Lyapunov method for unperturbed double integrator systems

In this paper a strong Lyapunov function is proposed, for the first time, for a parameterized family of homogeneous sliding mode based controllers. Indeed, from twisting algorithm, to the linear PD control law, to the uniformly stable control law, a general homogeneous family of control algorithms are considered. The strict locally Lipschitz homogeneous Lyapunov function allows the possibility to estimate an upper bound for the convergence time of the trajectories of the system to the equilibrium point, in finite-time, exponentially, or uniformly asymptotically, by exploiting the homogeneity properties of the system. Moreover, the introduction of Lyapunov function allows the analysis of the relationship between the control gains and its convergence time.