A set-valued nested sliding-mode controller

We propose a set-valued controller with a signum multifunction nested inside another one. We prove that the controller is well-posed and achieves robust ultimate boundedness in the presence of mismatched, non-vanishing disturbances. Even more, the selected output can be made arbitrarily small. Also, by applying the implicit Euler scheme introduced by Acary and Brogliato [2010], Acary et al. [2012] for matched disturbances, we derive a selection strategy for the discrete-time implementation of the set-valued control law. The discrete-time scheme diminishes chattering substantially