On forced and subharmonic oscillations under relay feedback

Abstract

The necessary and sufficient conditions for forced and subharmonic oscillations (FO and SO, respectively) in an externally driven single-loop relay feedback system (RFS) are analysed. It is shown that FO of any frequency will always occur in the RFS if and only if the amplitude of the external forcing signal is larger than some minimum. This minimum amplitude can be determined by graphical/numerical approaches. In contrast, the existence of SO is dependent on both the amplitude and frequency of the external signal. Interestingly, one may not be able to obtain any SO for arbitrary frequencies even if the amplitude of the external signal is large. Given this important fundamental difference, the study also sets out to determine the range of frequencies where SO can exists, along with the necessary minimum amplitude of the external signal required for the SO to occur. Simulation studies are given to illustrate the results of the analysis.