Time convergence estimation of a perturbed double integrator: Family of continuous sliding mode based output feedback synthesis

In this paper mechanical systems of relative degree two, i.e. a perturbed double integrator is under study. A sliding mode based algorithm is under study using strict non smooth Lyapunov functions, such that compensation of growing perturbations together with state variables is shown. Indeed, the well known twisting algorithm and a generalized smooth family of this algorithm are considered. A strict non-smooth Lyapunov function is proposed allowing to design tuning rules for the gains of a family of controllers such that global exact finite time stability of the origin is shown. The proposed methodology estimate an upper bound for convergence time of the closed loop system in spite of growing perturbation with respect to the state. To illustrate performance and robustness properties a numerical experiment is presented, using one-link pendulum as a test bed.