for the values K = 1, 2 ,3, ..., we find for the selected discreteness interval ∆t the maximum value of the function For the next discreteness interval ∆t2=2∆t, perform the above actions and find the maximum value As a result, we obtain a discrete sequence of maximum values of the function ) (  (6) represents the values of the upper bounds of the function: Using the obtained sequence (8)  As you know, in the case of non-stationary processes, their characteristics are determined by several independent implementations by finding the arithmetic mean of the results obtained for different implementations [8][9][10][11][12][13]. The length of the used realization in this case, the valley, significantly exceeds the value of the interval at which the characteristics of the process are determined [9][10][11][12][25][26][27].
Using the experimental results, for the process of methane concentration, it is possible to choose the implementation length equal to 1 day, since during this time at least one full production cycle has time to be completed in the longwall. The duration of this cycle significantly exceeds the value of the maximum measurement interval ∆tβ = 30 minutes, for which it is still advisable to determine the value of the function ) ( * t I С ∆ in practice.
Since the production processes at the site, carried out within one day, practically do not affect the nature of the aerogasdynamic process during the next day, the daily realizations of the methane concentration process can be considered independent [13][14][15]. The equivalent of several independent daily sales in this case is a multi-day implementation obtained in the normal operation of the site.  The results of the calculation according to the formulas (12)(13)(14) of the observation intervals (Table 1) indicate that to increase the accuracy of determining the estimates of the values of the unction ) ( , it is inappropriate to increase the observation interval, for example, to obtain 20-30% of the accuracy of the estimate, the required observation interval should be equal to 2-4 months [16] ( Table 2).
In the conditions of changing the parameters of the site over time, a simpler way to increase the accuracy of estimates of the measurement interval is the method of their refinement in the process of dispatch control and management. It is proposed to refine the measurement interval in the process of supervisory control and management according to the following method.
At a pace with the arrival of discrete control data, the values of the function ) (