Finite switching frequency effects in the sliding mode control of the double integrator system

Effects of finite switching frequency on the behaviour of a simple sliding mode control system are studied. We show that if the switching frequency is high enough then the system converges to a small neighborhood of the origin. Even for arbitrarily fast switchings there exist periodic orbits with infinitely long periods and complex aperiodic trajectories. We also prove that if the switching frequency is below a certain threshold, the system sustains periodic oscillations with arbitrarily large amplitudes