Analysis of a second-order sliding-mode algorithm in presence of input delays

In this note, a double integrator system under the action of a second-order sliding-mode control algorithm is considered, and the resulting closed-loop behavior in presence of an input delay is analysed. Due to the delay, in the limit the system trajectories are periodic. Whenever the control modulus is chosen to be constant, the amplitude and period of the resulting oscillations are fixed for any initial value. If the control behaves asymmetrically, it is shown that this is no more true, since the overall dynamical system can admit diverse limit cycles.