Invariant Sliding Sector for Variable Structure Control

A sliding sector has been defined as a subset of the state space, inside which a norm of state decreases with zero control input [1]. In this paper, an invariant sliding sector is defined to be such a kind of sliding sector inside which with a suitable control law, the norm keeps decreasing and the system state stays there since then after entering the sector, i.e. the sector defined in the paper is an invariant subset of the state space. Design algorithms of the invariant sliding sector and the Variable Structure (VS) controller with the sector are given. The proposed VS controller with the invariant sliding sector ensures that the system state moves from the outside to the inside of the sector in a finite time and stays inside it forever after being moved into it, and that the norm of state keeps decreasing in the state space. The resulted VS control system is quadratically stable with a smooth VS control law. The experimental results with the inverted pendulum show the invariance of the sector and the robustness of the proposed VS control system with the invariant sliding sector.