A new approach to solving the problem of instability of a system of linear algebraic equations (SLAE) with an ill-conditioned matrix describing a discrete model of the Fredholm integral equation of the sec-ond kind, which reduces the calculation by the method of secondary sources of three-dimensional static and quasi-stationary electromagnetic fields of any geometry in inhomogeneous and nonlinear media, is considered. The essence of the new approach is all about. There is a method for correctly compiling a description of an electrical circuit. In this method, for the first time, when describing an electrical cir-cuit, the parameters of a specific task are taken into account, but they are not taking into account in other methods. As a result, the solution to the problem is stable even in the case of a SLAE with an ill-conditioned matrix. The disadvantage of this method is the description of the electrical circuit in the form of a graph. The description of the discrete model of the integral equation is proposed to be trans-formed to a form of representation that satisfies the method of describing the electric circuit. To achieve this goal, the following tasks have been completed. The requirements of the method of correct compila-tion of the description, which the form of the description of the discrete model of the integral equation must satisfy, are formulated. The analysis of the linear discrete model of the integral equation is carried out, the graph of the discrete model is constructed, and the requirements for the method of transform-ing this graph to the graph that meets the requirements of the method are formulated. A technique for transforming a graph of a discrete model into a graph that meets the requirements of the method has been developed. Final result: a description of a discrete model of the Fredholm integral equation of the second kind, compiled by the method of secondary sources in the form of a graph, satisfying the re-quirements of the method is presented.

Housing and communal services in Kyiv have more than 10,000 multi-storey houses, and in Ky-iv region, there are more than 5,000. Reducing energy consumption for housing, heat and water supply, and lighting services should be considered as reducing energy costs. The trouble-free operation of the equipment is of the utmost importance. Its resource on many utilities has a minimum margin of safety. Centralized analysis of the equipment control, accounting of motor resources and, accordingly, timely preventive work allow extending the service life of many components of engineering systems in the utility sector, transport, and industry. These problems are solved by distributed systems of automated control of engineering equipment, including control systems of the hot water supply scheme for modern new buildings. In modern new buildings, there is used a scheme of hot water supply (HWS) with two circulating pumps and the HWS regulator. The control cabinet, developed in the IMMSP of the NAS of Ukraine, is made on the basis of the MRTP-7 controller with the corresponding software consisting of gen-eral and special purpose programs. The article considers the principles of building a control sys-tem for hot water supply, considers the software functions of the system, presents a mathemati-cal model of the HWS system, describes a laboratory test site for testing algorithms for HWP control and modeling the HWS control process. The scientific and technical solutions provided in the article can be used in the development of automated control systems for engineering equipment in the utility sector.

Modern automated control systems can not do without the special means of organizing commu-nication with humans. The main peculiarity of human-machine interaction is the combination of formalized information processes and structures with non-formalized (as a rule, creative) in-formation processing. The latter helps not only find ways to solve complex problems that can not be found in formal models (algorithms) but also organize the process of problem solving as a holistic system. Human-machine interaction can not be fully algorithmic (in the formal sense), but it is possible to develop technology for mutual activities of humans and technical systems for problem solving. The role of people in technical control systems is constantly changing. Previously, a human was the link that was mostly responsible for process manage-ment. With the development of automated systems that was supported by the ever-increasing use of computers, people are increasingly taking over the functions of the control unit, at the same time keeping the role of an observer of the process. However, in recent years the role of humans in control systems has changed. In modern technical systems, they again play a leading role, which fully reveals their abilities and uses its boundary capabilities to optimally distribute the functions between human and machine, human and process. Human-machine interaction is a form of mutual use of reserves of humans and automated systems that is ultimately focused on humans. The article considers the stages of human-machine automation systems development, the composition and distribution of functions in human-machine systems, the possibility of their implementation, emergencies in automated technological facilities, and some examples of human-machine control in ACS TP thick-sheet rolling mills.