Stability analysis of a continuous implementation of variable structure control

The authors study stability properties of a continuous implementation of variable-structure control in which a signum nonlinearity is approximated by a saturation nonlinearity. It is shown that the closed-loop system is globally uniformly ultimately bounded with respect to a compact set around the origin. This set can be made arbitrarily small by increasing the slope of the saturation function. Furthermore, under mild additional assumptions, it is shown that in the absence of persistent disturbance the system has a uniformly asymptotically stable equilibrium point at the origin, if the slope of the saturation function is sufficiently large