Uniform sliding mode controllers and uniform sliding surfaces

In this note, we propose two uniform sliding mode controllers (SMCs) for a second-order uncertain system. They provide convergence to an arbitrarily small vicinity centred at the origin in finite time, which can be bounded by some constant independent of the initial conditions and uncertainties. Towards this aim, a non-linear sliding surface is designed ensuring convergence, during the sliding motion, of every trajectory to an arbitrarily small vicinity of the origin in a finite time bounded by some constant, independent of the initial condition on the surface. A first-order SMC is designed providing convergence of every trajectory to the sliding surface in a finite time, bounded by another constant independent of the initial conditions and uncertainties/disturbances of a special class. In order to adjust chattering, a super twisting-based controller is suggested providing convergence of the trajectories to the sliding surface in a finite time bounded by some constant independent of any initial condition and uncertainties/disturbances of another class.