High-order sliding modes in control systems

The article considers the properties of a new type of solutions that arise in discontinuous dynamic systems. A specific feature of these solutions is the tangency of the phase velocity vectors to the manifold of the right-hand side discontinuities, instead of the transversal intersection of the manifold typically observed for ordinary sliding modes. The solutions identified in this case are high-order sliding modes, and the order of the mode is determined by the smoothness of tangency of the sliding manifold. Second-order sliding modes are considered in detail. Examples of systems with such modes are given; application of the theory to stabilization of uncertain dynamic systems is described. It is shown that the sensitivity of high-order sliding modes to small variations in the right-hand side of the discontinuous system is an order of magnitude higher than for ordinary sliding modes.