Sliding Mode Control: a Survey with Applications in Math

The paper presents a brief survey on Variable Structure Control Systems with Sliding Modes. Starting from a general case of sliding modes in dynamical systems with discontinuous right-hand side, classic approaches to sliding mode control systems are considered and some basic results about the control of uncertain systems are given. Then, Higher-Order Sliding Modes are presented as a tool to remove discontinuity from the control action, to deal with higher relative degree systems and to improve the accuracy of the real sliding mode behavior when the discrete time implementation is considered. Finally, three applications of the sliding mode control theory to applied math problems are presented: the numerical solution of constrained ODEs, the real-time differentiation, and the problem of finding the zeroes of nonlinear algebraic systems. The first is an almost straightforward application of the sliding mode control theory, while the last two are accomplished by computing the solution of properly defined dynamical systems. Some simulations are reported to clarify the approach.