On one problem of control of voltage during signal transmission in a long line

This article discusses the problem of controlling the process of electromagnetic oscillations in a long-distance transmission line. A long line is understood as an electrical line formed, in the simplest case, by two parallel current conductors, the length of which exceeds the wavelength of the transmitted electromagnetic waves, and the distance between the conductors is much less than the wavelength. Such a line is characterized by four distributed parameters, namely the ohmic resistance of the conductor, inductance, electrical capacity and leakage coefficient. A signal generator is connected to the left end of the line, and the right is grounded, but not of high quality. When transmitting a signal, the current and voltage make small oscillation. The control is the voltage at the left end of the long line, and the disturbance voltage at the right end, the values of which are limited. The boundaries of their permissible values are set. The magnitude of the aggregate of external disturbances acting on the conductor is not known exactly, but only its variation limits are given. Aim. The purpose of the control process is that at a given moment in time, the average value of the voltage value is in a given interval. This average is calculated using the specified function. Materials and methods. To solve the problem, the method of optimizing the guaranteed result was applied. Results. A transition was made to a new one-dimensional variable, with the help of which the considered problem was reduced to a control problem of the same type in the presence of interference. This made it possible to find the necessary and sufficient conditions under which it is possible to achieve the set goals with any admissible set of external forces and interference at the right end. A corresponding algorithm for constructing the law of voltage change at the left end of the conductor is proposed. An example is analyzed that clearly shows how management is built that guarantees the achievement of the set goal. Conclusion. If the found necessary and sufficient conditions are fulfilled, then it is always possible to construct such a law of voltage variation at the left end, which will lead to the achievement of the goal for any admissible interference.