In this paper, we present Lyapunov-based robust and adaptive Higher Order Sliding Mode (HOSM) controllers for nonlinear SISO systems with bounded uncertainty. The proposed controllers can be designed for any arbitrary sliding mode order. The uncertainty bounds are known in the robust control problem whereas they are partially known in the adaptive control problem. Both these problems are formulated as the finite time stabilization of a chain of integrators with bounded uncertainty. The controllers are developed from a class of nonlinear controllers which guarantee finite time stabilization of pure integrator chains. The robust controller establishes ideal HOSM i.e. the sliding variable and its r−1 time derivatives converge exactly to the origin in finite time. The adaptive controller establishes real HOSM, which means that the sliding variable and its r - 1 time derivatives converge to a neighborhood of the origin. Saturation functions are used for gain adaptation, which do not let the states exit the neighborhood after convergence. The effectiveness of these controllers is illustrated through simulations.
