Stability of the local set of limit cycles in digitally controlled relay systems

The widespread use of relay control in technical systems and the use of digital controllers for its implementation, requires the development of appropriate theoretical methods for studying such systems. It is known that time discretization in relay automatic systems leads to the appearance of many limit cycles in local regions of the phase space. Methods have been developed for distinguishing these regions and determining the parameters of periodic motions. Moreover, the issues of assessing the stability of these movements have not been sufficiently studied. In this paper, we develop a criterion that allows us to evaluate the convergence of phase trajectories to a local set of limit cycles. To do this, we consider the mapping of switching surfaces and use the Lyapunov functions. The practical application of the obtained theoretical results by a concrete example is considered.