A Lyapunov Approach to Barrier-Function Based Time-Varying Gains Higher Order Sliding Mode Controllers

In this paper, we present Lyapunov-based {color{black}time varying} controllers for {color{black}fast} stabilization of a perturbed chain of integrators with bounded uncertainties. We refer to such controllers as {color{black}time varying} higher order sliding mode controllers since they are designed for nonlinear Single-Input-Single-Output (SISO) systems with bounded uncertainties such that the uncertainty bounds are unknown. %{color{blue} OLD: Our main result states that, given any neighborhood $varepsilon$ of the origin, we determine a controller insuring, for every uncertainty bounds, that every trajectory of the corresponding closed loop system enters $varepsilon$ and eventually remains there. Furthermore, based on the homogeneity property, a new asymptotic accuracy, which depends on the size of $varepsilon$, is presented.} We provide a time varying control feedback law insuring verifying the following: there exists a family $(D(t))_{tgeq 0}$ of time varying open sets decreasing to the origin as $t$ tends to infinity, such that, for any unknown uncertainty bounds and trajectory $z(cdot)$ of the corresponding system, there exists a positive positve $t_z$ for which $z(t_z)in D(t_z)$ and $z(t)in D(t)$ for $tgeq t_z$. %enters convergence in finite time of all the trajectories to a time varying domain $D(t)$ shrinking to the origin and their maintenance there. Hence, since the function $eta(t)$ tends to zero, this leads the asymptotic convergence of all the trajectories to zero. The effectiveness of these controllers is illustrated through simulations.