Stability of the sliding mode in nonstationary automatic-control systems with variable structure

The paper addresses the stability of the sliding mode in nonstationary automatic-control systems of variable structure. The parameters of dynamic states are assumed to vary arbitrarily and continuously within a given bounded domain. The existence and stability conditions for the sliding mode on the switching hyperplane are established. They define nonstationary sets that contain phase velocity vectors directed away from the sliding hyperplane. The sets do not have common points with the switching hyperplane, except for the origin, for the known extreme values of the discontinuous coefficients of logical controls. This is the essence of the necessary and sufficient conditions for the stable sliding mode of the automatic-control system. The general results are used to analyze the stability conditions for the sliding mode in a nonstationary automatic-control system with a variable structure of the third order.