Stability of variable structure control systems

This paper studies asymptotic stability of a continuous implementation of variable structure control in which a signum function is approximated by a saturation nonlinear function. It is shown that the closed ring 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 gradient of the saturation function. Moreover, under slight additional suppositions, it is shown that in the absence of persistent disturbance the system has an uniformly asymptotically stable equilibrium point at the origin