A Lyapunov method analysis for double integrator with bounded perturbation

For a parameterized family of homogeneous sliding mode based controllers under the influence of bounded perturbation, a strong Lyapunov function is presented for the first time. The family of controllers range from twisting algorithm, the linear PD control law, to the uniformly stable control law. The strict homogeneous Lyapunov function proposed here allows the estimation of the ultimate bound of convergence of the trajectories of the system, in finite-time, exponentially, or uniformly, by exploiting the homogeneity properties of the system, even in the case when it is affected by bounded non-vanishing persistent external perturbation.