EVALUATION OF THE SYSTEM'S CONDITIONS OF ECONOMIC DYNAMICS BY PROJECTION METHODS AT PARTIALLY OBSERVED OUTPUT COORDINATES

Abstract

The problem of finding estimates of the state of systems of economic dynamics is quite common in the design of optimal continuous and discrete control systems in their stochastic and deterministic consideration. The solution of the problem in a stochastic sense is considered in the work “Prediction of the structure of dynamic systems”. It was based on the methods of factorization of correlation matrices of completely observable sets of output signals of dynamical systems and requires a significant number of assumptions about the properties of the matrices of the system. The article discusses the possibilities to solve individual problems of finding estimates and optimal controls by the method of designing multidimensional spaces on eigensubspaces. Here we will consider the problems of finding estimates and optimal controls in order of increasing complexity of the problems being solved. In the study of systems of economic dynamics in some cases, all output coordinates of the system allow direct measurement and observation. For linear systems of economic dynamics with such properties, the formation of an optimal control law as a function of state coordinates can be performed even if there are various deviations in the measurement. However, in engineering practice very often not all state coordinates allow observation and measurement. In these cases, the optimal control law is defined as a function of part of the best estimates of the state coordinates, determined by measuring the output signals of the system. Consequently, the problem of optimal control in a more general setting includes both the problem of finding the optimal estimate of the states of the system and the problem of optimal control. Based on the results of applying projection methods for assessing the state of the system of economic dynamics, it is concluded that the task is to find estimates for a multi-step process. As a result of this, estimates are successively found for all steps, and in each subsequent step, the found optimal solutions are used in the previous step, i.e. The principle of dynamic programming is implemented. Projection research methods also allow you to simultaneously and independently solve the problem of estimating the state vectors of the system of economic dynamics and finding the optimal control sequences.