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In this paper, we present Lyapunov-based adaptive controllers for thepractical (or real) stabilization of a perturbed chain of integrators withbounded uncertainties. We refer to such controllers as Adaptive Higher OrderSliding Mode (AHOSM) controllers since they are designed for nonlinear SISOsystems with bounded uncertainties such that the uncertainty bounds areunknown. Our main result states that, given any neighborhood N of the origin,we determine a controller insuring, for every uncertainty bounds, that everytrajectory of the corresponding closed loop system enters N and eventuallyremains there. The effectiveness of these controllers is illustrated throughsimulations.

Practical stabilization of perturbed integrator chains with unknown bounds