Forced oscillations conditions in relay feedback control systems

In the relay feedback system shown in Figure 1, when the magnitude of the forcing signal is larger enough, the phenomenon of forced oscillation occurs, that is, the system synchronizes itself automatically with the frequency of the external signal. However, the minimal magnitude of the forcing signal for forced oscillation occur can not be obtained easily. An aim of our research work was to determine the minimal magnitude of the external forcing signal in order to enforce forced oscillation in the loop. In this paper, we examined the necessary and sufficient conditions for forced oscillation and led to a solution. Simulation showed that the solution was the desired minimal magnitude of the external forcing signal for forced oscillation in a special limit range. Although the solution was not a general solution, it did give the accurate result in some degree and most real systems were within the special limit range then our result can be used.