Forced oscillations in first order systems

Forced oscillations is a phenomenon in which a nonlinear closed loop system is forced to operate at the same frequency as the external forcing sinusoid. This paper derives the necessary and sufficient conditions for which forced oscillations can be achieved when a first order linear system is placed in a closed loop relay feedback configuration. It is shown that a fundamental requirement for forced oscillations is the existence of a pair of switch points which are separated by exactly half the period of the external forcing signal. The conditions of forced oscillations for this class of plants are derived by ensuring that this basic requirement is satisfied. The results completely characterise the forced oscillations conditions for all first order plants with dead times. Simulation studies are given to illustrate these results.