On a sign controller for the triple integrator

It is well known that for a double integrator a controller using only the signs of the state variables, the Twisting controller, is able to bring all trajectories to the origin in finite time and robustly with respect to bounded matched perturbations. In this paper we analyze a triple integrator with a controller consisting of the sum of the signs of the state variables multiplied by constant gains. We show that, in contrast to the twisting controller, there is no set of gains for which the origin is a globally asymptotically stable equilibrium point. There is however a set of gains such that for almost all initial conditions the trajectories converge in finite time to the origin.