Control of delay systems with relay

We study the periodic oscillations of a first-order delayed linear system with relay output and proportional + integral feedback and describe the behaviour of the general solutions of the closed loop. For the system under study, we first exhibit a countable set of periodic limit cycles. We show that in the particular case where only proportional control is used, any solution tends in finite time towards one of the limit cycles (whose determination depends on the initial conditions). All the cycles are orbitally unstable except one of them, the only slowly oscillating one. We provide exact computations of their period and amplitude. We then show how these results may be used to identify the parameters of the plant and to tune the control-law parameters in order to control the amplitude and the period of the slowly oscillating limit cycle. Finally, we provide some well-posedness and ultimate boundedness results for a time-varying perturbed version of the system under study. The given estimates show that the proportional + integral feedback law permits rejection of various parametric perturbations.