Abstract
Within the framework of the article, the problem of statistical classification of states of a dynamic system is solved, which can be in two classes of states, in each of which its operation is described by its own system of autoregressive equations with a priori unknown parameters. It is assumed that the following conditions are fulfilled: a) two classes of states are described by the same sets of observed input and output variables; b) the output variables, both in the first and in the second class, are determined by different sets of regressors (input variables); c) the models of functioning in the first and second classes are different both in terms of coefficients and in the structure of autoregressive models; d) the covariance matrices of random variables in the functioning models and the observation models for the first and second classes are different. The rule of classification is constructed and its properties are investigated.The experience of successfully solving problems of detecting changes in the properties of dynamic systems based on regression equations in the work, where an approach to constructing mathematical models for monitoring the technical condition of power and power plants in long-term operation was proposed, shows the feasibility of applying this approach to solving problems of controlling the operation of rocket-space objects technology.The problem of classifying states of a dynamic system, which can be in two classes of states, is considered. The functioning of the system in classes is described by various systems of autoregressive equations. The rule of classification is constructed and its properties are investigated.
Highlights
Введем обозначенияГде O(n×h) – нулевая (n × h) -матрица; Σζ , Σε – заданные ковариационные (h × h) -матрицы в моделях функционирования и наблюдения соответственно.
Для оценивания неизвестных коэффициентов θ(k, q) , k, q = 1, 2,..., h по результатам наблюдения объекта (20) в [7]–[8] разработаны итерационные процедуры параметрической идентификации, в которых X(0).
Y – (N ×1) -вектор ненаблюдаемых значений; θ – (M ×1) -вектор неизвестных коэффициентов; ξ – (N ×1) -вектор ненаблюдаемых случайных аддитивных составляющих; R – объединённая (N × M ) -матрица регрессоров; N = n h ;.
Summary
Где O(n×h) – нулевая (n × h) -матрица; Σζ , Σε – заданные ковариационные (h × h) -матрицы в моделях функционирования и наблюдения соответственно. Для оценивания неизвестных коэффициентов θ(k, q) , k, q = 1, 2,..., h по результатам наблюдения объекта (20) в [7]–[8] разработаны итерационные процедуры параметрической идентификации, в которых X(0). Y – (N ×1) -вектор ненаблюдаемых значений; θ – (M ×1) -вектор неизвестных коэффициентов; ξ – (N ×1) -вектор ненаблюдаемых случайных аддитивных составляющих; R – объединённая (N × M ) -матрица регрессоров; N = n h ;. D(h) d(k, h) где для (M × N ) -матрицы C , состоящей из (h × h) блоков, выполнено. Где Σ ⊗ In – кронекеровское произведение матриц Σ и In ; Σε , Σζ – ковариационные (h × h) -матрицы в моделях наблюдения и функционирования, введённые в (17)–(18).
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