DefinePK

Abstract

Practically all real control objects are characterized by a certain spatial extent. In this regard, the controlled values of such objects depend not only on time, but also on the distribution over the spatial domain occupied by the object. The changes of controlled quantities, both in time and in space, are described by partial differential equations, integral, integro-differential equations or systems of equations of the most diverse nature. Compared to systems with lumped parameters, the class of control actions is expanding in principle. The number of control actions can include space-time controls, described by functions of several arguments - time and spatial coordinates. The structural representation of systems and the representation of distributed objects in the form of complex transfer coefficients for eigenfunctions greatly simplifies the problem of distributed systems analyzing, but can be used if there is a solution to a boundary value problem. To solve practical problems, most often, finite-dimensional approximation of distributed systems is used, which is based on finite-dimensional representations of partial derivatives based on the method of “grids” and “direct”, using Fourier series and Taylor series. The representation of discrete control actions in the form of delta functions allows one to investigate the class of systems with distributed parameters, for which there is a fundamental solution in the form of expansion in eigenvector functions of the object operator [1, 4, 9]. The study of the control actions discretization parameters influence on the regulation process allows the regulation of nonlinear discrete systems in the relay mode. The proposed work considers systems with distributed control objects. Three spatially distributed mathematical models are built. A closed-loop control system for the temperature field of a spatially distributed control object was synthesized. The frequency surfaces of all proposed regulators are constructed. A new method for analyzing the stability of spatially distributed control objects has been proposed.