The problem of inversion of dynamic systems has become widespread while solving the problems of control, identification, and measurement problems arising during the design and research of electrical and mechanical dynamic systems. Inverting is an effective way of implementing disturbance control processes, as well as in combined control systems with a predictive model. The analysis of information sources showed that in the practical solution of most inversion problems, a number of difficulties arise, which are associated with the high sensitivity of the results in relation to the accuracy of the parameters of the mathematical model of the control object, the instability of the inverse model of non-minimum-phase objects, and the violation of the conditions of physical feasibility. The work offers an effective method of inverting linear stationary dynamic systems, free from the mentioned shortcomings in many respects. The basis of the method is the presentation of input and output signals in the form of infinite linear combinations of their derivatives. A method of determining the sequence of matrix coefficients of linear representations of input and output signals is proposed. The main theoretical result is obtaining relationships between matrix coefficients of input and output signals. The work considers mathematical models of linear dynamic systems in the form of differential equations in the state space and in the equivalent "input-output" form. The considered systems must meet the conditions of asymptotic stability, as well as the condition of equal dimensions of the input and output vectors. Requirements for mathematical models of input and output signals are given, the fulfillment of which allows, instead of infinite sums representing signals, to be limited to a finite number of terms.

The time of information technology determines its priorities, which are a prerequisite for building a competitive production and economy. The ubiquitous spread of digitalization is one of the basic principles of new economy, a new type of socio–economic structure that is gradually being formed in the modern world through the introduction of scientific and technological progress and innovative methods of management, intellectualization and capitalization of human knowledge, the use of advanced new information and material technologies, accelerated development of knowledge-intensive sectors of the economy, the formation of creative, efficient, rational information and material production. Currently, at large foundries with mass and large-scale production of castings, the task of automating the control of technological processes using digital control systems was solved in general. They implement algorithms for controlling technological processes of casting in closed circuits (locally). The systems under consideration allow to implement optimal control strategies and automatically perform sequences of operations (start and stop of equipment; calculation and input of metal charge; calculation of formulations, dosing and mixing of molding and core mixtures) of multi-stage periodic casting processes. Digital transformation can significantly change the established practice of foundry production (from direct control and management of technological processes to business planning and document management). The transformation will have an impact on all parameters of the enterprise: economic efficiency of production (productivity, operating costs); reliability (operational readiness); safety (number of incidents); compliance with legislative norms on ecology. The technological criterion for success of the digital transformation of foundry production will be the release of a nomenclature of castings with a minimum level of defect, commercial – the release of a nomenclature of castings in demand on the market (machine parts and mechanisms), with a minimum self-cost, which is determined by the technological level of preparation of the production and its implementation and, as a consequence, low costs and optimal quality of molds, metal and castings.

Modern foundry production should be equipped with advanced technologies, equipment and materials, have a significant reserve for its development in the future, which is unthinkable without a transition to digital production, which includes integrated automation of technological processes, digital control of technology parameters, materials and quality indicators of the finished products by the non-destructive control methods, as well as the use of new foundry materials obtained with the help of nanotechnologies.

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