Invariant sliding sector for variable structure control

A sliding sector is defined as a subset of the state space, inside which some norm of state decreases with zero control input. In the paper, an invariant sliding sector is defined to be such a kind of sliding sector that with some control laws, the norm still keeps decreasing inside the sector and the sector is an invariant subset, i.e., the state will not move out of the sector if the initial state is inside it or after the state is moved into it by some control law. Design algorithms of the invariant sliding sector and the corresponding variable structure (VS) controller are given. The proposed VS controller with the invariant sliding sector ensures that the 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 especially ensures that the norm of state keeps decreasing in the state space with a smooth VS control law. The resulted VS control system is quadratic stable. The simulation result shows that the proposed VS controller has good control performance without chattering.