Design of variable structure control system by using the vector Lyapunov function method

The objective of this paper is to present a simple design method of a variable structure controller for systems with nonlinear uncertainty. The proposed method is based on a kind of a boundary layer method and can eliminate chattering. The control law is given by sum of a linear feedback and a saturation type nonlinear state feedback, that is, a linear feedback in the boundary layer and sum of an affine feedback outside it. So, it is a very simple control law. The proposed control law guarantees exponential stability of O solution if there is no disturbance. If disturbance exists, it guarantees ultimate boundness. The ultimately bounded region is inside the boundary layer, in which the control is a linear feedback. In the proposed design method, stability is analyzed by using the vector Lyapunov function approach