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Title: Application of numerical signal differentiation methods to determine statio-narity of A process
Authors: Maystrenko, Andrey Vasiljevich
Svetlakov, Anatoly Antonovich
Gandzha, Taras Viktorovich
Dmitriev, Vyacheslav Mikhaylovich
Aksenova, Nataliya Valerievna
Keywords: approximation; signal differentiation; derivative; stationary process; model; mathematical statistics; inversions test; pseudoinverse matrix; аппроксимация; дифференцирование; сигналы; производные; стационарные процессы; математические модели; статика; тесты
Issue Date: 2018
Publisher: Томский политехнический университет
Citation: Application of numerical signal differentiation methods to determine statio-narity of A process / A. V. Maystrenko [et al.] // Petroleum and Coal. — 2018. — Vol. 59, iss. 2. — [P. 311-318].
Abstract: Among important tasks requiring their solution when creating an integrated mathematical model of a major oil pipeline is a problem to determine stationary character of different processes in the pipeline. Importance of this task is defined by the fact that both mathematical description and analysis of stationary and non-stationary processes may significantly differ. Mathematical models of stationary processes (modes, objects) are as a rule significantly less complex than the models of nonstationary processes. For example, they may often be described by systems of linear equations which are easy to solve, while for nonstationary processes the researcher has to solve a system of differential equations of different orders, which a much more difficult and laborious task. It is obvious that there is a necessity to not just identify the process as a stationary or nonstationary, but also precisely determine the moment when it starts changing its stationary state to the opposite type. Meeting this requirement is linked to necessity to timely employ mathematical apparatus corresponding to the current process. In this paper, we propose a new method for the determination of stationary processes, based on the application of the algorithm of numerical differentiation of signals (NDS) using a moving quadratic approximation and pseudoinverse matrices. A wide spectrum of applications that need to identify the intervals of stationary regimes of observed and/or controlled processes, as well as a variety of conditions in which the mentioned intervals should be estimated, determine the relevance of improvement of existing and creation of new methods for solving the problem under discussion. The essence of the method lies in the fact that, during the analysis of stationarity of processes, we take into account not only the values of the signal itself, but also the values of its first and second derivatives. This approach opens up the opportunities to determine the boundaries of the regime of controlled process more precisely and to predict the behavior of process in time. We present the formulation of the real-time NDS problem and the description of the proposed algorithm for its solution, as well as the description of some of the results of research and a new method of determining intervals of stationary processes based on the proposed NDS algorithm, and comparison of the proposed method with well-known in mathematical statistics method for determining the stationary processes based on the use of the criterion of inversion. The proposed method allows to calculate precisely the values of derivatives, as well as to determine the modes of stationary processes of real objects. The method has significantly higher noise immunity in comparison with the methods based on the use of classical statistical tests of stationarity. We recommend applying the proposed method for the development of mathematical models of complex dynamic objects operating in real time
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