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.

[1]  Vadim I. Utkin,et al.  Sliding mode control in electromechanical systems , 1999 .

[2]  Mikhail Kryachkov,et al.  Finite-time stabilization of an integrator chain using only signs of the state variables , 2010, 2010 11th International Workshop on Variable Structure Systems (VSS).

[3]  Jaime A. Moreno,et al.  Construction of Lyapunov Functions for a Class of Higher Order Sliding Modes algorithms , 2012, 2012 IEEE 51st IEEE Conference on Decision and Control (CDC).

[4]  P. Olver.  Nonlinear Systems , 2013 .

[5]  B. Bandyopadhyay,et al.  Finite-time stabilization of uncertain triple integrator with only switch and gain , 2011, IECON 2011 - 37th Annual Conference of the IEEE Industrial Electronics Society.

[6]  A. Levant.  Sliding order and sliding accuracy in sliding mode control , 1993 .

[7]  Jean-Pierre Barbot,et al.  Sliding Mode Control In Engineering , 2002 .

[8]  Aleksej F. Filippov,et al.  Differential Equations with Discontinuous Righthand Sides , 1988, Mathematics and Its Applications.

[9]  Giorgio Bartolini,et al.  On the Finite-Time Stabilization of Uncertain Nonlinear Systems With Relative Degree Three , 2007, IEEE Transactions on Automatic Control.