Frequency analysis of relay feedback systems based on phase properties

Analyzes of the frequency variations of switching for relay feedback systems is considered in this paper. That is, the relationship between the frequency variations and the phase variations of the linear system, when used in relay feedback form, is derived. It will be shown that for linear systems that their phase angle does not crosses −180°, the switching frequency increases without bound and there is no equilibrium point. Furthermore, it will be shown that for systems with the relative degree greater than two, i.e. systems that their phase angle crosses −180°, the phase angle equal to −180° is an equilibrium point for the oscillations frequency. Moreover, the equation that indicates the relation between the switching frequency variations and characteristics of systems such as the phase and amplitude of the system will be derived.