Аннотация. В статье представлены и описаны время-частотные распределения класса Коэна, которые целесообразно использовать как математическое средство, позволяющее формировать удобное, с точки зрения информативности и семантической ясности-визуально-графическое отображение рабочих режимов разнохарактерных технологических процессов, в том числе процессов черной металлургии. Отмечено, что обычно процесс регулирования реализуется без синхронного визуального контроля каждой регулируемой скалярной (одномерной) координаты, однако наличие такого мониторинга является важным условием автоматизированного управления динамикой нестационарных технологических процессов. Для устранения этого недостатка предложено выполнять синхронный мониторинг с использованием многомерных время-частотных распределений класса Коэна, когда каждый измерительный скалярный сигнал специфически отображается посредством одного из таких распределений, например распределения Вигнера-Вилле. Приводится выражение для обобщенного распределения класса Коэна с наличием ядра распределения и функции неоднозначности. Последняя позволяет получать из материнской функции распределения разного типа. Приведены наиболее характерные представители время-частотных распределений этого класса с указанием их ядер. Доказана возможность возникновения на карте сигнального распределения интерференционных элементов, затрудняющих идентификацию контролируемых режимов. Рассмотрен случай формирования виртуальных элементов в составе распределения Вигнера-Вилле, отображающего двухкомпонентный одномерный сигнал. Поясняются условия возникновения паразитных элементов на карте распределения, получаемой, например, в ходе реализации процесса мультикомпонентного дозирования сыпучих шихтовых материалов при производстве доменного агломерата. Получено аналитическое выражение распределения Вигнера, отображающее многокомпонентный скалярный сигнал и содержащее информационную (полезную) и виртуальную (паразитную) части время-частотного распределения. Установлена связь между числом дозаторов сыпучих материалов в составе блока дозирующих устройств и количеством паразитных (виртуальных) элементов в распределении Вигнера. На примере процесса дозирования продемонстрирован эффект распространения шумовых компонент в распределении Вигнера. Приведен пример, иллюстрирующий проникновение шума в распределение Вигнера и возникновение в нем виртуальной части при отображении сигнальной осциллограммы с наличием зашумленной паузы и двух участков с разными частотами. Получено выражение для распределения Вигнера в виде решетчатой функции. Сделано заключение о параметрах периодичности распределения и необходимой частоте дискретизации измерительных сигналов. Ключевые слова: время-частотные распределения класса Коэна, распределение Вигнера-Вилле, ядро распределения, функция неоднозначности, интерференционные (виртуальные) элементы распределения, процессы черной металлургии, дозаторы непрерывного действия, сигнал дозирования, мультикомпонентный сигнал.

The article describes the development of a special approach based on using multidimensional wavelet distributions principle to monitor and control the feed dozing processes in the mix preparation unit. As a key component, this approach uses the multidimensional time-frequency Wigner-Ville distribution, which is the part of Cohen's class distributions. The research focuses on signals characterizing mass transfer processes in the form of material flow measuring signals in relevant points of the unit. Wigner-Ville distribution has been shown in time terms as Fourier transform of products of multiplied parts of the signal under consideration for past and future time moments; corresponding distribution for the frequency spectrum is shown as Fourier transform of the products of signal parts for high-frequency and low-frequency fragments of the signal spectrum. It has been noted that when using a complex model of a dozing signal, discrete values (samples) of the latter are considered as its real values. The description of the signal parameters (amplitude, phase, frequency) has been carried out with the help of Hilbert transform. In Cohen's class distributions which represent one-dimensional non-stationary flow signals, the concept of ‘instantaneous frequency’ has been introduced. A graphical explanation for the transformation of a process flow signal from a one-dimensional time domain to a time-frequency 2 D/ 3 D -space is presented. The technology of developing a multidimensional image in the form of Wigner distribution for one-dimensional signals of continuous spiral or screw-type feeders has been examined in detail. There have been considered the features to support Wigner distribution, which allow to guess the presence or absence of time-frequency distribution elements in the interval of signal recording. There has been demonstrated how Wigner distribution can be obtained for a continuous-intermittent feeding signal. It has been concluded that for a certain types of the signal for zero fragments of the latter, non-zero time-frequency elements (i.e. virtual, anomalous ones) appear on the distribution. In addition to Wigner distribution, two other distributions - of Rihachek and Page - are considered. They display the same signal and also contain virtual elements, but in different domains of the time-frequency space. A generalized multidimensional compound signal distribution with a so-called distribution kernel available in it is presented, which includes a correction parameter that allows controlling the intensity of the virtual signal energy.

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The article deals with the aspects of modal control in spaces of system states and wavelet time-frequency distributions. On the example of a mixture-producing aggregate mathematical models are formed to represent technological feeding material flow rate signals, and the relevant vector-matrix model based on the state space method is developed for the feeding devices unit (FDU). It is revealed that the signals generated by the feeders, are non-stationary in frequency and amplitude. They cannot be adequately interpreted by standard analysis techniques, e.g. such as the Fourier transform. To identify current operating modes of the feeding unit, it is proposed to use the apparatus of wavelet transforms, which allows you to identify local features of the flow signals in a combined time-frequency medium with essentially larger information richness and semantic clarity. For the aim of controlling current modes in FDU a closed automatic modal control system (CACS) with a full state vector feedback has been developed. In the modal control algorithm the feedback matrix is permanently recalculated in real time, which forms the specified localization of the poles constellation. Specified non-stationary feeding modes are demonstrated to represent the essence of modal controlling in the FDU.

The article presents and describes Cohen’s class time-frequency distributions which are expedient to use as a mathematical tool that allows to create a convenient – in terms of information content and semantic clarity – visual-graphical representation of the opera ting modes of various technological processes including processes of ferrous metallurgy. It was noted that a controlling process is usually implemented without simultaneous visual monitoring of each scalar (one-dimensional) coordinate that is under control, but the presence of such monitoring is an important condition for the computer-aided controlling of the dynamics of non-stationary technological processes. To eliminate this drawback, it was proposed to perform synchronous monitoring using the multidimensional Cohen’s class time-frequency distributions, when each measurement scalar signal is specifically represented through one of these distributions, for example, the Wigner-Ville distribution. An expression is given for the generalized distribution of Cohen’s class with a distribution kernel and an ambiguity function. This function allows receiving distributions of various types from the maternal function. The most typical representatives of time-frequency distributions forming this class are given with their available  kernels. The possibility of appearance of interference elements, which make it difficult to identify the controlled modes, on a signal distribution map is proved. Case of the formation of virtual elements within the Wigner-Ville distribution representing a two-component one-dimensional signal is considered. Te conditions are explained for the emergence of parasitic elements on the distribution map, obtained, for example, during realizing the process of multi-component feeding the bulk blast furnace charge materials in the production of sintering mixture. An analytical expression is obtained for the Wigner distribution, which displays a multi-component scalar signal and contains the information (useful) and virtual (parasitic) parts of the time-frequency distribution. A link between the number of bulk material feeders available in the feeding devices unit and the number of parasitic (virtual) elements in the Wigner distribution was determined. Using the dosing process as an example, the effect of the noise components propagation in the Wigner distribution is demonstrated. An example is given to illustrate the penetration of noise into the Wigner distribution and appearance of the virtual concentration in it when displaying a signal waveform with a noisy pause and two sections with different frequencies. An expression for the Wigner distribution in the form of a comb function is obtained. The conclusion was made about the parameters of the distribution periodicity and the required sampling frequency of measurement signals.